text stringlengths 8 1.01M |
|---|
Modify Your Results
Boldly go to worlds where no one has gone. Meet Chuck Lambert, who, though not exactly a fool, is guilty of letting his imagination get the best of his wits. That's because our young, naive Lambert wants his own planet. But rather than purchase one legally from the Interior Department of the Outer Galactic Control, he soon succumbs to the flashy advertising of an unsavory galactic swindler named Madman Murphy-- the purported King of Planetary Realtors.What Madman is the king of, is selling the unwary a planet that isn't quite right, a planet where one can't sit down because there's something the matter with its matter. And that's exactly what becomes the matter for our unlucky voyager, after Chuck toils for eleven grueling years to scrape together enough money to finally buy a planet of his own. "... this is a real corker, pulp fiction at its most entertaining."--Booklist* An International Book Awards Finalist
When a TV game show comes to Bayport, a crime wave begins. As Frank and Joe fight a grudge match with the current team champions, they are also trying to discover who on the staff of the show is the mastermind behind burglaries that seem to follow the traveling show.
Algebra 2 brings math to life with many real-life applications. Circling the globe are three key aspects of Algebra 2--the equations, graphs, and applications that you will use in this course. They will help you understand how mathematics relates to the world. As you explore the applications presented in the book, try to make your own connections between mathematics and the world around you!dougal Littell Middle School Math strengthens the students understanding and provides the tools that the students need to excel in mathematics. It has written lessons with frequent step-by-step examples which make even difficult math concepts and methods easier to understand |
Question 3 on Paper 1 covers these two topics, which are apparently unconnected. Not that it concerns us, but the topics become interwoven at university level maths. Anyway this question is one of the more popular on the first paper, and has often been the easiest question on the paper. There have been exceptions; in 1998, Question 3(c) was particularly difficult.
Complex numbers deal with the so-called 'imaginary unit', i, which stands for the square root of -1. It might at first appear as if this quantity has nothing to do with the world we live in, not existing in the way the number 3 'exists'. However, complex numbers have many applications in engineering, physics and science in general. On our course we discuss the properties of complex numbers, see how they help us to solve equations and investigate how to use complex numbers written in polar form.
Matrices were originally introduced to simplify the maths involved in transformation geometry, although in Question 3 on Paper 1 we do not see them being used for this purpose. We concentrate on the definitions associated with matrices and how we can perform the basic operations of addition, multiplication, etc. One interesting feature of matrices is the absence of division, and the use of the inverse of a matrix to overcome this problem.
Topic Structure: Complex Numbers
The study of Senior Cycle Complex Numbers can be divided into the following sections:
1. Definitions and Basic Operations
2. Complex Equations
3. Polar Form of a Complex Number
Topic Structure: Matrices
The study of Senior Cycle Matrices can be divided into the following sections:
The S.O.S. site on matrices is again aimed at students starting university, and so many of the questions refer to matrices of higher dimension than the 2 x 2 that we are used to.
This section covers a wide area of problems about complex numbers, with many well-worked examples provided at three different levels, along with good practice material. |
Language, notation and formulas
Communication is as vital in mathematics as in any language. This unit will help you to...
Communication is as vital in mathematics as in any language. This unit will help you to express yourself clearly when writing and speaking about mathematics. You will also learn how to answer questions in the manner that is expected by the examiner.
By the end of this unit you should be able to:
lay out and, where appropriate, label simple mathematical arguments;
understand the precise mathematical meaning of certain common English words;
Language, notation and formulas
Introduction
An integral part of learning mathematics involves communication.
Writing mathematics is a specific skill which needs to be developed and practised: there is a lot of difference between putting down a few symbols for your own use and writing a mathematical solution intended for someone else to read. In attempting mathematical questions, you may previously have written down very little, just enough, perhaps, to convince yourself that you could answer the questions. This may suffice now, but you may want to use your notes and solutions for revision so you will want them to be self-contained, able to stand on their own and easy to read. This will also be the case if you are writing mathematics for somebody else to read.
This unit is from our archive and is an adapted extract from Mathematics: a foundation course (MU120) |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
1.9 MB | 14 pages
PRODUCT DESCRIPTION
This single-lesson is intended for Pre-Calculus Honors students. Students will be able to use exponential growth, decay, and regression to model real-life problems. The lesson is complete with a SMART NOTEBOOK 11 presentation, an eight-page BOUND-BOOK STYLE FOLDABLE for a graphic organizer, DIRECTIONS TO ASSEMBLE THE FOLDABLE, and a PDF file of the completed lesson.
Average Ratings
Comments & Ratings
Product Questions & Answers
Be the first to ask Jean Adams1.70. |
Find a Griffith MathReal world applications are presented within the course content and a function's approach is emphasized. This course builds on algebraic and geometric concepts. It develops advanced algebra skills such as systems of equations, advanced polynomials, imaginary and complex numbers, quadratics, and includes the study of trigonometric functions.
...I would work with a student to isolate and determine the area in which there is a need for clarity and instruction and develop lessons that coincide with school-related texts and reflecting real-life scenarios. I have programmed in various Cobol versions from AcuCobol, RM-Cobol and M-F Cobol sin... |
Book Description: Olympiad mathematics is not a collection of techniques of solving mathematical problems but a system for advancing mathematical education. This book is based on the lecture notes of the mathematical Olympiad training courses conducted by the author in Singapore. Its scope and depth not only covers and exceeds the usual syllabus, but introduces a variety concepts and methods in modern mathematics. In each lecture, the concepts, theories and methods are taken as the core. The examples are served to explain and enrich their intension and to indicate their applications. Besides, appropriate number of test questions is available for reader's practice and testing purpose. Their detailed solutions are also conveniently provided. The examples are not very complicated so that readers can easily understand. There are many real competition questions included which students can use to verify their abilities. These test questions are from many countries, e.g. China, Russia, USA, Singapore, etc. In particular, the reader can find many questions from China, if he is interested in understanding mathematical Olympiad in China. This book serves as a useful textbook of mathematical Olympiad courses, or as a reference book for related teachers and researchers. Volume 1: Operations on Rational Numbers Linear Equations of Single Variable Multiplication Formulae Absolute Value and Its Applications Congruence of Triangles Similarity of Triangles Divisions of Polynomials Solutions to Testing Questions Volume 2: Congruence of Integers Decimal Representation of Integers Pigeonhole Principle Linear Inequality and System of Linear Inequalities Inequalities with Absolute Values Geometric Inequalities Solutions to Testing Questions |
Hi Virgil,
For general programming, I would pick some of the simpler problems from
sources like the following:
and ask the students to solve them in Mathematica.
For Mathematica specifically, in "Introduction to programming with
Mathematica" by Paul Wellin et al, there is a number of good and relatively
simple problems in the exercise sections. Perhaps other introductory
Mathematica books which I am not familiar with may also have some good
exercises. Some problems from the exercise sections of Michael Trott's
programming guidebook (or examples in the main text) may also do.
>From what comes straight to my mind, some recursive or rule-based (or plain
procedural) implementations of things like factorial, Fibonacci numbers,
GCD, sorting, merge sort, select according to some criteria, binary search,
or some problems like computing primes (sieve of eratosthenes etc),
frequencies of say characters in a string, testing a string for a
palindrome, and the like should give a good idea of students' ability. I
only mentioned programming problems since they are field - independent to a
large degree. For computational ones I would guess that any good multi-step
calculus problem done (say both analytically and numerically) entirely in
Mathematica will do what you need. One thing I would definitely do is to
separate programming and computational assignments and test these skills as
independently as possible.
But I have a strong feeling that most of the students will ordinarily have
the biggest
difficulty simply with getting the syntax right and being able to
explain/correct the unusual output (I certainly had this difficulty for
quite a long time myself). I am not even sure that this has so much to do
with the programming proper - rather with the (relatively complex for a
newbie) evaluation process and symbolic nature of Mathematica. Perhaps,
tests like "predict the output" or "what is broken in this example", or "get
this to work" will be at least as useful as programming assigments proper
(in Michael Trott's programming guidebook there are a number of such
examples, although perhaps many of them are too advanced for the average
student).
It also depends on how much of the core Mathematica programming language you
would like the students to use. If the projects are such that the procedural
subset of Mathematica is enough, and its value in this context is mostly in
a large number of ready-to-use built-in functions, it is one thing. If
patterns, rules and functional programming will be needed, it is another
story. If performance can be important / critical, this is yet another one.
I know that some people end up creating a simplified framework of some kind
on top of Mathematica (sort of an API) to "isolate" their students from the
core Mathematica language, since mastering it may take a course on its own,
and may not be directly relevant to the domain they are learning through
Mathematica (I personally don't like this option. If students are eager to
learn and are good in some other programming language, I would rather give
them a brief introduction to Mathematica at the beginning, and let them use
as much of the power of the language as they can. But I acknowledge that
often the "framework" way may be the right choice).
Hope this helps.
Regards,
Leonid
On Fri, Nov 6, 2009 at 1:17 PM, Virgil Stokes <vs at it.uu.se> wrote:
> I am a teacher and am often faced with the problem of how to determine
> the programming skill level for some of my students (3rd year
> undergraduates in Engineering and Physics). Knowledge of their skills in
> Mathematica can be very important for the design of student projects
> that require Mathematica.
>
> I would appreciate suggestions for small segments of Mathematica code
> (or perhaps a single segment of increasing complexity) that could be
> used to at least get an idea of their skills in Mathematica. You can
> assume they would be using Mathematica 7.
>
> All suggestions, examples, comments, etc. will be welcomed :-)
>
> --V. Stokes
> |
Intermediate Algebra, Fourth Edition was written to provide a solid foundation in algebra for students who might have had no previous experience in algebra. Specific care has been taken to ensure that students have the most up-to-date and relevant text preparation for their next mathematics course, as well as to help students to succeed in nonmathematical courses that require a grasp of algebraic fundamentals |
Online information:
or Office Hours: MW at 1, F at 11, and by arrangement.
Students who require accommodations and who have
documentation from Disability Services (874-2098) should make
arrangements with me as soon as possible.
Topics: vectors, matrices, linear systems, linear transformations,
vector spaces, determinants, eigenvalues, eigenvectors.
There are many applications of linear algebra to problems in many areas
of math and science. We will look at applications in order to
motivate the study of linear algebra.
Calculators: The recommended calculator is
TI-89 or TI-92. TI-83 and TI-86 could be also used.
Grading: Your grade will be based on three tests, a final exam, and
classwork as follows:
Three tests at 100 points each
300 points
Final exam
150 points
Classwork
150 points
Maple Projects
200 points
Total
800 points
Tests and the final exam: The tests
will be
given in the class. A comprehensive final exam will be
given during the final exam period. Time and place will be announced
later. The exams will reflect the variety of the homework problems. The
best way to prepare for the exams, and to develop confidence in your
ability to solve problems, is to work on the homework problems as
suggested. Your class may be slightly behind or ahead at any given
time. Some problems may be done in class or as homework, as your
instructor chooses.No makeups for exams will
be given unless you have a University sanctioned excuse.
Classwork: The distribution of the 150
points will be decided by your instructor. It will include quizzes and
class participation.
Makeups: No makeup quizzes will be
given. Instead, your lowest quiz grade will be dropped.
Homework: Homework plays a central role
in the class and in your understanding of the material. It is fair to
say
that most of the learning that you achieve during any math course is
from
your homework.
Read the textbook: An important part of your
mathematical education is acquiring the knack of learning mathematics
on your own, from books. You may not be used to reading
mathematics texts, but you
will be actively encouraged to read this one. By reading the text
before
class, even if you don't understand everything the first time, you will
have a better chance of making good use of your time in class.
Reading
the text after class is a good way of reinforcing the material in the
lecture,
nailing down what questions you need to ask in the next class, and
learning
material that was not gone over during class time. The text is
very
well written, with the beginning linear algebra student in mind. Linear algebra is much easier if you
keep up with the classes and homework. You also retain the
material longer and better if you review material frequently rather
than just studying at exam time. |
2. Relations and Functions : Ordered pairs, Cartesian product of sets, Number of elements in the Cartesian product of two finite sets. Cartesian product of the reals with itself ( upto R x R x R ). Definition of relation, pictorial diagrams, domain. Codomain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co – domain & range of a function. Real valued function of the real variable, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum and greatest integer functions with their graphs. Sum, difference, product and quotients of functions.
3. Trigonometric Functions : Positive and negative angles. Measuring angles in radians & in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin2 x + cos2 x = 1, for all x. Signs of trigonometric functions and sketch of their graphs. Expression sin ( x+y ) and cos ( x+y ) in terms of sin x, sin y, cos x & cos y. Deducing the identities like following :
II. Algebra
1. Principle of Mathematical Induction : Processes of the proof by induction, motivating the application of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications.
2. Complex Numbers and Quadratic Equations : Need for complex numbers, especially √−1, to be motivated by inability to solve every quadratic equation. Brief description of algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system.
3. Linear Inequalities : Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Solution of system of linear inequalities in two variables – graphically.
6. Sequences and Series : Sequence and Series. Arithmetic progression ( A. P. ). arithmetic mean ( A.M. ) Geometric progression G.P., general term of a G.P., sum of n terms of a G.P., geometric mean ( G.M. ), relation between A.M. and G.M. Sum to n terms of the special series Σn, Σn2 and Σn3.
III. Coordinate Geometry
1. Straight Lines : Brief recall of 2D from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line : parallel to axes, point – slope form, slope – intercept form, two – point form, intercepts form and normal form. General equation of a line. Distance of a point from a line.
2. Conic Sections : Sections of a cone : circles, ellipse, parabola, hyperbola, a point, a straight line and pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.
3. Introduction to Three – dimensional Geometry : Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula.
IV Calculus
1. Limits and Derivatives : Derivative introduced as rate of change both as that of distance function and geometrically, intuitive, idea of limit. Definition of derivative, relate it to slope of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
1. Matrices : Concept, notation, order, equality, types of matrices, zero matrix, transpose of a matrix, symmetric and skew symmetric matrices. Addition, multiplication and scalar multiplication of matrices, simple properties of addition, multiplication and scalar multiplication. Non – commutativity of multiplication of matrices and existence of nonzero matrices whose product is the zero matrix ( restrict to square matrices of order 2 ). Concept of elementary row and column operation. Invertible matrices and proof of the uniqueness of inverse, if it exists; ( Here all matrices will have real entries ).
2. Determinants : Determinant of a square matrix ( up to 3 x 3 matrices ), properties of determinants, minors, cofactors and applications of determinants in finding the area of a triangle, Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables ( having solution ) using inverse of a matrix.
2. Applications of Derivatives : Application of derivatives : rate of change, increasing / decreasing functions, tangents & normals, approximation, maxima and minima ( first derivative test motivated geometrically and second derivative test given as a provable tool ). Simple problems ( that illustrate basic principles and understanding of the subject as well as real – life situations ).
3. Integrals : Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, only simple integrals of the type
to be evaluated. Definite integrals as a limit of a sum, Fundamental Theorem of Calculus ( without proof ). Basic properties of definite integrals and evaluation of definite integrals.
4. Applications of the Integrals : Applications in finding the area under simple curves, especially lines, areas of circles / parabolas / ellipses ( in standard form only ), area between the two above said curves ( the region should be clearly identifiable ).
5. Differential Equations : Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation variables, homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type :
dy / dx + py = q, where p and q are functions of x.
IV Vectors and Three – Dimensional Geometry
1. Vectors : Vectors and scalars, magnitude and direction of a vector. Direction cosines / ratios of vectors. Types of vectors ( equal, unit, zero, parallel and collinear vectors ), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Scalar ( dot ) product of vectors, projection of a vector on a line, Vector ( cross ) product of vectors.
2. Three – dimensional Geometry : Direction cosines / ratios of a line joining two points. Cartesian and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between ( i ) two lines, ( ii ) two planes. ( iii ) a line and a plane. Distance of a point from a plane.
2. Kinematics : Frame of reference. Motion in a straight line : Position – time graph, speed and velocity, Uniform and non – uniform motion, average speed and instantaneous velocity. Uniformly accelerated motion, velocity – time position – time graphs, relations for uniformly accelerated motion ( graphical treatment ). Elementary concepts of differentiation and integration for describing motion. Scalar and vector quantities, Position and displacement vectors, general vectors and notation; Equality of vectors, multiplication of vectors by a real number; Addition and subtraction of vectors.
3. Laws of Motion : Intuitive concept of force. Inertia, Newton's first law of motion; momentum and Newton's second law of motion; impulse; Newton's third law of motion. Law of conservation of linear momentum and its applications. Equilibrium of concurrent forces. Static and Kinetic friction, laws of friction, rolling friction. Dynamics of uniform circular motion; Centripetal force, examples of circular motion ( vehicle on level circular road, vehicle on banked road )
4. Work, Energy and Power : Scalar product of vectors. Work done by a constant force and a variable force; Kinetic energy, work energy theorem, power. Notion of Potential energy, potential energy of a spring, conservative forces; conservation of mechanical energy ( Kinetic and potential energies ), Non – conservative forces; elastic and inelastic collisions in one and two dimensions.
5. Motion of System of Particles and Rigid Body : Centre of mass of a two-particle system, momentum conservation and centre of mass motion. Centre of mass of a rigid body; centre of mass of uniform rod. Vector product of vectors; moment of a force, torque, angular momentum, conservation of angular momentum with some examples.
Equilibrium of rigid bodies, rigid body rotation and equations of rotational motion, comparison of linear and rotational motions; Moment of inertia, radius of gyration. Values of moments of inertia for simple geometrical objects ( no derivation ). Statement of parallel and perpendicular axes theorems and their applications.
6. Gravitation : Keplar's Laws of planetary motion. The universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity of a satellite. Geo – stationary satellites.
8. Thermodynamics : Thermal equilibrium and definition of temperature ( zeroth law of thermodynamics ), Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics: reversible and irreversible processes. Heat engines and refrigerators.
9. Behaviour of Prefect Gas and Kinetic Theory : Equation of state of a perfect gas, work done on compressing a gas. Kinetic theory of gases – assumptions, concept of pressure. Kinetic energy and temperature rms speed of gas molecules; degrees of freedom, law of equipartition of energy ( statement only ) and application to specific heats of gases; concept of mean free path, Avogadro's number.
1. Electrostatics : Electric Charges; Conservation of charge, Coulomb's law – force between two point charges, forces between multiple charges; superposition principle and continuous charge distribution. Electric field, electric field due to a point charge, electric field lines; electric dipole, electric field due to a dipole' torque on a dipole in uniform electric field. Electric flux, statement of Gauss's theorem and its applications to find field due to infinitely long straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell ( field inside and outside ).
Electric potential, potential difference, electric potential due to a point charge, a dipole and system of charges; equipotential surfaces, electrical potential energy of a system of two point charges and of electric dipole in an electrostatic field. Conductors and insulators, free charges and bound charges inside a conductor. Dielectrics and electric polarisation, capacitors and capacitance, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, energy stored in a capacitor. Van de Graaff generator.
2. Current Electricity : Electric current, flow of electric charges in a metallic conductor, drift velocity, mobility and their relation with electric current; Ohm's law, electrical resistance, V – I characteristics ( linear and non – linear ) electrical energy and power, electrical resistivity and conductivity. Carbon resistors, colour code for carbon resistors; series and parallel combination of resistors; temperature dependence of resistance. Internal resistance of a cell, Potential difference and emf of a cell, combination of cells in series and in parallel.
Kirchoff's laws and simple applications. wheatstone bridge, metre bridge. Potentiometer – principle and its applications to measure potential difference and for comparing emf of two cells; measurement of internal resistance of a cell.
3. Magnetic Effects of Current and Magnetism: Concept of Magnetic field, Oersted's experiment. Bio – Savart law and its applications to current carrying circular loop. Ampere's law and its applications to infinitely long straight wire, straight and toroidal solenoids. Force on a moving charge in uniform magnetic and electric fields. Cyclotron. Force on a current – carrying conductor in a uniform magnetic field. Force between two parallel current – carrying conductors – definition of ampere.
Torque experienced by a current loop in uniform magnetic field; moving coil galvanometer – its current sensitivity and conversion to ammeter and voltmeter. Current loop as a magnetic dipole and its magnetic dipole moment. Magnetic dipole moment of a revolving electron. Magnetic field intensity due to a magnetic dipole ( bar magnet ) along its axis and perpendicular to its axis. Torque on a magnetic dipole ( bar magnet ) in a uniform magnetic field; bar magnet as an equivalent solenoid, magnetic field lines; Earth's magnetic field and magnetic elements. Para -, dia – and ferro – magnetic substances, with examples. Electromagnets and factor affecting their strengths. Permanent magnets.
6. Optics: Reflection of light, spherical mirrors, mirror formula. Refraction of light, total internal reflection and its applications, optical fibres, refraction at spherical surfaces, lenses, thin lens formula, lens – maker's formula. Magnification, power of a lens, combination of thin lenses in contact. Refraction and dispersion of light through a prism. Scattering of light – blue colour of the sky and reddish appearance of the sun at sunrise and sunset.
3. Classification of Elements and Periodicity in Properties : Significance of classification, brief history of the development of periodic table, modern periodic law and the present form of periodic table, periodic trends in properties of elements – atomic radii, ionic radii. Ionization enthalpy, electron gain enthalpy, electronegativity, valence.
Group 1 and Group 2 elements : General introduction, electronic configuration, occurrence, anomalous properties of the first element of each group, diagonal relationship, trends in the variation of properties ( such as ionisation enthalpy, atomic and ionic radii ), trends in chemical reactivity with oxygen, water, hydrogen and halogens; uses.
Preparation and properties of some important compounds : Sodium carbonate, sodium chloride, sodium hydroxide and sodium hydrogen carbonate, biological importance of sodium and potassium. CaO, CaCO3 and industrial use of lime and limestone, biological importance of Mg and Ca.
1. Solid State : Classification of solids based on different binding forces: molecular, ionic, covalent and metallic solids, amorphous and crystalline solids ( elementary idea ), unit cell in two dimensional and three dimensional lattices, calculation of density of unit cell, packing in solids, voids, number of atoms per unit cell in a cubic unit cell, point defects, electrical and magnetic properties.
4. Chemical Kinetics : Rate of a reaction ( average and instantaneous ), factors a affecting rates of reaction; concentration, temperature, catalyst; order and molecularity of a reaction; rate law and specific rate constant, integrated rate equations and half life ( only for zero and first order reactions ); concept of collision theory ( elementary idea, no mathematical treatment )
6. General Principles and Processes of Isolation of Elements: Principles and methods of extraction – concentration, oxidation, reduction electrolytic method and refining; occurrence and principles of extraction of aluminium, copper, zinc and Iron.
1. Diversity in Living World : Diversity of living organisms Classification of the living organisms ( five kingdom classification, major groups and principles of classification within each kingdom ). Systematics and binomial System of nomenclature Salient features of animal ( non – chordates up to phylum level and chordates up to class level ) and plant ( major groups; Angiosperms up to class ) classification, viruses, viroids, lichens, Botanical gardens, herbaria, zoological parks and museums.
2. Structural Organization in Animals and Plants : Tissues in animals and plants. Morphology, anatomy and functions of different parts of flowering plants: Root, stem, leaf, inflorescence, flower, fruit and seed. Morphology, anatomy and functions of different systems of an annelid ( earthworm ), an insect ( cockroach ) and an amphibian ( frog ).
General Concepts; Modular approach; Clarity and Simplicity of Expressions, Use of proper Names for identifiers, Comments, Indentation; Documentation and Program Maintenance; Running and Debugging programs, Syntax Errors, Run-Time Errors, Logical Errors; Problem Solving Methodology and Techniques: Understanding of the problem, Identifying minimum number of inputs required for output, Step by step solution for the problem, breaking down solution into simple steps, Identification of arithmetic and logical operations required for solution, Using Control Structure: Conditional control and looping ( finite and infinite );
Flow of control : Conditional statements : if – else, Nested if, switch..case..default, Nested switch..case, break statement ( to be used in switch..case only ); Loops: while, do – while , for and Nested loops;
Definition of a class, Members of a class – Data Members and Member Functions ( methods ), Using Private and Public visibility modes, default visibility mode ( private ); Member function definition: inside class definition and outside class definition using scope resolution operator ( :: ); Declaration of objects as instances of a class; accessing members from object ( s ), Array of type class, Objects as function arguments – pass by value and pass by reference; |
201308150 / ISBN-13: 9780201308150
Mathematics All Around
"Tom Pirnot" believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking ...Show synopsis"Tom Pirnot" believes that conceptual understanding is the key to a student's success in learning mathematics. He focuses on explaining the thinking behind the subject matter, so that students are able to truly understand the material and apply it to their lives. This textbook maintains a conversational tone throughout and focuses on motivating students and the mathematics through current applications. Ultimately, students who use this book will become more educated consumers of the vast amount of technical and mathematical information that they encounter daily, transforming them into mathematically aware citizens.Hide synopsis
Sound copy, mild reading wear. May have scuffs or missing...Good. Sound copy, mild reading wear. May have scuffs or missing DJ. May have some note, highlighting or underlining. Purchasing this item helps us provide vocational opportunities to people with barriers to employment.
Reviews of Mathematics All Around
I cannot believe that all of this was crammed into ONE 8 week college course. I'm still getting over the stress of trying to make it through this course. The book is ok if you understand math, but if you don't, you're just going to be more lost than you were before you started. Very confusing stuff |
Keith Devlin. You know him. You've read his columns in MAA Online, you've heard him on the radio, and you've seen his popular mathematics books. In between all those activities and his own research, he's been hard at work revising Sets, Functions and Logic, his standard-setting text that has smoothed the road to pure mathematics for legions of undergraduate... more...
Arnold's Problems' contains mathematical problems which have been brought up by Vladimir Arnold in his famous seminar at Moscow State University over several decades. In addition, there are problems published in his numerous papers and books. Many of these problems are still at the frontier of research today. more...
Mathematics does not exist in isolation but is linked inextricably to the physical world. At the 2003 International Congress of Industrial and Applied Mathematics, leading mathematicians from around the globe gathered for a symposium on the "Mathematics of Real World Problems," which focused on furthering the establishment and dissemination of those... more...
The physics of hot plasmas is of great importance for describing many phenomena in the Universe and is fundamental for the prospect of future fusion energy production on Earth. Non-trivial results of nonlinear electromagnetic effects in plasmas include the self-organization an self-formation in the plasma of structures compact in time and space. These... more...
Expanded coverage of essential math, including integral equations, calculus of variations, tensor analysis, and special integrals Math Refresher for Scientists and Engineers, Third Edition is specifically designed as a self-study guide to help busy professionals and students in science and engineering quickly refresh and improve the math skills needed |
With the same design and feature sets as the market leading Precalculus, 8/e, this addition to the Larson Precalculus series provides both students and instructors with sound, consistently structured explanations of the mathematical concepts. Designed for a two-term course, this text contains the features that have made Precalculus a complete solution for both students and instructorsMathematics Describing the Real World Precalculus and Trigonometry-Bruce H. Edwards AVI, XviD, 640x480, 29.97 fps | English, MP3@128 kbps , 2 Ch | ~36x30 mins | 10.82 GB The Teaching Company | 2011 | Course no. 1005 Trad... Filesonic, Fileserve, Uploading, Wupload, Uploadstation Links Engoy all members !!!...
TradClear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this text popular among students year after year. This latest edition of Swokowski and Cole's PRECALCULUS: FUNCTIONS AND GRAPHS retains these features. The problems have been consistently praised for being at just the right level for precalculus students like you. The book also provides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics.
As part of the market-leading Graphing Approach series by Larson, Hostetler, and Edwards, Precalculus Functions and Graphs
Get a good grade in your precalculus course with PrecalculusGet a good grade in your precalculus course with PRECALCULUSSuitable for either one or two semester college algebra with trigonometry or precalculus courses, Precalculus introduces a unit circle approach to trigonometry and includes a chapter on limits to provide students with a solid foundation for calculus concepts. The large number of pedagogical devices employed in this text will guide a student through the course. Integrated throughout the text, students and instructors will find Explore-Discuss boxes which encourage students to think critically about mathematical concepts. In each section, the worked examples are followed by matched problems that reinforce the concept being taught. In addition, the text contains an abundance of exercises and applications that will convince students that math is useful. A MathZone site featuring algorithmic exercises, videos, and other resources accompanies the text. |
0471938Consists of nine case studies on industrial applications, developed in collaboration with several European companies, including robotics, fluid mechanics, elastic geodesy, channel transmissions and electronic circuit development. Features an introduction to computer algebra and two contributions describing how numerical mathematics software can successfully interact with computer algebra. Demonstrates that the use of computer algebra in industry is both practical and cost effective |
Geometry topics presented with the aid of downloadable Geometer's Sketchpad (GSP) files; see, in particular, Japanese temple geometry. Two comprehensive Tutorials in Excel spreadsheet form are provided for intermediate...
To produce structures that are functional as well as models of architectural beauty, designers must apply principles of mathematics in their work. Scale drawings, commonly known as plans, are used as patterns in the...
The algebra index of the extensive S.O.S.Math site has lessons and reference material on units of conversion, complex numbers, equations, and much more. Each section features a concise review, notation, examples, and...
This site features Flash animations that illustrate how the Global Positioning System (GPS) works. The animations depict how GPS signals are derived, compare geostationary and polar orbits, and explain satellites,... |
Search form
You are here
MATH 121/6.0 - Differential and Integral Calculus
Chris Taylor
Calculus is a branch of mathematics that can describe precisely how one numerical output quantity changes in response to changes in one or more numerical input quantities. Two of its main aspects are differentiation, which concerns the instantaneous rate of change of one output variable with respect to one or more input variables; and integration, which deals with the cumulative change in a given output variable. For example, if a ball is thrown upwards, differentiation can tell us its vertical velocity (rate of change of height with time) at any point in time; and integration can tell us its net change in height over a given period. Many applications of calculus are found in science, engineering, commerce, and medicine; indeed, much of modern technology would be impossible without it. The development of calculus spans many centuries and cultures, but in Europe, calculus in something like its modern form first appeared in the seventeenth century, when Newton and Leibniz independently synthesized differential and integral calculus into a single powerful discipline. Many important developments and applications followed, although calculus was not made fully rigorous until the nineteenth century, with the emergence of the limit concept.
This is a general two-term calculus course, starting with a revision of high- school-level pre-calculus, and the basics of single-variable differentiation and integration, and then moving on to more advanced topics, such as multivariable calculus, differential equations, and various techniques of optimization. Students can take this course with or without high-school calculus experience. The course is not intended as a pure maths course, and so there is more of an emphasis on techniques and applications than on formal proofs.
The expected learning outcomes are as follows:
Conceptual understanding and technical mastery of the following main areas of calculus and pre-calculus:
Basics of algebra and arithmetic.
Functions and graphs.
Geometry and trigonometry.
Limits.
Differentiation.
Integration.
Differential equations.
Partial derivatives and vector calculus.
An ability to apply knowledge of the topics above to solve extended problems, both abstract and applied.
An ability to communicate and present such mathematical problem-solving skills in printed documents that combine explanatory English text with mathematical equations and graphs in a coherent and comprehensible way. |
This is the second semester course in a one-year sequence that is designed as a rigorous development of fundamentals of analysis for Mathematics majors. The following topics will be covered in this course: differentiation of functions, integration of functions, infinite series, and sequences and series of functions.
To prepare students with the theoretical foundation needed for further study in higher mathematics.
At a minimum, upon completion of this course, students will be able to present accurately mathematical definitions and concepts, and provide rigorous mathematical proofs of basic facts in the following topics:
The differentiations of functions. This includes:
A rigorous definition of the derivative of a function and rigorous proofs of rules and
theorems about derivatives that are presented in Calculus.
A special focus on the mean-value theorem and its popular applications.
At least 4 collections of home assignments will be given The students may discuss the assignments with each other only before they start their assignments. While they are doing their assignments the university rule on academic integrity applies.
GRADING STANDARDS- ASSESSMENT TOOLS
Participation
20%
Assignments
80%
The grading scale guideline:**
A
92-100%
A-
88-91%
B+
85-87%
B
82-84%
B-
78-81%
C+
75-77%
C
72-74%
C-
68-71%
D+
65-67%
D
58-64%
F
0-57%
UNIVERSITY POLICY
1) Attendance:
Lincoln University uses the class method of teaching, which assumes that each student has something to contribute and something to gain by attending class. It further assumes that there is much more instruction absorbed in the classroom than can be tested on examinations. Therefore, students are expected to attend all regularly scheduled class meetings and should exhibit good faith in this regard. For the control of absences, the faculty adopted the following regulations:
Four absences may result in an automatic failure in the course.
Three tardy arrivals may be counted as one absence.
Absences will be counted starting with whatever day is specified by the instructor but not later than the deadline for adding or dropping courses.
In case of illness, death in the family, or other extenuating circumstances, the student must present documented evidence of inability to attend classes to the Vice President for Student Affairs and Enrollment Management. However, in such cases the student is responsible for all work missed during those absences.
Students representing the University in athletic events or other University sanctioned activities will be excused from class (es) with the responsibility of making up all work and examinations. The Registrar will issue the excused format to the faculty member in charge of the off- or on-campus activity for delivery by the student(s) to their instructors.
2) Statement on Academic Integrity:
Students are responsible for proper conduct and integrity in all of their scholastic work. They must follow a professor's instructions when completing tests, homework, and laboratory reports, and must ask for clarification if the instructions are not clear. In general, students should not give or receive aid when taking exams, or exceed the time limitations specified by the professor. In seeking the truth, in learning to think critically, and in preparing for a life of constructive service, honesty is imperative. Honesty in the classroom and in the preparation of papers is therefore expected of all students. Each student has the responsibility to submit work that is uniquely his or her own. All of this work must be done in accordance with established principles of academic integrity.
An act of academic dishonesty or plagiarism may result in failure for a project or in a course. Plagiarism involves representing another person's ideas or scholarship, including material from the Internet, as your own. Cheating or acts of academic dishonesty include (but are not limited to) fabricating data, tampering with grades, copying, and offering or receiving unauthorized assistance or information.
3) The Student Conduct Code:
Students will be held to the rules and regulations of the Student Conduct Code as described in the Lincoln University Student Handbook. In particular, excessive talking, leaving and reentering class, phones or pagers, or other means of disrupting the class will not be tolerated and students may be asked to leave. Students who constantly disrupt class may be asked to leave permanently and will receive an F.
4) The Core Curriculum Learner Competencies:
All courses offered through the Department of Mathematics and Computer Science require students to meet at least the
following out of the 8 Core Curriculum Learner Competencies:
(6) Apply and evaluate quantitative reasoning through the disciplines of mathematics, computational science, laboratory
science, selected social sciences and other like-minded approaches that require precision of thought;
(8) Demonstrate positive interpersonal skills by adhering to the principles of freedom, justice, equality, fairness,
tolerance, open dialogue and concern for the common good.
Note:
* The instructor of a given section of the course may make some modifications to the evaluation as well as to the rest of the syllabi including but not limited to; the grade weights, number of tests, and test total points.
**The grading scale guideline includes a 2-point flexibility.
Please consult with the department chairperson for any program updates or corrections which may not be yet reflected on this page _ last updated 9/10/2007. |
This course explores the application of calculus towards the study of
higher-dimensional surfaces and their geometry. Topics include geodesics,
tangensors. Special topics (at the discretion of the instructor) may include
Lie groups, symmetric spaces, general relativity, cohomology, and complex
geometry. Students will be required to use a computer algebra system to
gain geometric intuition.
OBJECTIVES:
Students in this course will learn how the concepts of calculus can
be applied to understand the geometry of mathematical surfaces such as
planes, spheres, hyperbolic spaces and manifolds in general. Topics include
geodesics, tangensor. Special topics (at the discretion of the instructor) may include
Lie groups, symmetric spaces, general relativity, cohomology, and complex
geometry. Students will be required to use a computer algebra system to
solve problems and gain geometric intuition. |
Miller's Math for the ACT
Maximize Your Math Score on the ACT with Bob Miller! REA's newest ACT test prep helps high school students master math and get into the college of ...Show synopsisMaximize Your Math Score on the ACT with Bob Miller! REA's newest ACT test prep helps high school students master math and get into the college of their dreams! Bob Miller has taught math to thousands of students at all educational levels for 30 years. His proven teaching methods will help you master the math portion of the ACT and boost your score! Written in a lively and unique format that students embrace, "Bob Miller's Math for the ACT "prepares ACT test-takers with everything they need to know to solve the math problems that typify the math portion of the ACT. Unlike some dull test preps that merely present the material, Bob actually teaches and explains math concepts and ideas. His no-nonsense, no-stress style and decades of experience as a math teacher help students boost their ACT math score. In this new test prep, Bob breaks down math and puts it back together in an easy-to-follow, step-by-step format. Each chapter is devoted to a specific topic and is packed with examples and exercises that reinforce math skills. Some of the topics covered include: - Exponents- Square Roots- Algebraic Manipulations- Equations and Inequalities- Geometry and more! Packed with Bob Miller's engaging examples, ACT practice questions, plus test-taking tips and advice, this book is a must for any student preparing for the ACT! Remember, if you're taking the ACT and need help with math, Bob Miller's got your number |
So What Are Logarithms Good For, Anyway?
Summary: A teacher's guide to the applications of logarithmic functions.
As always, things get harder when we get into word problems. There are a few things I want them to take away here.
First—logs are used in a wide variety of real world situations.
Second—logs are used because they compress scales. In other words, because they grow so slowly, we use logarithmic scales whenever we want to work with a function that, by itself, grows too quickly. Or, to put it another way, we use logarithms whenever something varies so much that you don't care exactly what is, just what the power of 10 is. Don't say all this before they start working, but hopefully they will come up with something like this on #6.
Homework:
"Homework: What Are Logarithms Good For, Anyway?"
In addition to following up on the in-class work, the homework here also introduces the common and natural logs. It's a bit of a weak connection, but I had to stick them somewhere.
Time for Another Test!
The sample test is actually pretty important here. It pulls together a lot of ideas that have been covered pretty quickly.
The extra credit is just a pun. The answer is log cabin or, better yet, natural log cabin. Who says math can't be fun?
According to my reckoning, you are now approximately halfway through the curriculum. Mid-terms are approaching. If there are a couple of weeks before mid-terms, I would not recommend going on to radicals—spend a couple of weeks reviewing. Each topic (each test, really) can stand a whole day of review. It may be the most important time in the whole class |
Elementary Linear Algebra - 10th edition
Summary: When it comes to learning linear algebra, engineers trust Anton. The tenth edition presents the key concepts and topics along with engaging and contemporary applications. The chapters have been reorganized to bring up some of the more abstract topics and make the material more accessible. More theoretical exercises at all levels of difficulty are integrated throughout the pages, including true/false questions that address conceptual ideas. New marginal notes provide a fuller explanat...show moreion when new methods and complex logical steps are included in proofs. Small-scale applications also show how concepts are applied to help engineers develop their mathematical reasoning. ...show less
Ships same or next business day with delivery confirmation. Acceptable condition. Contains highlighting. Expedited shipping available.
$50.00 +$3.99 s/h
Good
txtbroker Murfreesboro, TN
10th Edition. Used - Good. Used books do not include online codes or other supplements unless noted. Choose EXPEDITED shipping for faster delivery! n
$53.95 +$3.99 s/h
Good
textbooknook Knoxville, TN
Awesome Text For the Price!!!
$50 |
Description
An easy-to-understand primer on advanced calculus topics
Calculus II is a prerequisite for many popular college majors, including pre-med, engineering, and physics. Calculus II For Dummies offers expert instruction, advice, and tips to help second semester calculus students get a handle on the subject and ace their exams.
It covers intermediate calculus topics in plain English, featuring in-depth coverage of integration, including substitution, integration techniques and when to use them, approximate integration, and improper integrals. This hands-on guide also covers sequences and series, with introductions to multivariable calculus, differential equations, and numerical analysis. Best of all, it includes practical exercises designed to simplify and enhance understanding of this complex subject.
Introduction to integration
Indefinite integrals
Intermediate Integration topics
Infinite series
Advanced topics
Practice exercises
Confounded by curves? Perplexed by polynomials? This plain-English guide to Calculus II will set you straightUser reviews
LibraryThing
LibraryThingAbout the author
Mark Zegarelli, a math tutor and writer with 25 years of professional experience, delights in making technical information crystal clear — and fun — for average readers. He is the author of Logic For Dummies and Basic Math & Pre-Algebra |
Nafoni: another math application that allows you to solve equations, simplify expressions, plot functions on a grid and much more. The application looks very good, but is perhaps more limited in terms of functionality than WolframAlpha. It does, however, look very well and manages to do all of it offline, as opposed to WolframAlpha.
Description
MathStep is a symbolic pocket calculator with a complete CAS (Computer Algebra System) built-in. That means that it can solve math problems for you! It has an expression editor with a live preview and even gives you hints to solving a problem by yourself for some operations. It can even work without an internet connection!
The following operations are currently supported:
IMC
Bit Fiddler
archxs
2 ratings
A visual approach to working with base 2 mathematics, BitFiddler is a binary calculator that provides a touch optimized UI for calculating the results of binary operations. Bit-Fiddler's visual approach to manipulating and displaying binary data...
Math Operations
ROM and Noodles
5 ratings
This application performs math operations such as evaluating functions with or without variables, the quadratic formula, simplifying square roots, solving derivatives, converting between bases, and graphing
MathGems
Nerlaska Studio
20 ratings
That will not rust the brain!!! Mathgems is an Intelligence Game where you have to prove your mental ability to apply basic mathematical calculations like addition, subtraction, multiplication and division. MathGems is a Intelligence Game where you...
Dyscalculator
Pind og Bjerre
6 ratings
Many challenges can be met when calculating. With the Dyscalculator, these challenges will become smaller and fewer. The Dyscalculator is a regular calculator AND a helpful tool that: helps you understand numbers. helps you choose the arithmetic...
Addition and Subtraction kids
Dreams app
39 ratings
Basic math game for kids lite version, let your children learn to add and subtract while playing. Pipo teach math to children with their addition and subtraction random, so operations are endless and the kids will not get bored by repeating.
This...
Easy Stocks
Startups Igniter
4 ratings
Easy Stocks allows you to easily follow your stocks portfolio. You can feed it in with the stocks you bought, and based on your purchase operations and real market data, it will provide you with information on your portfolio gain/loss. Do you...
Think Fast
Dejan Korez
3 ratings...
Quantitative Aptitude
Chourishi Systems
14 ratings
Good collection of Quantitative Aptitude Questions & Answers with explanation for competitive exams. The term quantitative refers to a type of information based in quantities or else quantifiable data. An aptitude is an innate component of |
CalcLabs with Maple for Single Variable CalMaple is a powerful software tool for mathematical computations and visualization. The goal of this manual is to introduce Maple to students who are taking first year calculus. As such, Maple is a tool to solve problems that are too difficult to solve by hand. In addition, students will improve their understanding of the concepts of calculus. The order of the material is organized by computational topic and should be suitable for most texts on Single Variable calculus. |
23 total
5 1,109
4 421
3 156
2 60
1 77
Nice app! Type in f(x)=2cos(100*pi*x) Out to the graph, zoom in and out to see the magic!Very few necessary functions This graphing programme might be useful for somebody doing GCSE or below. I opened it up and found that it only let's you put in graphs in the form f(x) and has very few functions I needed for graphs. The only reason to get a graphing programme is to get it to graph things you can't yet I believe I could do a far better job than this app. I was sorely disappointed in this app-Plot functions with n parameters, use the letters you prefer to represent parameters, words are not reconized as parameters. -Zoon in, Zoom out, enter the initiation of the graph, and the end of the graph. -Change the parameters, and see how the graph change. -Save graph to SD card as a png file with the name you choose. -Store your results in variables. -Plot patametric functions, polar functions, y=f(x) functions, x=f(y) functions
For functions y=f(x) you can -Evaluate functions. -Find the definite integral of a functions *In the interval selected: -Find intersections between two functions. -Find intersections between a function and x-axis(roots), or y-axis. -Find relative extreme points.
-Matrix operations You can enter matrix and use the following functions add(matrix1,matrix2) Add the matrix1 to the matrix2 sub(matrix1,matrix2) Substract the matrix2 to the matrix1 mult(matrix1,matrix2) Multiply the matrix2 with the matrix1 det(matrix) Obtain the determinant of the matrix inv(matrix) Obtain the inverse matrix of the matrix trans(matrix) Obtain the transpose matrix of the matrix print(matrix) print the matrix gaussj(matrix) Solve the system of linear equations of the matrix by the gauss jordan method esc(matrix,number) Mult the matrix with a number(escalar)\n You can store your answers in new matrices.
A powerful tool for building mathematical graphs. The perfect tool for students, teachers and anyone involved in math: - supports standard graphs in Cartesian coordinate system, parametric graphs, graphs in polar coordinates and parametric graphs in polar coordinates, point graphs; - simple and easy to use intuitive interface; - 46 mathematical functions, built-in constants; - any number of graphs on one screen; - save graphs as images with various options; - work in real time, gesture support, two virtual keyboard to choose, examples, help section, history, settings and moreFree version with ads: The best graphing and scientific calculator here.
As a scientific calculator cFunction supports functions like pow, square roots, trigonometric functions and the logarithm. As mathematical constants, the Euler number e and Pi (π) are supported
On top of that with this graphing calculator you can plot (multiple) functions, calculate derivatives, roots, extrema (maxima or minima of a function), inflection points, value table, certain values, definite integrals, intersections of functions and it can convert between degrees and radianGraphite Pro is the FREE professional scientific graphing calculator, with support for parsing advanced mathematical expressions including parentheses. Command history is kept to allow easy reproducibility of previous calculations.
This |
CSI: Pre-Algebra -- STEM Project -- Complete eBook
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
14.82 MB | 127 pages
PRODUCT DESCRIPTION
Nothing like a good criminal investigation to liven up number sense!
CSI: Pre-Algebra is a collection of nine different algebraically inspired mathematical puzzles with a little international pizazz. The nine puzzles intend to target specific Pre-Algebra units and add a little flair to the ordinary challenge question.
Save by buying the entire book! Each of the 9 projects is sold separately for $5. So at $25 you get over a 50% discount!
Each puzzle has 6 "scenes" which will uncover a mystery variable. These six mystery variables will be used to decode a cryptic text message and if everything is correct, the result will match one of the six suspects.
If you are teaching 7th or 8th Grade and are looking for fun review activities and enrichment problems, I have you covered. Student Satisfaction Guaranteed or I'll give you your money back.
I have used these puzzles with much success with a variety of students. Enrichment for advanced students. Review for state graduation exams. The puzzle solving hook engages many different students in solving traditionally mundane problems. Be the cool teacher :-)
In this Zip file you will receive 127 pages of documents including nine separate puzzles and their answer keys.
***ATTENTION DISTRICTS AND DEPARTMENTS*** If you are purchasing for your school's department, please buy the appropriate amount of licenses. If it is purchased with school funds, it belongs to the school. If you are a large school district and you are interested in a full district license, please message me and I can work out a quote that is cheaper than what you see.
I am very excited - have already got 'passports' for the students and will combine this with the people puzzle to do "math around the globe" in the fall. Our science teacher kicks off the year with a CSI unit so this is PERFECT!
My students loved these activities! I have used them with my 7th grade Pre Algebra students as review of material at the end of units. I am planning to use them at the end of the year with my 6th graders to prepare them for their final exams! The activities are engaging, fun, challenging, and accurate. We even made a world map and tracked the locations of the crimes, including print outs of the crime locations! I highly recommend this product!
I just wanted to leave a comment about the lesson on multiples and factors. When my students got to Scene #6, they figured out that since they already knew what M was not zero, the answer to Scene 6 must be that L=1. Therefore, there was no need for them to do any of the math on that page. I have really enjoyed doing other CSI units with them. I hope that you can correct this issue so the kids can't use logic to avoid practicing the math. Thanks!
Product Questions & Answers
On
September 5, 2013,
Buyer asked:
I just purchased the pre-algebra CSI set (8/30/13) and going through the packet I noticed there is info missing from most of the mysteries. Example: Unit 1 operations and expressions, PGA tour puzzle is missing the players names. Instead, it only has gradiated black and white blocks.
There are more puzzles with blocks instead of info but I won't list them here due to space. Is there a way I can get a correction? Do I need to list all the puzzles that I am missing info from?
No you definitely do not need a list :-) Some have had issues opening the files on applications other than Adobe Reader. If you use Adobe Reader and it is still a problem, email me at 21stcenturymathprojects@gmail.com and I will send it as a Google Doc.
You can certainly preview the files. You can see pretty much everything. I always find it hard to answer these types of questions because every class is different and they may need different things. There are projects I have posted here that I used in a 9th Grade class (with low skills) and I found out a 5th Grade teacher used it with her gifted class. So it all depends :-) Hope this helps!
For this item, the cost for one user (you) is $25.00.
If you plan to share this product with other teachers in your school, please add the number of additional users licenses that you need to purchase.
Each additional license costs only $15.00. |
Product Description
Microsoft Excel is the perfect partner for any mathematics curriculum. Applicable to several versions of Microsoft Excel, these books contain prepared templates and lessons that can be utilized to stimulate group discussions, put on-line as a supplemental tutorial, completed as a group visiting the computer lab, or set up as stations in a classroom. Aligned with NCTM & TEKS Standards. Grades 6 |
Having a rep that is easily contactable and very happy to help and advise us has been invaluable. It has saved us hours in trawling through the internet and catalogues trying to find connecting resources or resources we could do this with. Thank you!
Annie Hatton, Inclusion Manager, Park Primary School, Stratford
Essentials for Algebra
Interest Ages 11-16
Statistics
Geometry
Numbers
Algebra
Highly structured
maths
Provide the essential knowledge and focused maths lessons that under performing students need to prepare for secondary school exams
SRA Essentials for Algebra offers a unique progression for introducing and expanding problem types. When a new skill or operation is introduced, it is presented in a highly structured, step-by-step manner. As students advance from one lesson to the next, the teacher provides progressively less guidance.
Techniques used on new skills and problem types are developed in small increments from lesson to lesson. Students are not overwhelmed and receive the practice they need to become skilled at solving complex problems independently. |
GeoGebra ( is free dynamic mathematics software for all levels of education that brings together geometry, algebra, spreadsheets, graphing, statistics and calculus in one easy-to-use package. Interactive learning, teaching and evaluation resources created with GeoGebra can be shared and used by everyone at
GeoGebra is the world's favorite dynamic mathematics software, has received numerous educational software awards, and supports STEM education and innovations in teaching and learning worldwide.
Join us. Dynamic Mathematics for Everyone.
• Free to use software for learning, teaching and evaluation • Fully interactive, easy-to-use interface with many powerful features • Access to an ever-expanding pool of resources at • Available in many languages • A fun way to really see and experience mathematics and science • Adaptable to any curriculum or project • Used by millions of people around 728
4 205
3 133
2 67
1 135Not working correctly on samsung Galaxy Mega GT-I9205 Cannot draw lines or anything related to lines are invisible is there an update/fix for this machine?.wiki says only designed for tablets with bigger screens at moment.
Similar
It can be expensive to do science experiments. Physics Gizmo allows you to collect science data at school, home, or even in space using only your Android phone.
1) Turn on Bluetooth to enable all sensors 2) Select your sensor 3) Give your data a name or leave it with the time and date. 4) Set the timer. 5) Click start to collect data. 5) Upload your data to analyze laterperimeter and area of a square Area of an equilateral triangle area of a regular polygon with n sides
Square area Area of a rectangle Finding the side of a square surface Trapezoid area Parallelogram area Dalton area Scope Square Scope of the rectangle Square volume Triangle Area Scope of the triangle
Regular Shapes
the radius of the circle Perimeter of a circle, Perimeter of a square radius larger & smaller |
c.k.g.m. college
"...Department of Mathematics C.K.G.M. College, Perambra is conducting a state level seminar on 17th and 18th sep 2009. The aim of the seminar is to underline the importance of Python and other free software in Mathematics Education..." |
Product Details:
Numerical analysis is an increasingly important link between pure mathematics and its application in science and technology. This textbook provides an introduction to the justification and development of constructive methods that provide sufficiently accurate approximations to the solution of numerical problems, and the analysis of the influence that errors in data, finite-precision calculations, and approximation formulas have on results, problem formulation and the choice of method. It also serves as an introduction to scientific programming in MATLAB, including many simple and difficult, theoretical and computational exercises. A unique feature of this book is the consequent development of interval analysis as a tool for rigorous computation and computer assisted proofs, along with the traditional material.
Description:
This book provides professionals and students with a thorough understanding
of the interface between mathematics and scientific computation. Ranging from classical questions to modern techniques, it explains why numerical computations succeed or fail. The book is divided into four ... |
Prealgebra, Books a la Carte Edition, 4th Edition
Description
This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Books a la Carte also offer a great value—this format costs significantly less than a new textbook.
Addressing individual learning styles, Tom Carson presents targeted learning strategies and a complete study system to guide students to success. Carson's Study System, presented in the "To the Student" section at the front of the text, adapts to the way each student learns, and targeted learning strategies are presented throughout the book to guide students to success. Tom speaks to students in everyday language and walks them through the concepts, explaining not only how to do the math, but also where the concepts come from and why they work. |
This unusually well-written, skillfully organized introductory text provides an exhaustive survey of ordinary differential equations — equations which express the relationship between variables and their derivatives. In a disarmingly simple, step-by-step style that never sacrifices mathemati... read more
Differential Equations with Applications by Paul D. Ritger, Nicholas J. Rose Coherent introductory text focuses on initial- and boundary-value problems, general properties of linear equations, and differences between linear and nonlinear systems. Answers to most problems.
A First Look at Perturbation Theory by James G. Simmonds, James E. Mann, Jr. This introductory text explains methods for obtaining approximate solutions to mathematical problems by exploiting the presence of small, dimensionless parameters. For engineering and physical science undergraduates.
Ordinary Differential Equations by Jack K. Hale This rigorous treatment prepares readers for the study of differential equations and shows them how to research current literature. It emphasizes nonlinear problems and specific analytical methods. 1969 edition.
Ordinary Differential Equations by Edward L. Ince Among the topics covered in this classic treatment are linear differential equations; solution in an infinite form; solution by definite integrals; algebraic theory; Sturmian theory and its later developments; much more. "Highly recommended" — Electronics IndustriesA Second Course in Elementary Differential Equations by Paul Waltman Focusing on applicable rather than applied mathematics, this text is appropriate for advanced undergraduates majoring in any discipline. The author emphasizes basic real analysis as well as differential equations. 1986 edition. Includes 39 figures.Partial Differential Equations: Sources and Solutions by Arthur David Snider This newly updated text explores the solution of partial differential equations by separating variables, reviewing the tools for the technique, and examining the algorithmic nature of the process. 1999 edition.
Product Description:
This unusually well-written, skillfully organized introductory text provides an exhaustive survey of ordinary differential equations — equations which express the relationship between variables and their derivatives. In a disarmingly simple, step-by-step style that never sacrifices mathematical rigor, the authors — Morris Tenenbaum of Cornell University, and Harry Pollard of Purdue University — introduce and explain complex, critically-important concepts to undergraduate students of mathematics, engineering and the sciences. The book begins with a section that examines the origin of differential equations, defines basic terms and outlines the general solution of a differential equation-the solution that actually contains every solution of such an equation. Subsequent sections deal with such subjects as: integrating factors; dilution and accretion problems; the algebra of complex numbers; the linearization of first order systems; Laplace Transforms; Newton's Interpolation Formulas; and Picard's Method of Successive Approximations. The book contains two exceptional chapters: one on series methods of solving differential equations, the second on numerical methods of solving differential equations. The first includes a discussion of the Legendre Differential Equation, Legendre Functions, Legendre Polynomials, the Bessel Differential Equation, and the Laguerre Differential Equation. Throughout the book, every term is clearly defined and every theorem lucidly and thoroughly analyzed, and there is an admirable balance between the theory of differential equations and their application. An abundance of solved problems and practice exercises enhances the value of Ordinary Differential Equations as a classroom text for undergraduate students and teaching professionals. The book concludes with an in-depth examination of existence and uniqueness theorems about a variety of differential equations, as well as an introduction to the theory of determinants and theorems about Wronskians |
Advanced Calculus Textbooks
Advanced calculus textbooks examine complicated principles and applications of calculus. As a result, advanced calculus textbooks are used mostly at college level. Students often continue to find use in advanced calculus textbooks as their careers progress from pre-calculus textbooks. With this in mind, Textbooks.com has the most comprehensive and up-to-date advanced and applied calculus textbooks available |
Understanding Algebra is a textbook written by James Brennan of Boise State University. The entire contents of the textbook are located on this site, and a PDF version is also available through the author?s Website. B...
Using this Real World Learning Object, students will use data collected from the Global Sun ? Temperature Telecollaborative project to learn how ?linear functions, quadratic functions and other high-order pol...
The home of the World Wide Algebra project. This international megaproject has several principal parts, including Algebraic Cryptography, The World of Groups, and The World of Polynomials. These include lists of open...
About algebra, "a branch of maths where symbols are used to represent numbers," defining and describing the field and highlighting keywords used. All of the most common algebraic identities: Addition/Subtraction;... |
as a true teaching text, this book guides readers through clear, detailed explanations of the principles and practices of business mathematics. The Fifth Edition has been updated and improved to reflect the latest tools and techniques and to help users actively participate in their learning. Text covers whole numbers, fractions, decimal numbers, percent, bank records, payroll, insurance, financial statement analysis, the mathematics of buying and selling, simple interest, bank discount loans, using installment loans, using compound interest and present value, investments, real estate mathematics, inventory and overhead, depreciation, taxes and statistics and graphs. For business, real estate and other professionals requiring a review of principles and practices of business math at the arithmetic level. |
The fun and easy way to learn pre-calculus Getting ready for calculus but still feel a bit confused? Have no fear. Pre-Calculus For Dummies is an un-intimidating, hands-on guide that walks you through all the essential topics, from absolute value and quadratic equations to logarithms and exponential functions to trig identities and matrix operations. With this guide's help you'll quickly and painlessly get a handle on all of the concepts — ...
Titles in Barron's extensive Painless Series cover a wide range of subjects as they are taught on middle school and high school levels. These books are written for students who find the subjects unusually difficult and confusing--or in many cases, just plain boring. Barron's Painless Series authors' main goal is to clear up students' confusion and perk up their interest by emphasizing the intriguing and often exciting ways in which they ...
Now students have nothing to fear … Math textbooks can be as baffling as the subject they're teaching. Not anymore. The best-selling author of The Complete Idiot's Guide to Calculus has taken what appears to be a typical calculus workbook, chock full of solved calculus problems, and made legible notes in the margins, adding missing steps and simplifying solutions. Finally, everything is made perfectly clear. Students will be prepared ...
It used to take years or even decades for disruptive innovations to dethrone dominant products and services. But now any business can be devastated virtually overnight by something better and cheaper. How can executives protect themselves and harness the power of Big Bang Disruption? Just a few years ago, drivers happily spent more than $200 for a GPS unit. But as smartphones exploded in popularity, free navigation apps exceeded ...
From astrophysics to condensed matter theory, nearly all of modern physics employs the path integral technique. In this presentation, the developer of path integrals and one of the best-known scientists of all time, Nobel Prize–winning physicist Richard P. Feynman, presents unique insights into this method and its applications. Avoiding dense, complicated descriptions, Feynman articulates his celebrated theory in a clear, concise manner, ... But how does one exactly do data science? Do you have to hire one of these priests of the dark arts, the "data scientist," to ...
Your step-by-step solution to mastering precalculus Understanding precalculus often opens the door to learning more advanced and practical math subjects, and can also help satisfy college requisites. Precalculus Demystified , Second Edition, is your key to mastering this sometimes tricky subject. This self-teaching guide presents general precalculus concepts first, so you'll ease into the basics. You'll gradually master functions, graphs of ...
Prov ...
This is the first textbook on pattern recognition to present the Bayesian viewpoint. The book presents approximate inference algorithms that permit fast approximate answers in situations where exact answers are not feasible. It uses graphical models to describe probability distributions when no other books apply graphical models to machine learning. No previous knowledge of pattern recognition or machine learning concepts is assumed. Familiarity ...
This edition includes the most recent Integrated Algebra The book reviews all pertinent math topics, including sets, algebraic language, linear ...
When the numbers just don't add up… Following ... Algebra II For Dummies is the fun and easy way to get a handle on this subject and solve even the trickiest algebra ...
Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's ...
This edition includes the most recent Algebra 2/Trigonometry Regents tests through June 2013. These ever popular guides contain study tips, test-taking strategies, score analysis charts, and other valuable features. They are an ideal source of practice and test preparation. The detailed answer explanations make each exam a practical learning experience. In addition to practice exams that reflect the standard Regents format, this book reviews ... ...
Get the confidence and math skills you need to get started with calculus Are you preparing for calculus? This hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in the course. You'll get hundreds of valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every problem. You'll also memorize the most frequently used equations, see how to ...
Over the last thirty years, many influential church leaders and church planters in America have adopted various models for reaching unchurched people. An "attractional" model will seek to attract people to a local church. Younger leaders may advocate a more "missional" approach, in which believers live and work among unchurched people and intentionally seek to serve like Christ. While each of these approaches have merit, something is ...
People are hungry to make a difference in their community, yet most don't know where to start. In fact, "serving the least" is often one of the most neglected biblical mandates in the church. Barefoot Church shows readers how today's church can be a catalyst for individual, collective, and social renewal in any context. Whether pastors or laypeople, readers will discover practical ideas that end up being as much about the Gospel and ... |
Intermediate Algebra : Graphs and Models - 3rd edition
Summary: The Third Edition of the Bittinger Graphs and Models series helps students succeed in algebra by emphasizing a visual understanding of concepts. This latest edition incorporates a new Visualizing the Graph feature that helps students make intuitive connections between graphs and functions without the aid of a graphing calculator.
3.1 Systems of Equations in Two Variables 3.2 Solving by Substitution or Elimination 3.3 Solving Applications: Systems of Two Equations 3.4 Systems of Equations in Three Variables 3.5 Solving Applications: Systems of Three Equations 3.6 Elimination Using Matrices 3.7 Determinants and Cramer's Rule 3.8 Business and Economics Applications
0321416163 Instructors edition! Item has some cover wear but otherwise in good condition!!Used texts may not include supplemental matieral. All day low prices, buy from us sell to us we do it all!!10.892454 |
Algebra Study Guide
Overview
"Algebra Study Guide
FEATURES:
- Includes both Algebra I and II
- Clear and concise explanations
- Difficult concepts are explained in simple terms
- Illustrated with graphs and diagrams
- Search for the words or phrases
- Access the guide anytime, anywhere - at home, on the train, in the subway.
- Use your down time to prepare for an exam.
- Always have the guide available for a quick reference. |
plex Numbers, Vectors and Matrices
Summary: This module sets out to instruct about complex numbers: what they are, what they mean, how to manipulate them, and the different ways to describe them (i.e. polar form). The second half of this module proposes to introduce the characteristics of complex vectors and matrices and how they compare to the laws governing standard vectors and matrices.
Complex Numbers
A complex number is simply a pair of real numbers. In order to
stress however that the two arithmetics differ we separate the
two real pieces by the symbol
i.
More precisely, each complex number,
zz, may be uniquely expressed by
the combination
x+iyxy,
where xx and
yy are real and
i denotes
-1-1.
We call xx the real part and
yy the imaginary part of
zz. We now summarize the main
rules of complex arithmetic.
Complex Vectors and Matrices
A complex vector (matrix) is simply a vector (matrix) of
complex numbers. Vector and matrix addition proceed, as in the
real case, from elementwise addition. The dot or inner product
of two complex vectors requires, however, a little
modification. This is evident when we try to use the old
notion to define the length of a complex vector. To wit, note
that if:
z=(1+i1−i)z11
then
zTz=1+i2+1−i2=1+2i−1+1−2i−1=0zz12121211210
Now length should measure the distance
from a point to the origin and should only be zero for the
zero vector. The fix, as you have probably guessed, is to sum
the squares of the magnitudes of the
components of zz. This is
accomplished by simply conjugating one
of the vectors. Namely, we define the length of a complex
vector via:
z=z¯Tzzzz
(4)
In the example above this produces
|1+i|2+|1−i|2=4=2121242
As each real number is the conjugate of itself, this new
definition subsumes its real counterpart.
The notion of magnitude also gives us a way to define limits
and hence will permit us to introduce complex calculus. We say
that the sequence of complex numbers,
znn=12…n12…zn,
converges to the complex number
z0z0
and write
zn→z0znz0
or
z0=limit n→∞znz0nzn
when, presented with any
ε>0ε0
one can produce an integer NN for which
|zn−z0|<εznz0ε
when
n≥NnN.
As an example, we note that
i2n→02n0.
Examples
Example 1
As an example both of a complex matrix and some of the rules
of complex
arithmetic, let us examine the following matrix:
F=(11111i-1−i1-11-11−i-1i)F11111-11-11-11-1
(5)
Let us attempt to find
FF¯FF.
One option is simply to multiply the two matrices by brute
force, but this particular matrix has some remarkable
qualities that make the job significantly
easier. Specifically, we can note that every element not on
the diagonal of the resultant matrix is equal to
0. Furthermore, each element on the
diagonal is 4. Hence, we quickly arrive at the matrix
FF¯=(4000040000400004)=4iFF40000400004000044
(6)
This final observation, that this matrix multiplied by its
transpose yields a constant times the identity matrix, is
indeed remarkable. This particular matrix is an example of a
Fourier matrix, and enjoys a number of interesting
properties. The property outlined above can be generalized
for any
FnFn,
where FF refers to a Fourier
matrix with nn rows and
columns |
More About
This Textbook
Overview
From the ability to understand and use shop mathematics to the reading and interpreting of shop drawings, the editor's intent is to provide the information and know-how that students will need as they prepare themselves for jobs in metalworking industries. It includes material taken from Machinery's Handbook and other authoritative sources and is presented in as clear, accurate, and easy-to-follow form as possible. The reader will find a wide range of useful formulas and data together with extensive text. As a proven and affordable handbook covering those critical areas of interest commonly encountered by machinists, toolmakers, diemakers, drafters, and other shop and manufacturing personnel, it is an essential reference for students in vocational schools, technical institutes, and apprenticeship courses.
Discusses those aspects of applied mathematics needed on the job and covers the proper use of measuring instruments and methods.
Outlines the standard methods of presentation and the conventions used in preparing engineering drawings.
Offers detailed information on inch and metric standard tolerances, allowances, limits and fits, preferred numbers and sizes, as well as in-depth descriptions of the sizes, forms, dimensions of standard machine elements commonly encountered in and around the shop.
Provides machining methods and materials selection, recommended speeds and feeds for various kinds of machining operations on different materials, and the types and compositions of metals commonly used in machine construction.
Includes an extensive index that will enable the user to quickly and conveniently find the information and data that he or she requires.
"...a beginner's version of the 'Machinery's Handbook'... prepares students in vocational schools, technical institutes, & apprentice programs to use the Handbook effectively in their professional careers |
11
4 8
3 1
2 0
1 2Thank you for your suggestion! We've passed this along to our developers for review. In the meantime, if you're experiencing issues with any query while using the app, we encourage user feedback by emailing us at android-support@wolframalpha.com
SimilarNeed more than free videos to learn math? YourTeacher's AlgebraMy daughter is doing Algebra 1 in 8th Grade. She had been getting really low grades because they are moving through the material so quickly. She had a test 3 days after we bought your program and she got 94% (the highest score in the class) because we had her work through the modules over and. She really enjoys the program and her motivation is good again." Melanie
CHAPTER 4: INEQUALITIES, ABSOLUTE VALUE, FUNCTIONS, GRAPHING Solving and Graphing Inequalities Combined Inequalities The Coordinate System Domain and Range Definition of a Function Function and Arrow Notation Graphing within a Given Domain Graphing Lines The Intercept Method Graphing Inequalities in Two Variables
CHAPTER 5: LINEAR EQUATIONS Patterns and Table Building Word Problems and Table Building Slope as a Rate of Change Using the Graph of a Line to Find Slope Using Slope to Graph a Line Using Coordinates to Find Slope (Graphs and Tables) Using Coordinates to Find Slope Using Slope to Find Missing Coordinates Using Slope-Intercept Form to Graph a Line Converting to Slope-Intercept Form and Graphing Linear Parent Graph and Transformations Using Graphs and Slope-Intercept Form Using Tables and Slope-Intercept Form Direct Variation Applications of Direct Variation and Linear FunctionsGetting math help is incredibly easy and efficient with this app; you choose a problem, enter the conditions, and get the solution and systematic explanation RationalIf you like Khan Academy, you'll LOVE Algebra TestBank! Use with Ted, Quora, and Star Chart also.
DragonBox Algebra 12+ gives players a greater understanding of what mathematics is all about: objects and the relationships between objects.
This educational game targets children from the ages of 12 to 17 but children (or adults) of all ages can enjoy it. Playing doesn't require supervision, although parents can enjoy playing along with their children and maybe even freshen up their own math skills.
DragonBox Algebra 12+ introduces all these elements in a playful and colorful world appealing to all ages.
The player learns at his/her own pace by experimenting with rules that are introduced gradually. Progress is illustrated with the birth and growth of a dragon for each new chapter.
Dr. Patrick Marchal, Ph.D. in cognitive science, and Jean-Baptiste Huynh, a high school teacher, created DragonBox Algebra 12+ as an intuitive, interative and efficient way to learn algebra.
DragonBox Algebra 12+ is based on a novel pedagogical method developed in Norway that focuses on discovery and experimentation. Players receive instant feedback which differs from the traditional classroom setting where students can wait weeks for feedback. DragonBox Algebra 12+ creates an environment for kids where they can learn, enjoy and appreciate math.
Our previous educational game, DragonBox Algebra 5+ has received many distinctions including the Gold Medal of the 2012 Serious Play Award (USA), the Best Serious game at Bilbao´s Fun and Serious Game Festival and the Best Serious Game at the 2013 International Mobile Gaming Awards. It is also recommended by Common Sense Media where it won the Learn ON award.
Well paying careers demand skills like problem solving, reasoning, decision making, and applying solid strategies etc. and Algebra provides you with a wonderful grounding in those skills - not to mention that it can prepare you for a wide range of opportunities.
This is a COMPLETE Pre-Algebra guide to well over 325 rules, definitions and examples, including number line, integers, rational numbers, scientific notation, median, like terms, equations, Pythagorean theorem and much more!
Our guide will take you step-by-step through the basic building blocks of Algebra giving you a solid foundation for further studies in our easy-to-follow and proven format!
Algebra is a very unique discipline. It is very abstract. The abstractness of algebra causes the brain to think in totally new patterns. That thinking process causes the brain to work, much like a muscle. The more that muscle works out, the better it performs on OTHER tasks. In simple terms, algebra builds a better brain! Believe it or not algebra is much easier to learn than many of us think and this guide helps make it easier!
Like all our 'phoneflips', this lightweight application has NO ads, never needs an internet connection and wont take up much space on your phone! weird**REAL TEACHER TAUGHT LESSONS** Algebra This algebra course teaches basic number operations, variables and their applications. Gain a fundamental sense of equations, inequalities and their solutions. This course offers 11 full chapters with 6-8 lessons each chapter that present short easy to follow algebra videos. These 5 to 10 minutes videos take students through the lesson slowly and concisely. Algebra is taken by students who have gained skills like operation with number, rational numbers, basic equations and the basic coordinate plane the Materials |
research-backed learning system for preparatory chemistry that systematically reinforces both the fundamental concepts and math skills that are prerequisites for success in general chemistry.Recent research has documented that in K–12 programs over the past decade, instruction in math computation has been de-emphasized. Calculations in Chemistry addresses these gaps in student background, applying the findings of recent scientific research on cognition, fluency in applied mathematics, and reading comprehension, to prepare students for the rigor and pace of general chemistry. |
Measure Theory
Review: Measure Theory
User Review - Thomas - Goodreads
I read up to the Radon theorem. This book is actually a good introduction to Lebesgue measure and integration. The author provides plenty of examples so as to reinforce the theory introduced. It is not too terse nor is it too wordy.Read full review
References from web pages
Math 543 - Measure Theory Measure theory is the axiomatized study of areas and volumes. It is the basis of integration theory and provides the conceptual framework for probability. ... ~runde/ math543.html
JSTOR: Measure Theory ISBN 0-8176-3003-1 3-7643-3003-1 (Birkhauser) Measure theory is often viewed by undergraduates as one of the drier and least appealing areas of modem ... links.jstor.org/ sici?sici=0025-5572(199503)2%3A79%3A484%3C222%3AMT%3E2.0.CO%3B2-I
More
28: Measure and integration Measure theory and integration is the study of lengths, surface area, ... Measure theory is a meeting place between the tame applicability of real functions ... ~rusin/ known-math/ index/ 28-XX.html
Computable Measure Theory Computable measure theory studies computability of functions related to .... In order to introduce computability to measure theory, we use the concept of ... Programs/ infinity/ files/ ding1.pdf |
This is the perfect introduction for those who have a lingering fear of maths. If you think that maths is difficult, confusing, dull or just plain scary, then The Maths Handbook is your ideal companion.
Covering all the basics including fractions, equations, primes, squares and square roots, geometry and fractals, Dr Richard Elwes will lead you gently... more...
These simple math secrets and tricks will forever change how you look at the world of numbers. Secrets of Mental Math will have you thinking like a math genius in no time. Get ready to amaze your friends?and yourself?with incredible calculations you never thought you could master, as renowned ?mathemagician? Arthur Benjamin shares his techniques... more...
A is for Algebra-and that's the grade you'll pull when you use Bob Miller's simple guide to the math course every college-bound kid must take
With eight books and more than 30 years of hard-core classroom experience, Bob Miller is the frustrated student's best friend. He breaks down the complexities of every problem into easy-to-understandFrom the author of the highly successful The Complete Idiot's Guide to Calculus comes the perfect book for high school and college students. Following a standard algebra curriculum, it will teach students the basics so that they can make sense of their textbooks and get through algebra class with flying colors. more...
Whether you're a science major, an engineer, or a business graduate, calculus can be one of the most intimidating subjects around. Fortunately, Calculus for the Utterly Confused is your formula for success. Written by two experienced teachers who have taken the complexity out of calculus for thousands of students, this book breaks down tough concepts... more...
Tips for simplifying tricky operations Get the skills you need to solve problems and equations and be ready for algebra class,... more...
Master algebra from the comfort of home!
Want to ?know it all? when it comes to algebra? Algebra Know-It-ALL gives you the expert, one-on-one instruction you need, whether you're new to algebra or you're looking to ramp up your skills.
Providing easy-to-understand concepts and thoroughly explained exercises, math whiz Stan Gibilisco serves... more...
Master calculus from the comfort of home! Want to "know it ALL" when it comes to calculus? This book gives you the expert, one-on-one instruction you need, whether you're new to calculus or you're looking to ramp up your skills. Providing easy-to-understand concepts and thoroughly explained exercises, math whiz Stan Gibilisco serves as your own... more...
We want to help you succeed on the GMAT math section If math is the hardest part of the GMAT for you, we're here to help. McGraw-Hill's Conquering GMAT Math is packed with strategies for answering every kind of GMAT math question. You'll also get intensive practice with every question type to help you build your test-taking confidence. With... more... |
Algebra II
This course begins with a brief review of Algebra I and extends to include number systems, polynomials, rational expressions, linear equations and inequalities, systems of equations, elementary exponential and logarithmic functions, right-triangle trigonometry, and elementary probability and statistics. A graphing calculator is required. |
The Algebra 1: The Complete Course DVD Series will help students build confidence in their ability to understand and solve algebraic problems.
In this episode, students will learn to use concrete examples and practical applications to understand algebraic concepts. Students will learn about sets of natural and whole numbers, sets of integers, sets of rational numbers and sets of real numbers. Grades 5-9. 30 minutes on DVD. |
Algebra Intervention: Not Business as Usual
In this archive, In this webinar archive, Mardi A. Gale, Senior Research Associate at WestEd, shares research based, essential elements of algebraic intervention programs, citing portions of the Aim for Algebra program to illustrate each of these elements.
Use the archived online slideshow below or the MP4 or MP3 recordings on the right.
Presenters
Description
"Algebra for all students" has created the need for intervention and supplemental programs to support students through the rigors of algebraic content. This need raises the issue of how to evaluate and choose an intervention or readiness program for students. In this webinar, Mardi A. Gale, Senior Research Associate at WestEd, will share research based, essential elements of algebraic intervention programs, citing portions of the Aim for Algebra program to illustrate each of these elements. Aim for Algebra is an algebra intervention program that is standards-aligned, conceptually based, and supports students through modules that focus on common barriers to success in algebra. To enhance retention, the conceptual nature of this targeted curriculum is scaffolded in accordance with current learning theory.
Participants in this webinar will engage in an examination of essential elements for intervention and support curricula that will assist and guide them as they search for appropriate programs. Participants will also learn about an algebra intervention program appropriate for in-school use, as part of existing programs or as a stand-alone, or for use as an extended day program. This intervention program has potential for use as pre-algebra curriculum as well.
This webinar will be helpful for middle and high school teachers of mathematics, math coaches, math curriculum supervisors/consultants, administrators, and district personnel in charge of intervention/support/extended day programs. |
Discrete Mathematics for Computing presents the essential mathematics needed for the study of computing and information systems. The subject is covered in a gentle and informal style, but without compromising the need for correct methodology. It is perfect for students with a limited background in mathematics. This new edition includes: An expanded section on encryption Additional examples of the ways in which theory can be applied to problems in computing Many more exercises covering a range of levels, from the basic to the more advanced This book is ideal for students taking a one-semester introductory course in discrete mathematics - particularly for first year undergraduates studying Computing and Information Systems. PETER GROSSMAN has worked in both academic and industrial roles as a mathematician and computing professional. As a lecturer in mathematics, he was responsible for coordinating and developing mathematics courses for Computing students. He has also applied his skills in areas as diverse as calculator design, irrigation systems and underground mine layouts. He lives and works in Melbourne, Australia.
Author's Biography
PETER GROSSMAN has worked in both academic and industrial roles as a mathematician and computing professional. As a lecturer in mathematics, he was responsible for coordinating and developing of mathematics courses for Computer Science students. He is based in Australia and currently works in industry, in the areas of mathematical modelling and software development. |
Prepare for algebra with simple equations, powers, polynomials and much more. More than 120 hours of instruction and 139 lessons on 10 interactive CDs with lectures, problems, step-by-step solutions, tests and automated grading. Includes a 750-page student workbook, answer key and 17 chapter tests. Version 2.0. Windows and Mac compatible.
The new and improved edition includes:
Interactive lectures
Automated grading
Over a dozen more lessons and hundreds of new problems and solutions
Hints and second chance options for many problems
Animated buddies to cheer the student on
Non-Consumable Textbook w/ full lecture
Reference numbers for each problem so students and parents can see where a problem was first introduced
A digital gradebook that can manage multiple student accounts and be easily edited by a parent
Built in back up and gradebook transfer feature
Compatible with PC (Windows XP or later) and Mac (OS 10.4 or later)
Students watch video lectures on CD-ROM, do problems from the 600-page workbook. When they need help or review, you've got a printed Answer Key plus audiovisual step-by-step solutions to every homework and quiz problem.
Sample Pages
Sample Lectures
TheLecture & Practice CD'scontain 10-15 minute lectures for every lesson in the print textbook. They also feature multimedia step-by-step explanations to the 5 practice problems that accompany each lesson. This set of CDs is ideal for students who prefer listening and watching to reading.
Sample Solutions
TheSolution CD's contain a multimedia step-by-step explanation to every single one of the almost 3,000 homework problems in the textbook. If you're tired of having to help your child do half his homework, or if you simply want to give him access to a library of quality explanations which will undoubtedly supplement and reinforce his understanding, this is the tool you need.
by Deborah G on 2013-05-28 We used Singapore for my oldest dd until 7th grade. Then, per Sonlight's recommendation, we switched to TT. We've done Pre-Algebra and 1/2 a year of Algebra. And my daughter who does complex math games in her spare time for fun has started complaining about math. I finally sat down and talked to her about it. She feels TT is too easy. I then really looked at it and I totally agree. The exercises are basic "chew it up and spit it out" type questions, and much of the time any "meat" is already half digested. No critical thinking required. I cannot believe Sonlight recommends this product.
I am giving it two stars because dd#2 ended up with about three years of patchy math instruction as we dealt with life. However, in 6th grade, she is excelling in the Pre-Algebra course -- in spite of all the holes in her math learning. She's a smart cookie, but not that smart. But it is doing an excellent job at helping her patch up the holes. We'll be switching back to Singapore next year.
by Lisa L on 2013-02-13 I have enjoyed using Teaching Textbooks with my son. I find that, even though it is easy, my son is learning the concepts well. I love that each question tells them what lesson they first learned the concept on, so they can go back and review if need be.
by Rina F on 2011-06-22 TT is great. DO take the placement test to find the correct level for your child. My daughter went from hating math to enjoying math. I was worried about how much she was retaining since it was so much fun. However, we are required to take standardized tests each year and she did excellent! VERY satisfied!
by Shannon S on 2011-06-09 Very easy to implement. My daughter did this program mostly independently. When she would run into something she didn't understand (once every 10-20 lessons or so), I could figure out how to help her simply by briefly reading the text with her and going over the examples they presented.
by JESSICA P on 2010-09-09 TT has saved our homeschool. Yes, it is a bit easy, so DO the PLACEMENT TESTS and place your child in the right level. It may be easy but it is also thorough. It DOES teach everything it ought to if you stick with it through the years. Who says learning math has to be difficult? I want my dd to learn math, not be frustrated. My dd is finally loving math. I honestly could not homeschool without TT, now that I've had a taste of how wonderful it is. Thank you, Sonlight for being the first to tell me about this awesome program!
by CINDY S on 2010-04-26 I have not used this series, but the problem I see is that the exercises do not list the lesson where the problem concept can be found. I know the problems are worked out on the CDs, but I think it is important for the student to look up the information so he can figure it out for himself. JMHO!
by JENNIFER S on 2009-10-25 I was excited to try Teaching Textbooks because of the high reviews; however, we found the grade 7 program to be far below what our son was doing in 6th grade math in public school. I exchanged the 7th grade package for the pre-algebra package and was even more disappointed. The content was so elementary he could do the entire book without ever using the teaching program. I am going to return this and try Singapore Math. Otherwise, I fear he will be far behind in his math skills by the time he enters high school.
by KARIN B on 2008-08-20 Inspite of the high ratings this math program has received, I and my son found it disappointing. After having done 6th grade "Horizons" math, combined with another textbook, my son skipped the 1st ten chapters of this book because it was too easy. I also spoke with other homeschooling mothers who used it for Algebra 1 and believe it left their kids ill prepaired to go into higher math. Perhaps it is good for kids who have not mastered basic math skills, but it does not challenge students or help them further their logical thinking skills.
by JENNIFER B on 2008-04-23 I highly recommend for all students. Even my 10 year old was able to understand the concepts easily.
by Kelby H. on 2008-04-01 Teaching Textbooks has made quite a difference in this household. No longer am I a frustrated math teacher! No longer do I take it personally when my child misses a problem, nor do I have to rant and rave about the "simple" mistake he made. The videos completely, patiently, and thoroughly explain all problems without judgement on the user. The result is a less frustrated math mom and jubilant, confident kids. I have to say that all of our math-i-tudes have improved. The material is challenging and relevant. The word problems are a hoot. And the best and worst thing is, everbody was good at math before, but hated it. Now we are still good (if not better, because we aren't killing each other), and LOVE it. My youngest child will also be switching in the fall to Teaching Textbooks 6. |
Kaplan GMAT Math Foundations
Overview
The refresher guide for math skills tested on the GMAT.Kaplan GMAT Math Foundations is the ideal refresher course for the large number of GMAT test takers who have been out of school for more than 10 years, including the more than 10,000 people who apply for highly competitive Executive MBA programs each year. Since more than 70 percent of GMAT students are over the age of 25, a refresher on basic math concepts is crucial.Kaplan GMAT Math Foundationsfeatures:Comprehensive coverage of the arithmetic, algebra, and geometry concepts tested by the GMAT An intensive, back-to-basics, tutor-led approach to math review Hundreds of practice exercises to increase speed and accuracyKaplan GMAT Math Foundationsis a great study tool for both test takers who dread the Math section and those whose math skills are not their strength. This guide will give test takers the content review and skill building practice they need to feel confident on test day148470
There are no customer reviews available at this time. Would you like to write a review? |
Synopses & Reviews
Publisher Comments:
This book covers combinatorial data structures and algorithms, algebraic issues in geometric computing, approximation of curves and surfaces, and computational topology. Each chapter fully details and provides a tutorial introduction to important concepts and results. The focus is on methods which are both well founded mathematically and efficient in practice. Coverage includes references to open source software and discussion of potential applications of the presented |
Pages to are hidden for
"Tutorial Support Class Curricular Materials"
Please download to view full document
2155873292561
Tutorial Support Class Curricular Materials
Guiding Principles:
1) Students who demonstrate mastery of the CAHSEE math standards should not be
enrolled in a tutorial support class.
2) Main focus of Pre-Algebra/Algebra A tutorial support class will be number sense.
3) Main focus of Algebra B/Algebra C tutorial support class will be the CAHSEE
standards.
4) In general, class time should be divided as follows:
Approximately 25% reviewing work from regular math class.
Approximately 50% remediation of math skills
Approximately 25% pre-viewing work from regular math class
5) Teachers who have access to remedial computer software (Larson Learning, New
Century or Accelerated Math) are encouraged to use that software in addition to the
"Getting Ready for Algebra" (Pre-Algebra/Algebra A) and "CAHSEE Prep" (Algebra
B/Algebra C) materials.
6) Teachers in every tutorial support class should have a class set of the Kaplin Skills
review booklets, and the Kaplan Test-taking Strategies booklets |
Finding roots of functions in one variable: Sections 3.03, 3.04, and 3.05.
Solving systems of linear equations: Chapter 4.
Integration (quadrature): Chapter 7.
Differential equations: Chapter 8.
Interpolation (curve fitting): Chapter 5.
By each section listed in the table of contents you can click on "pdf" to bring up a pdf file of that
section, or on "more" to bring up various supplementary material including videos of lectures by the
author and self-tests you can take to see if you are understanding the material.
Online course notes: Numerical methods in JavaScript.
This is a work in progress, as the euphemism goes. There is also lots of material available on the
web on how to write JavaScript programs. I usually google "JavaScript", or something specific like
"JavaScript for loop". The w3schools JavaScript
tutorial is a good source.
Online graphing calculator: GCalc. It's often useful to
have a visual description of a function. Start with GCalc 2.
Online integration, root finding, and more:
Numerical methods. Use the Web Tools on the left
of the page.
Two quotes from Richard Hamming:
"The purpose of computing is insight, not numbers."
"In research, if you know what you are doing, then you shouldn't be doing it."
The first one is the more relevant for us. The key phrase in the second is
"in research". It's okay to know what your doing in this course.
Programs: You are expected to write at least one JavaScript program each week.
Tests: There will be eleven quizzes given on
Fridays: August 26, September 2, 9, 16, 23, and 30,
October 7, 14, 21, and 28, and November 4. Also two on Mondays,
November 14 and 21, and one on Wednesday, November 30.
The quizzes will be 25 minutes long.
Under no circumstances will a make-up quiz be given.
No electronic devices (calculators,
cell phones, computers, mp3 players) may be used during the tests. |
Summary: This manual is organized to follow the sequence of topics in the text, and provides an easy-to-follow, step-by-step guide with worked-out examples to help students fully understand and get the most out of their graphing calculator. Compatible models include the popular TI-83/84 Plus and MathPrint. This manual will be packaged with every textGibsons Bookstore mi Lansing, MI
2012 Paperback Good |
CFA Formula: Level 2 is a comprehensive list of all formulas required by the CFA Curriculum for Level 2 examination that allows you to review anywhere. Included are more than 250 formulas and their definitions for Level 2. CFA Formula was featured on the iTunes App store - please visit for a video demo.The current version is fully updated for the June 2011 curriculum.CFA Formula provides an efficient way to test your knowledge on-the-go, or quickly look up a formula during practice exams.Features:- Search through more than 250 formulas- Create custom flashcard lists- View formula description/definition- Learning Outcome Statement (LOS) reference- Formula Topic Area- Test your knowledge using flashcardsFor easier recognition, formulas are colour coded to their Topic Areas. Formulas are sorted by Name or Topic Area, and are fully searchable for quick reference. The included Flashcard feature is a great way to test your knowledge and improve knowledge retention. You can select the Topic Areas to be tested on and CFA Formula will randomly select a flashcard. Alternatively, create a custom flashcard playlist that the flashcards will be selected from, allowing you to focus only on the formulas you find particularly difficult.For comments and suggestions, please visit or send an e-mail to admin@quantum-guides.com
What's new in this version: - Fixed a bug that caused the application to crash on certain devices running iOS 4.2 and 4.3- Updated for retina display- Made minor content corrections and adjustments |
ALEX Lesson Plans
Title: Exponential Growth and Decay
Description:
ThisStandard(s): [MA2013] AL1 (9-12) 7: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] [MA2013] AL1 (9-12) 9: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* [A-SSE 36 [F-BF3 ALC (9-12) 3: Use formulas or equations of functions to calculate outcomes of exponential growth or decay. (Alabama) [MA2013] ALT (9-12) 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] [MA2013] ALT (9-12) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2 PRE (9-12) 25: Compare effects of parameter changes on graphs of transcendental functions. (Alabama)
Subject: Mathematics (9 - 12) Title: Exponential Growth and Decay Description: This
Title: Rags to Riches or Riches to Rags?
Description:
InStandard(s): 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1 3: Use formulas or equations of functions to calculate outcomes of exponential growth or decay 7: Interpret expressions that represent a quantity in terms of its context.* [A-SSE] (9) 7: Write in narrative, expository, and persuasive modes using figurative language and imagery, including simile and metaphor, when effective and appropriate. [ELA] (10) 7: Write in persuasive, expository, and narrative modes using an abbreviated writing process in timed and untimed situationsThinkfinity Lesson Plans
Title: Graph Chart
Description:
This reproducible transparency, from an Illuminations lesson, contains the answers to the similarly named student activity in which students identify the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs.
Standard(s): [MA2013] (7) 2: Recognize and represent proportional relationships between quantities. [7-RP2] [MA2013] (8) 7: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. [8-EE5] [MA2013] (8) 11: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.) [8-F1] [MA2013] (8) 12: Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). [8-F2] [MA2013] (8) 13: Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear. [8-F3 25: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). [F-IF1] [MA2013] AL1 (9-12) 26: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. [F-IF2 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] [MA2013] AL2 (9-12) 21: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1
Subject: Mathematics Title: Graph Chart Description: This reproducible transparency, from an Illuminations lesson, contains the answers to the similarly named student activity in which students identify the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Title: Make a Conjecture
Description:
In this lesson, one of a multi-part unit from Illuminations, students explore rates of change and accumulation in context. They are asked to think about the mathematics involved in determining the amount of blood being pumped by a heart.
Standard(s): [MA20131 (9-12) 40: Interpret the parameters in a linear or exponential function in terms of a context. [F-LE5 3: Use formulas or equations of functions to calculate outcomes of exponential growth or decay. (Alabama) [MA2013] ALC (9-12) 5: Determine approximate rates of change of nonlinear relationships from graphical and numerical data. (Alabama) [MA2013] ALC (9-12) 12: Create a model of a set of data by estimating the equation of a curve of best fit from tables of values or scatter plots. (Alabama) [MA2013] AL2 (9-12) 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1]2 AL2 (9-12) 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5 37: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6] [MA2013] AL2 (9-12) 38: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7] [MA2013] ALT (9-12) 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1 29: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.* [F-IF5] [MA2013] ALT (9-12) 37: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. [S-ID4 49: Evaluate reports based on data. [S-IC6] [MA2013] ALT (9-12) 41: (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6] [MA2013] ALT (9-12) 42: (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7] [MA2013] AL1 (9-12) 35: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.* [F-BF2]
Subject: Health,Mathematics Title: Make a Conjecture Description: In this lesson, one of a multi-part unit from Illuminations, students explore rates of change and accumulation in context. They are asked to think about the mathematics involved in determining the amount of blood being pumped by a heart. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 |
Matlab is a mathematics software package that is programmed in 'C' and 'C++'. Matlab
offers all of the standard math operations that come in other mathematics software
packages (such as Maple and Mathematica), in the form of functions (much like a 'C'
program). In addition, Matlab has its own high-level interpreted programming language,
which makes it a favorite among engineering professors and professional engineers.
Matlab is widely used in junior and senior level electrical and computer engineering
classes. It is also available at the Tiger-Lan locations in the engineering building.
The following links are a series of Matlab tutorials that may be helpful.
Matthew Bledsoe is a busy man. He is pursuing a dual major in Electrical and Computer Engineering and expects to graduate in the Spring of 2012. In addition, he is a non-comissioned officer (NCO) in the US Army where he is currently engaged in a major competition for NCO of the Year. More ... |
books.google.com - If... mathematical & physical formulas
If how to solve every kind of math and physics problem you're likely to encounter in school and business, and it explains simply and easily how to find answers fast, learn key formulas and definitions, study quickly and learn more effectively--from fundamental mathematical rules to physical definitions and constants. Presents all formulas, rules, and definitions precisely, simply, and clearly. Covers metric units of measurement, U.S. units of measurement (USCS), tables of equivalents metrics and USCS units. Reviews the fundamentals of algebra, geometry, trigonometry, and analytical geometry. Presents the application of differential equations and integral calculus. Solves problems concerning simple interest, compound interest, effective rate, annuity, amortization of loans, and sinking fund payment. Shows the comparative advantages of binomial distribution, standard distribution, Poisson distribution, and normal distribution. Includes most used definitions and formulas of kinematics, dynamics, statics, mechanics of fluids, thermal variable of state, thermodynamics, electricity and magnetism, light, and basic definition of atomic and nuclear physics. Offers most used fundamentals of physical constants.
From inside the book
22 pages matching resistance in this book
Page xviii
Review: Applied Mathematical and Physical Formulas Pocket Reference
User Review - Em Chitty - Goodreads
I copy-edited this book, actually--it's by a Bosnian refugee whom I helped resettle in Knoxville. It is a fantastic little book of mathematical formulas for every purpose under the sun. Everything you ever didn't want to know about math but needed to. Even has some mortgage calculations, etc.Read full review |
MatBasic 1.28
The MatBasic is the language of mathematical calculations. Strong mathematical base: full complex arithmetic's, linear algebra and operations, nonlinear methods and graphical visualization.
Advertisements
Description:
MatBasic is a calculating, programming and debugging environment using special high-level programming language designed for solving mathematical problems. MatBasic programming language allows execution of difficult mathematical calculations, involving an exhaustive set of tools for the purpose of creation of algorithmic programs. It also allows a user to abstract his mind from the type of working data which can be either real-valued, or complex numbers, or matrices, or strings, or structures, etc. The MatBasic supports both the text and the graphical data visualization.
MatBasic is fast language interpreter and its environment application field is wide: from solving the school problem to executing different engineering and mathematical computations. The MatBasic programming language combines; simplicity of BASIC language, flexibility of high-level languages such as C or Pascal and at the same time turns up to be a powerful calculation tool. By means of a special operating mode, Matbasic it is possible to use as the powerful calculator. Also the MatBasic can be used for educational purpose as a matter of studying the bases of programming and raising algorithmization skillsThis software utility can plot regular or parametric functions, in Cartesian or polar coordinate systems, and is capable to evaluate the roots, minimum and maximum points as well as the first derivative and the integral value of regular functions. |
Linear algebra is now included in the undergraduate curriculum of most universities. It is generally recognized that this branch of algebra, being less abstract and directly motivated by geometry, is easier to understand than some other branches and that because of the wide applications it should be taught as soon as possible. This book is an extension... more...
This book introduces the important concepts of finite-dimensional vector spaces through the careful study of Euclidean geometry. In turn, methods of linear algebra are then used in the study of coordinate transformations through which a complete classification of conic sections and quadric surfaces is obtained. more...
Primarily a textbook to prepare Sixth Form students for public examinations in Hong Kong, this book is also useful as a reference for undergraduate students since it contains some advanced theory of equations beyond the sixth form level. more...
This book provides students of mathematics with the minimum amount of knowledge in logic and set theory needed for a profitable continuation of their studies. There is a chapter on statement calculus, followed by eight chapters on set theory. more...
This book provides students of mathematics with the minimum amount of knowledge in logic and set theory needed for a profitable continuation of their studies. There is a chapter on statement calculus, followed by eight chapters on set theory. more... |
Calculus 1
Calculus can be thought of as the mathematics of CHANGE. Because everything in the world is changing, calculus helps us track those changes. Algebra, by contrast, can be thought of as dealing with a large set of numbers that are inherently CONSTANT. Solving an algebra problem, like y = 2x + 5, merely produces a pairing of two predetermined numbers, although an infinite set of pairs. Algebra is even useful in rate problems, such as calculating how the money in your savings account increases because of the interest rate R, such as Y = X0+Rt, where t is elapsed time and X0 is the initial deposit. With compound interest, things get complicated for algebra, as the rate R is itself a function of time with Y = X0 + R(t)t. Now we have a rate of change which itself is changing. Calculus came to the rescue, as Isaac Newton introduced the world to mathematics specifically designed to handle those things that change.
Calculus is among the most important and useful developments of human thought. Even though it is over 300 years old, it is still considered the beginning and cornerstone of modern mathematics. It is a wonderful, beautiful, and useful set of ideas and techniques. You will see the fundamental ideas of this course over and over again in future courses in mathematics as well as in all of the sciences (e.g., physical, biological, social, economic, and engineering). However, calculus is an intellectual step up from your previous mathematics courses. Many of the ideas you will gain in this course are more carefully defined and have both a functional and a graphical meaning. Some of the algorithms are quite complicated, and in many cases, you will need to make a decision as to which appropriate algorithm to use. Calculus offers a huge variety of applications and many of them will be saved for courses you might take in the future.
This course is divided into five learning sections, or units, plus a reference section, or appendix. The course begins with a unit that provides a review of algebra specifically designed to help and prepare you for the study of calculus. The second unit discusses functions, graphs, limits, and continuity. Understanding limits could not be more important, as that topic really begins the study of calculus. The third unit introduces and explains derivatives. With derivatives, we are now ready to handle all of those things that change mentioned above. The fourth unit makes visual sense of derivatives by discussing derivatives and graphs. The fifth unit introduces and explains antiderivatives and definite integrals. Finally, the reference section provides a large collection of reference facts, geometry, and trigonometry that will assist you in solving calculus problems long after the course is over.
This course provides students the opportunity to earn actual college credit. It has been reviewed and recommended for 4 credit hours by The National College Credit Recommendation Service (NCCRS). While credit is not guaranteed at all schools, we have partnered with a number of schools who have expressed their willingness to accept transfer of credits earned through Saylor. You can read more about our NCCRS program here.
The chapters and sections of the original text have been reorganized and carefully aligned with the course subunits so that while solving any mathematical problem, the theory you might need to solve the problem is only a page or two away. The best way to proceed through the course is to read the assigned section in the order it is presented. You may download each assigned reading as you work through each subunit. If you prefer to download the entire text for the course, remember to refer to the reading titles, rather than the section or page numbers as these have been revised for the purpose of this course. Be advised that, depending upon your Internet speed, the file can take a couple of minutes to download, as it contains 329 pages and more than 300 megabytes of file size.
Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. Pay special attention to units 1 and 2, as these lay the groundwork for understanding the more advanced, exploratory material presented in the latter units. You will also need to complete problem sets in each unit and the final exam.
Note that you will only receive an official grade on your final exam. However, in order to adequately prepare for this exam, you will need to work through the problems presented for solution.
In order to pass this course, you will need to earn a 70% or higher on the final exam. Your score on the exam will be tabulated as soon as you complete it. If you do not pass the exam, you may take it again.
Time Commitment: We recommend that you dedicate approximately 2 or 3 hours of work every weeknight and 6 or 7 hours each weekend if you expect to perform highly in this course.
This course should take you a total of approximately 130.75 hours to complete. Each unit includes a time advisory that lists the amount of time you are expected to spend on each subunit. These should help you plan your time accordingly. It may be useful to take a look at these time advisories, determine how much time you have over the next few weeks to complete each unit, and then set goals for yourself. For example, Unit 1 should take you 7.75 hours. Perhaps you can sit down with your calendar and decide to complete Subunit 1.1 and Subunit 1.2 (a total of 2.5 hours) on Monday night; Subunit 1.3 and Subunit 1.4 (a total of 3.5 hours) on Tuesday night; Subunit 1.5 and the unit assessment (a total of 1.75 hours) on Wednesday night; and so forth.
Tips/Suggestions: Calculus takes time. Most people who fail a calculus course do so because they are unwilling, or unable, to devote the necessary time to the course.
Do not skip topics. The understanding of calculus is typically sequential. It is very difficult to understand one topic after lightly skipping over a preceding topic.
Test yourself. You are testing yourself when you follow the procedure of always solving a problem independently BEFORE looking at a solution of the same.
Work on details. Focus on the parts you missed. Determine what you did not understand before moving on.
This course has been developed through a partnership with the Washington State Board for Community and Technical Colleges. Unless otherwise noted, all materials are licensed under a Creative Commons Attribution 3.0 Unported License. The Saylor Foundation has modified some materials created by the Washington State Board for Community and Technical Colleges in order to best serve our users.
This course features a number of Khan Academy™ videos. Khan Academy™ has a library of over 3,000 videos covering a range of topics (math, physics, chemistry, finance, history and more), plus over 300 practice exercises. All Khan Academy™ materials are available for free at
While a first course in calculus can strike you as something new to learn, it is not comparable to learning a foreign language where everything seems different. Calculus still depends on most of the things you learned in algebra, and the true genius of Isaac Newton was to realize that he could get answers for this something new by relying on simple and known things like graphs, geometry, and algebra. There is a need to review those concepts in this unit, where a graph can reinforce the adage that a picture is worth one thousand words. This unit starts right off with one of the most important steps in mastering problem solving: Have a clear and precise statement of what the problem really is about.
Instructions: Read the brief text after the heading "Problems for Solution" on page 4, and then work through the odd-numbered problems 1 - 7 on page 5 and page 6. Once you have completed the problem set, check your answers for the odd-numbered questions here.
Instructions: Read Section 1.5 on pages 1 - 5 for an introduction to mathematical language, then work through practice problems 1 - 4. For the solutions to the practice problems, see page 7 and page 8.
Reading this section and completing the practice problems should take approximately 30 minutes.
Instructions: You are now ready to complete "Problem Set 1 30 minutes.
Unit 2: Functions, Graphs, Limits, and Continuity
The concepts of continuity and the meaning of a limit form the foundation for all of calculus. Not only must you understand both of these concepts individually, but you must understand how they relate to each other. They are a kind of Siamese twins in calculus problems, as we always hope they show up together.
A student taking a calculus course during a winter term came up with the best analogy that I have ever heard for tying these concepts together: The weather was raining ice - the kind of weather in which no human being in his right mind would be driving a car. When he stepped out on the front porch to see whether the ice-rain had stopped, he could not believe his eyes when he saw the headlights of an automobile heading down his road, which ended in a dead end at a brick house. When the car hit the brakes, the student's intuitive mind concluded that at the rate at which the velocity was decreasing (assuming continuity), there was no way the car could stop in time and it would hit the house (the limiting value). Oops. He forgot that there was a gravel stretch at the end of the road and the car stopped many feet from the brick house. The gravel represented a discontinuity in his calculations, so his limiting value was not correct.
Instructions: Read Section 2.1 on pages 1 - 7 for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1 - 4. For the solutions to these practice problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: Read Section 2.2 on pages 1 - 7 for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1 - 4. For solutions to these practice problems, see page 10.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: Watch this video on finding limits algebraically. Be warned that removing x - 4 from the numerator and denominator in Step 4 of this video is only legal inside this limit. The function (x - 4)/(x - 4) is not defined at x = 4; however, when x is not 4, it simplifies to 1. Because the limit as x approaches 4 depends only on values of x different from 4, inside that limit (x - 4)/(x - 4) and 1 are interchangeable. Outside that limit, they are not! However, this kind of cancellation is a key technique for finding limits of algebraically complicated functions.
Watching this video and pausing to take notes should take approximately 30 minutes.
Instructions: Watch this video on limits as the slopes of tangent lines.
An earlier Khan Academy video (not used in this course) defined the limit that gives the slope of the tangent line to a curve as y = f(x) at a point x = a and called it the derivative of f(x) at a. Your text will introduce this term in Unit 3.
Watching this video and taking notes should take approximately 30 minutes.
Instructions: You are now ready to complete "Problem Set 2 2.4 on pages 1 - 11 for an introduction to what we mean when we say a function is continuous. Work through practice problems 1 and 2. For solutions to these problems, see page 16.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: You are now ready to complete "Problem Set 3 1 hour.
Unit 3: Derivatives
In this unit, we start to see calculus become more visible when abstract ideas such as a derivative and a limit appear as parts of slopes, lines, and curves. Then, there are circles, ellipses, and parabolas that are even more geometric, so what was previously an abstract concept can now be something we can see. Nothing makes calculus more tangible than to recognize that the first derivative of an automobile's position is its velocity and the second derivative of that position is its acceleration. We are at the very point that started Isaac Newton on his quest to master this mathematics, what we now call calculus, when he recognized that the second derivative was precisely what he needed to formulate his Second Law of Motion F = MA, where Fis the force on any object, Mis its mass, and A is the second derivative of its position. Thus, he could connect all the variables of a moving object mathematically, including its acceleration, velocity, and position, and he could explain what really makes motion happen.
Instructions: Read Section 3.1 on pages 1 - 5 to lay the groundwork for introducing the concept of a derivative. Work through practice problems 1 - 5. For solutions to these problems, see page 8 and page 9.
Reading this section and completing the practice problems should take approximately 45 minutes.
Instructions: You are now ready to complete "Problem Set 4 3.7 on pages 1 - 7 to learn to connect derivatives to the concept of the rate at which things change. Work through practice problems 1 - 3. For solutions to these problems, see page 12 and page 13.
Reading this section and completing the practice problems should take approximately 45 minutes.
Instructions: Read Section 3.9 on pages 1 - 10 to learn how linear approximation and differentials are connected. Work through practice problems 1 - 10. For the solutions to these problems, see page 14 and page 15.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Instructions: You are now ready to complete "Problem Set 5 4: Derivatives and Graphs
A visual person should find this unit extremely helpful in understanding the concepts of calculus, as a major emphasis in this unit is to display those concepts graphically. That allows us to see what, so far, we could only imagine. Graphs help us to visualize ideas that are hard enough to conceptualize - like limits going to infinity but still having a finite meaning, or asymptotes - lines that approach each other but never quite get there.
Graphs can also be used in a kind of reverse by displaying something for which we should take another mathematical look. It is hard enough to imagine a limit going to infinity, and therefore never quite getting there, but the graph can tell us that it has a finite value, when it finally does get there, so we had better take a serious look at it mathematically.
Instructions: Read Section 4.1 on pages 1 - 9 to learn about maximums, minimums, and extreme values for functions. Work through practice problems 1 - 5. For solutions to these problems, see page 13 and page 14.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: Read Section 4.3 on pages 1 - 8 to learn how the first derivative is used to determine the shape of functions. Work through practice problems 1 - 9. For the solution to these problems, see pages 10 - 12.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Instructions: Read Section 4.4 on pages 1 - 6 to learn how the second derivative is used to determine the shape of functions. Work through practice problems 1 - 9. For solutions to these problems, see page 8 and page 9.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Instructions: Watch this video on the second derivative test. This video describes a way to identify critical points as minima or maxima other than the first derivative test, using the second derivative.
Watching this video and taking notes should take approximately 30 minutes.
Instructions: You are now ready to complete "Problem Set 6 4.5 on pages 1 - 6 to learn how to apply previously learned principles to maximum and minimum problems. Work through practice problems 1 - 3. For solutions to these problems, see page 15 and page 16. There is no reading review for this section; instead, make sure to study the problems carefully to become familiar with applied maximum and minimum problems.
Reading this section and completing the practice problems should take approximately 45 minutes.
Instructions: Read Section 4.6 on pages 1 - 10 to learn how to use and apply infinite limits to asymptotes. Work through practice problems 1 - 8. For solutions to these problems, see page 13 and page 14.
Reading this section and completing the practice problems should take approximately 1 hour and 30 minutes.
Instructions: You are now ready to complete "Problem Set 7 5: The Integral
While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with the development of the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the notion of an area.In fact, we will be able to extend the notion of the area and apply these more general areas to a variety of problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of Calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.
Instructions: Read Section 5.4 on pages 1 - 8 to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1 - 5. For solutions to these problems, see page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: Read Section 5.5 on pages 1 - 6 to learn about the relationship among areas, integrals, and antiderivatives. Work through practice problems 1 - 5. For solutions to these problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: You are now ready to complete "Problem Set 8 5.7 on pages 1 - 9 to see how one can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1 - 4. For solutions to these problems, see page 12 and page 13.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: Read Section 5.8 on pages 1 - 8 to see how some applied problems can be reformulated as integration problems. Work through practice problems 1 - 4. For solutions to these problems, see page 10 and page 11.
Reading this section and completing the practice problems should take approximately 1 hour.
Instructions: Read Section 5.9 on pages 1 - 3 to learn how to use tables to find antiderivatives. See the following "Calculus Reference Facts" for the table of integrals mentioned in the reading. Work through practice problems 1 - 5. For solutions to these problems, see page 6.
Reading this section and completing the practice problems should take approximately 45 minutes.
Instructions: You are now ready to complete "Problem Set 9 6: Appendix
By reviewing and having access to this unit, you will have an invaluable list of references to assist you in solving future calculus problems after this course has ended. It is a standard experience, when solving calculus problems on your own, to react to the new problem with the following: "We did not solve that kind of problem in the course." Ah, but we did, in that the new problem is often a combination, or composition, of two problem types that were covered.
The course could not cover all possible trigonometric functions you will encounter. If you encounter a need for the derivative of tan(x),it is sufficient to recall that tan(x) = sin(x)/cos(x)and that sine and cosine were covered. You can eventually become so good at this that future calculus problems can almost seem to be little more than plugging into formulas.
Instructions: There are neither readings nor problems associated with this section. Rather, it consists of two pages of formulas that can be useful to you in your further explorations of calculus, including the final exam. You should be able to quickly print out these two pages or save them where they can be easily located as a quick reference when needed
Instructions: The above linked exam has been specially created as part of our National College Credit Recommendation Service (NCCRS) review program. Successfully passing this exam will make students eligible to receive a transcript with 4 hours of recommended college credit.
Please note that because this exam has the possibility to be a credit-bearing exam, it must be administered in a proctored environment, and is therefore password protected. Further information about Saylor's NCCRS program and the options and requirements for proctoring, can be found here. Make sure to read this page carefully before attempting this exam.
If you choose to take this exam, you may want to first take the regular, certificate-bearing MA005 Final Exam as a practice test, which you can find above. |
...
Show More environmental essays that open each chapter connect algebra to real world problem solving and can be used to stimulate class discussions and promote collaborative learning. Functions and graphing are introduced early (Chapter Three) and are then integrated throughout the rest of the text. This approach allows for visual interpretation of the mathematical concepts, which in turn encourages students to develop an intuitive understanding of equations and their graphs. Also by introducing these topics early, students become familiar and comfortable with concepts that are critical to their success in future math courses. Intermediate Algebra includes marginal notes and examples that indicate how technology can enhance the study of algebra through exploration, visualization, and geometric interpretation. These examples fall at the end of section discussions and may be omitted if a graphing tool is not being used. The text is written in a clear, concise style with numerous examples which are connected by thoughtful transitions that either reinforce the student's understanding of the previous concepts or prepare them for the next example. Each example is followed by a "Check Yourself' exercise that facilitates the student's active involvement in the learning process |
What's the sure road to success in calculus? The answer is simple: Precalculus.
Traditionally studied after Algebra II, this mathematical field covers advanced algebra, trigonometry, exponents, logarithms, and much more. These interrelated topics are essential for solving calculus problems, and by themselves are powerful methods for describing the real world, permeating all areas of science and every branch of mathematics. Little wonder, then, that precalculus is a core course in high schools throughout the country and an important review subject in college.
Unfortunately, many students struggle in precalculus because they fail to see the links between different topics?between one approach to finding an answer and a startlingly different, often miraculously simpler, technique. As a result, they lose out on the enjoyment and fascination of mastering an amazingly useful tool box of problem-solving strategies.
And even if you're not planning to take calculus, understanding the fundamentals of precalculus can give you a versatile set of skills that can be applied to a wide range of fields?from computer science and engineering to business and health care.
Mathematics Describing the Real World: Precalculus and Trigonometry is your unrivaled introduction to this crucial subject, taught by award-winning Professor Bruce Edwards of the University of Florida. Professor Edwards is coauthor of one of the most widely used textbooks on precalculus and an expert in getting students over the trouble spots of this challenging phase of their mathematics education.
"Calculus is difficult because of the precalculus skills needed for success," Professor Edwards points out, adding, "In my many years of teaching, I have found that success in calculus is assured if students have a strong background in precalculus."
A Math Milestone Made Clear
In 36 intensively illustrated half-hour lectures, supplemented by a workbook with additional explanations and problems, Mathematics Describing the Real World takes you through all the major topics of a typical precalculus course taught in high school or college. Those who will especially benefit from Professor Edwards's lucid and engaging approach include
* high school and college students currently enrolled in precalculus who feel overwhelmed and want coaching from an inspiring teacher who knows where students stumble;
* parents of students, who may feel out of their depth with the advanced concepts taught in precalculus;
* those who have finished Algebra II and are eager to get a head start on the next milestone on the road to calculus;
* beginning calculus students who want to review and hone their skills in crucial precalculus topics;
* anyone motivated to learn precalculus on his or her own, whether as a home-schooled pupil or as an adult preparing for a new career.
35. Elementary Probability
What are your chances of winning the lottery? Of rolling a seven with two dice? Of guessing your ATM PIN number when you?ve forgotten it? Delve into the rudiments of probability, learning basic vocabulary and formulas so that you know the odds.
36. GPS Devices and Looking Forward to Calculus
In a final application, locate a position on the surface of the earth with a two-dimensional version of GPS technology. Then close by finding the tangent line to a parabola, thereby solving a problem in differential calculus and witnessing how precalculus paves the way for the next big mathematical adventure.
HOme: |
Mathematical Models in Biology: An Introduction is an introductory textbook in discrete mathematical modeling covering a wide variety of biological topics: dynamic models of population growth, models of molecular evolution, the construction of phylogenetic trees, genetics, and infectious disease modeling. The authors, Elizabeth S. Allman and John A. Rhodes, describe their target audience as undergraduates "having a strong interest in biological science and a mathematical background sufficient to study calculus." While they do not assume any training in calculus or beyond, they do include (well-marked) supplemental exercises that give students who have had calculus an opportunity to pursue the connection between the discrete and the continuous. The authors have made some MATLAB programs available to accompany the text "to give readers hands-on experience with the mathematical models developed." However, technology does not play a leading role in the text. Depending on the availability of software packages and the chapters to be covered, I can imagine a situation where a professor might choose to use a computer algebra system like Maple or Mathematica instead of MATLAB. However, this would require some additional effort and would probably not be a reasonable option if Chapters 4 and 5 are covered, since the MATLAB files supporting these chapters rely more heavily on MATLAB's numerical capabilities.
The biggest strength of the text is that it touches on a variety of topics from a wide range of mathematical subdisciplines (e.g., mathematical modeling, discrete dynamical systems, linear algebra, and probability,) while following a coherent and logical pathway through an interesting set of biological topics. Hence an undergraduate studying with this text gains a broader view of mathematics as well as a feel for how mathematical models are constructed and used in biology.
Chapter 1 (Dynamical Modeling with Difference Equations) covers discrete models of single populations, starting with the exponential growth model and proceeding on to the logistic growth model, before discussing the different types of long-term behavior (equilibrium points, n-cycles, and chaos.) The final sections of the chapter cover variations of the logistic growth model and include a few closing comments about discrete models versus continuous models. There are plenty of exercises supporting each section, but many of the exercises involve the analysis of a model that is already set up for the student. I would prefer more exercises requiring the students to construct population models for themselves — given certain assumptions and conditions. I was also surprised by the scarcity of graphs in this chapter as I find that first- and second-year students gain a great deal from visualization.
Chapter 2 (Linear Models of Structured Populations) focuses primarily on discrete models of a single population that is partitioned into subpopulations. For example, the first example in the chapter divides a population of insects into three groups: eggs, larva, and adult. Given that death rates differ across these different stages in insect development, it makes sense to break the population up in this way. The authors extend the models from Chapter 1 by introducing some matrix algebra. They introduce the Leslie population model, mentioning it is a Markov model, and then finish up the chapter with a section on eigenvalues and eigenvectors. Because of the level of mathematical maturity required to tackle the material, I do not think that this last section is accessible to the typical college student who has not taken calculus. A more realistic audience would be a math student at the sophomore level. Nonetheless, I liked the progression of topics. The material builds nicely upon Chapter 1.
Chapter 3 (Nonlinear Models of Interaction) introduces the dynamics of interacting populations, starting with a simple predator-prey model. There is a nice coverage of phase plane plots, equilibria and stability before the authors introduce other examples of interacting population models. The final section considers three different situations: Competition (where two species are competing for resources), Immune system vs. infective agent (where a disease-causing agent infects an organism resulting in an interaction that is detrimental for both), and Mutualism (where the interaction is positive to both parties.) The authors provide numerous exercises throughout the chapter. Once again, however, most of the exercises require an analysis on models that are already given to the student.
Chapter 4 (Modeling Molecular Evolution) presents mathematical models that describe the process of DNA mutation. The first section provides a clear and informative presentation of the necessary background information on DNA. Because mutations are random events, the models of molecular evolution require a basic knowledge of probability theory. To this end, the authors include sections on introductory probability and conditional probabilities. The final two sections develop and analyze the mathematical models, all of which are Markov models. The authors start with the Jukes-Cantor one-parameter model and then present the Kimura 2-parameter model. The chapter closes with a discussion of phylogenetic distances, which quantify the amount of mutation that has occurred, allowing scientists to make meaningful comparisons. I found this chapter to be delightful! The authors are successful in explaining the preliminary information about DNA without getting too caught up in jargon, and the models seem to be quite accessible. The MATLAB files that are available are also helpful here in that students can experiment with the models and observe the results. Biology students (and others) are certain to find this material interesting.
Chapter 5 (Constructing Phylogenetic Trees) considers methods of constructing phylogenetic trees that best reflect the ancestral history of an organism or a group of organisms. The first section provides preliminary information about trees, defining topological trees (in which only the branching structure matters) and metric trees (in which the branch length represents the amount of mutation that occurred between splittings of the lineage.) In section 2, the authors introduce two distance methods for constructing phylogenetic trees: the unweighted pair-group method with arithmetic mean (UPGMA) and the Fitch-Margoliash algorithm. While neither of these distance methods is typically used in practice, the authors present them to motivate the Neighbor Joining algorithm, which "has become the distance method of choice for tree construction." Section 4 covers a tree construction method known as the method of Maximum Parsimony, and Section 5 provides a brief discussion about other methods (Maximum Likelihood and Bootstrapping.) The chapter closes with a list of applications and some suggestions for further reading. At the end of the chapter, the authors also provide a nice set of exercises that make good use of technology, as well as a project titled "Dental transmission of HIV" that requires students to construct phylogenetic trees to determine clustering patterns that will allow them to identify patients that were infected with the HIV virus by their dentist. The project uses DNA sequences mined from a paper that appeared in Science in 1992. The project looks fascinating! As with Chapter 4, I found this chapter to be a delight.
Chapter 6 (Genetics) presents a brief introduction to genetics while providing an extension of the discussion on probability in Chapter 4. Section 1 focuses on Mendelian genetics, describing Mendel's groundbreaking work in understanding the genetics of pea plants. Section 2 discusses probability distributions in genetics (in particular, the binomial distribution and the Chi-squared distribution.) Numerous exercises follow this section. The authors then discuss linked genes and gene mapping, and the chapter ends with a section on gene frequencies in populations. In this final section, the authors discuss the Hardy-Weinberg equilibrium, fitness and selection, and genetic drift. Other than the probability theory introduced in Chapter 4, this chapter seems to stand alone.
Chapter 7 (Infectious Disease Modeling) revisits the population models covered in Chapter 3, extending the ideas further to model the dynamics of infectious disease. The chapter begins with the famous SIR-model, in which a population is divided into three categories: the susceptibles, the infected, and the removed. The authors present and analyze the model — considering threshold values and critical parameters — before presenting two variations of the model: the SI-model and the SIS-model. The former reflects a situation where a member infected by the disease never recovers (as in the case of untreated head lice.) In the latter model members can recover, but are then susceptible again (as in the case of syphilis and gonorrhea.) The chapter closes with an introduction to differentiated infectivity models. These models can account for the fact that certain subgroups of a population are more (or less) likely to become infected with a disease. The authors introduce the differentiated model using a "sexy" example that explores the spread of gonorrhea among a population.
Chapter 8 (Curve Fitting and Biological Modeling) appears to represent the authors' recognition that statistics plays an important role in biological modeling. Although their textbook focuses on mathematical models in biology, the authors want to note that statistical methods are necessary in both developing meaningful models from real data and in assessing the validity of models. To this end, the authors provide a quick overview of curve fitting. They introduce semilog and log-log graphs, the method of least squares, and finally polynomial curve fits. (They also include an appendix that touches on the basic analysis of numerical data.) It is worth noting that Chapter 8 does not build upon the models covered earlier in the text, but rather serves as a stand-alone overview of statistical approaches to biological modeling. As such, I don't see how I would use this chapter when teaching the course, and I imagine that many biologists would prefer that statistics play a more vital and central role in a textbook about biological modeling. That said, I understand why one would want to include this chapter, and I think that the authors' approach is reasonable for what they have set out to accomplish in this book.
Again, the most positive aspect of the text is the fact that it covers a diverse set of mathematical topics while pursuing an interesting and coherent study of biological models. Chapters 5 and 6 are particularly interesting, and they appear to be unique to this textbook. Given the importance of bioinformatics in scientific research today, these chapters are certain to raise enthusiasm among students and faculty alike.
Regarding the authors' claim that this textbook is suited for the student "having a strong interest in biological science and a mathematical background sufficient to study calculus," I have some doubts. While I agree that the content of the text does not rely on calculus, I do believe that parts of the text require a level of mathematical sophistication that isn't typically seen in students who have not taken calculus. For this reason, I suspect that a more realistic target audience would be students who have already had a year of calculus. Professors should be aware of this and plan accordingly by advertising the course to a sophomore-level audience.
I have already touched upon the negative comments I would make about the text. First, I would prefer to see more exercises that require the students to construct models for themselves. Second, I thought that the authors could have placed a greater emphasis on visualization throughout Chapters 1-3. Much of the material relating to population models can be reinforced with meaningful graphs, yet there were very few graphs in those sections. Finally, I felt that the role of technology could have been greater, but perhaps individual faculty can address this issue when teaching the course.
Overall, I found this to be a very nice textbook. It would be appropriate for an undergraduate course in mathematical modeling for math majors, or it could serve as a course designed specifically for students interested in pursuing mathematical biology (e.g., biology majors having a mathematics minor.) Given the growing interest in the field of mathematical biology, I imagine that such a course would be attractive to students.
Judy Holdener is currently an Associate Professor at Kenyon College in Gambier, Ohio. She can be reached at holdenerj@kenyon.edu. |
Geometry
Description: The theorems and principles of basic geometry are clearly presented in this workbook, along with examples and exercises for practice. All concepts are explained in an easy-to-understand fashion to help students grasp geometry and form a solidMore...
The encourage students to enjoy working the pages while gaining valuable practice in geometry |
From a review of the first edition:
This book is suitable for those who lack confidence … It is particularly useful as a revision guide... Tasks and practical exercises are included throughout the book. Worth buying a copy' - Primary Mathematics
This task-driven text emphasizes strategies and processes and is very different from the usual style of mathematics textbooks. For example, algebra is treated as a way of thinking mathematically, rather than merely manipulating symbols. Each of the sections is designed to stand alone so that they can be studied in any order or dipped into as needed.
The Second Edition has been updated to meet the needs of anyone wanting to refresh their knowledge and understanding of mathematics to GCSE level. There is an emphasis on learning and doing mathematics, with an expanded section on measures and proportion, and statistics.
Trainees and students have reported that the book has not only helped them to raise their knowledge and understanding to the required standard but also greatly improved their mathematical confidence.
This is a set book for the Open University Course, 'Ways of Knowing: language, mathematics and science in the early years'. |
Math Tutor educational software series
high school and middle school math tutorial software programs.
Schools seeking
curriculum based interactive math learning software
for Grades 6-12 will find the series to be ideal. Parents in need of
homeschool math learning and tutorial software
will also find the series to be ideal. Schools, families, and individuals alike have found the Math Tutor series to offer the
best math educational teaching software
available anywhere. The series is also effective for
remedial and developmental math instruction.
The subject of pre-algebra serves as a critical foundation for the math subjects that follow. Our
pre-algebra tutorial software program
will help students master the concepts of pre-algebra and build proficiency in problem solving.
It will give students outstanding preparation for the math courses to come.
Each of the subjects that follow pre-algebra - algebra 1, geometry, algebra 2, trigonometry, and calculus - is given outstanding, comprehensive treatment by other programs in the series. For example, Algebra 1 in the "Concepts and Skills Series" will help students learn the concepts of elementary algebra, and basic problem solving. Algebra 1 in the "Practice and Review Series" will help students build mastery in solving problems of intermediate and advanced difficulty.
The Math Tutor series features self-paced lessons suitable for students at all levels of ability. The series helps students gain the confidence so critical for success in math.
Additional titles in the Math Tutor series include SAT/ACT Math, and Learning Business Math.
The award-winning INTELLIGENT TUTOR™ series is the ideal resource for Grades 6-12 math learning. The series helps students build math skills, gain confidence, and develop a solid foundation for future math success, in the classroom and beyond.
The programs in the INTELLIGENT TUTOR™ series use graphics and animation to help students learn the concepts, ideas, and methods of math. Interactive lessons are designed to help students at all levels to build confidence and mastery at their own pace.
The INTELLIGENT TUTOR™ series covers pre-algebra, algebra 1 and 2, geometry, trigonometry, calculus, SAT/ACT math, and business math. The Concepts and Skills series features interactive, self-paced tutorials that help students master concepts. The Practice and Review series helps students build problem solving skills using the concepts they have learned.
Outstanding curriculum-based series designed by math educators.
Ideal both as a supplement to classroom instruction, and for self-study.
The INTELLIGENT TUTOR™ software series for Grades 6-12 math is ideal for homeschooling families, self-study, and independent learning environments. Each of the titles in the series is an interactive software program designed for students and adult learners who wish to learn the concepts of math and build problem solving skills.
Self-paced tutorials allow students to learn and master each concept before moving on to the next.
Students and adult learners will find that the programs in the INTELLIGENT TUTOR™
series truly bring math to life. Graphics and animation are used throughout the lessons to help students
visualize the key concepts of math. Even students who have struggled will
find themselves learning math with ease.
Award-winning software series designed for self-paced, independent study |
Solution There are as many words as there are ways of filling in 4 vacant places ... Solution There are 12 letters, of which N appears 3 times, E appears 4 times and D ... From a class of 25 students, ...
learning of math would continue beyond the 140 periods. 3. ... Class V, which overlap very often, not only with each other but also with themes developed in other subjects that are being learnt simultaneously. 4. ... 12. The purpose is not ...
The students of Elective Math will necessarily also study the Core Math module. ... NCERT approved textbooks may be used as most of the core topics are covered in ... Acquire basic skills relating to comprehension and solution of problems by analysis of data, ...
recommendations of the National Curriculum Framework developed by NCERT, ... 12. To make a right circular cylinder of given height and circumference of base ... solution (intersecting lines); some have infinitely many solutions (coincident lines) ...
... ICSE & NCERTCLASS-XII Unit I : Relations and Functions ... students must learn mathematical notation. In math-ematical notation, one common symbol for the derivative of function is an apostro- ... Class-XII-9 12. The three roots of the equation 3 7 2 2 0 7 6 x x x
needed problems matching the experience and needs of the children of her class. 12. There should be continuity of the presentation within a chapter and across the chapters. Opportunities should be taken to give students the feel for need of a topic, which may follow later.
Class12 Hardware Concepts, IT Tools for Life . ... Math Lab is completely structured on NCERT guidelines. It enables teacher to demonstrate, explain and reinforce ... solution that revolutionises the teaching and learning of subjects like
NCERT TEXT BOOK Chapters Concepts Number of Questions for revision Total ... Detail of the concepts to be mastered by every child of class XII with exercises and examples of NCERT Text Book. ... ** Q. 3,6,12 pg 406 Solution of differential equation of the type dy/dx +py=q where
• Successfully complete Class12 ... NCERT. 19 Life Outcomes: What can they do and who can they be? Only 6% married All after the legal age of 18 88% Graduation Rate ... If our technology solution is successful for scale up, it can be used
only the topics of class12, these topics remain neglected. ... Now we study the NCERT mathematics textbooks of class XI of four representative years (1988, 1995, ... This requires collaboration with technology solution
... to transform a class into a dynamic learning space. S. CHAND HARCOURT ... Math is learned, as the NCERT and NCF 2005 recommend, ... to make best use of a seamless integrated blended learning solution for classes K-12.
removes the math's phobia from the young minds right from class 6th & enables them to learn easily and effectively. ... click providing a different and time smart solution for which they are trained through different preparation ... Not only NCERT, ...
presenting and explaining mathematics to the whole class. It can be ... 1/6, 1/12 family . TOPIC Straight line graphs • Straight line graphs represent linear functions. • The general equation of a straight line graph is y = mx + c.
of the subject from NCERT curriculum are divided into two years conveniently in Maharashtra State. ... Classification of phylum chordata upto class level with three to five salient features and two ... 12 Study of external morphology of earthworm, cockroach and frog through models.
ncert/biology/NCERT_Biology_Class_12_ and its ... Applied mathematics/Intro to the Math. and Stat. Foundations of Econometrics - H ... pure/algebr a ...
sured the state government into removing the mandatory pass requirement in math-ematics for the school exit examination in the year 2010. ... and the solution of simple equations in one unknown. ... a general way" (Mathematics: Text book for class VI 2006, p. 221). It then provides
students of class 6th to 12th covering CBSE, ... different and time smart solution for which they are trained through different preparation programs in one year duration packages. ... Complete solved NCERT ...
Refer to NCERT Mathematics Book of Class VII. ... Group 2: Roll No. 7 to 12 Topic: Multiplication of Integers using patterns Group 3:Roll No. 13 to 18 Topic: Related Angles (definition and their properties) ... Prepare a solution of soap and then add a few drops of turmeric solution to it. |
Take away number in all things and all things perish.
Take calculation from the world and all is enveloped in dark ignorance,
nor can he who does not know the way to reckon be distinguished from the
rest of the animals.
—St. Isidore of Seville
The Mathematics Minor is a great way to enhance your education, increase your set of skills, learn interesting mathematics, and see how it's used in your field. From a professional standpoint a Mathematics Minor can help your degree stand out to prospective employers. It will allow you to work in many computer-related jobs. It is also good preparation if you plan to take the Engineer in Training exam.
Anyone considering entering the teaching profession will find their employment prospects are proportional to the number of mathematics courses taken.
The tradition of Aristotle and St. Thomas sees that mathematics is the science of abstract quantity, a science which arises directly or by analogy from a consideration of quantity as found in the physical world, which has the fundamental property of "having part outside of part." The two branches of mathematics, Geometry and Algebra, arise out of the observation that the parts can have common boundaries (continuous quantity) or no common boundary (discrete quantity).
This discipline, delightful to know in itself, is also an essential part of a liberal education: the unique simplicity of its subject matter allows its students to practice logical thought in a realm in which truth is readily apparent; and its instrumental use opens insights into the nature of physical reality.
Christendom offers one elementary course in Euclidean Geometry and another in the historical development and philosophical aspects of mathematics. Both courses help the student understand the place of mathematics in man's understanding of the world around him. College Algebra and several more advanced courses deepen a student's mathematical knowledge, as well as preparing him for programs in business, engineering, mathematics, or science. Any of the courses in mathematics fulfills the one course requirement of the core curriculum.
Requirements for
the Mathematics Minor
A student may obtain a minor in mathematics by completing 18 credit hours of 200 or above level mathematics courses (General Physics can also be used to complete the mathematics minor). The minor generally corresponds to the first two years of an undergraduate degree in mathematics. Courses are for 3 credit hours unless otherwise noted. A course grade of at least C-minus is necessary for a course to fulfill the department's requirements for a minor.
Mathematics
MATH 101 Introduction to Mathematical Thought This course focuses
on our changing conception of the notion of extension leading to the rise
of the various branches of mathematics and the application of mathematics
to describing the universe.
MATH 103 Euclidean
Geometry A study of selected books from Euclid's Elements. Topics
covered include plane geometry, theory of proportions, and classical arithmetic.
Students will also investigate the relation between mathematics and more
comprehensive philosophical issues.
MATH 153 Computer
Programming An introduction to problem solving methods and algorithm
development. Programming in a high-level language including how to design,
code, debug, and document programs using techniques of good programming
style. Prerequisite: MATH 105 or equivalent.
MATH 201 Calculus
I Basic course in differential calculus with an introduction to integration.
Topics covered include limits and continuity, the notion of the derivative,
techniques of differentiation, the definite and indefinite integral, and
the fundamental theorem of calculus. Prerequisite: MATH 105 or equivalent
or permission of the instructor. (4 credit hours)
MATH/PHIL 353 Symbolic
Logic Introduction to symbolic logic and the theory of formal systems.
Topics include the traditional logic of categorical sentences, truth functional
logic, the first order predicate calculus, higher order systems, the notions
of decidability and completeness, and some typical applications, among them
a brief look at the design of digital computing machinery. Prerequisite:
PHIL 102 or equivalent. (Cross-listed in Philosophy)
MATH/PHIL 354 Modal
Logic An introduction to the structure and techniques of the logic
of necessity and possibility from an axiomatic standpoint. Topics include
sentential modal logic and the systems T, S4, and kS5; validity; decision
procedures and completeness; and quantified modal logic. Prerequisite:
MATH 353 or permission of the instructor. (Cross-listed in Philosophy)
MATH 355 Mathematical
Logic Development of the principal topics of mathematical logic. Through
an axiomatic approach, the course treats the foundations of mathematics
and illustrates the power as well as the limitations of mathematical reasoning.
Topics include propositional and quantificational logic from an axiomatic
standpoint; formal number theory; recursive functions, Gödel's theorem,
and recursive undecidability; and an introduction to axiomatic set theory.
Prerequisite: MATH 353 or permission of the instructor.
MATH 361 Differential
Equations This course covers the basic techniques for solution of
ordinary differential equations. Topics include first and second order
linear equations, non-linear equations, systems of linear equations, the
fundamental matrix, series solutions of differential equations, numerical
methods and introduction to stability theory. Prerequisite: MATH 202 or
equivalent. (4 credit hours)
MATH 490-99 Special
Topics or Directed Studies in Mathematics A topic chosen according
to the interests of the students and the instructor, such as nonparametric
statistics, linear programming, set theory, numerical analysis, and complex
variables.
Natural
Science
In his Physics Aristotle laid the foundations for a philosophical knowledge of the natural, changeable world, but he failed to fully develop what modern scientists, beginning with Galileo and Newton, have exploited, the potential of mathematics to describe and systematize our knowledge of the natural world. Yet the latter approach also has its limits; because it relies so heavily on mathematics, which deals entirely in abstract quantity, it fails to account for form and purpose in physical objects. Christendom=s approach to natural science integrates the best of both traditions.
The College offers one introductory course dealing with the historical and philosophical principles of science, and another concentrating on the first quantified natural knowledge, Descriptive Astronomy. The more advanced courses, the two semester sequence in General Physics, deepen the student=s understanding of the nature of physical reality while not neglecting philosophical questions. Any of the science courses satisfies the core requirement in science.
SCIE 102 Introduction
to Scientific Thought This course focuses on our changing conception
of the universe, the rise of the various physical sciences, and the development
of the scientific method.
SCIE 104 Descriptive
Astronomy A study of astronomy beginning with its historical roots
and leading to our current understanding of the universe. Major developments
are placed in their historical and philosophic context by appropriate
study of original works. Students also study the night sky and methods
used by astronomers, by means of activities outside the classroom.
SCIE 204 General
Physics I Introduction to mechanics and thermodynamics. Topics in
mechanics include Newton's laws of motion; physical concepts of mass,
velocity, acceleration, motion, energy, and work; conservation laws, and
application of mechanics to simple problems. Topics in thermodynamics
include the four laws, the concepts of temperature and entropy, and the
kinetic theory of gases. Prerequisite: MATH 201 or permission of the instructor.
SCIE 205 General
Physics II Continuation of SCIE 204. Topics include oscillatory motion
and wave motion, the nature of light and optical phenomena, geometric
optics, electricity and magnetism, and an introduction to special relativity
and quantum physics.
SCIE 204L-205L
Laboratory for General Physics I & II Students conduct experiments
illustrating the physics discussed in the classroom and learn and practice
principles of data acquisition and data analysis. (Required with SCIE
204-205) (1 credit hour per semester) |
Browse Results
Modify Your Results
Algebra Essentials and Applications is focused, organized, and easy to follow. The program shows your students how to read, write, and understand the unique language of mathematics, so that they are prepared for every type of problem-solving and assessment situation.
The book broadly covers selections from major forms of literature fiction, poetry, drama, biographies of authors among others. Each section contains a detailed critical introduction to each form, brief biographies of the authors, and a clear, concise editorial apparatus.
The textbook provides the student/learner with eight categories of academic standards for reading, writing, and speaking and listening. It also helps the student/learner study the Writer's Models and learn to write short stories, editorials, and more.
Holt Geometry provides foundation concepts for high school mathematics. As presented in grades k-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics.
Have you ever wondered...why some places are deserts while other places get so much rain? What makes certain times of the year cooler than others? Why do some rivers run dry? Maybe you live near mountains and wonder what processes created them. You will learn about this and so much more |
The authors are experts in test preparation with extensive classroom experience in teaching SAT math Includes crucial strategies for using calculators to solve problems efficiently Gives students five sample SAT math sections with complete solutions for every question more...
Get ready to master basic arithmetic subjects, principles, and formulas! Master Math: Basic Math and Pre-Algebra is a comprehensive reference guide that explains and clarifies mathematic principles in a simple, easy-to-follow style and format. Beginning with the most basic fundamental topics and progressing through to the more advanced, Master Math:... more...
Perfect for revision, this Study Guide concisely covers all the syllabus topics in a digestible format. With lots of opportunity to practise, examiner hints and past exam questions, it will fully prepare students for exams. more...
Fully covering the revised 2012 syllabus and addressing the new focus on applications and the GSC, this text has over 600 pages of guided explanation and exercises to ensure your students achieve the best results. An eBook with extensive digital material gives even more support, with interactive worked solutions, conceptual animations and more. |
Lectures on Finite Precision Computations
For the advanced undergraduate and beginning graduate, this book combines techniques from engineering and mathematics to describe the theory of computability in finite precision. In the challenging cases of nonlinear problems, theoretical analysis is supplemented by the use of MATLAB. |
Guide to the Math and Math/Stats Major
The Carleton Mathematics department offers two tracks within the major, Mathematics or Mathematics/Statistics which are designed to reflect the depth and diversity of modern mathematics or statistics. We seek to provide you with an accurate picture of the nature of mathematics or statistics itself, its applications, and its connections to other disciplines. Specifically, in our courses, seminars, colloquia, and other activities, you will:
learn a broad range of mathematical topics;
increase your ability to read and write mathematical proofs;
become a competent mathematical problem solver;
be exposed to the mathematical literature available in our excellent library;
learn how mathematics or statistics is applied to real problems;
become aware of the connections between the various branches of mathematics, and between mathematics and other disciplines.
In addition, students on the statistics track will gain experience with statistical software.
We offer two tracks, Mathematics or Mathematics/Statistics. Both options include requirements that majors (a) complete the senior integrative exercise and (b) accumulate a total of eight talk-credits during the junior and senior years by attending the comps talks of their colleagues. In addition, we encourage our majors to participate in the numerous activities that take place around the department.
Mathematics major
The basic calculus/linear algebra sequence (Math 121, 211, and 232) and the course on mathematical structures (Math 236) are required of all majors. Also, each major must take at least 6 courses (36 credits) numbered above 236 chosen from at least three of the following areas. Consult the catalog for course descriptions.
Mathematics/Statistics major
A Carleton student will be able to obtain a major in mathematics following a statistics track by taking the following courses above Calculus:
Math 232 - Linear Algebra
Math 236 - Structures
Math 245 - Applied Regression Analysis
Math 265 - Probability
Math 275 - Statistical Inference
Math 315 - Topics in Probability and Statistics
plus two additional Mathematics courses above math 236 (one of which may be Computer Science 324), at least one of which must be taken outside of the Applied Mathematics area.
CS 111 (Introduction to Computer Science) is also recommended.
It is strongly recommended that students on this track engage in some data analysis learning experience outside the classroom such as an internship involving data analysis, a research experience with a statistician, either on or off campus, or a comps project that is explicitly statistical in nature. Students interested in graduate school in statistics are advised to take Math 321 (Real Analysis I). |
Complex Analysis for Mathematics, Science, and Engineering
This book provides a comprehensive introduction to complex variable theory and its applications. The Second Edition features a revised and updated ...Show synopsisThis book provides a comprehensive introduction to complex variable theory and its applications. The Second Edition features a revised and updated presentation that reflects the latest theories and their applications to current engineering problems |
Book Summary of S.Chand Mental Mathematics For Class I
(Paperback)
The series of Mental Mathematics books I to V has been designed to provide extensive practice of all types of problems. Skill tests containing objective type questions have been added at the end of each book to prepare the child for Talent Search and Olmpiad Tests. * A Useful Practice Book For All types Of Questions. * A Perfect Guide For Olympiads And Talent Search Tests.
Details Of Book :
S.Chand Mental Mathematics For Class I
Book:
S.Chand Mental Mathematics For Class I
Author:
Vikas Aggarwal
ISBN:
8121931258
ISBN-13:
9788121931250
Binding:
Paperback
Publisher:
SChand |
Geometry : concepts and applications by Cindy J Boyd(
Book
) 42
editions published
between
1999
and
2012
in
English
and held by
301 WorldCat member
libraries
worldwide
Geometry: Concepts and Applications is designed to help you discover, learn, and apply geometry. You will be challenged to make connections from concrete examples to abstract concepts. The real-world photographs and realistic art will help you see geometry in your world. You will also have plenty of opportunities to review and use algebra concepts as you study geometry. And for those of you who love a good debate, you will find plenty of opportunities to flex your logical muscles. - p. iii
Algebra 1 by John A Carter(
Book
) 38
editions published
between
2003
and
2014
in
English
and held by
300 WorldCat member
libraries
worldwide
Understanding art by Gene A Mittler(
Book
) 8
editions published
between
1992
and
2007
in
English
and held by
291 WorldCat member
libraries
worldwide
For middle school/junior high/Includes index
Pre-Algebra by Carol Malloy(
Book
) 41
editions published
between
2003
and
2012
in
English and Spanish
and held by
278 WorldCat member
libraries
worldwide
One program, all learners. Flexibility: print and digital resources for your classroom today and tomorrow. Appropriate for students who are approaching, on or beyond grade level. Differentiation: Integrated differentiated instruction support that includes Response to Intervention (RtI) strategies. A complete assessment system that monitors student progress from diagnosis to mastery. More in-depth and rigorous mathematics, yet meets the needs of all students. 21st century success: preparation for student success beyond high school, in college or at work. Problems and activities that use handheld technology, including the TI-84 and the TI-Nspire. A wealth of digital resources such as eStudent Edition, eTeacher Edition, animations, tutorials, virtual manipulatives and assessments right at your fingertips. - Publisher
Arttalk by Rosalind Ragans(
Book
) 6
editions published
between
1988
and
2005
in
English
and held by
272 WorldCat member
libraries
worldwide
ArtTalk has expanded its coverage of art history, strengthened its technology integration features, and placed more emphasis on the performing arts - all while maintaining its focus on a media approach to the elements and principles of art. ArtTalk integrates lessons in Perception, Creative Expression, Historical and Cultural Heritage, and Evaluation to form a comprehensive approach to art that helps every student - regardless of their learning style - think more creatively, make better decisions, even learn the art of self discipline. - Publisher
Glencoe literature : the reader's choice by Beverly Ann Chin(
Book
) 20
editions published
between
2000
and
2009
in
English
and held by
264 WorldCat member
libraries
worldwide
Unit one. The Anglo-Saxon period and the Middle Ages 449-1485 -- unit two. The English Renaissance 1485-1650 -- unit three. From puritanism to the enlightenment 1640-1780 -- unit four. The triumph of romanticism 1750-1837 -- unit five. The Victorian Age 1837-1901 -- unit six. The Modern Age 1901-1950 -- unit seven. An international literature 1950-present -- Reference section
Chemistry : matter and change(
Book
) 15
editions published
between
2002
and
2013
in
English
and held by
260 WorldCat member
libraries
worldwide
Chemistry: Matter and Change is a comprehensive chemistry course of study designed for a first-year high school chemistry curriculum. The program incorporates features for strong math support and problem-solving development. The content has been reviewed for accuracy and significant enhancements have been made to provide a variety of interactive student- and teacher-driven technology support. - Publisher
Algebra 2 by John A Carter(
Book
) 25
editions published
between
2003
and
2012
in
English
and held by
218 WorldCat member
libraries
worldwide
Introducing art by Gene A Mittler(
Book
) 8
editions published
between
1999
and
2007
in
English
and held by
218 WorldCat member
libraries
worldwide
Earth science : geology, the environment, and the universe by Mcgraw-Hill(
Book
) 10
editions published
between
2002
and
2013
in
English
and held by
196 WorldCat member
libraries
worldwide
Earth Science: Geology, the Environment, and the Universe is designed for complete concept development and supported with riveting narrative to clarify understanding. Challenging with engaging hangs-on labs, this complete program provides results that you and your students will appreciate. - Publisher
Geography : the world and its people by Richard G Boehm(
Book
) 14
editions published
between
1996
and
2002
in
English
and held by
184 WorldCat member
libraries
worldwide
Geography is the study of the earth in all of its variety. When you study geography, you learn about the earth's land, water, and plant and animal life. You analyze where people are, how they live, and what they do and believe. You especially look at places people have created and try to understand how and why they are different. Geographers study the earth as the home of people. Five geographic themes -- location, place, human/environment interaction, movement, and region -- are used here to help you think like a geographer. - p. 21
Glencoe science by Alton Biggs(
Book
) 57
editions published
between
2002
and
2008
in
English
and held by
178 WorldCat member
libraries
worldwide
"Strong content coverage is integrated with a wide range of hands-on experiences, critical thinking opportunities, and real-world applications. The modular approach allows teachers to mix and match books to meet their curricula."--Publisher's website
Glencoe health by Mary H Bronson(
Book
) 14
editions published
between
2005
and
2011
in
English
and held by
177 WorldCat member
libraries
worldwide
Glencoe world geography by Richard G Boehm(
Book
) 13
editions published
between
2000
and
2012
in
English
and held by
177 WorldCat member
libraries
worldwide
If you think that geography means memorizing a list of states and their capitals, think again. Geography is a broad and ever-changing subject. It includes the study of Earth's physical features, as well as the countless and fascinating ways that humans, animals, and plants interact with the world around them. This textbook covers six essential elements of Geography learning standards: 1. The world in spatial terms, 2. Places and regions, 3. Physical systems, 4. Human systems, 5. Environment and society, 6. The uses of geography. - Publisher
Biology : the dynamics of life by Alton L Biggs(
Book
) 16
editions published
between
2000
and
2005
in
English
and held by
167 WorldCat member
libraries
worldwide
General biology text with National Geographic features in each unit and test-taking tips written by the Princeton Review
Exploring art by Gene A Mittler(
Book
) 7
editions published
between
1999
and
2007
in
English
and held by
163 WorldCat member
libraries
worldwide
Physics : principles and problems by James T Murphy(
) 9
editions published
between
2005
and
2009
in
English
and held by
163 WorldCat member
libraries
worldwide
2005 State Textbook Adoption
Marketing essentials by Lois Farese(
Book
) 11
editions published
between
2002
and
2009
in
English
and held by
158 WorldCat member
libraries
worldwide
Considered the nation's number one marketing program, Marketing Essentials is the essential text for introducing students to the skills, strategies, and topics that make up the ever-changing world of marketing. It effectively captures the excitement of this fast-paced discipline with engrossing narrative, engaging graphics, and real-life case studies. - Publisher
The stage and the school by Katharine Anne Ommanney(
Book
) 4
editions published
between
1999
and
2005
in
English
and held by
155 WorldCat member
libraries
worldwide
Outlines the history of drama and aspects of dramatic interpretation and production |
Introduction to Diophantine Equations A Problem-Based Approach
Description: This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methodsMore...
This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The material is organized in two parts: Part I introduces the reader to elementary methods necessary in solving Diophantine equations, such as the decomposition method, inequalities, the parametric method, modular arithmetic, mathematical induction, Fermat's method of infinite descent, and the method of quadratic fields; Part II contains complete solutions to all exercises in Part I. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions.An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants '" including Olympiad and Putnam competitors '" as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques |
Phase II is a 42,000 square foot addition that will provide a new home for Physics, Math and Computer Sciences as well as the new Science Center for Integrated Learning. Ground breaking for Phase 2 is scheduled for March 2014, with a completion date of May 2015. This is the second of a three-phase science building initiative repesenting an investment of more than $30 million.
Combining logic and precision with intuition and imagination
At Earlham, we strive to teach our students mathematical fundamentals and problem-solving skills that they can apply in a variety of disciplines or in further study of mathematics. Mathematics students may participate in weekly "mathophiles" seminars and informal lunches, attend regional meetings of professional mathematicians, and participate in mathematically related off-campus programs during the academic year or the summer.
The Association for Women in Mathematics (Careers That Count) describes mathematics as "… a powerful tool for solving practical problems and a highly creative field of study, combining logic and precision with intuition and imagination. The basic goal of mathematics is to reveal and explain patterns — whether the pattern appears as electrical impulses in an animal's nervous system, as fluctuations in stock market prices, or as fine detail of an abstract geometric figure."
Special Learning Opportunities
We have close and continued faculty-student collaboration, both in class and in settings like working together in the Math Studio.
Students can participate in summer internships at the Centers for Disease Control, the National Laboratories, the National Institute for Standards and Technology, NASA, the NSA and major universities.
Math students have also studied and modeled heat loss from College buildings and houses, and for the City of Richmond designing improved trash collection routes.
Student-faculty research projects have studied such topics as global warming, pattern formation in animal coats and the spread of ideas during the "Arab Spring."
Quant House — a student organized off-campus house — is not only a place to live but also a source of tutoring and lectures.
Tutoring for Calculus and Elementary Statistics is available. Tools like smartpens and Kurzweil readers are available at the Academic Enrichment Center.
Outcomes
Earlham math majors have gone on to graduate school in mathematics, physics, economics, finance, music, geosciences, and psychology.
Alumni have pursued a wide variety of careers, including finance, agents both for the NSA and for the FBI, actuaries, aspects of computing and secondary teaching.
A student recently presented her work at a national joint meeting of the Mathematical Association of America and American Mathematical Society.
Math Champ
Jacob LaChance says math is simple, and the key is not to think about math but rather to think with math. As a sophomore, he was part of the Earlham duo that placed first in the Michigan Autumn Take Home Challenge.
It is easy for Earlham students to design and participate in projects that explore connections between math and other interests as a class project, independent study or as a double major.
Earlham students have participated in the Budapest Semesters in Mathematics, studying math in one of the world's great centers of mathematical research and discovering Hungarian culture |
S.O.S. MATHematics
S.O.S. MATHematics is your free resource for math review material from Algebra to Differential Equations!The perfect study site for high school, college students and adult learners. Get help to do your homework, refresh your memory, prepare for a test, ….Browse our more than 2,500 Math pages filled with short and easy-to-understand explanations. Click on one of the following subject areas: Algebra, Trigonometry, Calculus, Differential Equations, Complex Variables, Matrix Algebra, or Mathematical Tables.
This entry was posted on November 22, 2007 at 8:17 am and is filed under Curriculum, Mathematics |
My World, Inside and Out
Algebra is always necessary
A New York Times opinion article came out recently stating how Algebra is a failing point in the American education system. The author states statistics and polls showing how the US is just not that good at math. Statements like
It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.
do not line up at all. Stating that mathematic education needs to be less theoretical and more functional is a good point. I agree with him when he writes
I hope that mathematics departments can also create courses in the history and philosophy of their discipline, as well as its applications in early cultures. Why not mathematics in art and music — even poetry — along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet.
But math has little to no value without the basis that needs to be taught. How do we expect students to be able to handle statistics, harmonics, physics, economics, trends, or anything relatively advanced with just a basic knowledge in arithmetic? As a math minor, it would be great to see classes focused on math in the world around us in a purely mathematical sense. Right now, they are subdivided amongst their specialized majors, and are only accessible to students studying these fields. It would be ideal for students to have access to these classes, but without an underlying core understanding of algebra, what is the point? you cannot write a sentence without words, and words without letters. Mathematics is compounding as well: arithmetic, algebra, algebra ii, geometry, trigonometry, calculus, calculus ii, calculus iii, calculus iv, calculus v, advanced calculus, abstract algebra, linear algebra. These classes listed are pure math classes focused on the theoretical aspect. The part of math that one can only truly picture in their mind, and it is understandable when students clam that there is no real life application. The entire argument of a physics major is math in the real world, math in motion, math that a user can see.
Algebra starts by teaching us abstraction. We learn to deal with ideal environments, ones that the real world tries to emulate, but cannot due to the limitations of existence. But the ability to think abstractly is a skill that all people can use. To say that learning algebra made someone dumber is unheard of. Harder, yes, but not dumber. Stating that
It's true that students in Finland, South Korea and Canada score better on mathematics tests. But it's their perseverance, not their classroom algebra, that fits them for demanding jobs.
is just insulting. Do American students not try hard enough? Does that not imply that they should be taught to work a little harder. To persevere, to learn perseverance? It may seem like a slap to American parents and teachers; that the kids we raise are not outdoing children from other countries.
It is essentially contrary to what Hacker writes:
This need not involve dumbing down. Researching the reliability of numbers can be as demanding as geometry. More and more colleges are requiring courses in "quantitative reasoning." In fact, we should be starting that in kindergarten.
His statement is not true at all: to remove algebra from the required school curriculum is to admit defeat. It is admitting that math is too hard for our next generation, and only the "few" who are good at it should even bother.
Yes, it is silly to make every student take calculus, but it is not required for graduation. Yes, it would be silly to make Linear Algebra a core requirement for graduation as well, but algebra is necessary. We use it in our every day lives, to calculate milage, cost, weight change, calculating tips, understanding coupons, and a million and a half other day to day calculations that we just do not think about. Algebra is necessary, to stop teaching it will only perpetuate the decline in the American public school system. |
Introduction to Maximize/Minimize Quadratic Functions
The techniques to maximize/minimize quadratic functions can be applied to problems outside of mathematics. We can maximize the enclosed area, minimize the surface area of a box, maximize revenue, and ...
Introduction to The Change of Base Formula
There are countless bases for logarithms but calculators usually have only two logarithms—log and ln. How can we use our calculators to approximate log 2 5? We ...
Introduction to Applications of Logarithm and Exponential Equations
Now that we can solve exponential and logarithmic equations, we can solve many applied problems. We will need the compound growth formula for an investment earning interest rate r , ...
Introduction to Finding the Growth Rate
We can find the growth rate of a population if we have reason to believe that it is growing exponentially and if we know the population level at two different times. We will use the first population level as n
Introduction to Radioactive Decay
Some radioactive substances decay at the rate of nearly 100% per year and others at nearly 0% per year. For this reason, we use the half-life of a radioactive substance to describe how fast its radioactivity decays. For ... |
Note: this article is intended for undergraduate math majors, preferably seniors, but freshman will benefit from the look ahead, at what is expected to be known by them when they are seniors, as well as entering graduate students.
I am readying myself for the GRE math subject exam in the late Spring, and therefore reviewing all four years of undergraduate mathematics at this time.
In what follows, I will be summarizing every major area of undergraduate mathematics, as follows:
Geometry: Plane, Elliptic, Hyperbolic, and Affine
Linear Algebra
Vector Analysis
Real Analysis
Complex Analysis
Topology
Ordinary Differential Equations
Fourier Analysis, Lebesgue Integration and General Transform Theory
Probability Theory
Abstract Algebra
Graph Theory
Combinatorics and Algorithmic Complexity
Set Theory and Transfinite Arithmetic
Basic & Analytic Number Theory
Partial Differential Equations
Differential Geometry
If you are a student of any subject, then I hope you have already asked yourself at times the question "What does all of this mean? What is it for?" if you are as I am, a mathematics major, then I believe that this question is particularly important. For one can too often get lost in the forest, standing amid the numberless trees, each one a bit different from its neighbors yet all oddly familiar, somehow similar, like wondering through a waking dream.
This is neither an idle nor a superfluous question. Call it "the big picture." Call it what you will. It is important to know the gist and the connections between different areas of ones subject, and to know each area for what it really is about.
———————————————————————————————————————————- Linear Algebra
Let's look at linear algebra. The matrix is the lingua franca tool of linear algebra, and so linear algebra is the study of vector spaces and their transformations using matrices. After a first course, one should know the axioms that define a vector space, the algebra of matrices, how to express the structure of vector spaces using matrices, and especially how, given the basis of a vector space, to represent any transformation of that vector space using a matrix, and how to use and calculate eigenvalues and eigenvectors. Lastly, the key theorem of linear algebra is to know the 8 distinct conditions that each alone can guarantee a matrix is inevitable. And that is it, the heart of an entire semester of undergraduate linear algebra summarized in one paragraph. Again, in a nutshell: basic linear algebra is the study of the transformation of vector spaces into one another and understanding equivalent conditions under which a matrix is invertible.
What is linear algebra used for? Everything, and almost everywhere, is the short answer. Anytime you seek a first approximation to some problem, it is likely that you will be using linear algebra. In graph theory matrices are used to represent a graph's structure in a precise and concise manner. In abstract algebra, matrices are used in the representation of groups and other algebraic structures. And linear algebra comes up in both ordinary and partial differential equations, differential geometry, knot theory, number theory, and almost everywhere else. Learn it well, know it well if you are going to be doing research in pure or applied mathematics.
Share this:
Like this:
Note: this article is for someone who has had considerable exposure to calculus, especially integration problems involving the trigonometric functions. There is no preamble or hand-holding beforehand. And, as there is little theory in this article, this is more of a practicum. I will write extensively on Real and Complex analysis in upcoming articles that will contain voluminous proofs. -=KuRt=-
So, let's evaluate some integrals.
Here, we immediately see the need to use substitution. Therefore, let us set the inner expression in our integral to
Then, we will have that
or simply that so now we may perform our full substitution, so that our integral from in terms of becomes
then, we immediately have the result
and, so, substituting our original value for u back into this answer, we have that our integral is:
and we may also generalize this to finding all similar integrals, namely
where this holds for and we simply substituted such that and then proceeded identically as in our specific integral to evaluate our general integral.
Next, we shall evaluate each of the following integrals, in order:
The general integral substitution formula
A definite integral:
And the Gamma Function
Definition
which is an instance where the evaluation of an integral yields an entirely new function.
Like this:
Euler's Characteristic Formula V-E+F = 2
How is it that for thousands of years the best minds in mathematics did not see the fundamental relationship that, in any regular polyhedron, the sum of the vertices and faces minus the edges equals two? Although this question is interesting, no attempt will be made to answer it. Yet the question, merely by being asked, serves to highlight the tremendous stature of Leonard Euler.
Euler worked from first principles, digging into a topic by performing calculations to get a feel for the shape and edges of a problem, then developed conjectures and proofs based on such research. And so, Euler must have tabulated the edges, faces and vertices of shapes such as the platonic solids to thereby notice this relationship. Perhaps it took an almost childlike playfulness to discover this relationship. Yet it is all speculation. The only fact is that Euler did discover that where V, E, and F are, respectively, the vertices, edges and faces of a regular polyhedron
I first learned of Euler's formula in a senior course on graph theory taught by the Polish graph theorist Dr. Siemion Fajtlowicz. Therefore, let me provide a few definitions before offering a compact proof that using basic graph theoretical methods.
Definition. A graph consists of a non-empty set of vertices and a set of edges, possibly empty. If an edge E exists, then it will be related to an unordered pair (a,b) of vertices in V. We may write to denote the graph.
A graph is finite if both the vertex set and edge set are finite. A graph that is not finite is infinite. The size of a graph is the number of vertices. The order of a graph is the number of edges. All graphs discussed in this article are finite.
Immediately note that if we are given one edge, then there exist two vertices — because an edge must connect to something and that something will always be a vertex. Two vertices are incident if they share a common edge and said to be adjacent. An edge with identical ends is called a loop while an edge with distinct ends is called a link. A graph is simple if it has no loops.
The degree of a vertex in a graph G is equal to the number of edges in G incident with v where each loop counts as two edges. Thus, a vertex with two edges has degree 2 because the degree of the other two vertices would be one each, and so the sum of the degrees of all the vertices would be 4. From this we have immediately the following two theorems.
Theorem 1.The sum of the degree of the vertices of a graph equals twice the number of edges. That is
proof. If a graph has no edges, then e=0 and the theorem is true because the sum of the degree of the vertices of a graph with no edges must, by definition, be zero. If a given vertex has an edge, then there must be a second vertex connected to the first vertex by the given edge and so the sum of the degree of the vertexes will be 2, even if the edge were a loop beginning and returning to a single vertex the degree of that vertex due to the loop would still be 2.
Now, each additional edge added to the graph will increase either the degree of the vertex it is connect to and add one additional vertex or else it will add one degree each to two vertices that already exist. Thus, each time an edge is added, the degree count of the vertices increases by 2. Therefore, the sum of the degree of all the vertices will always be simply twice the number of edges present.
Theorem 1.1The number of vertices in a graph of odd degree is even.
proof. Let be sets of vertices where the is of odd degree and is of even degree. Then, from Theorem 1, we must have that
but we have already established that
and therefore we must have
but
so
is even. Thus, we have that the sum of the vertices in a graph of odd degree is even. Now, we will need the following definitions to proceed.
Definition. A cycle is a closed chain of edges. A connected graph that contains no cycles is a tree.
Definition. A spanning tree of a graph G is one that uses every vertex of G but not all of the edges of G.
Every connected graph G contains a spanning tree T as a subgraph of G.
Definition. A planer graph is one that can be drawn in the plane without crossing any edges.
Definition. A face in the plane consists of both unbounded and bounded regions.
The last definition allows us to say that the number of "faces" of a finite line segment in the plane is 1, but that the number of faces of an infinite line is 2, one for each side that the line divides the plane into, and that the number of faces of a circle in the plane is 2, one corresponding to the inside and the other to the outside of the circle one being the unbounded and the other the bounded region.
Now we are ready to prove Euler's formula as it may be stated in graph theory.
Theorem 2. If G is a connected planer graph with vertices v, edges e, and faces f, then
proof. Let T be a spanning tree of a graph G. Then the number of faces f=1 and the number of edges e= v – 1 is true for the spanning tree T of G and so we have and our formula is true. Now, G may be constructed from T by adding edges to T, but each time we do so we are adding a new face to T. Therefore, with the addition of each new edge, e will increase by 1 and f will increase by 1. Therefore, all of these additions cancel out and we have our formula.
Now, a homomorphism exists between polyhedron and graphs in that a connected plane graph can be uniquely associated with a polyhedron by making any face the flat unbounded part of the plane. Therefore, Euler's formula is true for polyhedron. We can therefore immediately prove:
Theorem 3. Exactly five platonic solids exist.
proof. Let the number of edges and faces be n and the number of edges each vertex is incident to be d. If we multiply nf, we get twice as many edges because each edge belongs to two faces, so nf is the number of faces multiplied by the number of edges on each face. Likewise, with dv each edge is incident to two faces so that we have so that we have the equality
or
and
Now, placing these into Euler's formula, we get
or
But both v and n are positive integers and for to be true, we must have that
or
and so there are just five possibilities for the values of n and d and each of these corresponds to one of the five platonic solids, so that we have
is a tetrahedron is a hexahedron also known as a cube is a octahedron is a dodecahedron is a icosahedron
Drowned in the honk-squeal, I can almost hear the waves
sweep in, their soft susurration in the tip of their lips breaking the sand between my curled toes.
Noise is now everywhere I want to be
without it. Cars swoosh past Galveston beach roaring their inept monstrous lungs. I can barely
breathe. Or think. Why do trees and blades of every green thing shudder?
Because we are a hyper-intelligent insidious poison? Cats and dogs cling to us in shock and awe.
Ninety-five percent of a car's energy goes towards moving simply itself not the passengers.
Or rather that's 2,500 pounds of wastefulness before the crux of tissue steering the steel.
In Hermann Memorial Park a yellow-blue finch tries to sing and fails
in the roar and wall of sound the cars shed in their wake on the I-10 adjoining the beige greenery.
I nod off under a canker tree. A whale whistles out of its water fountain, breathing.
I roll under such plushness, floating with barnacles and sticky ambergris. So glued are our dream's illogical logic.
I am a sticky carbuncle tearing through the earth's thin breathability. It's afternoon in Houston.
I shower again. I scrunch into a starched shirt. I rope my throat with a dead worm's shiny excrement.
2
My right ear is dead. When I was three
German measles like dappled freckles grew in me, killing the nerve.
Now thirteen, my left ear still good, I hear pretty well
the unprettiness in my parents voices when they divorce
and light fades as I listen in, on the mosquito bitten dark
roof above the living room window, then roll on my back
to swallow insignificance in the drifting milky way above.
Now the frogs have started up. A few ducks quack. A splash
might be catfish come to nibble at the stars
tangled in a cheap tabloid suspended on the pond's scum.
I grasp, as we watch you climbing our mistakes.
While sipping coffee, I read shit the whole of our English alphabet.
The paths in the park my mother walks me in
crisscross, and cut the grass into Latin squares
or rectangles into which scarf-like flowers bloom
and rooted angels grow, staring up at the sky
with empty alabaster eyes. I chase the wind, turning
stone-edged corners as fast as I can
without falling into the arms of an angel
crouching to take flight. I watch them, their bodies
like sleek athletes, ready to pace the sky
for a brief time, before earth might pull them back
or rain ground them. I climb a boulder's perch, and jump
spread-armed, trying to stay up in the air
longer and longer each time, to no avail.
But hanging upside down on the monkey bars
I float far above the pitiless earth
as if I spend my days walking the clouds,
and all the angels look up at me, surprised
Kneeling to untangle my dog's leg from its leash,
how did I get here, walking a pit bull in the dark
under the sour leaves of drought resistant Texas oaks?
How have these years colluded to put me
with a woman who doesn't like to be touched
as if my life were still attached
to a former life, lived in felt robes, kneeling,
questioningly, before God's dead silence?
Why do I sometimes whisper beatitudes in Latin
when grinding roasted coffee beans for breakfast?
Why can't a fuck be just a fuck like breathing
or the necessary forward movement of starlight
entering my eyes from Polaris when I look up?
Why is my life so intertwined that it folds me
into fractal compartments that expand, as if
from each decision, outward, new enclosures grip me
as I venture forward, faster than any logic I can conjure?
Should I kill politicians to address society's wrongs?
Or open a shop and sell cracked imported Chinese
Chia Pets? Or get to the lunar surface to erase
the names of loved ones astronauts left behind?
How can this sticky motion of salt and water
hoisted on these dry branches of bone
discern a purpose, lost among thin pricks of starlight
that amble like ancient animals into the night?
Tucked into the rear-end of a sagging neighborhood
of immodest upscale homes, I attend
Jeb Stuart High on Peace Valley Lane
in Falls Church, Virginia. My sophomore year
there's this guy, a senior, named Ferenc Molnár.
"Ferenc, what?", I said. Classmates whisper
he'd gotten a perfect SAT. "Hmm,"
and I let go of trying to say his name.
That summer at the local library
I find Ferenc
scrawled in pencil on the checkout cards
of books that no one else but he had deigned
to date in years. I skedaddle
to the shelf of the first one
Ferenc courted,
a hard blue bound book from Oxford
abundant with English poets.
It opens to pencil-lined pages and lands
on Shelly. I read. In the library's stacks,
sand begins to blow and cover my blue jean legs
and rubber flip-flops. The long hair of my face
burns in a blond sun. I see
a chiseled sneer erect on a pedestal
of bone, and ornate letters that tell
what a bad ass the owner was,
and watch the hurled tiny intricate skeletons
of shells from an ancient ocean
rub out all traces of his high-on-himself. I'm left
alone with the wind-thrown sand
sticking to the sky. And nothing
but nothing remains
Share this:
Like this:
This week, Professor Siemion Fajtlowicz assigned two home work problems:
1. How many graphs with vertices 1 … n are there?
2. Up to isomorphism, how many graphs are there with n vertices?
3. If you invite 6 random people to a party, show that 3 of them will know each other or 3 of them will be mutual strangers — and show that this is guaranteed to always be the case — but only if you invite a minimum of 6 random people to the party. It will not be the case if we only invite 5 random people or 4 random people, et cetera.
____________________________________________________________________________________________________________________________
Question 1 and 2 may appear to be identical questions, but they are not at all identical though they are related.
Question 1 is asking for the number of elements of S. Question 2 is asking for the number of elements of the quotient G\S, a very different and much more difficult question.
Theorem. There are
graphs with vertex set .
Proof. The question is easily answered. Given n vertexes, if we start from vertex 1 and connect an edge to each of the remaining vertexes, so that 1 goes to 2, 1 goes to 3, and so on until 1 goes to (n-1) , and 1 goes finally to n. Now, including the null edge or simply 1 itself connected to nothing, which is a legitimate graph, we have that there are n graphs. Repeating this for each additional vertex, and taking into account that the edge may originate at either one of the vertexes making for two graphs — if allowing full duplicates — then we will have that there are exactly,
possible graphs raised to the power of 2 to account for starting at either vertex, which gives us:
Therfore, the number of graphs with vertices 1 to n are:
and this is simply a more compact way of writing the RHS of the previous result using binomial coefficients. QED
The answer to the question is much more difficult, because the answer involves some hefty but basic machinery from Group Theory and Combinatorics involving Burnside's Lemma and Polya's Enumeration Theorem such that this question may be reworded in those terms, so that it becomes:
How large is the set where denotes the symmetric group on n letters andis the set of graphs with vertex set ?
At any party with 6 guests, either 3 are mutual friends or else 3 are mutual strangers. That is, the symmetric Ramsey Number R(3,3) = 6
If we consider just one person, then the other five must fall into one of the two classes of being either a friend or a stranger. This follows immediately from the pigeonhole principle, namely that, if m pigeons occupy n holes where , then at least one hole contains:
pigeons, where is the greatest integer function. The proof of this follows from the fact that the largest multiple of n that divides into m is the fractional part left over after n divides m-1 or
and so for n pigeons, we get
pigeons that could be put into holes. But we have m pigeons, and so there must be more than this many pigeons in the holes.
Now, for our problem, we have two classes, friends and strangers. If we choose one person, then that leaves 5 people with whom that person is either a friend or a stranger. And so, by the pigeonhole principle, for five objects going into two classes we must have at least:
members in one of the classes. In terms of a graph theoretical viewpoint, we have, starting with one vertex extended out to the other five "people" vertexes, that:
where the red edges represent friends and the blue edges represent strangers. We easily see that, regardless of how we choose one of our 2 colors, 3 of them must be blue or else 3 of them must be red, for if we have five red, then we have met the required condition that at least 3 be red, and likewise if 4 are red, we again fulfill the requirement that at least 3 be red; and so, this whole sequence of argument applies if we swap blur for red. Therefore, we are always guaranteed that there are 3 mutual friends or else that there are 3 mutual strangers in any group of 6 people. QED
In terms of Ramsey Numbers, the statement would be written .
This is utterly fascinating, because what this is really telling us it that there is a type of structure built into any random finite set. In this case, for any binary operation or else any question or property that has two values, we have it that any finite set of 6 is sufficient to support there being 3 of one or 3 of the other of that property, and that one of the two sets of 3 always exists inside of the 6 items.
This is a glimpse into a type of "deep structure" embedded within the fabric of finite sets. This is more than merely surprising, as one should not really be expecting to find any such structure whatsoever in a random set.
____________________________________________________________________________________________________________________________
holds for n<=20 in a completely trivial and weak algebraic sense that the Ramsey Numbers in this range do indeed appear in this formula.
However, there is no natural or reasonable theoretical connection whatsoever between this formula and Ramsey Numbers other than the search for generating functions and sequence matching studies that I conducted. So, for n = 3 to 20 this formula yields:
which all fall nicely into current best-known intervals for these numbers. But these values are essentially nonsense. However, it is currently believed that rather than 49. But there is no polynomial expression that yields 43 and the other lower already known numbers in the sequence of Ramsey Numbers evident from running long and large searches for such. Therefore, I believe that any meaningful approximate or asymptotic formula must be non-polynomial.
Therefore, my intuition says that there may well exist an exponential, non-polynomial expression for the Ramsey Numbers — perhaps similar to the P(n) function of Rademacher such as
where
But then I may be dreaming here because there need be no sequential connection between one of these numbers and the other — if each is a unique value in it's own problem space.
What I find frustrating with papers on Ramsey Numbers that I have read is their lack of a more probing approach. We know that calculating Ramsey Numbers is NP-hard. One paper even suggested that this is Hyper-NP hard — but did not specify in what manner they meant this to be true. Most likely they were referring to the absurdly rapid exponential growth of the possible solution space. But where are the more basic insights into the nature of these numbers? Most of what we have now is not much beyond Paul Erdős work in the 1930′s!
In a recent paper on Ramsey Numbers, the physicist Kunjun Song, said: "Roughly speaking, Ramsey theory is the precise mathematical formulation of the statement: Complete disorder is impossible. or Every large enough structure will inevitably contain some regular substructures. The Ramsey number measures how large on earth does the structure need to be so that the specied (sic.) substructures are guaranteed to emerge."
I think that we have yet to ask ourselves the deeper questions to get further along here. I am now researching the different ways in which the same questions may be asked, such as via Shannon limits of graphs and quantum algorithms to see what pure mathematical insight might be gleamed from these approaches. I am looking for good questions that, if properly phrased, should provide a road-map for further fruitful research.
Share this:
Like this:
This was the first week of my senior-level class with Professor Siemion Fajtlowicz, MATH 4315 – Graph Theory, and it was a blast!
The central question put to the class is when are two graphs isomorphic. There is nothing easy nor trivial about this question. It can be challenging to even distinguish that two simple graph representations are of different graphs, let alone of the same graph.
Also, we have been assigned the task of showing how one is to interpret the historically notable problem, first presented to and solved by Leonhard Euler in 1735, of the Seven Bridges of Königsberg in a graph theoretic manner. Euler resolved this question in the negative, but there is a lot more to it than that, as we will see in this article.
Along with this problem, we have also been assigned the Knight's Tour problem which we are also to interpret in a graph theoretic fashion.
Share this:
Like this:
I pity the pitifulgrasp, as we watch you climbing our mistakes our English alphabet.
I was awake, but closed my eyes for a few
seconds, and then opened them and continued
reading the words on a page about chromatic numbers,
how hard they are to calculate, how slippery
the ideas connecting to them, that we really can't
put much of a dent in the problem. Then I turned
back to the world around me, how the dishes
need doing. The dog lays its head in my lap
asking to be walked. The bronze curtains
blow about in a chill gust, prefiguring storm.
I lift the mountain that is my body , that excavates
a tunnel through the room, a zigzag of patterns arriving
at the door shore of my entry, and with dog on leash,
stumble out into the dark certainty of night.
Like this:
I'm sipping coffee and manureBack from dropping the children safely ensconced at school,
a speckled man with freckles on his egg shaped head sits
sipping coffee with in both hands, his elbows
crushed into the sunflower and dandelion placemat
on his kitchen table. Behind him, his wife
rinses shiny grease and grits from a plastic plate, then drips and tips it, sets it into formation
to dry among others in the draining board.
The woman hasn't spoken and the man says nothing.
In front of him, amid clutter gracing the table, a glass bottle of Heinz ketchup,
three-quarters full, smeared with peanut butter around its svelte neck
catches and throws a shadow of his wife working.
In a moment he will kill her with it.
They've not had sex in months
or so the plot thickens, as my wife asks
why I read so many books, and the day still
wide open for questions to which I have no answers.
At first I couldn't remember the name
you spat me with, that clung to my face, dripping
that smelled of beer and cigarettes
as I wiped off the brown spittle
with the back of my right hand, my left
still clutching the onion sheaves imprinted
with the thin stalks of penned equations
from my lecture I believe I saw you at
leaning forward in the front row
staring at my Palestinian skin.
Take this dead hat. Now hold it
up into the sunlight. Notice
the sweet Cuban tobacco that drifts
into your nose, if you wander
too close. Pay particular attention
to the three thin white lines of cloth
circling the broad black ribbon
encompassing the whole of its felt parts.
Now turn it over. Look into the depths
like venturing at night to the edges
of a smoldering volcano, where only your feet
slipping tells you you are on its clipped edges.
Share this:
Like this:
The work here is actual assigned homework from Dr. Mary Flagg's Number Theory, Math 4383 – Section 19842, going on now at the University of Houston this Spring 2012 semester. This is from homework assignment 2. The problems are taken from Elementary Number Theory, Seventh Edition, by David. M. Burton.
However, I will be going beyond much of this homework by adding additional material as it strikes me as being relevant to the topic at hand.
Basic Divisibility Theory of the Integers
2.2 – 5. For , prove thatis an integer.
Proof. By the division algorithm, if , we may write
where . Therefore, since r may only assume integer values, we have that may only assume any one of the following six forms, corresponding to each value of the remainder so that n must be in one of the following forms:
, , , , ,
Therefore, we need only substitute each of these in our expression and simplify to see what results. So, proceeding on a case by case basis, we have, for the first case:
for we get
which is an integer, as there are no fractions left on the RHS. Now, in the second case:
for we get
which is also clearly an integer. And, in the third case:
for we get
which is also clearly an integer. And, in the fourth case:
for we get
which is also clearly an integer. And, in the fifth case:
for we get
which is also clearly an integer. And, in the sixth case:
for we get
and this too is an integer. Therefore, for all values of , we have established that
is an integer for all . QED
Note. Of course, this expression is also the sum of the squares of the first n integers, or
for all and so, because the square of an integer is an integer, so would be a sum of squares of integers, and so this may be developed out into another proof, as a corollary, by obtaining and proving the summation formula itself it would follow that it is an integer.
2.2 – 11. If n is an odd integer, show thatis of the form 16k.
Proof. Assume that n has the form so that n is odd. Then, by direct substitution and upon simplification, we have
where
.
Therefore, is of the form 16K for all odd positive integers. QED
2.3 – 3. Prove or disprove: if, then eitheror .
Proof. By counterexample, we have
but false and is false.
So, without loss of generality, we may say that,
if and but , then it is true that:
or more fully, that and we see that and are both false. QED.
2.3 – 4. For , use mathematical induction to establish the following divisibility statement that
.
Proof. Using the PMI, we have for ,
or
and therefore it is true that
when n=1. Assume that for some positive integer k that
is true and note that this also means that:
or that
or
for some multiple q of 15. But the LHS factors into:
which implies that:
or
and now this implies that, either:
or else
or else
and likewise for 5, that either:
or else
or else
one of which must be true and for each of the integers 3 and 5. Now, for (k+1) we have that:
or
or
where q is the quotient and r must equal 0 if 15 is to divide exactly into . Now, by factoring, we have that:
so that we have also:
or
but each of these factors is simply the same as the assumed case n=k when n=k+1. And it is already true, by assumption, that
or
Therefore, by the PMI, we have that,
for all
, . QED
2.3 – 13 Given two integers a and b, prove the following: there exists integers x and y for whichif and only if
Proof. This is almost the converse of theorem 2.4 on page 23, namely that, if a and b are both integers not equal to zero, then a and b are relatively prime if and only if there exist integers x and y such that . From this we have that, if , then:
but from corollary 1 to the theorem 2.4, both on page 23, we have that if then and this implies that:
but multiplying both sides of this by d and setting d=c yields:
but this means that but c|a and c|b and therefore because . QED
2.3 – 13(b) If there exist integers x and y for which, then .
Proof. This is theorem 2.4 where the terms are inverted. We have, by theorem 2.4, that a and b are relatively prime if and only if there exists integers x and you such that . Now, suppose that for some choice of x and y, such that . Because and , by theorm 2.2 we get that or but this last condition forces d to be 1. Therefore, must be true. QED
2.3 – 20(e). Confirm the following properties of the greatest common divisor. Ifandand , then .
Proof. We are given that a and b are relatively prime, which means that:
for some linear combination of a and b their exist such integers x and y that this is true. Now, if and means that and for some q, q', r, r'. Therefore, we must have that d divides either a or c, given that , and likewise, d must divide either b or c, given that .
Assume, in the first case, that d divides a but not c. And assume that, in the second case, that d divides b but not c also. Then we would have and or subtracting these from the other set, that:
or
and
or
but a and b are relatively prime implies, if d dives both a and b, then:
or
but since it is also true that and that each implies that and that the respectably, which is a contradiction. Therefore, d must divide c if and . QED
2.3 – 21. (a) Prove that if, then .
Proof. First off, if , then there exists a q and r such that:
where and so, we are being asked to prove that:
where
Now, from first principles, because , we must have that:
or else there could be no division by d into n in integer terms, and so therefore, we also have:
for some positive integer because . Note that this implies that . Now, subtracting one from both sides yields:
And, because implies that , and this implies that , upon division of the RHS by the LHS, yields:
, then , and so, if the , then
for some . Now, without loss of generally, taking and
substituting, we have:
but this implies that
or
or dividing both sides by b
which implies that
and likewise argument if we go substituting but this is just and therefore, if , then . QED
2.4 – 9. Prove that the greatest common divisor of two positive numbers a and b divides their least common multiple, especially that .
Proof. By merging the definitions of the gcd and lcm of two integers, a and b, we have that the gcd of a and b is the positive integer d satisfying the following:
(a) and
(b) if and , then
and the lcm of two positive integers a and b is the positive integer m satisfying:
(a) and
(b) If and , with , then
but and , therefore and because and . Again, we have that and , therefore and because and . Therefore, we see that the gcd does indeed divide the lcm, and we have established that
The venue for the reading was the second floor of the marvelous Avant Garden. As you can see from the photo above, the place exuded a very beat-like 1950′s atmosphere and even had a trio downstairs on stage performing melancholy soundscapes on cello, piano, and guitar across from the open bar.
First happening at the reading was a presentation and warm welcome for the amazing artist, Lindsey Slavin, whose work is prominently featured in the current issue. Then followed introductions of the readers, prior to each getting on stage. The readers, in order of reading, were Chris Oidtmann, Justin Carter, and Kurt Lovelace.
Below are listed their poems in the order in which they were read. Also, the full text of the poems that Chris Oidtmann and Kurt Lovelace read are included here in their entirety. Some of Justin Carter's pieces are pending publication elsewhere, so he was not able to make them available here, but the titles of what he read are nonetheless included.
Ghost
I Help You Create An eHarmony Account
Poem For A Blind Friend
Walmart Sestina
I Hope The Motion-Detecting Cameras Did Not See Our Faces
After Hoagland's Color of the Sky
We Discovered We Were Chewable
Rita Repulsa
Lord Zedd
I remember your black shoes with the silver buckle
Leaping across rocks and along jettys
Sending up splashes of salty water shining
In the moonlight like sparks from a bottle rocket
That same night you left your wallet by the bed,
And pulsed against the wall until it broke
Into a milky way of beige and fluorescent beige.
Sidney can't sing any more. He's dead.
Isn't that how this began?
Drops of still gin fizz, so sloe
They fell into our laps and covered
The little boy's overalls with mud?
Fuck the rabbit hole. He hit the concrete.
Great forces won't come to our aid.
They know better than to hide under our
Pillows while we sleep so they can swap stories
And return to their rightful owners in Bakersfield
Where you gagged my mouth so we could kick,
Kick, and kick again until the headboard fell.
He was with us in that abandoned apartment
Looking for something to make it home.
One eye pressed against a crack in the door
Surveying the vacant parking lot for
Whatever left him behind by mistake.
You asked me if I wanted you
and I did, so said I did,
and you asked
me which you
I wanted, so I told you.
The you who looked at me
with stained glass eyes
under stained glass stars,
into strained black eyes
weighted down by glasses
broken under the rose colored
weight of a thousand petals
pressed into books we read
when we chose thorns over petals
and pricked our feet on thorn
after thorn after thorn, but drew
no blood because we were covered
with roses that shielded our feet
from shards of glass and kept
the pains from breaking our stride.
When we left the Cathedral Rock
you asked me if I loved you
and I did
so I asked if you loved me.
You said you loved
the water in my eyes
and the boat in my mind
that carried us across a sea
where took to the shore
and saw reflections of ourselves
walking over sandy rocks
Because you said I was your rock
when you need to be strong
and I was the white sand
when you need to be weak
so weeks won't become months
of seclusion hidden in jars
in the cupboard next to jam
and pickles, and peaches.
We agreed that the boat
would carry us across the ocean
until some lighthouse lamp
hits the panes of a church window
and reflects off the water,
the glory of stained glass
One Night Stand
A man in the corner catches my eye
He rejected me last month.
Coiffed admirers dangle from his every word.
Scenester bastards in skinny jeans.
Everyone who's anyone is here tonight.
A festering pile of tweaked-out skanks.
It's great when old friends get together again.
Heard Sam gave Ryan herpes.
Laser lights illuminate naked, nubile torsos.
I see you're still snorting your inheritance.
Bodies pressed together in warm embraces
Your friend was much cuter.
Tongues intertwined for the very first time.
The acrid taste of liquor and cigarettes.
I wonder if he's "the one".
What the fuck is your name?
Fumbling hands unlock the apartment door
I went home with your neighbor last month
Hands gently caress soft cheeks
This apartment smells like cat piss.
Tender fingers trace down a shoulder and up a spine
If he has backne I'm leaving.
Two bodies fall gently onto clean linen sheets
Did I wear underwear tonight?
Tangled limbs move in unison to a single heartbeat.
Get your knee off my shin.
An arcane glimpse of the universe. We're decoding the secret.
Your moans sound like the loose timing belt on my car
He falls asleep, head resting gently on my chest
I like him better when he's quiet
His slow breathing matches mine
God, he looks so peaceful.
I can feel his breath in the hair on my chest
That sort of tickles.
My eyes are getting heavy, but I can't look away yet
I wonder if he goes home with a lot of guys
What does he thinks of me?
Am I special, or does this happen all the time?
Did he see me spill that drink earlier?
He's talking in his sleep.
I hope I didn't seem too drunk.
Damn, he has great skin.
Maybe I shouldn't have tried to be witty.
Didn't he say he liked Siouxsie?
I should have gotten a manicure.
I haven't been this comfortable in a while.
Was my cologne too strong?
I like a guy with strong hands
Maybe we can get breakfast tomorrow morning.
Those eyes were pretty amazing
Where would I take him?
The clearest blue I've ever seen.
I'll let him choose.
I.
Before dawn she woke me
drove us to the race track
in the El Camino, 1976
with the windows down.
We parked behind the concession stand
so I could meet you
shortly after your first heart attack.
Asphalt pebbles shattered
under the weight of the well worn boots
that held you up.
You realized your mortality
and banished it to the ticket booth.
Place another wager.
Pray for a superfecta.
We sat and watched the horses run trials
with a stopwatch in my hand
held inside your hand.
II.
Again, I rode to meet you at the track
on a motorbike with blue trim,
bought with court-ordered money
you never sent
My leather jacket reeked of pot
your denim smelled of Maker's Mark and cigarettes.
We sat together on the ground
pretending we had no senses
You hugged me in that parking lot.
I turned my head
towards an abandoned truck
run down, full of scrap metal
hoping you didn't crush the joint in my breast pocket.
III.
My sister called at 7am
while I shaved for work
two weeks ago you died at last
I hadn't known
I stared at my calloused hands, and thought of a child
leaving home in a starched white shirt
and black cotton trousers
running around a red dirt track
whipping red welts into pink skin
until a checkered flag signaled the end of the race.
I don't even smoke, but I will tonight.
sitting in the black vinyl passenger seat
as I laugh and wish I wasn't there
listening to a Cocteau Twins tape he plays
because I said I liked it after he said hello
and I said hey and he bought me a drink
and offered me a smoke, and his eyes
slowly traced from the tip of my new heels,
up my stocking covered thighs.
When we get to his place I wonder why
men are so fucking clumsy with straps
and garters, why they always bite my lip
and softly pant in my ear "you like it, don't you"
because this sort of thing should remain unspoken,
like the clock on his wall that moves in silence.
His hands moves across my breast and
I wonder why a blowjob is called a blowjob
because no one actually blows on anything.
We just move in and out of each other's
lives and that's the new handshake we learn
in charm school after they teach us how to hide
our text messages, and email accounts, and lists
of partners we've been with because the number
is too high and private and won't get us laid
by a man we might want to marry one day.
When he sleeps, I look through his cabinets,
and his desk drawers, and his address book
next to his prescription for Dexedrine
and imagine the man back in Georgia
who makes my body crackle and hum
and accept the substitute sleeping next to me
and reach for the man I'm with
while the one I want
bounces through the sky like a positive charge
looking for my negative to make us whole again.
Some of Justin Carter's pieces are pending publication elsewhere, so he was not able to make them available here, but the titles of what he read are nonetheless included above. [back to the top]
Kurt Lovelace
The last reader of the evening, Kurt Lovelace, read the following poems in the order that they appear here. Kurt addressed the audience, and his comments appear below in the green text.
"Hello, everyone, I am glad that you were all able to come out tonight. I'd like to make a few comments before I begin reading. Having just returned to school last year after a nearly 30 year career as a software engineer, I feel like a time traveler, being back where I was 30 years ago. I took Kevin Prufer's workshop along with Chris and Justin last semester, and so, it is a real privilege to be up here reading with my two classmates."
I'll begin with a few older poems written when I was in my early twenties, and I begin with these simply because they may be of interest in themselves and because they have never been heard by anyone before -- so it's a starting point in introducing you to my work, which now spans over 35 years. Then I will move on to my current work. The first of the three older pieces is called 'Taradiddle Smile' followed by 'Good Morning Captain Kangaroo' and concluded by 'Hunger' -- written at a time when I was perhaps too fond of John Ashbery, but they contain some fun and hopefully entertaining language in them."
Dislocate that chagrin with a taradiddle smile
texturing into laughter. Realize that these tweed gearings
are success's necessary dress, de rigueur
in the legato of getting ahead. You are commodity enough for your greed
rat-tailing through some echelon of employment, up-stepping
like a slinky down-steps. Your profit is
pragmatic redundancy, plus the intangibles of involvement:
no why for a house pregnant with family. Asking would
flabbergast you into a gypsy barreling over Niagara
for the gloat, thrill of it, a runaway reaction
like Caligula, claiming it made as much sense as
hiccups or Halloween, feeling your life like Sampson's haircut, unfair,
yet you in fools-gold accord with its TV shine.
I started out believing in everything:
the open field, plow in hand, horse
waiting to be worked, words
hedged in the furrow, irises open
to the moment of opening.
Perhaps I can ask you about it someday
and you'll tell me everything I've ever wanted
was within reach, if only I would have put
out my hand, wide palms like bells ringing.
Say again, what?
Put your fears in a little box, and smoke it,
not this warm interrogatory weather we've been having
that no one really wants to talk about, that peels
shirts from bodies with an utter unconcern that's neither
here nor there.
The cost of involvement is you get involved and there she is
and your her painting garden her kitchen things on her desk at the office and
she's looking how it's all arranged
your colors the smell of your herbs
why your dishes
aren't put away are your pencils
sharpened
then she sees
carrots need planting, the rhubarb
must go, suddenly
you need new dishes.
Then you start drunk serious writing poems about the cost of involvement
how her lips are not cherries but
red angry commandments painted in a delightful rouge
to elicit your obeisance
so softly
her requests patter
and when her such and such of such words fall
your obeisance
yawns lifts up its arms at her
smiling
Minding the green chain on our cockatoo,
my mother has coffee and oranges, spends
Sunday sitting in her sunny chair. Ripe fruit falls
from our plum bushes and banana trees, yet our boughs
hang heavy in the clearest blue. My father says:
"We're here to track the rockets." I am seven. Soon,
one man is going to walk upon the moon.
The beaches go on and on the water even more.
What about the man already in the moon?
As the Shipley brothers, and Buddy, and I
yank coral piled by the Shipley's rusting Chevy
that hunkers, helpless in Bahamian sea-spray:
scorpions skedaddle, stick sticky, skitter or scatter
if we let them, but we leave them twitching.
Hammocked, I wake naked to the naked sun:
nose itching in salt-spray, head stuck out
the A-frame's attic window. Sand-tufted plum fields
sway by seaweed bedraggled beaches, splashed
and resplashed and splashing in ambergris
waves that glint and pop with turquoise and white
bubbling froth the crabs scurry-up in, till waves pull back
leaving their wet-shaved arms on the shore
shiny, and smooth and new, invites us to play.
2
I eat hyacinths, their wet, red lips
flopped or folded open, sticky witch's doors:
my tongue feels the ridges of her floor,
unswept, gritty with the bones of children, before
I swallow them, then pocket three gleaming cat's
eyes. I've won the dare of Buddy Bogus
the 3rd. His mother holds me by my hair
hanging, just off the ground, above the dirt road
that her paneled wagon had banged against
from Freeport, with five of us screaming. But when
I said "shit", she slammed the car into its own dust,
stopping. She shook when she said: "We do not swear!"
That's when she'd yanked me out the door,
fisted my crew-cut, then lifted my brain
an astronaut, at T-minus zero, not counting,
pushed back by gravity: around me the hushed jungle
peed in it's pants. But I sat on her hat
all the way home, naming the stars
just coming-on: Ursa Major, Andromeda,
Hydra, Cygnus, Draco, Vulpecula
poking through, their thin pinpricks of light
slow-moving against one fast satellite of ours.
3
My parents go to the Missile Base to dance
leaving me asleep. I wake up and run
breathless outside crying on sprinkled grass.
Back in, I pull out the TV's boney on-switch:
a soda jerk on The Twilight Zone tips-up
his white hat: a third eye looks out.
Something scratches on the screen door.
I pull my hair under my mother's milk sheets
and squeeze my eyes tight, and whisper: "Make it go, make it go, make it go!" Asleep,
I fall into the same sticky hole, night
after night, I grab its edges but keep slipping
over into it only to slip over into it again.
Papaya grow with phosphorescent slugs
a breadcrumb's throw behind the Blue Lagoon
Apartments where then we live, and two older boys
make me dance naked till I scream, pounding
through palm trees towards home, unable
to speak for a week. But once, I stole the night
pulling it tight like an enormous sheet
of black with dotted lights, and naked, swam
back into the sea from where I came;
and later sit on the shore with the storied moon,
wiggling my toes, squishy, in the midnight tides
pulling back drowned voices; I think I almost know
the sound of drowning.
4
Just in from Germany, my English played-out,
Dad tutors me, three months at the kitchen table.
I spell everything exactly as it sounds:
"Witch witch wood u bee?" At Saint Mary's Star
Of the Sea, the nuns thought me dyslexic,
till someone told them I was bilingual
and could recite the Lords Prayer in Latin: Pater noster, qui es in caelis, sanctificetur
nomen tuum. It made the nuns flutter, holding
white habits as if they might take flight, be
the sea foam that floats inland on the wind
as at noon, they air their prayers, the angelus: Et verbum caro factum est. Et habitavit in nobis.
(and the word, made flesh, dwelt among us.)
And I kissed Eve, that morning's milk break,
under stiff pines, the sea gulls shreeing, we ran
back, caught. Sister Anne ruled my knuckles raw,
beating their boney nubs on top her desk.
5
Dad floats a bright brochure at me: green eyes
burn in a panther's face, peering out from a jungle
that fans its Stygian felt. "Let's go," he says,
and opens our white battled Ford. For half an hour
we bump the pebbled roads towards Freeport
to see a Brazilian emerald dealer in his shop
tucked away, in an alley of hallways going up
and down, around corners, and then two doors
both locked with slapping bolts, opened. He unrolls
black velvet in a curtained room, one bright light
shines down to show how black the velvet is. He
lays seven stones out, their rough edges "Uncut,"
he says, "from Santa Muerto," and rolls them burning
in the blood between the tips of his fingers.
What was it then? What is it now? Let's ask,
what measurements for us? If we stretch out our arms
the edge of ocean along the sinking sun
seems a dimming thing, encompassed
by the milk of the visiting moon. We bounce
off of the porch and walk toward the beach.
A stick bug extends its manifold hand,
and boysenberries ripen under pricking cactus;
in-between driftwood, ashore, a hermit crab
discards its shell, and in the shallows a leopard ray
wiggles underneath the sand, its spotted wings. |
These suggestions are intended to make your life easier. Some of them
may seem like extra work, but really they cause you less work in the
long run.
This is a preliminary document, i.e. a work in progress.
1.
Be tidy and systematic. Whenever you have data in tables,
especially when place-value is important, be sure to keep things
lined up in neat columns and neat rows. See section 4 for
an example of how this works in practice.
2.
If your columns are wobbly, get a pad of graph
paper and see if that helps. If you don't have a supply of
store-bought graph paper, you can make graph paper on your computer
printer. There are freeware programs that do a very nice job of
this.
If that's too much bother, start with a plain piece of paper and
sketch in faint guidelines when necessary.
3.
Paper is cheap. If you find that you are
running out of room on this sheet of paper, get another sheet of
paper, rather than squeezing the calculation into a smaller space.
See section 4 for an example of using extra paper to
achieve a better result.
4.
The whole calculation should be structured as a succession
of true statements. The first statement is true, the next statement
is true, and the statement after that, et cetera. Each statement is
a consequence of the previous statements (in conjunction with known
theorems, and the "givens" of the problem). Finally, we get to the
bottom line, and we know it is true. An example of this can be seen
in reference 1.
5.
Sometimes, especially for long and/or complex calculations, it
helps to organize your calculation in two columns. An example of
this can be seen in reference 1. In the left column, you
write an equation. In the right column, make a note as to how you
derived that equation. (Numbering all your equations makes this
easier.) If there isn't enough width to do this easily, turn the
paper 90 degrees, so it becomes wider and less tall. (Writing paper
isn't suitable for this, so use graph paper ... or lacking that,
plain white paper.)
6.
Avoid writing down an un-named number like 17, or an un-named
expression like a+17 ... because you might forget the meaning
thereof. Instead, whenever possible, write down an equation
such as b=a+17. That way you can point to every item on the page
and say that's true, that's true, that's true... in accordance with
the strategy described in item 4.
If the meaning of variables such as a and b is not obvious, write
down a legend somewhere (like the legend of a map), explaining in a
sentence or two the meaning of each variable. That is, a name is not
the same as an explanation. Do not expect the structure of a name or
symbol to tell you everything you need to know. Most of what you
need to know belongs in the legend. The name or symbol should allow
you to look up the explanation in the legend.
7.
It is important to be able to go back and check the
correctness of the calculation you have done. See
section 4 and/or reference 1 for examples of
what this means in practice.
8.
As a corollary of item 7: Don't perform
surgery on your equations. That is, once you write down a correct
equation, don't start crossing out terms (or, worse, erasing terms)
and replacing them with other expressions. Such a substitution may
be "mathematically" correct, if the replacement is equal to the
thing being replaced ... but it is a bad strategy, because it makes
it hard for you to check your work. Instead, write a new equation.
Leave the old equation as it is. Paper is cheap. An example of this
can be seen in reference 1.
9.
Keep track of the units for each expression. For example,
the statement "x=2.5 inches" means something rather different
from "x=2.5 meters" ... and if you shorthand it as "x=2.5"
you're just asking for trouble.
Sometimes the penalty for getting this wrong is 328 million dollars,
as in the case of the Mars Climate Orbiter
(reference 2).
Most computer languages do not automatically keep track of the units,
so you will have to do it by hand, in the comments. If the
calculation is nicely structured, it may suffice to have a
legend somewhere, spelling out the units for each of the
variables. If variables are re-used and/or converted from one set of
units to another, you need more than just a legend; you will need
comments (possibly quite a lot of comments) to indicate what units
are being used at each point in the code.
One policy that is sometimes helpful (but sometimes risky) is to
convert everything to SI units as soon as it is read in, even in
fields where SI units are not customary. Then you can do the
calculation in SI units and convert back to conventional units (if
necessary) immediately before writing out the results. (This is
problematic when writing an "intermediate" file. Should it be SI
or customary? How do you know the difference between an
"intermediate" result and a final result?)
It is certainly possible for computer programs to keep track of the
units automatically. A nice example is reference 3. It is
a shame that such features are not more widely available
10.
The factor-label method is a convenient and
powerful way of converting units when necessary. The correctness of
this method is a direct consequence of the axioms of algebra, since
it starts by multiplying by unity, which is allowed by the axioms.
11.
Haste makes waste, especially with multi-step processes. If
you work methodically, you'll get the right answer the first time,
and that's all there is to the story. If you try to do it twice as
fast, you'll get the wrong answer, and then you'll have to do it over
again ... and again ... and again.
Here's a classic example: The task is to add 198 plus 215. The
easiest way to solve this problem in your head is to rearrange it as
(215 + (200 − 2)) which is 415 − 2 which is 413. The small point
is that by rearranging it, a lot of carrying can be avoided.
One of the larger points is that it is important to have multiple
methods of solution. This and about ten other important points are
discussed in reference 4.
The classic "textbook" diagram of an inequality uses shading to
distinguish one half-plane from the other. This is nice and
attractive, and is particularly powerful when diagramming the
relationship between two or more inequalities, as shown in
figure 2.
Obviously the hatched depiction is not as beautiful as the shaded depiction,
but it is good enough. It is vastly preferable on cost/benefit
grounds, for most purposes.
Some refinements:
I recommend hatching the excluded half-plane, so that the
solution-set remains unhatched rather than hatched. This is
particularly helpful when constructing the conjunction (logical
AND) of multiple inequalities.
I recommend using solid lines versus dashed lines to distinguish
"≥" inequalities from ">" inequalities. If you're going to
hatch the excluded half-plane, use solid lines for the ">"
inequalities, to show that the boundary itself is excluded.
As suggested in figure 4, for linear
inequalities you can do the hatching faster and more accurately with
the help of a ruler or straightedge, which makes it easy to ensure
that none of the hatches stray into the wrong region.
Short multiplication refers to any multiplication problem where you
just memorize the answer. You must memorize the multiplication
table for everything from 0×0 through 9×9. You get the
next step (up to 10×10) practically for free, and it is often
worthwhile to keep going up to 12×12.
Long multiplication refers to multiplying larger numbers. This works
by breaking the numbers down into their individual digits, then
multiplying on a digit-by-digit basis (using the short multiplication
facts for zero through nine) and then combining all the results with
due regard for place value.
We now discuss a nice way to do long multiplication.
The first steps are shown in figure 5.
There are two parts to the figure, representing two successive
stages of the work. Anything shown in black is something you actually
write down, whereas anything shown in color is just commentary, put
there to help us get through the explanation the first time.
We start with the leftmost part of the figure. This is just the
statement of the problem, namely 4567×321. The important point
here is to line up the numbers as shown, so that the ones' place of
the top number lines up with the ones' place of the bottom number, et
cetera. Keeping things aligned in columns is crucial, since the
colums represent place value. If you have trouble keeping things
properly lined up, use grid-ruled paper. You can see such a grid in
figure 5. If you don't have grid-ruled paper, you
can always sketch in some guide lines. As mentioned in item 1,
tidiness pays off.
You may omit the multiplication sign (×) if it is obvious from
context that this is a multiplication problem (as opposed to an
addition problem).
Tangential remark: Some people attempt to
call one of these numbers the multiplier and the other the
multiplicand, but since multiplication is commutative the
distinction is meaningless. People often use the terms in ways
inconsistent with the supposed definition. I call both of
them multiplicands, which is more-or-less Latin for "thing being
multiplied". In figure 6 we have two things being
multiplied. Note that since multiplication is associative, you
could easily have many things being multiplied, as in
12×32×65×99, in which case it again makes sense to
call each of them multiplicands, and it is obviously hopeless to
attempt to distinguish "the" multiplier from "the"
multiplicand. As a related issue, there are also holy wars as to
whether 4567×321 means 4567 "times" 321, or 4567
"multiplied by" 321. Again the distinction is meaningless.
Don't worry about it.
If one of the multiplicands is longer than the other, it will usually
be more convenient to place the longer one on top. That's not
mandatory, but it makes the calculation slightly more compact.
We now proceed digit by digit, starting with the rightmost digit in
the bottom multiplicand, which in this case is a 1. We multiply this
digit into each digit of the upper multiplicand, working in order
right-to-left, which makes sense because it is the direction of
increasing place value (even though it is opposite to the direction of
reading normal text).
We place these results in order in row c, with due regard for place
value. The 1×7 result goes in the ones' place, the 1×6
result goes in the tens' place, and the 1×5 result goes in the
hundreds' place, and so forth. Actually, multiplication by 1 is so
easy that you could just copy the whole number 4567 into row c
without thinking about it very hard.
You may wish to leave a little bit of space above the
numbers in row c, for reasons that will become apparent later.
That is all we need to do with the low-order digit of the bottom
multiplicand. We now move on to the next digit, working
right-to-left. In this case it is a 2.
Again we multiply this digit into each digit of the upper
multiplicand. The result of the first such multiplication is
2×7=14, which we place in row d. This is most clearly seen in
the middle column of the figure 6. Alignment is crucial
here. The 14 must be aligned under the 2 as shown. That's because it
"inherits" the place value of the 2.
Next we multiply 2×6=12. You might be tempted to write this in
row d, but there is no room for it there, so it must go on row e,
as shown in the middle column of the figure. Again alignment is
critical. The 12 is shifted one place to the left of the 14, because
it inherits additional place value. It inherits some from the 2 and
some from the 6. Since we are working systematically right-to-left,
you don't need to think about this too hard; just remember that each
of these short-division products must be shifted one place to the left
of the previous one.
We have now more than half-way finished. We have reached the stage
shown in figure 5.
The next step is the 2×5=10 multiplication. There is room for
this on row d, which is a good place to put it. Next we do the
2×4=8 multiplication. There is room for this on row e, which
is a good place to put it. Note the pattern of placing successive
short-division results on alternating rows. This is guaranteed to
work, because the product of two one-digit numbers can never have more
than two digits.
At this point (or perhaps earlier), if you are not using grid-ruled
paper, you should lightly sketch in some vertical guide lines, as
shown by the dashed lines in figure 7. The tableau has
become large enough that there is some risk of messing up the
alignment, i.e. putting things into the wrong columns, if you don't
put in guide lines.
That's all for the "2" digit in the bottom multiplicand. We now
progress to the "3" digit. The work proceeds in the same fashion.
All the short-multiplication results are put in rows f and g.
In the tableau, you can see where everything comes from. The
color-coded background indicates which digit of the upper multiplicand
was involved, and the row indicates which digit of the bottom
multiplicand was involved.
At this point you can draw a line under row g as shown in the
figure. All that remains is a big addition problem, adding up rows
c through g inclusive. You can use the space above row c to
keep track of carries if you wish, but this is not mandatory. (There
are never very many carries, so keeping track of them is easy, no
matter how you do it. Some people just count them on their fingers.)
The result of the addition is the result of the overall multiplication
problem, as shown on row h.
Let's do another example, as shown in figure 7, which
illustrates one more wrinkle.
This example calls attention to the situation where some of the
short-multiplication products have one digit, while others have two.
You can see this on row c of the figure, where we have 3×3=9
and 3×8=24. In most cases it is safer to pretend they all have
two digits, which is what we have done in the figure, writing 9 as 09.
Similarly on line d we write 6 as 06. This makes the work fall into
a nice reliable pattern. It helps you keep things lined up, and
makes the work easier to check.
In some cases, such as 345×1 or 432×2, all the
short-multiplication products have one digit, so you can write them
all on a single line, if you wish. This saves a little
bit of paper. On the other hand, remember the advice in
item 3: paper is cheap. You may find
it helpful to write the short-multiplication products
as two digits even if you don't have to.
Mathematically speaking, writing one-digit products as two digits is
unconventional, but it is entirely correct. It has the advantage of
making the algorithm more systematic, and therefore easier to check.
In any case, the result of the addition is the result of the overall
multiplication problem, as shown on row g of figure 7.
That's all there is to it.
This algorithm uses two rows of intermediate results for each digit in
the bottom multiplicand (except when the digit is zero or one). This
has two advantages: First of all, you don't need to do any adding or
carrying as you go along; you just write down the short-multiplication
results "as is". Secondly, it makes it easy to check your work.
Each of the short-multiplication results is sitting there in an
obvious place, almost begging to be checked.
This differs from the old-fashioned "textbook" approach, which uses
only one row per digit, as shown below. The old-fashioned approach
supposedly uses less paper – but the advantage is slight at best, and
if you allow room keeping track of "carries" throughout the tableau,
the advantage becomes even more dubious.
What's worse is that the old-fashioned approach is significantly more
laborious. It may look more compact, but it involves more work. You
have to do the same number of short multiplications, and a
greater number of additions. It requires you to do additions
(including carries) as you go along.
Last but not least, it makes it much harder to check your work.
4
5
6
7
×
3
2
1
4
5
6
7
9
1
3
4
1
3
7
0
1
1
4
6
6
0
0
7
The old-fashioned approach.
Not recommended.
Remember, paper is cheap (item 3) and it is important
to be able to check your work (item 7).
The usual "textbook" instructions for how to do long division are
both unnecessarily laborious and unnecessarily hard to understand.
There's another way to organize the calculation that is much less
mysterious and much less laborious (especially when long multi-digit
numbers are involved).
Note that as discussed above, keeping things lined up in columns
is critical. It may help to use grid-ruled paper, or at least to
sketch in some guidelines.
After writing down the statement of the problem, and before doing any
actual dividing, it helps to make a crib, as shown in the lower
left of the figure. This is just a multiplication table, showing all
multiples of 13 (or, more generally, all multiples of the assigned
divisor). It is super-easy to construct such a table, since no
multiplication is required. Successive addition will do the job. We
need all the multiples from ×1 to ×9, but you might as
well calculate the ×10 row, by adding 117+13, as a check on
all the additions that have gone before.
The first step is to consider the leading digit of the dividend (which
in this case is a 7). Since this is less than the divisor (13), there
is no hope of progress here, so we proceed to the next step.
Now consider the first two digits together, namely the 7 and the 5. Look at
the crib to find the largest entry less than or equal to 75. It is
65, as in 5×13=65. Write copy this entry to the division
problem, on row b, directly under the 75. Since this came from row
5 of the crib, write a 5 on the answer line, aligned with the 75 and
the 65, as shown. Check the work, to see that 5 (on the answer line)
times 13 (the divisor, on line a), equals 65 (on line b).
Now do the subtraction, namely 75−65=10, and write the result
on line c as shown.
We now shift attention from the first column to the middle column of
figure 8. Bring down the 2 from the dividend, as shown by
the red arrow in the middle column of the figure. So now the number
we are working on is 102, on line c.
The steps from now on are a repeat of earlier steps.
Look at the crib to find the largest entry less than or equal to 102.
It is 91, as in 7×13=91. Write copy this entry to the division
problem, on row d, directly under the 102. Since this came from row
7 of the crib, write a 7 on the answer line, aligned with the 102 and
the 91, as shown. Check the work, to see that 7 (on the answer line)
times 13 (the divisor, on line a), equals 91 (on line d). Do the
subtraction.
As a check on the work, when doing this subtraction,
the result should never be less than zero, and should never be
greater than or equal to the divisor. Otherwise you've used the
wrong line from the crib, or made an arithmetic error.
We now shift attention to the rightmost column of figure 8.
Bring down the 7, look in the crib to find the largest entry less
than or equal to 117, which is in fact 117, as in 9×13=117.
Since this came from line 9 of the crib, write a 9 on the answer line,
properly aligned.
The final subtraction yields the remainder on line g. The remainder is
zero in this example, because 13 divides 7527 evenly.
Perhaps the crib's most important advantage, especially when people
are first learning long division, is that the crib removes the mystery
and the guesswork from the long division process. This is in contrast
to the "trial" method, where you have to guess a quotient digit, and
you might guess wrong. Using the crib means we never need to do a
short division or trial division; all we need to do is skim the table
to find the desired row.
We have replaced trial division by multiplication and table-lookup.
Actually we didn't even need to do any multiplication, so it would be
better to say we have replaced trial division by addition.
Another advantage is efficiency, especially when the dividend has many
digits. That's because you only need to construct the crib once (for
any given divisor), but then you get to use it again and again, once
for each digit if the dividend. For long dividends, this saves a
tremendous amount of work. (This is not a good selling point
when kids are just learning long division, because they are afraid
of big multi-digit dividends.) Setting up the crib is so fast that
you've got almost nothing to lose, even for small dividends.
Another advantage is that it is easy to check the correctness of the
crib. It's just sitting there begging to be checked.
When bringing down a digit, you
may optionally bring down all the digits. For instance, in
the middle column of figure 8, if you bring down all the
digits you get 1027 on row c. One advantage is that 1027 is a
meaningful number, formed by the expression
7527−13×5×102. This shows how the steps of the
algorithm (and the intermediate results) actually have mathematical
meaning; we are not not blindly following some mystical mumbo-jumbo
incantation. I recommend that if you are trying to
understand the algorithm, you should bring down all the digits a few
times, at least until you see how everything works.
A small disadvantage is that bringing down all the digits requires
more copying. The countervailing small advantage is that it may
help keep the digits lined up in their proper columns. Whether the
advantages outweigh the cost is open to question, and probably
boils down to a question of personal preference.
Another remark: Division is the "inverse function" of
multiplication. In a profound sense, for any function that can be
tabulated, you can construct the inverse function – if it exists –
by switching columns in the table. That is, we interchange abscissa
and ordinate: (x,y)↔(y,x). That's why we are able to
perform division using what looks like a multiplication table; we just
use the table backwards.
The modern numeral system is based on place value. As we understand
it today, each numeral can be considered a polynomial in powers of
b, where b is the base of the numeral system. For decimal
numerals, b=10. As an example:
Given two expressions such as (a+b+c) and (x+y), each of which has
one or more terms, the systematic way to multiply the expressions is
to make a table, where the rows correspond to terms in the first
expression, and the rows correspond to terms in the second expression:
In the special case of multiplying a two-term expression by another
two-term expression, the mnemonic FOIL applies. That stands for
First, Outer, Inner, Last. As shown in figure 9, we start with
the First contribution, i.e. we multiply the first term from in each
of the factors. Then we add in the Outer contribution, i.e. the first
term from the first factor times the last term from the last factor.
Then we add in the Inner contribution, i.e. the last term from the
first factor times the first term from the last factor. Finally we
add in the Last contribution, i.e. we multiply the last terms from
each of the factors.
Most square roots are irrational, so they cannot be represented
exactly in the decimal system. (Decimal numerals are, after all,
rational numbers.) So the name of the game is to find a decimal
representation that is a sufficiently-accurate approximation.
We start with the following idea: For any nonzero x we know that
x÷√x is equal to √x. Furthermore, if s1 is
greater than √x it means x/s1 is less than √x. If
we take the average of these two things, s1 and x/s1, the
average is very much closer to √x. So we set
and then iterate. The method is very powerful; the number of digits
of precision doubles each time. It suffices to use a rough estimate
for the starting point, s1. In particular, if you are seeking the
square root of an 8-digit number, choose some 4-digit number as the
s1-value.
This is a special case of a more general technique called
Newton's method, but if that doesn't mean anything to you, don't
worry about it.
Note that the square of 1.01 is very nearly 1.02. Similarly, the
square of 1.02 is very nearly 1.04. Turning this around, we find the
general rule that if x gets bigger by two percent, then √x
gets bigger by one percent ... to a good approximation.
We can illustrate this idea by finding the square root of 50. Since
50 is 2% bigger than 49, the square root of 50 is 1% bigger than 7
... namely 7.07. This is a reasonably good result, accurate to better
than 0.02%.
If we double this result, we get 14.14, which is the square root of
200. That is hardly surprising, since we remember that the square
root of 2 is 1.414, accurate to within roundoff error.
Sine and cosine are transcendental functions. Evaluating them will
never be super-easy, but it can be done, with reasonably decent
accuracy, with relatively little effort, without a calculator.
In particular:
You can always start with a zeroth-order approximation: For
angles near zero, the sine will be near zero. For angles near
90∘, the sine will be near 1. For angles near 30∘, the sine
will be near 0.5, et cetera.
You can draw a graph, using the anchor points in equation 4 as a guide. You can then use graphical interpolation to
obtain values for any angle.
A simple Taylor series gives a result accurate to 2.1% or
better using only a couple of multiplications. Remember to express
the angle in radians before using these formulas.
The following facts serve to "anchor" our knowledge of the sine
and cosine:
Actually, that hardly counts as "remembering" because if you ever
forget any part of equation 4 you should be able to
regenerate it from scratch. The 0∘ and 90∘ values are
trivial. The 30∘ is a simple geometric construction. Then the
60∘ and 45∘ values are obtained via the Pythagorean theorem.
The value for 45∘ should be particularly easy to remember,
since √2 = 1.414 and √½ = ½√2.
The rest of this section is devoted to the Taylor series. A low-order
expansion works well if the point of interest is not too far from the
nearest anchor.
For angles between −10∘ and +10∘, the approximation
sin(x)≈x is accurate to better than 0.51%. This is a
one-term Taylor series. Let's call it the Taylor[1] approximation.
It is super-easy to evaluate, since it involves no additions and no
multiplications, or at most one multiplication if we need to convert
to radians from degrees or some other unit of measure. See
equation 5c and the blue line near x=0 in figure 10.
For angles from 25∘ to 65∘, all the points are with a
few degrees of one of the anchor points in equation 4.
This means the first-order Taylor series is accurate to better than 1
percent in this region. We can call this the Taylor[0,1]
approximation. It requires knowing the sine and the cosine at the
nearest anchor point ... which we do in fact know from equation 4. See equation 9 and the blue line in
figure 10.
For a rather broad range of angles near the top of the sine,
from 65∘ to 115∘, the approximation sin(x)≈1−x2/2
is accurate to better than half a percent. This is a second-order
Taylor series with only two terms, because the linear term is zero.
Let's call this the Taylor[0,2] approximation. See equation 5e and the green line near x=90∘ in figure 10.
This leaves us with a region from 10∘ to 25∘ that
requires some special attention. Options include the following:
For most purposes, the best option is to use the Taylor[1,3]
approximation anchored at zero. This requires a couple more
multiplications, but the result is accurate to better than 0.07%.
If you really want to minimize the number of multiplications, we can
start by noting that the Taylor[1] extrapolation coming up from zero
is better than the Taylor[0,1] extrapolation coming down from
30∘, so rather than using the closest anchor we use the 0∘
anchor all the way up to 20∘ and use the 30∘ anchor above
that. This has the advantage of minimizing the number of
multiplications. Disadvantages include having to remember an obscure
fact, namely the need to put the crossover at 20∘ rather than
halfway between the two anchors. The accuracy is better than 2.1%,
which is not great, but good enough for some applications. The error
is shown in figure 11.
If you can maintain even a vague memory of the form of equation 7, you can easily reconstruct the exact details. Use the
fact that it has to be symmetric under exchange of a and b (since
addition is commutative on the LHS). Also it has to behave correctly
when b=0 and when b=π/2.
If we assume b is small and use the small-angle approximations from
equation 5, then equation 7 reduces to the
second-order Taylor series approximation to sin(a+b).
You can use the Taylor series to interpolate between the values given
in equation 4. Since every angle in the first quadrant
is at least somewhat near one of these values, you can find the sine
of any angle, to a good approximation, as shown in figure 10. |
According to the author, "We study fundamental algebraic structures, namely groups, rings, fields and modules, and maps...
see more
According to the author, "We study fundamental algebraic structures, namely groups, rings, fields and modules, and maps between these structures. The techniques are used in many areas of mathematics, and there are applications to physics, engineering and computer science as well. In addition, I have attempted to communicate the intrinsic beauty of the subject. Ideally, the reasoning underlying each step of a proof should be completely clear, but the overall argument should be as brief as possible, allowing a sharp overview of the result.״Each chapter of the book is downloadable as a separate pdf file.
״Abstract Algebra: Theory and Applications is an open-source textbook written by Tom Judson that is designed to teach the...
see more
״Abstract Algebra: Theory and Applications is an open-source textbook written by Tom Judson that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.״ |
explanation
This is a very complete book: ita has good explanations for people who cannot understand the topics that are being covered, but there are also challeging exercises for the poeple who is good at calculus. |
Details: Course content will include the real number system, similarity and proportional reasoning, number theory, measurement, probability and data analysis. Prerequisite: A grade of C or better in MATH 246. (F, Sp) |
Description of Lifepac Math: Grade 8: Unit 7: Integers by Alpha Omega
Repetition, drills and application ensure mastery of computational skills with Lifepac Math. Students will progress to higher-level cognitive reasoning and analysis as their problem solving ability increases.
Perfect for students who flourish in a self-paced, individualized learning format, each consumable LIFEPAC combines lessons, exercises, projects, reviews and tests.
Product:
Lifepac Math: Grade 8: Unit 7: Integers
Vendor:
Alpha Omega
Binding Type:
Paperback
Media Type:
Book
Minimum Grade:
8th Grade
Maximum Grade:
8th Grade
Weight:
0.5 pounds
Length:
11 inches
Width:
8.5 inches
Vendor Part Number:
MAT 0807
Subject:
Math Math: Grade 8: Unit 7: Integers.
Average Rating
Parent Rating
Comments
Nice that it is broken up into seperate LifePacs, allowing the feeling of progress for the student. |
Math Workbook
This self-teaching workbook offers extensive preparation and brush-up in math for all who plan to take the GED High School Equivalency Test. A ...Show synopsisThis self-teaching workbook offers extensive preparation and brush-up in math for all who plan to take the GED High School Equivalency Test. A diagnostic test with answers is presented to help test takers assess their strengths and weaknesses. The math review that follows is supplemented with hundreds of exercises. All GED math topics are covered, including measurement, geometry, algebra, number relations, and data analysis. Four practice tests with answers reflect questions and question types found on the actual GED GED Math Workbook (Barron's Math Workbook for the Ged)...Good. GED Math Workbook (Barron's Math Workbook for the Ged |
I am currently a first year undergraduate majoring in mathematics. I'm taking an introductory analysis course and find it very hard compared to other math couses. I know that the topics covered in the ... |
more details
Inequalities continue to play an essential role in mathematics. Perhaps, they form the last field comprehended and used by mathematicians in all areas of the discipline. Since the seminal work Inequalities (1934) by Hardy, Littlewood and Pã³lya, mathematicians have laboured to extend and sharpen their classical inequalities. New inequalities are discovered every year, some for their intrinsic interest whilst others flow from results obtained in various branches of mathematics. The study of inequalities reflects the many and various aspects of mathematics. On one hand, there is the systematic search for the basic principles and the study of inequalities for their own sake. On the other hand, the subject is the source of ingenious ideas and methods that give rise to seemingly elementary but nevertheless serious and challenging problems. There are numerous applications in a wide variety of fields, from mathematical physics to biology and economics. This volume contains the contributions of the participants of the Conference on Inequalities and Applicationsnbsp;held in Noszvajnbsp;(Hungary) in September 2007. It is conceived in the spirit of the preceding volumes of the General Inequalities meetings held in Oberwolfach from 1976 to 1995 in the sense that it not only contains the latest results presented by the participants, but it is also a useful reference book for both lecturers and research workers. The contributions reflect the ramification of general inequalities into many areas of mathematics and also present a synthesis of results in both theory and practice |
Common Core Standards: Math
Math.CCSS.Math.Content.HSA-CED.A.1
1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Students should be able to interpret word problems and form equations and inequalities in order to solve the problem. That means translating a word problem to an algebraic equation.
Let's be real, here. Math is another language, just like Spanish, Japanese, or Icelandic. When you start learning a language, you don't start by translating words like "absquatulate" or "loquacious" or "pneumonoultramicroscopicsilicovolcanoconiosis" (and yes, that is a real word).
It's better to start easier, with words like "cat" and "girl" and slowly work your way up. Just the same, if you use simple linear equations that are familiar to students, they can focus on the translation process and it'll all go a lot smoother.
Translation is a useful analogy in and of itself because it emphasizes that the algebraic equation is the same as the word problem, just presented in a different way. In addition to helping students to understand the process, the translation analogy can also help reassure struggling learners and encourage practice.
After they've gotten a hang of the basics, students can start learning quadratic, rational, and exponential functions to address all aspects of this standard. Once students are familiar with these operations individually, they should be asked to distinguish them from each another.
As students gain experience, there are additional strategies that should be introduced. One experienced problem solver strategy is to read the question twice before beginning. It's a useful piece of advice in general, actually.
Writing a list of what is known and a list of what needs to be calculated is also an excellent strategy. Such lists are especially useful when sorting out unnecessary information, identifying an appropriate formula to utilize, or constructing a proof. These strategies should be suggested and shown to students after they are proficient with the basic translation process.
To start off, the chart below may be presented as a dictionary to support word to symbol translation. Students can also add to the chart as they find other key words or phrases.
Algebra Symbol
Key Words
= equals
all
equals
gives
is, are, was, were, will be
results
same
yields
< is less than
below
less than
≤ is less than or equal to
maximum of
not more than
> is greater than
greater than
more than
over
≥ is greater than or equal to
at least
minimum of
not less than
+ addition
add
and
combine
increase
more
plus
raise
sum
together
total
– subtraction
decrease
difference
fewer
less
lose
minus
reduce
× multiplication
directly proportional
double(× 2), triple(× 3), etc.
group of
linear
multiplied
product
times
/ division
average
cut
divided by/into
each
inversely proportional
out of
per
pieces
quotient
ratio
share
split
xn power
power
square (n = 2), cube (n = 3), etc.
nx exponential
decays
doubles (n = 2), triples (n = 3), quadruples (n = 4), etc.
grows
rate of n per x
Drills
There are 60 students going on a field trip to the chocolate factory. The students are from three different classes. Mrs. Hooper's class has 24 students and Mr. Gomez's class has 18 students. Which of the equalities correctly describes the students and could be used to solve for how many students are from Mr. Anderson's class? (Let A = the number of students in Mr. Anderson's class.)
The relation Hooper's class + Gomez's class + Anderson's class = 60 students going on a field trip becomes an equation by changing the written descriptions into numbers and variables. Mrs. Hooper's class has 24 students and Mr. Gomez's class has 18 students giving 24 + 18 + Anderson's class = 60 students going on a field trip. The number of students in Anderson's class is the unknown and must be represented by a variable like A for Anderson. That means 24 + 18 + A = 60.
There are 60 students going on a field trip to the chocolate factory. The students are from three different classes. Mrs. Hooper's class has 24 students and Mr. Gomez's class has 18 students. How many students are from Mr. Anderson's class?
(A) 16 students(B) 18 students(C) 20 students(D) 22 students
Correct Answer:
18 students
Answer Explanation:
The equation 24 + 18 + A = 60 (where A = students in Anderson's class) needs to be solved to isolate A. First, simplify 24 + 18 to 42 to get 42 + A = 60. Then, subtract 42 from both sides to get A = 60 – 42. This gives A = 18. There are 18 students from Anderson's class.
There are six chaperones going on a field trip. There are two buses for the trip. The chaperones divide so there is the same number of chaperones on each bus. Which of the equations could be utilized to find the number of chaperones on the first bus? (Let c = number of chaperones on first bus.)
(A)(B)(C) 6 × 2 = c(D) 6 × c = 2
Correct Answer:
Answer Explanation:
The relation becomes an equation by changing the written descriptions into numbers and variables. There are six chaperones on the trip giving . There are two buses, giving . The number of chaperones on the first bus is an unknown represented by the variable c. This results in the equation .
There are six chaperones going on a field trip. There are two buses for the trip. The chaperones divide so there is the same number of chaperones on each bus. How many chaperones are there on the first bus?
(A) 2(B) 3(C) 6(D) 12
Correct Answer:
3
Answer Explanation:
The equation = c (where c = number of chaperones on first bus) needs to be solved for the variable c. This gives 3 = c or c = 3. There are 3 chaperones on the first bus.
A total of 66 people attended a field trip to a chocolate factory for a tour. A maximum of 15 people are allowed to tour at one time. Which equation correctly describes how many tour groups to organize? (Let g = the number of groups.)
(A)(B)(C)(D)
Correct Answer:
Answer Explanation:
The relation becomes an equation by changing the written descriptions into numbers and variables. The 66 and 15 become numbers directly giving . The number of tour groups is an unknown so we need to make a variable g = number of tour groups. The resulting equation is .
A total of 66 people attended a field trip to a chocolate factory for a tour. A maximum of 15 people are allowed to tour at one time. What is the minimum number of tour groups that can be formed?
(A) 4(B) 5(C) 15(D) 66
Correct Answer:
5
Answer Explanation:
The equation (where g = number of tour groups) needs to be solved for the variable g. Multiplying each side by g to remove it from the denominator gives 66 ≤ 15 × g. Dividing each side by 15 then gives which becomes 4.4 ≤ g. Reversing this gives g ≥ 4.4. The actual number of tour groups formed must be a whole number since there cannot be fractions of tours. The smallest whole number that is greater than or equal to 4.4 is 5. A minimum of 5 tour groups must be formed. which of these equations could be utilized to solve for the number of chocolates in the small box (2x)? how many chocolates are in the small box (2x)?
(A) 3(B) 4(C) 5(D) 6
Correct Answer:
6
Answer Explanation:
The equation 4x2 + 3x2 + 2x = 69 can be solved for 2x, the number of chocolates in the small box. This gives 7x2 + 2x – 69 = 0 which is factored to (7x + 23)(x – 3) = 0. The solutions are and x = 3. These give 2x = -6.572 and 2x = 6, respectively. The negative solution does not make sense because you cannot have a negative number of chocolates in a box. So, the correct solution must be 2x = 6. There are 6 chocolates in the small box.
On the day of the class field trip, the chocolate factory produced three times as many plain chocolate bars as crispy bars. They produced 50 more nutty bars than crispy bars. The ratio of plain chocolate bars produced to nutty bars produced was 2 to 1. Which of the equations below could be utilized to solve for the number of crispy bars produced on the day of the field trip?
(A) 3c + 2c = 50(B)(C) 2 × 3c = 1 × (c + 50)(D)
Correct Answer:
Answer Explanation:
The key word "ratio" indicates division such that . Then, you need number of plain chocolate bars in terms of crispy bars, which is 3c from the key word "times." And, you need number of nutty bars in terms of crispy bars, which is c + 50 from the key word "more." Combining these gives the equation .
On the day of the class field trip, the chocolate factory produced three times as many plain chocolate bars as crispy bars. They produced 50 more nutty bars than crispy bars. The ratio of plain chocolate bars produced to nutty bars produced is 2 to 1. How many crispy bars were produced?
(A) 50(B) 100(C) 125(D) 175
Correct Answer:
100
Answer Explanation:
The equation equates the 2 to 1 ratio with 3c (the number of plain chocolate bars) to c + 50 (the number of nutty bars). To solve for c, the number of crispy bars, first multiply both sides by c + 50 to get 3c = 2c + 100. Then, subtract 2c from both sides to get c = 100. So, 100 crispy bars were produced Which equation could be utilized to calculate the height of the box in inches?
The initial relation for this problem requires knowing that the volume of the box will be length × width × height. Since each chocolate has a volume of 1 in3, the total volume of the box with 144 chocolates will be 144 in3. So, the basic relationship is length × width × height = 144. We do not know length, but we do know that it is twice the width. And, we know the width is three times the height. The key words "twice" and "times" both indicate multiplication so we have width = 3 × height = 3h and length = 2 × width = 2 × 3h. And the height is, of course, just h. So, the overall equation becomes (2 × 3h) × 3h × h = 144 What is the height of the box in inches?
(A) 1 in(B) 2 in(C) 6 in(D) 12 in
Correct Answer:
2 in
Answer Explanation:
Given the relation length × width × height = volume, the resulting equation is (2 × 3h) × 3h × h = 144. This simplifies to 18h3 = 144. First, each side is divided by 18 to get h3 = 8. Then, taking the cubed root of both sides gives h = 2. The height of the box is 2 in.
The candy company's total revenue this year was 1.1 times the revenue last year. If they experience the same growth every year, what equation describes how many years will it take before the revenue is more than double?
(A)(B)(C) 1.1y ≥ 2(D) 1.1y ≥ 2
Correct Answer:
1.1y ≥ 2
Answer Explanation:
The basic relation here is for year revenue to be at least initial revenue. That is yearly revenue ≥ 2 × initial revenue. But there is no information given about the total initial revenue or the total year revenue. Instead, we know that the first year the revenue will be 1.1 × initial revenue. The second year, it will be 1.1 × first year revenue, or 1.1 × 1.1 × initial revenue, or 1.12 × initial revenue. For any given year, the revenue is 1.1y × initial revenue, where y is the number of years. Going back into the inequality gives 1.1y × initial revenue ≥ 2 × initial revenue. Canceling the initial revenue from both sides gives the equation 1.1y ≥ 2.
The candy company's total revenue this year was 1.1 times the sales last year. If they experience the same growth every year, how many years will it take before the revenue is more than double?
(A) 8(B) 7(C) 3(D) 2
Correct Answer:
8
Answer Explanation:
For any given year, the revenue will be 1.1y times the initial revenue. More than double means 1.1y ≥ 2. Taking the log10 of both sides gives y × log101.1 ≥ log102. Using a calculator for the base 10 log gives log101.1 = 0.0414 and log102 = 0.301. The equation becomes 0.0414y ≥ 0.301. Dividing both sides by 0.0414 gives y ≥ 7.27. Looking at whole numbers of years, the first year with more than double the current revenue is 8 years. |
Intermediate Mathematics 1
This course is an intermediate level mathematics course which covers the introductory concepts as in EPMATH134 Introductory Mathematics 1 as well as further skills in algebra and functions and further practice in problem solving. The course content includes skills in numeracy, algebra, linear and non-linear functions, graphing, exponential and logarithmic theory. The course aims to provide a sound foundation in a wide range of basic mathematical skills and in their application to problem solving.
Recommended for degrees in Business, Commerce, Information Technology and the Sciences, including degrees in Health Science, Social Science and Behavioural Science.
Available in 2014
By the end of the course the student will: * have obtained a background in number theory, algebra, functions, graphing and exponential and logarithmic theory. * demonstrated a broad understanding of the subject area, which together with EPMATH235 Intermediate Mathematics 2, is suitable for undergraduate studies in Business, Commerce, Science and the Health Sciences and other areas with a mathematical content. |
over 175 worked examples and more than 500 practice problems and quiz questions to help students develop and practice their problem solving ...Show synopsisProvides over 175 worked examples and more than 500 practice problems and quiz questions to help students develop and practice their problem solving |
Sign in to YouTube
The video shows basics of using the TI83 and TI84 series graphing calculators. On and off, entering functions, using the window view, graphing functions, using table view, how to press keys to get different inputs, using TRACE, using ANS, using the last ENTRY feature, adjusting contrast of the viewscreen. |
Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. It defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, focusing on the simplest cases of linear parallel transport in a v... read more
Customers who bought this book also bought:
Our Editors also recommend:
Differential Geometry by Heinrich W. Guggenheimer This text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; more. Includes exercises and 62 figures.
Lectures on Classical Differential Geometry: Second Edition by Dirk J. Struik Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, envelopes, more. Many problems and solutions. Bibliography.
Differential Geometry by Erwin Kreyszig An introductory textbook on the differential geometry of curves and surfaces in 3-dimensional Euclidean space, presented in its simplest, most essential form. With problems and solutions. Includes 99 illustrations.
Product Description:
Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. It defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, focusing on the simplest cases of linear parallel transport in a vector bundle. The treatment opens with an introductory chapter on fiber bundles that proceeds to examinations of connection theory for vector bundles and Riemannian vector bundles. Additional topics include the role of harmonic theory, geometric vector fields on Riemannian manifolds, Lie groups, symmetric spaces, and symplectic and Hermitian vector bundles. A consideration of other differential geometric structures concludes the text, including surveys of characteristic classes of principal bundles, Cartan connections, and spin structures |
Mathematics for 3D Game Programming and Computer Graphics, Third Edition
BOOK DESCRIPTION
This updated third edition illustrates the mathematical concepts that a game developer needs to develop 3D computer graphics and
game engines at the professional level. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more
advanced topics in 3D programming such as illumination and visibility determination. Particular attention is given to derivations of key results, ensuring that
the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review
the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series.
ABOUT THE AUTHOR
Eric Lengyel is a veteran of the computer games industry with over 16 years of experience writing game engines. He has a PhD in Computer Science from the
University of California at Davis and an MS in Mathematics from Virginia Tech. Eric is the founder of Terathon Software, where he currently leads ongoing development
of the C4 Engine.
TABLE OF CONTENTS
Preface
What's New in the Third Edition
Contents Overview
Notational Conventions |
Buy Used Textbook
Buy New Textbook
eTextbook
180 day subscription
$83.40
More New and Used from Private Sellers
Starting at $12 College Algebra, 5th Edition is designed to make algebra interesting and relevant to the student. The text adopts a problem-solving approach that motivates readers to grasp abstract ideas by solving real-world problems. The problems lie on a continuum from basic algebraic drills to open-ended, non-routine questions. The focus is shifted from learning a set of discrete mathematical rules to exploring how algebra is used in the social, physical, and life sciences. The goal of Explorations in College Algebra, 5th Edition is to prepare students for future advanced mathematics or other quantitatively based courses, while encouraging them to appreciate and use the power of algebra in answering questions about the world around us.
Table of Contents
An Introduction to Data and Functions
Describing Single-Variable Data
Visualizing Single-Variable Data
Numerical Descriptors: What is "Average" Anyway?
An Introduction to Algebra Aerobics
An Introduction to Explore and Extend
Describing Relationships between Two Variables
Visualizing Two-Variable Data
Constructing a "60-Second Summary"
Using Equations to Describe Change
An Introduction to Functions
What is a Function?
Representing Functions: Words, Tables, Graphs and Equations
Input and Output: Independent and Dependent Variables
When is a Relationship Not a Function?
The Language of Functions
Function Notation
Domain and Range
Visualizing Functions
Is There a Maximum or Minimum Value?
When is the Output of the Function Positive, Negative or Zero?
Is the Function Increasing or Decreasing?
Is the Graph Concave Up or Concave Down?
Getting the Big Idea
Chapter Summary
Check Your Understanding
Chapter 1 Review: Putting it all Together
Exploration 1.1 Collecting, Representing, and Analyzing Data
Rates of Change and Linear Function
Average Rates of Change
Describing Change in the U.S. Population over Time
Defining the Average Rate of Change
Limitations of the Average Rate of Change
Change in the Average Rate of Change
The Average Rate of Change is a Slope
Calculating Slopes
Putting a Slant on Data
Slanting the Slope: Choosing Different End Points
Slanting the Data with Words and Graphs
Linear Functions: When Rates of Change are Constant
What if the U.S. Population Had Grown at a Constant Rate?
Real Examples of a Constant Rate of Change
The General Equation for a Linear Function
Visualizing Linear Functions
The Effect of b
The Effect of m
Finding Graphs and Equations of Linear Functions
Finding the Graph
Finding the Equation
Special Cases
Direct Proportionality
Horizontal and Vertical Lines
Parallel and Perpendicular Lines
Breaking the Line: Piecewise Linear Functions
Piecewise Linear Functions
The absolute value function
Step functions
Constructing Linear Models for Data
Fitting a Line to Data: The Kalama Study
Reinitializing the Independent Variable
Interpolation and Extrapolation: Making Predictions
Looking for Links between Education and Earnings: Using Regression Lines |
Introduction to :
On the first day of school, you walked into geometry class with a smile on your face. Not because you like math or geometry, but just because you knew that algebra was truly done and over with. No more quadratic equations or polynomials or solving for x.
Unfortunately, that only lasted for a good two minutes. You quickly realized that geometry, as different as it might seem, actually relies heavily on algebraic concepts. If you don't know how to work with exponents, simplify expressions, or solve for that aggravating x, you might end up hating geometry just as much as algebra.
Now, we're not saying you'll need to remember every tiny detail about algebra. It's all right to refresh your memory. In fact, we highly encourage you to go back and look at your notes just to wipe the cobwebs off those concepts. |
Copyright C> 1967 by Morris Kline.
All rights reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30
Lesmill Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd.,
10 Orange Street, London WC2H 7EG.
This Dover edition, first published in 1985, is an unabridged re
publication of the work first published by Addison-Wesley Publishing
Company, Inc., Reading, Massachusetts, in 1967 under the tide Matht
matiafor Lihtral Arts. The Instructor's Manual published with the origi
nal edition, containing additional answers and solutions to the problems
in the text, has been added to this edition.
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Library of Congress Cataloging in Publication Data
Kline, Morris, 1908-
Mathematics for the nonmathematician.
Reprint. Originally published: Mathematics for liberal ans. Reading,
Mass.: Addison-Wesley, c> 1967. (Addison-Wesley series in introductory
mathematics)
Includes bibliographies and index.
1. Mathematics-I961- I. Tide.
QA37.2.K6 1985 510 84-25923
ISBN 0-486-24823-2
MATHEMATICS FOR THE
NONMATHEMATICIAN
MORRIS KLINE
Professor of Mathematics, Emeritus
Courant Institute of Mathematical Sciences
New York University
DOVER PUBLICATIONS, INC.
NEW YORK
COPYRIGHT @ 1967 BY ADDISON-WESLEY PUBLISHING COMPANY, INC. ALL RIGHTS RESERVED.
THIS BOOK, OR PARTS THEREOF, MAY NOT BE REPRODUCED IN ANY FORM WITHOUT WRITTEN
PERMISSION OF THE PUBLISHER. PRINTED IN THE UNITED STATES OF AMERICA. PUBLISHED
SIMULTANEOUSLY IN CANADA. LIBRARY OF CONGRESS CATALOG CARD NO. 67-12831.
PREFACE
" ... I cunsider that withum understanding as much of the abstruser part of
geometry, as Archimedes or Apollunius, Im£ may understand enough to be assisted
by it in the contemplatiun of nature; and that une needs not !mow the profoundest
mysteries of it to be able to discern its usefulness.. .. I have often wished that
I had employed about the speculative part of geometry, and the cultivatiun of the
specious [symbolicJ algebra I had been taught very young, a good part of that time
and industry that I spent about surveying and fortificatiun. .. .
"
ROBERT BOYLE
I believe as firmly as I have in the past that a mathematics course addressed to
liberal arts students must present the scientific and humanistic import of the sub
ject. Whereas mathematics proper makes little appeal and seems even less pointed
to most of these students, the subject becomes highly significant to them when it
is presented in a cultural context. In fact, the branches of elementary mathematics
were created primarily to serve extra-mathematical needs and interests. In the
very act of meeting such needs each of these creations has proved to have inesti
mable importance for man's understanding of the nature of his world and himself.
That so many professors have chosen to teach mathematics as an integral
part of Western culture, as evidenced by their reception of my earlier book,
Mathematics: A Cultural Approach, has been extremely gratifying. That book will
continue to be available. In the present revision and abridgment, which has been
designed to meet the needs of particular groups of students, the spirit of the original
text has been preserved. The historical approach has been retained because it is
intrinsically interesting, provides motivation for the introduction of various topics,
and gives coherence to the body of material. Each topic or branch of mathematics
dealt with is shown to be a response to human interests, and the cultural import
of the technical development is presented. I adhered to the principle that the level
of rigor should be suited to the mathematical age of the student rather than to
the age of mathematics.
As in the earlier text, several of the topics are treated quite differently from
what is now fashionable. These are the real number system, logic, and set theory.
I tried to present these topics in a context and with a level of emphasis which I
believe to be appropriate for an elementary course in mathematics. In this book,
v
VI PREFACE
the axiomatic approach to the real numbers is formulated after the various types
of numbers and their properties are derived from physical situations and uses.
The treatment of logic is confined to the fundamentals of Aristotelian logic. And
set theory serves as an illustration of a different kind of algebra.
The changes made in this revision are intended to suit special groups. Some
students need more review and drill on elementary concepts and techniques than
the earlier book provides. Others, chiefly those preparing for teaching on the
elementary level, need to learn more about elementary mathematics than their
high school courses covered. Teachers of twelfth-year high school courses and
one-semester college courses often found the extensive amount of material in
Mathematics: A Cultural Approach rather disconcerting because it offered so much
more than could be covered.
To meet the needs of these groups I have made the following changes:
I. Four of the chapters devoted entirely to cultural influences have been dropped.
The size of the original book has thereby been reduced considerably.
2. A few applications of mathematics to science have been omitted, primarily
to reduce the size of the text.
3. Some of the chapters on technical topics, Chapter 3 on logic and mathematics,
Chapter 4 on number, Chapter 5 on elementary algebra, and Chapter 21 on
arithmetics and their algebras have been expanded.
4. Additional drill exercises have been added within a few chapters, and a set
of review exercises providing practice in technique has been added to each of
a number of chapters.
5. Improvements i n presentation have been made in a number of places.
With respect to use in courses, it is probably true of the present text, as it is of the
earlier one, that it contains more material than can be covered in some courses.
However. many of the chapters as well as sections in chapters are not essential
to the logical continuity. These chapters and sections have been starred (*).
Thus Chapter !O on painting shows historically how mathematicians were led to
projective geometry (Chapter II), but from a logical standpoint, Chapter10 is
not needed in order to understand the succeeding chapter. Chapter 19 on musical
sounds is an application of the material on the trigonometric functions in Chapter 18
but is not essential to the continuity. The two chapters on the calculus are not
used in the succeeding chapters. Desirable as it may be to give students some
idea of what the calculus is about, it may still be necessary in some classes to omit
these chapters. The same can be said of the chapters on statistics (Chapter 22) and
probability (Chapter 23).
As for sections within chapters, Chapter 6 on Euclidean geometry may well
serve as an illustration. The mathematical material of this chapter is intended as
a review of some basic ideas and theorems of Euclidean geometry and as an intro
duction to the conic sections. Some of the familiar applications are given in
Section 6-3 (see the Table of Contents) and probably should be taken up. How-
PREFACE Vll
ever the applications to light in Sections 6-4 and 6-6 and the discussion of cultural
influences in Section 6-7 can be omitted.
Some of the material, whether or not included in the following recommenda
tions for particular groups, can be left to student reading. In fact, the first two
chapters were deliberately fashioned so that they could be read by students. The
objective here, in addition to presenting intrinsically important ideas, was to
induce students to read a mathematics book, to give them the confidence to do so,
and to get them into the habit of doing so. It seems necessary to counter the
students' impression, resulting no doubt from elementary and high school instruc
tion in mathematics, that whereas history texts are to be read, mathematics texts
are essentially reference books for formulas and homework exetcises.
For C(JUTSeS emphasizing the number concept and its extension to algebra, it is
possible to take advantage of the logical independence of numerous chapters and
use Chapters 3 through 5 on reasoning, arithmetic, and algebra and Chapter 2 1
on different algebras. To pursue the development of this theme into the area of
functions one can include Chapters 1 3 and 15.
C(JUTSes emphasizing geometry can utilize Chapters 6, 7, 1 1 , 12, and 20 on
Euclidean geometry, trigonometry, projective geometry, coordinate geometry,
and non-Euclidean geometry respectively. Some algebra, that reviewed in Chapter 5,
is involved in Chapters 7 and 12. If knowledge of the material of Chapter 5 cannot
be presupposed, this chapter must precede the treatment of geometry.
The essence of the two preceding suggestions may be diagrammed thus:
Chapters 3, 4, 5
I
1 1
Algebra Geometry
Chapters 2 1 , 1 3 , 1 5 Chapters 6, 7, 1 1 , 1 2 , 20
Of course, starred sections in these chapters are optional.
For a one-semester liberal arts C(fUrse, the basic content can be as follows:
Chapter 2 on a historical orientation,
Chapter 3 on logic and mathematics,
Chapters
4 and 5 on the number system and elementary algebra,
Chapter 6 through Section 6-5, on Euclidean geometry,
Chapter 7 through Section 7-3, on trigonometry,
Chapter 12 on coordinate geometry,
Chapter 13 on functions and their uses,
Chapter 14 through ,Section 14-4. on parametric equations.
Chapter 1 5 through Section 15-10, on the further use of functions In
SCience,
Chapter 20 on non-Euclidean geometry,
Chapter 21 on different algebras.
VllI PREFACE
Any additional material would enrich the course but would not be needed for
continuity.
Though the teacher's problem of presenting material outside the domain
of mathematics proper is far simpler with this text than with the earlier one, it
may still be necessary to assure him that he need not hesitate to undertake this
task. The feeling that one must be an authority in a subject to say anything about
it is unfounded. We are all laymen outside the field of our own specialty, and
we should not be ashamed to point this out to students. In contiguous areas we
are merely giving indications of ideas that students may pursue further in other
courses or in independent reading.
I hope that this text will serve the needs of the groups of students to which
it is addressed and that. despite the somewhat greater emphasis on technical matters,
it will convey the rich significance of mathematics.
I wish to thank my wife Helen for her critical scrutiny of the contents and her
conscientious reading of the proofs. I wish to express, also, my thanks to members
of the Addison-Wesley staff for very helpful suggestions and for their continuing
support of a culturally oriented approach to mathematics.
New York,1967 M.K.
CONTENTS
1 Why Mathematics?
2 A Historical Orientation
2-1 Introduction . II
2-2 Mathematics in early civilizations. II
2-3 The classical Greek period 14
2-4 The Alexandrian Greek period 17
2-5 The Hindus and Arabs 19
2-6 Early and medieval Europe 20
2-7 The Renaissance 20
2-8 Developments from 1550 [Q 1800 22
2-9 Developments from 1800 to the present 24
2-10 The human aspect of mathematics 27
3 Logic and Mathematics
3-1 Introduction. 30
3-2 The concepts of mathematics 30
3-3 Idealization 38
3-4 Methods of reasoning 39
3-5 Mathematical proof 45
3-6 Axioms and definitions 50
3-7 The creation of mathematics 51
4 Number: the Fundamental Concept
4-1 Introduction . 58
4-2 Whole numbers and fractions 58
4-3 Irrational numbers 65
4-4 Negative numbers 72
4-5 The axioms concerning numbers 75
* 4-6 Applications of the number system 82
5 Algebra, the Higher Arithmetic
5-1 Introduction . 94
5-2 The language of algebra 94
5-3 Exponents 97
5-4 Algebraic transformations 102
5-5 Equations involving unknowns 106
5-6 The general second-degree equation 112
* 5-7 The history of equations of higher degree 119
ix
X CONTENTS
6 The Nature and Uses of Euclidean Geometry
6-1 The beginnings of geometry 123
6-2 The content of Euclidean geometry 125
6-3 Some mundane uses of Euclidean geometry 131
* 6-4 Euclidean geometry and the study of light 138
6-5 Conic sections 142
* 6-6 Conic sections and light 144
* 6-7 The cultural influence of Euclidean geometry 149
7 Charting the Earth and the Heavens
7-1 The Alexandrian world 153
7-2 Basic concepts of trigonometry 158
7-3 Some mundane uses of trigonometric ratios 163
* 7-4 Charting the earth 165
* 7-5 Charting the heavens 171
* 7-6 Further progress in the study of light. 176
8 The Mathematical Order of Nature
8-1 The Greek concept of nature 187
8-2 Pre�Greek and Greek views of nature 188
8-3 Greek astronomical theories 190
8-4 The evidence for the mathematical design of nature. 192
8-5 The destruction of the Greek world 194
*9 The Awakening of Europe
9-1 The medieval civilization of Europe 197
9-2 Mathematics in the medieval period 199
9-3 Revolutionary influences in Europe 200
9-4 New doctrines of the Renaissance 202
9-5 The religious motivation in the study of nature 206
* 10 Mathematics and Painting in the Renaissance
10-1 Introduction 209
10-2 Gropings toward a scientific system of perspective 210
10-3 Realism leads to mathematics 213
10-4 The basic idea of mathematical perspective 215
10-5 Some mathematical theorems on perspective drawing 219
10-6 Renaissance paintings employing mathematical perspective 223
10-7 Other values of mathematical perspective 229
11 Projective Geometry
11-1 The problem suggested by projection and section 232
11-2 The work of Desargues 234
11-3 The work of Pascal 239
11-4 The principle of duality 242
11-5 The relationship between projective and Euclidean geometries 247
12 Coordinate Geometry
12-1 Descartes and Fermat. 250
12-2 The need for new methods in geometry. 253
12-3 The concepts of equation and curve 256
CONTENTS Xl
12-4 The parabola 264
12-5 Finding a curve from its equation 269
12-6 The ellipse . 27 1
* 12-7 The equations of surfaces 273
* 12-8 Four-dimensional geometry 275
12-9 Summary 277
13 The Simplest Formulas in Action
13-1 Mastery of nature 280
13-2 The search for scientific method 281
13-3 The scientific method of Galileo . 284
13-4 Functions and formulas 290
13-5 The formulas describing the motion of dropped objects 293
13-6 The formulas describing the motion of objects thrown downward 299
13-7 Formulas for the motion of bodies projected upward 300
14 Parametric Equations and Curvilinear Motion
14-1 Introduction 307
14-2 The concept of parametric equations 308
14-3 The motion of a projectile dropped from an airplane 310
14-4 The motion of projectiles launched by cannons . 31 3
* 14-5 The motion of projectiles fired at an arbitrary angle 318
14-6 Summary 323
15 The Application of Formulas to Gravitation
15-1 The revolution in astronomy 326
15-2 The objections w a heliocentric theory 330
15-3 The arguments for the heliocentric theory 331
15-4 The problem of relating earthly and heavenly motions 334
15-5 A sketch of Newton's life 336
15-6 Newton's key idea . 337
15-7 Mass and weight 340
15-8 The law of gravitation 341
15-9 Further discussion of mass and weight 343
15-10 Some deductions from the law of gravitation 346
* 15-11 The rotation of the earrh 352
* 15-12 Gravitation and the Keplerian laws 355
* 15-13 Implications of the theory of gravitation. 359
* 16 The Differential Calculus
16-1 Introduction 365
16-2 The problems leading to the calculus 365
16-3 The concept of instantaneous rate of change 367
16-4 The concept of instantaneous speed 368
16-5 The method of increments 371
16-6 The method of increments applied to general functions 374
16-7 The geometrical meaning of the derivative 379
16-8 The maximum and minimum values of functions 382
� 17 The Integral Calculus
17-1 Differential and integral calculus compared 388
17-2 Finding the formula from the given rate of change 389
xii CONTENTS
17-3 Applications to problems of motion 390
17-4 Areas obtained by imegration 394
17-5 The calculation of work . 397
17-6 The calculation of escape velocity 401
17-7 The integral as the limit of a sum 404
17-8 Some relevant history of the limit concept 409
17-9 The Age of Reason 412
18 Trigonometric Functions and Oscillatory Motion
18-1 Introduction 416
18-2 The motion of a bob on a spring. 417
18-3 The sinusoidal functions 418
18-4 .-\cccleration in sinusoidal motion 427
18-5 The mathematical analysis of the motion of the bob 429
18-6 Summary 434
* 19 The Trigonometric Analysis of Musical Sounds
19-1 Introduction 436
19-2 The nature of simple sounds . 438
19-3 The method of addition of ordinates. 442
19-4 The analysis of complex sounds 445
19-5 Subjective properties of musical sounds 448
20 Non·Euclidean Geometries and Their Significance
20-1 Introduction 452
20-2 The historical background 452
20-3 The mathematical content of Gauss's non-Euclidean geometry 458
20-4 Riemann's non-Euclidean geometry 460
20-5 The applicability of non-Euclidean geometry 462
20-6 The applicability of non-Euclidean geometry under a new interpretation
of line 464
20-7 Non-Euclidean geometry and the nature of mathematics 471
20-8 The implications of non-Euclidean geometry for other branches of our
culture 474
21 Arithmetlcs and Their Algebras
21-1 Introduction 478
21-2 The applicability of the real number system 478
21-3 Baseball arithmetic. 481
21-4 Modular arithmetics and their algebras 484
21-5 The algebra of sets 491
21-6 Mathematics and models 497
* 22 The Statistical Approach to the Social and Biological Sciences
22-1 Introduction 499
22-2 A brief historical review 500
22-3 Averages 502
22-4 Dispersion 503
22-5 The graph and the normal curve 505
22-6 Fitting a formula to data 511
22-7 Correlation 516
22-8 Cautions concerning the uses of statistics 518
CONTENTS xiii
<Ie 23 The Theory of Probability
23-1 Introduction 522
23-2 Probability for equally likely outcomes 524
23-3 Probability as relative frequency. 529
23-4 Probability in continuous variation 530
23-5 Binomial distributions. 53l
23-6 The problems of sampling 518
24 The Nature and Value. of Mathematics
24--1 Introduction 541
24--2 The structure of mathematics. 541
24--l The values of mathematics for the study of nature . 546
24-4 The aesthetic and intellectual values. 550
24--5 Mathematics and rationalism 552
24-6 The limitations of mathematics 553
Table 01 Trigonometric Ratios 557
Answers to Selected and Review Exercises 559
Additional Answers and Solutions 569
Index 633
CHAPTER 1
WHY MATHEMATICS?
In mathematics I can report no deficience, except it be that men do not
sufficiently understand the excellent use of the Pure Mathematics. ...
FRANCIS BACON
One can wisely doubt whether the study of mathematics is worth while and
can find good authority to support him. As far back as about the year 400 A.D.,
St. Augustine, Bishop of Hippo in Africa and one of the great fathers of
Christianity, had this to say:
The good Christian should beware of mathematicians and all those who make
empty prophecies. The danger already exists that the mathematicians have made
a covenant with the devil to darken the spirit and to confine man in the bonds
of Hell.
Perhaps St. Augustine, with prophetic insight into the conflicts which were to
arise later between the mathematically minded scientists of recent centuries and
religious leaders, was seeking to discourage the further development of the
subject. At any rate there is no question as to his attitude.
At about the same time that St. Augustine lived, the Roman jurists ruled,
under the Code of Mathematicians and Evil-Doers, that "to learn the art of
geometry and to take part in public exercises, an art as damnable as mathe
matics, are forbidden."
Even the distinguished seventeenth-century contributor to mathematics,
Blaise Pascal, decided after studying mankind that the pure sciences were not
suited to it. In a letter to Fermat written on August 10, 1660, Pascal says: "To
speak freely of mathematics, I find it the highest exercise of the spirit; but at
the same time I know that it is so useless that I make little distinction between
a man who is only a mathematician and a common artisan. Also, I call it the
most beautiful profession in the world; but it is only a profession; and I have
often said that it is good to make the attempt [to study mathematics], but not
to use our forces: so that I would not take two steps for mathematics, and I am
confident that you are strongly of my opinion." Pascal's famous injunction
was, "Humble thyself, impotent reason."
I
2 WHY MATHEMATICS?
The philosopher Arthur Schopenhauer, who despised mathematics, said
many nasty things about the subject, among others that the lowest activity of
the spirit is arithmetic, as is shown by the fact that it can be performed by a
machine. Many other great men, for example, the poet Johann Wolfgang
Goethe and the historian Edward Gibbon, have felt likewise and have not
hesitated to express themselves. And so the student who dislikes the subject
can claim to be in good, if not living, company.
In view of the support he can muster from authorities, the student may
well inquire why he is asked to learn mathematics. Is it because Plato, some
2300 years ago, advocated mathematics to train the mind for philosophy? Is it
because the Church in medieval times taught mathematics as a preparation for
theological reasoning? Or is it because the commercial, industrial, and scien
tific life of the Western world needs mathematics so much' Perhaps the sub
ject got into the curriculum by mistake, and no one has taken the trouble to
throw it out. Certainly the student is justified in asking his teacher the very
question which Mephistopheles put to Faust:
Is it right, I ask, is it even prndence,
To bOTe thyself and bore the students?
Perhaps we should begin our answers to these questions by pointing out
that the men we cited as disliking or disapproving of mathematics were really
exceptional. In the great periods of culture which preceded the present one,
almost all educated people valued mathematics. The Greeks, who created the
modern concept of mathematics, spoke unequivocally for its importance. Dur
ing the Middle Ages and in the Renaissance, mathematics was never challenged
as one of the most important studies. The seventeenth century was aglow not
only with mathematical activity but with popular interest in the subject. We
have the instance of Samuel Pepys, so much attracted by the rapidly expanding
influence of mathematics that at the age of thirty he could no longer tolerate
his own ignorance and begged to learn the subject. He began, incidentally,
with the multiplication table, which he subsequently taught to his wife. In
1 681 Pepys was elected president of the Royal Society, a post later held by
Isaac Newton.
In perusing eighteenth-century literature, one is struck by the fact that
the journals which were on the level of our Harper's and the Atlantic Monthly
contained mathematical articles side by side with literary articles. The edu
cated man and woman of the eighteenth century knew the mathematics of
their day, felt obliged to be au courant with all important scientific develop
ments, and read articles on them much as modern man reads articles on politics.
These people were as much at home with Newton's mathematics and physics
as with Pope's poetry.
The vastly increased importance of mathematics in our time makes it all
the more imperative that the modern person know something of the nature and
WHY MATHEMATICS? 3
role of mathematics. It is true that the role of mathematics in our civilization
is nor always obvious, and the deeper and more complex modern applications
are not readily compr.ehended even by specialists. But the essential nature and
accomplishments of the subject can still be understood.
Perhaps we can see more easily why one should study mathematics if we
take a moment to consider what mathematics is. Unfortunately the answer
cannot be given in a single sentence or a single chapter. The subject has many
facets or, some might say, is Hydra-headed. One can look at mathematics as a
language, as a particular kind of logical structure, as a body of knowledge
about number and space, as a series of methods for deriving conclusions, as the
essence of our knowledge of the physical world, or merely as an amusing in
tellectual activity. Each of these features would in itself be difficult to describe
accurately in a brief space.
Because it is impossible to give a concise and readily understandable defi
nition of mathematics, some writers have suggested, rather evasively, that
mathematics is what mathematicians do. But mathematicians are human beings,
and most of the things they do are uninteresting and some, embarrassing to
relate. The only merit in this proposed definition of mathematics is that it
points up the fact that mathematics is a human creation.
A variation on the above definition which promises more help in under
standing the nature, content, and values of mathematics, is that mathematics is
what mathematics doel-. If we examine mathematics from the standpoint of
what it is intended to and does accomplish, we shall undoubtedly gain a truer
and clearer picture of the sunject.
Mathematics is concerned primarily with what can be accomplished by
reasoning. And here we face the first hurdle. Why should one reason? It is
not a natural activity for the human animal. It is clear that one does not need
reasoning to learn how to eat or to discover what foods maintain life. J\1an
knew how to feed, clothe, and house himself millenniums before mathematics
existed. Getting along with the opposite sex is an art rather than a science
mastered by reasoning. One can engage in a multirude of occupations and
even climb high in the business and industrial world without much use of
reasoning and certainly without mathematics. One's social position is harctl y
elevated by a display of his knowledge of trigonometry. In fact, civiliz?tions
in which reasoning and mathematics played no role have endured and even
flourished. If one were willing to reason, he could readily supply evidence to
prove that reasoning is a dispensable activity.
Those who are opposed to reasoning will readily point out other methods
of obtaining knowledge. Most people are in fact convinced that their senses
are really more than adequate. The very common assertion "seeing is believ
ing" expresses the common reliance upon the senses. But e\'eryone should
recognize that the senses are limited and often fallible and, even where ac
curate, must be interpreted. Let us consider, as an example, the sense of sight.
4 WHY MATHEMATICS?
How big is the sun? Our eyes tell us that it is about as large as a rubber ball.
This then is what we should believe. On the other hand, we do not see the
air around us, nor for that matter can we feel, touch, smell, or taste it. Hence
we should not believe in the existence of air.
To consider a somewhat more complicated situation, suppose a teacher
should hold up a fountain pen and ask, What is it? A student coming from
some primitive society might call it a shiny stick, and indeed this is what the
eyes see. Those who call it a fountain pen are really calling upon education
and experience stored in their minds. Likewise, when we look at a tall build
ing from a distance, it is experience which tells us that the building is tall.
Hence the old saying that "we are prone to see what lies behind our eyes,
rather than what appears before them."
Every day we see the sun where it is not. For about five minutes before
what we call sunset, the sun is actually below the geometrical horizon and
should therefore be invisible. But the rays of light from the sun curve toward
us as they travel in the earth's atmosphere, and the observer at P (Fig. 1-1)
not only "sees" the sun but thinks the light is coming from the direction O'P.
Hence he believes the sun is in that direction.
Apparent position
01 -.;,'U.
/I�
__ --
- - - - - - - - ----
o::;�--����
True position
Fig. 1-1. Deviation of a ray by the earth's atmosphere.
The senses are obviously helpless in obtaining some kinds of knowledge,
such as the distance to the sun, the size of the earth, the speed of a bullet
(unless one wishes to feel its velocity), the temperature of the sun, the predic
tion of eclipses, and dozens of other facts.
If the senses are inadequate, what about experimentation or, in simple cases,
measurement? One can and in fact does learn a great deal by such means. But
suppose one wants to find a very simple quantity, the area of a rectangle. To
obtain it by measurement, one could lay off unit squares to cover the area and
then count the number of squares. It is at least a little simpler to measure the
lengths of the sides and then use a formula obtained by reasoning, namely, that
the area is the product of length and width. In the only slightly more com
plicated problem of determining how high a projectile will go, we should
certainly not consider traveling with the projectile.
As to experimentation, let us consider a relatively simple problem of mod
ern technology. One wishes to build a bridge across a river. How long and
WHY MATHEMATICS? 5
how thick should the many beams be? What shape should the bridge take?
If it is to be supported by cables, how long and how thick should these be? Of
course one could arbitrarily choose a number of lengths and thicknesses for the
beams and cables and build the bridge. In this event, it would only be fair that
the experimenter be the first to cross this bridge.
It may be clear from this brief discussion that the senses, measurement,
and experimentation, to consider three alternative ways of acquiring knowl
edge, are by no means adequate in a variety of situations. Reasoning is essential.
The lawyer, the doctor, the scientist, and the engineer employ reasoning daily
to derive knowledge that would otherwise not be obtainable or perhaps obtain
able only at great expense and effort. Mathematics more than any other human
endeavor relies upon reasoning to produce knowledge.
One may be willing to accept the fact that mathematical reasoning is an
effective procedure. But just what does mathematics seek to accomplish with
its reasoning) The primary objective of all mathematical work is to help man
study nature, and in this endeavor mathematics cooperates with science. It
may seem, then, that mathematics is merely a useful tool and that the real
pursuit is science. We shall not attempt at this stage to separate the roles of
mathematics and science and to evaluate the relative merits of their contribu
tions. We shall simply state that their methods are different and that mathe
matics is at least an equal partner with science.
We shall see later how observations of nature are framed in statements
called axioms. Mathematics then discloses by reasoning secrets which nature
may never have intended to reveal. The determination of the pattern of mo
tion of celestial bodies, the discovery and control of radio waves, the under
standing of molecular, atomic, and nuclear structures, and the creation of
artificial satellites are a few basically mathematical achievements. Mathematical
formulation of physical data and mathematical methods of deriving new con
clusions are today the substratum in all investigations of nature.
The fact that mathematics is of central importance in the study of nature
reveals almost immediately several values of this subject. The first is the prac
tical value. The construction of bridges and skyscrapers, the harnessing of the
power of water, coal, electricity, and the atom, the effective employment of
light, sound, and radio in illumination, communication, navigation, and even
entertainment, and the advantageous employment of chemical knowledge in
the design of materials, in the production of useful forms of oil, and in medi
cine are but a few of the practical achievements already attained. And the
future promises to dwarf the past.
However, material progress is not the most compelling reason for the study
of nature, nor have practical results usually come about from investigations so
directed. In fact, to overemphasize practical values is to lose sight of the
greater significance of human thought. The deeper reason for the study of
nature is to try to understand the ways of nature, that is, to satisfy sheer in-
6 WHY MATHEMATICS?
tellectual curiosity. Indeed, to ask disinterested questions about nature is one of
the distinguishing marks of mankind. In all civilizations some people at least
have tried to answer such questions as: How did the universe come about?
Howald is the universe and the earth in particular? How large are the sun
and the earth? Is man an accident or part of a larger design? Will the solar
system continue to function or will the earth some day fall into the sun? What
is light? Of course, not all people are interested in such questions. Food,
shelter, sex, and television are enough to keep many happy. But others, aware
of the pervasive natural mysteries, are more strongly obsessed to resolve them
than any business man is to acquire wealth and power.
Beyond improvement in the material life of man and beyond satisfaction
of intellectual curiosity, the study of nature offers intangible values of another
sort, especially the abolition of fear and terror and their replacement by a
deep, quiet satisfaction in the ways of nature. To the uneducated and to those
uninitiated in the world of science, many manifestations of nature have ap
peared to be agents of destruction sent by angry gods. Some of the beliefs in
ancient and even medieval Europe may be of special interest in view of what
happened later. The sun was the center of all life. As winter neared and the
days became shorter, the people believed that a battle between the gods of
light and darkness was taking place. Thus the god Wodan was supposed to be
riding through heaven on a white horse followed by demons, all of whom
sought every opportunity to harm people. When, however, the days began to
lengthen and the sun began to show itself higher in the sky each day, the peo
ple believed that the gods of light had won. They ceased all work and cele
brated this victory. Sacrifices were offered to the benign gods. Symbols of
fertility such as fruit and nuts, whose growth is, of course, aided by the sun,
were placed on the altars. To symbolize further the desire for light and the
joy in light, a huge log was placed in the fire to burn for twelve days, and
candles were lit to heighten the brightness.
The beliefs and superstitions which have been attached to events we take
in stride are incredible to modern man. An eclipse of the sun, a threat to the
continuance of the light and heat which causes crops to grow, meant that the
heavenly body was being swallowed up by a dragon. Many Hindu people
believe today that a demon residing in the sky attacks the sun once in a while
and that this is what causes the eclipse. Of course, when prayers, sacrifices,
and ceremonies were followed by the victory of the sun or moon, it was clear
that these rituals were the effective agent and so had to be pursued on every
such occasion. In addition, special magic potions drunk during eclipses insured
health, happiness, and wisdom.
To primitive peoples of the past, thunder, lightning, and storms were
punishments visited by the gods on people who had apparently sinned in some
way. The stories in the Old Testament of the flood and of the destruction of
Sodom and Gomorrah by fire and brimstone are examples of such acts of
WHY MATHEMATICS? 7
wrath by the God of the Hebrews. Hence there was continual concern and
even dread about what the gods might have in mind for helpless humans. The
only recourse was to propitiate the divine powers, so that they would hring
good fortune instead of evil.
Fears, dread, and superstitions have been eliminated, at least in our Western
civilization, by just those intellectually curious people who have studied na
ture's mighty displays. Those "seemingly unprofitable amusements of specula
tive brains" have freed us from serfdom, given us undreamed of powers, and,
in fact, have replaced negative doctrines by positive mathematical laws which
reveal a remarkable order and uniformity in nature. Man has emerged as the
proud possessor of knowledge which has enabled him to view nature calmly
and objectively. An eclipse of the sun occurring on schedule is no longer an
occasion for trembling but for quiet satisfaction that we know nature's ways.
We breathe freely, knowing that nature will not be willful or capricious.
Indeed, man has been remarkably successful in his study of nature. History
is said to repeat itself, but, in general, the circumstances of the supposed repe
tition are not the same as those of the earlier occurrence. As a consequence,
the history of man has not been tOO effective a guide for the future. Nature is
kinder. When nature repeats herself, and she does so constantly, the repetitions
are exact facsimiles of previous events, and therefore man can anticipate na
ture's behavior and be prepared for what will take place. We have learned to
recognize the patterns of nature and we can speak today of the uniformity of
nature and delight in the regularity of her behavior.
The successes of mathematics in the study of inanimate nature have in
spired in recent times the mathematical study of human nature. Mathematics
has not only contributed to the very practical institutions such as banking,
insurance, pension systems, and the like, but it has also supplied some substance,
spirit, and methodology to the infant sciences of economics, politics, and so
ciology. Number, quantitative studies, and precise reasoning have replaced
vague, subjective, and ineffectual speculations and have already given evidence
of greater values to come.
As man turns to thoughts about himself and his fellow man, other questions
occur to him which are as fundamental as any he can ask. Why is man bo�n?
What purposes does he serve or should he serve? What future awaits him'
The knowledge acquired about our physical universe has profound implications
for the origin and role of man. Moreover, as mathematics and science have
amassed increasing knowledge and power, they have gradually encompassed
the biological and psychological sciences, which in turn have shed further light
on man's physical and mental life. Thus it has come about that mathematics
and science have profoundly affected philosophy and religion.
Perhaps the most profound questions in the realm of philosophy are, What
is truth and how does man acquire it? Though we have no final answer to
these questions, the contribution of mathematics toward this end is paramount.
8 WHY MATHEMATICS?
For twO millenniums mathematics was the prime example of truths man had
unearthed. Hence all investigations of the problem of acquiring truths neces
sarily reckoned with mathematics. Though some startling developments in the
nineteenth century altered completely our understanding of the nature of
mathematics, the effectiveness of the subject, especially in representing and
analyzing natural phenomena, has still kept mathematics the focal point of all
investigations into the nature of knowledge. Not the least significant aspect of
this value of mathematics has been the insight it has given us into the ways and
powers of the human mind. Mathematics is the supreme and most remarkable
example of the mind's power to cope with problems, and as such it is worthy
of study.
Among the values which mathematics offers are its services to the arts.
Most people are inclined to believe that the arts are independent of mathe
matics, but we shall see that mathematics has fashioned major styles of painting
and architecture, and the service mathematics renders to music has not only
enabled man to understand it, but has spread its enjoyment to all corners of
our globe.
Practical, scientific, philosophical, and artistic problems have caused men
to investigate mathematics. But there is onc other motive which is as strong
as any of these - the search for beauty. Mathematics is an art, and as such
affords the pleasures which all the arts afford. This last statement may come
as a shock to people who are used to the conventional concept of the true arts
and mentally contrast these with mathematics to the detriment of the latter.
But the average person has not thought through what the arts really are and
what they offer. All that many people actually s�e in painting, for example,
are familiar scenes and perhaps bright colors. These qualities, however, are not
the ones which make painting an art. The real values must be learned, and a
genuine appreciation of art calls for much study.
Nevertheless. we shall not insist on the aesthetic values of mathematics. It
may be fairer to rest on the position that just as there are tone-deaf and color
blind people, so' may there be some who temperamentally are intolerant of
cold argumentation and the seemingly overfine distinctions of mathematics.
To many people, mathematics offers intellectual challenges, and it is well
known that such challenges do engross humans. Games such as bridge, cross
word puzzles, and magic squares are popular. Perhaps the best evidence is the
attraction of puzzles such as the following: A wolf, a goat, and cabbage are
to be transported across a river by a man in a boat which can hold only one
of these in addition to the man. How can he take them across so that the wolf
does not eat the goat or the goat the cabbage? Two husbands and two wives
have to cross a river in a boat which can hold only two people. How can they
cross so that no woman js in the company of a man unless her husband is also
present' Such puzzles go back to Greek and Roman times. The mathematician
Tartaglia, who lived in the sixteenth century, tells us that they were after
dinner amusements.
\VHY MATHEMATICS? 9
People do respond to intellectual challenges, and once one gets a slight
start in mathematics, he encounters these in abundance. In view of the addi
tional values to be derived from the subject, one would expect people to spend
time on mathematical problems as opposed to the more superficial, and in some
instances cheap, games which lack depth, beauty, and importance. The tan
talizing and compelling pursuit of mathematical problems offers mental absorp
tion, peace of mind amid endless challenges. repose in activity, battle without
conflict, and the beauty which the ageless mountains present to senses tried by
the kaleidoscopic rush of events. The appeal offered by the detachment and
objectivity of mathematical reasoning is superbly described by Bertrand Rus
sell.
Remote from human passions, remote even from the pitiful facts of nature, the
generations have gradually created an ordered cosmos, where pure thought can
dwell as in its natural home and where one, at least, of our nobler impulses can
f!scape from the dreary exile of the actual world.
The creation and contemplation of mathematics offer such values.
Despite all these arguments for the study of mathematics, the reader may
have justifiable doubts. The idea that thinking about numbers and figures leads
to deep and powerful conclusions which influence almost all other branches of
thought may seem incredible. The srudy of numbers and geometrical figures
may not seem a sufficiently attractive and promising enterprise. Not even the
founders of mathematics envisioned the potentialities of the subject.
So we start with some doubts about the worth of our enterprise. We
could encourage the reader with the hackneyed maxim, nothing ventured,
nothing gained. We could call to his attention the daily testimony to the
power of mathematics offered by almost every newspaper and journal. But
such appeals are hardly inspiring. Let us proceed on the very weak basis that
perhaps those more experienced in w hat the world has to offer may also have
the wisdom to recommend worth-while studies.
Hence, despite St. Augustine, the reader is invited to tempt hell and dam
nation by engaging in a study of the subject. Certainly he can be assured that
the subject is within his grasp and that no special gifts or qualities of mind are
needed to learn mathematics. It is even debatable whether the creation of
mathematics requires special talents as does the creation of music or great
paintings, but certainly the appreciation of what others have done does not
demand a "mathematical mind" any more than the appreciation of art requires
an "artistic mind." Moreover, since we shall not draw upon any previously
acquired knowledge, even this potential source of trouble will not arise.
Let us review our objectives. We should like to understand what mathe
matics is, how it functions. what it accomplishes for the world, and what it
has to offer in itself. We hope to see that mathematics has content which
serves the physical and social scientist, the philosopher, logician, and the artist;
content which influences the doctrines of the statesman and the theologian;
10 WHY MATHEMATICS?
content which satisfies the curiosity of the man who surveys the heavens and
the man who muses on the sweetness of musical sounds; and content which has
undeniably, if sometimes imperceptibly, shaped the course of modern history.
In brief, we shall try to see that mathematics is an integral part of the modern
world, one of the strongest forces shaping its thoughts and actions, and a body
of living though inseparably connected with, dependent upon, and in turn
valuable to all other branches of our culture. Perhaps we shall also see how by
suffusing and influencing all thought it has set the intellectual temper of our
times.
EXERCISES
I. A wolf, a goat, and a cabbage are to be rowed across a river in a boat holding
only one of these three objects besides the oarsman. How should he carry them
across so that the goat should not cat the cabbage or the wolf devour the goat?
2. Another hoary teaser is the following: A man goes to a tub of water with two
jars, one holding 3 pt and the other 5 pt. How can he bring back exactly 4 pt?
3. Two husbands and two wives have to cross a river in a boat which can hold
only two people. How can they cross so that no woman is in the company of a
man unless her husband is also present?
Recommended Reading
RUSSELL, BERTRAND: "The Study of Mathematics," an essay in the collection entitled
Mysticism and Logic, Longmans, Green and Co., New York, 1925.
WHITEHEAD, ALFRED NORTH: "The Mathematical Curriculum," an essay in the col
lection entitled The Aims of Education, The New American Library, New York,
1949.
WHITEHEAD, ALFRED NORTH: Science and the Modern World, Chaps. 2 and 3,
Cambridge University Press, Cambridge, 1926.
CHAPTER 2
A HISTORICAL ORI ENTATION
An educated mind is, as it were, composed of all the minds of preceding
ages.
LE BOV) ER DE FONTENELLE
2-1 INTRODUCTION
Our first objective will be to gain some historical perspective on the subject of
mathematics. Although the logical development of mathematics is not mark
edly different from the historical, there are nevertheless many features of
mathematics which are revealed by a glimpse of its history rather than by an
examination of concepts, theorems. and proofs. Thus we may learn what the
subject now comprises, how the various branches arose, and how the character
of the mathematical contributions made by various civilizations was condi
tioned by these civilizations. This historical survey may also help us to gain
some provisional understanding of the nature, extent, and uses of mathematics.
Finally, a preview may help us to keep our bearings. In studying a vast sub
ject, one is always faced with the danger of getting lost in details. This is
especially true in mathematics, where one must often spend hours and even
days in seeking to understand some new concepts or proofs.
2-2 MATHEMATICS IN EARLY CIVILIZATIONS
Aside possibly from astronomy, mathematics is the oldest and most contin
uously pursued branch of human thought. Moreover, unlike science, philos
ophy, and social thought, very little of the mathematics that has ever been
created has been discarded. Mathematics is also a cumulative development,
that is, newer creations are built logically upon older ones, so that one must
usually understand older results to master newer ones. These facts recommend
that we go back to the very origins of mathematics.
As we examine the early civilizations, one remarkable fa�t emerges im
mediately. Though there have been hundreds of civilizatio:1s, many with
great art, literature, philosophy, religion, and social institutions, very few
possessed any mathematics worth talking about. Most of these civilizations
hardly got past the stage of being able to count to five or ten.
11
12 A HISTORICAL ORIENTATION
In some of these early civilizations a few steps in mathematics were taken.
In prehistoric times, which means roughly before 4000 B.C., several civilizations
at least learned to think about numbers as abstract concepts. That is, they
recognized that three sheep and three arrows have something in common, a
quantity called three, which can be thought about independently of any phys
ical objects. Each of us in his own schooling goes through this same process
of divorcing numbers from physical objects. The appreciation of "number"
as an abstract idea is a great, and perhaps the first, step in the founding of
mathematics.
Another step was the introduction of arithmetical operations. It is quite
an idea to add the numbers representing two collections of objects in order
to arrive at the total instead of counting the objects in the combined collec
tions. Similar remarks apply to subtraction, mult.iplication, and division. The
early methods of carrying out these operations were crude and complicated
compared with ours, bur the ideas and the applications were there.
Only a few ancient civilizations, Egypt, Babylonia, India, and China, pos
sessed what may be called the rudiments of mathematics. The history of
mathematics, and indeed the history of Western civilization, begins with what
occurred in the first two of these civilizations. The role of India will emerge
later, whereas that of China may be ignored because it was not extensive and
moreover had no influence on the subsequent development of mathematics.
Our knowledge of the Egyptian and Babylonian civilizations goes back to
about 4000 B.C. The Egyptians occupied approximately the same region that
now constitutes modern Egypt and had a continuous, stable civilization from
ancient times until about 300 B.C. The term "Babylonian" includes a succession
of civilizations which occupied the region of modern Iraq. Both of these
peoples possessed whole numbers and fractions, a fair amount of arithmetic,
some algebra, and a number of simple rules for finding the areas and volumes
of geometrical figures. These rules were but the incidental accumulations of
experience, much as people learned through experience what foods to eat.
Many of the rules were in fact incorrect bur good enough for the simple
applications made then. For example, the Egyptian rule for finding the area
of a circle amounts to using 3.16 times the square of the radius; that is, their
value of Tr was 3.16. This value, though not accurate, was even better than the
several values the Babylonians used, one of these being 3, the value found in
the Bible.
What did these early civilizations do with their mathematics? If we may
judge from problems found in ancient Egyptian papyri and in the clay tablets
of the Babylonians, both civilizations used arithmetic and algebra largely in
commerce and state administration. to calculate simple and compound interest
on loans and mortgages, to apportion profits of business to the owners, to buy
and sell merchandise, to fix taxes, and to calculate how many bushels of grain
would make a quantity of beer of a specified alcoholic content. Geometrical
rules were applied to calculate the areas of fields, the estimated yield of pieces
MATHEMATICS IN EARLY CIVILIZATIONS 13
of land, the volumes of structures, and the quantity of bricks or stones needed
to erect a temple or pyramid. The ancient Greek historian Herodotus says
that because the annual overflow of the Nile wiped out the boundaries of the
farmers' lands, geometry was needed to redetermine the boundaries. In fact,
Herodotus speaks of geometry as the gift of the Nile. This bit of history is a
partial truth. The redetermination of boundaries was undoubtedly an applica
tion, but geometry existed in Egypt long before the date of 1400 B.C. men
tioned by Herodotus for its origin. Herodotus would have been more accurate
to say that Egypt is a gift of the Nile, for it is true today as it was then that
the only fertile land in Egypt is that along the Nile; and this because the river
deposits good soil on the land as it overflows.
Applications of geometry, simple and crude as they were, did play a large
role in Egypt and Babylonia. Both peoples were great builders. The Egyptian
temples, such as those at Karnak and Luxor, and the pyramids still appear to be
admirable engineering achievements even in this age of skyscrapers. The
Babylonian temples, called ziggurats, also were remarkable pyramidal struc
tures. The Babylonians were, moreover, highly skilled irrigation engineers,
who built a system of canals to feed their hot dry lands from the Tigris and
Euphrates rivers.
Perhaps a word of caution is necessary with respect to the pyramids. Be
cause these are impressive structures, some writers on Egyptian civilization
have jumped to the conclusion that the mathematics used in the building of
pyramids must also have been impressive. These writers point out that the
horizontal dimensions of any one pyramid are exactly of the same length, the
sloping sides all make the same angle with the ground, and the right angles are
right. However, not mathematics but care and patience were required to ob
tain such results. A cabinetmaker need not be a mathematician.
Mathematics in Egypt and Babylonia was also applied to astronomy. Of
course, astronomy was pursued in these ancient civilizations for calendar reck
oning and, to some extent, for navigation. The motions of the heavenly bodies
give us our fundamental standard of time, and their positions at given times
enable ships to determine their location and caravans to find their bearings in
the deserts. Calendar reckoning is not only a common daily and commercial
need, but it fixes religious holidays and planting times. In Egypt it was also
needed to predict the flood of the Nile, so that farmers could muve property
and cattle away beforehand.
It is worthy of note that by observing the motion of the sun, tho Egyptians
managed to ascertain that the year contains 365 days. There is a conjecture
that the priests of Egypt knew that 365t was a more accurate figure but kept
the knowledge secret. The Egyptian calendar was taken over much later by
the Romans and then passed on to Europe. The Babylonians, by contrast,
developed a lunar calendar. Since the duration of the month as measured from
new moon to new moon varies from 29 to 30 days, the twelve-month year
adopted by the Babylonians did not coincide with the year of the seasons.
14 A HISTORICAL ORIENTATION
Hence the Babylonians added extra months, up to a total of seven, in every
19-year cycle. This scheme was also adopted by the Hebrews.
Astronomy served not only the purposes just described, but from ancient
times until recently it also served astrology. In ancient Babylonia and Egypt
the belief was widespread that the moon, the planets, and the stars directly
influenced and even controlled affairs of the state. This doctrine was gradually
extended and later included the belief that the health and welfare of the in
dividual were also subject to the will of the heavenly bodies. Hence it seemed
reasonable that by studying the motions and relative positions of these bodies
man could determine their influences and even predict his future.
When one compares Egyptian and Babylonian accomplishments in mathe
matics with those of earlier and contemporary civilizations, one can indeed
find reason to praise their achievements. But judged by other standards,
Egyptian and Babylonian contributions to mathematics were almost insignifi
cant, although these same civilizations reached relatively high levels in religion,
art, architecture, metallurgy, chemistry, and astronomy. Compared with the
accomplishments of their immediate successors, the Greeks, the mathematics of
the Egyptians and Babylonians is the scrawling of children just learning how
to write as opposed to great literature. They barely recognized mathematics
as a distinct subject. It was a tool in agriculture, commerce. and engineering,
no more important than the other tools they used to build pyramids and zig
gurats. Over a period of 4000 years hardly any progress was made in the
subject. Moreover, the very essence of mathematics, namely. reasoning to es
tablish the validity of methods and results, was not even envisioned. Experi
ence recommended their procedures and rules, and with this support they were
content. Egyptian and Babylonian mathematics is best described as empirical
and hardly deserves the appellation mathematics in view of what, since Greek
times, we regard as the chief features of the subject. Some flesh and bones of
concrete mathematics were there. but the spirit of mathematics was lacking.
The lack of interest in theoretical or systematic knowledge is evident in
all activities of these two civilizations. The Egyptians and Babylonians must
have noted the paths of the stars, planets, and moon for thousands of years.
Their calendars, as well as tables which are extant, testify to the scope and
accuracy of these observations. But no Egyptian or Babylonian strove, so far
as we know, to encompass all these observations in one major plan or theory of
heavenly motions. Nor does one find any other scientific theory or connected
body of knowledge.
2-3 THE CLASSICAL GREEK PERIOD
We have seen so far that mathematics, initiated in prehistoric times, struggled
for existence for thousands of years. It finally obtained a firm grip on life in
the highly congenial atmosphere of Greece. This land was invaded about
THE CLASSICAL GREEK PERIOD 15
1 000 B.C. by people whose oflgms are not known. By about 600 B.C. these
people occupied not only Greece proper but many cities in Asia Minor on the
Mediterranean coast, islands such as Crete, Rhodes, and Samos, and cities in
southern Italy and Sicily. Though all of these areas bred famous men, the chief
cultural center during the classical period, which lasted from about 600 B.C.
to 300 B.C., was Athens.
Greek culture was not entirely indigenous. The Greeks themselves ac
knowledge their indebtedness to the Babylonians and especially to the Egyp
tians. Many Greeks traveled in Egypt and in Asia Minor. Some went there
to study. Nevertheless, what the Greeks created differs as much from what
they took over from the Egyptians and Babylonians as gold differs from tin.
Plato was too modest in his description of the Greek contribution when he
said, "Whatever we Greeks receive we improve and perfect." The Greeks
not only made finished products out of the raw materials imported from Egypt
and Babylonia, but they created totally new branches of culture. Philosophy,
pure and applied sciences, political thought and institutions, historical writings,
almost all our literary forms (except fictional prose), and new ideals such as
the freedom of the individual are wholly Greek contributions.
The supreme contribution of the Greeks was to call attention to, employ,
and emphasize the power of human reason. This recognition of the power of
reasoning is the greatest single discovery made by man. Moreover, the Greeks
recognized that r"eason was the distinctive faculty which humans possessed.
Aristotle says, "Now what is characteristic of any nature is that which is best
for it and gives most joy. Such to man is the life according to reason, since it
is that which makes him man."
It was by the application of reasoning to mathematics that the Greeks
completely altered the nature of the subject. In fact, mathematics as we
understand the term today is entirely a Greek gift, though in this case we
need not heed Virgil's injunction to fear such benefactions. But how did the
Greeks plan to employ reason in mathematics? Whereas the Egyptians and
Babylonians were content to pick up scraps of useful information through
experience or trial and error, the Greeks abandoned empiricism and undertook
a systematic, rational attack on the whole subject. First of all, the Greeks saw
clearly that numbers and geometric forms occur everywhere in the heavens
and on earth. Hence they decided to concentrate on these important concepts.
Moreover, they were explicit ahout their intention to treat general abstract
concepts rather than particular physical realizations. Thus they would consider
the ideal circle rather than the boundary of a field or the shape of a wheel.
They then observed that certain facts about these concepts are both obvious
and basic. It was evident that equal numbers added to or subtracted from equal
numbers give equal numbers. It was equally evident that two right angles are
necessarily equal and that a circle can be drawn when center and radius are
given. Hence they selected some of these obvious facts as a starting point and
16 A HISTORICAL ORIENTATION
called them axioms. Their next idea was to apply reasoning, with these facts
as premises, and to use only the most reliable methods of reasoning man pos
sesses. If the reasoning were successful, it would produce new knowledge.
Also, since they were to reason about general concepts, their conclusions
would apply to all objects of which the concepts were representative. Thus if
they could prove that the area of a circle is 7T times the square of the radius,
this fact would apply to the area of a circular field, the floor area of a circular
temple, and the cross section of a circular tree trunk. Such reasoning about
general concepts might not only produce knowledge of hundreds of physical
situations in one proof, but there was always the chance that reasoning would
produce knowledge which experience might never suggest. All these advan
tages the Greeks expected to derive from reasoning about general concepts on
the basis of evident reliable facts. A neat plan, indeed!
It is perhaps already clear that the Greeks possessed a mentality totally
different from that of the Egyptians and Babylonians. They reveal this also
in the plans they had for the use of mathematics. The application of arithmetic
and algebra to the computation of interest, taxes, or commercial transactions,
and of geometry to the computation of the volumes of granaries was as far
from their minds as the most distant star. As a matter of fact, their thoughts
were on the distant stars. The Greeks found mathematics valuable in many
respects, as we shall learn later, but they saw its main value in the aid it
rendered to the study of nature; and of all the phenomena of nature, the
heavenly bodies attracted them most. Thus, though the Greeks also studied
light, sound, and the motions of bodies on the earth, astronomy was rheir chief
scientific interest.
Just what did the Greeks seek in probing nature? They sought no mate
rial gain and no power over nature; they sought merely to satisfy their minds.
Because they enjoyed reasoning and because nature presented the most im
posing challenge to their understanding, the Greeks undertook the purely in
tellectual study of nature. Thus the Greeks are the founders of science in the
true sense.
The Greek conception of nature was perhaps even bolder than their con
ception of mathematics. Whereas earlier and later civilizations viewed nature
as capricious, arbitrary, and terrifying, and succumbed to the belief that magic
and rituals would propitiate mysterious and feared forces, the Greeks dared
to look nature in the face. They dared to affirm that nature was rationally and
indeed mathematically designed, and that man's reason, chiefly through the
aid of mathematics, would fathom that design. The Greek mind rejected
traditional doctrines, supernatural causes, superstitions, dogma, authority, and
other such trammels on thought and undertook to throw the light of reason
on the processes of nature. In seeking to banish the mystery and seeming
arbitrariness of nature and in abolishing belief in dreaded forces, the Greeks
were pioneers.
THE ALEXANDRIAN GREEK PERIOD 17
For reasons which will become clearer in a later chapter, the Greeks
favored geometry. By 300 B.C., Thales, Pythagoras and his followers, Plato's
disciples, notably Eudoxus, and hundreds of other famous men had built up an
enormous logical structure, most of which Euclid embodied in his Elements.
This is, of course, the geometry we still study in high school. Though they
made some contributions to the study of the properties of numbers and to the
solution of equations, almost all of their work was in geometric form, and so
there was no improvement over the Egyptians and Babylonians in the repre
sentation of, and calculation with, numbers or in the symbolism and techniques
of algebra. For these contributions the world had to wait many more centuries.
But the vast development in geometry exerted an enormous influence in suc
ceeding civilizations and supplied the inspiration for mathematical activity in
civilizations which might otherwise never have acquired even the very concept
of mathematics.
The Greek accomplishments in mathematics had, in addition, the broader
significance of supplying the first impressive evidence of the power of human
reason to deduce new truths. In every culture influenced by the Greeks, this
example inspired men to apply reason to philosophy, economics, political the
ory, art, and religion. Even today Euclid is the prime example of the power
and accomplishments of reason. Hundreds of generations since Euclid's days
have learned from his geometry what reasoning is and what it can accomplish.
Modern man as well as the ancient Greeks learned from the Euclidean docu
ment how exact reasoning should proceed, how to acquire facility in it, and
how to distinguish correct from false reasoning. Although many people de
preciate this value of mathematics, it is interesting nevertheless that when these
people seek to offer an excellent example of reasoning, they inevitably turn to
mathematics.
This brief discussion of Euclidean geometry may show that the subject is
far from being a relic of the dead past. It remains important as a stepping
stone in mathematics proper and as a paradigm of reasoning. With their gift
of reason and with their explicit example of the power of reason, the Greeks
founded Western civilization.
2-4 THE ALEXANDRIAN GREEK PERIOD
The intellectual life of Greece was altered considerably when Alexander the
Great conquered Greece, Egypt, and the Near East. Alexander decided to
build a new capital for his vast empire and founded the city in Egypt named
after him. The center of the new Greek world became Alexandria instead of
Athens. Moreover, Alexander made deliberate efforts to fuse Greek and Near
Eastern cultures. Consequently, the civilization centered at Alexandria, though
predominantly Greek, was strongly influenced by Egyptian and Babylonian
contributions. This Alexandrian Greek civilization lasted from about 300 B.C.
to 600 A.D.
18 A HISTORICAL ORIENTATION
The mixture of the theoretical inrerests of the Greeks and the practical
outlook of the Babylonians and Egyptians is clearly evident in the mathematical
and scientific work of the Alexandrian Greeks. The purely geometric investi
gations of the classical Greeks were continued, and twO of the most famous
Greek mathematicians, Apollnnius and Archimedes, pursued their studies dur
ing the Alexandrian period. In fact, Euclid also lived in Alexandria, but his
writings reflect the achievements of the classical period. For practical applica
tions, which usually require quantitative results, the Alexandrians revived the
crude arithmetic and algebra of Egypt and Babylonia and used these empir
ically founded tools and procedures, along with results derived from exact
geometrical studies. There was some progress in algebra, but what was newly
created by men such as Nichomachus and Diophantus was still short of even
the elementary methods we learn in high school.
The attempt to be quantitative, coupled with the classical Greek love for
the mathematical study of nature. stimulated two of the most famous astron
omers of all time, Hipparchus and Ptolemy, to calculate the sizes and distances
of the heavenly bodies and to build a sound and, for those times, accurate
astronomical theory, which is still known as Ptolemaic theory. Hipparchus
and Ptolemy also created the chief tool they needed for this purpose, the
mathematical subject known as trigonometry.
During the centuries in which the Alexandrian civilization flourished, the
Romans grew strong, and by the end of the third century D.C. they were a
world power. After conquering Italy, the Romans conquered the Greek main
land and a number of Greek cities scattered about the Mediterranean area.
Among these was the famous city of Syracuse in Sicily, where Archimedes
spent most of his life, and where he was killed at the age of 75 by a Roman
soldier. According to the account given by rhe noted historian Plutarch, rhe
soldier shouted to Archimedes to surrender, but the latter was so absorbed in
studying a mathematical problem that he did not hear the order, whereupon
the soldier killed him.
The contrast between Greek and Roman cultures is striking. The Romans
have also bequeathed gifts to Western civilization, but in the fields of mathe
matics and science their influence was negative rather than positive. The
Romans were a practical people and even boasted of their practicality. They
sought wealth and world power and were willing to undertake great engineer
ing enterprises, such as the building of roads and viaducrs, which might help
them to expand, control, and administer their empire, but they would spend
no time or effort on theoretical studies which might further these activities.
As the great philosopher Alfred North Whitehead remarked, "No Roman
ever lost his life because he was absorbed in the contemplation of a mathe
matical diagram."
Indirectly as well as directly, rhe Romans brought about the destruction
of the Greek civilization at Alexandria, directly by conquering Egypt and
THE HINDUS AND ARABS 19
indirectly by seeking to suppress Christianity. The adherents to this new
religious movement, though persecuted cruelly by the Romans, increased in
number while the Roman Empire grew weaker. In 3 1 3 A.D. Rome legalized
Christianity and, under the Emperor Theodosius ( 379-395), adopted it as the
official religion of the empire. But even before this time, and certainly after it,
the Christians began to attack the cultures and civilizations which had opposed
them. By pillage and the burning of books, they destroyed all they could
reach of ancient learning. Naturally the Greek culture suffered, and many
works wiped out in these holocausts are now lost to us forever.
The final destruction of Alexandria in 640 A.D. was the deed of the Arabs.
The books of the Greeks were closed, never to be reopened in this region.
2-S THE HINDUS AND ARABS
The Arabs, who suddenly appeared on the scene of history in the role of
destroyers, had been a nomadic people. They were unified under the leader
ship of the prophet Mohammed and began an attempt to convert the world to
Mohammedanism, using the sword as their most decisive argument. They
conquered all the land around the Mediterranean Sea. In the Near East they
took over Persia and penetrated as far as India. In southern Europe they oc
cupied Spain, southern France, where they were stopped by Charles Martel,
southern Italy and Sicily. Only the Byzantine or Eastern Roman Empire was
not subdued and remained an isolated center of Greek and Roman learning.
In rather surprisingly quick time as the history of nations goes, the Arabs
settled down and built a civilization and culture which maintained a high level
from about 800 to 1 200 A.D. Their chief centers were Bagdad in what is now
Iraq, and Cordova in Spain. Realizing that the Greeks had created wonderful
works in many fields, the Arabs proceeded to gather up and study what they
could still find in the lands they controlled. They translated the works of
Aristotle, Euclid, Apollonius, Archimedes, and Ptolemy into Arabic. In fact,
Ptolemy's chief work, whose title in Greek meant "Mathematical Collection,"
was called the Almagest (The Greatest Work) by the Arabs and is still known
by this name. Incidentally, other Arabic words which are now common
mathematical terms are algebra, taken from the title of a book written by AI
Khowarizmi, a ninth-century Arabian mathematician, and algorithm, now
meaning a process of calculation, which is a corruption of the man's name.
Though they showed keen interest in mathematics, optics, astronomy, and
medicine, the Arabs contributed little that was original. It is also peculiar that,
although they had at least some of the Greek works and could therefore see
what mathematics meant, their own contributions, largely in arithmetic and
algebra, followed the empirical, concrete approach of the Egyptians and
Babylonians. They could on the one hand appreciate and critically review the
precise, exact, and abstract mathematics of the Greeks while, on the other, offer
20 A HISTORICAL ORIENTATION
methods of solving equations which, though they worked, had no reasoning to
support them. During all the centuries in which Greek works were in their
possession, the Arabs manfully resisted the lures of exact reasoning in their
own contributions.
We are indebted to the Arabs not only for their resuscitation of the Greek
works but for picking up some simple bur useful ideas from India, their neigh
bor on the East. The Indians, too, had built up some elementary mathematics
comparable in extent and spirit with the Egyptian and Babylonian develop
ments. However, after about 200 A.D., mathematical activity in India became
more appreciable, probably as a result of contacts with the Alexandrian Greek
civilization. The Hindus made a few contributions of their own, such as the
use of special number symbols from 1 to 9, the introduction of 0, and the use
of positional notation with base ten, that is, our modern method of writing
numbers. They also created negative numbers. These ideas were taken over
by the Arabs and incorporated in their mathematical works.
Because of internal dissension the Arab Empire split into two independent
parts. The Crusades launched by the Europeans and the inroads made by the
Turks further weakened the Arabs, and their empire and culture disintegrated.
2-6 EARLY AND MEDIEVAL EUROPE
Thus far Europe proper has played no role in the history of mathematics. The
reason is simple. The Germanic tribes who occupied central Europe and the
Gauls of western Europe were barbarians. Among primitive civilizations. theirs
were primitive indeed. They had no learning, no art, no science, not even a
system of writing.
The barbarians were gradually civilized. While the Romans were still
successful in holding the regions now called France, England, southern Ger
many, and the Balkans, the barbarians were in contact with, and to some extent
influenced by, the Romans. When the Roman Empire collapsed, the Church,
already a strong organization, took on the task of civilizing and converting the
barbarians. Since the Church did not favor Greek learning and since at any
rate the illiterate Europeans had first to learn reading and writing, one is not
surprised to find that mathematics and science were practically unknown in
Europe until about 1 100 A.D.
2-7 TH E RENAISSANCE
Insofar as the history of mathematics is concerned, the Arabs served as the
agents of destiny. Trade with the Arabs and such invasions of the Arab lands
as the Crusades acquainted the Europeans, who hitherto possessed only frag
ments of the Greek works, with the vast stores of Greek learning possessed by
the Arabs. The ideas in these works excited the Europeans, and scholars set
about acquiring them and translating them into Latin. Through another acci-
THE RENAISSANCE 21
dent of history another group of Greek works came to Europe. We have
already noted that the Eastern Roman or Byzantine Empire had survived the
Germanic and the Arab aggrandizements. But in the fifteenth century the
Turks captured the Eastern Roman Empire, and Greek scholars carrying
precious manuscripts fled the region and went to Europe.
We shall leave for a later chapter a fuller account of how the European
world was aroused by the renaissance of the novel and weighty Greek ideas,
and of the challenge these ideas posed to the European beliefs and way of
life.· From the Greeks the Europeans acquired arithmetic, a crude algebra,
the vast development of Euclidean geometry, and the trigonometry of Hip
parchus and Ptolemy. Of course, Greek science and philosophy also became
known in Europe.
The first major European development in mathematics occurred in the
work of the artists. Imbued with the Greek doctrines that man must study
himself and the real world, the artists began to paint reality as they actually
perceived it instead of interpreting religious themes in symbolic styles. They
applied Euclidean geometry to create a new system of perspective which
permitted them to paint realistically. Specifically, the artists created a new
style of painting which enabled them to present on canvas, scenes making the
same impression on the eye as the actual scenes themselves. From the work
of the artists, the mathematicians derived ideas and problems that led to a new
branch of mathematics, projective geometry.
Stimulated by Greek astronomical ideas, supplied with data and the
astronomical theory of Hipparchus and Ptolemy, and steeped in the Greek
doctrine that the world is mathematically designed, Nicolaus Copernicus
sought to show that God had done a better job than Hipparchus and Ptolemy
had described. The result of Copernicus' thinking was a new system of
astronomy in which the sun was immobile and the planets revolved around
the sun. This heliocentric theory was considerably improved by Kepler. Its
effects on religion, philosophy, science, and on man's estimations of his own
importance were profound. The heliocentric theory also raised scientific and
mathematical problems which were a direct incentive to new mathematical
developments.
Just how much mathematical activity the revival of Greek works might
have stimulated cannot be determined, for simultaneously with the translation
and absorption of these works, a number of other revolutionary developments
altered the social, economic, religious, and intellectual life of Europe. The
introduction of gunpowder was followed by the use of muskets and later
cannons. These inventions revolutionized methods of warfare and gave the
newly emerging social class of free common men an important role in that
domain. The compass became known to the Europeans and made possible
long-range navigation, which the merchants sponsored for the purpose of
• See Chapter 9.
22 A HISTORICAL ORIENTATlON
finding new sources of raw materials and better trade routes. One result was
the discovery of America and the consequent influx of new ideas into Europe.
The invention of printing and of paper made of rags afforded books in large
quantities and at cheap prices, so that learning spread far more than it ever had
in any earlier civilizations. The Protestant Revolution stirred debate and doubts
concerning doctrines that had been unchallenged for 1 500 years. The rise of a
merchant class and of free men engaged in labor in their own behalf stimulated
an interest in materials, methods of production, and new commodities. All of
these needs and influences challenged the Europeans to build a new culture.
2-8 DEVELOPMENTS FROM 1550 TO 1800
Since many of the problems raised by the motion of cannon balls, navigation,
and industry called for quantitative knowledge, arithmetic and algebra became
centers of attention. A remarkable improvement in these mathematical fields
followed. This is the period in which algebra was built as a branch of mathe
matics and in which much of the algebra we learn in high school was created.
Almost all the great mathematicians of the sixteenth and seventeenth centuries,
Cardan, Tartaglia, Vieta, Descartes, Fermat, and Newton, men we shall get to
know better later, contributed to the subject. In particular, the use of letters to
represent a class of numbers, a device which gives algebra its generality and
power, was introduced by Vieta. In this same period, logarithms were created
to facilitate the calculations of astronomers. The history of arithmetic and
algebra illustrates one of the striking and curious features of the history of
mathematics. Ideas that seem remarkably simple once explained were thousands
of years in the making.
The next development of consequence, coordinate geometry, came from
two men, both interested in Ipethod. One was Rene Descartes. Descartes is
perhaps even more famous as a philosopher than as a mathematician, though
he was one of the major contributors to our subject. As a youth Descartes was
already troubled by the intellectual turmoil of his age. He found no certainty
in any of the knowledge taught him, and he therefore concentrated for years
on finding the method by which man can arrive at truths. He found the clue
to this method in mathematics, and with it constructed the first great modern
philosophical system. Because the scientific problems of his time involved
work with curves, the paths of ships at sea, of the planets, of objects in motion
near the earth, of light, and of projectiles, Descartes sought a better method of
proving theorems about curves. He found the answer in the use of algebra.
Pierre de Fermat's interest in method was confined to mathematics proper, but
he too appreciated the need for more effective ways of working with curves
and also arrived at the idea of applying algebra. In this development of
coordinate geometry we have one of the remarkable examples of how the
times influence the direction of men's thoughts.
DEVELOPMENTS FROM 1 550 TO 1 800 23
We have already noted that a new society was developing in Europe.
Among its features were expanded commerce, manufacturing, mining, large
scale agriculture, and a new social class-free men working as laborers Of
as independellt artisans. These activities and interests created problems of
materials, methods of production, quality of the product, and utilization of
devices to replace or increase the effectiveness of manpower. The people
involved, like the artists, had become aware of Greek mathematics and
science and sensed that it could be helpful. And so they too sought to employ
this knowledge in their own behalf. Thereby arose a new motive for the
study of mathematics and science. Whereas the Greeks had been content to
study nature merely to satisfy their own curiosity and to organize their
conclusions in patterns pleasing to the mind, the new goal, effectively pro
claimed by Descartes and Francis Bacon, was to make nature serve man.
Hence mathematicians and scientists turned earnestly to an enlarged program
in which both understanding and mastery of nature were to be sought.
However, Bacon had cautioned that nature can be commanded only when
one learns to obey her. One must have facts of nature on which to base
reasoning about nature. Hence mathematicians and scientists sought to acquire
facts from the experience of artists, technicians, artisans, and engineers. The
aIIiance of mathematics and experience. was gradually transformed into an
alliance of mathematics and experimentation, and a new method for the pursuit
of the truths of nature, first clearly perceived and formulated by Galileo
Galilei ( 1564--1642) and Newton, was gradually evolved. The plan, perhaps
oversimply stated, was that experience and experiment were to supply basic
mathematical principles and mathematics was to be applied to these principles
to deduce new truths, just as new truths are deduced from the axioms of
geometry.
The most pressing scientific problem of the seventeenth century was the
study of motion. On the practical side, investigations of the motion of
projectiles, of the motion of the moon and planets to aid navigation, and of
the motion of light to improve the design of the newly discovered telescope
and microscope, were the primary interests. On the theoretical side, the new
heliocentric astronomy invalidated the older, Aristotelian laws of motion and
called for totally new principles. It was one thing to explain why a ball fell
to earth on the assumption that the earth was immobile and the center of the
universe, and another to explain this phenomenon in the light of the fact that
the earth was rotating and revolving around the sun. A new science of motion
was created by Galileo and Newton, and in the process two brand-new
developments were added to mathematics. The first of these was the notion
of a function, a relationship between variables best expressed for most purposes
as a formula. The second, which rests on the notion of a function but repre
sents the greatest advance in method and content since Euclid's days, was the
calculus. The subject matter of mathematics and the power of mathematics
24 A HISTORICAL ORIENTATION
expanded so greatly that at the end of the seventeenth century Leibniz could
say,
Taking mathematics from the beginning of the world to the time when Newton
lived, what he had done was much the better half.
With the aid of the calculus Newton was able to organize all data on
earthly and heavenly motions into one system of mathematical mechanics
which encompassed the motion of a ball falling to earth and the motion of
the planets and stars. This great creation produced universal laws which not
only united heaven and earth but revealed a design in the universe far more
impressive than man had ever conceived. Galileo's and Newton's plan of
applying mathematics to sound physical principles not only succeeded in one
major area but gave promise, in a rapidly accelerating scientific movement, of
embracing all other physical phenomena.
We learn in history that the end of the seventeenth century and the
eighteenth century were marked by a new intellectual attitude briefly described
as the Age of Reason. We are rarely told that this age was inspired by the
successes which mathematics, to be sure in conjunction with science, had
achieved in organizing man's knowledge. Infused with the conviction that
reason, personified by mathematics, would not only conquer the physical
world but could solve all of man's problems and should therefore be employed
in every intellectual and artistic enterprise, the great minds of the age
undertook a sweeping reorganization of philosophy, religion, ethics, literature,
and aesthetics. The beginnings of new sciences such as psychology, economics,
and politics were made during these rational investigations. Our principal
intellectual doctrines and outlook were fashioned then, and we still live in the
shadow of the Age of Reason.
While these major branches of our culture were being transformed,
eighteenth-century scientists continued to win victories over nature. The
calculus was soon extended to a new branch of mathematics called differential
equations, and this new tool enabled scientists to tackle more complex problems
in astronomy, in the study of the action of forces causing motions, in sound,
especially musical sounds, in light, in heat (especially as applied to the develop
ment of the steam engine), in the strength of materials, and in the flow of
liquids and gases. Other branches, which can be merely mentioned, such as
infinite series, the calculus of variations, and differential geometry, added to
the extent and power of mathematics. The great names of the Bernoullis,
Euler, Lagrange, Laplace, d' Alemberr, and Legendre, belong to this period.
2-9 DEVELDPMENTS FRDM 1800 TO THE PRESENT
During the nineteenth century, developments in mathematics came at an ever
increasing rate. Algebra, geometry and analysis, the last comprising those
DEVELOPMENTS FROM 1 800 TO THE PRESENT 25
subjects which stem from calculus, all acquired new branches. The great
mathematicians of the century were so numerous that it is impractical to list
them. We shall encounter some of the greatest of these, Karl Friedrich Gauss
and Bernhard Riemann, in our work. We might mention also Henri Poincare
and David Hilbert, whose work extended into the twentieth century.
Undoubtedly the primary cause of this expansion in mathematics was the
expansion in science. The progress made in the seventeenth and eighteenth
centuries had sufficiently illustrated the effectiveness of science in penetrating
the mysteries of the physical world and in giving man control over nature, to
cause an all the more vigorous pursuit of science in the nineteenth century.
In that century also, science became far more intimately linked with engi
neering and technology than ever before. Mathematicians, working closely
with the scientists as they had since the seventeenth century, were presented
with thousands of significant physical problems and responded to these
challenges.
Perhaps the major scientific development of the century, which is typical
in its stimulation of mathematical activity, was the study of electricity and
magnetism. While still in its infancy this science yielded the electric motor,
the electric generator, and telegraphy. Basic physical principles were soon
expressed mathematically, and it became possible to apply mathematical tech
niques to these principles, to deduce new information just as Galileo and
Newton had done with the principles of motion. In the course of such
mathematical investigations, James Clerk Maxwell discovered electromagnetic
waves of which the best known representatives are radio waves. A new world
of phenomena was thus uncovered, all embraced in one mathematical system.
Practical applications, with radio and television as most familiar examples,
soon followed.
Remarkable and revolutionary developments of another kind also took
place in the nineteenth century, and these resulted from a re-examination of
elementary mathematics. The most profound in its intellectual significance
was the creation of non-Euclidean geometry by Gauss. His discovery had
both tantalizing and disturbing implications: tantalizing in that this new field
contained entirely new geometries based on axioms which differ from Euclid's,
and disturbing in that it shattered man's firmest conviction, namely that
mathematics is a body of truths. With the truth of mathematics undermined,
realms of philosophy, science, and even some religious beliefs went up in
smoke. So shocking were the implications that even mathematicians refused to
take non-Euclidean geometry seriously until the theory of relativity forced
them to face the full significance of the creation.
For reasons which we trust will become clearer further on, the devastation
caused by non-Euclidean geometry did not shatter mathematics but released
it from bondage to the physical world. The lesson learned from the history
of non-Euclidean geometry was that though mathematicians may start with
26 A HISTORICAL ORIENTATION
axioms that seem to have little to do with the observable behavior of nature,
the axioms and theorems may nevertheless prove applicable. Hence mathe
maticians felt freer to give reign to their imaginations and to consider abstract
concepts such as complex numbers, tensors, matrices, and n-dimensional
spaces. This development was followed by an even greater advance in mathe
matics and. surprisingly, an increasing use of mathematics in the sciences.
Even before the nineteenth century, the rationalistic spirit engendered by
the success of mathematics in the study of nature penetrated to the social
scientists. They began to emulate the physical scientists, that is, to search
for the basic truths in their fields and to attempt reorganization of their
subjects on the mathematical pattern. But these attempts to deduce the laws
of man and society and to erect sciences of biology, economics, and politics
did not succeed, although they did have some indirect beneficial effects.
The failure to penetrate social and biological problems by the deductive
method, that is, the method of reasoning from axioms, caused social scientists
to take over and develop further the mathematical theories of statistics and
probability, which had already been initiated by mathematicians for various
purposes ranging from problems of gambling to the theory of heat and
astronomy. These techniques have been remarkably successful and have given
some scientific methodology to what were largely speculative domains.
This brief sketch of the mathematics which will fall within our purview
may make it clear th:lt mathematics is not a closed book written in Greek
times. It is rather a living plant that has flourished and languished with the
rise and fall of civilizations. Since about 1600 it has been a continuing
development which has become steadily vaster, richer, and more profound.
The character of mathematics has been aptly, if somewhat floridly, described
by the nineteenth-century English mathematician James Joseph Sylvester.
Mathematics is not a book confined within a cover and bound between brazen
clasps, whose contents it needs only patience to ransack; it is not a mine, whose
treasures may take long to reduce into possession, but which fill only a limited
number of veins and lodes; it is not a soil, who�e fertility can be exhausted by the
yield of successive harvests; it is not a continent or an ocean, whose area can be
mapped out and its contour defined; it is as limitless as the space which it finds
too narrow for its aspirations; its possibilities are as infinite as the worlds which
are forever crowding in and multiplying upon the astronomer's gaze,. it is incapable
of being restricted within assigned boundaries or being reduced to definitions of
permanent validity as the consciousness, the life, which seems to slumber in each
monad, in every atom of matter, in each leaf and bud and cell and is forever ready
to burst forth into new forms of vegetabLe and animal existence.
Our sketch of the development of mathematics has attempted to indicate
the major eras and civilizations in which the subject has flourished, the
variety of interests which induced people to pursue mathematics, and the
branches of mathematics that have been created. Of course, we intend to
THE HUMAN ASPECf OF MATHEMATICS 27
investigate more carefully and more fully what these creations are and what
values they have furnished to mankind. One fact of history may be noted
by way of summary here. Mathematics as a body of reasoning from axioms
stems from one source, the classical Greeks. All other civilizations which
have pursued or are pursuing mathematics acquired this concept of mathe
matics from the Greeks. The Arab and Western European were the next
civilizations to take over and expand on the Greek foundation. Today
countries such as the United States, Russia, China, India, and Japan are
also active. Though the last three of these did possess some native mathe
matics, it was limited and empirical as in Babylonia and Egypt. Modern
mathematical activity in these five countries and wherever else it is now
taking hold was inspired by Western European thought and actually learned
by men who studied in Europe and returned to build centers of teaching in
their own countries.
2-10 THE HUMAN ASPECT OF MATHEMATICS
One final point about mathematics is implicit in what we have said. We have
spoken of problems which gave rise to mathematics, of cultures which empha
sized some directions of thinking as opposed to others, and of branches of
mathematics, as though all these forces and activities were as impersonal as
the force of gravitation. But ideas and thinking are conveyed by people.
Mathematics is a human creation. Although most Greeks did believe that
mathematics existed independently of human beings as the planets and moun
tains seem to, and that all that human beings do is discover more and more of
the structure, the prevalent belief today is that mathematics is entirely a human
product. The concepts, the axioms, and the theorems established are all
created by human beings in man's attempt to understand his environment, to
give play to his artistic instincts, and to engage in absorbing intellectual
activity.
The lives and activities of the men themselves are also fascinating. While
mathematicians produce formulas, no formula produces mathematicians. They
have come from all levels of society. The special talent, if there is such,
which makes mathematicians has been found in Casanovas and ascetics, among
business men and philosophers, among atheists and the profoundly religious,
among the retiring and the worldly. Some, like Blaise Pascal and Gauss, were
precocious; Evariste Galois was dead at 2 1 , and Niels Hendrik Abel at 27.
Others, like Karl Weierstrass and Henri Poincare, matured more normally and
were productive throughout their lives. Many were modest; others extremely
egotistical and vain beyond toleration. One finds scoundrels, such as Cardan,
and models of rectitude. Some were generous in their recognition of other
great minds; others were resentful and jealous and even stole ideas to boost
their own reputations. Disputes about priority of discovery abound.
28 A HISTORICAL ORIENTATION
The point in learning about these human varIations, aside from satisfying
our instinct to pry into other people's lives, is that it explains to a large extent
why the progress of the highly rational subject of mathematics has been highly
irrational. Of course, the major historical forces, which we sketched above,
limit the actions and influence the outlook of individuals, but we also find in
the history of mathematics all the vagaries which he have learned to associate
with human beings. Leading mathematicians have failed to recognize bright
ideas suggested by younger men, and the authors died neglected. Big men and
little men made unsuccessful attempts to solve problems which their successors
solved with ease. On the other hand, some supposed proofs offered even by
masters were later found to be false. Generations and even ages failed to note
new ideas, despite the fact that all that was needed was not a technical achieve
ment but merely a point of view. The examples of the blindness of human
beings to ideas which later seem simple and obvious furnish fascinating insight
into the working of the human mind.
Recognition of the human element in mathematics explains in large measure
the differences in the mathematics produced by different civilizations and the
sudden spurts made in new directions by virtue of insights supplied by genius.
Though no subject has profited as much as mathematics has by the cumulative
effect of thousands of workers and results, in no subject is the role of great
minds more readily discernible.
EXERCISES
1. Name a few civilizations which contributed to mathematics.
2. What basis did the Egyptians and Babylonians have for believing in their mathe-
matical methods and formulas?
3. Compare Greek and pre-Greek understanding of the concepts of mathematics.
4. What was the Greek plan for establishing mathematical conclusions?
5. What was the chief contribution of the Arabs to the development of mathe
matics?
6. In what sense is mathematics a creation of the Greeks rather than of the Egyp
tians and Babylonians?
7. Criticize the statement "Mathematics was created by the Greeks and very little
was added since their time."
Topics for Further Investigation
To write on the following topics use the books listed under Recommended Reading.
I. The mathematical contributions of the Egyptians or Babylonians.
2. The mathematical contributions of the Greeks.
THE HUMAN ASPECT OF MATHEMATICS 29
Recommended Reading
BALL, W. W. ROUSE: A Short Account of the History of Mathematics, Dover Pub
lications, Inc., New York, 1960.
BELL, ERIC T.: Men of Mathematics, Simon and Schuster, New York, 1937.
CHILDE, V. GORDON: Man Makes Himself, The New American Library, New York,
1951.
EVES, HOWARD: An Introduction to the History of Mathematics, Rev. ed., Holt,
Rinehart and Winston, Inc., New York, 1964.
NEUGEBAUER, OTTO: The Exact Sciences in Antiquity, Princeton University Press,
Princeton, 1952.
SCOTT, J. F.: A History of Mathematics, Taylor and Francis, Ltd., London, 1958.
SMITH, DAVID EUGENE: History of Mathematics, Vol. I, Dover Publications, Inc.,
New York, 1958.
STRUlK, DIRK J.: A Concise History of Mathematics, Dover Publications, Inc., New
York, 1948.
CHAPTER 3
LOGIC AND MATHEMATICS
Geometry will draw the soul toward truth and create the spirit of
philosophy.
PLATO
3-1 INTRODUCTION
Mathematics has its own ways of establishing knowledge, and the understand
ing of mathematics is considerably promoted if one learns first just what those
ways are. In this chapter we shall study the concepts which mathematics treats;
the method, called deductive proof, hy which mathematics establishes its con
clusions; and the principles or axioms on which mathematics rests. Study of
the contents and logical structure of mathematics leaves untouched the subject
of how the mathematician knows what conclusions to establish and how to
prove them. We shall therefore present a brief and preliminary discussion of
the creation of mathematics. This topic will recur as we examine the subject
matter itself in subsequent chapters.
Since mathematics, as we conceive the subject today, was fashioned by the
Greeks, we shall also attempt to see what features of Greek thought and
culture caused these people to remodel what the Egyptians and Babylonians
had pursued for several thousand years.
3-2 THE CONCEPTS OF MATHEMATICS
The first major step which the Greeks made was to insist that mathematics
must deal with abstract concepts. Let us see just what this means. When we
first learn about numbers we are taught to think about collections of particular
ohjects such as two apples, three men, and so on. Gradually and rather sub
consciously we begin to think about the numbers 2, J, and other whole
numbers without having to associate them with physical objects. We soon
reach the more advanced stage of adding, subtracting, and performing other
operations with numbers without having to handle collections of objects in
order to understand these operations or to see that the results agree with
experience. Thus we soon become convinced that 4 times 5 must be 20,
whether these numbers represent quantities of apples, horses, or even purely
30
THE CONCEPTS OF MATHEMATICS 31
imaginary objects. By this time we are really dealing with concepts or ideas,
for the whole numbers do not exist in nature. Any whole number is rather an
abstraction of a property which is common to many different collections or
sets of objects.
The whole numbers then are ideas, and the same is true of fractions such
as �, f, and so on. In the latter case, too, the formulation of the physical
relationship of a part of an object to the whole, whether it refers to pies,
bushels of wheat.. or to a smaller monetary value in relation to a larger one.
again leads to an abstraction. Mathematicians formulate operations with
fractions. that is. combining parts of an object. taking one part away from
the other, or taking a part of a part, in such a way that the result of any
operation on abstract fractions agrees with the corresponding physical occur
rence. Thus the mathematical process of, say adding i and S, which yields ii,
expresses the addition of ! of a pie and ! of a pie, and the result tells us how
many parts of a pie one would actually have.
Whole numbers, fractions, and the various operations with whole numbers
and fractions are abstractions. Although this fact is rather easy to understand,
we tend to lose sight of it and cause ourselves unnecessary confusion. Let us
consider an example. A man goes into a shoe store and buys 3 pairs of shoes
at 10 dollars per pair. The storekeeper reasons that 3 pairs times 10 dollars is
30 dollars and asks for 30 dollars in return for the 3 pairs of shoes. If this
reasoning is correct, then it is equally correct for the customer to argue that
3 pairs times 10 dollars is 30 pairs of shoes and to walk out with 30 pairs of
shoes without handing the storekeeper one cent. The customer may end up in
jail, but he may console himself while he languishes there that his reasoning
is as sound as the storekeeper's.
The source of the difficulty is, of course, that one cannot multiply shoes
hy dollars. One can multiply the number 3 hy the number 10 and obtain the
numher 30. The practical and no doubt obligatory physical interpretation of
the answer in the above situation is that one must pay 30 doIlars rather than
walk out with 30 pairs of shoes. We see, therefore, that one must distinguish
between the purely mathematical operation of multiplying 3 by 10 and the
physical objects with which these numhers may be associated.
The same point is involved in a slightly different situation. Mathematically
t is equal to !. But the corresponding physical fact may not be true. One may
he willing to accept 4 half-pies instead of 2 whole pies, but no woman would
accept 4 half-dresses in place of 2 dresses or 4 half-shoes in place of 1 pair of
whole shoes.
The Egyptians and Babylonians did reach the stage of working with pure
numhers dissociated from physical objects. But like young children of our
civilization, they hardly recognized that they were dealing with abstract
entities. By contrast, the Greeks not only recognized numbers as ideas but
emphasized that this is the way we must regard them. The Greek philosopher
32 LOGIC AND MATHEMATICS
Plato, who lived from 428 to 348 B.C. and whose ideas are representative of the
classical Greek period, says in his famous work, the Republic,
We must endeavor that those who are to be the principal men of our State go
and learn arithmetic, not as amateurs, but they must carry on the study until
they see the nature of numbers with the mind only; . . . arithmetic has a very
great and eLevating effect, compelling the soul to reason about abstract number,
and rebelling against the introduction of visible or tangible objects into the
argument.
The Greeks not only emphasized the distinction between pure numbers
and the physical applications of such numbers, but they preferred the former
to the latter. The study of the properties of pure numbers, which they called
arithmetica, was esteemed as a worthy activity of the mind, whereas the use
of numbers in practical applications, iogistica, was deprecated as a mere skill.
Geometrical thinking prior to the classical Greek period was even less
advanced than thinking about numbers. To the Egyptians and Babylonians
the words "straight line" meant no more than a stretched rope or a line traced
in sand, and a rectangle was a piece of land of a particular shape. The Greeks
began the practice of treating point, line, triangle, and other geometrical
notions as concepts. They did of course appreciate that these mental notions
are suggested by physical objects, but they stressed that the concepts differ
from the physical examples as sharply as the concept of time differs from the
passage of the sun across the sky. The stretched string is a physical object
illustrating the concept of line, but the mathematical line has no thickness, no
color, no molecular structure, and no tension.
The Greeks were explicit in asserting that geometry deals with abstractions.
Speaking of mathematicians, Plato says,
And do you not know also that although they make use of the visible forms and
reason about them, they are thinking not of these, but of the ideals which they
resemble; not of the figures which they draw, but of the absoLute square and the
absolute diameter . . . they are really seeking to behold the things the'l11seives,
which can be seen only 'u..'ith the eye of the mind?
On the basis of elementary abstractions, mathematics creates others which
are even more remote from anything real. Negative numbers, equations
involving unknowns, formulas, and other concepts we shall encounter are
abstractions built upon abstractions. Fortunately, every abstraction is ultimately
derived from, and therefore understandable in terms of, intuitively meaningful
objects or phenomena. The mind does play its part in the creation of mathe
matical concepts, but the mind does not function independently of the outside
world. Indeed the mathematician who treats concepts that have no physically
real or intuitive origins is almost surely talking nonsense. The intimate connec
tion between mathematics and objects and events in the physical world is
THE CONCEPTS OF MATHEMATICS 33
reassuring, for it means that we can not only hope to understand the mathe
matics proper, but also expect physically meaningful and valuable conclusions.
The use of abstractions is not peculiar to mathematics. The concepts of
force. mass, and energy, which are studied in physics, are abstractions from
real phenomena. The concept of wealth, an abstraction from material pos
sessions such as land, buildings, and jewelry, is studied in economics. The
concepts of liberty. justice, and democracy are familiar in political science.
Indeed, with respect to the use of abstract concepts, the distinction between
mathematics on the one hand and the physical and social sciences on the other
is not a sharp one. In fact, the influence of mathematics and mathematical
ways' of thinking on the physical sciences especially has led to ever increasing
use of abstract concepts including some, as we shall see, which may have no
direct real counterpart at all. any more than a mathematical formula has :t
direct real counterpart.
The very fact that other studies also engage in abstractions raises an
important question. Mathematics is confined �o some abstractions, numbers
and geometrical forms, and to concepts built upon these basic ones. Abstrac
tions such as mass, force, and energy belong to physics, and still other
abstractions belong to other subjects. Why doesn't mathematics also treat
forces, wealth, and justice' Certainly these concepts are also worthy of study.
Did the mathematicians make an agreement with physicists, economists, and
others to divide the concepts among themselves? The restriction of mathe
matics to numbers and geometrical forms is partly a historical accident and
partly a deliberate decision made by the Greeks. Numbers and geometrical
forms had already been introduced by the Egyptians and Babylonians, and
their utility in daily life was established. Since the Greeks learned the rudi
ments of mathematics from these civilizations, the sheer weight of tradition
might have caused them to continue the practice of regarding mathematics as
the study of numbers and geometrical figures. But people as original and bold
in thought as the Greeks would not have been bound merely by tradition, had
they not found in numbers and geometrical forms sharp and clear notions
which appealed to their delight in the processes of exact thinking. However,
an even more compelling reason was their belief that numerical and geometrical
properties and relationships were basic, that they underlay the phenomena of
the physical world and the design of the entire universe. Hence to under
stand the world one should seek this mathematical essence. The brilliance and
depth of their conception of the universe will be revealed more and more as
we proceed.
When one compares the pre-Greek and Greek understanding of the con
cepts of mathematics and notes the sharp transition from the concrete to the
abstract, another question presents itself. The Greeks eliminated the physical
substance and retained only the idea. Why did they do it' Surely it is more
difficult to think about abstractions than about concrete things. Also it would
34 LOGIC AND MATHEMATICS
seem that an attempt to study nature by concentrating on just a few aspects of
physical objects rather than on the objects themselves would fall far short of
effectiveness.
Insofar as the emphasis on abstractions is concerned, the Greeks saw at
once what any thinking people would see sooner or later. One advantage of
treating abstractions is the gain in generality. When a child learns that
5 +5 = 10, he acquires in one swoop a fact which applies to hundreds of
situations. Likewise a theorem proved about the abstract triangle applies to a
triangular piece of land, a musical percussion instrument, and a triangle
determined by three heavenly bodies at any instant of time. It has been said
that the process of abstraction amounts to giving the same name to different
things, but this very recognition that different objects possess the common
property named in the abstraction carries with it the implication that anything
true of the abstraction will apply to the several objects. Part of the secret of
the power of mathematics is that it deals with abstractions.
Another advantage of abstraction was also clear to the Greeks. Abstract
ing from a physical situation just those properties which are to be studied frees
the mind from burdensome and irrelevant details and enables one to concen
trate on the features of interest. When one wishes to determine the area of a
piece of land, only shape and size are relevant, and it is desirable to think only
about these and not about the fertility of the soil.
The emphasis on mathematical abstractions by the classical Greeks was
part and parcel of their outlook on the entire universe. They were concerned
with truths, and leading philosophical schools, notably the Pythagoreans and
the Platonists, maintained that truths could be established only about abstrac
tions. Let us follow their argument. The physical world presents various
objects to the senses. But the impressions received by the senses are inexact,
transitory, and constantly changing; indeed, the senses may be even deceived,
as by mirages. However, truth, by it� very meaning, must consist of perma
nent, unchanging, definite entities and relationships. Fortunately, the intelli
gence of man excited to reflection by the impressions of sensible objects may
rise to higher conceptions of the realities faintly exhibited to the senses, and so
man may rise to the contempla"ion of ideas. These are eternal realities and the
true goal of thought, whereas mere "things are the shadows of ideas thrown on
the screen of experience."
Thus Plato would say that there is nothing real in a horse, a house, or a
beautiful woman. Tht reality is in tht universol type or idea of a horse, a
home, or a woman. The ideas, among w�ich Plato emph1sized Beauty, Justice,
Intelligence, Goodness, Perfection, and the Sute, are independent of the super
ficial appearances of things, of the /lux of life, and of the biases and warped
desires of man; they arc in fact constant and invariabie, and knowledge con
cerning them is firm acd indestructible. Real and eternal knowledge concerns
these ideas, rather than sensuous objects. This distinr:tion between the intelli
gible world and the world revealed by the senses is all-important in Plato.
THE CONCEPTS OF MATHEMATICS 35
Fig. 3-1.
Polyclitus: Spear·bearer (Oaryphorus). National Museum,
Naples.
To put Plato's doctrine in everyday language, fundamental knowledge
does not concern itself with what John ate, Mary heard, or William felt.
Knowledge must rise above individuals and particular objects and tell us about
broad classes of objects and about man as a whole. True knowledge must
therefore of necessity concern abstractions. Plato admits that physical or sen
sible objects sugges: the ideas just as diagrams of geometry suggest abstract
geometrical concepts. Hence there is a point to studying physical objects, but
one must not lose himself in trivial and confusing minutiae.
The abstractions of mathematics possessed a special importance for the
Greeks. The philosophers pointed out that, to pass from a knowledge of the
world of matter to tho world of ideas, man must train his mind to grasp the
36 LOGIC AND I\lATHEi\lATICS
FIG. 3-2.
Bust of Caesar. Vatican.
ideas. These highest realities blind the person who is not prepared to contem
plate them. He is, to use Plato's famous simile, like one who lives continuously
in the deep shadows of a cave and is suddenly brought out into the sunlight.
The study of mathematics helps make the transition from darkness to light.
Mathematics is in fact ideally suited to prepare the mind for higher forms of
thought because on the one hand it pertains to the world of visible things and
on the other hand it deals with abstract concepts. Hence through the study of
mathematics man learns to pass from concrete figures to abstract forms; more
over, this srudy purifies the mind by drawing it away from the contemplation
of the sensible and perishable and leading it to the eternal ideas. These latter
abstractions are on the same mental level as the concepts of mathematics.
Thus, Socrates says, "The understanding of mathematics is necessary for a
sound grasp of ethics."
To sum up Plato's position we may say that while a little knowledge of
geometry and calculation suffices for practical needs, the higher and more
advanced portions tend to lift the mind above mundane considerations and
enable it to apprehend the final aim of philosophy, the idea of the Good.
Mathematics, then, is the best preparation for philosophy. For this reason
Plato recommended that the future rulers, who were to be philosopher-kings,
be trained for ten years, from age 20 to 3D, in the study of the exact sciences,
arithmetic, plane geometry, solid geometry, astronomy, and harmonics (music) .
The oft-repeated inscription over the doors of Plato's Academy, stating that
no one ignorant of mathematics should enter, fully expresses the importance
THE CONCEPTS OF MATHEMATICS 37
Fig. 3-3. Parthenon, Athens.
he attached to the subject, although modern Critics of Plato read into these
words his admission that one would not be able to learn it after entering. This
value of mathematical training led one historian to remark, "Mathematics
considered as a science owes its origins to the idealistic needs of the Greek
philosophers, and not as fable has it, to the practical demands of Egyptian
economics."
The preference of the Greeks for abstractions is equally evident in the art
of the great sculptors, Polyclirus, Praxiteles, and Phidias. One has only to
glance at the face in Fig. 3-1 to observe that Greek sculpture of the classical
period dwelt not on particular men and women but on rypes, ideal types.
Idealization extended to standardization of the ratios of the parts of the body
to each other. Polyclitus believed, in fact, that there were ideal numerical ratios
which fix the proportions of the human body. Perfect art must follow these
ideal proportions. He wrote a book, The Canon, on the subject and con
structed the "Spear-bearer" to illustrate the thesis. These abstract types con
trast sharply with what is found in numerous busts and starues of private
individuals and military and political leadets made by Romans (Fig. 3-2 ) .
Greek architecrure also reveals the emphasis o n ideal forms. The simple
and austere buildings were always rectangular in shape; even the ratios of
the dimensions employed were fixed. The Parthenon at Athens (Fig. 3-3)
is an ex"mple of the style and proportions found in almost all Greek temples.
38 LOGIC AND MATHEMATICS
EXERCISES
1. Suppose 5 trucks pass by with 4 men in each. To answer the Question of how
many men there are in all the trucks, a person reasons that 4 men times 5 trucks
is 20 men. On the other hand, if there ar.e 4 men each owning 5 trucks, the total
number of trucks is 20 trucks. Hence 4 men times 5 trucks yields 20 trucks.
How do you know that the answer is 20 men in one case and 20 trucks in the
other?
2. If the product of 2 5 ¢ and 2 5 ¢ is obtained by multiplying 0.25 by 0.25 the result
is 0.0625 or 6 t ¢ . Does it pay to multiply money?
3. Can you suggest some abstract political or ethical concepts?
4. Suppose 30 books are to be distributed among 5 people. Since 30 books divided
by 5 people yields 6 books, each person gets 6 books. Criticize the reasoning.
5 . A store advertises that it will give a credit of $ 1 for each purchase amounting to
$ 1 . A man who spends $6 reasons that he should receive a credit of $6 times $ 1 ,
or $6. But $ 6 is 600¢ and $ 1 is 100¢. Hence 600¢ times IOO¢ is 60,000¢, or $600.
It would seem that it is more profitable to operate with the almost worthless cent
than with dollars! What is wrong?
6. What does the statement that mathematics deals with abstractions mean?
7. Why did the Greeks make mathematics abstract?
3-3 IDEALIZATIDN
The geometrical notions of mathematics are abstract in the sense that shapes
are mental concepts which actual physical objects merely approximate. The
sides of a rectangular piece of land may not be exactly straight nor would each
angle be exactly 90°. Hence, in adopting such abstract concepts, mathematics
does idealize. But in studying the physical world, mathematics also idealizes
in another sense which is equally important. Very often mathematicians under
take to study an object which is not a sphere and yet choose to regard it as
such. For example, the earth is not a sphere but a spheroid, that is, a sphere
flattened at the top and bottom. Yet in many physical problems which are
treated mathematically the earth is represented as a perfect sphere. In problems
of astronomy a large mass such as the earth or the sun is often regarded as
concentrated at one point.
In making such idealizations, the mathematician deliberately distorts or
approximates at least some features of the physical situation. Why does he
do it? The reason usually is that he simplifies the problem and yet is quite
sure that he has not introduced any gross errors. If one is to investigate, for
example, the motion of a shell which travels ten miles, the difference between
the assumed spherical shape of the earth and the true spheroidal shape does not
matter. In fact, in the study of any motion which takes place over a limited
region, say one mile, it may be sufficient to treat the earth as a Rat surface. On
the other hand, if one were to draw a very accurate map of the earth, he would
METHODS OF REASONING 39
take into account that the shape is spheroidal. As another example, to find the
distance to the moon, it is good enough to assume that the moon is a point in
space. However, to find the size of the moon, it is clearly pointless to regard
the moon as a point.
The question does arise, how does the mathematician know when idealiza
tion is justified' There is no simple answer to this question. If he has to solve
a series of like problems. he may solve one using the correct figure, and
another, using a simplified figure, and compare results. If the difference does
not matter for his purposes, he may then retain the simpler figure for the
remaining problems. Sometimes he can estimate the error introduced by using
the simpler figure and may find that this error is too smaIl to matter. Or the
mathematician may make the idealization and use the result because it is the
best he Can do. Then he must accept experience as his guide in deciding
whether the result is good enough.
To idealize by deliberately introducing a simplification is to lie a little,
but the lie is a white one. Using idealizations to study the physical world does
impose a limitation on what mathematics accomplishes, but we shaH find that
even where idealizations are employed, the knowledge gained is of immense
value.
EXERCISES
1. Distinguish between abstraction and idealization.
2. Is it correct to assume that the lines of sight to the sun from two places A and B
on the earth's surface are parallel?
3. Suppose you wished to measure the height of a flagpole. Would it be wise to
regard the flagpole as a line segment?
3-4 METHODS OF REASONING
There are many ways, more or less reliable, of obtaining knowledge. One can
resort to authority as one often does in obtaining historical knowledge. One
may accept revelation as many religious people do. And one may rely upon
experience. The foods we eat are chosen on the basis of experience. No one
determined in advance by careful chemical analysis that bread is a healthful
food.
We may pass over with a mere mention such sources of knowledge as
authority and revelation, for these sources cannot be helpful in building
mathematics or in acquiring knowledge of the physical world. It is true that
in the medieval period of Western European culture men did contend that
all desirable knowledge of nature was revealed in the Bible. However in no
significant period of scientific thought has this view played any role. Experi
ence, on the other hand, is a useful source of knowledge. But there are
40 LOGIC AND MATHEMATICS
difficulties in employing this method. We should not wish to build a fifty
story building in order to decide whether a steel beam of specified dimensions
is strong enough to be used in the foundation. Moreover, even if one should
happen to choose workable dimensions, the choice may be wasteful of ma
terials. Of course, experience is of no use in determining the size of the earth
or the distance to the moon.
Closely related to experience is the method of experiment which amounts
to setting up and going through a series of purposive, systematic experiences.
It is true that experimentation fundamentally is experience, but it is usually
accompanied by careful planning which eliminates extraneous factors, and
the experience is repeated enough times to yield reliable information. How
ever, experimentation is subject to much the same limitations as experience.
Are authority, revelation, experience, and even experimentation the only
methods of obtaining knowledge? The answer is no. The major method is
reasoning, and within the domain of reasoning there are several forms. One
can reason by analogy. A boy who is considering a college career may note
that his friend went to college and handled it successfully. He argues that
since he is very much like his friend in physical and mental qualities, he too
should succeed in college work. The method of reasoning just illustrated is
to find a similar situation or circumstance and to argue that what was true for
the similar case should be true of the one in question. Of course, one must be
able to find a similar situation and one must take the chance that the differences
do not matter.
Another common method. of reasoning is induction. People use this
method of reasoning every day. Because a person may have had unfortunate
experiences in dealing with a few department stores, he concludes th�t all
department stores are bad to deal with. Or, for example, experimentation
would show that iron, copper, brass, oil, and other substances expand when
heated, and one consequently concludes that all substances expand when
heated. Inductive reasoning is in fact the common method used in experimen
tation. An experiment is generally performed many times, and if the same
result is obtained each time, the experimenter concludes that the result will
always follow. The essence of induction is that one observes repeated occur
rences of the same phenomenon and concludes that the phenomenon will
always occur. Conclusions obtained by induction seem well warranted by the
evidence, especially when the number of instances observed is large. Thus the
sun is observed so often to rise in the morning that one is sure it has risen even
on those mornings when it is hidden by clouds.
There is still a third method of reasoning, called deduction. Let us consider
some examples. If we accept as basic facts that honest people return found
money and that John is honest, we may conclude unquestionably that John
will return money that he finds. Likewise, if we start with the facts that no
mathematician is a fool and that John is a mathematician, then we may con-
METHODS OF REASONING 41
c1ude with certainty that John is not a fool. In deductive reasoning we start
with certain statements, called premises. and assert a conclusion which IS a
necessary or inescapable consequence of the premises.
All three methods of reasoning, analogy, induction, and deduction, and
other methods we could describe, are commonly employed. There is one
essential difference, however, between deduction on the one hand and all other
methods of reasoning on the other. Whereas the conclusion drawn by analogy
or induction has only a probability of being correct, the conclusion drawn by
deduction necessarily holds. Thus one might argue that because lions are
similar to cows and cows eat grass, lions also eat grass. This argument by
analogy leads to a false conclusion. The same is true for induction: although
experiment may indeed show that two dozen different substances expand when
heated, it does not necessarily follow that all substances do. Thus water, for
example, when heated from 0° to 4° centigrade· does not expand; it contracts.
Since deductive reasoning has the outstanding advantage of yielding an
indubitable conclusion, it would seem obvious that one should always use
this method in preference to the others. But the situation is not that simple.
For one thing analogy and induction are often easier to employ. In the case
of analogy, a similar situation may be readily available. In the case of induction,
experience often supplies the facts with no effort at all. The fact that the sun
rises every morning is noticed by all of us almost automatically. Furthermore,
deductive reasoning calls for premises which it may be impossible to obtain
despite all efforts. Fortunately we can use deductive reasoning in a variety of
situations. For example, we can use it to find the distance to the moon. In
this instance, both analogy and induction are powerless, whereas, as we shall
see later, deduction will obtain the result quickly. It is also apparent that
where deduction can replace induction based on expensive experimentation,
deduction is preferred.
Because we shall be concerned primarily with deductive reasoning, let us
become a little more familiar with it. We have given several examples of
deductive reasoning and have asserted that the conclusions are inescapable
consequences of the premises. Let us consider, however, the following ex
ample. We shall accept as premises that
All good cars are expensive
and
All Locomobiles are expensive.
We might conclude that
All Locomobiles are good cars.
• In scientific texts, "celsius" is considered to be the more precise term.
42 LOGIC AND MATHEMATICS
Expensive Learned
objects people
Good Intelligent
cars people
Fig. 3-4 Fig. 3-5
The reasoning here is intended as deductive; that is, the presumption in draw
ing this conclusion is that it is an inevitable consequence of the premises.
Unfortunately, the reasoning is not correct. How can we see that it is not
correct? A good way of picturing deductive arguments which enables us to
see whether or not they are correct is called the circle test.
We note that the first premise deals with cars and expensive objects. Let
us think of all the expensive objects in this world as represented by the points
of a circle, the largest circle in Fig. 3-4. The statement that all good cars are
expensive means that all good cars are a part of the collection of expensive
objects. Hence we draw another circle within the circle of expensive objects,
and the points of this smaller circle represent all the good cars. The second
premise says that all Locomobiles are expensive. Hence if we represent all
Locomobiles by the points of a circle, this circle, too, must be drawn within the
circle of expensive objects. However we do not know, on the basis of the two
premises, where to place the circle representing all Locomobiles. It can, as far
as we know, fall in the position shown in the figure. Then we cannot conclude
that all Locomobiles are good cars, because if that conclusion were inevitable,
the circle representing Locomobiles must fall inside the circle representing
good cars.
Many people do conclude from the above premises that all Locomobiles
are good cars and the reason that they err is that they confuse the premise " All
good cars are expensive" with the statement that "All expensive cars are good."
Were the latter statement our first premise then the deductive argument would
be valid or correct.
Let us consider another example. Suppose we take as our premises that
All professors are learned people
and
Some professors are intelligent people.
May we necessarily conclude that
Some intelligent people are learned?
METHODS OF REASONING 43
It may or may not be obvious that this conclusion is correct. Let us use the
circle test. We draw a circle representing the class of learned people (Fig 3-5 ) .
Since the first premise tells us that all professors are learned people, the
circle representing the class of professors must fall within the circle represent
ing learned people. The second premise introduces the class of intelligent
people, and we now have to determine where to draw that circle. This class
must include some professors. Hence the circle must intersect the circle of
professors. Since the latter is inside the circle of learned people, some intelli
gent people must fall within the class of learned people.
These examples of deductive reasoning may make another point clear. In
determining whether a given argument is correct or valid, we must rely only
upon the facts given in the premises. We may not use information which is
not explicitly there. For example, we may believe that learned people are
intelligent because to acquire learning they must possess intelligence. But
this belief or fact, if it is a fact, cannot enter into the argument. Nothing that
one may happen to know or believe about learned or intelligent people is to
be used unless explicitly stated in the premises. In fact, as far as the validity
of the argument IS concerned, we might just as well have considered the
premises
All x's are y's,
Some x's are z's,
and the conclusion, then, IS
Some z's are y's.
Here we have used x for professor, y for learned person, and z for intelligent
person. The use of x, y, and z does make the argument more abstract and
more difficult to retain in the mind, but it emphasizes that we must look only
at the information in the premises and avoids bringing in extraneous informa
tion about professors, learned people, and intelligenr people. When we 'Nrite
the argument in this more abstract form, we also see more clearly that what
determines the validity of the argument is the fonn of the premises rather than
the meaning of x, y, and z.
A great deal of deductive reasoning fails into the patterns we have been
illustrating. There are, however, variations that should be noted. It is quite
customary, especially in the geometry we learn in high school, to state
theorems in what is called the "if . . . then" form. Thus one might say, if a
triangle is isosceles, then its base angles are equal. One could as well say, all
isosceles triangles have equal base angles; or, the base angles of an isosceles
triangle are equal. All three versions say the same thing.
Connected with the "if . . . then" form of a premise is a related statement
which is often misunderstood. The statement "if a man is a professor, he is
learned" offers no difficulty. As noted in the preceding paragraph, it is
equivalent to "all professors are learned." However the statement "only if a
44 LOGIC AND MATHEMATICS
man is a professor, is he learned" has quite a different meaning. It means that
to be learned one must be a professor or that if a man is learned, he must be
a professor. Thus the addition of the word only has the significance of inter
changing the "if" clause and the "then" clause.
We shall encounter numerous instances of deductive reasoning in our
work. The subject of deductive reasoning is customarily studied in logic, a
discipline which treats more thoroughly the valid forms of reasoning. How
ever, we shaIl not need to depend upon formal training in logic. In most cases,
common experience will enable us to ascertain whether the reasoning is or is
not valid. When in doubt, we can use the circle test. Moreover, mathematics
itself is the superb field from which to learn reasoning and is the best exercise
in logic. The laws of logic were in fact formulated by the Greeks on the
basis of their experiences with mathematical arguments.
EXERCISES
1. A coin is tossed ten times and each time it falls heads. What conclusion does
inductive reasoning warrant?
2. Characterize deductive reasoning.
3. What superior features does deductive reasoning possess compared with induc
tion and analogy?
4. Can you prove deductively that George Washington was the best president of
the United States?
5. Can one always apply deductive reasoning to prove a desired statement?
6. Can you prove deductively that monogamy is the best system of marriage?
7. Are the following purportedly deductive arguments valid?
a) All good cars are expensive. A Daffy is an expensive car. Therefore a Daffy
is a good car.
b) All New Yorkers are good citizens. All good citizens give to charity. There
fore all New Yorkers give to charity.
c ) All college students are clever. All young boys are clever. Therefore all
young boys are college students.
d) The same premises as in (c), but the conclusion: All college students are
young boys.
e) It rains every Monday and it is raining today; hence today must be Monday.
£) No decent people curse; Americans are decent; therefore Americans do
not curse.
g) No decent people curse; Americans curse; therefore some Americans are
not decent.
h) No decent people curse; some Americans are not decent; therefore some
Americans curse.
i) No undergraduates have a bachelor-of-arts degree; no freshmen have a
bachelor-of-arts degree. Therefore all freshmen are undergraduates.
MATHEMATICAL PROOF 45
8. If someone gave you a valid deductive argument but the conclusion was not
true, where would you look for the difficulty?
9. Distinguish between the validity of a deductive argument and the truth of the
conclusion.
3-5 MATHEMATICAL PROOF
We have seen so far in our discussion of reasoning that there are several
methods of reasoning and that all are useful. These methods can be applied to
mathematical problems. Let us suppose that one wished to determine the sum
of the angles of a triangle. He could draw on paper many different triangles
or construct some out of wood or metal and measure the angles. In each
case he would find that the sum is as close to 180° as the eye and hand can
determine. By inductive reasoning he could conclude that the sum of the
angles in every triangle is 1 80°. As a matter of fact, the Babylonians and
Egyptians did in effect use inductive reasoning to establish their mathematical
results. They must have determined by measurement that the area of a triangle
is one-half the base times the altitude and, having used this formula repeatedly
and having obtained reliable results, they concluded that the formula is correct.
Fig. 3-6.
The mid· points of parallel
chords lie on a straight
(a) (b) line.
To see that reasoning by analogy can be used in mathematics, let us note
fitst that the centers of a set of parallel chords of a circle lie on a straight line
(Fig. 3-6a). In fact this line is a diameter of the circle. Now an ellipse
(Fig. 3-6b) is very much like a circle. Hence one might conclude that the
centers of a set of parallel chords of an ellipse also lie on a straight line.
Deduction is certainly applicable in mathematics. The proofs which one
learns in Euclidean geometry are deductive. As another illustration we might
consider the following algebraic argument. Suppose one wishes to solve the
equation x 3 = 7. One knows that equals added to equals give equals. If we
-
added 3 to both sides of the preceding equation, we would be adding equals to
an equality. Hence the addition of 3 to both sides is justified. When this is
done, the result is x = l a, and the equation is solved.
Thus all three methods are applicable. There is a lot to be said for the use
of induction and analogy. The inductive argument for the sum of the angles
of a triangle can be carried out in a matter of minutes. The argument by
analogy given above is also readily made. On the other hand, finding deduc-
46 LOGIC AND MATHEMATICS
tive proofs for these same conclusions might take weeks or might never be
accomplished by the average person. As a matter of fact, we shall soon en
counter some examples of conjectures for which the inductive evidence is
overwhelming but for which no deductive proof has been thus far obtained
even by the best mathematicians.
Despite the usefulness and advantages of induction and analogy, mathe
matics does not rely upon these methods to establish its conclusions. All
mathematical proofs must be deductive. Each proof is a chain of deductive
arguments, each of which has its premises and conclusion.
Before examining the reasons for this restriction to deductive proof, we
might contrast the method of mathematics with those of the physical and
social sciences. The scientist feels free to draw conclusions by any method of
reasoning and, for that matter, on the basis of observation, experimentation,
and experience. He may reason by analogy as, for example, when he reasons
about sound waves by observing water waves or when he reasons about a
possible cure for a disease affecting human beings by testing the cure on
animals. In fact reasoning by analogy is a powerful method in science. The
scientist may also reason inductively: if he observes many times that hydrogen
and oxygen combine to form water, he will conclude that this combination
will always form water. At some stages of his work the scientist may also
reason deductively and, in fact, even employ the concepts and methods of
mathematics proper.
To contrast further the method of mathematics with that of the scientist
-and perhaps to illustrate just how stubborn the mathematician can be-we
might consider a rather famous example. Mathematicians are concerned with
whole numbers, or integers, and among these they distinguish the prime
numbers. A prime is a number which has no integral divisors other than itself
and 1. Thus 1 1 is a prime number, whereas 1 2 is not because it is divisible by
2 for example. Now by actual trial one finds that each of the first few even
numbers can be expressed as the sum of two prime numbers. For example,
2 1 + 1 ; 4 2 + 2 ; 6 3 + 3; 8 3 + 5 ; 1 0 3 + 7; . . . . If one investi
� � � � �
gates larger and larger even numbers, one finds without exception that every
even number can be expressed as the sum of two primes. Hence by inductive
reasoning one could conclude that every even number is the sum of two prime
numbers.
But the mathematician does not accept this conclusion as a theorem of
mathematics because it has not been proved deductively from acceptable
premises. The conjecture that every even number is the sum of two primes,
known as Goldbach's hypothesis because it was first suggested by the
eighteenth-century mathematician Christian Goldbach, is an unsolved problem
of mathematics. The mathematician will insist on a deductive proof even if it
takes thousands of years, as it literally has in some instances, to find one�
However a scientist would not hesitate to use this inductively well supported
conclusion.
MATHEMATICAL PROOF 47
Of course, the scientist should not be surprised to find that some of his
conclusions are false because, as we have seen, induction and analogy do not
lead to sure conclusions. But it does seem as though the scientist's procedure
is wiser since he can take advantage of any method of reasoning which will
help him advance his knowledge. The mathematician by comparison appears
to be narrow-minded or shortsighted. He achieves a reputation for certainty,
but at the price of limiting his results to those which can be established
deductively. How wise the mathematician may be in his insistence on deduc
tive proof we shaIl learn as we proceed.
The decision to confine mathematical proof to deductive reasoning was
made by the Greeks of the classical period. And they not only rejected all
other methods of proof in mathematics, but they also discarded all the
knowledge which the Egyptians and Babylonians had acquired over a period
of four thousand years because it had only an empirical justification. Why
did the Greeks do it'
The intellectuals of the classical Greek period were largely absorbed in
philosophy and these same men, because they possessed intellectual interests,
were the very ones who developed mathematics as a system of thought. The
lonians, the Pythagoreans, the Sophists, the Platonists, and the Aristotelians
were the leading philosophers who gave mathematics its definitive form. The
credit for initiating this step probably helongs to one school of Greek
philosopher-mathematicians, known as the Ionian school. However, if credit
can be assigned to any one person, it belongs to Thales, who lived about 600
B.C. Though a native of Miletus, a Greek city in Asia Minor, Thales spent
many years in Egypt as a merchant. There he learned what the Egyptians had
to offer in mathematics and science, but apparently he was not satisfied, for he
would accept no results that could not be established by deductive reasoning
from clearly acceptable axioms. In his wisdom Thales perceived what we
shaIl perceive as we follow the story of mathematics, that the obvious is far
more suspect than the abstruse.
Thales probably supplied the proof of many geometrical theorems. He
acquired great fame as an astronomer and is believed to have predicted an
eclipse of the sun in 585 B.C. A philosopher-astronomer-mathematician might
readily be accused of being an impractical stargazer, but Aristotle tells us
otherwise. In a year when olives promised to be plentiful, Thales shrewdly
cornered all the oil presses to be found in Miletus and in Chios. When the
olives were ripe for pressing, Thales was in a position to rent out the presses
at his own price. Thales might perhaps have lived in history as a leading
businessman, but he is far better known as the father of Greek philosophy and
mathematics. From his time onward, deductive proof became the standard in
mathematics.
It is to be expected that philosophers would favor deductive reasoning.
Whereas scientists select particular phenomena for observation and experimen
tation and then draw conclusions by induction or analogy, philosophers are
48 LOGIC AND MATHEMATICS
concerned with broad knowledge about man and the physical world. To
establish universal truths, such as that man is basically good, that the world is
designed, or that man's life has purpose, deductive reasoning from acceptable
principles is far more feasible than induction or analogy. As Plato put it in his
Republic, "If persons cannot give or receive a reason, they cannot attain that
knowledge which, as we have said, man ought to have."
There is another reason that philosophers favor deductive reasoning. These
men seek truths, the eternal verities. We have seen that of all the methods of
reasoning only deductive reasoning grants sure and exact conclusions. Hence
this is the method which philosophers would almost necessarily adopt. Not
only do induction and analogy fail to yield absolutely unquestionable conclu
sions, but many Greek philosophers would not have accepted as facts the data
with which these methods operate, because these are acquired by the senses.
Plato stressed the unreliability of sensory perceptions. Empirical knowledge,
as Plato put it, yields opinion only.
The Greek preference for deduction had a sociological basis. Contrary to
our own society wherein bankers and industrialists are respected most, in
classical Greek society, the philosophers, mathematicians, and artists were the
leading citizens. The upper class regarded earning a living as an unfortunate
necessity. Work robbed man of time and energy for intellectual activities,
the duties of citizenship and discussion. These Greeks did not hesitate to
express their disdain for work and business. The Pythagoreans, who, as we
shall see, delighted in the properties of numbers and applied numbers to the
study of nature, derided the use of numbers in commerce. They boasted that
they sought knowledge rather than wealth. Plato, too, maintained that knowl
edge rather than trade was the goal in studying arithmetic. Freemen, he
declared, who allowed themselves to become preoccupied with business should
be punished, and a civilization which is concerned mainly with the material
wants of man is no more than a "city of happy pigs." Xenophon, the famous
Greek general and historian, says, "What are called the mechanical arts carry
a social stigma and are rightly dishonored in our cities." Aristotle wanted an
ideal society in which citizens would nOt have to practice any mechanical arts.
Among the Boeotians, one of the independent tribes of Ancient Greece, those
who defiled themselves with commerce were by law excluded from state
positions for ten years.
Who did the daily work of providing food, shelter, clothing, and the other
necessities of life? Slaves and free men ineligible for citizenship ran the busi
nesses and the households, did unskilled and technical work, managed the in
dustries, and carried on the professions such as medicine. They produced
even the articles of refinement and luxury.
In view of this attitude of the Greek upper class towards commerce and
trade, it is not hard to understand the classical Greek's preference for deduction.
People who do not "live" in the workaday world can learn little from experi-
MATHEMATICAL PROOF 49
ence, and people who will not observe and use their hands to experiment will
not have the facts on which to base reasoning by "nalogy or induction. In
fact the institution of slavery in classical Greek society fostered a divorce of
theory from practice and favored the development of speculative and deduc
tive science and mathematics at the expense of experimentation and practical
applications.
Over and above the various cultural forces which inclined the Greeks
toward deduction were a farsightedness and a wisdom which mark true genius.
The Greeks were the first to recognize the power of reason. The mind was a
faculty not only additional to the senses but more powerful than the senses.
The mind can survey all the whole numbers, but the senses are limited to
perceiving only a few at a time. The mind can encompass the earth and the
heavens; the sense of sight is confined to a small angle of vision. Indeed the
mind can predict future events which the senses of contemporaries wiII not
live to perceive. This mental faculty could be exploited. The Greeks saw
clearly that if man could obtain some truths, he could establish others entirely
by reasoning, and these new truths, together with the original ones, enabled
man to establish still other rruths. Indeed the possibilities would multiply at an
enormous rate. Here was a means of acquiring knowledge which had been
either overlooked or neglected.
This was indeed the plan which the Greeks projected for mathematics.
By starting with some truths about numbers and geometrical figures they
could deduce others. A chain of deductions might lead to a significant new
fact which would be labeled a theorem to call attention to its importance.
Each theorem added to the stock of truths that could serve as premises for
new deductive arguments, and so one could build an immense body of knowl
edge about the basic concepts.
Although the Greeks may have been guilty of overemphasizing the power
of the mind unaided by experience and observation to obtain truths, there is
no doubt that in insisting on deductive proof as the sole method, they rose
above the practical level of carpenters, surveyors, farmers, and navigators. At
the same time they elevated the subject of mathematics to a system of thought.
Moreover the preference for reason which they exhibited gave this faculty the
high prestige which it now enjoys and permitted it to exercise its true powers.
When we have surveyed some of the creations of the mind that succeeding
civilizations building on the Greek plan contributed, we shall appreciate the
true depth of the Greek vision.
EXERCISES
I. Compare Greek and pre-Greek standards of proof in mathematics. Reread the
relevant parts of Chapter 2.
2. Distinguish science and mathematics with respect to ways of establishing con
clusions.
50 LOGIC AND MATHEMATICS
3. Explain the statement that the Greeks converted mathematics from an empirical
science to a deductive system.
4. Are the following deductive arguments valid?
a) All even numbers are divisible by 4. Ten is an even number. Hence 10 IS
divisible by 4.
b) Equals divided by equals give equals. Dividing both sides of Jx = 6 by IS
dividing cquals by equals. Hence x = 2.
5. Does it foIlow from the fact that the square of any odd number is odd that the
square of any even number is even?
6. Criticize the argument:
The square of every even number is evcn because 22 = 4, 42 = 16, 62 = 36, and
it is obvious that the square of any larger even number also is even.
7. If we accept the premises that the square of any odd number is odd and the
square of any even number is even. does it follow deductively that if the square
of a number is even. the number must be even?
8. Why did the Greeks insist on deductive proof in mathematics?
9. Let us take for granted that if a triangle has two equal sides, the opposite
angles are equal and that we have a triangle in which all three sides are equal.
Prove deductively that all three angles are equal in the triangle under considera
tion. You may also use the premise that things equal to the same thing are equal
to one another.
10. How did the Greeks propose to obtain new truths from known ones?
3-6 AXIOMS AND DEFINITIONS
From our discussion of deductive reasoning we know that to apply such
reasoning we must have premises. Hence the question arises, what premises
does the mathematician use? Since the mathematician reasons about numbers
and geometrical figures, he must of course have facts about these concepts.
These cannot be obtained deductively because then there would have to be
prior premises. and if one continued this process backward, there would be no
starting point. The Greeks readily found premises. It seemed indisputable, for
example, that two points determine one and only one line and that equals
added to equals give equals.
To the Greeks the premises on which mathematics was to be built were
self-evident truths, and they called these premises axioms. Socrates and Plato
believed, as did many later philosophers, that these truths were already in our
minds at birth and that we had but to recall them. And since the Greeks be
lieved that axioms were truths and since deductive reasoning yielded unques
tionable conclusions, they also believed that theorems were truths. This view
is no longer held, and we shall see later in this book why mathematicians
abandoned it. We now know that axioms are suggested by experience and
observation. Naturally, to be as certain as we can of these axioms we select
those facts which seem clearest and most reliable in our experience. But we
THE CREATION OF MATHEMATICS 51
must recognize that there is no guarantee that we have selected truths about
the world. Some mathematicians prefer to use the word assumptions instead
of axioms to emphasize this point.
The mathematician also takes care to state his axioms at the outset and to
be sure as he performs his reasoning that no assumptions or facts are used
which were not so stated. There is an interesting story told by former Presi
dent Charles W. Eliot of Harvard which illustrates the likelihood of intro
ducing unwarranted premises. He entered a crowded restaurant and handed
his hat to the doorman. When he came out, the doorman at once picked Eliot's
hat out of hundreds on the racks and gave it to him. He was amazed that the
doorman could remember so well and asked him, "How did you know that was
my hat?" "I didn't," replied the doorman. "Why, then, did you hand it to
me?" The doorman's reply was, "Because you handed it to me, sir."
Undoubtedly no harm would have been done if the doorman had assumed
that the hat he returned to President Eliot belonged to the man. But the
mathematician interested in obtaining conclusions about the physical world
might be wasting his time if he unwittingly introduced an assumption that he
had no right to make
There is one other element in the logical structure of mathematics about
which we shall say a few words now and return to in a later chapter (Chap
ter 20). Like other studies mathemarics uses definitions. Whenever we have
occasion to use a concept whose description requires a lengthy statement, we
introduce a single word or phrase to replace that lengthy statement. For
example, we may wish to talk about the figure which consists of three dis
tinct points which do not lie on the same straight line and of the line segments
joining these points. It is convenient to introduce the word triangle to repre
sent this long description. Likewise the word circle represents the set of all
points which are at a fixed distance from a definite point. The definite point
is called the center, and the fixed distance is called the radius. Definitions
promote brevity.
EXERCISES
1 . What belief did the Greeks hold about the axioms of mathematics?
2. Summarize [he changes which the Greeks made in the nature of mathematics.
3. Is it fair to say that mathematics is the child of philosophy?
3·7 THE CREATION OF MATHEMATICS
Because mathematical proof is strictly deductive and merely reasonable or
appealing arguments may not be used to establish a conclusion, mathematics
has been described as a deductive science, or as the science which derives
necessary conclusions, that is, conclusions which necessarily or inevitably
52 LOGIC AND MATHEMATICS
follow from the axioms. This description of mathematics is incomplete.
Mathematicians must also discover what to prove and how to go about estab
lishing proofs. These processes are also part of mathematics and they are
not deductive.
How does the mathematician discover what to prove and the deductive
arguments that lead to the conclusions? The most fertile source of mathemat
ical ideas is nature herself. Mathematics is devoted to the study of the physical
world, and simple experience or the more careful scrutiny of nature suggests
idea after idea. Let us consider here a few simple examples. Once mathema
ticians had decided to devote themselves to geometric forms, it was only
natural that such questions should arise as, what are the area, perimeter, and
sum of the angles of common figures? Moreover, it is even possible to see
how the precise statement of the theorem to be proved would follow from
direct experience with physical objects. The mathematician might measure
the sum of the angles of various triangles and find that these measurements all
yield results close to 1 80°. Hence the suggestion that· the sum of the angles
in every triangle is 1800 occurs as a possible theorem. To decide the question,
which has more area, a polygon or a circle having the same perimeter, one
might cut out cardboard figures and weigh them. The relative weights would
suggest the statement of the theorem to be proved.
After some theorems have been suggested by direct physical problems,
others are readily conceived by generalizing or varying the conditions. Thus
knowing the problem of determining the sum of the angles of a triangle, one
might ask, What is the sum of the angles of a quadrilateral, a pentagon, and so
forth? That is, once the mathematician begins an investigation which is sug
gested by a physical problem, he can easily find new problems which go be
yond the original one.
In the domains of arithmetic and algebra direct calculation with numbers,
which is analogous to measurement in geometry, wiIl suggest possible theorems.
Anyone who has played with integers, for example, has doubtless observed the
following facts:
1 = I,
+ 3 = 4 = 2 2,
+ 3 + 5 9 = 3 2,
+ 3 + 5 + 7 16 = 4 2,
We note that each number on the right is the square of the number of odd
numbers appearing on the left; thus in the fourth line, there are four numbers
on the left side, and the right side is 42• The general result which these cal
culations suggest is that if the first " odd numbers were on the left side, then
the sum would be ,,2. Of course, this possible theorem is not proved by the
THE CREATION OF MATHEMATICS 53
above calculations. Nor could it ever be proved by such calculations, for no
mortal man could make the infinite set of computations required to establish
the conclusion for every n. The calculations do, however, give the mathe
matician something to work on.
These simple illustrations of how observation, measurement, and calcu
lation suggest possible theorems are not too striking or very profound. We
shall see in the course of later work how physical problems suggest more
significant mathematical theorems. However, experience, measurement, cal
culation, and generalization do not include the most fertile source of possible
theorems. And it is especially true in seeking methods of proof that more
than routine techniques must be utilized. In both endeavors the most im
portant source is the creative act of the human mind.
IJ � C Fig. 3-7
Let us consider the matter of proof. Suppose one has discovered by
measurements that the sum of the angles of various triangles is 1 800• One
must now prove this result deductively. No obvious method will do the job.
Some new idea is required, and the reader who remembers his elementary
geometry will recall that the proof is usually made by drawing a line through
one vertex (A in Fig. 3-7) and parallel to the opposite side. It then turns out
as a consequence of a previously established theorem on parallel lines that the
angles I and 2 are equal, as are the angles 3 and 4. However the angles I, 3,
and the angle A of the triangle itself do add up to 1 80", and so the same is true
for the angles of the triangle. This method of proof is not routine. The idea
of drawing the line through A must be supplied by the mind. Some methods
of proof seem so devious and artificial that they have provoked critical com
ments. The philosopher Arthur Schopenhauer called Euclid's proof of the
Pythagorean theorem "a mouse-trap proof" and "a proof walking on stilts,
nay, a mean, underhand proof."
The above example has been offered to emphasizc the fact that ingenious
mathematical work must be done in finding methods of proofs even after the
question of what to prove is disposed of. In the search for a method of proof,
as in finding what to prove, the mathematician must use audacious imagination,
insight, and creative ability. His mind must see possible lines of attack where
others would not. In the domains of algebra, calculus, and advanced analysis
especially, the first-rate mathematician depends upon the kind of inspiration
that we usually associate with the creation of music, literature, or art. The
composer feels that he has a theme which when properly developed will pro-
54 LOGIC AND j\JATHEi\IATICS
duce true music. Experience and a knowledge of music aid him in arriving at
this conviction. Similarly, the mathematician surmises that he has a conclusion
which will follow from the axioms of mathematics. Experience and knowledge
may guide his thoughts into the proper channels. Modifications of one sorr or
another may he required before a correct proof and a satisfactory statement of
the new theorem are achieved. But essentially both mathematician and com
poser are moved by an afflatus which enables them to see the final edifice
before a single stone is laid.
We do not know just what mental processes may lead to correct insight
any more than we know how it was possible for Keats to write fine poetry
or why Rembrandt was able to turn out fine paintings. One might say of
mathematical creation what P. W. Bridgman, the noted physicist, has said of
scientific method. that it consists of "doing one's damnedest with one's mind,
no holds barred." There is no logic or infallible guide which tells the mind
how to think. The vcry fact that many great mathematicians have tackled a
problem and failed and that another comes along and solves it shows that the
mind has something to contribute.
The preceding discussion of the creation of mathematics should correct
several mistaken popular impressions. When creating a mathematical proof,
the mind does not see the cold, ordered arguments which one reads in texts,
but rather it perceives an idea or a scheme which when properly formulated
constitutes the deductive proof. The formal proof, so to speak, merely sanc
tions the conquest already made by the intuition. Secondly, the deductive
proof is not the preferable form by which to grasp the idea or method em
ployed. In fact the deductive argument often conceals the idea because the
logical form is not perspicuous to the intuition. At the very least the details
of the arguments obscure the main threads. The value of the deductive or
ganization of the proof is that it enables the creator and the reader to test the
arguments by the standards of exact reasoning. Thirdly, there is the prevalent
but mistaken notion that scientists and mathematicians must keep their minds
open and unbiased in pursuing an investigation. They are not supposed to
prejudge the conclusion. Actually the mathematician must first decide what
to prove, and this conclusion not only does bur must precede the search for
the proof, or else he would not know where to head. This is not to say that
the mathematician may not sometimes make a false conjecture. If he does, his
search for a proof will fail or in the course of the search he will realize that
he cannot prove what he is after, and he will correct his conjecture. But in
any case he knows what he is trying to prove.
EXERCISES
I. Consider the parallelogram ABeD (Fig. 3-8). By definition, the opposite sides
are parallel. Now introduce the diagonal BD. Does observation suggest a
possible theorem relating the triangles ABD and BDC?
THE CREATION OF MATHEJ\lATICC;; 55
Ar-------� B A _---;;;E�_:_-,U
F
H
---'
/),""'---c D L- -- -- � -- -- � C
-- � -- -- --
G
--
Fig. 3-8 Fig. 3-9
2. Consider any quadrilateral ABeD (Fig. 3-9) and the figure formed by joining
the mid-points E, F, G, H of the sides of the quadrilateral. Does observation or
intuition suggest any significant fact about the quadrilateral EFGH?
3. The formula n2 - n + 41 IS supposed to yield primes for various values of n.
Thus when n = I,
1' - 1 + 41 = 41,
and this is a prime. When n = 2,
2' - 2 + 41 = 43,
and this is a prime. Test the formula for n = 3 and n = 4. Are the resulting
values of the formula primes? Have you proved, then, that the formula always
yields primes?
4. Can you specify conditions under which two yuadrilaterals will be congruent,
that is, have the same size and shape?
5. The following lines show some calculations with the sum of the cubes of whole
numbers:
13 = I,
1 3 + 23 = 1 + 8 = 9 = 32 = (I + 2) 2,
1 3 + 23 + 3 3 = 1 + 8 + 27 = 36 = 6 2 = (1 + 2 + 3) 2
13 + 2 3 + 3 3 + 43 = 1 + 8 + 27 + 64 = 100 = 10 2
= (1 + 2 + 3 + 4) 2.
What generalization do these few calculations suggest?
REVIEW EXERCISES
1. What basis did the Egyptians and Babylonians have for believing in the cor
rectness of their mathematical conclusions?
2. Compare Greek and pre-Greek understanding of the concepts of mathematics.
3. What was the Greek plan for establishing mathematical conclusions?
56 LOGIC AND MATHEMATICS
4. ]n what sense is mathematics a creation of the Greeks rather than of the
Egyptians and the Babylonians?
5. Suppose we accept the premises that all professors are intelligent people and
all professors are learned people. Which of the following conclusions is
validly deduced?
a ) Some intelligent people are learned.
b) Some learned people are intelligent.
c ) All intelligent people are learned.
d ) All leamed people are intelligent.
6. Suppose we accept the premises that all college srudents are wise, and no pro
fessors are colIege students. Which of the following conclusions is validly
deduced?
a ) No professors are wise.
b) Some professors are wise.
c ) All professors are wise.
7. ]s the foIIowing argument valid?
All parallelograms are quadrilaterals, and figure ABCD is a quadrilateral.
Hence figure ABCD is a parallelogram.
S. What conclusion can you deduce from the premises,
Every successful srudent must work hard,
and
John does not work hard?
9. Smith says,
If it rains I go to the movies.
If Smith went to the movies, what can you conclude deductively?
10. Smith says,
] go to the movies only if it rains.
If Smith went to the movies, what can you conclude deductively?
Topics for Further Investigation
To pursue any of these topics use the books listed below under Recommended
Reading.
1. The life and work of the Pythagoreans
2. The life and work of Euclid
Recommended Reading
BELL, ERIC T.: The Development of Mathematics, 2nd ed., Chaps. 2 and 3, McGraw
Hill Book Co., N.Y., 1945.
BELL, ERIC T.: Men of Mathematics, Simon and Schuster, New York, 1937.
THE CREATION OF MATHEMATICS 57
CLAGETT, MARSHALL: Greek Science in Antiquity, Chap. 2, Abelard-Schuman, Inc.,
New York, 1955.
COHEN, MORRIS R. and E. !·..J"AGEL: An Introduction to Logic and Scientific Method,
Chaps. 1 through 5, Harcourt Brace and Co., New York, 1934.
COOLIDGE, j. L.: The Mathematics of Great Amateurs, Chap. 1, Dover Publications,
Inc., New York, 1963.
HAMILTON, EDITH: The Greek Way to Western Civilization, Chaps. 1 through 3,
The New American Library, New York, 1948.
JEANS, SIR JAMES: The Growth of Physical Science, 2nd ed., Chap. 2, Cambridge
University Press, Cambridge, 195 1 .
NEUGEBAUER, OTTO: The Exact Sciences in Antiquity, Princeton University Press,
Princeton, 1952.
SMITH, DAVID EUGENE: History of Mathematics, Vol. 1., Dover Publications, Inc.,
New York, 1958.
STRUIK, D IR K j.: A Concise History of Mathematics, Dover Publications, Inc., New
York, 1948.
TAYLOR, HENRY OSBORN: Ancient Ideals, 2nd ed., Vol. I, Chaps. 7 through 13, The
Macmillan Co., New York, 1913.
WEDBERG, ANDERS: PLato's Philosophy of Mathematics, Almqvist and Wiksell, StOck
holm, 1955 ( for students of philosophy ) .
CHAPTER 4
N U M BER: THE FUNDAM ENTAL CONCEPT
A marvelous neutrality have these things Mathematical, and also a strange
participation between things supernatural, immortal, intellectual, simple,
and indivisible, and things natural, mortal, sensible, compounded and
divisible.
JOHN DEE ( 1 527-1608)
4-1 INTRODUCTION
Just as we are inclined to accept the sun, moon, and stars as our birthright and
do not appreciate the grandeur, the mystery, and the knowledge which can be
gleaned from the contemplation of the heavens, so are we inclined to accept
our number system. There is, however, this difference. Many of us would
not claim the latter and would gladly sell it for a mess of pottage. Because we
ace forced to learn about numbers and operations with numbers while we are
still too young to appreciate them-a preparation for life which hardly excites
our interest in the future-we grow up believing that numbers are drab and
uninteresting. But the number system warrants attention not only as the basis
of mathematics, but because it contains weighty and beautiful ideas which lend
themselves to powerful applications.
Among past civilizations, the Greeks best appreciated the wonder and
power of the concept of number. They were, of course, a people with great
intellectual perception, but perhaps because they viewed numbers abstractly,
they saw more clearly their true nature. The very fact that one can abstract
from many diverse collections of objects a property such as "fiveness" struck
the Greeks as a marvelous discovery. If one may use the ridiculous to ac
centuate the sublime, one may say that the Greek delight in numbers was the
rational counterpart of the hysteria which many young and old Americans
experience when they encounter numbers in the form of baseball scores and
batting averages.
4-2 WHOLE NUM BERS AND FRACTIONS
The first Greeks who, to our knowledge, expressed their satisfaction with
numbers and propounded a philosophy based on numbers which is extremely
58
WHOLE NUMBERS AND FRACTIONS 59
alive and vital today were the Pythagoreans. This group was founded in the
middle of the sixth century B.C. by Pythagoras. We know rather little that is
certain about this man. However, it seems very likely that he was born in
569 B.C. in a Greek settlement on the island of Samos in the Aegean Sea. Like
many other Greeks he traveled to Egypt and to the Near East to learn what
these older civilizations had to offer, and then settled in Croton, another Greek
city in southern Italy. Pythagoras and his followers were among the early
founders of the great Greek civilization, and so it is not surprising to find that
the rational attitude which characterizes the Greeks was still surrounded in his
times with mystical and religious doctrines prevalent in Egypt and its eastern
neighbors. In fact the Pythagoreans were a religious sect as well as students
of philosophy and nature.
Membership in the group was restricted, and the members were pledged
to secrecy. Among their religious doctrines was the belief that the soul was
tainted by the body. To purify the soul they maintained celibacy; their
religious practices were also supposed to be efficacious in purifying the soul.
At death the soul was reincarnated in another human or an animal. Like most
mystics they observed certain taboos. They would not touch a white cock,
walk on the highways, use iron to stir a fire, or leave the marks of ashes on a
pot.
The secrecy of the group, its aloofness, and an attempted interference in
the political affairs of Croton finally aroused the people of this city to drive
out the Pythagoreans. We do not know for certain what happened to Pythag
oras. One story has it that he fled to Metapontum, another Greek city in
southern Italy, and was murdered there. However, the Pythagoreans con
tinued to be influential in Greek intellectual life. One of their notable mem
bers was the philosopher Plato.
The Pythagoreans were impressed with numbers and, because they were
mystics, attached to the whole numbers meanings and significances which we
now regard as childish. Thus, they considered the number "one" as the es
sence or very nature of reason, for reason could produce only one consistent
body of doctrines. The number "two" was identified with opinion, clearly
because the very meaning of opinion implies the possibility of an opposing
opinion, and t:ms of at least two. "Four" was identified with justice because
it is the first number which is the product of equals. Of course, one can also
be thought of as 1 times 1, but to the Pythagoreans one was not a number in
the full sense because it did not represent quantity. The Pythagoreans repre
sented numbers as dots in sand or by pebbles, and for each number the dots
or pebbles had a special arrangement. Thus the number "four" was pictured as
four dots suggesting a square, and so the square and justice were also linked.
Foursquare and square shooter still mean a person who acts justly. "Five"
signified marriage because it was the union of the first masculine number, three,
and the first feminine number, two. (Odd numbers were masculine and even
60 NUMBER: THE FUNDAMENTAL CONCEPT
numbers feminine.) The number "seven" represented health and "eight,"
friendship or love.
We shall not pursue all the ideas which the Pythagoreans developed about
numbers. What is significant about their work is that they were the first to
study properties of whole numbers. As we shall see in a later chapter, they also
possessed the vision of deep mystics and saw that numbers could be used to
represent and even embody the essence of natural phenomena.
The speculations and results obtained by the Pythagoreans about whole
numbers and ratios of whole numbers, or fractions as we prefer to call them,
were the beginning of a long and involved development of arithmetic as a
science as opposed to arithmetic as a tool for daily applications. During the
2500 years since the Pythagoreans first called attention to the importance of
numbers, man has not only learned to better appreciate the idea but has in
vented excellent methods of writing quantity and of performing the four
operations of arithmetic, i.e., ambition, distraction, uglification, and derision,
as Lewis Carroll called them. While these methods of writing and operating
with numbers are largely familiar, there are a few facts which are worthy of
comment.
One of the most important members of our present number system is the
mathematical representation of no quantity, that is, zero. We are accustomed
to this number and yet usually fail to appreciate two facts about it. The first
is that this member of our number system came rather late. The idea of using
zero was conceived by the Hindus and, like other of their ideas, reached
Europe through the Arabs. It had not occurred to earlier civilizations, even
to the Greeks, that it would be useful to have a number which represents the
absence of any objects. Connected with this late appearance of the number is
the second significant fact, namely, that zero must be distinguished from noth
ing. Undoubtedly it was the inability of earlier peoples to perceive this dis
tinction which accounts for their failure to introduce the zero. That zero
must be distinguished from nothing is easily seen from several examples. A
student's grade in a course he never rook is no grade or nothing. He may,
however, have the grade of zero in a course he has taken. If a person has no
account in a bank, his balance is nothing. If he has a bank account, he may
very well have a balance of zero.
Because zero is a number, we may operate with it; for example, we may
add zero to another number. Thus 5 + 0 5 . By contrast 5 + nothing is
=
meaningless or nothing. The only restriction on zero as a number is that one
cannot divide by zero. Division by zero does, so to speak, produce nothing.
Because so many false steps in mathematics result from division by zero, it is
well to understand clearly why we cannot do this. The answer to a problem
of division, say !, is some number which when multiplied by the divisor
yields the dividend. In our example, 3 is the answer because 3 . 2 = 6. Hence
WHOLE NUMBERS AND FRACfIONS 61
the answer to � should be a number which when multiplied by 0 gives 5.
However, any number multiplied by 0 gives 0 and not 5. Thus, there is no
answer to the problem of �. In the case of * the answer should be some
number which when multiplied by 0 yields the dividend O. However, any
number may then serve as a quotient because any number multiplied by 0 gives
O. But mathematics cannot tolerate such an ambiguous situation. If � arises
and any number may serve as an answer, we do not know what number to
take and hence are not aided. It is as if we asked a person for directions to
some place and he replied, Take any direction.
With the availability of zero, mathematicians were finally able to develop
our present method of writing whole numbers. First of all we count in units
and represent large quantities in tens, tens of tens, tens of tens of tens, etc.
Thus we represent two hundred and fifty-two by 252. The left-hand 2 means,
of course, twO tens of tens; the 5 means 5 times 10; and the right-hand 2 means
2 units. The concept of zero makes such a system of writing quantities prac
tical since it enables us to distinguish 22 and 202. Because ten plays such a
fundamental role, our number system is called the decimal system, and ten is
called the base. The lise of ten resulted most likely from the fact that man
counted on his fingers and, when he had used the fingers on his two hands,
considered the number arrived at as a larger unit.
Because the position of an integer determines the quantity it represents,
the principle involved is called positional notation. The decimal system of
positional notation is due to the Hindus; however, the same scheme was used
two millenniums earlier by the Babylonians, but with base 60 and in more
limited form since they did not have zero.
The operations of arithmetic, addition, subtraction, multiplication, and
division, are of course familiar to us, but it is perhaps not recognized that
these operations are quite sophisticated and remarkably efficient. They date
back to Greek times and gradually evolved, as improvements in the methods
of writing numbers and the concept of zero were introduced. The Europeans
picked up the methods from the Arabs. Previously the Europeans had used
the Roman system of writing numbers, and the operations were based on that
system. Partly because these latter methods were relatively cumbersome and
partly because education was limited to a few people, those who acquired the
art of calculation were regarded as skilled mathematicians. In fact the pro
cesses defied the average man so much that it seemed to him that those pos
sessed of the ability must have magical powers. Good calculators were called
practitioners of the "Black Art."
To appreciate the efficiency of our present methods we would have to
learn the older ones and even acquire some facility in them, to make the
comparison a fair one. But we cannot spare the time and effort. Perhaps the
one point we should emphasize is how much our methods of arithmetic depend
62 NUMBER: THE FUNDAMENTAL CONCEPT
upon positional notation. This can be seen even in a simple problem of ad
dition. To add 387 and 3 59 say, the written work is
387
359
746
However, in performing this work, we think as follows. We add the units
7 and 9, the "tens" quantities 8 and 5, and the "hundreds" 3 and 3, separately.
When we add the 7 and 9, we obtain 16. We recognize that 16 is 1 · 10 + 6,
and so we add the I · 1 0 to the 1 3 · 1 0 already obtained from the 8 and 5. We
say that we "carry" the I · 1 0 over, and instead of I l · 10 we obtain 14 · 10.
However, 14 · 10 is ( 1 0 + 4) · 1 0 or 1 · 10' + 4 · 10. Thus we write 4 in the
tens' column, add the I · 10' to the 6 · 10' already obtained from the 3 and 3,
and arrive at 7 · 10'. All these steps are usually executed rather mechanically
by writing the appropriate numbers in the units', tens', and hundreds' places
and by using the process called carrying. Were we to analyze the processes
of subtraction, multiplication, and division, we would again see how the steps
which we learn mechanically in elementary school are just the skeletal pro
cesses of thinking suited to positional notation in base ten.
A word about fractions may also be in order. The natural method of
writing fractions, for example, ! or �, to express parts of a whole presents no
difficulties of comprehension. However the operations with fractions do seem
to be somewhat arbitrary and mysterious. To add � and t, say, we go through
the following process:
i + t = H + H = H'
What we have done is to express each fraction in an equivalent form such
that the denominators are now alike, and then add the numerators. We are
not required by law to add fractions in this manner. It would, of course, be
much simpler if we agreed to add fractions by adding the numerators and
adding the denominators so that
�+t= l
As a matter of fact, when we multiply two fractions, we do multiply the
numerators and multiply the denominators so that it does seem as though the
mathematicians prefer to be unnecessarily complicated about the addition of
fractions.
The explanation of this seeming mathematical idiosyncrasy is simple: the
operations with fractions are formulated to fit experience. When one has t
of a pie and t of a pie, he has in all not ! but H of a pie. In other words, if
mathematical concepts and operations are to fit experience, the nature of the
operations is forced upon us. In the case of multiplication of fractions it is
WHOLE NUMBERS AND FRACTlONS 63
correct that multiplying the numerators and multiplying the denominators will
yield the fraction which represents the physical result. Thus suppose we had
to find � of t, that is � + We think of i as 2 . }. Now
1 , 7 _ L ll _ -'L
"3 '5 - "3 15 -
IS'
Then
1 5 7 _
2 . "3 . 1. - 2 . 15 - ll
n'
The same result is obtained by multiplying the original numerators and multi
plying the original denominators.
The operation of dividing one fraction by another presents a little more
difficulty, To see how we arrive at the correct process let us start with some
simple examples. Suppose we had to answer the question of how many one
thirds of a pie are in 2 pies. Mathematically this question is formulated as how
much is
We should note that one bar is larger than the other and the longer bar sepa
rates the numerator, the 2, from the denominator � Now, we know on phys
..
ical grounds that we can obtain 6 one-thirds from 2 pies. We can obtain this
answer arithmetically hy inverting the denominator ! and multiplying the
inverse into the numerator 2. That is,
2
f'f = t = 6.
!
Now let us complicate the problem slightly. How many two-thirds of a
pie are contained in 2 pies? Again this question is formulated mathematically
as
2
We know on physical grounds that there are 3 two-thirds of a pie in 2 pies.
We can obtain this answer arithmetically by inverting the denominator and
multiplying this inverse fraction into the numerator. Thus,
2
- = f'4 = ! = 3.
�
Now let us comRlicate the problem still more. We would certainly agree
that 2 pies are the same as .If pies. If thetefore we had to answer the question
of how many two-thirds of a pie are contained in .1.f pies, we would know
64 NUMBER: THE FUNDAMENTAL CONCEPT
from the preceding example that the answer is 3. How could we obtain this
answer directly? The question is, how much is
Let us invert the denominator and multiply the inverse into the numerator.
Thus
Again we see that the process of inverting the denominator and multiplying it
into the numerator gives the result which we know on physical grounds is
correct.
The significant point, then, is that the rule "to divide one fraction by an
other, invert the denominator and multiply this inverse into the numerator"
is designed to make the mathematical operation give a result which fits ex
perience. This is, of course, the same principle which applies to the other
operations. Logically, we may say that we define the operations to be what
we have just illustrated for addition, multiplication and division, and in our
purely mathematical definitions we do not say anything about agreement with
physical facts. Bur, of course, the definitions would be pointless if they did
not give physically correct results.
Fractions, like the whole numbers, can be written in positional notation.
Thus
_ _
2_
1 _ _� _ _ "-"- + ---'--- 2 + 5 .
'4 - T01)" - 1lITi - 10 TIm
_
1lITi
If we now agree to suppress the powers of 10, that is 10, 100, and higher
powers where they occur, then we can write :l 0.25. The decimal point
=
reminds us that the first number is really �, the second Th, and so forth.
The Babylonians had employed positional notation for fractions, but they
used base 60 rather than base 10, just as they had for whole numbers. The
decimal base for fractions was introduced by sixteenth-century European
algebraists. Of course, the operations with fractions can also be carried out in
decimal form.
The disappointing feature of the decimal representation of fractions is that
some simple fractions cannot be represented as decimals with a finite number
of digits. Thus when we seek to express l as a decimal, we find that neither
0.3, nor 0.33, nor 0.3 3 3, and so on, suffices. All one can say in this and similar
cases is that by carrying more and more decimal digits one comes closer and
closer to the fraction, but no finite number of digits will ever be the exact
answer. This fact is expressed by the notation
i = 0. 3 3 3 . . . ,
IRRATIONAL NUMBERS 65
where the dots indicate that we must keep on adding threes to approach the
fraction * more and more closely.
From the standpoint of applications the fact that some fractions cannot
be expressed as decimals with a finite number of digits does not matter because
we can always carry enough digits to obtain an answer as accurate as the
application requires.
EXERCISES
1. What is the principle of positional notation?
2. Why is the number zero almost indispensable in the system of positional notation?
3. What is the meaning of the statement that zero is a number?
4. What two methods are there of representing fractions?
5. What principle determines the definitions of the operations with fractions?
4-3 IRRATIONAL NUMBERS
The Pythagoreans, as we noted earlier, were the first to appreciate the very
concept of number, and sought to employ numbers to describe the basic
phenomena of the physical and social worlds. Numbers to the Pythagoreans
were also interesting in and for themselves. Thus they liked square numbers,
that is, numbers such as 4, 9, 16, 25, 36, and so on, and observed that the sums
of cerrain pairs of square numbers, or perfect squares, are also square numbers.
For example, 9 + 16 25, 25 + 144 169, and 36 + 64 lOa. These relation
= = =
ships can also be written as
3 2 + 42 = 5 2, 5 2 + 1 22 = [ J 2, and 6 2 + 82 = ! O2 .
The three numbers whose squares furnish such equalities are today called
Pythagorean triples. Thus 3, 4, 5 constitute a Pythagorean triple because
3 2 + 4' = 5'.
The Pythagoreans liked these triples so much
n
because, among other features, they have an inter
esting geometrical interpretation. If the two smaller
numbers are the lengths of the sides or arms of a
right triangle, then the third one is the length of the 3
hypotenuse (Fig. 4-1 ) . Just how the Pythagoreans 13 12
knew this geometrical fact is not clear, but assert it
they did. They also claimed that in any right tri
angle, the square of the length of one arm added to
the square of the length of the other gives the square
of the length of the hypotenuse. This more general 5
assertion is still called the Pythagorean theorem and Fig. 4-1
66 NUMBER: THE FUNDAMENTAL CONCEPT
a proof of it, such as we learn in high-s�hool geometry, was given about 200
years later by Euclid. Pythagoras is said to have been so overjoyed with this
theorem that he sacrificed an ox to celebrate its discovery.
This theorem proved to be the undoing of a central doctrine in the
Pythagorean philosophy and caused woe and misery to many mathematicians.
But before we pursue this story, we should look into a few simple properties
of the whole numbers which are embodied in the following exercises.
EXERCISES
I. Prove that the square of any even number is an even number. [Suggestion: By
definition every even number contains 2 as a factor.]
2. Prove that the square of any odd number is an odd number. [Suggestion: Every
odd number ends in I, 3, 5, 7, or 9.]
3. Let a stand for a whole number. Prove that if a2 is even, then a is even. [Sug
gestion: Use the result in Exercise 2.]
4. Establish the truth or falsity of the assenion that the sum of any two square
numbers is a square number.
There are tragedies in mathematics also, and one of these struck the very
group of mathematicians who deserved a better fate. The Pythagoreans had
constructed, a� least to their own satisfaction, a philosophy which asserted that
all natural phen0mena and all social and ethical concepts were in essence just
whole numbers or relationships among whole numbers. But one day it oc
curred to a member of the group to examine the seemingly simplest case of the
Pythagorean theorem. Suppose each arm of a right triangle (Fig. 4-2) is 1
unit in length; how long, he asked, is the hypotenuse? The Pythagorean
theorem says that the square of (the length of) the hy-
potenuse equals the sum of the squares of the arms.
Hence if we call c the unknown length of the hypote-
nuse, then the theorem says that v2
(' = 1 ' + 1 '
or
Fig. 4-2
Now 2 is not a square number, that is, a perfect square, and so c is not a
whole number. But it certainly seemed reasonable to this Pythagorean that
c should be a fraction; that is; there should be a fraction whose square is 2.
Even the simple fraction .g. comes close to being the correct value because
(t )' = H, and this is almost 2. However, simple trial does not easily yield a
fraction whose square is 2. Hence this Pythagorean became worried, and he
decided to investigate the question of whether there is a fraction whose square
IRRATIONAL NUMBERS 67
is 2. We shall examine his reasoning which, as far as we know, is the same as
that given in Euclid's famous work on geometry, the Elements.
The number c which we seek to determine is one whose square is 2. Let
us denote it by Vi. All we mean by this symbol is that it represents a number
whose square is 2. And now let us suppose that Vi is a fraction a/ b, where a
and b are whole numbers. Moreover, to make matters simpler, let us suppose
that any factors common to a and b are cancelled. Thus if a/ b were t, for
example, we would cancel the common factor 2 and write it as !. Hence we
have assumed so far that
(I )
and that a and b have no common factors.
If equation ( 1 ) is correct, then by squaring both sides, a step which utilizes
the axiom that equals multiplied by equals give equals (because we multiply
the left side by' Vi and the right side by a/ b), we obtain
a2
2 =
b2
Again by employing the axiom that equals multiplied by equals yield equals,
we may multiply both sides of this last equation by b2 and write
2b2 = a2• (2)
The left side of this equation is an even number because it contains 2 as a
factor. Hence the right side must also be an even number. But if a2 is even,
then, according to Exercise 3 above, a must be even. If a is even, it must
contain 2 as a factor. That is, a 2d, where d is some whole number. If we
=
substitute this value of a in (2) we obtain
2b2 = ( ld) 2 = 2d · 2d = 4d2. (3)
Since then
2b2 = 4d2,
we may divide both sides of this equation by 2 and obtain
b2 = 2d2• (4)
We now see that b2 is an even number and so, by again appealing to the result
in Exercise 3, we find that b is an even number.
What we have shown in the above argument is that if Vi a/ b, then a =
and b must be even numbers. But at the very outset we had cancelled any
common factors in a and b; yet we find that a and b still contain 2 as a com-
68 NUMBER: THE FUNDAMENTAL CONCEPT
mon factor. This result contradicts the fact that a and b have no common
factors.
Why do we arrive at a contradiction? Since our reasoning is correct, the
only possibility is that the assumption that V2 equals a fraction is not correct.
In other words. VI cannot be a ratio of two whole numbers.
This proof is so ncat that one can almost believe the legend that Pythag
oras sacrificed an ox in honor of its creation. But there are at least two reasons
for discrediting this tale. The first is that if all the legends telling of Pythagoras
sacrificing an ox were true, he could not have had time for mathematics. The
second reason is that the above proof was not a triumph for the Pythagoreans
but a disaster. The symbol V2 is a number because it represents the length
of a line, namely the hypotenuse of the triangle in Fig. 4--2 . But this number
is not a whole number or a fraction. The Pythagoreans had, however, devel
oped an embracing philosophy which asserted that everything in the universe
reduced to whole numbers. Clearly, then, this philosophy was inadequate.
Indeed the existence of numbers such as V2 was such a serious threat to the
Pythagorean philosophy that another legend, more credible, states that the
Pythagoreans. who were at sea when the above discovery was made, threw
overboard the member who made it, and pledged to keep the discovery secret.
But secrets will out, and later Greeks not only learned that V2 is neither
a whole number nor a fraction, but they discovered that there is an indefinitely
large collection of other numbers which are not whole numbers or fractions.
Thus YJ, ys, 0, and, more generally, the square root of any number which
is not a perfect square. the cube root of any number which is not a perfect
cuhe, and so on, are numbers which are not whole numbers or fractions. The
number 7f, which is the ratio of the circumference of any circle to its diameter,
is also neither a whole number nor a fraction. All these new numbers are called
irrational numbers, the word "irrational" now meaning that these numbers
cannot be expressed as ratios of whole numbers, although in Pythagorean times
it meant unmentionable or unknowable.
If these irrational numbers are really so common and represent lengths of
sides of triangles and circumferences of circles in terms of the diameters, why
weren't they encountered before? Didn't the Babylonians and Egyptians run
across them? They did. But since they were concerned only with having
numbers serve their practical purposes, they used convenient approximations.
Thus, when they encountered a length such as yfi, they were content to use a
value such as 1 .4 or 1.41. For 7f, as we noted in :?on earlier chapter, they used
values even as crude as 3. Not only did these peoples use such approximations,
but they never realized that the most complicated fraction or decimal could
never represent an irrational number exactly. The Egyptians and Babylonians
treated irrational numbers and their mathematics in general rather lightheart
edly. We may hail their blithe spirits, but mathematicians they never were.
IRRATIONAL NUMBERS 69
The Greeks, as we know, were of a different intellectual breed and could
not be content with approximations, but they also exhibited a weakness. Al
though they recognized that quantities exist which are neither whole numbers
nor fractions, they were so convinced that the concept of number could not
comprise anything else than whole numbers or fractions that they did not
accept irrationals as numbers. Instead they thought of such quantities only as
geometrical lengths or areas. Thus the Greeks never did develop an arithmetic
of irrational numbers. In their astronomical work, for example, they used only
whole numbers and fractions. The difficulty which the Greeks experienced
also baffled all mathematicians up to modern times. The greatest mathema
ticians refused to accept irrationals as numbers and followed the Greek pro
cedure of thinking about such quantities as lengths or areas. All these people
wished that the Pythagoreans had thrown all irrational numbers overboard
rather than the man who discovered them.
But the needs of society often oblige even mathematicians to face un
pleasantnesses. In the seventeenth century, science began to develop at an
amazing rate, and science needs quantitative results. It may be nice to know
that Vi is a certain length and that Vi, v'l is an area, but this knowledge does
not suffice when one needs numerical results. And so finally mathematicians
had to accept the fact that if they were to treat numerically all the quantities
that arise in scientific work, they must handle irrational numbers as numbers.
The mathematicians' refusal over centuries to grant irrationals the status of
numbers illustrates one of the surprising features of the history of mathematics.
New ideas are often as unacceptable in this field as they are in politics, religion,
and economics.
The situation, then, which must be faced squarely is that there are other
numbers besides whole numbers and fractions. It is, of course, quite under
standable that whole numbers and fractions should have been created and used
first, for these numbers arise in the simplest physical situations man encounters.
The irrational numbers on the other hand are not commonly encountered.
Only the application of a theorem such as the Pythagorean theorem brings
them to our attention, and even then one must go through a proof such as that
examined above, to see that they are not whole numbers or fractions. But the
fact that irrational numbers are late-comers does not mean that they are less
acceptable or less genuine numbers. Just as we gradually add to our knowledge
of the varieties of human beings and animals which exist in our physical world,
so must we broaden our knowledge of the varieties of numbers and with true
liberality accept these strangers on the same basis as the already familiar
numbers.
However. if we are to use irrational numbers, we must know how to
operate with them, that is, how to add, subtract, multiply, and divide them.
We have already noted with whole numbers and fractions that if we wish
70 NUMBER: THE FUNDAMENTAL CONCEPT
the operations to fit experience, we must formulate the operations accord
ingly. So it is with the irrational numbers. We could define addition, multi
plication, and the other operations as we please. But if we wish these opera
tions to represent physical situations, we must define them properly. However,
there is no real difficulty here. Since irrational numbers are quantities, as are
whole numbers and fractions, we may use the latter as a guide to the proper
operations with irrational numbers.
Let us consider a few examples which will be sufficient to indicate the
general principles. Should we say that
V2 + v'J = 0?
To answer this question let us consider the analogous Question: May we say
that
y'4 + V9 = VTJ?
It is clear in the latter case that 2 + l does not equal v'fi, for v'fi is certainly
less than 4. Hence we should not add the radicands, that is, the 2 and the l,
in the preceding equation. One might then ask, How much is Vi + 0? Since
both summands are numbers, the sum is also a number, but it cannot be written
more compactly than Vi + 0. This inability to combine the summands is
not something new or troublesome. When we add 2 and t, for example, the
answer continues to be 2 + t. We usually omit the plus sign and write 2t,
but the summands are really not combined.
Let us consider next whether
Here too we shall see what the analogous operation with whole numbers sug
gests. Is it true that
y'4 . V9 = V36?
The answer is clearly yes, and so we shall agree that to multiply square roots
we shall multiply the radicands. That is,
V2 . v'J = 0.
The definitions of the operations of subtraction and division are also readily
determined. Thus 0 - Vi yields a definite number, but the difference cannot
be written any more compactly than 0 - Vi.
For division, say 0/0, the procedure. as in the case of multiplication. is
suggested by observing that
IRRATIONAL NUMBERS 71
for this equation simply says that � � l Hence we shall agree that
The general principle which these examples illustrate is that operations
with irrational numbers are defined so as to agree with the same operations on
whole numbers when the latter are expressed as roots. We could state our
definitions in general form. but there is no need [Q do so.
In applications we often approximate irrational numbers by fractions or
decimals because actual physical objects cannot be constructed exactly anyway.
Thus if we had to construct a length which strictly should be 0, we would
approximate "ff. Since ( 1 .4) 2 = 1.96 and 1 .96 is nearly 2, we could approx
imate 0 by 1.4. If we desired a more accurate approximation, we might de
termine to the nearest hundredth the number whose square approximates 2.
Thus, since
(1.4 1 ) 2 = 1 .988 and (1 .42) 2 = 2 .0 1 6,
we see that 1.41 is a good two-decimal approximation of "ff. We could, of
course, improve still more on the accuracy of the approximation. We should.
however, realize that no matter how many decimal places we employed, we
would never obtain a number which is exactly V2 because any decimal with
a finite number of digits or a whole number plus such a decimal is just another
way of writing a fraction, whereas 0, as the above proof showed, can never
equal a quotient of two whole numbers.
The fact that we often approximate an irrational number when we wish to
construct something raises a question which merits an answer. The question
is, Why don't we approximate irrational numbers wherever they arise and
forget about operations with irrationals as such? For example. to calculate
0 · 0, we could approximate 0 by, say 1.41, approximate 0 by 1.73, and
then multiply 1.41 by 1.73. The answer is 2.44, and since (2.44)' is 5.95, we
see that we have a good approximation to V6. If we wanted a more accurate
answer, we could approximate 0 and Yl more closely and then multiply.
One reason we do not approximate in mathematics proper is that mathematics
is an exact science. It insists on reasoning as rigorous as human beings can
perform. We pay a price for this rigor by expending more thought and effort,
but we shall see that mathematics has made its contributions just because it
insists on exactness.
There is also a practical advantage in working with irrational numbers as
such. ' Let us suppose that some problem required us to calculate (Yl) 4, that
is, 0 · 0 · 0 . 0. The person who insists on approximating would now
approximate Yl to some number of decimal places, for example, 1.732, and
then calculate ( 1.732)4 While the practical person takes an hour to calculate
72 NUMBER: THE FUNDAMENTAL CONCEPT
and check his arithmetic, the mathematician would see at once that
0 · 0 · 0 · 0 = ( 0 · 0) (0 · 0) = 3 · 3 = 9,
and could spend the rest of the hour in refreshing sleep. Moreover, the mathe
matician's answer is exact, whereas the practical man's answer is not accurate
even to the four demical places with which he started, because the product of
two approximate numbers is less accurate than either facror. To achieve an
answer accurate to four decimal places. the practical man would have to use
an approximation of 0 containing seven decimal places and then multiply.
The irrational number is the first of many sophisticated ideas which the
mathematician has introduced to think about and cope with the real world.
The mathematician creates these concepts, devises ways of working with
them which fit real situations, and then uses his abstractions to think about
the phenomena to which the ideas apply.
EXERCISES
1. Express the answers to the following problems as compactly as you can:
a) 0 + V5 b) 0+0 c) 0 + v'7 d) v7 + v'7
e) 0 · v'7 f) 0 · .;IS g) 0 · V'4 h) Vi1 . vI
V5 .) VB
i) k) �
v2 J v2 o
2. Simplify the following:
a) v'sO b) v200 c) v05
[Sugg,stion: V 50 = v25 . 2 = VTI · V2.]
3. Criticize the foIIowing argument: No irrational number can be expressed as a
decimal with a finite number of decimal places. The number i cannot be ex
pressed as a decimal with a finite number of decimal places. Hence i is an
irrational number.
4-4 NEGATIVE NUM BERS
One more addition to the number system which has considerably extended the
power of mathematics comes from far-off India. Numbers are commonly used
to represent an amount of money, in particular the amount of money which a
person owns. Perhaps because the Hindus were in debt more often than not, it
occurred to them that it would also be useful to have numbers which represent
the amount of money one owes. They therefore invented what are now
called negative numbers, while the previously known numbers are called
NEGATIVE NUMBERS 73
positive numbers. Thus numbers which we denote by - 3, - � and -Vi came
into existence. Where necessary to distinguish clearly positive from negative
numbers or to emphasize what is positive as opposed to what is negative, one
writes + 3 or + ! instead of 3 or ! .
Ir is not necessary, incidentally, to use such symbols as - 3 to represent the
negative counterpart to 3. Modern banks and large commercial corporations,
which deal with negative numbers continually, often write these in red ink,
whereas positive numbers are written in black ink. However, we shall find
that placing the minus sign in front of a number to indicate a negative number
.
IS a convemence.
.
The use of positive and negative numbers is not limited to the representa
tion of assets and debts. One represents temperatures below 0° as negative
temperatures, while temperatures above 0° are positive. Likewise heights
above and below sea level can be represented by positive and negative numbers,
respectively. It is sometimes convenient to represent time after and before a
specified event by positive and negative numbers. For example, using the birth
of Christ as the event, the year 50 B.C. can just as well be described as the
year - 50.
To derive more use from the concept of negative numbers it must be
possible to operate with them just as we operate with positive numbers. The
operations with negative numbers and with negative and positive numbers
together are easy to understand if one keeps in mind the physical significance
of these operations. For example, suppose a man has assets of 3 dollars and
debts of 8 dollars. What is his net wealth? Clearly the man is 5 dollars in
debt. The same calculation is represented in terms of positive and negative
numbers by stating that the amount 8 dollars must be taken from 3 dollars, that
is, 3 - 8, or that a debt of 8 dollars must be added to assets of 3 dollars, that
is, +3 + ( - 8 ) . The answer is obtained by subtracting the smaller numerical
value (that is, the smaller number without regard to sign) from the larger
numerical value and giving the answer the sign attached to the larger numerical
value. That is, we subtract 3 from 8 and call the answer negative because the
larger numerical value, namely 8, has the minus sign attached to it.
Since negative numbers represent debts and subtraction usually has the
physical meaning of "taking away" or "removing," then the subtraction of a
negative number means the removal of a debt. Thus, if a person has assets of,
say 3 dollars, but this figure already takes into account a debt of 8 dollars, the
removal or cancellation of the debt leaves the person with assets of 1 1 dollars.
Mathematically we say + 3 - ( - 8 ) + 1 1 . In words, to subtract a negative
=
number we add the corresponding positive number.
Suppose a man goes into debt at the rate of 5 dollars per day. Then in
3 days after a given date he will be 15 dollars in debt. If we denote a debt of
5 dollars as - 5, then going into debt at the rate of 5 dollars per day for 3 days
can be stated mathematically as 3 · ( - 5 ) - 1 5. That is, the multiplication of
=
74 NUMBER: THE FUNDAMENTAL CONCEPT
a positive and a negative number yields a negative number whose numerical
value is the product of the two given numerical values.
]n the very same situation in which a man goes into debt at the rate of 5
dollars per day, his assets three days before a given date are 1 5 dollars more
than they are at the given date. If we represent time before the given or zero
date by J and the loss per day as - 5, then his relative financial position J
-
days ago can be expressed as - J . ( - 5 ) + 1 5 ; that is, to consider his assets
=
three days ago, we would multiply the debt per day by - J, whereas to calcu
late the financial status three days in the future, we multiply by + J . Hence
the result is + 1 5 in the former case compared to 1 5 in the latter.
-
There is one more definition concerning negative numbers which is readily
seen to be sensible. For the positive numbers and zero we say for obvious
reasons that 3 is greater than 2, that 2 is greater than i, and that any positive
number is greater than zero. The negative numbers are said to be less than the
positive numbers and zero. Moreover, we say that - 5 is less than - 3, or that
- J is greater than -5. If one thinks of these various numbers as representing
people's wealth, then the agreement concerning their order fits our usual
understanding of relative wealth. A person whose financial status is - 3 is
wealthier than one whose status is - 5; one is better off to be J dollars than 5
dollars in debt. Incidentally, the symbol > is used to denote "greater than" as
in 5 > J, and the symbol < denotes "less than" as in - 5 < - J.
The relative position of the various positive and negative numbers and zero
is readily remembered if one visualizes these numbers as points on a line as
shown in Fig. 4-J. The figure is really not different from that obtained by
moving a thermometer scale into a horizontal position.
-2 -1
3
-4 -3 o 2 2 3 4 5
I I I I I I I I Fig. 4-3
The above situations, which illustrate how the definitions of the operations
with positive and negative numbers were suggested, are of course by no means
the only ones in which positive and negative numbers are employed. Indeed
the usefulness of negative numbers would hardly be great were this the case.
However, these simple financial transactions show not only how mathemati
cians arrived at the definitions, but that there is no more mystery about nega
tive numbers than about positive ones. The definitions represent in abstract
form what takes place physically, and, as with all numbers, we can think in
terms of the abstractions to arrive at a knowledge of physical happenings.
It may be of some comfort to the reader to know that the concept of nega
tive numbers, like the concept of irrational numbers, was resisted by mathema
ticians for several hundred years. The history of mathematics illustrates the
rather significant observation that it is more difficult to get a truth accepted
than to discover it. The mathematicians to whom "number" meant whole
THE AXIOMS CONCERNING NUMBERS 75
numbers and fractions found it hard to accept negative numbers as true num
bers. They, too, failed to realize for centuries that mathematical concepts are
man-made abstractions which can be introduced at will if they can serve
useful purposes.
EXERCISES
1. Suppose a man has $3 and incurs a debt of $5. What is his net worth?
2. Suppose a man owes $5 and then incurs a new debt of $8. Use negative numbers
to calculate his financial condition.
3. Suppose a man owes $5 and earns $8. Use positive and negative numbers to
calculate his net worth.
4. Suppose a man owes $ 1 3, and a debt of $8 is cancelled. Use negative numbers
to calculate his net worth.
5. A man loses money in business at the rate of $100 per week. Let us denote this
change in his assets by - 100 and let us denote time in the furore by positive
numbers and time in the past by negative numbers. How much will the man lose
in 5 weeks? How much more was the man worth 5 weeks ago?
4-5 THE AXIOMS CONCERNING NUM BERS
In the preceding chapter we said that mathematics proceeds by deductive
reasoning from explicitly stated axioms. Yet thus far in this chapter we have
said nothing about axioms. The reason is simply that the axioms concerning
numbers are such obvious properties that we use them automatically without
realizing that we are doing so.
This situation may perhaps be better understood by means of an analogy.
Whenever a child at play throws a ball up into the air, he expects that the ball
will come down. He is really assuming that all balls thrown up will come
down. Of course, this assumption is well founded in experience; nevertheless,
the child's expectation that the ball will come down is a deduction from the
assumption just stated and the additional premise that he is throwing a ball up
into the air. Recognition of the fact that he has made an assumption makes
clear the reasoning, conscious or unconscious, behind the act.
To understand the deductive process in the mathematics of numbers, as
well as in geometry, we must recognize the existence and use of the axioms. We
do not hesitate to say that 275 + 384 384 + 275. Surely we did not add 384
=
objects to 275, count the total, then add 275 objects to 384, count that total, and
check that the two totals agree. Rather, whenever in our experience we co�
bined two groups of objects, we found that we obtained the same total collec
tion regardless of whether we put the first group with the second or the
second with the first. Of course, our evidence to the effect that the order of
addition is immaterial is limited to a small number of cases, whereas in practice
76 NUMBER; THE FUNDAMENTAL CONCEPT
we use this fact with all numbers. Hence, we are really making an assumption,
namely, that for any two numbers a and b, integral, fractional, irrational, and
negative, the order of addition will not affect the result. Thus our assumption
also includes the affirmation that 0 + V5 V5 + 0. It is important for
�
another reason to recognize that this assumption is being made. Numbers are
not apples or cows. They are abstractions from physical situations. Mathema
tics works with these abstractions in order to deduce information about
physical situations. However, if the axioms are not well chosen, the deductions
will not apply. Hence it is well to note what assumptions are being employed
and to ascertain that they are well founded in experience.
Let us, therefore, note the axioms which we have been using and will
continue to use. The first axiom is the one discussed in the preceding para
graph:
AXIOM I. For any two numbers a and b,
a + b = b + a.
The aXIOm is called the commutative axiom of addition because it says
that we can commute or interchange the order of the two numbers to be
added. We note that subtraction is not commutative, that is, 3 5 does nOt
-
equal 5 3 .
-
If we had to calculate 3 + 4 + 5 , we could 'first add 4 to 3 and then add 5
to this result, or we could add 5 to 4, and then add this result to 3 . Of course,
the result is the same in the two cases, and this is exactly what our second
aXIOm says.
AXIOM 2. For any numbers a, b, and c,
(a + b) + c = a + (b + c) .
This axiom is called the associative axiom of addition because we can
associate the three numbers in two different ways in performing the addition.
The two axioms we have just discussed have their analogues for the
operation of multiplication.
AXIOM 3. For any two numbers a and b,
a ' b = b · a.
This axiom is called the commutative axiom of multiplication. Inciden
tally, the dot which is used to denote multiplication is omitted if there is no
danger of misunderstanding. Thus, we could as well write ab ba. The �
axiom is clearly a property of numbers; yet we sometimes fail to recognize
that it is applicable. Many a student hesitates to write 5 · a instead of a · 5.
But the commutative axiom says that the two expressions are equal. We
THE AXIOMS CONCERNING NUMBERS 77
might note in this context that the operation of division is not commutative,
for 4 + 2 does not equal 2 + 4.
AXIOM 4. For any three numbers a, b, and c,
(ab) e = a (be) .
This aXIOm IS called the associative axiom of multiplication. Thus
( 3 · 4)5 3 ( 4 · 5).
=
We also find in our work with numbers that it is convenient to use the
number O. To recognize formally that there is such a number and that it has
the properties which its physical meaning requires, we state another axiom.
AXIOM 5. There is a unique number 0 such that
a) 0 + a a for every number a,
=
b) o · a 0 for every number a,
=
c) if ab 0 then either a 0 or b
= = = 0 or both are O.
The number 1 is another whose properties are somewhat special. Again,
we know from the physical meaning of I just what its properties are, but if
one is to justify the operations with I by appealing to axioms rather than to
physical meaning, there must be a statement which tells us just what these
properties are. In the case of 1. it is sufficient to specify a sixth axiom.
AXIOM 6. There is a unique number I such that
I'a = a
for every number a.
In addition to adding and multiplying any two numbers, we also have
physical uses for the operations of subtraction and division. We know that
given any two numbers a and b, there is a number c which results when b is
subtracted from a. From a practical standpoint, it is helpful to recognize that
subtraction is the inverse operation to addition. What this means is simply
that if we have to find the answer to 5 - 3 we can and, in fact, do ask ourselves
what number added to 3 gives 5. If we know addition, we can then answer
the subtraction problem. Even if we obtain the answer by a special subtrac
tion process, and we do in the case of large numbers, we check it by adding
the result to what we subtracted to see if it gives the original number or
minuend. Hence a subtraction problem such as 5 - 3 x really asks for what
=
number x added to 3 gives 5, that is, x + 3 = 5.
In our logical development of the number system we wish to affirm that
we can subtract any number from any other number and we phrase this
statement so that the meaning of subtraction is precisely what it is, the inverse
of addition.
78 NUMBER: THE FUNDAMENTAL CONCEPT
AXIOM 7. If a and b are any two numbers, there is a unique number x
such that
a = b + x.
Of course, the quantity x is what we usually denote by a - b.
The relation of division to multiplication is also that of an inverse opera
tion. When we seek the answer to ! we may happen to know directly from
experience that the answer is 4. But jf we don't, we can reduce the division
problem to a multiplication problem and ask what number, x, multiplied by 2
gives 8, and if we know multiplication we can find the answer. Here too, as
in the ·case of subtraction, even if we use a special division process such as long
division to find the answer, we check the answer by multiplying the divisor by
the quotient to see if the product is the dividend. The reason for doing this
is simply that the basic meaning of a/b is to find some number x such that
bx = a.
In our logical development of the number system we affirm that we can
divide any number by any other number (except 0) and we phrase the asser
tion so that the meaning of division is precisely what it actually is, the inverse
of multiplication.
AXIOM 8. If a and b are any two numbers, except that b "" 0, then there
is a unique number x such that
bx = a.
Of course x is the number usually denoted by a/b.
The next axiom is not quite so obvious. It says, for example, that
3 · 6 + 3 · 5 3 (6 + 5). In this example we can perform the calculation to
=
see that the left and right sides are equal, but this is really not necessary.
Suppose we had 157 cows in one herd and 379 in another, and each herd in
creased sevenfold. The total number of cattle is then 7 · 157 + 7 · 379. But if
the original two herds were one herd with 157 + 379 cows, and this single
herd increased sevenfold, we would have 7 ( 157 + 379) cows. It is physically
clear that we have the same number of cows now as before, that is, that
7 . 157 + 7 . 379 = 7 ( 1 57 + 379). Stated in general terms, the axiom is:
AXIOM 9. For any three numbers a, b, and c,
ab + ac = a(b + c) .
This axiom, called the distributive axiom, is very useful. For example, to
calculate 571 . 36 + 571 · 64 we can apply the axiom to state that this quantity
is 571 ( 36 + 64) or 5 7 1 · 100 or 57,100. We say often that we have factored
the quantity 571 out of the sum 571 . 36 + 5 7 1 · 64.
THE AXIOMS CONCERNING NUMBERS 79
We should note that from
ab + ac = a(b + c)
we Can also state that
ba + ca = (b + c)a,
because in each term of the first equation we can apply the commutative axiom
of multiplication to change the order of the factors.
We often use the second form of the distributive axiom. Thus, suppose a
is some number and we wish to calculate Sa + 7a. We can replace this sum
by (5 + 7)a, and obtain 1 2a.
The distributive axiom is also applicable in the following situation. Sup-
pose we have to calculate
296 + 148
296
One might be tempted to cancel the two numbers 296. But this is incorrect.
The given fraction means
$(296 + 148),
and the distributive axiom tells us that we may write instead
$ ' 296 + $ ' 148, or 1 + t.
In addition to the above axioms, we have the following evident properties
of numbers:
AXIOM 10. Quantities equal to the same quantity are equal to each other.
AXIOM 1 1 . If equal quantities are added to, subtracted from, multiplied
with, or divided into equal quantities, the results are equal. However,
division by zero is not permitted.
The set of axioms we have just given is not complete; that is, it does not
form the logical basis for all of the properties of the positive and negative
whole numbers, fractions, and irrational numbers. However, the set does pro
vide the logical basis for what is usually done with numbers in ordinary algebra.
Moreover, it does give some idea of what the axiomatic basis for mathematical
work with numbers amounts to.
Now that we have the axioms, what do we do with them? We can prove
theorems about numbers. Let us consider a few examples. Negative numbers
were introduced to represent physical happenings such as debts or time before
a given event. When we examined the physical situation in which we wished
to use these numbers, we found that if the numbers are to be useful then we
80 NUMBER: THE FUNDAMENTAL CONCEPT
should agree that, for example,
- 2 ' 3 = - 6 and - 2 ' ( - 3) = 6.
There we agreed to operate with positive and negative numbers so as to make
the results fit the physical situation. In the deductive approach to numbers
we prove on the basis of our axioms that certain theorems are correct. Let us
prove that a positive number times a negative number is negative.
Let a and b be positive numbers. Then - b is a negative number; for
example, -b = -5. We shall prove that a( -b) -abo We know by Axiom
=
7, wherein we let it be 0, that if b and a are given, then there is a number x
such that b + x = O. This number x is denoted by 0 - b or -b. Then
b + ( - b) = O.
Now we can multiply both sides of this equation by a, and since equals multi
plied by equals give equals, we have
a[b + ( - b) ] = a · O.
Now a · 0 O · a by Axiom 3 , and O · a 0 by Axiom 5. By applying the
= =
distributive axiom, Axiom 9, to the left-hand side of our equation, we obtain
a ' b + a( - b) = O, (\)
and now we see that a( -b) is the number which added to ab gives O. But
Axiom 7 says that given ab and 0 (these are the a and b of Axiom 7 ) , there
is a unique number x such that
ab + x = 0. (2)
This number x is denoted by ° - ab or -abo But equation ( \ ) says that a( -b)
is the number which added to ab gives O. Since there is just one such number
which when added to ab gives 0, and that number we know is -ab, it must be
be that a( -b) = -abo
The proof is now complete and yet may not be convincing. The reason
is simply that we are so accustomed to operating with numbers on the basis of
physical arguments and experience with them that we have not accustomed
ourselves to reasoning with numbers on an axiomatic basis.
Let us consider another proof. In Section 4-2 we gave a physical argument
to show that if we divide alb by cld then we can get the answer by inverting
cld to obtain dlc and then multiplying; that is,
a
b a d ad
-= - .- =
e b e be
d
THE AXIOMS CONCERNING NUMBERS 81
We can prove, on the basis of our axioms, that this rule of inverting the de
nominator and then multiplying is correct.
To divide alb by eld is to find a number x such that
a c
x · _ ·
(3)
b d
Now Axiom 8 tells us that there is a unique number x which satisfies such an
equation. We do know that
because if we cancel common factors in the numerator and denominator of the
right side we obtain alb. Hence one number which can serve as the x which
satisfies equation ( 3 ) is adlbe. But since the value of x is unique, x = adlbe.
Thus the result of dividing alb by eld is adlbe. We note that the answer
adlbe is obtained by multiplying alb by the inverse of eld. Hence to divide
one fraction by another, we invert the denominator and multiply.
This proof, like the preceding one, may not be convincing, and the reason
is the same. We are not accustomed to reasoning about numbers on an axiom
atic basis. Rather, we have relied upon the physical meaning of numbers and
operations. Historically, the mathematicians did the same thing. They learned
to operate with numbers by noting the uses to which numbers were put, and
they constructed the axiomatic basis long afterward, just to satisfy themselves
that deductive proofs of the properties of numbers could be made.
Since we, too, are accustomed to the properties and operations with
numbers since childhood, and we are sure of these properties, we shall not
often cite axioms to justify our steps. Thus if we write 3a in place of a · 3, we
shall not cite the commutative axiom of multiplication as the justification for
this step. In fact, it would be pedantic to do so. The axioms are useful, rather,
in helping us to determine what is correct when our experience fails us or
leaves us in doubt. However, we should not lose sight of the fact that the
mathematics built upon the number system is a deductive system. This point
needs emphasis because we begin to learn arithmetic at an early age by rote
and thereafter we tend to operate with numbers mechanically without per
ceiving that we are constantly using axioms of numbers.
EXERCISES
1. Do you believe that
256(437 + 729) = 256 ' 437 + 256 ' 729?
Why?
82 NUMBER: THE FUNDAMENTAL CONCEPT
2. Is it correct to assert that
a(b - c) = ab - ac?
[Suggestion: b - c = b + (-c).J
3. Perform the operations called for in the following examples:
a) 3a + 9a b) a ' 3 + a ' 9 c) a(5 + 0) d) 7a - 9a
e) 3(2a + 4b) f) (40 + 5b)7 g) a(a + b) h) a(a - b)
i) 2 (8a) j) a(ab)
4. Carry out the multiplication:
(a + 3)(a + 2).
[Suggestion: Regard (a + 3) as a single Quantity and apply the distributive axiom.]
5. Calculate (n + I ) ( n + I ) .
6. I f 3x = 6, is x = 2? Why?
7. 1f 3x + 2 = 7, is 3x = 5? Why?
8. Is the equality x' + xy = x(x + y) correct?
9. Is it correct to assert that
a + (be) = (a + b) (a + c) ?
* 4-6 APPLICATIONS OF THE NUM BER SYSTEM
Something of the power, methodology, and subtlety of mathematical reasoning
can already be seen in the applications which have been made of the several
types of number. Indeed, we shall see that these resulted in significant scien
tific discoveries.
Let us begin with some rather simple matters. Suppose a man drives a car
for one mile at 60 miles per hour and for another mile at 120 miles per hour.
What is his average speed? We tend to answer this question by applying the
common procedure for finding an average. Thus, if a man buys one pair of
shoes for $5 and another for $10, the average price is $5 + $10 divided by 2,
or $7.50. Hence it would seem as though the average speed in the above
problem should be 6 0 + 120 divided by 2, or 90 miles per hour. However, this
answer is not correct. The number 90 is an average in the arithmetic sense, but
it is not the average we seek. The average speed should be that speed which
would enable the man to drive the two miles in the same time as it took him to
drive that distance at the two different speeds. Now, it took the man I minute
to drive the first mile and it took him � minute to drive the second mile.
Hence it took him I ! minutes to drive 2 miles. We now ask, what average
* Starred sections and chapters can be omitted without disrupting the mathematical
continuity.
APPLICATIONS OF THE NUMBER SYSTEM 83
speed maintained for I! minutes would cover 2 miles? Since the average speed
multiplied by the total time should give the total distance, the average speed
is the total distance divided by the total time, that is,
2 4
average speed = -
� J
The average speed is then ! miles per minute or 80 miles per hour.
The point of this example, not a momentous one to be sure, is merely that
the unthinking, blind application of arithmetic does not produce the correct
result. The notion of average speed serves a physical purpose, and unless we
are clear about what average speed is supposed to mean, we shall not profit by
the use of arithmetic.
EXERCISES
1. A man can row a boat in still water at 6 mi;hr. He plans to row upstream for
12 mi and then back in a river whose current flows at 2 mi;hr. Thus his speed
upstream is 4 mi;hr and his speed downstream is 8mi;hr. He reasons that his
average �peed is 6 mi/hr and that the entire trip of 24 mi should therefore take
4 hr. ]s this reasoning correct?
2. Suppose that a merchant sells apples at a price of 2 for 5 ¢ and oranges at 3 fo r
5,. To make his arithmetic simpler he decides to sell any 5 pieces of fruit for to,
or at the average price of 2, per piece. Thus, if he sells 2 apples and 3 oranges, he
sells 5 pieces of fruit at 2, each and receives the same to, as if he had sold them at
the original separate prices. ]s the merchant's average price correct? [Suggestion:
Consider what results if he sells 1 2 apples and 12 oranges.]
3. Suppose the merchant wishes to sell a apples and b oranges to some customer at
the prices given in Exercise 2. What should the average price be?
4. Given the data of Exercise 2, is there an average price which would be correct
no matter how many apples and how many oranges are sold?
5. One man can dig a certain ditch in 2 days and another can dig the same ditch in
3 days. What is their average rate of ditch-digging per day?
Let us consider next an application of simple arithmetic to genetics.
Suppose we have before us 2 red aces and 2 red kings from the usual deck
of 52 cards. How many different pairs consisting of one ace and one king can
be put together' Since each ace can be paired with either of 2 kings, there are
2 different pairs for any one ace. Since we have 2 aces, there are 2 · 2 or 4
different pairs.
Now let us suppose that we have 2 red aces, 2 red kings, and 2 red queens.
How many different sets consisting of one ace, one king, and one queen can
we form with the given cards? We saw above that there are 4 different pairs
of aces and kings. With each of these 4 pairs we can place 2 different
84 NUMBER: THE FUNDAMENTAL CONCEPT
queens. Hence there are 4 · 2 or 8 different sets of J cards. We note that
4 · 2 2 2 . 2 23.
= . =
If we have 2 red aces, 2 red kings, 2 red queens, and 2 · red jacks, the
number of different sets, each consisting of one ace, one king, one queen, and
one jack, can also be readily calculated. Each of the 8 choices of ace, king, and
queen can be paired with each of the 2 jacks. Hence there are 8 · 2 or 16
choices in all. Now,
8.2 4'2 .2= 2'2.2.2 = 2 4. =
Clearly, if we had 1 0 different pairs of cards and had to make all possible
choices of 10 cards, one from each pair, the number of all possible sets of 10
cards would be
2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 = 2 '0 = 1024.
This simple reasoning about cards has an important application to genetics.
The reproductive cells <as well as ordinary cells) of the human male contain
24 pairs of chromosomes. When a sperm cell is formed from the reproductive
cell, it contains 24 chromosomes, each coming from one of the 24 pairs.
Hence a sperm cell can be formed in 224 possible combinations. The reproduc
tive cells of the human female also contain 24 pairs of chromosomes. An ovum
formed from the female reproductive cell contains 24 chromosomes, each
coming from one of the 24 pairs of the reproductive cell. Hence there are
224 possible ways in which an ovum can be formed. In conception, any one
sperm joins, or fertilizes, any one ovum. Since there are 224 possible sperms
and 2 24 possible ova, the number of possible chromosome combinations for
the fertilized ovum is then
2 24 . 2 2 4 = 1 6,777,2 1 6 ' 16,777,216 = 281 ,474,976,710,656.
This is the number of possible variations in the genetic make-up of any one
child a man and wife may have. Actually, the number of variations is some
what larger. Each chromosome contains genes, and these determine the
hereditary qualities. Biologists have found that any two paired chromosomes
in a reproductive cell may exchange some genes, and this exchange gives rise
to new varieties of sperm cells and ova.
EXERCISES
1. The usual deck of 52 cards contains 4 different aces and 4 different kings. How
many different pairs of cards, each pair consisting of one ace and one king, can
be formed from the aces and kings?
2. A manufacturer offers his automobile in 3 different colors, with or without .a
heater, and with or without a radio. How many different choices can a purchaser
make?
APPLICATIONS OF THE NUMBER SYSTEM 85
3. A girl has 3 hats. 2 dresses. and 2 pairs of shoes. How many different costumes
does she have?
4. There are six numbers on a die (singular of dice). How many different pairs of
numbers can show up on a throw of a pair of dice? The twO dice are to be
marked so that a throw of a 2 on one die (say A ) and of a 5 on the other (say B )
can be distinguished from the reverse arrangement ( 5 on A and 2 on B ) .
We have already discussed the fact that our method of writing quantities
uses the idea of positional notation in base ten (see Section 4--2 ) . However,
some civilizations used other numbers as a base. For example, the Babylonians.
for reasons that are obscure. selected 60. This system was taken over by the
Greek astronomers and was used in Europe for many mathematical and all
astronomical calculations as late as the seventeenth century. It still survives in
our practice of dividing hours and angles into 60 minutes and 60 seconds. In
adopting ten as a base, Europe followed the practice of the Hindus. Let us
challenge history and see whether we can derive some advantage from a
change to a new base.
We shall choose base six. The quantities from zero to five would be
designated by the symbols 0, 1, 2, 3, 4, 5, as in base ten. The first essential
difference comes up when we wish to denote six objects. Since six is to be the
base, we would no longer use the special symbol 6, but place the 1 in a new
position to denote 1 times the base. just as in base ten the I in 10 denotes one
times the base, or the quantity ten. Hence, to write six in base six, we would
write 10, but now the symbols 10 means 1 times six plus O. Thus the symbols
10 can denote two different quantities, depending upon the base employed.
Seven in base six would be written 1 1 , because in base six these symbols mean
I times six + I, just as I I in base ten means I times ten + I . Again the symbols
\ I represent different quantities, depending upon the base implied. As another
example. to denote twenty-two in base six we write 34. because these symbols
now mean 3 times six + 4.
In base ten, to write numbers larger than ninety-nine, we use a third
position, the hundreds' place, to indicate tens of tens. Similarly in base six,
when we reach numbers larger than thirty-five, we use a third position to
denote sixes of sixes. Thus thirty-eight would be written in base six as 102,
wherein the one means I times six times six, the 0 means 0 times six, and the
2 denotes just 2 units. To express very large numbers we would use four
place numbers, five-place numbers, and so forth.
We can perform the usual arithmetic operations in base six. However, we
would have to learn new addition and multiplication tables. For example, in
base ten we write 5 + 3 8, whereas in base six, eight must be written 1 2 .
=
Hence our addition table would have to state that 5 + 3 12. Likewise, our
=
new multiplication table would have to list 3 · 5 2 3 , because fifteen is 2 · 6 + 3
=
or, in base six, 23. When we learned to use base ten, we had to memorize the
86 NUMBER; THE FUNDAMENTAL CONCEPT
result of adding each number from 0 to 9 to every number from 0 to 9, and
the result of multiplying each number from 0 to 9 with every number from
o to 9. For base six we would have to learn to add (and multiply) only
numbers from 0 to 5 to (or with) the numbers of this set. Thus our addition and
multiplication tables would be shorter, and we would learn arithmetic sooner
as youngsters. We might even pass the hurdle of arithmetic so easily that we
might get to like mathematics. The only disadvantage of base six would be
that to represent large quantities we would have to use more digits. For
example, the quantity fifty-four, written as 54 in base ten, must be written as
1 30 in base six, because fifty-four equals I 6' + 3 . 6 + O.
.
There are people who campaign for the adoption of base twelve, because
it offers special advantages. For one thing, more fractions can be written as
finite decimals in base twelve. Thus i must be written as the unending decimal
0.333 . . . in base ten, but can be written as 0.4 in base twelve, since in this base
0.4 means n' Also, since the English system of denoting length calls for 12
inches in one foot, we could, for example, express 3 feet and 6 inches as 36
inches in base twelve, whereas to express this number of inches in base ten, we
must first calculate 3 · 12 + 6 and then write 42 in base ten. To a limited
extent we could use base twelve in our method of recording time. In the United
States the day has two sets of twelve hours, and in base twelve the hours of
the day would run from 0 to 20. Whereas determining what 7 hours after 6
o'clock will be requires at present some computation, under the addition table
for base twelve we would state at once that 7 + 6 I ! . However, base ten is
=
now so widely used that a change to another base for ordinary daily use or
commerce is hardly likely.
In the subject of bases we have an idea that was pursued for centuries,
largely as an interesting and amusing speculation, but which suddenly became
highly important in science and even in the commercial world. For several
centuries mathematicians worked on the design of machines which would
perform arithmetical computations quickly and thus remove a good deal of
the drudgery of arithmetic. Although some types of computing machines
were invented and used, mathematicians saw their golden opportunity in the
electronic devices developed by modern radio engineers. The key is the radio
vacuum tube, which can be made to pass current by applying a voltage to it
or can be kept inactive. Two maneuvers are thus possible. In base two all
numbers require only two symbols, the 0 and the 1 . A typical number would
be lOll, which means
I . 23 + 0 . 2 2 + I . 2 + 1 .
This number can be recorded by the machine by employing four tubes, one
for the units' place, another for multiples of 2, a third for multiples of 2', and
a fourth for multiples of 23. To record lOll, the first, second, and fourth
tubes can be activated, and the third, which records the third place in the
APPLICATlONS OF THE NUMBER SYSTEM 87
number, kept inactive. The currents passed by the tubes which "fire" are
recorded by the machine in special circuits. Another number can then be fed
into the machine. Let us suppose that it is to be added to the first number. The
result of having two l's in the same place means, in base 2, that the sum is to
be 0 and a I carried over to the next place. This operation is readily performed
by the circuits. While this description of an electronic computer certainly
doesn't begin to present the ingenious ideas which engineers and mathemati
cians have incorporated, we may perhaps see that the workings of a vacuum
tube are ideally suited to operations in base two.
To take advantage of the fact that computers can perform calculations
in base two, the numbers to be worked on are converted beforehand from
base ten to base two and then fed to the machine along with other instruc
tions. The machine then operates in this latter base. The result is, of course,
reconverted to base ten.
Because computers work with microsecond speed, they are exceedingly
valuable in any commercial or industrial organization which must process a
lot of numerical data. Calculations in banks, insurance companies, and in
industry, which used to require an immense amount of human labor, are now
performed by machines. Computers are the first in a new series of machines
which keep track of great quantities of data, select information from millions
of cards on which data are recorded, plan factory operations, direct machinery,
and may soon provide translations of foreign-language publications.
Electronic computing machines are an enormous boon to science and
mathematics also. The arithmetic required to extract concrete information
from mathematical formulas is often so lengthy that it would take years to
perform these calculations. Computing machines do such work in hours.
Moreover, mathematicians no longer hesitate to work on problems which will
lead to extensive computations, because they now know that their work will
not be in vain.
Computing machines may help us to learn more about the action of the
human brain. According to biologists, the nerve cells in the nerve chains and
the cells in our brains respond to electrical impulses much as a vacuum tube
does. Just as a tube will "fire" when it receives electrical current beyond a
certain minimum value and remain inactive otherwise, so do the nerve cells
in the nerve chains transmit an electrical impulse to whatever organ they may
lead when this impulse exceeds a thresh hold value; otherwise they are inactive.
Computing machines also have a memory; that is, partial results of calcula
tions are stored aUtOmatically in a special device, called the memory. When
these results are needed, the memory device releases them. Thus the result of
an addition process might be stored in the memory device until the result of
some multiplication process is obtained and then, if so instructed, the machine
will add these two results. The machine's memory device, then, functions
somewhat like the human memory. Hence, in tWO respects at least, electronic
88 NUMBER: THE FUNDAMENTAL CONCEPT
computing machines simulate the actions of human nerves and memory.
Though machines are, in speed, accuracy, and endurance, superior to the
human brain, one should not infer, as many popular writers are now suggest
ing, that machines will ultimately replace brains. Machines do not think. They
perform the calculations which they are directed to perform by people who
have the brains to know what calculations are wanted. Nevertheless, we un
doubtedly have in the machine a useful model for the study of some functions
of the human brain and nerves.
EXERCISES
1. Construct an addition table for base six.
2. Construct a multiplication table for base six.
3. Construct an addition and multiplication table for base two.
4. The following numbers are in base ten:
9, 10, 12, 36, 48, 100.
Write the respective quantities in base six.
5. The following numbers are in base six:
5, 10, 12, 20, 100.
Write the respective quantities in base ten.
6. Write the fraction ! as a decimal in base six.
7. The Quantity 0.2 is in base six. Write the corresponding quantity in base ten.
S. The number 101 is written in some unknown base and equals ten. What is the
base?
9. Find the least number of weights needed to weigh. to the nearest pound. objects
weighing from 0 to 63 lb. The scale to be used contains twO pans and the weights
are to be put in one pan. [Suggestion: Consider the problem of representing all
numbers from 0 to 63 in base 2.]
Some of the most remarkable uses of numbers, which have led to profound
discoveries, are found in the study of the structure of matter. During the
early and middle part of the nineteenth century, certain basic experimental
facts about the varieties of matter found in nature led, after some inevitable
fumbling on the parr of John Dalton, Amadeo Avogadro, and Stanislav
Cannizzaro, to the theory that all matter is made up of atoms. Thus hydrogen,
oxygen, chlorine, copper, aluminum, gold, silver, and all other varieties of
matter are composed of atoms. Experimental techniques were developed to
measure the relative weights of the atoms of different elements. The con
venient unit of weight chosen was T�the weight of the oxygen atom, so that
the weight of the atom of oxygen is 16. The weight of the hydrogen atom
then proved to be 1.0080, that of copper 63.54, that of gold 197.0, and so on.
APPLICATIONS OF THE NUMBER SYSTEM 89
By this time, too, a number of chemical properties of these various elements,
such as their melting temperatures, boiling temperatures, and their ability to
combine with other elements to form compounds, had been determined.
The question which had begun to stir the chemists was whether there
existed any law or principle which utilized the atomic weights of these
elements. The crowning discovery was made in 1 869 by Dimitri Ivanovich
Mendeleev ( 1834-1907). He found, as he began to arrange the elements in
order of increasing atomic weight, that every eighth element, starting from a
given one, had chemical properties similar to the first one. Thus, the gases
fluorine and chlorine, the latter the eighth element starting from fluorine, both
combine readily with metals. However, as he continued to place each of the
63 different elements known in his time in the eighth position after the one
with similar chemical properties, he saw that he had to leave blank spaces.
Mendeleev was so much impressed with the periodicity of chemical properties
that he did not hesitate to leave the blank spaces and to affirm that there must
be elements to fill these spaces. Since each of these missing elements should
have chemical properties similar to those of the element found in the eighth
position preceding or succeeding the missing element, he could even predict
some of the properties of the unknown elements. Mendeleev described the
properties of three of the missing elements, and his immediate successors dis
covered them. These are now called scandium, gallium, and germanium. Still
later, others were found. The interesting fact about Mendeleev's work from
the mathematical standpoint is that he had no physical explanation of why
elements eight positions removed from one another should have similar chemi
cal properties. He knew only that the number eight was the key to the
arrangement, and he followed this mathematical guide faithfully. Long after
Mendeleev's time, other elements, for example helium, were found, which do
not fit into this arrangement, but his periodic table is still the basic one which
all students of chemistry learn today.
Simple arithmetic continued to play a leading role in subsequent develop
ments of atomic theory. The continuing study of the atomic weights of
various elements and their chemical properties showed that elements formerly
regarded as pure were really not so. Thus, there are two kinds of hydrogen.
These have similar chemical properties, but different atomic weights; in fact,
one is twice as heavy as the other. Since both were previously called hydrogen,
and since they do, in any case, have similar chemical properties, these two
forms of hydrogen are called isotopes of hydrogen. Likewise, there is not one
substance, oxygen, but there are three, of atomic weights 16, 17, and 1 8.
Uranium, a very important element today, has two isotopes of atomic weights
238 and 235.
The startling fact which emerged from the discovery of isotopes is that
when all isotopes are distinguished and the relative weights of the distinct
elements determined, the weight of any one element is within 1 % of a whole
90 NUMBER: THE FUNDAMENTAL CONCEPT
number. Such a fact can hardly be accidental. The explanation would seem
to be that all these elements are really multiples of a single element, namely
the lighter isotope of hydrogen, which has the least weight of all elements.
In other words, the various elements which previously appeared to be entirely
different substances. now were seen to be just smaller or larger collections of
the same element, but arranged in special ways peculiar to the substance.
(Strictly speaking, the fundamental building block is not the lighter isotope of
hydrogen, bur what is now called the proton. The lighter hydrogen isotope
also has an electron whose weight is insignificant by comparison.)
If all of the different elements are really just aggregates of the lighter
hydrogen atom, it should be possible to remove some atoms and convert one
substance into another. Thus, we should be able to convert mercury, which
is the next heavier element after gold, into gold. And we can. What the
medieval alchemists hoped to do on mystical and superficial grounds, we can
now do on the basis of far better scientific knowledge. Unfortunately, the
cost of converting mercury into gold is too great to make it worth while. But
we do have uses for the transmutation of elements which are, in our age, more
valued, and which we shall describe in a moment.
A scientist who has a theory cannot afford to overlook even one detail,
trivial as it may seem, which does not square with his theory. If all elements
are merely combinations of the lighter hydrogen atom, then their weights
should be exact multiples of the weight of this atom instead of having values
within 1 % of such weights. (The electrons in the atoms do not account for
the difference. ) This discrepancy must be explained. The lightest isotope of
oxygen had, somewhat arbitrarily, been given weight 16. With this rather
arbitrary standard, the lighter hydrogen atom has weight 1.008 rather than
exactly 1 . But helium, which consists of 4 hydrogen atoms, proves to have
weight 4.0028. However, if it consists of 4 hydrogen atoms, its atomic weight
should be 4 times 1.008, or 4.032. The difference, 4.032 - 4.0028, or about
0.03, is the discrepancy which muSt be accounted for. Now it so happens that
Einstein, working in an entirely different field, the theory of relativity, had
already shown that mass can be converted into energy. Energy can take
different forms. It can be the heat created by burning coal or wood, or it can
be radiation such as comes to us from the sun. At the moment the precise
form of it does not matter. What does matter is the thought which occurred
to scientists that perhaps, when 4 hydrogen atoms are fused to form helium,
the missing 0.03 of matter is converted into energy in the process. Hence the
fusion of elements should release energy. And experiments showed that this is
indeed what happens. The energy which is released is called the binding
energy, and it is this energy which is released when a hydrogen bomb is
exploded.
In this brief account of the role of arithmetic in chemistry and atomic
theory, we have said almost nothing abour the great thinking and brilliant
APPLICATIONS OF THE NUMBER SYSTEM 91
experiments which physicists and chemists contributed. Our interest has
been to show how the use of simple numbers supplies scientists with a power
ful tool. Of course, the mathematics of numbers remains to be developed, and
we shall learn how much more can be accomplished with slightly more ad
vanced tools. But we can already see something of what the Pythagoreans
envisioned when they spoke of numbers as the essence of reality.
REVIEW EXERCISES
I. Calculate:
a) �+� b) %-� c) f-�
. ,
e) j - f-,; f) � - rt h) -
b
+-
d
I I
i) �
b
- �
d
o k) -+-
x 2
2. Calculate:
a) %.* b) �.-
: c) =,!- ' � d) (-�) . (-t)
, , • b • -c
f) - . •
. . - -
e) - ' g) '
h)
b d b b • b d
i) i + .g. k) � + -to-
• c
m) - +- n) � + 5� 0) - 8 + - 2
b d
3. Calculate:
.
a) (2 ' 5) (2 7) b) 2. · 2b c) 2. · l b
d) 2x ' ly e) 2x ' ly . 4z
4. Calculate:
l + 6a + 6b
l . -'-
b) c) - -
l
4x + 8y c) .b + .c
d)
2 •
5. Calculate:
a) v'49 b) Vill c)
v1 d) VR
92 NUMBER: THE FUNDAMENTAL CONCEPT
v'4
v'2
e) vJ v'J f) v'2 v'8 g) h) v'2 V1
6. Simplify:
a) v32 b) v'48 c) vn d) v'8
I)
v'¥ h) v'¥
e) v1 v'Jj g)
7. Write as a fraction:
a) 0.294 b) 0.3742 c) 0.08 d) 0.003
8. Approximate by a number which is correct to 1 decimal place:
a) v'l b) v'5 c) v'7
9. Are the following equations true for all values of a and b? [Suggestion: If you
wish to disprove a general statement. it is sufficient to show one instance where
it does not hold.)
a) 2 (a + b) = 2a + 2b b) 2ab = 2a ' 2b
a+ b � � ab a b
c) = + d) - = - ' -
2 2 2 2 2 2
e) va2 + b2 = v'Q2 + v'b2 f) va2 + b2 = a+ b
g) v'iib = Va vb h) va2 - b2 = a- b
i) Va + b = Va + vb
to. Write the following numbers in positional notation but in base 2. The only
digits one can use in base 2 are 0 and 1.
a) I b) c) 5 d) 7 e) 8 f) 1 6 g) 19
1 1 . The following numbers are in base 2. Write the corresponding quantities in
base 10.
a) I b) 101 c) 1 1 0 d) 1 101 e) 1001
Topics for Further Investigation
1 . The Egyptian method of writing whole numbers and fractions.
2. The Babylonian method of writing whole numbers and fractions.
3. The Roman method of writing whole numbers and fractions.
4. The fundamental arithmetical laws of atomic theory. (Use the references to
Holton and Roller and to Bonner and Phillips).
5. Pythagorean number theory,
APPLICATIONS OF THE NUMBER SYSTEM 93
Recommended Reading
BALL, W. W. ROUSE: A Short Account of the History of Mathematics, Chaps. I and
2, Dover Publications, Inc., New York, 1960.
BONNER, F. T. and M. PHILLIPS: Principles of Physical Science, Chap. 7, Addison
Wesley Publishing Co., Inc., Reading, Mass., 1957.
CoLERUS, EGMONT: From Simple Numbers to the Calculus, Chaps. 1 through 8, Wm.
Heineman Ltd., �ondon, 1954.
DANTZIG, TOBIAS: Number, the Language of Science, 4th ed., Chaps. I through 6,
The Macmillan Co., New York, 1954 (also in a paperback edition) .
DAVIS, PHILIP J.: The Lore of Large Numbers, Random House, New York, 196 1 .
EVES, HOWARD: An Introduction to the History o f Mathematics, Rev. ed., pp. 29-64,
Holt, Rinehart and Winston, Inc., New York, 1964.
GAMOW, GEORGE: One Two Three . . . Infinity, Chap. 9, The New American
Library, New York, 1953.
HOLTON, G. and D. H. D. ROLLER: Foundations of Modern Physical Science, Chaps.
22 and 23, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958.
JONES, BURTON W.: Elementary Concepts of Mathematics, Chaps. 2 and 3, The
Macmillan Co., New York, 1947.
SMITH, DAVID EUGENE: History of Mathematics, Vol. I, pp. 1-75, Vol. II, Chaps. 1
through 4, Dover Publications, Inc., New York, 1953.
CHAPTER 5
ALGEBRA, T H E HIGHER ARITH M ETIC
Algebra is the intellectual instrument which has been created for rendering
clear the quantitative aspect of the world.
ALFRED NORTH WHITEHEAD
5-1 INTRODUCTION
Mathematics is concerned with reasoning about certain special concepts, the
concepts of number and the concepts of geometry. Reasoning about numbers
-if one is to go beyond the simplest procedures of arithmetic-requires the
mastery of two facilities, vocabulary and technique, or one might say, vocab
ulary and grammar. In addition, the entire language of mathematics is charac
terized by the extensive use of symbolism. In fact, it is the use of symbols
and of ' reasoning in terms of symbols which is generally regarded as marking
the tra nsition from arithmetic to algebra, though there is no sharp dividing
line.
The task of learning the vocabulary and techniques of algebra may be
compared with that which faces the prospective musician. He must learn to
read music and he must develop the technique for playing an instrument.
Since our goal in mathematics is far more the acquisition of an understanding
than the attainment of professional competence, the problem of learning the
vocabulary and techniques will hardly be a severe one.
5-2 THE LANGUAGE OF ALGEBRA
The nature and use of the language of algebra are readily illustrated, although
the illustration is at the moment a trivial one. Most readers have encountered
parlor number games, of which the following is an especially simple example.
The leader of the game says to any member of the group: Take a number; add
10; multiply by 3 ; subtract 30; and give me your answer. And now, says the
leader, I shall tell you the number you chose originally. To the amazement of
the audience he does so immediately. The secret of his method is absurdly
simple. Suppose the subject chooses the number a. Then adding 10 yields
a + 10. Multiplication by 3 means 3 (a + 1 0 ) . By the distributive axiom, this
quantity is 3a + 30. Subtraction of 30 yields 3a. The leader has only to divide
by 3 the number given to him to tell the subject what his original choice was.
94
THE LANGUAGE OF ALGEBRA 95
If the leader wishes to be especially impressive, he can ask the subject to
perform many more computations that will yield a simple and known multiple
of the original number, and he can give the original number just as readily.
By representing in the language of algebra the operations which he asks the
subject to perform and by noting what the operations amount to, the leader
can easily see how the final result is related to the original number chosen.
The language of algebra involves more than the use of a letter to represent
a number or a class of numbers. The expression J(a + 10) contains, in addi
tion to the usual plus sign of arithmetic, the parentheses which denote that
the J multiplies the entire quantity a + 10. The notation b' is a shorthand
expression for b . b and is read b-square. The word square enters here because
Ii' is the area of a square whose side is b. Likewise, the notation b3 means b · b . b
and is read b-cube. The word cube is suggested by the fact that b3 is the
volume of a cube whose side is b. The expression (a + b)' means that the
entire quantity a + b is to be multiplied by itself. An expression such as Jab'
means J times some quantity a and that product multiplied by the quantity b'.
In addition, the notation uses the convention that numbers and letters following
one another with no symbol in between any two are to be multiplied together.
Another important convention stipulates that if a letter is repeated in an
expression, it stands for the same number throughout. For example, in a2 + ab
the value of a must be the same in both terms. Thus algebra uses many
symbols and conventions to represent quantities and operations with quantities.
Why do mathematicians bother with such special symbols and conven
tions? Why must they place hurdles in the way of would-be students of
their subject? The answer is not that mathematicians are trying to introduce
hurdles; nor are they seeking to impress people by making their subject look
awesome. Rather the symbolism of algebra and the symbolism of mathematics
in general are an unfortunate necessity. The most weighty reason is compre
hensibility. Symbolism enables the mathematician to write lengthy expressions
in a compact form so that the eye can see quickly and the mind can retain
what is being said. To describe in words even the simple expression Jab' + abc
would require the phrase, "The product of J times a number multiplied by a
second number which is multiplied into itself added to the product of the first
number, the second number. and still a third number." It is unfortunate that
our eyes and minds are limited. The long and complicated sentences that
would be required if ordinary language were used could not be remembered
and, in fact, can be so involved as to be incomprehensible.
In addition to comprehensibility, there is the advantage of brevity. The
expression in ordinary language of what is covered in typical texts on mathe
matics would require tomes of two to ten or fifteen times the customary
size of such books.
Still another advantage is clarity. Ordinarily English or, for that matter,
any other language is ambiguous. The statement, "I read the newspaper," can
96 ALGEBRA, THE HIGHER ARITHMETIC
mean that one reads newspapers regularly, once in a while, or often, or that
one has read the newspaper, presumably the paper of the day. One must
judge by the context just what this sentence means. Such ambiguity is in
tolerable in exact reasoning. By using symbols for specific ideas mathematics
avoids ambiguity or, to put the matter positively, each symbol has its own
precise meaning, and so the resulting expressions are clear.
Symbolism is one of the sources of the remarkable power of algebra. Sup
pose that one wished ro discuss equations of rhe form 2x + 3 0, 3x + 7 0,
=
=
4x 9 - =0, and the like. The particular numbers which appear in these
equations do not happen to be important in the discussion; in fact, one wishes
to include all equarions in which the product of some number and x is added
to some other number. The way to represent all possible equations of this
form is
ax + b = 0. (1 )
Here a stands for any number, and so does b. These numbers are known,
but their precise value is not stated. The letter x stands for some unknown
number. By reasoning about the general form ( 1 ) the mathematician covers
the millions of separate cases which arise when a and b have specific values.
Thus, by means of symbolism, algebra can handle a whole class of problems
in one bit of reasoning.
Of course, it is unfortunate that one must learn the elements of a new
language to master some mathematics. But one could with much justice
complain that the French people insist on their language, the Germans on
theirs, and so on. Obviously English is the best language, and the French and
Germans are exhibiting provincialism by insisting on holding on to their re
spective languages. The language of mathematics has the additional merit of
being universal.
There are justifiable criticisms of the symbolism of algebra, although they
are hardly major ones. Mathematicians are greatly concerned about the ac
curacy of their reasoning, but pay little attention to the aesthetics or appropri
ateness of their symbolism. Very few symbols suggest their meaning. The
signs + , , -
=
,varc easy to write, but they are historical accidents. No
mathematician has bothered to replace these by at least prettier ones, perhaps 0
for plus. The seventeenth-century mathematician Gottfried Wilhelm Leibniz,
who did spend days on the choice of symbols in an effort to make them sug
gestive, was an exception. There are even inconsistencies in symbolism which,
once recognized, fortunately do not impair the clarity. For example, when
two letters, such as ab, are written together with no symbol between them,
then it is understood that multiplication is meant. However two numbers,
such as 3t, with no symbol between them, mean 3 + t.
Symbolism entered algebra rather late. The Egyptians, Babylonians,
Greeks, Hindus, and Arabs knew and applied a great deal of the algebra which
EXPONENTS 97
we learn in high schoo!. But they wrote out their work in words. Their
algebraic style is in fact called rhetorical algebra because, except for a few
symbols, they used ordinary rhetoric. It is significant that symbolism entered
mathematics in the sixteenth and seventeenth centuries when pressure to im
prove the efficiency of mathematics was applied by science. The idea of using
symbols was no longer new, but mathematicians were undoubtedly stimulated
to extend the application of symbolism and to adopt it readily.
EXERCISES
1. Why does mathematics use symbols?
2. Criticize the statement that all men are created equal.
3. In the following symbolic expressions the letters stand for numbers. Write out
in words what the expressions state.
a) a + b b) a(a + b) c) a(a2 + ab) d) 3x2y
f) x + 3
e) (x + y)(x -- y) g) t(x + 3)
7
4. Does
x+ 3
-- = t(x + 3 ) ?
7
[Suggestion: What do these symbolic expressions say in words?]
5. Write in symbols: (a) three times a number plus four; (b) three times the square
of a number plus four.
5-3 EXPONENTS
One of the simplest examples of the convenience of algebraic symbolism or
algebraic language is found in the use of exponents. We have already used
frequently such expressions as 52. In this expression the number 2 is an expo
nent, and 5 is called the base. The exponent is placed above and to the right
of the base to indicate that the quantity to which it applies, 5 in this example,
is to be multiplied by itself, so that 5 ' 5 . 5. Of course, there would be no
=
great value in the use of exponents if their use were limited to such instances.
Suppose, however, that we wished to indicate
5 . 5 . 5 . 5 . 5 . 5.
Here 5 occurs as a factor six times. We can indicate this quantity hy means of
exponents thus: 5 6, That is, when the exponent is a positive whole number it
indicates how many times the quantity to which it is applied occurs as a
factor in a product of this quantity and itself. In such instances as 5' the use
of exponents saves a lot of writing and counting of factors.
98 ALGEBRA, THE HIGHER ARITHMETIC
Exponents are even more useful than we have thus far indicated. Suppose
we wished to write
5 . 5 . 5 . 5 . 5 . 5 times 5 . 5 . 5 . 5 .
With exponents we can write
5 6 . 5 '.
Moreover, the original product calls for 5 multiplied by itself to a total of
10 factors. This product can be written as 510. We see, however, that if we
add the exponents in 56 . 5\ we also get 510• That is, it is correct to write
5 6 . 5' =
5 6+' =
5 10.
More generally, when m and 11 are positive whole numbers,
This statement is really a theorem on exponents. Its proof is trivial. All that
the theorem says is that if one quantity contains a as a factor m times, and
another quantity contains a as a factor 11 times, the product of these two
quantities contains a as a factor m + 11 times.
And now suppose we wished to write
5.5.5.5.5.5
5.5.5.5
With the use of exponents we can write
56
5'
Moreover, if we were to calculate the value of the original quotient, we know
that we could cancel 5's in the numerator and denominator. We would be
left with
5.5
or
We can obtain the same result if in 5 '/54 we subtract the 4 from the 6 and so
arrive at 52, Here. too, as in the case of multiplication, the exponents keep
track of the number of 5's which occur in the numerator and denominator, and
the subtraction of the 4 from the 6 tells us the net number of 5's remaining as
factors.
EXPONENTS 99
]n more general language, we can say that if m and 11 are positive whole
numbers, and if m is greater than n, then
This result, too, is a theorem on exponents, and the proof is again trivial be
cause all the theorem states is that if we cancel the a's common to numerator
and denominator, we shall have m 11 factors left over.
-
We might find that we have to deal with
5'5.5.5
5
5 .5.5.5.5
.
In exponent form this quotient is
This time, if we cancel the 5 's common to the numerator and denominator, we
are left with 5 occurring twice in the denominator; that is, we are left with
We can obtain this result at once by subtracting the exponent 4 in the numera
tor from the exponent 6 in the denominator. ]n general form we have the
theorem: If m and 11 are positive whole numbers and if n is greater than m,
then
There is the possibility of encountering
5.5.5.5
5.5.5.5
In exponent form this quotient is written
It would be nice to be able to simplify this expression, too, by using exponents.
However, here, unlike the two previous cases, the twO exponents are equal. If
100 ALGEBRA, THE HIGHER ARITHMETIC
we were to, say, subtract the exponent 4 of the denominator from the exponent
4 of the numerator, we would have
5'
5 '- ' =
5°
5'
Now 5° has no meaning. However we know that 5'/54 has the value 1 . If we
agree to give a meaning to a zero exponent and, in fact, agree that a number
to the 0 exponent is to be I , then we can use the symbol 5°. Further, with this
meaning we can properly write
5'
- =
5 4 -' = 5° = 1 .
5'
]n general, if m is a positive whole number,
EXERCISES
1 . Simplify the following expressions by using the theorems on exponents:
5' f) 10'
a) 5 " 5 6 b) 63 . 6' c) 105 . 10' d) x2 . x3 e) -
5' 104
x4 1 04 5' 7' 57 . 5 2
g) - h) i) 10 ' 10' j) 5' k) 1) --
2
x 10' 74 58
2. Could we apply the above theorems on exponents to negative numbers as bases?
Then, is it true that (- 3 )'(- 3 )4 = ( _ 3 )9?
3 . Which of the following equations are correct?
a) 32 + 3' = 36 b) 32 • 34 = 36
e) ) 4 + ) ' = ) 8
We can use exponents even more effectively than has thus far been in
dicated. Suppose that in the course of algebraic work, there occurred the
expression
53 ' 5 3 ' 53 ' 5 3 •
• It is necessary to add that a must not be 0, because then the original quotient has
no meaning.
EXPONENTS 101
Could we write this more briefly ' By the very meanmg of an exponent we
certainly can write
5 3 . 5 3 . 53 . 5 3 = ( 53) 4 .
We can go further. The left-hand side of this equation contains 5 as a factor
1 2 times. We can recognize the same fact if we multiply the two exponents on
the right-hand side. That is,
This example is the essence of another theorem on exponents. namely, if m and
n are positive integers, then
There is one more commonly useful theorem on exponents. Suppose we
wished to denote
2.2.2.2.3.3. 3.3
briefly, by taking advantage of exponents. We certainly could write this
quantity as
24 • 3 4•
However, we know that the order in which we multiply numbers does not
matter. Hence it is correct that
2 ' 2.2 .2. 3 .3.3. 3 = 2'3.2.3.2. 3 . 2 . 3,
and now if we use exponents, we can say that
24 ' 3 4 = (2 ' 3) 4 .
What this fact amounts to, in general terms, is that if m is a positive whole
number, then
EXERCISES
1 . Use the theorems on exponents to simplify the following expressions:
a) 3 4 ' 3 4 ' 34 b) ( 34)3 c) (5 4) , d) 10' . 10' . 10'
e) ( 1 04) 3 f) 54 . 24 g) 37 . 3 3 h) 1 04 . 34
2. Calculate the values of the following Quantities:
24 . 3'" 45 . 2 5 (ab) 3 a3b3
a) 25 . 5 5 b) c)
(ab) 2
c) d)
63 S" a'
102 ALGEBRA, THE HIGHER ARITHMETIC
3. Which of the following equations are correct?
a) (3 ' 10)' = 3' · 10' b) (3 ' 102)3 = 33 . 10" c) (3 + 10)' = 3' + 10'
d) (3 2 . 53)' =
38 . 5 " e) (3')3 = 3' f) (32) 3 = 39
All the above theorems deal with positive integral (whole-numbered)
exponents or zero as an exponent. Though we shall not deal with other types
of numbers as exponents, it is significant to know that the exponent notation
can be more valuahle than we have thus far indicated. Let us consider y3. We
know that
VJ · VJ = 3.
Let us suppose that we would like to investigate whether the exponent notation
could be used to simplify work with irrational numbers. (Of course, this is
the kind of problem that one takes up when he has nothing better to do.) Now
the right-hand side of the ahove equation can be written as 3 1 • No matter what
exponent notation we do adopt for '1/3. say 3a, the equation would have to read
3a . 3" = 3 I.
Moreover, we would like, if possible, to maintain the validity of our previous
theorems on exponents. In the present case, we would like to be able to say
that
and since pa 3', we would have to have Za = 1 or a
= = 1. What these ex
ploratory thoughts suggest is that if we denote
VJ by 3 1 / 2,
then we would be able to use at least the first theorem on exponents to state
that
3 1/2 . 3 1/2 = 3 1/2+1f2 31 3. = =
As a matter of fact, this example typifies what is done. Thus we use the
notation
yl 3 1/2=
,
and so on.
5-4 ALGEBRAIC TRANSFORMATIONS
Symbolism is a means to an end. The function of algebra is not to display
symbols hut to convert or transform expressions from one form to another
which may be more useful for the problem in hand.
ALGEBRAIC TRANSFORMATIONS 103
Let us consider an example. Suppose that in the course of some mathe
matical work we encounter the expression
(x + 4) (x + 3 ) . (2)
The letter x in this expression may stand for some number whose value we
do or do not happen to know, or it may stand for any one of some class of
numbers. What matters is that x stands for a number. If x is a number, then
x + 4 is a number. We may now apply the distributive axiom, which states
that for any numbers a, h, and c,
a(b + c) = ab + ac. (3)
If we compare ( 2 ) and ( 3 ) we see that (2) has the form of ( 3 ) if we think
of x + 4 in (2) as the a of ( 3 ) . Then by applying the distributive axiom to
(2) we may assett that
(x + 4) (x + 3) = (x + 4)x + (x + 4) 3 . (4)
We also know that there is another form of the distributive axiom, namely
(b + c)a = ba + ca.
If we apply this axiom to each of the terms on the right side of (4), we see
that
(x + 4).x = x2 + 4x and (x + 4) 3 = 3x + 1 2 .
If we substitute these last two results on the right side of (4), w e have
(x + 4) (x + 3) = x2 + 4x + 3x + 1 2 = x 2 + (4 + 3)x + 1 2 (5)
or
(x + 4) (x + 3) = x2 + 7x + 1 2 . (6)
Before discussing what this example illustrates, let us note that we do not
usually carry out the multiplication of (x + 4) by (x + 3 ) in this long and
rather cumbersome fashion. Instead we write
x + 4
x+ 3
3x + 1 2
x2 + 4x
x2 + 7x + 1 2.
The partial product 3x + 12 results from multiplying x + 4 by 3, and the
104 ALGEBRA, THE HIGHER ARITHMETIC
partial product x' + 4x results from multiplying x + 4 by x. The two partial
products are then added. This manner of carrying out the multiplication is
faster, but fails to indicate explicitly that we have used the distributive axiom
several times.
The main point of the above example is that we have transformed the
expression (x + 4) (x + 3 ) into the expression x2 + 7x + 1 2 . We do not
maintain that the latter expression is more attractive than the former, but it
may be more useful in a particular mathematical application. On the other
hand, we might, in some situation, find ourselves with the expression x2 +
7x + 12 and, by recognizing that it is equal to (x + 4) (x + 3 ) , be able to
make progress toward some significant conclusion. In this latter transformation
we say that we have factored x2 + 7x + 1 2 into (x + 4) (x + 3 ) . Which of
the two forms is more useful depends upon the application in hand. At the
moment we should merely see that algebra is concerned with the technique of
such transformations, and that a skilled mathematician should be able to per
form them rapidly. Since we shall not become too involved in complicated
technical processes, we shall not spend much time in developing skills.
The problem of factoring to which we referred in the preceding paragraph
does arise reasonably often. For example, one usually starts with an expression
such as x' + 6x + 8 and seeks to transform it into a product of factors of the
form (x + a) (x + b ) . The original expression is said to be of second degree
because it contains x' but no higher power of x. The factors are first-degree
expressions because each contains x but no higher power of x. The problem
is to find the correct values of a and b so that the product ( x + a) (x + b)
will equal the original expression. We know from our work on multiplication
[see equation ( 5 ) 1 that
x' + (a + b)x + ab = (x + a) (x + b).
Hence to factor the second-degree expression, we should look for two num
bers a and b whose sum is the coefficient of x and whose product is the con
stant. Thus to factor x2 + 6x + 8, we look for two numbers whose sum is 6
and whose product is 8. By mere trial of the possible factors of 8 we see that
a 4 and b 2 will meet the requirement; that is:
= =
x2 + 6x + 8 = (x + 4) (x + 2 ) .
EXERCISES
1. Transform to an equal expression:
a) 3x ' 5x b) (x + 4) (x + 5) c) (3x + 4) (x + 5)
d) (x - 3) (x + 3) e) (x + �)(x + �) f) (x + �) (x - �)
ALGEBRAIC TRANSFORMATIONS 105
2. Factor the following expressions. Experiment with numbers to find the correct
factors.
a) x' + 9x + 20 b) x' + 5x +6 c) x' - 5x + 6
d) x' - 9 e) x' - 16 f) x' + 7x - 1 8
3. Prove that x(x' + 7x) = x" + 7x'.
4. Can you think of a way of testing or verifying (not proving) that
x' + 5xy + 6y' = (x + 3y) (x + 2y)
for all values of x and y?
5. Write out in words the equivalent of
(x - 3 ) (x + 3 ) = x' - 9.
6. A high school girl had to simplify (a2 - b2 )/(a b). She reasoned that a'
-
divided by a gives a. Minus divided by minus gives plus. And b2 divided by b
gives b. Hence the answer is a + b. Is the answer correct? Is the argument
correct?
7. There is a well known "proof" that 2 I. The proof runs as follows.
=
Suppose a and b are two numbers such that
a = b.
We may multiply both sides of this equation by a and obtain
a2 = abo
Now we may subtract b2 from both sides and obtain
By factoring we may replace the left and right sides of this equation by
(a - b) (a + b) = b(a - b).
Division of both sides of this equation by a - b yields
a+ b = b.
Since a = b, we may as well write
2b = b.
But now we can divide both sides of this last equation by b, and there results
2 = I.
Find the flaw in this proof.
106 ALGEBRA, THE HIGHER ARITHMETIC
5-5 EQUATIONS INVOLVING UNKNOWNS
The study of algebraic transformations as such is not very interesting. It is
much like the grammar of a language. The significant uses of these transforma
tions occur in larger investigations which we shall undertake later. However,
a direct use of the processes of algebra does arise in the problem of finding
unknown quantities, a problem not without some interest in itself and one
which also arises in the course of broader investigations.
A somewhat practical, though by no means vital, example is the following.
The radiator of a car contains 1 0 gallons of liquid 20 per cent of which is
alcohol. The owner wishes to draw off a quantity of liquid and replace it by
pure alcohol so that the resulting mixture contains 50 per cent alcohol. How
many gallons of liquid should he draw off?
Now the very practical person who "refuses to use mathematics can handle
this situation very readily. He can draw off 5 gallons of the mixture and re
place it by 5 gallons of alcohol. Then the mixture will certainly contain at
least 50 per cent alcohol because even the remaining 5 gallons contain some
alcohol. However, if the final mixture need contain only 50 per cent alcohol,
then the practical person has wasted alcohol and therefore money. If he draws
off 4 gallons, 6 gallons will be left, and since 20 per cent of this is alcohol, the
alcoholic content is It gallons. If he now adds 4 gallons of alcohol he will
have 5t gallons of alcohol, or more than 50 per cent, in the 10 gallons. On the
other hand, if he draws off only J gallons, 20 per cent of the remaining 7
gallons is t gallons of alcohol, and the addition of J more will yield 4i gallons
of alcohol out of 10, or less than 50 per cent. The correct answer lies some
where between 3 and 4, but where? Instead of continuing to guess let's use a
little algebra.
Let x be the number of gallons of the mixture to be drawn off and to be
replaced by an equal amount of pure alcohol. Then the number of gallons
remaining of the original mixture is 1 0 - x. Of this 20 per cent, or t, is alcohol,
so that of the 10 - x gallons, (i) ( I O - x) is alcohol. After the x gallons are
replaced with pure alcohol, the amount of alcohol in the tank will be ( ! ) ( 10 -
x) + x. We should like to fix x so that the amount of alcohol should be 50
per cent of 10 gallons, or 5 gallons. Hence we seek the value of x which
satisfies the equation
HIO - x) + x = 5. (7)
Now we can apply the distributive axiom to start off our transformations and
write
! . 10 - tx + x = 5 . (8)
The terms - !x + x or x - tx amount to !x. Hence (8) is equivalent to
2 + !x = 5. (9)
EQUATIONS INvOLVING UNKNOWNS 107
If we now subtract 2 from both sides of this equation, the result will still be
an equality because equals subtracted from equals give equals. Then
!x = l.
We now multiply both sides of this equation by �, and since equals multiplied
by equals give equals, we have
x = 3 · :t. (10)
Hence the answer is that the owner of the car should draw off J! gallons of
the original liquid. We knew before we applied algebra that the answer lies
between 3 and 4, and we now know exactly where.
The more significant point made by this example, however, is that we
started with equation (7) which expresses the condition to be satisfied by the
unknown quantity x and that, by executing a series of almost mechanical steps
justified by axioms about numbers, we arrived at a new equation, ( 10), which
tells us what we wish to know. In other words, we performed a series of
transformations which carried us from one equation to another and we profited
thereby. The answer is not sensational, but we see how the manipulation of
symbols gives us new information.
There is another point which the above example illustrates, at least in a
minor way. Once we formulate equation (7) we forget all about the physical
situation and concentrate solely on the equation. Nothing that is not relevant
to the problem, i.e., to the problem of determining the number x, interferes
with our thinking. Ernst Mach, a famous scientist of the late nineteenth cen
tury, said that mathematics is characterized by "a total disburdening of the
mind," and we can now see what he meant. The make of the car, the shape
of the radiator, the fact that the owner may be concerned with protecting the
liquid in the radiator from freezing, and any other facts which have nothing
to do with determining x can be forgotten. We disburden our minds of every
thing but the quantitative facts expressed in equation (7), and proceed to
handle quantitative relationships only.
Equation (7) is rather simple. It is called a linear or first-degree equation
because the unknown x occurs to the first power only. Let us consider a
second example which will again illustrate the transformation value of algebra,
but which also has other interesting features. Suppose that one ship is at A
(Fig. 5-1) and another is at B, exactly 10 miles north of A. The ship at B is
steaming east at the rate of 2 miles per hour. The ship at A is capable of
traveling at a speed of 5 miles per hour and wishes to intercept the other ship.
To set his course properly the captain of the ship at A must know where the
two will meet.
Let us suppose that C is the point where they will meet. If the captain
can determine the distance BC, he will head along the hypotenuse of a right
108 ALGEBRA, THE HIGHER ARITHMETIC
triangle whose arms are AB and BC. Let us therefore denote the distance BC
by x. Now that we seem to have labeled all relevant quantities, we encounter
the first puzzling aspect of this problem, namely, that we do not have any
equation to find x. Without this, of course, we can only sit and do some wish
ful thinking. Vet we do have enough information to set up such an equation.
What we have overlooked is a physical fact which is implied by the given
information: The time that the ship at B will take to travel to C must be the
same as the time it will take the ship at A to reach C. Since the ship at B
travels at 2 miles per hour, it will take xl2 hours to reach C. To calculate
the time required by the ship at A to reach C, we need the distance AC. We
do not know AC, bur we can at least express its value by means of the
Pythagorean theorem of geometry. This theorem says in the present instance
that
AC ' = 100 + x' .
Then Brr--"x�--, C
AC = VTOO x' .+
The time required for the ship at A to travel the dis- 10
tance AC at 5 miles per hour is
ylOO + "
5 A Fig. 5-1
We next equate the time required for the ship at B to travel the distance
BC and the time required for the ship at A to travel the distance AC. This
equation is
x ylOO + "
(II)
2 5
We now have an equation to work with. Let us see whether we can trans
form it so that it will yield a value for x. Since the square root is annoying,
let us square both sines, i.e., multiply the left side by itself and the right side
by itself. Since the left side equals the right side we are in effect multiplying
equals hy equals, and so the step is justified. Squaring both sides, we obtain
" 100 + "
( 1 2)
4 25
Since fractions are also annoying, let us multiply both sides by lOa. We choose
100 because both 25 "nd 4 divide evenly into 100. Thus
" 100 + x'
100 ' - = 100 .-::c:-
- -
4 25
EQUATIONS INVOLVING UNKNOWNS 109
We may apply our operations with fractions to write
25x' = 4(100 + x') .
Application o f the distributive axiom yields
( 1 3)
Now we subtract 4x2 from both sides, and because equals subtracted from
equals yield equals, we obtain
2 1 x' = 400. (14)
Division of both sides by 2 1 , which is a division of equals by equals, yields
2
400
x = _ . (1 5)
21
Now we ask ourselves what number squared yields 400/21. Certainly
v'400/21 is one possibility. But a negative number squared or multiplied by
itself is also positive. Hence there are two possible answers:
x = v'400/21 and x = - v'400/2 1 . (16)
Let us accept both of these for the moment and dispose first of a purely
arithmetical question. How much is v'400/21? Well, we can divide 2 1 into
400 and obtain 19.05 to two decimal places. We must now find v'19.05.
There is an arithmetic process for finding the square root of a number, but
for our purposes it will be sufficient to estimate the answer. Clearly 4 is toO
small and 5 is too large. By sheer trial we find that (4. 3 ) ' = 1 8.49 and
(4.4) ' = 19.36. Hence the correct value lies between 4.3 and 4.4. If we
wished to have a more accurate answer, we could now try 4. 3 1 , 4.32, and so
on, until we found a result which came as close to 19.05 as possible, and so
obtain an answer to the nearest hundredths' place. We shall accept 4.4 as
good enough for our purposes and thus we may say that
x = 4.4 and x = -4.4. (17)
And now we have more than we want; we have two answers, whereas we
sought only one. Of course, we wish to use the positive answer because the x
we seek stands for a length which is positive. This is the value which has the
proper physical meaning in our problem. But the question, How did the
negative value of x get into the picture, remains open. The answer involves
a rather important point about the nature of mathematics and its relation to
the physical world. The mathematician starts with concepts and axioms which
1 10 ALGEBRA, THE HIGHER ARITHMETIC
express some idealized facts about the world, and proceeds to apply these
concepts and axioms to solve physical problems. In the present case the
methods used lead to two solutions. Hence the methods may involve new
elements which arc not present in the physical world, even though the intent
was to stay close to it. Thus, squaring both sides of equation ( I I ) , a justifiable
mathematical step, introduced a new solution, for, if our original equation had
been
x VIOO + X2
(I 8)
2 5
we would have obtained the same equation, ( 12) and everything we did
thereafter would have applied to ( 1 8 ) as well as ( I I ) . Hence, in this case,
we can see specifically where mathematics departs from the physical situation.
The main point to be noted is then that, although mathematical concepts
and operations are formulated to represent aspects of the physical world,
mathematics is not to be identified with the physical world. However, it tells
us a g<'>od deal about that world if we are careful to apply it and interpret it
properly. We shall find that this point, which eluded the best thinkers until
the late nineteenth century, will acquire increasing importance as we proceed.
There is another valuable lesson to be learned from the solution of the
problem we have just examined. When we arrived at step ( 1 3 ) , we combined
terms in x' and then proceeded to find x. The subsequent work led to a fair
amount of arithmetic. An engineer working with the same problem and
perhaps satisfied with an approximate answer might argue that the term 4x'
is small compared with the term 25x' and so disregard it. Instead of our next
equation, ( 1 4), his new equation would then read
25x2 = 400,
and by dividing both sides of this equation by 25, he would obtain
x2 = 1 6.
It now follows that
x = 4 and x = -4.
Thus 4 is an approximate answer. Engineers often are satisfied with such
approximations because, in constructing actual objects of wood and steel, they
cannot meet a specified value exactly. Not only can't one measure exactly,
but tools and machines also introduce errors. By neglecting 4x' in ( IJ ) the
engineer gained the advantage of finding the approximate answer much more
readily than we were able to determine the correct answer even to one decimal
place only.
EQUATIONS INVOLVING UNKNOWNS III
In the present problem the saving is trivial, but approximation may make
a lot of difference in more difficult problems. Whereas the mathematician,
who seeks exact answers, will work months and years on a problem, the
engineer will often settle for an approximate answer and obtain it far more
easily. The point we are making is not that the engineer is smarter. To get
on with his job the engineer must arrive at an answer quickly, whereas the
mathematician's job is to obtain a correct answer, no matter how long it takes.
Both are true to the objectives and spirit of their own work. Moreover, in
making approximations, the engineer raises a question which he may not be
able to answer. How good is his approximation? After all, while physical
constructions and measurements are not exact, beams must fit. Hence the
engineer should really ascertain that the approximation is good enough for
his purposes. If he can tolerate an error of only 0.1 of an inch, he must make
sure that his approximations do not introduce a larger error.
In really difficult problems the engineer will make approximations and,
usually with the aid of a mathematician, determine the error introduced. If
he cannot do so, he will often overdesign; that is, if the approximate result
shows that a beam supporting a building need be only one inch thick, he may
make it two inches thick and thereby hope that he has more than allowed for
the error. Is he certain even with this precaution that his beam will hold up?
No. Big bridges have collapsed because such calculations and additional pre
cautionary measures were not enough. A recent example was the Tacoma
bridge in the State of Washington. The bridge did not withstand the force of
the wind and collapsed.
EXERCISES
I. The speed of sound in an iron rod is 16,850 ft/sec, and the speed in air is 1 100
ft/sec. If a sound originating at one end of the rod is heard one second sooner
through the rod than through the air, how long is the rod?
2. A bridge AB is 1 mi (5280 ft) long in winter and expands 2 ft in the summer.
For simplicity suppose that the shape in summer is the triangle ACB shown in
Fig. 5-2. How far does the center of the bridge drop in summer, that is, how long
is CD? Before calculating the answer, estimate it. To calculate, use the Pythag
orean theorem and estimate the square root to the nearest foot.
A D B
~ C Fig. 5-2
3. An airplane which can fly at a speed of 200 mi;hr in still air flies a distance of 800
mi with the wind in the same time as it flies 640 mi against the wind. What is the
112 ALGEBRA, THE HIGHER ARITHMETIC
speed of the wind? [Suggestion: If x is the speed of the wind, then the speed of
(he plane when flying with the wind is 200 + x; the speed of the plane when
flying against the wind is 200 x.]
-
4. The population of town A is 10,000 and is increasing by 600 each year. The
population of town B is 20,000 and is increasing by 400 each year. After how
many years will the two towns have the same population?
5. A rope hanging from the top of a flagstaff is 2 ft longer than the staff. When
pulled out taut, it reaches a point on the ground 18 ft from the foot of the staff.
How high is the staff?
6. A publisher finds that the cost of preparing a book for printing and of making
the plates is $5000. Each set of 1000 printed copies cOstS $1000. He can sell the
books at $5 per copy. How many copies must he sell to at least recover his costs?
7. We may certainly say that
! dollar= 25 cents.
We take the square root of both sides and obtain
! dollar 5 cents.
=
What is wrong?
8. A glass which is half full certainly contains as much liquid as a glass which is
half empty. Then
! f = ! empty.
ull
If we multiply both sides by 2 we obtain
I full = I empty.
or a full glass contains as much as an empty glass. What is wrong?
5-6 THE GENERAL SECOND·DEGREE EQUATION
Our discussion of the solution of equations in the preceding section dealt with
two types of equations, first-degree equations illustrated by equation ( 7 ) and
second-degree equations illustrated by equation ( 14). No difficulties can arise
in the process of solving first-degree equations, i.e., equations which, by proper
algebraic operations, can be expressed in the form
ax + b = O, (19)
where a and b are definite numbers and x is the unknown. Equation ( 19) can
readily be solved for x.
The case of second-degree equations is not so simple. We were fortunate
that equation ( 14) led to ( 1 5 ) . and that by taking the square root of both
sides we obtained the two solutions, or roots as they are called. However we
might have to solve an equation such as
x2 - 6x + 8 = O. (20)
THE GENERAL SECOND-DEGREE EQUATION 113
This equation is more complicated than ( 14) because (20) also contains the
first-degree term in x.
In solving equation (20), we still do not encounter much trouble. We
know from our work on transforming algebraic expressions that the left-hand
side of (20) can be factored; that is, the equation can be written as
(x - 2) (x - 4) = o. (21)
We now see that when x = 2, the left side is zero because
(2 - 2) (2 - 4) = O.
When x = 4, the left side is again zero because
(4 - 2) (4 - 4) = O.
Hence the solutions or roots are
x = 2 and x = 4.
Now suppose we had to solve the second-degree equation
,,' + lOx + 8 = O. (22)
This time it is not possible to find simple factors of the left side. Equations
such as (22) do arise in real problems. Hence the mathematician considers
the question, Is there a method which will solve such second-degree equations'
Naturally he studies those he can solve to see whether they furnish any clue
to such a method.
Examination of equation (20) reveals an interesting fact. The roots are
2 and 4. The sum of these two numbers is 6, and the coefficient, or multiplier,
of x is -6. The product of 2 and 4 is 8, and 8 is the constant term, t," is,
,t
the term free of x. These facts might be a coincidence, and so the mathema
tician would investigate whether they hold for other simple equations. Con
sider the very simple equation:
'
x - 4 = O. (23)
Here the roots are +2 and - 2. Their sum is 0, and we note that the term
in x is missing. which means it is 0 · x. The product of the roots is - 4 pre
,
cisely the constant term in ( 2 3 ) . Presumably we have some facts about the
roots, but how can we use them?
Equations of the form ( 2 3 ) are easy to solve, since one only has to take a
square root. Perhaps the method we should seek is one which reduces all
equations of the type (20) to the type ( 2 3 ) . But how do we do this? The
sum of the roots in ( 2 3 ) is zero. The sum of the roots in (20) is 6, and this is
1 14 ALGEBRA. THE HIGHER ARITHMETIC
the negative of the coefficient of x. If we added to each root of (20) one-half
the coefficient of x, that is, 3, the sum of the roots would be zero. What
-
this suggests, then, is to form a new equation whose roots are the roots of the
old one, each increased" by one-half the coefficient of x. Since the coefficient
of x is 6 we let
- ,
y = x + ( - 3) = x - 3
or
x = y + 3. (24)
If we substitute this value of x in (20), we obtain
(y + 3) 2 -
6(y + 3) + 8 = O.
We now calculate the square in the first term, carry out the multiplication in
the second term, and find that
y2 + 6y + 9 6y - 1 8 + 8 = 0
or
y2 _ 1 = O.
Then
y2 =
and
y = and y = -1.
But from (24) we see that
and x = -1 + 3
or
x = 4 and x = 2.
Thus we obtain without factoring the very same roots of equation (20) that
we found previously by factorization.
Now let us reconsider equation (22), namely
x2 + lOx + 8 = O. (22)
Since the roots cannot be obtained by any apparent method of factoring, let
us see whether the idea just tried works here also; that is. let us form a new
equation whose roots are the roots of (22) increased by one-half the coeffi-
•We use the term "increased" here, even though in the example we add a negative
quantity to each root and really decrease the value of the roots.
THE GENERAL SECOND-DEGREE EQUATION 115
cient of x. The roots of (22) are represented by x. Then we shall form a new
equation whose roots y are:
y= x + ¥ = x + 5. (25)
From (25) we have
x = y - 5. (26)
We substitute this value of x in (22) and obtain
(y - 5)' + 1O(y - 5) + 8 = O.
We perform the indicated multiplications and obtain
y ' - l Oy + 25 + lOy - 50 + 8 = O.
By combining terms we find that
y' 17 = 0
or
y ' = 17.
Then
y = vu and y= -
VU .
We now use (26) to state that
x = VU - 5 and x = -VU - 5. (27)
We have found the two roots of (22) without factoring.
EXERCISES
1. Find the roots of the following equations by factoring the left-hand side:
a) x' - 8x + 1 2 = 0 b) x' + 7x - 18 =0
2. Find the roots of each of the equations in Exercise 1 by forming a new equation
whose roots are "larger" than those of the original equation by one-half the
coefficient of x.
3. Solve the following equations by the method of forming a new equation whose
roars are "larger" than those of the original equation by one-half the coefficient
of x.
a) x' + 12x + 9 = 0 b) x' - 12x + 9 = 0
The method of solving second-degree equations by forming a new equa
tion seems to work, but we have no proof that it will always work. To secure
116 ALGEBRA, THE HIGHER ARrTHMETIC
a general proof we shall use one of the basic devices of algebra; that is, instead
of working with particular equations, we shall consider the general second
degree equation
x2 + px + q = O. (28)
Here p and q are letters, each of which can stand for any given real number.
The use of the letters p and q must be distinguished from the use of x to stand
for the specific unknown rOOts of the equation. Now we follow the method
employed to solve equations (20) and (22); that is, we form a new equation
whose roots are the roots of (28), each increased by one-half the coefficient
of x. This means that we introduce the expression
Then
(29)
We substitute this value of x in (28) and obtain
By squaring the first term and multiplying through by p in the second one,
we obtain
2 2
y2 _
py + L + py _ f!.::. + q = O.
4 2
The terms involving py cancel. Moreover, p2/4 - p2/2 = -p'/4. Hence
2
y2 _ L+ q =
O.
4
By adding p'/4 to both sides and subtracting q from both sides, we obtain
p2
Y2 = - - q.
4
Hence
y = �p: - q and y = - l: - q.
In this general case, we cannot determine the numerical value of the square
THE GENERAL SECOND-DEGREE EQUATION 117
root, but we can leave the result in this form. We now see from equation
(29) that
x= I'
p q e and (30 )
'V 4
_ _
2
This result is remarkable.· We have shown that the roots of any equation
of the form (28) (that is, no matter what p and q are) are given by the ex
pressions ( 30 ) .
We really have accomplished more than we sought to accomplish. We
sought a method of solving an equation such as ( 2 2 ) . We not only have
found such a method, but, since the result ( 30) holds for any such equation,
we do not have to go through the entire process each time; we proceed by
simply substituting the proper value of p and q in ( 30) . Thus if we compare
equations (22) and (28), we see that the p in (22) is 10 and q is 8. Hence
let us substitute [0 for p and 8 for q in ( 30). We find
x= v� - 8 - V- and x = - v� - 8 - V- ,
or
x = Vl7 - 5 and x = - Vl7 - 5. ( J l)
This is exactly the result obtained in (27).
By working with the general form x' + px + q = 0 instead of equations
with specific numbers as coefficients, we have shown how to solve any second
degree equation. This general result could never be derived from equation,
with numerical coefficients because there are infinitely many such equations,
and one could not investigate them all. Thus the use of letters to represent
any one of a class of numbers gives mathematics a power and generality
which achieves what could not be accomplished in many lifetimes of effort
with particular equations. Of course, to people who do not care to solve one
quadratic the ability to solve all is no boon. But even these people have
benefited indirectly. The preceding theory illustrates how the mathematician,
when called upon to solve the same type of problem repeatedly, seeks a general
method which will handle all of them.
• In many books a method is given for solving the general second-degree equation
ax' + hx + c = O. If we divide this equation by a, we obtain x' + (hla ) x + (cia ) = o.
This equation is now of the same form as (28), where p = bja and q = cia. If we
enter these values of p and q in ( 30), we get the roots
b -v!b, -'- ,:
" ::
;;- 4a b -v!"'---- 4a,
b, -;; -:::
-
'
- - - -'---'- """
- '''
x = - - + -=-= ---' and x=
2a 2a 2a 2a
1 18 ALGEBRA, THE HIGHER ARITHMETIC
The use of letters such as p and q, which has made an enormous difference
in the effectiveness of mathematics, seems like a small idea once understood,
and yet it is a rather recent development. From the time of the Babylonians
and Egyptians to about 1550, all the equations solved had numerical coefficients.
Although many algebraists realized that the method they used for one set of
numerical coefficients would work for any other, they had no general proof.
The idea of employing general coefficients in algebraic equations, an idea
which, as we shall see, was taken over into other domains of mathematics, is
due to Fran�ois Vieta ( 1540--1603), a great French mathematician. The re
markable fact about Vieta is that he was a lawyer who worked for the kings
of France. Mathematics was just a hobby to him, but one at which he
"worked" extensively. Vieta was ful1y conscious of what he had done by in
troducing literal coefficients. He said that he was introducing a new kind of
algebra which he called logistica speciosa, that is calculation with whole species,
as opposed to the numerical work of his predecessors which he called logistica
llumerosa.
We could consider other examples of how the processes of algebra permit
us to solve equations involving unknowns, but we shall not devote more time
to the subject. What is important is the recognition that by means of algebra
we can extract information from some given facts. ] t is also important to see
how readily and mechanically the processes of solving equations yield the
desired information. In fact, one of the curious things about mathematics that
clearly emerges even from our brief work in algebra is that mathematics which
is concerned with reasoning nevertheless creates processes which can be applied
almost mechanically, that is, without reasoning. The thinking is, so to speak,
mechanized and this mechanization enables us to solve complicated problems
in no time. We think up processes so that we don't have to think.
It may be necessary to caution the reader again that while the techniques
of transformations are necessary to perform useful and interesting mathe
matical work, they are not the substance of mathematics. If all that one learns
in mathematics is the ability to execute these techniques, however quickly and
accurately, he will not see the real purpose, nature, and accomplishments of
mathematics. To a large extent, techniques are a necessary evil, like practicing
scales on a piano, in order to be able to play grand and beautiful compositions.
Naturally those who wish to be professional mathematicians must learn as
many of these techniques as possible.
EXERCISES
I . Solve by means of ( 30) the following equations:
a) x2 - 8x + 10 = 0 b) x2 + 8x + 10 = 0 c) x2 - 6x - 9 = 0
d) 2x2 + 8x + 6 = 0 e) x2 - 8x + 16 = 0
HISTORY OF EQUATIONS OF HIGHER DEGREE 1 19
* 5-7 THE HISTORY OF EQUATIONS OF HIGHER DEGREE
The search for generality in mathematics began in the sixteenth century. One
type of generality became possible when Vieta showed how to treat a whole
class of equations by means of literal coefficients. Another direction which
the search for generality took was the investigation of equations of degree
higher than the second.
The first of the notable mathematicians to pursue the mathematics of
equations of higher degree and certainly the greatest combination of mathe
matician and rascal is Jerome Cardan. He was born in Pavia, italy, in 1501
to somewhat disreputable parents, although his father was a lawyer, doctor,
and minor mathematician. Cardan had no upbringing worth speaking about
and was sickly during the first half of his life. Despite these handicaps, he
studied medicine and became so celebrated a physician that he was invited to
treat prominent people in many countries of Europe. At various times he was
professor of medicine, and he also lectured on mathematics at several Italian
universities.
He was aggressive, high-tempered, disagreeable, and even vindictive, as
if anxious to make the world suffer for his early deprivations. Because illnesses
continued to harass him and prevented him from enjoying life, he gambled
daily for many years. This experience undoubtedly helped him to write a
now famous book, 0" Games of Chance, which treats the probabilities in
gambling. He even gives advice on how to cheat, which was also gleaned from
expeflence.
A product of his age in many respects, Cardan collected and published
prolifically legends, false philosophical and astrological doctrines, folk cures,
methods of communion with spirits, and superstitions. Apparently he himself
believed in spirits and in astrology. He cast horoscopes, many of which proved
to be false. Toward the end of his life he was imprisoned for casting the
horoscope of Christ, but was soon pardoned, pensioned by the Pope, and lived
peacefully until his death in 1576. In his Book of My Life, an autobiography,
he says that despite his years of trouble he has to be grateful, for he had
acquired a grandson, wealth, fame, learning, friends, belief in God, and he still
had fifteen teeth.
Part of Cardan's rascality concerns our present subject. The mathema
ticians of the sixteenth century had undertaken to solve higher-degree equa
tions, for example, equations of the third degree such as
x3 - 6x = 8.
Among them was another famous man. Nicolo of Brescia, better known as
Tartaglia ( 1499-1 557), whom we shall meet occasionally in other contexts.
Tartaglia had discovered a method for solving third-degree equations, and
Cardan wished to publish this method in a book he was writing on algebra,
120 ALGEBRA, THE HIGHER ARITHMETIC
which later appeared under the title Ars Magna, the first major book on
algebra in modern times. After refusing to divulge the method, Tartaglia
finally acquiesced, but asked Cardan to keep it secret. However, Cardan
wished his book to be as important as possible and so published the method,
though acknowledging that it was Tartaglia's. From this book, which appeared
in 1545, the mathematical world learned how to solve third-degree equations.
In this same book Cardan also published a method of solving fourth-degree
equations discovered by one of his own pupils, Lodovico Ferrari ( 1522-1 56 5 ).
Although general coefficients were not in use as yet, it was clear that all third
and fourth-degree equations could be solved. In other words, the solutions
could be expressed in terms of the coefficients by means of the ordinary
operations of algebra, i.e., addition, subtraction, multiplication, division. and
roots (though not necessarily square roots) , in just about the manner in which
( 30) expresses the solutions of a second-degree equation in terms of the co
efficients p and q.
And now the mathematicians' interest in generality rook over. Since the
general equations of the first, second, third, and fourth degree could be solved,
what about fifth-, sixth- and higher-degree equations? It seemed certain that
these equations could also be solved. For three hundred years many mathe
maticians worked on this basic problem and made almost no progress. And
then a young Norwegian mathematician, Niels Henrik Abel ( 1 802-1829),
showed at the age of 22 that fifth-degree equations could not be solved by the
processes of algebra. Another youth, Evariste Galois ( 18 1 1-1832), who failed
twice ro pass the entrance examinations for the fcole Polytechnique and spent
just one year at the f.cole Normale, demonstrated that all general equations of
degree higher than the fourth cannot be solved by means of the operations of
algebra. In a letter he wrote the night before he was killed in a duel, Galois
explained his ideas and showed how a new and general theory of the solution
of equations could be developed. Galois' ideas gave algebra a totally new turn.
Instead of being a tool, a series of techniques for the transformation of ex
pressions into more useful ones, it became a beautiful body of knowledge which
can be of interest in itself. Unfortunately we cannot undertake to study Galois'
ideas, or the Galois theory as it is called, because there are more basic things to
be learned first.
This brief account of the search for generality in the solution of equations
has been given here because it illustrates many important features of mathe
matics. One is the persistence, stubbornness if you will, of mathematicians
over hundreds of years. Another is the experience that the search for gen
erality leads to new and important developments, even though at the outset
the generality is sought for its own sake. Today, the solution of higher-degree
equations is a most practical marter, and we owe to Galois the most revealing
insight into this subject. We also find in this history of the theory of equations
a major example of how mathematicians find prohlems on which to work,
HISTORY OF EQUATIONS OF HIGHER DEGREE 121
problems of significance drawn from other problems which have humble and
practical origins such as simple equations involving unknowns.
REVIEW EXERCISES
I. Carry out the indicated multiplication:
a) 3 (2x + 6) b) (x + 3 ) (x + 2) c) (x + 7)(x -- 2)
d) (x + 3)(x -- 3) e) (x + �) (x + �) f) (2x + 1 ) (x + 2)
g) (x + y) (x -- y)
2. Factor the following expressions. You may have to experiment to find the
correct factors.
a) x2 - 9 b) x' -- 16 c) x2 - a2
d) a2 _ b2 e) x' + 6x + 9 f) x' + 7x + 6
g) x' + 5x + 4 h) x' -- 6.;; + 9 i) x' -- 7x + 6
j) x' -- 5x + 4 k) x' -- 7x + 12 1) x' + 6x -- 16
m) x' + 6x -- 27
3 . If 2x + 7 = 5, what does 2x equal, and what does x equal?
4. Solve the following equations. State what you do in each step.
a) 2x + 9 == 1 2 b) 2x + 1 2 == 9 c) -t + 3 == 4
o d) -t + �
o ==
�
e) fo + ! == � f) 3x + fa == ! g) ¥ + � = 6 h) ax + 2 == b
i) ax -- b == c
5. A solution of acid and water contains 75% water. How many grams of acid
would you add to 50 grams of the solution to make the percentage of water
60%?
6. A student has grades of 60 and 70 on two examinations. What grade must he
earn on a third examination to attain an average of 75%?
7. Solve the following equations by factoring:
a) x' -- 6x + 5 == 0 b) x' -- 6x -- 7 == 0 c) x' -- 7x + 6 == 0
d) x' + 6x -- 27 == 0 e) " -- 7x + 1 2 == 0 f) x' -- 5x -- 14 == 0
8. Solve the following equations by the method of forming a new equation whose
roots are "larger" than those of the original equation by one-half the coefficient
of x.
a) x' + lOx + 9 == 0 b) x' -- lOx + 9 == 0 c) x' + lOx + 6 == 0
122 ALGEBRA, THE HIGHER ARITHMETIC
d) x' - lOx + 6 = O e) x' - 12x + 1 5 = 0 f) x' + 12x + 15 = 0
9. Solve the following equations by applying formula (30) of the text:
a) x' + 12x + 6 = 0 b) x' - I 2x + 6 = 0 c) x' + 12x - 6 = 0
d) x' - 12x - 6 = 0 e) 2x' + 1 2x + 6 = 0 f) 3%' + 27x + 1 5 = 0
g) t' + lOt = 8
10. In Section 5-5 of the text, we solved a problem wherein one ship sets its course
properly so as to overtake another ship. To set up the equation which solved
the problem, equation ( 1 1 ) , we started by letting x be the distance which the
ship traveling east covers. Solve the same problem by letting t be the time that
both ships travel until they meet. Then x = 2t. The algebra of this alternative
solution is easier to handle. However it is not so obvious that we should let our
unknown be the time of travel.
Topics for Further Investigation
1. The rise of symbolism in algebra.
2. The history of the solution of equations.
Recommended Reading
BALL, W. W. ROUSE: A Short Account of the History of Mathematics, pp. 201-243.
Dover Publications Inc., New York, 1960.
COLERUS, EGMONT: From Simple Numbers to the Calculus, Chaps. 9 through 13.
Wm. Heinemann Ltd., London, 1954.
ORE, OYSTEIN: Cardano, The Gambling Scholar, Chaps. 1 through 5, Princeton
University Press, Princeton, 1953.
SAWYER, W. W.: A Mathematician's Delight, Chap. 7. Penguin Books Ltd., Har
mondsworth, England. 1943.
SMITH, DAVID E.: History of Mathematics, Vol. II, pp. 378-470, Dover Publications
Inc., New York, 1958.
WHITEHEAD, ALFRED N.: An Introduction to Mathematics, Chaps. V and VI, Holt,
Rinehart and Winston, Inc., New York, 1939 ( also in paperback) .
CHAPTER 6
THE NATURE AND USES OF EUCLIDEAN GEOM ETRY
Circles to square and cubes to double
Would give a man excessive trouble.
MATTHEW PRIOR
6-1 THE BEGINNINGS OF GEOMETRY
Just as the study of numbers and its extensions fo algebra arose out of the very
practical problems of keeping track of property, trading, taxation, and the like,
so did the study of geometry develop from the desire to measure the area of
pieces of land (or geodesy in general), to determine the volumes of granaries,
and to calculate the dimensions and amount of material needed for various
structures.
The physical origin of the basic figures of geometry is evident. Not only
the common figures of geometry bur the simple relationships, such as per
pendicularity, parallelism, congruence, and similarity, derive from ordinary
experiences. A tree grows perpendicular to the ground, and the walls of a
house are deliberately set upright so that there will be no tendency to fall.
The banks of a river are parallel. A builder constructing a row of houses ac
cording to the same plan wishes them to have the same size and shape, that is,
to be congruent. A workman or machine producing many pieces of a particu
lar item makes them congruent. Models of real objects are often similar to the
object represented, especially if the model is to be used as a guide to the con
struction of the object.
The science of geometry, indeed, the science of mathematics, was founded
by the Greeks of the classical period. We have already described the major
steps: the recognition that there afe abstract concepts or ideas such as point,
line, triangle, and the like, which are distinct from physical objects, the adop
tion of axioms which contained the surest knowledge about these abstractions
man can obtain, and the decision to prove deductively any other facts abour
these concepts. The Greeks converted the disconnected, empirical, limited
geometrical facts of the Egyptians and Babylonians into a vast, systematic, and
thoroughly deductive structure.
12l
124 NATURE AND USES OF EUCLIDEAN GEOMETRY
Although the Greeks also studied the properties of numbers, they favored
geometry. The reasons are pertinent. First of all, the Greeks liked exact
thinking, and found that this faculty was more readily applied to geometry.
Possible theorems are rather easily gleaned from the visualization of geometri
cal configurations. The neat correspondence between deductively established
conclusions and intuitive understanding further increases this appeal of geom
etry. That one can draw pictures to represent what one is thinking about in
geometry has its drawbacks. One is prone to confuse the abstract concept with
the picture and to accept unconsciously properties of the picture. Of course,
the idea of a triangle must be distinguished from the triangle drawn in chalk
or pencil, and no properties of the picture may be used unless they are con
tained in the axioms or in some previously proved theorem. The Greeks were
careful to make this distinction.
Secondly, the Greek philosophers who founded mathematics were in
trigued with the design and structure of the universe, and they studied the
heavens. certainly the most impressive spectacle in nature. to fathom the
design. The shapes and paths of the heavenly bodies and the over-all plan
of the solar system were of interest. On the other hand, they hardly saw any
value in the ability to describe the exact locations of the moon, sun. and
planets and to predict their precise locations at a given time, information of
importance in calendar reckoning and in navigation.
Thirdly, since commerce and daily business were handled in large part by
slaves, and were in any case in low regard. the study of numbers, which
served such purposes, was subordinated. Why worry abour the uses of num
bers for measurement and trade if one does not measure or trade? One does
not need the dimensions of even one rectangle to speculate about the properties
of all rectangles.
The Greek philosophers emphasized an aspect of reality which is today,
at least in scientific circles, neglected. To the Greeks of the classical period
the reality of the universe consisted of the forms which matter possessed.
Matter as such was formless and therefore meaningless. But an object in the
shape of a triangle was significant by the very fact that it was triangular.
Finally, there were purely mathematical grounds for the Greek emphasis
on geometry. The Greeks were the first to recognize that quantities such as
VI. V3, iVl, etc., are neither whole numbers nor fractions, but they failed
to recognize that these were new types of numbers, and that one could reason
with them. To handle all types of quantities, they conceived the idea of
treating them as line segments. As line segments, the hypotenuse of a right
triangle (Fig. 4--2) and the arms have the same character, despite the fact that
if the arms are each 1 unit long, the hypotenuse has the irrational length Vi,
To execute their plan of treating all quantities geometrically, the Greeks
converted the algebraic processes developed in Egypt and Babylonia into
geometrical ones. We could illustrate how the Greeks solved equations geo-
THE CONTENT OF EUCLIDEAN GEOMETRY 125
metrically, but their methods are no longer favored. For science and engineer
ing, the knowledge that a certain line segment solves an equation is not nearly
so useful as a numerical answer which can be calculated to as many decimal
places as needed. But the classical Greeks, who regarded exact reasoning as
paramount in importance and who deprecated practical applications, found the
solution of their difficulty in geometry and were conte.nt with this solution.
Geometry remained the basis for all exact mathematical reasoning until the
seventeenth century, when the needs of science forced the shift to number
and algebra and the ultimate recognition that these could be built up as
logically as geometry. In the intervening centuries arithmetic and algebra
were regarded as practical disciplines.
Of course. the Greek conversion of exact mathematics to geometry was,
from our present viewpoint, a backward step. Not only are the geometrical
methods of performing algebraic processes insufficient for science, engineer
ing, commerce, and industry, bur they are by comparison clumsy and lengthy.
Moreover. because Greek geometry was so complete and so admirable, mathe
maticians following in the Greeks' footsteps continued to think that exact
mathematics must be geometrical. As a consequence, the development of
algebra was unnecessarily delayed.
6-2 THE CONTENT OF EUCLIDEAN GEOMETRY
The major book on geometry of the classical Greek era is Euclid's Elements,
a work on plane and solid geometry. Written about 300 B.C., it contains the
best results produced by dozens of fine mathematicians during the period from
600 to 300 B.C. The work of Thales, the Pythagoreans, Hippias, Hippocrates,
Eudoxus, members of Plato's Academy, and many others furnished the material
which Euclid organized. His text was not the first to be written, but un
fortunately we do not have copies of the earlier ones. It is quite certain that
the particular axioms one finds in the Ele111ents, the arrangement of the the
orems, and many of the proofs are all due to Euclid. The geometry texts used
in high schools today in essence reproduce Euclid's work, although these
contemporary versions usually contain only a small part of the 467 theorems
and many corollaries found in the Elements. Euclid's version is so marvelously
knit together that most readers are amazed to see so many profound theorems
deduced from the few self-evident axioms.
Though the reader may already be familiar with the basic theorems of
Euclidean geometry, we shall take a few moments to review some features of
the subject and the nature of the accomplishment. We might nOte first the
structure of Euclid's Elements. He begins with some definitions of the basic
concepts: point, line, circle, triangle, quadrilateral, and the like. Although
modern mathematicians would make some critical comments about these
definitions, we shall not discuss them at present. (See Chapter 20.)
1 26 NATURE AND USES OF EUCLIDEAN GEOMETRY
Euclid then states ten axioms on which all subsequent reasoning is based.
We shall note these merely to see that they do indeed describe apparently
unquestionable properties of geometric figures. The first five axioms are:
AXIOM 1 . Two points determine a unique straight line.
AXIOM 2. A straight line extends indefinitely far in either direction.
AXIOM 3 . A circle may be drawn with any given center and any given
radius.
AXIOM 4. All right angles are equal.
P m
Fig. 6-1.
The parallel axiom.
AXIOM 5. Given a line I (Fig. 6- 1 ) and a point P not on that line, there
exists in the plane of P and I and through P one and only one line m, which
does not meet the given line I.
In a separate defil1ition Euclid defines parallel lines to be any two lines in
the same plane which do not meet, that is, do not have any points in common.
Thus, Axiom 5 asserts the existence of parallel lines.
The remaining five axioms are:
AXIOM 6. Things equal to the same thing are equal to each other.
AXIOM 7. If equals be added to equals, the sums are equal.
AXIOM 8. If equals be subtracted from equals, the remainders are equal.
AXIOM 9. Figures which can be made to coincide are equal (congruent).
AXIOM 10. The whole is greater than any part.
The formulations of these axioms are not quite the same as those pre
scribed by Euclid. Axiom 5 is, in fact, different from Euclid's, but is stated
here in the form which is most likely to be familiar to the reader. The differ
ences between Euclid's versions and those introduced by later mathematicians
are not important for our present purposes, and so we shall not take time now
to note them. (See Chapter 20.)
After stating his axioms, Euclid proceeded to prove theorems. Many of
these theorems are indeed simple to prove and obviously true of the geo
metrical figures involved. But Euclid's purpose in proving them was to play
safe. As we shall see in later chapters, many a conclusion seems obvious but
is false. Of course, the major proofs are those which establish conclusions that
are not at all obvious and, in some cases, even come as a surprise.
THE CONTENT OF EUCLIDEAN GEOMETRY 127
Partly to refresh our memories about some theorems of Euclidean geom
etry and partly to note once again the deductive procedure of mathematics,
let us review one or two proofs. A basic theorem of Euclidean geometry
asserts the following:
THEOREM I. An exterior angle of a triangle is greater than either re-
mote interior angle of the triangle.
B
,� C
Fig. 6-2.
An exterior angle of a triangle is greater
than either remote interior angle.
Before proving this theorem, let us be clear about what it says. Angle D,
in Fig. 6-2, is called an exterior angle of triangle ABC because it is outside
the triangle and is formed by one side, Be, and an extension of another side,
AC. With respect to angle D, angles A and B are remote interior angles of
triangle ABC, whereas angle C is an adjacent interior angle. Hence we have
to prove that angle D is larger than angle A and larger than angle B. Let us
prove that angle D is larger than angle B.
A �' C
Fig. 6-3
The problem before us is a tantalizing one because, while it does seem
visually obvious that angle D is greater than angle B, there is no apparent
method of proof. An idea is needed, and this is supplied by Euclid. He tells
us to bisect side BC (Fig. 6-3 ), to join the mid-point E of BC to A , and to
extend AE to the point F, so that AE EF. He then proves that triangle
=
AEB is congruent to triangle CEF, that)s, that the sides and angles of one
triangle are equal, respectively, to the sides and angles of the other. This
congruence is easy to prove. Euclid had previously proved that vertical angles
are equal, and we see from Fig. 6--3 that angles 1 and 2 are vertical angles.
Further, the fact that E is the mid-point of BC means that BE = EC. More
over, we constructed EF to equal AE. Hence, in the two triangles in question,
two sides and the included angle of one triangle are equal to two sides and the
included angle of the other. But Euclid had previously proved that two tri
angles are congruent if merely two sides and the included angle of one are
128 NATURE AND USES OF EUCLIDEAN GEOMETRY
equal to two sides and the included angle of the other. Since these facts are
true of our triangles, the two triangles must be congruent.
Because triangles AEB and CEF are congruent, angle B of the first triangle
equals angle 3 of the second one. We know that angle 3 is the angle to choose
in the second triangle as the angle which corresponds to B, because angle B
is opposite AE, and angle 3 is opposite the corresponding equal side EF. The
proof is practically finished. Angle D is larger than angle 3 because the whole,
angle D in our case, is greater than the part, angle 3 . Hence angle D is also
greater than angle B because angle B has the same size as angle 3 .
We have now proved a major theorem, and we should see that a series of
simple deductive arguments leads to an indubitable result.
And now let us prove another, equally important theorem which will
exhibit one or two other features of Euclid's work:
THEOREM 2. If two lines are cut by a transversal so as to make alternate
interior angles equal, then the lines are parallel.
Fig. 6-4
Again let us see what the theorem means before we consider its proof.
In Fig. 6-4, AB and CD are two lines cut by the transversal EF. The angles
1 and 2 are called alternate interior angles, and we are told that they are equal.
The theorem asserts that, under this condition, A B must be parallel to CD. As
in the case of the preceding theorem, the assertion is seemingly correct, and
yet the method of proof is by no means apparent.
Here Euclid uses what is usually called the indirect method of proof;
that is, he supposes that AB is not parallel to CD. Two lines that are not
parallel must, by definition, meet somewhere. Thus AB and CD meet, let us
say, in the point G. But now EG, GF, and FE form a triangle. Angle 2 is an
exterior angle of this triangle and angle I is a remote interior angle. Since
we have the theorem that in any triangle an exterior angle is greater than
either remote interior angle, it follows that angle 2 must be greater than angle
1. But, in the above figure, we were given as fact that angle 2 equals angle 1 .
We have arrived at a contradiction which. if we did not make any mistakes
in reasoning. has only one explanation: somewhere we introduced a false
premise. We find that the only questionable fact is the assumption that AB
is not parallel to CD. But there are only two possibilities, namely, that AB
is parallel to CD or that it is not parallel to CD. Since the latter supposition
THE CONTENT OF EUCLIDEAN GEOMETRY 129
led to a contradiction, it must be that A B is parallel to CD. Thus the theorem
is proved.
We should be sure to note that the indirect method of proof is a deductive
argument. The essence of the argument is that if AB is not parallel to CD,
then angle 2 must be greater than angle I. But angle 2 is not greater than
angle 1 . Hence it is not true that AB is nOt parallel to CD. But A B is or is
not parallel to CD. If :lonparallelism is not true then parallelism must hold.
Though we shall use a few other theorems of Euclidean geometry in sub
sequent work, we shall not present their proofs. We are now reasonably
familiar with the nature of proof in geometry, and so we shall merely state
the theorems when we wish to use them.
Perhaps one other point about the contents of the Elements warrants
attention. A superficial survey of the many different theorems may leave one
with the impression that the Greek geometers proved what they could and
produced merely a melange. But there are broad themes in Euclidean geom
etry, and these are pursued systematically. The first major theme is the study
of conditions under which geometric figures must be congruent. This is a
highly practical subject. Suppose, for example, that a surveyor has twO tri
angular pieces of land and wishes to show that they are equal or congruent.
Must he measure all the sides and all the angles of the first piece and show that
they are of the same size, respectively, as the sides and angles of the second
piece? Not at all' There are several Euclidean theorems which can aid the
surveyor. If he can show, for example, that two angles and the included side
of the first triangle equal, respectively, the two angles and the included side
of the second one, then Euclid's theorem tells him that the triangular pieces of
land must be equal.
A second major theme in Euclid's work is the similarity of figures, that
is, figures with the same shape. We have already mentioned that models of
houses, ships, and other large structures are often built to assist in planning.
One may wish to know what conditions will guarantee the similarity of the
model and the actual structure. Let us suppose that the model or some part
of it is triangular in shape. One of Euclid's theorems tells us that if the cor
responding sides of two triangles have the same ratio, then the two triangles
will be similar. Thus, if the model is constructed so that each side of the model
is n\o of the corresponding side of the actual structure, we know that the
model will be similar to the structure. This similarity is useful because, by
definition, twO triangles are similar if the angles of one equal the corresponding
angles of the other. Hence, an engineer can measure the angles of the model
and know precisely what the angles of the actual structure will be.
Suppo,e that two figures are neither congruent nor similar. Could they
have some other significant property in common? One answer. clearly, is
area. And so Euclid considers conditions under which two figures may have
the same area, or, in Euclid's language, be equivalent.
1 30 NATURE AND USES OF EUCLIDEAN GEOMETRY
-- _
- - - .-1.. ..... ..... �
_
Tetrahedron Cube Octahedron
Fig. 6-5.
Dodecahedron Icosahedron
The five regular polyhedra.
There are many other themes in Euclid, such as interesting proper'ties of
circles, quadrilaterals, and regular polygons. He also considers the common
solid figures such as pyramids, prisms, spheres, cylinders, and cones. Finally,
Euclid devotes considerable space to a class of figures which all Greeks favored,
the regular polyhedra (Fig. 6--5 ) .
EXERCISES
1. What essential fact distinguishes axioms from theorems?
2. Why were the Greeks willing to accept the statements 1 through 10 above as
.
aXIOms.,
3. Use the indirect method of proof to show that if two angles of a triangle are
equal, then the opposite sides are equal. [Suggestjon: Suppose that angle A
(Fig. 6-<\) equals angle C, but that BC is greater than BA. Lay off BC' = BA
and draw AC'. Use the theorem that the base angles of an isoceles triangle are
equal and Theorem 1 above.]
G
B --
A B
c D
A ..::=----',C Fig. 6-6 F Fig. 6-7
SOME MUNDANE USES OF EUCLIDEAN GEOMETRY 131
4. Use the indirect method of proof to show that if two lines are parallel, alternate
interior angles must be equal. [Suggestion: Suppose angle 1 in Fig. 6-7 is greater
than angle 2. Then draw GH so that angle l' equals angle 2. Now use Theorem
2 and Axiom 5.]
5. ]n Section 3-7, we have briefly outlined the proof that the sum of the angles
of a triangle is 180°. Write out the full proof.
6. Under what conditions would two parallelograms be congruent?
7. What conditions would ensure the similarity of two rectangles?
8. A right triangle has an arm 1 mi long and a hypotenuse 1 mi plus 1 ft long.
How long is the other arm? Before you apply mathematics, use your imagina
tion to estimate the answer. To work out the problem, use the Pythagorean
theorem which says that the square of the hypotenuse equals the sum of the
squares of the arms.
9. A farmer is offered two triangular pieces of land. The dimensions are 25, 30,
and 40 ft and 75, 90, and 120 ft, respectively. Since the dimensions of the second
one are 3 times the dimensions of the first, the two triangles are similar. The
price of the larger piece is 5 times the price of the smaller one. Use intuition,
measurement, or mathematical proof to decide which is the better buy in the
sense of price per square foot.
Fig. 6-8
to. Suppose a roadway is to be built around the earth and each point on the surface
of the roadway is to be 1 ft above the surface of the earth (Fig. 6-8 ) . Given
that the radius of the earth is 4000 mi or 2 1 , 120,000 ft, estimate by how much
the length of the roadway would exceed the circumference of the earth. Then
use the fact that the circumference of a circle is 27T times the radius and calculate
how much longer the rGadway would be.
1 1. Criticize the statemen�: Euclid assumes that two parallel lines do not meet.
6-3 SOME MUNDANE USES OF EUCLIDEAN GEOMETRY
The creation of Euclidean geometry was motivated by the desire to learn the
properties of figures in the world about us. Let us see now whether the
knowledge can be applied to the world to good ad,·anrage.
Suppose a farmer has 100 feet of fencing at his disposal and he wishes to
enclose a rectangular piece of land. Since the perimeter will be 100 feet, the
132 NATURE AND USES OF EUCLIDEAN GEOMETRY
farmer can enclose a piece of land 10 feet by 40 feet, 1 5 feet by 35 feet, 20
feet by 30 feet, or of still other dimensions, all of which yield a perimeter of
100 feet. The farmer plans to garden in the enclosed plot and therefore
wishes the enclosed area to be as large as possible. He notes that the dimen
sions 10 by 40 would yield an area of 400 square feet; the dimensions 1 5 by
3 5 enclose 5 2 5 square feet; and the dimensions 20 by 30 enclose 600 square
feet. Evidently the area can vary considerably despite the fact that the
perimeter in each case is 100 feet. The question then arises, What dimensions
would yield the maximum area?
Our first task in seeking to answer this question is to make some reason
able conjecture about these dimensions. We might then be able to prove
that the conjecture is correct. Since in the present instance it is easy to play
with the numbers involved, let us make a little table of dimensions (always
yielding a perimeter of 100 feet) and the corresponding area.
Dimensions, in feet Area, in square teet
I by 49 49
5 by 45 225
10 by 40 400
I S by 3 5 525
2 0 b y 30 600
Study of the table suggests that the more nearly equal the dimensions are,
the larger is the area. Hence one might readily conjecture that if the dimen
sions were equal, that is, if the rectangle were a square, the area would be a
maximum.
We can see at once that the dimensions 2 5 by 2 5 give an area of 625
square feet, and this area is larger than any of the areas in the table. So far
our conjecture is confirmed. However, we could not be sure that some other
dimensions, perhaps 24k by 25t, would not do even better. Moreover, even
if we could be certain that the square furnishes the largest area among all
rectangles with a perimeter of 100 feet, the question would arise whether the
square would continue to be the answer for some other perimeter. Hence,
let us see whether we can prove the general theorem that ot all rectangles
with the same perimeter, the square has maximum area.
Figure 6-9 shows the rectangle ABCD. Since this rectangle is not a square,
let us erect on the longer side a square which has the same perimeter. Thus,
the square EFGD has the same perimeter as ABCD. We now denote equal
segments by the same letters. The perimeter of the rectangle is then 2x + 2u +
2 y, and the perimeter of the square is 2x + 2v + 2 y. Since the two figures
have the same perimeter. we have
2x + 2v + 2y = 2x + 2u + 2y.
SOME MUNDANE USES OF EUCLIDEAN GEOMETRY 133
If we subtract 2x and 2y from both sides of this equation and then divide both
sides by 2, we obtain
v = u. (I)
Moreover, because the square has equal sides,
y = x + v. (2)
If we now multiply the left side of equation (2) by the left side of equation
( I ) , and do the same for the right sides, the results must be equal. Hence,
yv = u(x + v),
or, by the distributive axiom,
yv = ux + uv.
Since yv = ux plus an additional area, it must be that yv is greater than ux.
Now yv is area B in the figure, and !IX is area A. Thus B is greater than A,
and so B + C is greater than A + C. But B + C is the area of the square, and
A + C is the area of the rectangle. Hence the square has more area than the
rectangle.
We have proved that a square has more area than a rectangle of the same
perimeter, no matter what this perimeter may be. A little thinking proves in
a few minutes what may have taken man hundreds of years to learn through
trial and error.
y
F G
v B
u y
B c
x A x
C
x
Fig. 6-9.
A D Of all rectangles of the same perimeter the
U
E y square has the greatest area.
The result is far more useful than may appear at first sight. Suppose a
house is to be built. The major consideration is to have as much floor area
or living space as possible. Now the perimeter of the lIoor determines the
number of feet of wall that will be needed and hence the cost of the walls.
To obtain the maximum floor area for a given cost of walls, the shape of the
lIoor should be square.
A farmer who seeks the rectangle of maximum area with given perimeter
might, after finding the answer to his question, turn to gardening, but a
mathematician who obtains such a neat result would not stop there. He might
1 34 NATURE AND USES OF EUCLIDEAN GEOMETRY
ask next, Suppose we were free to utilize any quadrilateral rather than just
rectangles, which one of all quadrilaterals with the same perimeter has max
imum area? The answer happens to be a square, though we shall not prove it.
The mathematician might then consider the question, Which pentagon of all
pentagons with the same perimeter has maximum area? One can show that
the answer is the regular pentagon, that is, the pentagon whose sides are all
equal and whose angles are all equal. Now the square also has equal sides and
equal angles. Hence it would seem that if one compares all polygons of the same
perimeter and same number of sides, then the one with equal sides and equal
angles, i.e., the regular polygon, should have maximum area. This general
result can also be proved.
But now an obvious question comes to the fore. The square has maximum
area among all quadrilaterals of the same perimeter. The regular pentagon
has maximum area among all pentagons of the same perimeter. Suppose that
we compared the regular pentagon with the square of the same perimeter.
Which would have more area? The answer, perhaps surprising, is the regular
pentagon. And now the conjecture seems reasonable that of two regular
polygons with the same perimeter, the one with more sides will have more
area. This is so. Where does this result lead? One can form regular polygons
of more and more sides, which all have the same perimeter. As the number
of sides increases, the area increases. But as the number of sides increases, the
regular polygon approaches the circle in shape. Hence the circle should have
more area than any regular polygon of the same perimeter. And since the
regular polygon has more area than an arbitrary polygon, the circle has more
area thall any poly gOIl with the same perimeter. This result is a famous
theorem.
Now the sphere, among surfaces, is the analogue of the circle among
curves. Hence, a reasonable conjecture would be that the spherical surface
bounds more volume than any other surface with the same area. This conjec
ture can be proved. Nature obeys this mathematical theorem. For example, if
one blows up a rubber balloon, the balloon assumes a spherical shape. The
reason is that the rubber must enclose the volume of air blown into the balloon
and the rubber must be stretched. But rubber contracts as much as possible.
The spherical figure requires less surface area to contain a given volume of gas
than does any other shape. Hence, with the spherical shape, the rubber is
stretched as little as possible.
The problem of bounding the greatest possible area with a perimeter of
given length has a variation whose solution shows how ingenious mathematical
reasoning can be. Suppose that a person has a fixed amount of fencing at his
disposal and wishes to enclose as much area as possible along a river front in
such a way that no fencing is required along the shore itself. The question
now is, What should the shape of the boundary curve be? According to a
legend, which may or may not have a factual basis, this problem was solved
SOME MUNDANE USES OF EUCLIDEAN GEOMETRY 135
thousands of years ago by Dido, the founder of the city o f Carthage on the
Mediterranean coast of Africa. Dido, the daughter of the king of the Phoeni
cian city of Tyre, ran away from home. She took a fancy to this land on the
Mediterranean, and made an agreement to pay a definite sum of money for as
much land as "could be encompassed by a bull's hide." Dido thereupon took a
bull's hide, cut it up into thin long strips, tied the strips together, and used this
length to "encompass land." She chose an area along the shore, because she was
smart enough to realize that no hide would be needed along the shore. But
there still remained the question of what shape to use for the boundary formed
by the hide, that is, for ABC of Fig. 6-10. Dido decided that the most favor
able shape was a semicircle, enclosed that shape, and built a city there.
B
c
Fig. 6-10 Fig. 6-11
A sequel to this story, which has nothing to do with the mathematics of
Dido's problem, is not without relevance to the history of mathematics.
Shortly after she founded Carthage, Aeneas, a refugee from Troy, intent on
getting to Italy to found his own city, was blown ashore along with his com
patriots. Dido took a fancy to Aeneas also, and did her best to persuade him
to remain at Carthage, but despite the best of hospitality, Aeneas could not be
diverted from his plan, and soon sailed away. Rejected and scorned, Dido was
so despondent that she threw herself on a blazing pyre just as Aeneas sailed
out of the harbor. And so an ungrateful and unreceptive man with a rigid
mind caused the loss of a potenrial mathematician. This was the first blow to
mathematics which the Romans dealt.
Dido's fate was a tragic end to a brilliant beginning, for her solution to the
geometrical problem described above was correct. The answer is a semicircle.
We do not know how Dido found the answer, but it can be obtained very
neatly. The way to prove it is by complicating the problem. Suppose that,
instead of bounding an area on one side of the seashore, which we idealize as
the line AC (Fig. 6-1 1 ) , we try to solve the problem of enclosing an area on
both sides of AC with double the length of hide Dido had for one side, i.e.,
now we seek to solve the problem by determining the maximum area which
can be completely enclosed by a perimeter of given length. The answer to this
problem is a circle. If, therefore, we choose a semicircle for arc ABC, it will
contain maximum area on one side of the shore. For if there were a more
favorable shape than the semicircle, the mirror image in AC of that shape
1 36 NATURE AND USES OF EUCLIDEAN GEOMETRY
would, together with the original, do better than the circle and yet have the
same perimeter as the circle. But this is impossible.
Our last few pages have dealt with problems which grew out of deter
mining the rectangle of maximum area with given perimeter. We can see from
the lines of thought pursued how the mathematician can raise one question
after another on this same theme of figures with maximum area and given
perimeter and will find the answers to these questions. Moreover. many of
these answers prove to be applicable to physical problems.
Fig. 6-12.
Eratosthenes' method of deducing the
circumference of the earth.
The first reasonably accurate calculation of the size of the earth was made
by a simple application of Euclidean geometry. One of the most learned men
of the Alexandrian Greek world, Eratosthenes (275-194 B.C.), a geographer,
mathematician, poet, historian, and astronomer. used the following plan. At
the summer solstice, the sun shone directly down into a well at Syene (C in
Fig. 6-1 2 ) . As Eratosthenes well appreciated, this meant that the sun was
directly overhead. At the same time, at the city of Alexandria, 500 miles north
of Syene, the direction of the sun was AS', whereas the overhead direction was
OAD. Now the sun is so far away that the lines AS' and CS could be taken to
be parallel. Eratosthenes measured the angle DAS' and found it to be 7to.
But this angle equals the vertical angle OAE, and the latter and angle AOC are
alternate interior angles of parallel lines. Hence angle AOC is also 7tO, or
7t/360, or 1/48 of the entire angle at O. Then arc AC is 1/48 of the entire
circumference. Since AC is 500 miles, the entire circumference is 48 · 500 or
24,000 miles.
Strabo, a Greek geographer who lived in the first century B.C., tells us that
after Eratosthenes obtained this result, he realized that one might sail from
Greece past Spain across the Atlantic Ocean to India. This is. of course,
what Columbus attempted. Fortunately or unfortunately, the geographers
who lived after Eratosthenes, notably Poseidonius (first century B.C.) and
Ptolemy (second century A.D. ) , gave other results which were interpreted by
Columbus (because of some uncertainty about the units of distance used by
these early scientists) to mean that the circumference of the earth is 17,000
miles. Had he known the correct value, he might never have undertaken to
sail to India because the greater distance might have daunted him.
SOME MUNDANE USES OF EUCLIDEAN GEOMETRY 1 37
EXERCISES
I. Suppose that DF (Fig. 6-1 3 ) is the course of a railroad, and A and B are two
towns. It is desired to build a station somewhere on DF so that the station will
be equally distant from A and B. Where should the station be built? One draws
the line AB and, at its mid-point, erects the perpendicular CEo The point E on
DF is equidistant from A and B. Prove this statement.
F
B c
E
A
D
c A D
Fig. 6-15
B
Fig. 6-13
J foot
Fig. 6-14
I \ �.1'
O ,--
i � D'
----, B'
2. A pinhole camera is a practical device if a long exposure time is possible. In
fact, one of the best pictures of the scene following the explosion of the first
atomic bomb was made with a pinhole camera. The principle involves similar
triangles. The object AB being photographed ( Fig. 6-14) appears on the film
inside the box as A'B'. If one draws OD perpendicular to AB, the extension of
OD to D' will be perpendicular to A'B'. Then triangles OAD and OA'D' are
similar. Now suppose the sun. whose radius is AD, is photographed. We know
that OD is 93,000,000 mi. Suppose that OD', rhe width of the box, is 1 ft. The
length A'B' is readily measured and is found to be 0.009 ft. What is the radius
of the sun?
3. A farmer has 400 yd of fencing and wishes to enclose a rectangle of maximum
area. What dimensions should he choose?
4. A farmer has p yd of fencing and wishes to enclose a rectangle of maximum
area. What dimensions should he choose?
5. A farmer plans to enclose a rectangular piece of land alongside a lake; no
fencing is required along the shoreline AD ( Fig. 6-1 5 ) . He has toO ft of fence
and wishes the area of the rectangle to be as large as possible. What dimensions
should he choose?
138 NATURE AND USES OF EUCLIDEAN GEOMETRY
6. Of any two numbers whose sum is 12, the product is
x
greatest for 6 and 6; that is, 6 · 6 is greater than 5 · 7,
4 8, 3!
.
' 8!, and so forth. Can you explain why this is
so? [Suggestion: Think in geometrical terms.]
7. Suppose h is the known height of a mountain, and R
is the radius of the earth (Fig. 6-16). How far is it
from the top of the mountain to the horizon; that is,
how long is x.? [Suggestion: Use the fact that the line
of sight from the top of the mountain to the horizon
is tangent to the circle shown, and that a radius of a
circle drawn to the point of tangency is perpendicular Fig. 6-16
to the tangent.]
8. Having obtained the exact answer to Problem 7, can you suggest a good
approximate answer which would suffice for many applications and yet make
calculation easier?
9. A boy stands on a cliff i mi above the sea. How far away is the horizon?
10. Knowing that of all rectangles with the same perimeter, the square has maximum
area, prove that of all rectangles with the same area, the square has the least
perimeter. [Suggestion: Use the indirect method of proof. Suppose, then, that
the square has more perimeter than the rectangle of the same area and consider
the square which has the same perimeter as the rectangle.]
* 6-4 EUCLIDEAN GEOM ETRY AND THE STUDY OF LIGHT
Light is certainly a pervasive phenomenon. Man and the physical world are
subject daily to the light of the sun, and the process of vision of course is
dependent upon light. Hence it is to be expected that the Greeks, the first
great students of nature, would investigate this phenomenon. Plato and
Aristotle had much to say on the nature of light, and the Greek mathemati
cians also tackled the subject. It has continued to be a primary concern of
mathematicians and physicists right down to the present day. Despite man's
continuous experience with light, the nature of this occurrence is stilI largely
a mystery. Through mathematics and through Euclidean geometry in par
ticular, man obtained his first grip on the subject. Two books by Euclid were
the beginning of the mathematical attack.
In ordinary air, light is observed to travel along straight lines. This
preference of light for the simplest and shortest path is in itself of significance.
But Euclid proceeded beyond this point to study the behavior of light under
reflection in a mirror, and discovered a now famous mathematical law of light.
Suppose light issuing from A (Fig. 6-1 7) takes the path AP to the point P
on the mirror m. As we all know, the light is reflected and takes a new direc
tion, PA'. The significant fact about this reflection, which was pointed out by
Euclid, is that the reflected ray, i.e., the line PA' along which the reflected
light travels, always takes a direction such that angle 1 equals angle 2 . Angle
EUCLIDEAN GEOMETRY AND THE STUDY OF LIGHT 1 39
Q
A'
m
Fig. 6-17.
The law of reflection of light.
1 is called the angle of incidence and angle 2 the angle of reflection.· It is, of
course, very obliging of light to follow such a simple mathematical law. As a
consequence, we are able to prove other facts rather readily.
Assume there is a source of light at A (Fig. 6-18), and rays of light spread
out in all directions from A. Many of these will strike the mirror. But through
a definite point A ' only one of these rays will pass, namely the ray PA' for
which angle 1 equals angle 2 . To prove that only one ray from A will pass
through A ', let us suppose that another ray, AQ, is also reflected to A'. Now
angle 2 is an exterior angle of triangle A'QP. Hence
L2 > L4.
Angle 3 is an exterior angle of triangle AQP, and so
L3 > Ll.
Since angle 1 equals angle 2 , we see from the two preceding inequalities that
L 3 > L4.
Then QA' cannot be the reflected ray corresponding to the incident ray AQ
because the reflected ray must make an angle with the mirror which equals
angle 3.
.�
�, :;
4 1
;P
2
;
m
z Fig. 6-18
The more interesting point, which was first observed and proved by the
Greek mathematician and engineer Heron ( first century A.D. ) , is that the
• It is more common to introduce the perpendicular PQ to the mirror and to call
angle 3 the angle of incidence and angle 4 the angle of reflection. However, if angle
1 equals angle 2, then angle 3 equals angle 4.
140 NATURE AND USES OF EUCLIDEAN GEOMETRY
A
.1 '
c 2 m
Fig. 6-19.
The shortest path from A to A' is
B the one for which L 1 = L2.
unique ray from A (Fig. 6-(9) which does reach A' after reflection in the
mirror travels the shortest possible path in going from A to the mirror and
then to A'. In other words, AP + PA' is less than AO + OA', where 0 is
any point on the mirror other than P, the point at which the angle of incidence
equals the angle of reflection.
How can we prove this theorem? Nature not only sets problems for US,
but often solves them too, if we are but keen enough in our observations. If
a person at A' sees in the mirror the reflection of an object at A, he must be
looking in the direction A'P and actually sees the image of A at B. Hence,
perhaps we should bring B into our thinking. Closer observation shows that
the mirror image of an object is on the perpendicular from A to the mirror
and, moreover, seems to be as far behind the mirror as the object is in front.
That is, AB seems to be perpendicular to the mirror and AC seems to equal CB.
Let us use this suggestion. We construct the perpendicular from A to the
mirror, thus obtaining AC, and extend AC by its own length to B. Now it
is not hard to see that triangles ACO and BCO are congruent because OC is
common to both triangles, the angles at C are right angles, and AC CB. =
Hence, A O BO, because they are corresponding parts of congruent triangles.
=
Likewise triangles ACP and BCP are congruent and AP BP. We wish to
=
prove that
(AP + PA') < (AQ + OA').
But now, since AP = BP and AO = BO, it will be enough to prove that
(BP + PA') < (BQ + QA'.) (3)
Well, we have exchanged one difficulty for another, but perhaps this
second one is easier to overcome. Physically one looks directly along A'P
and sees B. If we could prove that BPA' is a straight line, then, of course, the
inequality ( 3 ) would be proved because BO and OA' are the other two sides
EUCLIDEAN GEOMETRY AND THE STUDY OF LIGHT 141
of triangle A'BQ, and the sum of these two sides must be greater than the
third side. Our goal, then, is to prove that A'PB is a straight line.
We know that
L 1 + L J + L2 = 180° (4)
because m is a straight line. But angle 1 equals angle 4 because triangle PCA
and PCB are congruent. Also, according to the law of reflection, angle 2
equals angle 1 . If, therefore, in ( 4 ) we replace angle 1 by angle 4 and angle
2 by angle 1, we have
L4 + L J + L I = 180°. (5)
Hence, A'PB is a straight line and the inequality ( 3 ) is proved. Then the light
ray, in going from A to m to A', really travels the shortest path.
This behavior of light rays is striking. It seems to show that nature is
interested in accomplishing its ends by the most efficient means. We shall
find this theme to be a recurring one, and it will be seen to have broad
applicability.
We have proved a theorem about light rays, bur we have also proved
somewhat more. As far as the mathematics is concerned, the lines AP and
PA' are any lines which make equal angles with m, and the fact that they are
light rays plays no role. What we have proved, then, is a theorem of geometry,
namely:
Of all tbe broken line paths from a point A to a point on a line and then
to a point A' on the same side as A , the sbortest patb is tbe one fixed by
tbe point P on m for whicb AP and A'P make equal angles with m.
This theorem has applications in quite different domains (see the exercises).
It is worth noting how the study of light gives rise to purely mathematical
theorems. The converse of this theorem is, incidentally, equally true and is
presented in the exercises.
EXERCISES
1. Where is the mirror image of a point A which is in from of a plane mirror?
2. Suppose that m (Fig. 6--2 0) is the shore of a river and a pier is to be built
somewhere along m so that merchandise can be trucked from the pier to two
inland towns, A and A'. Where should the pier be built so that the total trucking
distance from the pier to A and from the pier to A' is a minimum?
A
•
A'
•
m Fig. 6-20
142 NATURE AND USES OF EUCLIDEAN GEOMETRY
Fig. 6-21
l. A billiard player wishes to hit the ball at A (Fig. 6-2 1 ) in such a way that it will
strike side 111 of the table and then hit the ball at A'. Now billiard balls behave
like light rays, that is, the angle of reflection equals the angle of incidence. At
what point on m should the billiard player aim?
4. A billiard ball starting from a point A on the table (Fig. 6--2 1 ) strikes two suc
cessive sides and then travels along the table. What can you say about the final
in relation to the original direction of travel?
.1
A'
5, I n the text we proved that if angle 1 equals angle 2 ( Fig. 6-22), then A P + PA'
is the shortest path from A to any point on the mirror to A'. Prove the converse,
namely, that if AP + PA' is the shortest path, then L 1 must equal L 2. rSug
gestion: Use the indirect method of proof. If L 1 does not equal L 2, then one can
find another point, P', on 111 for which the angles made by AP' and A'P' with 111
are equa1.]
6-5 CONIC SECTIONS
The Elements of Euclid dealt with plane figures which can be built up with
line segments and circles, with the corresponding solid figures which can be
huilt up with pieces of a plane, such as prisms and the regular polyhedra, and
with the sphere. But the classical Greeks also studied another class of curves
which they called conic sections because they were originally obtained by
slicing a cone with a plane, The resulting curves, the parabola, ellipse, and
hyperbola, were treated by Euclid in a separate book. Unfortunately, no
copies of this book have survived, But a little after Euclid's time another
famous Greek geometer, Apollonius, wrote a book entitled Conic Sections,
which is known to us and which is about as exhaustive in its treatment of these
curves as the Element,- are about figures formed by lines and circles,
CONIC SECTIONS 143
I) f--r---�r,p
p
Axis
F F' F
d
Fig. 6-23. Fig. 6-24.
The parabola. The ellipse.
Conic sections were introduced, as already noted, by cutting a conical
surface with a plane. However, the curves themselves can be considered apart
from the surface on which they lie. For example, the circle is also one of the
conic sections. Vet we know that the circle can be defined as the set of all
points which are at a fixed distance from a given point, and this definition does
not involve the cone at all. Indeed, insofar as properties and applications of
these curves are concerned, it is far more convenient to disregard the conical
surface and concentrate on the curves themselves.
Let us consider, therefore, the direct definitions of conic sections. To
define the parabola, we start with a fixed point F and a fixed line d (Fig. 6-2 3 ) .
We then consider the set of all points, each of which is equally distant from F
and d. Thus the point P in Fig. 6-23 is such that PF PD. The collection of
=
all points, each of which is equidistant from F and d, fills out a curve called
the paf3bola. The point F is called the focus of the parabola, and the line d is
called the directrix.
Each choice of a point F and line d determines a parabola. Hence there
are infinitely many different parabolas. The general shape of all such curves
is, however, about the same. Each is symmetric about the line which passes
through F and is perpendicular to d. This line is called the axis of the
parabola. Each parabola passes between its focus and directrix and opens out
as it extends farther and farther from the directrix.
The direct definition of the ellipse is also simple. We start with two fixed
points F and F' (Fig. 6-24) and consider any constant quantity greater than
the distance F to F'. If, for example, the distance from F to F' is 6, we may
choose 10 as the constant quantity. One then determines all points for each of
which the distance from F and the distance from F' add up to 10. This collec
tion of points is called an ellipse. Thus, if P is a point for which PF + PF'
equals 10, then P lies on the ellipse determined by F, F', and the quantity 10.
The points F and F' are called the foci of the ellipse.
By changing the distance FF' or the quantity 10, one obtains another
ellipse. Some ellipses are long and narrow; others are almost circular. All are
144 NATURE AND USES OF EUCLIDEAN GEOMETRY
symmetric about the line FP and about the line perpendicular to and midway
between F and P.
The direct definition of the hyperbola also calls for choosing two fixed
points F and F', called foci, and a constant quantity which, however, must be
less than the distance from F to F'. If FP is 6, then the constant quantity can,
for example, be 4. We now consider any point P for which the difference
PF' - PF equals 4. All such points lie on the right-hand portion of Fig. 6-2 5 ,
whereas the points for which PF - PF' � 4 lie on the left-hand portion of the
figure. The two portions together are the hyperbola; each portion is a branch
of the hyperbola.
F'
Fig. 6-25.
The hyperbola.
As for the ellipse, each choice of the distance FF' and the constant quantity
determines a hyperbola. Here, too, the curve is symmetric about the line FF'
and about a line perpendicular to and midway between F and F'. One branch
opens to the right and the other to the left.
We shall not prove that the curves we have defined by means of focus
and directrix or by means of foci and constant quantities are the same as those
obtainable by slicing a conical surface. In our future work we shall use the
direct definitions.
EXERCISES
1. Since the circle is also a conic section, it should be included among one of the
three types-parabola, ellipse, and hyperbola. From the shapes of these curves it
would appear that the circle falls among the ellipses. Can you see how the circle
may arise as a special kind of ellipse?
2. Suppose that we have an ellipse for which F'F is 6 and the constant quantity is
10. If the point P of the ellipse lies on the line F'F to the right of F, how much
is PF?
3. For the ellipse, why must the constant quantity be chosen greater than the
distance F'F?
4. Given a parabola for which the distance from focus to directrix is 10, how
far from the focus is that point on the parabola which lies on the axis?
* 6-6 CONIC SECTIONS AND LIGHT
Next to straight line and circle, conic sections are the most valuable curves
mathematics has to offer for the study of the physical world. We shall examine
here the uses of the parabola in the control of light.
CONIC SECfIONS AND LIGHT 145
/2 V
D,p---�Q�--�P�--7Q:-'
Axis Axis
F
d
d
Fig. 6-26. Fig. 6-27
The reflecting property ofthe parabola.
Let P be any point on the parabola (Fig. 6-26). By the tangent to the
parabola at P we mean the line through P which meets the parabola in just that
one point and lies entirely outside the curve. From the standpoint of the con
trol of light, the curve possesses a most pertinent property. If P is any point
on the curve and F is the focus, then the line FP and PV, the line through P
parallel to the axis, where V is any point on this parallel, make equal angles
with the tangent t at P. That is, angle 1 equals angle 2.
Before proving the geometrical property just stated, let us see why it is
significant. If a light ray issues from some source of light at F and strikes a
parabolic mirror at. P, it will be reflected in accordance with the law that the
angle of incidence equals the angle of reflection. The curve acts at P as though
it had the direction of the tangent. Then angle 1 is the angle of incidence.
Because angle 1 equals angle 2, the reflected ray will be PV. Hence the re
flected ray will travel out parallel to the axis of the parabola. Now, P is any
point on the parabola. Hence any ray leaving F and striking the parabola will,
after reflection, travel out parallel to the axis of the parabola, and the reflected
light will form a powerful beam in one direction. We thus obtain a concen
tration of light.
Let us now prove that PF and PV make equal angles with the tangent at P.
We shall prove first that every point outside of the parabola is farther from
the focus than from the directrix, and every point inside is closer to the focus
than to the directrix. Consider the point Q (Fig. 6-27) outside the parabola.
We wish to show QF > QD, where QD is the distance from Q to the directrix.
We continue the line QD until it strikes the parabola at P. Now
QF > PF - PQ
because any side of a triangle is greater than the difference of the other two
146 NATURE AND USES OF EUCLIDEAN GEOMETRY
sides. Since P i, on the parabola, by the very definition of the curve, PF = PD.
then
QF > PD - PQ = QD.
We may use the same figure to show that Q', any point inside the parabola,
is closer to F than to the directrix, that is, that Q'F < Q'D. First,
Q'F < PF + PQ'
because any side of a triangle is less than the sum of the other two sides.
Since PF = PD,
Q'F < PD + PQ' = Q'D.
And now let us prove that PF and PV of Fig. 6-26 make equal angles with
the tangent t at P. We shall invert our approach just to make the proof
easier. Let us draw a line t through the point P (Fig. 6-28), which makes
equal angles with PF and PV, and we shall prove that this line is the tangent
to the parabola at P. We shall make the proof by showing that any point Q
on this line lies outside the parabola. Since this line does have one point P in
common with the parabola, it must, by the definition of tangent, be the
tangent.
-f---t-�---- .\xis
d Fig. 6-28
Consider the triangles PDQ and PFQ. We know that PD = PF because P
is a point on the parabola. Further, since L I = L 2 by the very choice of the
line t, and since L 2 L J because they are vertical angles, then L I L J.
= =
Finally, PQ is common to the two triangles. Then the triangles are congruent
and QD = QF because they are corresponding sides of the congruent triangles.
Now QE is the distance from Q to the directrix, and QE < QD because the
hypotenuse of a right triangle is longer than either arm. Then QF > QE.
According to the preceding proof, Q must lie outside the parabola. Since Q is
CONIC SECTIONS AND LIGHT 147
any point on t (except P, of course) , the line t must be the tangent at P. Thus
the line through P which makes equal angles with PF and P V is the tangent
at P.
We now know, then, that any light ray issuing from F and striking the
parabola at P will be reflected along PV, that is, parallel to the axis. The
parabolic mirror's power to concentrate light in one direction is very useful.
The commonest application is found in automobile headlights. In each head
light there is a small bulb. Surrounding this bulb is a surface (Fig. 6-29),
called a paraboloid, which is fonned by rotating a parabola about its axis. (The
surface is, of course, silvered so that it will reflect.) Light issuing in millions of
directions from the bulb, which is placed at the focus of the paraboloidal mirror,
strikes the mirror, is reflected along the axis of the paraboloid, and illuminates
strongly whatever lies in that direction. The effectiveness of this arrangement
may be judged from the fact that the light thrown forward by bulb and mirror
is about 6000 times as intense as that thrown in the same direction by the bulb
alone. The reflecting property of the paraboloidal mirror is also utilized in
searchlights and flashlights.
The reflecting property of a paraboloidal mirror can be used in reverse.
If a beam of parallel light rays enters such a mirror while traveling parallel to
the axis, each ray will be reflected by some point on the surface in accordance
with the law of reflection. But since FP (Fig. 6-29) and VP make equal angles
with the tangent, the reflected ray will travel along PF and all reflected rays
will arrive at the focus F. Hence there will be a great concentration of light
at F.
Fig. 6-29.
The paraboloidal mirror.
This concentration of light is used effectively in telescopes. The light
emitted by stars is so faint that it is necessary to collect as much as possible in
order to obtain a clear image. The axis of the telescope is therefore directed
toward the star, and, because this source is so far away, the rays enter the tele
scope practically parallel to the axis, travel down the telescope to a paraboloidal
mirror at the back, and are reflected to the focus of the mirror.
Radio waves behave very much like light rays. Hence paraboloidal
reflectors made of metal are used to concentrate radio waves issuing from a
small source into a powerful beam. Conversely, a paraboloidal antenna can
148 NATURE AND USES OF EUCLIDEAN GEOMETRY
pick up faint radio signals and produce a relatively strong signal at the focus.
Since radio is used today for hundreds of purposes, the paraboloidal radio
antenna is a very common instrument.
We see from this brief account that the conic sections are immensely
valuable. Some of the most momentous applications have yet to be described
and will be taken up in later chapters.
How did the Greeks come to study these curves? As far as we know, the
conic sections were discovered in attempts to solve the famous construction
problems of Euclidean geometry, i.e., to trisect any angle, to construct a square
equal in area to a given circle, and to construct the side of a cube whose volume
is twice that of a cube of given side. The constructions were to be performed
subject to the restriction that only a straight edge (not a ruler) and a compass
be used. Having obtained the curves, the Greeks continued to work on them,
partly because they were interested in geometrical forms and partly because
they discovered the uses of these curves in the control of light. Apollonius
himself wrote a book entitled On Burning Glasses, whose subject was the
parabola as a means of concentrating light and heat, and there is a story that
Archimedes constructed a huge paraboloid which focused the sun's rays on the
Roman ships besieging his city of Syracuse, and thus set them on fire.
We see in the history of conic sections one more example of how mathe
maticians, pursuing a subject far beyond the immediate problems which give
rise to it, come to make important contributions to science.
EXERCISES
l. Let Q be any point outside of an ellipse ( Fig. 6-30). Prove that F2Q + FI Q is
greater than a, where a is the sum of the distances of any point on the ellipse
from the foci. [Suggestion: Introduce the point P where F2Q cuts the ellipse.]
Q
p
'P
�---"'---
F,
Fig. 6-30 Fig. 6-31
2. Let t be the tangent at any point P of an ellipse (Fig. 6-3 1 ). Let F2 and Fl be
the foci. Prove that F2P and FIP maKe equal angles with t. [Suggestion: Use the
result of Exercise 1 and Exercise 5 of Section 6-4.]
3. In view of the result of Exercise 2. what do you expect to happen to the light
rays issuing from a source placed at the focus F2 of the ellipse?
CULTURAL INFLUENCE OF EUCLIDEAN GEOMETRY 149
4. When the distance between the two foci F2 and F I of an ellipse approaches 0,
the ellipse approaches a circle in shape. What do the lengths F2P and FIP become
when F2 and FI coincide? What theorem about circles follows as a special case
of the result in Exercise 2?
* 6-7 THE CULTURAL INFLUENCE OF EUCLIDEAN GEOMETRY
If the development of mathematics had ceased with the creation of Euclidean
geometry, the contribution of the subject to the molding of Western civiliza
tion would still have been enormous, for Euclidean geometry was and still is
an overwhelming demonstration of the power and effectiveness of our reason
ing faculty. The Greeks loved to reason and applied it to philosophy, political
theory, and literary criticism. But philosophy breaks down into philosophies
whose relative merits become the object of much dispute between the adher
ents of one school and those of another. Plato's Republic may indeed be the
perfect answer to the quest for a satisfactory political system, but we must still
be convinced of this fact. And literary criticism certainly does not lead to
universally accepted standards and the creation of universally acclaimed
literature. In Euclidean geometry, however, the Greeks showed how reason
ing which is based on just ten facts, the axioms, could produce thousands of
new conclusions, mostly unforeseen, and each as indubitably true of the
physical world as the original axioms. New, unquestionable, thoroughly
reliable, and usable knowledge was obtained, knowledge which obviated the
need for experience or which could not be obtained in any other way.
The Greeks, therefore, demonstrated the power of a faculty which had not
been put to use in other civilizations, much as if they had suddenly shown the
world the existence of a sixth sense which no one had previously recognized.
Clearly, then, the way to build sound systems of thought in any field was to
start with truths, apply deductive reasoning carefully and exclusively to these
basic truths, and thus obtain an unquestionable body of conclusions and new
knowledge.
The Greeks themselves recognized this broader significance of Euclidean
geometry, and Aristotle stressed that the Euclidean procedure must be the
aim and goal of all sciences. Each science must start with fundamental prin
ciples relevant to its field and proceed by deductive demonstrations of new
truths. This ideal was taken over by theologians, philosophers, political
theorists, and the physical scientists. We shall see later on how widely and
how deeply it influenced subsequent thought.
By teaching mankind the principles of correct reasoning, Euclidean
geometry has influenced thought even in fields where extensive deductive
systems could not be or have not thus far been erected. Stated otherwise,
Euclidean geometry is the father of the science of logic. We pointed out in
Chapter 3 that certain ways of combining statements lead to unquestionable
1 50 NATURE AND USES OF EUCLIDEAN GEOMETRY
conclusions, provided the original premises are unquestionable. These ways
are called principles or methods of deductive reasoning. Where did we get
these principles? The answer is that the Greeks learned to recognize them in
their work on Euclidean geometry and then appreciated that these principles
apply to all concepts and relationships. If one argues from the premises that
all bankers are wealthy and some bankers are intelligent to the conclusion that
some intelligent men are wealthy, he is using a principle of valid reasoning
discovered in the work on Euclidean geometry. The indirect method of proof
which we applied earlier in this chapter owes its recognition to the same source.
Toward the end of the classical Greek period, Aristotle formulated the valid
principles of reasoning and created the science of logic. In particular, he
called attention to some basic laws of logic, such as the principle of contradic
tion, which says that no proposition can be both true and false, and the prin
ciple of the excluded middle, which states that any proposition must be either
true or false.
It is because Euclidean geometry applies these principles of reasoning so
clearly and so repeatedly that this subject is often taught as an approach to
reasoning. The Greeks themselves stressed the value of mathematics as a
preparation for the study of philosophy. Whether this is the best way of
learning to reason may perhaps be disputable, but there is no doubt that his
torically this is the way in which Western man learned. And it is pertinent that
even current texts on logic use mathematical examples quite freely because
these illustrate the principles clearly, unobscured by irrelevant implications or
by vagueness in the concepts and relations employed.
The most portentous fact about Euclidean geometry is that it inspired a
large-scale mathematical investigation of nature. From the outset the geomet
rical studies were an investigation of nature. But as the Greeks proved more
and deeper theorems and these theorems continued to agree perfectly with
observations and measurements, the Greeks became convinced that through
mathematics they were learning some of the secrets of the design of this world.
It became clear that mathematics was the instrument for this investigation, and
the results fostered the expectation that the further application of mathematics
would reveal more and more of that design. Just how far the Greeks were
emboldened to carry this venture will be apparent in the next twO chapters.
From the Greeks the Western world learned that mathematics was the extra
ordinarily powerful instrument with which to explore nature.
REVIEW EXERCISES
1. One of the basic theorems of Euclidean geomecry is that the base angles of an
isoceles triangle are equal. Euclid's proof proceeded thus. Given triangle ABC
(Fig. 6-3 2 ) with AB AC, prolong AB to D and AC to E so that BI) CEo
= =
CULTURAL INFLUENCE OF EUCLIDEAN GEOMETRY 151
A
,�,
3
A -- -- � � --
-- --4 1 -- -B
�
"I
I .... ....
........ ...-"' ...-
...- ...- ><
...-
\
\
C ----�2�----D
\
.... .... ....
/�...- "' ''' ...- ..... .... .... ....\
D o<- - - - - - - - --- --"'E Fig. 6-32 Fig. 6-33
Now draw DE, BE, and DC. Complete the proof by first proving that BE DC =
and then that t. CBD = t. BCE.
2. In the text we proved that if the alternate interior angles 1 and 2 of Fig. 6-33 are
equal, the lines AB and CD are parallel. Prove
a) if the corresponding angles 2 and 3 are equal, the lines are parallel;
b) if the angles 2 and 4 are supplementary, that is, if their sum is 1 80°, then the
lines are parallel.
3. Suppose that 111 in Fig. 6-34 represents a road, and a telephone central is to be
built somewhere on this road to serve towns at A and A'. Where along the road
should the central be built to minimize the total distance from the central to A
and the central to A'?
• A'
m Fig. 6-34 m Fig. 6-35
4. A ship must pass between the guns of a fort at A ( Fig. 6-35) and equally power
ful guns along the shore 111. What path should it take to be as safe as possible
from all the guns?
Fig. 6-36
5. Let C and /) be two fixed circles (Fig. 6-36) with radii c and d, respectively. and
c > d. Moreover D and C are tangent internally at P. Now let T be a third
circle which is tangent externally to D and internatly to C. Show that the posi
tions of the centers of all possible circles T is an ellipse whose foci are the centers
of C and D.
152 NATURE AND USES O F EUCLIDEAN GEOMETRY
Topics for Further Investigation
1 . The use of Euclidean geometry in the design of spherical mirrors. Use the first
reference to Kline or the reference to Taylor below or any college physics text.
2. The use of Euclidean geometry in the design of optical lenses. Use the references
to Taylor below or any college physics text.
3. The contents of Euclid's Elements. Use the reference to Heath.
4. Euclidean geometry as a manifestation of Greek culture. Use the second refer
ence to Kline.
Recommen!led Reading
BALL, W. W. R.: A Short A ccount of the History Of Mathematics, pp. 1 3-63,
Dover Publications, Inc., New York, 1960.
Boys, C. VERNON: Soap Bubbles, Dover Publications, Inc., New York, 1959.
COURANT, R. and H. ROBBINS: What is Mathematics?, pp. 329-338, pp. 346-361,
Oxford University Press, New York, 1941.
EVES, HOWARD: An Introduction to the History of Mathematics, Rev. ed., pp. 52-
130, Holt, Rinehart and Winston, Inc., New York, 1964.
HEATH, SIR THOMAS L.: A Manual of Greek Mathematics, Chaps. 8, 9 and 10, Dover
Publications, Inc., New York, 1963.
KLINE, MORRIS: Mathematics: A Cultural Approach, Sections 6-8, Addison-Wesley
Publishing Co., Reading, Mass., 1962.
KLINE, MORRIS: Mathematics and the Physical World, Chaps. 6 and 17, T. Y. Crowell
Co., New York, 1959. Also in paperback, Doubleday and Co., N.Y., 1963.
SAWYER, W. W.: Mathematician's Delight, Chaps. 2 and 3, Penguin Books, Har
mondsworth, England, 1943.
ScOTT, J. F.: A History of Mathematics, Chap 2, Taylor and Francis, Ltd., London,
1958.
SMITH, DAVID EUGENE: History of Mathematics, Vol. I., Chap. 3, Vol. II, Chap. 5,
Dover Publications, Inc., New York, 1958.
TAYLOR, LLOYD WM.: Physics, The Pioneer Science, Chaps. 29-32, Dover Publica
tions, Inc., New York, 1959.
CHAPTER 7
CHARTING THE EARTH AND THE H EAVENS
Thrice happy souls! to whom 'twas given to rise
To truths like these, and scale the spangled skies!
Far distant stars to clearest view they brought,
And girdled ether with their chains of thought.
So heaven is reached:-not as old they tried
By mountaim piled on mountains in their pride.
OVID
7-1 THE ALEXANDRIAN WORLD
The course of mathematics is very much dependent upon the caprices of man.
What more the classical Greeks might have produced had they been able
to continue their way of life uninterruptedly, we shall never know. In 352 H.C.
Philip II of Macedonia, a province to the north of Athens and outside the pale
of Greek culture, started out to conquer the world. He defeated Athens in
338 B.C. In 336 B.C. Alexander the Great, Philip's son, took over the Mace
donian armies, completed the conquest of Greece, conquered Egypt, and
penetrated Asia as far east as India, and Africa as far south as the cataracts of
the Nile. For a new capital he chose a site in Egypt which was central in his
empire. Too hig a man to be hampered by modesty, he called the capital
Alexandria. Alexander drew up plans for the city and for populating it, and
the work was begun. Alexandria did become the center of the Hellenistic
world, and even 700 years later was still called the noblest of all cities.
Alexander was the most cosmopolitan of men and sought to break down
barriers of race and creed. Hence he encouraged and invited Greeks, Egyp
tians, Jews, Romans, Ethiopians, Arabs, Indians. Persians, and Negroes to settle
in the city. At that time the Persian culture was flourishing, and so Alexander
made special efforts to fuse Greek and Persian ways of life. He himself married
Statira, daughter of Darius, in 32 5 B.C. and compelled 100 of his generals and
10,000 of his soldiers to marry Persians. After his death written orders were
found to transport large groups of Asians to Europe and vice versa.
Alexander died while still engaged in reconstructing the world, and his
empire split into three parts. Of these Egypt proved to be the most significant
153
1 54 CHARTING THE EARTH AND THE HEAVENS
from the standpoint of mathematical progress. Alexander had indeed chosen
a good site for his capital. Located at the junction of Asia, Africa, and Europe,
it became the center of trade, which brought wealth to the city. The succes
sors of Alexander who ruled Egypt and who adopted the title of Ptolemy were
wise men. They appreciated the cultural greatness of classical Greece and
decided to make Alexandria a great cultural center. Under their direction parr
of the wealth was used to beautify the city with splendid buildings, baths,
parks, theaters, temples, libraries, and a national archive. They also erected a
famous building devoted to the Muses of literature, arr, and science, called the
Museum, and adjacent to it, an enormous library to house manuscripts. At its
height this library was said to contain 750,000 works, an enormous number in
view of the fact that in those days "books" were written and reproduced by
hand. The Ptolemies invited scholars from all over the world to work there
and supported them. Euclid and Eratosthenes, to speak for the moment of men
we have already met, lived and worked at this center; Apollonius was educated
there; and we shall meet other luminaries shortly. These men, coming from all
over the world, brought knowledge of their lands, people, animals, and vege
tation to Alexandria, and this in itself helped to make Alexandria cosmopolitan.
The scholars set to work in the fields of mathematics, science, philosophy,
philology, astronomy, history, geography, medicine, jurisprudence, natural
history, poetry, and literary criticism. Fortunately, Egyptian papyrus, cheaper
than parchment, was available for books, and so many more works could not
only be written but copied. Alexandria became in fact the center of the book
copying trade of the ancient world. The scholars undertook not only to create
and write, bur they sent expeditions all over the world to gather knowledge.
At Alexandria they built a huge zoological garden and a botanical garden to
house the species of animals and plants brought back by these expeditions.
Alexander had planned to fuse cultures in his new empire, and at Alex
andria his goal was realized. The culture which developed there was indeed
different from that of classical Greece, for reasons that are of interest because
they account for the kind of mathematics the Alexandrians produced. First
of all, the rather sharp segregation between free men and slaves which existed
in Athens was destroyed. The scholars came from all parts of the world and
from all economic levels and took a natural interest in the scientific, commer
cial, and technical problems of commerce, industry, engineering, and naviga
tion. Although Athens also was primarily a sea power and lived on trade,
Alexandrian commerce and navigation were far more widespread. Hence
there developed an intense interest in astronomy and in geography, i.e., in the
subjects enabling man to tell time, navigate over land and sea, build roads, and
determine boundaries of the empire. Free men engaged in commerce arc
naturally more concerned with materials, methods of production, and new
ventures. Fin.lll', though the nucleus of the scholars gathered at Alexandria
was Greek, it was exposed to the influence of the practical Egyptians to whom
THE ALEXANDRIAN WORLD 155
mathematics, to the extent that it was used in ancient Egypt, was a tool for
engineering, commerce, and state administration.
The results of the new outlook and interests are readily detected. First of
all, there was a sharp increase in mechanical devices, which of course aid men
in their work. Even training schools to educate young people in mechanics
were established. Pulleys, wedges, tackles, geared devices, and a mileage
measuring instrument such as is found in the modern automobile were invented.
Archimedes, the greatest intellect of the Alexandrian world, constructed a
planetarium which reproduced the motions of the heavenly bodies and de
signed a pump for raising water from a river to land. He used pulleys to
launch a heavy galley for King Hiero of Syracuse. Instruments to improve
astronomical measurements were also invented.
Another science whose beginnings may be found in Alexandria is the
study of gases. The Alexandrians, notably Heron (about first century A.D. ) ,
a famous mathematician and engineer, learned that the steam created by heating
water seeks to expand and that compressed air can also exert force. Heron is
responsible for many inventions which used these forces. Temple doors
opened automatically when a coin was deposited. Inside the temple another
coin inserted in a machine blessed the donor by automatically sprinkling holy
water upon him. Fires lit under the altar created steam, and the mystified and
awe-struck audience observed gods who raised their hands to bless the wor
shippers, gods shedding tears, and statues pouring out libations. Doves rose
and descended under the unobservable action of steam. Guns similar to the
toy bee-bee gun were operated by compressed air. Steam power was used to
drive automobiles in the annual religious parade along the streets of Alexandria.
The Alexandrians also studied water power and applied it. They invented
improved water clocks (used in the courts to limit the time allowed to
lawyers), fountains in which figures moved under water pressure, pumps to
bring water from wells and cisterns, musical organs worked by water pressure,
and a water-spraying device operating on exactly the same principle as that
applied in contemporary lawn-sprinklers.
The study of sound and light was intensified. We have already mentioned
Euclid's and Heron's studies on the reflection of light by mirrors. Books on
optics were written not only by Euclid and Apollonius but also by Heron, the
astronomer Ptolemy (whom we shall discuss shortly), and others. Indeed the
Alexandrians were the first to concern themselves with a second basic phe
nomenon of light, refraction, which we shall encounter in this chapter.
Chemical and medical skills, if not a science of chemistry, show a marked
advance in Alexandria. The Egyptians had previously acquired some knowl
edge in these areas, as we know from their ability to embalm. However,
metallurgical studies, including the first text on the subject, and the investiga
tion of chemicals, including poisons and their uses, wefe essentially new
developments. Dissection of bodies, forbidden in classical Greece, was per-
1 56 CHARTING THE EARTH AND THE HEAVENS
mitted, and the Alexandrian world produced the beginnings of anatomy and
the most famous doctor of the ancient world, Galen.
Where was mathematics in this scheme of things' The Greeks brought to
Alexandria a fully formed, mature, and philosophically oriented mathematics
which had little bearing on practical problems. Although the great Alexan
drian mathematicians continued to display the Greek genius for theory and
abstraction. they combined with that an interest in the world about .them and
in practical problems. To the classical Greek concern with qualitative prop
erties such as congruence. the Alexandrians added a new theme. quantitative
results which are useful in a variety of ways.
To illustrate the combination of old and new we might note that while
Euclid chronologically belongs to the Alexandrian period, his mathematical
work is in essence a recapitulation of the work done in the classical period.
Thus Euclid tells us, for example, that the ratio of the area of any circle to
the square of its radius is the same for all circles. In symbols, if A is the area
of any circle of radius r, then
A
2 = k,
T
where k is the same number for al/ circles. But now suppose that we wish to
find the area of a particular circle. Does Euclid's theorem help us? Not
directly. We know from the preceding equation that for any circle
A = kr',
where k is a constant. But how much is k? This quantity which we usually
denote by 7r is an irrational number. It is not readily computed and, because
it is irrational, can be expressed as a decimal only approximately. One of
Archimedes' great achievements, which also illustrates the interest in quanti
tative knowledge, is his determination that '11' lies between 3t and 3*. The
achievement is all the more remarkable because neither the classical nor the
Alexandrian Greeks had an efficient system for writing and operating with
numbers.
As a matter of fact, Archimedes (287-2 1 2 B.C.) is the man whose work
best illustrates the character of Alexandrian Greek mathematics. He derived
many formulas for the areas and volumes of geometric figures, and his results,
as opposed to those of Euclid and Apollonius, made actual computations
possible. At the same time, Archimedes also pursued the classical Greek
interest in proof and in beautiful mathematical results. In this area, he was
proudest of his proof that the ratio of the volume of a sphere inscribed in a
cylinder (Fig. 7-1) is to the volume of the cylinder as 2 is to 3. He also
proved that the same ratio holds for the areas of the sphere and the cylinder.
Archimedes was so pleased with this result that he asked that it be inscribed
THE ALEXANDRIAN WORLD 157
on his tomhstone. After Archimedes was killed by a Roman soldier during
the Roman conquest of Syracuse, the Romans built an elaborate tomb on
which they inscribed this theorem. It was this inscription which enabled
Cicero to recognize the tomb on a visit to Syracuse two hundred years later.
Fig. 7-1.
The volume of a sphere inscribed
in a cylinder is two-thirds the volume
of the cylinder.
Even in his physical studies Archimedes displayed this combination of
theoretical and practical interests. He took up the subject of the lever, a
device which had been used in Egypt and Babylonia for thousands of years.
Like a true Greek, he produced a scientific work, 0" the Lever, along the lines
of Euclidean geometry; that is, he started from axioms and proved theorems
ahout the lever. He did the same with the subjects of floating hodies and
centers of gravity of various surfaces and volumes. To these achievements
must be added his inventions, of which we have already spoken.
The work of some other giants of the Alexandrian civilization also il
lustrates the combination of theoretical and practical interests. Eratosthenes
(273 H.c.-192 B.C.) , director of the library at Alexandria, was distinguished in
mathematics, poetry, philology, philosophy, and history. He was the first
outstanding mathematical geographer and geodesist. The calculation of the
circumference of the earth, which we studied in the preceding chapter, is one
of his great achievements. He collected and integrated all available geograph
ical knowledge, introduced methods of surveying, made maps, and compiled
all of this information in his Geographica.
Eratosthenes was also an astronomer. He constructed some new instru
ments, made many astronomical measurements, and, among other applications,
used his astronomical knowledge to improve the calendar. As a result of his
work, an old Greek calendar based on a year of 1 2 months each containing 30
days was replaced by the Egyptian year of 365 days, to which Eratosthenes
added an extra day every fourth year. This calendar was adopted by the
Romans when Julius Caesar called in Sosigenes, an Alexandrian, to reform the
calendar. Julius contributed his name. The Julian calendar was taken over by
the Western world with the slight modification that we omit the leap year in
three out of every four century years.
158 CHARTING THE EARTH AND THE HEAVENS
The work in geography and astronomy, continued by such famous men as
Strabo (ca. 63 B.C.-ca. 1 5 B.C. ) , Poseidonius ( first century B.C.), and many
others, was crowned by the achievements of two of the greatest men of the
Alexandrian world, Hipparchus and Ptolemy. Hipparchus (second century
B.C.), about whom we know rather little, lived at Rhodes, but was in close
touch with the developments in Alexandria. After criticizing Eratosthenes'
Geographica, he refined the method of locating places on the earth by system
atically employing latitude and longitude. He improved astronomical instru
ments, measured irregularities in the moon's motion, catalogued about 1000
stars, and estimated the length of the solar year as 365 days, 5 hours, and 5 5
minutes, i.e., he overestimated by about 6!- minutes. One o f his notable astro
nomical discoveries was the precession of the equinoxes, a slow change in the
time of occurrence of the spring and fall equinoxes. Hipparchus is the creator
of the most famous and most useful astronomical theory of antiquity, about
which we shall learn more later.
The work of Hipparchus is known to us largely through the writings
of the mathematician, astronomer, geographer. and cartographer Claudius
PtOlemy. Ptolemy, who is believed to be Egyptian-he was no relation to the
Greek rulers of Egypt-lived from about 100 to 178 A.D. One of his influ
ential achievements was his Guide to Geography, or Geographica, the most
comprehensive work of antiquity on this subject. This book, which contains
the latitude and longitude of 8000 places, almost every place on the earth then
known, estimates of the size and extent of the habitable world, and methods of
map-making, summarized the geographical knowledge of the ancient world
and became the standard atlas for over a thousand years. Better known is
Ptolemy's great work on astronomy, the Mathematical Syntaxis or The Mathe
matical Col/ection, which the Arabs called Al Megiste (an Arabic and Greek
combination meaning "the greatest")-Iater Anglicized as The Almagest. This
book contains the full development of Hipparchus' and Ptolemy's astronomical
theory, generally known as Ptolemaic theory, which dominated astronomy
until about 1600 A.D. when it was superseded by the work of Copernicus and
Kepler.
7-2 BASIC CONCEPTS OF TRIGON OM ETRY
The theoretical sciences of geography and astronomy require their own mathe
matical tOol, trigonometry. Hipparchus and Ptolemy created this branch of
mathematics whose first presentation is found in PtOlemy's Almagest. With this
simple branch of mathematics it is possible to calculate the sizes and distances
of the heavenly bodies as easily as one calculates the area of a rectangle. In
presenting the trigonometry of Hipparchus and Ptolemy, we shall not use
their notation and proofs; however, the modern approach is not essentially
different.
BASIC CONCEPTS OF TRIGONOMETRY 1 59
B
Fig. 7-2.
A L---Ll C A ' L-------� C' Simi lar right triangles.
Let us consider the two right triangles shown in Fig. 7-2 and let us sup
pose that angle A equals angle A'. Since all right angles are equal, angle C
equals angle C. One of the key theorems in Euclidean geomerry states that
the sum of the angles in any triangle is 1 80". Since all three angles in each
triangle add up to rhe same amount and two angles of one are equal to two of
the other, the third angles must be equal; i.e., angle B equals angle B'.
Now another theorem of Euclidean geometry states that if two triangles
are similar, the ratio of any two sides in one equals the ratio of the correspond
ing sides in the other. Thus, for example,
BC We'
AB A'B'
Let us note here that triangle A'B'C is any other right triangle which has an
acute angle, A', equal to angle A. Hence for any such triangle the ratio
B'CIA'B' must equal BCI AB. Therefore, if we could compute this ratio-
it is a number of course-for any one right triangle containing a given angle
A, we would know it for all right triangles having an acute angle equal to A.
Before we pursue this idea, let us observe that what we said about the
ratio BCIAB applies to any other ratio of two sides of triangle ABC. Of the
many ratios we can form three are especially useful and are gIVen names.
These ratios are:
side opposite angle A BC
sine A = - ,
hypotenuse AB
side adjacent to angle A - -
AC ,
cosine A =
hypotenuse AB
side opposite angle A BC
tangent A - _ .
side adjacent to angle A AC
160 CHARTING THE EARTH AND THE HEAvENS
B"
Fig. 7-3.
The variation of sin A
with angle A.
The angle A is written alongside the name of each ratio. This practice is
necessary not only because the very use of such words as opposite and adjacent
depends upon which angle of the triangle we are talking about, but also
because the values of the ratios depend upon the size of the angle. It is very
common to abbreviate these names as sin, cos, and tan, respectively.
Since we intend to employ these ratios, our first task should be to see
whether we can compute them for angles of various sizes. First of all let us
get some general notion of how these ratios vary with the angle. Let us
consider sin A as an example. We have already pointed out that the values of
these ratios for a given angle A are the same in any right triangle containing
A. To study the variation of sin A as A changes, we can then take right
triangles whose hypotenuse is I . We know from the very definition of sin A
that it is the ratio of the side opposite angle A to the hypotenuse. Since sin A
equals BC/AB and AB = 1, then sin A Be. When A is small (Fig. 7-3) ,
=
BC or sin A is small. We should expect, then, that for an angle close to 0",
the sine of that angle should be close to O. On the other hand, as Fig. 7-3
shows, when angle A increases and the hypotenuse is kept one unit in length,
the opposite side must increase; hence the sine ratio must increase. When angle
A is very close to 90°, as in triangle AC"B", the side R"C" is almost as large as
A B"; hence sin A must be close to I . When angle A is 90", it can no longer be
an acute angle of a right triangle, but because sin A approaches 1 as A ap
proaches 90", it is agreed that in this special case we shall take sin A to be I .
Likewise, we take sin 0" to be O . The general point of this discussion is that
sin A varies from 0 to 1 as A varies from 0° to 90°.
Next let us take a particular angle and let us see whether we can calculate
the three ratios. We shall choose 30". Consider the equilateral triangle ABD
(Fig. 7-4). We know that in such a triangle each angle is 60". If we now
draw the angle bisector AC, then angle BAC is 30". Moreover, triangle ACB
is a right triangle because triangles ACB and ACD are congruent, and hence
the twO angles at C must be equal. Since the sum of these two angles is 180",
each must be 90". Triangle ACB is, then, a right triangle containing an acute
angle of 30".
Now it does not matter how long we take AB to be, for we saw earlier
that we may compute the ratios in any right triangle containing the given
BASIC CONCEPTS OF TRIGONOMETRY 161
acute angle. Let us therefore choose a convenient number, say 2, for the
length of AB. Since ABD is equilatetial, side BD 2 . But because triangles
=
ACB and ACD are conguent, CB CD. Hence CB = 1. We now find the
=
length of AC. The Pythagorean theorem says that
B
(AC) ' + (CB) ' = (AB) ' ;
2
therefore
(AC) ' = (AB) ' - (CB) ' .
Since AB = 2 and CB = 1,
(AC) ' = 4 I,
D
or
AC = 0. Fig. 7-4
We can now use the definitions of sine, cosme, and tangent to state at
once that
side opposite
sin 30° = = - ,
hypotenuse 2
side adjacent .
0,
-
cos 30° =
hypotenuse 2
side opposite 1
tan 30° = '
0
=
side adjacent
As a dividend for our patience we get more information from the above
reasoning than we sought. Let us note that angle B, which is 60", is also an
acute angle in a right triangle, and we know the lengths of the sides. Hence,
since sin B is the side opposite angle B divided by the hypotenuse, we have
. 0
sm 60° = - '
2
Similarly, by applying the definitions of cosine and tangent we obtain
cos 60° = t, tan 60° = vl.
We must admit that in undertaking to find the ratios belonging to 30" we
selected a simple case. For most angles the ratios are not so easily found, and
a good deal of geometry must be applied. The process of determining the
ratios for angles from 0" to 90" is not particularly fascinating. Fortunately
these values were obtained by Hipparchus and Ptolemy and compiled in a
table to be found in Ptolemy's Almagest. (These tables were checked and
162 CHARTING THE EARTH AND THE HEAVENS
extended by many later mathematicians.) Hence let us take over their results
which appear in the "Table of Trigonometric Ratios" (in the Appendix).
The table gives the sine, cosine, and tangent values for each angle from
0" to 90". For angles from 0" to 45" we use the left-hand column and the
headings across the top of the page. For example, alongside of 30" and under
tangent we find 0.5774. This number is the approximate decimal value of
lh!J. To find the sine, cosine, or tangent of an angle from 45" to 90" we use
the right-hand column and the column designations at the bottom of the page.
For example, to find sin 60" we look for 60" in the right-hand column and
above the word sine we find 0.8660. This number is the approximate decimal
value of Vfl 2.
Our table does not give the ratios for angles which contain minutes and
seconds as well as degrees. There are tables which do so, but we shall not
bother with them because the idea is the same. Where we need the value of
a ratio for an angle not in the table, it will be supplied in the text proper.
Let us note that we can use these tables in reverse. If, for example, we are
given tan A 1.7 3 2 1 , we can look down one tangent column and up the other
=
until we come to 1.7 3 2 1 . We can then look to the left (or to the right, de
pending upon where we locate this number) , and find the angle which has
the given tangent value. In the present case we must choose the angle at the
right, namely 60°. If the table does not contain the exact value given, it will
suffice, for our purposes, to choose the one nearest to it.
EXERCISES
1. Use the isosceles right triangle shown in Fig. 7-5 to compute sin 45°, cos 450,
and tan 45°.
2. Use the Table of Trigonometric Ratios to find
a) sin 20° b) sin 70" c) cos 3 5"
d) cos 55" e) ran 1 5° f) tan 80°
3. Use Fig. 7-3 in the text to determine the range of cosine values as angle A varies
from 0° to 900.
4. Use Fig. 7-3 in the text to determine the range of tangent values as angle A varies
from 0" ro 90".
5. Show that when A and B are the two acute angles of a right triangle, then sin
A = cos B and cos A = sin B.
6. Prove that cos (90" - A ) sin A and that sin (90" - A ) cos A.
= =
7. Show that sin2A + cos2A = 1 for any acute angle A. Here the notation sin2A
means (sin A )(sin A ) or the square of sin A. Can the result be used to compute
trigonometric ratios? If so, how?
8. State the definitions of sine, cosine, and tangent of angle D in terms of the sides
DE, EF, and FD of the triangle shown in Fig. 7-6.
SOME MUNDANE USES OF TRIGONOMETRIC RATIOS 163
B
D
FL-------LJ E
Fig. 7-5 Fig. 7-6 Fig. 7-7
7-3 SOME MUNDANE USES DF TRIGONOMETRIC RATIOS
Before we venture onto vast stretches of the earth's surface or into the heavens.
let us see what we can do with trigonometric ratios in rather simple. homely
situations. Suppose we had to find the height of the cliff BC in Fig. 7-7. Of
course, we could climb the cliff, let a rope down from point B until it just
reaches C, pull up the rope, and measure the length which stretched from B
to C. There is, however, an easier method which is especially recommended to
people who do not like heights.
Instead of climbing the cliff, one can walk along the ground from C to any
convenient point A. The distance from C to A is then measured; let us suppose
it proves to be 1 50 feet. At A, a person measures the angle between the hori
zontal AC and the line of sight from A to B. A surveyor would use a transit
for this purpose, but there are simpler devices, called protractors, which one
can carry in his pocket. Suppose that angle A turns out to be 40". We are
interested in side BC and we know side AC. The fact that these two sides are
the side opposite angle A and the side adjacent to angle A suggests that we use
the tangent ratio and write
BC
tan 400 =
1 50
This equation involves numbers, and we can therefore apply the axiom that
equals multiplied by equals give equals, to justify multiplying both sides by
150. We obtain
150 (tan 40°) = Be.
Now tan 40" can be found in the table which Hipparchus and Ptolemy so
considerately prepared, and proves to be 0.8391. Hence
Be = 1 50(0.8391) = 1 26.
The answer, then, is 1 26 feet. We ignore the decimals because the gIven
information is presumably accurate only to the nearest foot.
164 CHARTING THE EARTH AND THE HEAVENS
EXERCISES
1. To measure the width BC of a canyon (Fig. 7-8 ), a surveyor at C walks along
the edge (preferably alongside the edge) to some convenient point A. He then
measures AC and the angle A. Suppose AC is 300 ft and angle A is 56°. How
large is Be?
2. At some point on the ground, located at a distance from the Empire State
Building in New York City, an observer finds that the angle between the
horizontal and the line of sight to the top is 5° ( Fig. 7-9). The building is 1248
ft high. How far away is the observer?
B 5°
}
Fig. 7-9
Fig. 7-8
58°
c
Fig. 7-10
1
3. A railroad line is being planned which must rise to 1000 ft (Fig. 7-10) at a
"grade" of 5D• How long must the line be?
4. A lighthouse beacon is 400 ft above sea level (Fig. 7-1 1 ) , and the sea around it
is obstructed by rocks extending as far as 300 ft from the base of the lighthouse.
A sailor on a ship's deck 20 ft above sea level measures the angle between his
horizontal and the line of sight to the top of the beacon and finds it to be 500.
Is his ship clear of the rocks?
5. The Alexandrian Greek mathematician and engineer Heron showed how one
could dig a tunnel under a mountain by working from both ends simultaneously
and have the borings meet. He chose a convenient point A on one side, a con-
400
D
t-i--- 500
Fig. 7-11 Fig. 7-12
CHARTING THE EARTH 165
venient point B on the other, and finally point C for which angle ACB is 90°
(Fig. 7-12 ) . He next measured AC and BC and found their lengths to be 100 ft
and 75 ft, respectively. Now, said Heron, it is possible to calculate angles A and
B. He then instructed the workers at A to follow a line which made the calcu
lated angle with AC, and gave analogous directives to the workers at B. How
did he calculate angle A and angle B?
6. Trigonometric ratios can be used to compute the radius of the earth, whence,
of course, the circumference can be determined by plane geometry. The method
is an alternative to Eratosthenes' procedure. From a point A which is 3 mi above
the surface of the earth (A can be the top of a mountain or an airplane) , an
observer looks to the horizon. His line of sight. A C A
in Fig. 7-13, is just tangent to the earth's surface.
According to a theorem of Euclidean geometry, the
radius OC of the earth is perpendicular to the tangent
c
at C. Hence triangle ACO is a right triangle. Sup T
pose that the size of angle A is 87° 46'. Let us denote T
the length of OC by T. Then OD is also T. We can
now say that o
sin 87°46' = __ T .
r + 3
Given that sin 87° 46' is 0.99924, calculate r. Fig. 7-13
* 7-4 CHARTING THE EARTH
We have already related that geography was one of the major interests of
the Alexandrians. Here Hipparchus and Ptolemy, helped by the trigonometry
they had created, made great strides. Let us see how they determined the loca
tions of important places and how they calculated the distances between such
places.
Hipparchus proceeded by employing systematically an idea already ad
vanced prior to his time, namely the scheme of latitude and longitude. The
earth is, of course a sphere. Let us consider circles with center 0, which is
the center of the earth, each going through the North and South Poles, N
and S in Fig. 7-14. Thus NWS is one half of such a circle; the other half
runs in back of our figure and is therefore invisible. Likewise N VS is one
half of another such circle. Obviously we can think of such a circle through
N, S, and any other point on the earth's surface. Each half circle from N to
S is called a longitude line or a meridian of longitude.
To distinguish among these many lines, we introduce another circle,
XWVU, which is perpendicular to the longitude lines and halfway between
the two poles. This circle is called the equator. Now one of the longitude
lines, say NWS, is chosen as the starting line, so to speak. (Today this line goes
through the city of Greenwich, England.) We consider next any other line,
166 CHARTI!\"G THE EARTH A:\,D THE HEAVEl\'S
N
T
x
-- .... ..... ....
v
\
Bquator
Fig. 7-14.
s latitude and longitude.
such as N VS in our figure. The angle VOW formed at the earth's center, 0,
by the lines VO and O W is called the longitude of any point on N VS. Thus
longitude is an angle. To distinguish the meridians of longitude on the left of
'!\' lVS from those on the right, we use the term "west longitude" to designate
the angles determined by the former, and apply the term "east longitude" to
those formed hy the latter.
Thus any point on the earth's surface has a definite longitude. However
all points on the half circle N VS have the same longitude. How shall we
distinguish any one of these points from the others? The answer is: by intro
ducing horizontal circles going around the earth. The equator is one such
circle, and the circle TPQR of our figure is another. Clearly we can introduee
many such circles lying in planes parallel to the equator. These circles are
called circles of latitude. Again we have the problem of distinguishing among
these circles. This is solved by introducing angles formed at the center of the
earth, such as PO V of Fig. 7-14, where P is any point on a circle of latitude, 0
is the center of the earth, and V is on the equator and on the meridian of
longitude through P. The angle PO V is called the latitude of P. If P is north
of the equator, it is said to have north latitude; if it is south of the equator, it
is said to have south latitude. Thus points on the same meridian of longitude are
distinguished b,' their differing latitudes.
The point P is a typical point on the earth's surface, and its position is now
described by its latitude and longitude. For example, it might have 30° north
latitude and 50° west longitude. In this case, angle PO V is 30°, and angle
VO W is 50°. Any point north or south of P, that is, on the same meridian,
",ill have the same longitude as P but a different latitude. Any point east or
west of P, that is, on the same circle of latitude, will have the same latitude as
P but a different longitude.
We have described what is meant by the latitude and longitude of any
point on the earth's surface, but how do ,ve determine the latitude and longi
tude for any given point P' (After all, we cannot penetrate to the center of
CHARTING THE EARTH 167
N
---
--
Z -+---.."f----'--"'o II
,--
Equator
Equator
v
s
Fig. 7-15.
The determination of latitude at a point on Fig. 7-16
the earth's surface.
the earth to measure the angles POV and VOW.) There are numerous
methods available. We shall describe a simple one just to see that the latitude
and longitude of places on the earth can be determined. Suppose we seek the
latitude of some point P (Fig. 7-15). On the day of the spring equinox, that
is about March 2 1 , the sun is in the plane of the equator and, at noon on that
day, it is also in the plane of the meridian of longitude. For a person ,I t P, the
overhead direction is PA, and the direction to the sun is PZ'. Now the sun is
so far away that PZ' and VZ can be taken to be parallel lines. Then angle 2
equals the angle of latitude PO V because they are alternate interior angles of
parallel lines. But angle 1 equals angle 2 because they arc verticlc angles.
Hence angle 1 equals the latitude of P. But angle 1 can be measured. It is the
angle between the direction of the sun and the overhead direction at P. Thus
the latitude of P can be determined.
There are other methods of measuring latitude as well as methods for
finding the longitude of places on the earth. It is of interest that the methods
of measuring latitude are more readily applied. The prohlem of determining
longitude accurately aboard a ship at sea was not resolved until the middle of
the eighteenth century. We shall have more to say about this later.
We may suppose, then, that the latitude and longitude of places on the
earth can be determined. Can we now determine how far apart two places
are? We can and we shall illustrate the process. Suppose P (Fig. 7-16) is
New York City, which has a north latitude of 41° and a west longitude of 74°.
Hence angle POV is 41°, and angle VOW is 74°. Let us answer first the
question, How far north of the equator is New York City? This question is
easy to answer. The distanee we seek is the arc PV. But POV is 41° and arc
PV is the arc of a circle whose radius is the radius of the earth. Hence arc PV
is that part of the circumference of the earth which 41 ° is of 360°; that is, if
168 CHARTING THE EARTH AND THE HEAVENS
we take the circumference of the earth to be 25,000 miles, then
PV = Nr; ' 25,000 = 2847.
Thus New York City is 2847 miles north of the equator.
Now let us calculate how far west New York City is of the point Q which
has the same latitude and has longitude 0". This point Q is actually the location
of Morella, Spain, a small town about 200 miles east of Madrid. Since the
longitude of New York City is 74", angle VOW is 74". But the distance we
seek is not arc V W but arc PQ. Now arc PQ is on the circle of latitude
through P. This circle has its center at 0' on the straight line through NS,
and its radius, O'P, is not the radius of the earth. If we could calculate O'P,
then we could calculate the circumference of the circle of latitude, and since
angle PO'Q is also 74", we could calculate arc PQ.
Our problem then reduces to finding O'P. We can find it. The radius
O'P is a side of the triangle OO'P. Moreover, 00' is perpendicular to O'P.
Hence we have a right triangle. Since O'P and a v are parallel, angle O'PO
equals the latitude of P because O'PO and PO V are alternate interior angles of
parallel lines. Hence in triangle O'PO,
O'P
cos 41° =
OP
or
O'P OP cos 41°.
=
Now OP is the radius of the earth, or 4000 miles. From our table we find that
cos 41" is 0.7547. Hence
O'P = 4000 ' 0.7547 = 3019.
We may now calculate arc PQ. This arc is � of the circumference of the
circle whose radius is 3019 miles. Hence
PQ = Ho ' 2". ' 3019.
Using the approximate value of 3.14 for 11' , we find that
PQ = 3897.
Thus New York City is 3897 miles west of Morella, Spain.
We have computed the distance between two points on the same meridian
of longitude and the distance berween twO points on the same circle of lati
tude. We could investigate how to calculate the distance between two points
on the earth's surface which have neither the same longitude nor the same
latitude. However, we have seen enough of the method to comprehend how
CHARTING THE EARTH 169
the trigonometric ratios can be used. Only one point may be worthy of note
here. Suppose that P and Q (Fig. 7-1 7 ) are two points on the surface of the
earth. and we now consider the question. What is the distance between them?
We cannot mean the straight-line distance between P and Q because this does
not lie on the earth's surface. The distance along the surface of the earth from
P to Q must then be an arc of a curve. Which shall we choose? If we choose
a circle whose center 0 is the center of the earth and which passes through P
and Q, then we shall have what is called a great circle. The shorter of the two
arcs from P to Q along this great circle is the shortest distance from P to Q
along the surface of the sphere. This theorem of spherical geometry, which
we shall not prove, is noteworthy because it tells us what route ships and
planes should take if they are to save time and expense.
N
Fig. 7-17.
A great circle on the
s earth's surface.
Let us consider this theorem in connection with travel by the shortest
route between two points such as New York City and Morella, Spain, which
are on the same circle of latitude. Although in this case, one wishes to reach a
point due east or west (depending on the direction of the trip) of a given
point, the circle of latitude is not the shortest route because it is not a great
circle. We saw in fact that the circle of latitude has 0' as its center (Fig.
7-16), whereas the center of the earth is O.
Determining the latitude and longitude of places on the earth and their
distances apart is valuable not only for navigation but for map-making. Both
Hipparchus and Ptolemy made maps of the ancient world. Although we
shall not describe their mathematical methods, we would like to call artention
to the problem of making a map. A map is supposed to be a reproduction on
flat paper of the relative locations of places on the earth. Now the earth is a
sphere, and the one deficiency of this most prized figure of the Greeks is that
one cannot take a sphere, cut it open. and lay it fiat without creasing, folding,
stretching, or tearing the material. One can see this readily if he peels an
orange and then tries to flatten out the skin.
1 70 CHARTING THE EARTH AND THE HEAVENS
Since it is not possible to flatten a sphere without distorting it, any attempt
to reproduce on flat paper the relationships that exist on the sphere must
involve a distortion of areas, or the relative directions of one place from
another, or distances. Hipparchus and Ptolemy therefore invented several
methods of map making each of which has features useful for one or more
purposes. Thus some methods preserve area, others direction, and still others
project great circles into straight lines so that the shortest distance on the sphere
between two points is represented by the shortest distance on the map. No
map can be a true representation in all respects.
North Star
N
Fig. 7-18
N s N
Equator
Eqmltol'
Fig. 7-19 s Fig. 7-20 s
EXERCISES
1. To determine the latitude of a point P (Fig. 7-18) on the surface of the earth,
an observer at P measures the angle between the horizontal at P and the direction
of the North Star. He finds this angle to be 10'. What is the latitude of P?
2. As one travels north along a meridian from the South Pole to the North Pole,
how does his latitude change?
3. Suppose that one travels west from some point on the 0° meridian. How does
his longitude change?
CHARTING THE HEAVENS 171
4. Of the two circles of latitude, 300 nonh and 400 north, which has the larger
radius?
5. If a man changes his latitude by 2 D in traveling along a meridian (Fig. 7-19),
how far does he travel?
6. Suppose a man travels due west along the 41 ° circle of latitude and changes his
longitude by 5° (Fig. 7-20) . How far does he travel?
7. In one day (24 hr) the earth rotates through 360°. Hence a person has in effect
moved around in a complete circle. How far has a person traveled who is at
410 latitude?
* 7-5 CHARTING THE HEAVENS
From the determination of the latitude and longitude of places on the earth's
surface and of distances between places, Hipparchus and Ptolemy proceeded to
the far more ambitious problem of calculating the sizes and distances of the
heavenly bodies. The classical Greeks had indeed speculated about these sizes
and distances, but since they relied far more upon aesthetically pleasing prin
ciples than upon keen observation, measurement of angles, and numerical
calculation. their conclusions were often absurd.
The Alexandrian Greeks made the decisive step in quantitative astronomy.
They were, as we have noted, more disposed to measure. Moreover many of
them, including Hipparchus himself, had improved the astronomical instruments
and the sundials and water clocks which helped to fix more accurately the time
at which observations were made. Hipparchus and Ptolemy also had at their
disposal in Alexandria a wealth of astronomical data which the Egyptians,
Babylonians, and Alexandrians had compiled over many centuries. Let us see
how these men "triangulated" the heavens. We shall not reproduce their exact
procedures but merely show the essential principles.
,
,,
,
.. - ---- +.:
------ ==-M
E P
Fig. 7-21.
Equator Finding the distance to the moon.
We shall consider first how one can find the distance to the moon. Suppose
that P and Q (Fig. 7-2 1) are two points on the earth's equator which are
chosen to satisfy the following conditions: The moon is to be directly overhead
at P; that is, the moon, M, regarded as a point, is to be on the line from the
center of the earth, E, through P. The moon is in this position at certain times
each month. The point Q is chosen such that the moon is just visible from it.
This means that the moon is clearly visible from points closer to P but not
172 CHARTING THE EARTH AND THE HEAVENS
visible from points farther away from P and along the equator. Another way
of saying the same thing is that the line MQ is tangent to the equator at Q .
Let us draw the line EQ. The angle EQM is a right angle because the radius
of a circle drawn to the point of contact of a tangent is perpendicular to
the tangent.
We now have a right triangle. Moreover, EQ is the radius of the earth
and this is known. The angle at E is the difference in longitude between the
points P and Q, and since the longitudes of places on the earth are known, so
is angle E. A modern value for it is 89° 4', a value far more accurate than
Hipparchus or Ptolemy could have obtained with their instruments. The
calculation of EM is now child's play, for
EQ
cos E = - ·
EM
The value of cos E or cos 89' 4', taken from a larger trigonometric table than
ours, is 0.0163. Moreover EQ is 4000 miles. Then
4000 .
0.0163 =
EM
If we multiply both sides of this equation by EM and then divide by 0.0163,
we obtain
4000
EM = -- = 245 000 .
0.01 63 '
Our data yield EM 245,000 miles, and if we now subtract EP, the radius
=
of the earth, we find that PM, the distance from the surface of the earth to
the moon, is 241 ,000 miles. Hipparchus arrived at the figure of about 280,000
miles because his angular measure of E was nOt so accurate.·
Precisely the same method can be used to find the distance to the sun.
The point M (Fig. 7-2 1 ) would now represent the sun. However, because
the distances PM and QM are much larger in the case of the sun, angle E is
larger and very close to 90°. Moreover, the angle must be measured very
accurately because a small error in the angle will cause a large error in the
value of PM. For this reason the result of Hipparchus and Ptolemy, of the
order of millions of miles, was, as they realized, not very accurate (see
Exercise 1 ) .
Let us now find the radius of the moon. Whereas in the preceding calcu
lation we regarded the moon as a point, this idealization will obviously not
• The distance of the earth from the moon varies over the year. The above value is
about an average value.
CHARTING THE HEAVENS 173
Fig. 7-22.
Determining the radius
of the moon.
do in finding the radius. Instead let us regard the moon as a small sphere with
center M and radius MR (Fig. 7-22). At a point E on the earth's surface, one
measures the angle between the line EM which has the direction from E to
the center of the moon, and the line ER which is tangent to the moon's
surface. This angle proves to be 1 5'. We know the distance from earth to
moon, at least when the moon is idealized as a point. Let us use this distance,
even though it is not exactly EM in our figure. We shall see that the error
introduced is minor. Hence EM for us is 241,000 miles. We shall use again
the Euclidean theorem that a radius of a circle drawn to the point of tangency
of a tangent is perpendicular to the tangent. For our figure this theorem says
that MR is perpendicular to ER. Then in the right triangle EMR we have
. MR
sm E = -_ .
EM
Now angle E 1 5' and sin 15', taken from a tahle giving sine values for angles
=
in minutes, is 0.0044. Moreover EM is 241,000. Hence
MR
0.0044 =
241,000
Then
MR = 241,000 · 0.0044 = 1060.
Thus the radius of the moon is 1060 miles. We can see now that the error
introduced by using 241,000 miles as the distance to the center of the moon
cannot be great because the radius of the moon is only 1060 miles. The
distance of 241,000 miles is really the distance EM' since, in determining the
distance to the moon, we could observe only the surface. (For a more accurate
calculation of the moon's radius see Exercise 3.) It is of interest that Hip
parchus obtained the result of 1 , 3 3 3 miles; his measurement of angle E was
not as accurate as the modern one.
The method just used to find the radius of the moon can also be applied
to find the radius of the sun. The point M in Fig. 7-22 becomes the center
of the sun, and the distance EM becomes the distance to the sun (see Exer
cise 2). Angle E is about the same for this case as for the moon, as one might
174 CHARTING THE EARTH AND THE HEAVENS
Ca) Cb)
Fig. 7-23.
Determining the d i stance of Venus from the sun.
expect from the fact that when the moon is between the earth and the sun,
the moon just about eclipses the sun.
We can find the distances to the moon and sun and the radii of the moon
and sun by making measurements on the surface of the earth. But now sup
pose that we wish to calculate the distance from Venus to the sun. If we
were to use the preceding methods we should have to make measurements on
the surface of V cnus. Of course, we all expect to be able to make the rrip
to Venus shortly, and can then make the measurements. In the meantime, to
satisfy our curiosity, we shall employ a somewhat less direct method.
Let us regard all three bodies, the earth, the sun, and Venus, as points and
let us suppose that the paths of earth and Venus are circular. At any time
the three bodies are the vertices of triangle ESV' in Fig. 7-23 (a). From the
earth we can observe the size of angle E, which, of course, changes as the
earth and Venus move around the sun.
A neat fact, which emerges from a study of Fig. 7-2 3 (b), is that when
angle E is a maximum, then the line from earth to Venus is tangent to the
path of Venus around the sun. For when the angle at E is a maximum, the
line from the earth to Venus is farthest from ES and still meets the circle on
which Venus travels. But such a line must be tangent to the circle. A tangent
to a circle is perpendicular to the radius drawn to the point of contact. Hence
the radius SV (Fig. 7-23b) is perpendicular to EV. What we should do, then,
is measure the angle E at various times of the year and find out when it is
largest. At this time EV is perpendicular to SV.
Measurements show that the largest value of angle E is 47". If in Fig.
7-2 3(b), we use 47" for E and the fact that angle V must then be a right
angle, we have
. " SV
s," 47 = - '
ES
CHARTING THE HEAVENS 175
From our tables we find that sin 47° = 0.73 14. The distance ES is 93,000,000
miles. Hence
SV .
0.7314 =
9 3 ,000,000
Then
SV = 93,000,000 · 0. 7 3 1 4 = 68,000,000.
Thus the distance from Venus to the sun is 68,000,000 miles.
We can begin to see from these examples how Hipparchus and Ptolemy
gave mankind its first reasonable values for the dimensions of our solar system.
The figures they produced were staggering to the Greeks because these people
believed that our solar system and universe were far smaller.
The crowning achievement of Hipparchus and Ptolemy was the creation
of a new astronomical theory which described the paths of the heavenly bodies
and enabled man to predict their positions. We shall consider their theory in the
next chapter.
EXERCISES
I. Let us use the method given in the text to find the distance to the sun. We
know that in Fig. 7-24, QE is the radius of the earth, or 4000 mi. The angle at E
is the difference in longitude between P and Q and in our case is 89° 59' 51".
Given that cos E 0.00004 J, find ES.
=
E
Fig. 7-24 Fig. 7-25
2. Let us apply the method of the text to find the radius of the sun (Fig. 7-2 5 ) .
The distance to the sun, ES, is 93,000,000 mi. Angle E is measured and found to
be about 16'. Given that sin 16' 0.0046, find the radius SR.
=
3. In the text (Fig. 7-22) we found the radius of the moon without considering the
distance M'M. By a slight bit of extra work we can take this radius into account.
Let us denote M'M, which equals RM, by r. Then, since EM' is 241,000 miles and
angle E equals 1 5', we have
T
sin 1 5' :-::- -,--
= -:-:- :-:- -
241,000 + T
Use the value of sin IS' given in the text to find T.
176 CHARTING THE EARTH AND THE HEAVENS
4. Use the method in the text to find the distance from Mercury to the sun. The
relevant angle E in this case is 2 3 0 •
* 7-6 FURTHER PROGRESS IN THE STUDY DF LIGHT
We saw in the preceding chapter that Euclid had already formulated one basic
law for the behavior of light, namely the law of reflection. The Alexandrians
undertook to study a second basic phenomenon of light, namely the change in
the direction of light as it passes from one medium to another.
i
Air
Witter
Fig. 7-26.
Refraction of light.
We often recognize that something strange does happen when light goes
from air to water, say, because a straight rod when partially immersed in water
seems to bend sharply at the water level. Also if one should shine a flashlight
beam into water, he would observe the sudden change in the direction of the
beam as it enters the water. This bending of light is called refraction. The
Alexandrians sought to determine the extent of this change in direction. Spe
cifically, if i is the angle (Fig. 7-26) which the direction of the incident light
ray makes with the perpendicular to the surface which separates the two media,
say air and water, and if T is the angle which the refracted light makes with
this same perpendicular, then what the Alexandrians sought is the relationship
between angle i and angle T. But the Alexandrians and Ptolemy, in particular,
who worked very hard on this problem, were baffied. They did observe that as
i increased, r increased, but the increase did not occur in any simple manner.
Moreover, the r which corresponds to a given i is not the same for any two
different media. Thus, if the first medium should be air, then for the same i,
the value of T for glass would be different from that for water.
Ptolemy did not succeed in arriving at the correct law, but he developed
the mathematical tool which finally enabled the Dutchman Willebrord Snell
and the Frenchman Rene Descartes to discover and express it. It was found
in the seventeenth century that light travels with a finite velocity and that this
velocity is different in different media. Let us suppose that we have two media
bordering each other as shown in Fig. 7-26, and let VI be the velocity of light
in the upper medium and V, the"velocity in the lower medium. Then Snell and
FURTHER PROGRESS IN THE STUDY OF LIGHT 177
Descartes demonstrated by arguments we shall not reproduce that
sm 1 Vl
-- = _ .
(I)
Thus it is the sine ratio which proves to be the key to this phenomenon of
light. We see here, as we shall see many times later, how-as more mathe
matical ideas are placed at our disposal-we can take hold of more natural
phenomena.
To familiarize ourselves with the law of refraction, we shall consider a
concrete example. Let us suppose that the two media in question are air and
water. The ratio of VI to V2 in this case is 4 to 3. If we are then given a
value of i, say 30", we can find r. Since sin 30" = t, it follows from ( I ) that
!
-- = t·
sm T
Therefore
t = ! sin T.
If we multiply both sides of this equation by i, we obtain
sin r = i = 0.3750.
We have now but to find the angle whose sine value is 0.3750. From our
table we see that r = 22° to the nearest degree. Hence, as the light enters
the water, the angle between its direction and the perpendicular will change
from 30" to 22".
We now know how light refracts. Can we put this knowledge to use? Let
us assume that the sun is close to the horizon. Of course, light from the sun
streams out in all directions, but some rays will travel horizontally. Suppose
that the surface over which the light travels is the surface of a large body of
water (Fig. 7-27). Then some rays will enter the water at a very large angle
of incidence i, in fact, so close to 90" that we shall consider angle i to be 90".
The question we shall discuss is, How large is angle r in this case? To obtain
an answer, we follow the procedure described in the preceding paragraph.
This time, angle i is 90", and sin 90" is I . If we substitute this value in For
mula ( I ) , we have
Sin T
or
sin r = !.
Reference to the table shows that r= 49". Here the change 10 direction is
178 CHARTING THE EARTH AND THE HEAVENS
P'
Q
Sun Air Air
Water Wilier
,
/'
Fig. 7-27. Fig. 7-28
The fish-eye view of the world.
considerable. Before we draw any further conclusions, let us note that 90°
is the largest angle of incidence possible. If angle i should be less than 90',
angle r will be less than 49'. Hence any light entering the water will take a
direction which makes an angle between 0' and 49' with the perpendicular.
Now suppose that a person located at the point P (Fig. 7-27 ) in the water
sees the light ray OP coming toward him. Since the light travels along the
direction OP, he will conclude that the sun is located above the water in the
direction PO. Moreover, if light from any source enters the water at an angle
of incidence less than 90', the angle of refraction will be even less than 49'.
In such cases a person in the water observing this light will conclude that the
light comes from a source whose angle of incidence is less than 49'. The point
of this discussion is that a person at P will believe that all objects in the air
are situated within a 49'-angle from the perpendicular because the light from
them will seem to come from a direction within this range. The region in the
air extending in all directions within 49' of the perpendicular at 0 is the
interior of a cone. Hence to a person in the water all objects seem to lie
within this cone. We have of course presumed that the person in the water
does not know the refractive effect of light or that he at least does not know
how much light is bent under refraction. It is fairly certain that fish do not
know mathematics. and so the inference that all ohjects above water must be
within the 49'-cone around the perpendicular is called the fish-eye view of
the world.
To be a little clearer about the possible error into which one may be led,
suppose that light comes to a person in a submarine at P (Fig. 7-28) and that
the direction of the light is OP. The object which emits the light will then
appear to lie along the line PO. If, to hit the object, he shoots a bullet in the
direction PO, the bullet will enter the air and follow the direction POP'. But
the object at which he believes to be shooting is located along the direction
OQ.
FURTHER PROGRESS IN THE STUDY OF LIGHT 179
Let us now reverse the roles of air and water and let us suppose that the
light originates in the water. Assume, in fact, that a beam of light is shot in
the direction of QO of Fig. 7-29. The angle of incidence is now the angle
marked i in the figure. The angle of refraction is the angle T. Since the ratio
of the twO velocities of light, that is VI/V" is now to the law of refraction
( 1 ) becomes
s�n , f
=
Sin T
or
sin T = ! sin i. (2)
We see that sin T is greater than sin i and that therefore T must be greater
than i. This is as it should be because light in going from water to air will
bend away from the perpendicular.
Let us now suppose that angle i is greater than 49", say 60", and let us
seek to determine the corresponding angle of refraction, T. Then, knowing that
sin 60" = 0.8660, we find by (2) that
sin r = !(0.8660) = 1.1 55.
We see then that sin r is greater than 1. Unfortunately, there is no angle
whose sine value is greater than I. and so there is no angle of refraction. Thus
mathematics predicts that the light cannot leave the water. Is this the case
physically' Well, no decently behaving light ray would wish to disobey the
mathematical law of refraction. And none does. The light remains in the
water. But what does it do' The answer is that the light is reflected from
the boundary between air and water. Since the light must return to the water,
it may as well do what it has already learned how to do under the process of
reflection, namely, be reflected at an angle equal to the angle of incidence
which in the present case is 60" (Fig. 7-30).
r
Air Air
o
Wat('r Water
Q
Fig. 7-29. FIg. 7-30.
Refraction from water into air. Tota l reflection.
1 80 CHARTING THE EARTH AND THE HEAVENS
Thus we have discovered that if light seeks to pass from one medium to
a second one in which the velocity is greater, then for all angles larger than
a certain angle (49" for water and air), the light is not refracted but reflected.
This particular angle, the largest for which refraction is still possible, is called
the critical angle, and the phenomenon that for all greater angles of incidence
the light is reflected is called total reflection.
This phenomenon is indeed a surprising one. It means that a surface, such
as the surface of the water in the above example, serves as a mirror for some
angles of incidence. Now mirrors are very useful devices. Generally they
are made by silvering the back of a glass plate. Would there be any use for
the phenomenon of total reflection in view of the fact that here too we en
counter a reflecting surface' As a matter of fact, the phenomenon is put to
use in a number of familiar instruments.
0-----1
C ""----.,.,,-"-A
Fig. 7-31.
Parallel displacement of light by
means of tota l reflection in two
prisms.
Let us consider the following situation (Fig. 7-3 1 ) , where ACB and A'C'B'
are two prism-shaped pieces of glass with the faces AC and A'C' parallel to
each other. Both prisms are shaped as isosceles right triangles. Suppose that
OP is a light ray which first strikes the face BC perpendicularly. Here the
angle of incidence is 0". Hence the angle of refraction is also 0", and the light
therefore goes through unchanged in direction. The light ray strikes the
face BA at an angle of 45". Now if the prisms are made of flint glass, the
critical angle is 37". Thus the light ray strikes the face BA at an angle of
incidence greater than the critical angle. In accordance with the phenomenon
of total reflection, the light is reflected at an angle of 45" to AB and follows
the direction PO. The light strikes the faces AC and A'C' at an angle of
incidence of 0" and so goes right through unchanged. It then strikes the face
A'B' at an angle of 45". Since this angle of incidence also is greater than the
critical angle, the light is again totally reflected at an angle of 45" with B'A'
FURTHER PROGRESS IN THE STUDY OF LIGHT 181
and takes the direction RO'. Thus the final ray, RO', has the same direction
as the original ray, OP, but is displaced by the distance PRo
We might well ask, Does this combination of prisms have any practical
value? One application is the periscope. The two prisms are at opposite ends
of a long vertical tube. Now OP is the light received above water and RO'
is the light received below. One could very well use two silvered mirrors at
BA and A'B' and obtain the same result. But silvered mirrors tarnish with
age and lose their effectiveness. Moreover, well-made glass prisms reflect
almost all the light that falls on a face such as BA, whereas a silvered mirror
reflects only about 70% of the incident light; the rest is absorbed or scattered
in all directions. Hence the prism not only outlasts the silvered mirror but
is much more efficient.
Another application of the above combination of two prisms is made in
binoculars. The two tubes which first receive the light are deliberately placed
rather far apart so that the field of vision is large. But the eye pieces of the
binoculars cannot be farther apart than the distance between a person's eyes.
In each half of a binocular, the incident light is displaced as OP is displaced
to RO'. Then the two incoming rays, one in each of the main tubes, can be
far apart, whereas the two emerging rays are no farther apart than the eyes
of a person.
0, II,
0, II,
P S
Fig. 7-32.
0, II, Refraction by a
lens.
Total reflection is but one phenomenon of the refractive effect of light.
The most common use of the refractive effect of light is in lenses. If light
streams out from an object at P (Fig. 7-32) in all directions, some of the
rays will strike the lens at points such as 0" 0" and 0,. There their direc
tions will change because they are entering glass. Thus the rays PO " PO"
and PO, may take the directions O,R" O,R" and O,R" respectively. At the
right-hand surface of the glass, the rays re-enter the air and, since the medium
in which the light is traveling changes, the light rays bend again. By properly
shaping the lens surfaces, that is 0,0,0, on the left and R,R,R, on the right,
the light from P may be made to concentrate at S. All optical instruments,
such as telescopes. microscopes, binoculars, and cameras, contain lenses of this
kind.
The eye itself is a complicated refracting device. When light enters the
eye (Fig. 7-33 ) , it passes through a liquid (denoted by A in the figure),
182 CHARTING THE EARTH AND THE HEAVENS
called the aqueous humor, then through the lens, L, which is made of a fibrous
jelly, and finally it enters another liquid, V, called the vitreous humor. Al
though all three media have some refractive effect upon the light, most of the
refraction occurs when it encounters the aqueous humor. To be perceived,
light rays that enter the eye must strike the retina, R, in the rear. The eye
has a ciliary muscle which changes the shape of the lens and therefore the
direction of the light rays passing through the eye so that the rays are directed
toward the retina. Eyes which for one reason or another cannot direct the
rays to the retina must be aided by additional lenses in eyeglasses. Clearly the
science of medicine profits immensely from the mathematical and physical
knowledge acquired about the action of the eye.
In the camera, the lens or lenses are fixed in shape. The film acts as does
the retina in the eye. Since the shapes of the lenses are fixed, the distances
of the lenses from the film can be varied to enable the refracted light to reach
the proper places on the film.
Fig. 7-33.
A sketch of the eye.
We have been discussing the law of refraction and some of the remarkable
effects which take place at a sharp boundary between the two media. But the
refractive effect of light is equally striking and important when there is a
gradual change in the nature of the medium through which the light passes.
Let us consider the passage of light through air, which is not a uniform me
dium. Generally it is more dense near the ground and thinner at higher alti
rudes. Hence, when light comes to a person at P (Fig. I-I) from the sun at
0, the light ray follows a curved path as it travels through the earth's atmos
phere because it is continually refracted. For the observer at P the direction
of the incoming light is O'P, and hence he thinks that the source lies along the
direction PO'. This is the reason that we are often deceived about the true
position of the sun (see Chapter I ) .
The refractive effect of light is, as we can see, a peculiar phenomenon.
Why does light behave this way? We do not understand what light is and
so cannot analyze the substance itself to learn why it refracts, but we have
another kind of explanation which sheds light on nature's operations. The
clue lies in the law of refraction. We note that refraction depends upon the
velocity of light in the medium. The seventeenth-century mathematician
Pierre de Fermat, whom we shall meet again, pondered on this fact and, after
analyzing the law of refraction, found an important principle. Suppose that
FURTHER PROGRESS IN THE STUDY OF LIGHT 183
p
Air
Waler
Fig. 7-34.
IJ light takes the path requiring
least time.
light travels from the point P in air (Fig. 7-34) to the point Q in water and
bends at 0 in accordance with the law of refraction. Were the light to follow
the straight-line path from P to Q instead of the broken-line path POQ, it
would travel a shorter distance. Let us note, however, that the distance O'Q
in water would be longer than OQ. Because the velocity in water is smaller
than in air, the light might lose more time in traveling the path O'Q instead
of OQ than it might save by traveling the shorter distance PO' instead of PO
in air. By a mathematical argument Fermat showed that light takes the path
which requires least time.
But is this fact true for other phenomena of light? When light travels
from one point to another in a uniform medium, it takes the straight-line path.
It would seem as though in this case light chooses the criterion of shortest
path and not that of least time. But in a uniform medium the velocity of light
is constant, and so the shortest path requires the least time. Let us consider
next what happens when light goes from a point P to a mirror and then to a
point Q. We proved in Chapter 6 that light takes the shortest path. But here
too the light travels in one medium and. because the medium is uniform, the
velocity is constantj hence the shortest path again means least time. It would
appear from Fermat's analysis that nature l'i wise. It knows mathematics and
employs it in the interest of economy.
We have gotten a little ahead of our story by presenting the mathematical
law of refraction and Fermat's analysis of the deeper implications of this law.
The Alexandrian Greeks had grappled with the phenomenon of refraction and,
as we noted earlier. supplied the key in the concept of the trigonometric ratios,
but did not attain the law itself or see its meaning in terms of least time. But
by providing these ratios and by charting the earth and heavens, the Alex
andrians extended enormously man's mathematical understanding of the phys
ical world. The power of mathematics to describe and analyze nature's ways
was advanced well beyond the stage at which Euclid and Apollonius had left
it. The crowning achievement of the Alexandrians is yet to be related.
184 CHARTING THE EARTH AND THE HEAVENS
EXERCISES
1. Given that the ratio of the velocity of light in air to that in water is 4 to 3 and
that the angle of incidence of a light ray originating in the air and striking the
surface of the water is 450, what is the angle of refraction?
2, Suppose a light ray traveling in glass strikes the boundary of the glass and seeks
to enter the air beyond the boundary. The velocity of light in the glass is rwo
thirds of its velocity in air. What angles of incidence can the light ray have and
still penetrate into the air?
Air
;jI ,I
)Glass
, /('
Air
Fig. 7-35
3. Prove that a light ray passing through a plate of glass (Fig. 7-35 ) emerges
parallel to its original direction but is somewhat displaced.
4. Suppose that one measures the angle of incidence, i, and the angle of refraction,
T, for a light ray passing from air into a plate of glass, and assume that angle i
proves to be 500 and angle r, 450• The velocity of light in air is 186,000 mi/sec.
What is the velocity of light in the glass?
5. What is the mathematical theme of this chapter?
'
6, Is it correct to say that the trigonometry of the Alexandrian Greeks is an exten
sion of Euclidean geometry?
7. Contrast the classical and the Alexandrian Greek activities in mathematics.
REVIEW EXERCISES
I. Use the Table of Trigonometric Ratios to find the angle
a) whose sine is 0.3256,
b) whose tangent is 0.5 3 1 7.
c) whose cosine is 0.3256,
d) whose tangent is 1.8807.
2, It is possible to find the sine, cosine, and tangent of 450 in somewhat the same
manner as we found the corresponding values of 30° and 600, Take a right
triangle whose arms are each 1 . Calculate the length of the hypotenuse by
means of the Pythagorean theorem. Now write the values of sin 450, cos 45°,
and tan 45°.
FURTHER PROGRESS IN THE STUDY OF LIGHT 1 85
3. Find the sine, cosine, and tangent of the acute angle A of a right triangle
a) when the opposite side is 5 and the hypotenuse is 13,
b) when the opposite side is 12 and the adjacent side is 5,
c ) when the opposite side is v'3 and the adjacent side is 2,
d) when the opposite side is V3 and the adjacent side is v'6,
e ) when the opposite side is 1 and the hypotenuse is v'iO.
4. If sin A = t, find cos A and tan A.
5. If cos A t , find sin A and tan A.
=
6. If tan A = t, find sin A and cos A.
7. To find the width AB of a river, a line segment AC perpendicular to AB is
measured along one bank and found to be 100 ft. By sighting along CA and
CB, the angle ACB is found to be 40°. How wide is the river?
8. The shadow on the horizontal ground of a vertical pole is 1 5 ft. At the end of
the shadow the angle between the horizontal and the line of sight to the top of
the pole is 20·. Find the height of the pole.
9. A wire 60 ft long reaches from the top of a 4O-ft pole to the ground. What
angle does the wire make with the pole?
10. From the top of a lighthouse 60 ft high, the angle between the vertical and the
line of sight to a ship at sea is 35°. How far is the ship from the foot of the
lighthouse?
I I . An observer in an airplane 2000 ft directly above a gun observes that the angle
between his vertical and the line of sight to an enemy target is 500. How far
is the target from the gun?
12. Find the radius and the circumference of the circle of latitude 230 north.
13. Suppose a man changes his longitude by 5 ° while traveling along the circle of
latitude 23° north. How far does he travel?
14. Find the radius of the circle of latitude 670 north.
1 5. Suppose a light ray traveling in air strikes the water at an angle of incidence
of 45°. What is the angle of refraction?
16. A ray of light starts from a point P in water and strikes the surface at an angle
of incidence of 300 and emerges into air. What is the angle of refraction of the
light ray?
Topics for Further Investigation
I . The mathematics of lenses. Use the references to Taylor, or to Sears and
Zemansky. or look up any elementary physics book.
2. The mathematics of map-making. Use the references to Brown. Raisz, Deetz, or
Chamberlin.
3. The history of mathematics during the Alexandrian period. Use the references
to Smith. Ball, Eves. or Scott.
4. The creation of trigonometry. Use the references to Aaboe.
5. The life and work of Archimedes. Use any history.
186 CHARTING THE EARTH AND THE HEAVENS
Recommended Reading
AABOE, ASGER: Episodes from the Early History of Mathematics, Chap. 4, Random
House, New York, 1964.
BALL, W. W. ROUSE: A Short A ccount of the History of Mathematics, 4th ed.,
Chaps. 4 and 5, Dover Publications, Inc., New York, 1960.
BROWN, LLOYD A.: The Story of Maps, Little, Brown and Co., Boston, 1944.
CHAMBERLIN, WELLMAN: The Round Earth on Flat Paper, National Geographic
Society. Washington, D.C., 1947.
DEETZ, CHARLES H. and OSCAR S. ADAMS: Elements of Map Projection, pp. 1-52.
U.S. Department of Commerce, Special Publication No. 68, 1938.
GREENHOOD, DAVID: Mapping, The University of Chicago Press, Chicago, 1964.
HEATH, SIR THOMAS L.: A Manual of Greek Mathematics, Chap. 14, Dover Publi
cations Inc., New York. 1963.
PARSONS, EDWARD A.: The Alexandrian Library, The Elsevier Press, Amsterdam,
1952.
RAISZ, E.: General Cartography, McGraw-Hill Book Co., New York, 1948.
SAWYER, W. W.: Mathematician's Delight, Chap. 13, Penguin Books, Harmonds
worth, England, 1943.
ScOTT, J. F.: A History of Mathematics, Chap. 3, Taylor and Francis, Ltd.,
London, 1958.
SEARS, FRANCIS W. and MARK ZEMANSKY: University Physics, 3rd ed., Chaps. 39-43,
Addison-Wesley Publishing Co., Inc., Reading, Mass., 1964.
SMITH, DAVID E.: History of Mathematics, Vol. I, Chap. 4, Dover Publications, Inc.,
New York, 1958.
TAYLOR, LI,OYD W.: Physics, The Pioneer Science, pp. 442-470, Dover Publications,
Inc., New York, 1959.
CHAPTER B
THE MATHEMATICAL ORDER OF NATURE
Great men! elevated above the common standard of human nature, by
discovering the laws which celestial occurrences obey, and by freeing the
wretched mind of man from the fears which the eclipses inspired.
PLINY
8-1 THE GREEK CONCEPT OF NATURE
The Greeks, as we now know, molded the nature of mathematics, constructed
Euclidean geometry and trigonometry, and applied their theoretical results to
objects in space, to the behavior of light, to mapping the earth, and to deter
mining the sizes and distances of heavenly bodies. But these extensive and
magnificent achievements within mathematics proper and in its applications do
not exhibit the full greatness of the Greek genius, and are indeed dwarfed by
the Greeks' grand conception of the universe itself.
Possessed with insatiable curiosity and courage, they asked and answered
the questions which occur to many, are tackled by few, and are resolved only
by individuals of the highest intellectual caliber. Is there any plan under
lying the workings of the entire universe? Are planets, men, animals, plants,
light, and sound merely physical accidents or are they part of a grand plan?
Because they were dreamers enough to arrive at new points of view, the
Greeks fashioned a conception of the universe which has dominated all sub
sequent Western thought. They affirmed that nature is rationally and indeed
mathematically designed. All phenomena apparent to the senses, from the
motions of planets in the heavens to the stirrings of leaves on a tree, can be
fitted into a precise, coherent, intelligible pattern. The Greeks were the first
people with the audacity to conceive of such law and order in the welter of
phenomena and the first with the genius to uncover a pattern to which nature
conforms. They dared to ask for and they found a design underlying the
greatest spectacle man beholds, the motion of the brilliant sun, the changing
shapes of the many-hued moon, the piercing shafts of the planets, the broad
panorama of lights from the canopy of stars, and the seemingly miraculous
eclipses of the sun and moon.
187
188 THE MATHEMATICAL ORDER OF NATURE
8-2 PRE·GREEK AND GREEK VIEWS OF NATURE
To appreciate the originality and boldness of the steps which the Greeks took
in this direction, one must compare their attitude with what preceded. To all
pre· Greek civilizations and later ones which lay beyond the Greek pale, nature
appeared arbitrary, capricious, mysterious, and even terrifying. The ancient
Egyptians and Babylonians did note the periodic motions of the sun and moon.
But the motions of the planets made no sense at all. These bodies moved with
varying speeds at different times of the year; at times they stood still; and
often they reversed their courses. They appeared and disappeared. The few
regularities which were observed in these motions were beclouded by the
many irregularities.
If these two ancient peoples had any expectation at all that the universe
would continue to function in the future as it had in the past, it was because
they believed that sun, moon, and planets were gods who would most likely
behave in a gentlemanly and beneficent manner. In the complex actions of
nature, they saw no glimpse of plan, order, or law. They scarcely dreamed
of design and certainly conceived no embracing theories.
Even the Greeks of abour 1000 B.C. accepted fanciful accounts of the
universe, accounts which are found in Homer and Hesiod. There were many
gods, each of whom played some role in the creation and maintenance of the
universe. Indeed the names Jupiter, Saturn, Venus, Mercury, and Mars are
merely the Roman names for the Greek gods, and the Greek names, such as
Aphrodite for Venus and Hermes for Mercury, were replacements for Baby
lonian names. These gods not only determined but even intervened in the
affairs of man.
Rather suddenly, or so at least our knowledge of history indicates, rational
accounts of the structure of the universe and of the motions of heavenly
bodies appeared in the Greek city of Miletus located in Ionia, a region of
Asia Minor. There is the theory that the Miletans, far from home and there
fore free of the tyranny of beliefs which a society imposes on its members
and yet repelled by the strange doctrines they encountered among the peoples
of the Near East, were propelled into thinking for themselves. Certainly from
600 B.C. onward rational views dominate the picture. These Greeks and their
successors were the first to reveal the passionate desire for knowledge, the
love of reason, and the conviction that nature not only is rational but that an
examination of nature's ways would reveal the order inherent in the physical
world. The new thesis is proclaimed by the Ionian Anaxagoras: "Reason rules
the world." The early rational theories are crude from a modern standpoint,
but the new outlook is evident.
The decisive step leading to the construction of precise and verifiable
scientific theories in place of vague and largely speculative accounts was the
involvement of mathematics. This step was made by the Pythagoreans. We
have already noted the prepossession of these people with the concept of
PRE-GREEK AND GREEK VIEWS OF NATURE 189
number, though admixed with mystical and religious doctrines. In their philos
ophy of nature the Pythagoreans began with the principle that number is the
essence of all substance. Unlimited space furnishes the material for particular
forms of matter. But to the Pythagoreans any form was a pattern of discrete
points arranged, as small pebbles might be, to build up the form. Hence the
forms reduced to numbers. Since number is the essence of any object, the
explanation of natural phenomena could be achieved only through number.
The natural philosophy of the Pythagoreans is hardly very substantial.
Aesthetic principles commingled with an obsession to find number relation
ships certainly led to assertions transcending observational evidence. Nor did
the Pythagoreans develop any one branch of physical science very far. One
can justifiably call their theories superficial. But whether by a lucky stroke
or by intuitive genius the Pythagoreans did hit upon two doctrines which later
proved to be all important. The first is that nature is built in accordance with
mathematical principles, and the second that number relationships reveal the
order in nature. They underlie and unify the seeming diversity exhibited by
nature. The Pythagoreans said in fact that numbers and number relationships
are the essence of nature. This statement will assume deeper meaning when
we get to modern times.
Perhaps because mathematics developed considerably in the intervening
century, the principle that nature is mathematically designed emerged more
sharply and was applied more substantially in Plato's time. Plato was indeed
a Pythagorean but a master in his own right who influenced Greek thought
in a most important century, the fourth century B.C. He was the founder of
an academy in Athens, a university which attracted the leading thinkers of his
day and which, in fact, endured for nine hundred years.
Plato's own doctrines were extreme. Reality to him was not to be found
in the physical world but in a system of ideas and in an ideal plan of the
universe which God himself had created and contemplates. The visible and
sensible world is just a vague, dim, and imperfect realization of these ideas.
Moreover, the ideas were perfect and eternal, whereas the physical world is
imperfect and decays. One might say that, unlike the Pythagoreans, Plato did
not wish to comprehend the physical world through mathematics but aimed
at understanding the mathematical plan itself which observation of the physical
world suggested very imperfectly.
For example, Plato describes the real science of astronomy. The visible
figures in the heavens are far inferior to the true objects, namely those objects
that are to be apprehended by reason and mental conceptions. The varied
configurations which the sky presents to the eye are to be used only as
diagrams to assist in the study of higher truths. We must treat astronomy,
like geometry, as a series of problems suggested by visible things. True
astronomy deals with the laws of motion of true stars in a mathematical
heaven of which the visible heaven is but an imperfect expression. True
190 THE MATHEi\lATlCAL ORDER OF NATURE
astronomy must leave the actual heavens alone. It is clear, incidentally, that
Plato, like the classical Greeks in general, was indifferent to the practical
problems of navigation, calendar reckoning, and the measurement of time.
Although the planets, at least as seen from the earth, do not appear to
follow any regular course (the word "planet" means in fact "wanderer," and
the planets were referred to as the vagabonds of the sky), Plato was sure
because "God eternally geometrizes"-that there was a mathematical pattern
underlying and governing the motions of all heavenly bodies. Plato's own
attempts to find such a plan were crude, largely because he would not devote
himself to a careful study of the actual motions. Bur he did pose to his col
leagues and students the problem of devising a mathematical scheme that
would call for regular motions and yet account for the irregular motions we
see, the problem he described as "saving the appearances."
8-3 GREEK ASTRONOMICAL THEORIES
One of Plato's pupils, Eudoxus (408-355 B.C. ) , who later became one of the
most famous of Greek mathematicians, did take on this problem and, by cre
ating the first major astronomical theory known to history, made one of the
great and ingenious contributions to the demonstration of the mathematical
design of nature. We shall not present the details of his theory. It was con
structed before Hipparchus and Ptolemy calculated the sizes and distances of
the heavenly bodies, and so Eudoxus did not have the data on which to build
an accurate system. The defects in the theory were soon recognized.
The problem of finding the design of planetary motions continued to
engage the minds of the Greeks, possibly because they were not distracted
by the "heavenly" stars of stage, screen, and radio with whom many modern
minds seem to be preoccupied. One of the solutions advanced but rejected is
worthy of mention. Aristarchus, who lived about 270 B.C. and who had made
many estimates of the sizes and distances of heavenly bodies, though with
methods cruder than those developed later by Hipparchus and Ptolemy, pro
posed the theory that the planets move in circles about the sun. Aristarchus.
to our knowledge, did not attempt to show that such a theory would fit the
data known to his time. But the theory was not acceptable to his contem
poraries and successors because it was totally at variance with Greek concep
tions of the universe and Greek physics. For one thing, the Greeks already
knew that simple circular motion would not do because the distance of the
earth from the sun was known not to be constant. One piece of evidence was
that the apparent diameter of the sun varied with the seasons. Another objec
tion to Aristarchus' plan arose from the knowledge that the earth consisted
of heavy matter; it was inconceivable that such a heavy body could be in
motion. The planets, on the other hand, were supposed to be made of some
GREEK ASTRONOJ\llCAL THEORIES 191
light substance and so their motion was feasible. This distinction between the
physical constitution of the earth and that of the planets was almost universally
accepted up to the seventeenth century. Moreover, if the earth were in mo
tion, why did objects on the earth not fall behind? Greek physics had no
answer to this argument.
The supreme achievement of all Greek efforts aimed at exhibiting the
mathematical design of the universe is the astronomical theory of Hipparchus
and Ptolemy. These two men, as we noted in the preceding chapter, had
created the mathematical method that enabled them to determine the sizes and
distances of the sun, moon, and several planets, the method which, as Ptolemy
put it, gave them the tool needed to base astronomy "on the incontrovertible
ways of arithmetic and geometry." They also had older Egyptian and Baby
lonian observations at their disposal as well as innumerable others made by
Hipparchus himself at Rhodes and by the observatory in Alexandria. They
tackled the plan of organizing all this knowledge into one comprehensive
scheme.
-
, Q_--+--�/
...::!:
t
oE Q p
Fig. 8-1.
A planet moves on its epicycle, which in turn moves around the deferent.
In the astronomy of Hipparchus and Ptolemy, which we now refer to as
the Ptolemaic theory, the earth is the center of the universe and stationary.
To account for the motion of a planet P (Fig. 8-1 ) , these men assumed that
P moves at a constant speed along a circle whose center is Q. At the same
time that P moves around Q, Q is supposed to be moving in a circle and at
192 THE MATHEMATICAL ORDER OF NATURE
a constant speed around the earth, E. The circle on which P moves is called
an epicycle, and the circle on which Q moves is called the deferent. Hip
parchus and Ptolemy could, of course, choose the radii of the two circles and
the speeds at which P and Q move on their respective circles so that the
motion of P agreed with the observed positions of the particular planet. For
each planet the choice of radii and speeds was different.
Actually the above scheme did not give these men enough latitude. Hence
their astronomical system contained also some minor devices which enabled
them to fit a system of such circles to the motion of any one heavenly body, but
the essential principle is the use of deferent and epicycle. It should be noted
that the motion of a planet as viewed from the earth is actually quite compli
cated and yet, by the above scheme, is readily understood in terms of a combi
nation of circular motions. This theory accounted for planetary motions
within the accuracy of observations attained in Alexandrian times. From the
time of Hipparchus an eclipse of the moon could be predicted to within an hour
or two. Predictions of the sun's motion were not so precise, but we must recall
here a point made in the preceding chapter, namely, that calculations of the sun's
distances at various times were not exact because the requisite angles were too
small to be measured accurately.
The scheme we have just described is contained in Ptolemy's Almagest,
the book mentioned earlier (Chapter 7 ) . This theory was quantitatively so
precise that it was accepted as the true design of the heavens until the work
of Copernicus and Kepler displaced it. It is significant, however, that Ptolemy
at least laid no claims to truth. He had constructed a mathematical scheme
which accounted for the motions of the celestial bodies, a theory which
worked, but he did not profess that God had so designed the universe. Un
fortunately people's confidence in the truth of a doctrine increases with the
length of time it holds sway, and since Ptolemaic theory was accepted for
about 1500 years, people came to regard it as an absolute and unchallengeable
truth. No other product of the entire Greek era rivals the Almagest in the
profound influence it exerted on conceptions of the universe and none, except
Euclid's Elements, achieved such unquestioned authority.
The theory of Hipparchus and Ptolemy is the final Greek answer to Plato's
problem of rationalizing the appearances in the heavens and is the first really
great scientific synthesis. Whereas the Greeks of the classical period were
convinced on philosophical and intuitive grounds that nature was rationaIly
designed, Ptolemaic theory provided overwhelming, concrete evidence.
8-4 THE EVIDENCE FOR THE MATHEMATICAL DESIGN OF NATURE
Let us look back for a moment to see the total evidence which the Greeks
could muster for their momentous doctrine that nature is mathematically
designed. The astronomical theory of Hipparchus and Ptolemy was certainly
EVIDF:NCE FOR THE MATHEMATICAL DESIGN OF NATURE 193
the most impressive evidence not only because it dealt with the grandest natural
spectacle but because it showed design in a maze of phenomena whose out
ward appearances scarcely suggested design. To this achievement we must
add Euclidean geometry. We have already pointed out the larger significance
of this body of knowledge; it demonstrated that the shapes and sizes of earthly
figures conform to a reasoned system of doctrines. One might very well prove
on the basis of self-evident axioms and of reasoning that satisfies the mind that
the sum of the angles of a triangle is 1 80". But when one constructs triangle
after triangle for various purposes and finds in every case that the sum is
indeed 180", one cannOt escape the implication that this and the other theorems
of Euclidean geometry express essential principles of nature. Moreover, be
cause these principles are all part of one reasoned body of knowledge, it seems
clear that nature is designed in accordance with a reasoned plan.
In the domains of light and sound (music) , the progress made by the
Greeks was not nearly so impressive, but they had produced the law of re
flection and they did know and use the properties of curved mirrors to con
centrate light. The Greeks were sure that further investigation would reveal
additional laws, and almost every Greek mathematician worked on light.
Many, among them Euclid, Archimedes, Apollonius, Heron, and Ptolemy,
wrote mathematical books on the subject. The development of a mathematical
theory of musical sounds was initiated by the Pythagoreans and, as in the case
of light, pursued by many later Greeks.
The Greeks also applied mathematics to various other classes of natural
phenomena and found the mathematical laws applicable. Archimedes wrote
a still famous book on the mathematical laws of the lever. Another of his
works investigated the weight and stability of various shapes placed in water.
It was primarily motivated by the experience that a ship whose shape is not
well chosen may readily overturn in water. Still another study dealt with the
centers of gravity of various shapes, an important bit of knowledge if bodies
are to be balanced or remain upright.
Phenomena of motion were also studied by the Greeks. Here too they
adopted what seemed to be self-evident principles and made deductions which
fitted their limited experience. In the Aristotelian theory of matter all objects
were composed of lightness. heaviness, wetness, and dryness. Those in which
lighmess dominated (for example, fire) always sought to rise. Those in which
heaviness dominated (for example, metals) sought to fall. Every object had
a natural place and, when not hindered, sought it. Thus the natural place of
light objects was a region near the moon, whereas heavy objects tended to
congregate at the center of the universe which was, of course, the center of
the earth. Force is required to set an object in motion, and a measure of this
force was the product of the weight and the velocity given to the body. Also,
a force must constantly be applied to keep a body in motion or else the motion
194 THE MATHEMATICAL ORDER OF NATURE
would cease. Forces are transmitted by material agents. Thus one body must
strike another to transmit motion to the latter." The Greeks made progress
in other scientific fields such as geography and geodesy which we discussed
somewhat in the preceding chapter.
In all of the fields discussed above mathematics was, at the very least,
considerably involved. In fact, in the classical period mathematics meant
arithmetic, geometry, astronomy, and music and, by the end of the Alexan
drian period. it had come to mean, in addition. mechanics (motion, the lever,
the hydrostatics of Archimedes), optics, geodesy, and logistics (practical
arithmetic) .
From these scientific investigations one major fact stood forth: the uni
verse is mathematically designed. Mathematics is immanent in nature; it is the
truth about its structure, or, as Plato would have it, the reality of the physical
world. Moreover, human reason could penetrate the divine plan and reveal
the mathematical structure of nature. Almost all of the mathematical and
scientific research which has taken place since Greek times has been inspired
by the conviction that there is law and order in the universe, and that mathe
matics is the key to this order.
The Greek miracle has not been rivaled, not even by our modern civiliza
tion. A relative handful of people produced in a few hundred years supreme
works not only in mathematics and science but in literature, art, music, logic,
and in many branches of philosophy.
8-5 THE DESTRUCTION OF THE GREEK WORLD
It is accurate to say of the Greeks that God proposed them but man disposed
of them. We have already related in Chapter 2 that the Romans conquered
the Greek lands and that Roman practicality affected adversely the theoretical
studies in Alexandria. We have also mentioned the rise of Christianity and
that the Christian reaction to Roman persecution was to condemn and forbid
all pagan learning, though, of course, the new religion did absorb some Greek
philosophic doctrines, notably Aristotle's. The destruction of what remained
at Alexandria, Christian and pagan, was completed by the Mohammedans. The
Arabs had been inspired by Mohammed to adopt a new religion. Mohammed
died in 632 A.D., but his successors undertook to convert the world by the
sword. They conquered Alexandria in 646 and burned the Museum on the
ground that if the books there contained anything contrary to the teachings
of Mohammed, they were wrong, and if in agreement, superfluous. With this
stroke the dusk settled on Alexandria.
Although the Museum was destroyed and the scholars dispersed, Greek
learning did ultimately become an integral part of European civilization and
• See also Section 1 3-5.
THE DESTRUCTION OF THE GREEK WORLD 195
culture. Just how the Greek creations found a new home in Western Europe
through one of the quirks of history has already been indicated briefly in
Chapter 2, and we shall say more about it in later chapters.
EXERCISES
I . What essential differences can you find between the pre-Greek and the Ptolemaic
view of the heavens?
2. What is the Pythagorean doctrine concerning the essence of reality?
3. What is the meaning of the statement that Ptolemaic theory is a geocentric
theory?
4. Describe the basic idea in Ptolemaic theory.
5. Suppose a planet moves on an epicycle at twice the speed with which the center
of the epicycle moves on the deferent. Suppose, further, that the radius of the
deferent is three times the radius of the epicycle. Sketch the path of the planet
around the earth.
6. What is meant by the rationality of nature?
7. How does Ptolemaic theory support the belief in the mathematical design of
nature?
8. How does Euclidean geometry tend to establish the mathematical design of
nature?
Topics for Further Investigation
1. The mathematical doctrines of the Pythagoreans. Use the references on the
history of mathematics in Chapter 7.
2. The accomplishments of Greek physical science.
3. The astronomical theory of Eudoxus.
4. The astronomical theory of Aristarchus.
5. The astronomical theory of Ptolemy.
6. Pre-Greek views of the universe. Use Dreyer in the references below.
Recommended Reading
CLAGETI, MARSHALL: Greek Science in Antiquity, Abelard-Schuman, Inc., New
York, 1955.
DAMPIER-WHETHAM, WM. C. D.: A History of Science, Chap. 1, Cambridge Uni
versity Press, Cambridge, 1929.
DREYER, J. L. E.: A History of Astronomy, 2nd ed., Chaps. 1 through 9, Dover
Publications, Inc., New York, 1953.
FARRINGTON, BENJAMIN: Greek Science, 2 voIs., Penguin Books, Harmondsworth,
England, 1944 and 1949.
196 THE MATHEMATICAL ORDER OF NATURE
JEANS, SIR JAMES: The Growth of Physical Science, 2nd ed., Chaps. 1 through 3,
Cambridge University Press, Cambridge, 1951.
JEANS, SIR JAMES: Science and Music, pp. 160-190, Cambridge University Press,
Cambridge, 1947.
KUHN, THOMAS S.: The Copernican Revolution, Chaps. 1 through 3, Harvard
University Press, Cambridge, 1957.
SAMBURSKY, S.: The Physical World of the Greeks, Routledge and Kegan Paul,
London, 1956.
SARTON, GEORGE: A History of Science, Vols. I and II, Harvard University Press,
Cambridge, 1952 and 1959.
SINGER, CHARLES: A Short History of Science, Chaps. 1 through 4, Oxford Univer
sity Press, London, 1953.
CHAPTER 9*
THE AWAKENING OF EUROPE
Solicit not thy thoughts with matters hid,
Leave them to God, Him serve and fear.
. . . . . . . . . . . . be lowly wise;
Think only what concerns thee and thy being.
JOHN MILTON
9-1 THE MEDIEVAL CIVILIZATION OF EUROPE
It is perhaps a comfort after reading about the destruction of the Greek civili
zation to turn to a new one-the civilization of western Europe. We know
that Europe did acquire the Greek creations and built upon them a vast, scien
tifically oriented civilization. How did this come about? To answer this
question and to understand the special nature of subsequent developments in
Europe, we must note a few historical facts.
The Germanic tribes, who have occupied western and central Europe as
far back as history goes and who are the forefathers of most Americans, were
barbarians. We know very litt.!e about their early history because they had
no writing and hence no records were kept. From Roman historians, notably
Tacitus (first century A.D.), we know that the Germanic tribes possessed a
very primitive civilization. Tacitus describes them as honest, hospitable, hard
drinking, hating peace, and proud of the loyalty of their wives. Their dwell
ings were huts of timber and straw located in woods and surrounded by crude
fortifications. Animal skins and coarse linens served for clothes, while herds
of cattle, hunting, and the cultivation of grain crops provided food. Industry
was unknown; just enough iron was mined to provide crude weapons. Trade
was effected through barter and supplemented by plundering other tribes and
more civilized regions. There were no arts, no science, and no learning. The
chief activities were eating, sleeping, carousing, and fighting other tribes. Since
such activities are also characteristic of peoples we call civilized, we may say
that to that extent the Germanic tribes were civilized.
Although the Romans won many battles with the Germanic tribes, the
Empire grew weaker for a variety of reasons which we cannot survey here,
and the barbarians finally conquered it. Barbarians became kings of Rome and
197
198 THE AWAKENING OF EUROPE
what was left of the Empire. Only a small region around Constantinople,
which we call the Eastern Roman or Byzantine Empire, managed to remain
independent and isolated. The Eastern Roman Empire, incidentally, also
withstood the Mohammedans, who in the seventh century conquered Egypt,
the Near East, and the lands bordering the Mediterranean Sea.
By the time that the Roman Empire collapsed in the fifth century A.D.,
the Catholic Church had become a strong organization with good leadership.
It gradually converted the heathens to Christianity, established schools in
Europe, and taught reading, writing, and ethics. Moreover, it perpetuated and
imposed the legal and political organization of Rome. The Christian influence
was certainly beneficial in that it produced a more stable state of affairs and
even induced the barbarians to remain at peace for longer periods of time, a
restraining influence which the barbarians did not resent because they soon
learned that civilization had its advantages. With a little thought they found
that peaceful interludes permitted them to develop methods of mass destruc
tion and so do as much killing at intervals as previously in constant warfare.
Cities and small states governed by powerful leaders were established in
Europe. Trade between cities developed, producing the wealth necessary to
support scholarship. But study was almost entirely confined to understanding
the word of God as fostered, expounded, and dictated by the Fathers of the
Church. Those Greek works which had survived destruction by Romans,
Christians, and Mohammedans lay almost unnoticed in neglected public build
ings, in private libraries, or in the isolated, beleaguered Eastern Roman Empire.
What little knowledge of nature was deemed necessary in the life pre
scribed by the Church was derivable, so the Christian leaders said, from the
Bible. St. Augustine ( 3 54-430), a man learned in Greek and Christian thought,
even declared that the authority of the Scriptures is greater than the capacity
of the human mind. Unfortunately the Biblical statements about the nature
and structure of the physical world are of Babylonian origin and hence
decidedly inferior to the knowledge acquired by the Greeks.
Of course, some of the actual phenomena of nature were observed, and
questions raised about them. The medieval intellectuals who pursued such
matters offered a kind of explanation which is satisfying to some minds. They
believed that natural processes were mainly means to an end, i.e., they adopted
what is called a teleological viewpoint. Thus rain existed to nourish the crops.
Crops and animals existed to provide food for man. Sickness was a punishment
from God. Plagues and earthquakes were expressions of God's anger. In
general, all explanations focused on the phenomenon's value to, or effect on,
man. Man was the center of the universe not merely geographically but also
in terms of the ultimate purposes served by nature.
Although nature existed to serve man, man himself existed on this earth
only to serve an apprenticeship during which he prepared his soul for a life
in heaven with God-or elsewhere. Life on earth was but an unimportant
MATHEMATICS IN THE MEDIEVAL PERIOD 199
prelude, to be endured but not enjoyed. To prepare for the afterlife man
had to wrest his soul from a stubborn flesh which was guilty of original sin.
Participation in the bounty of nature, food, clothing, and sex, tainted the soul
and so had to be severely restricted. Medieval man, certain of his sins and
doubtful of salvation, had to bend all his efforts to attain redemption. By
earning divine grace man could escape from this foul earth to the divine
empyrean.·
9-2 MATHEMATICS IN THE MEDIEVAL PERIOD
We see that a new civilization did arise in Europe, but from the standpoint of
the perpetuation of mathematical learning or the creation of mathematics, it
was totally ineffective. Although this civilization did spread ethical teachings,
fostered Gothic architecture and great religious paintings, no scientific, tech
nical, or mathematical concept gained any foothold. In none of the civiliza
tions which have contributed to the modern age was mathematical learning
reduced to so Iow a level.
Superficially mathematics did seem to play an important role. In the
medieval schools the standard curriculum consisted of seven subjects, the
quadrivium and the trivium. The quadrivium comprised arithmetic, the science
of pure numbers; music as an application of numbers; geometry, or the study
of magnitudes such as length, area, and volume at rest; and astronomy. the
study of magnitudes in motion. But the scope of these studies was terribly
limited. Even the first universities of Europe, which began to function about
1 100 A.D., offered merely a minimum of arithmetic and geometry. Arithmetic
consisted of simple calculations mingled with complex superstitions. Geometry
was confined to the first part of Euclid, far less than we learn in high-school
courses today. The most advanced point reached in some of these institutions
of learning was the very elementary theorem that the base angles of an isosceles
triangle are equal.
The little mathematics kept alive in the schools served various purposes in
the medieval period. Some astronomy was pursued to keep the calendar.
Here a minimum of arithmetic and geometry sufficed for the accuracy needed,
just as it did in ancient Egypt and Babylonia. This work was usually per
formed by monks because the clergy was the most learned class. Astronomy
and therefore elementary mathematics played a larger role in medieval life in
that they provided the factual information needed for astrology, which was
regarded as a science.
One more medieval use of mathematics is worthy of mention. Plato's
belief that the study of mathematics trains the mind for philosophy was taken
over by the Church which, however, substituted theology for philosophy .
• A term of medieval cosmology referring to the "highest heaven" or paradise.
200 THE AWAKENING OF EUROPE
Clearly the interest here was not in mathematics as such but as a preparation
for grasping the subtle reasoning which the Church employed to build and
strengthen the foundations of religious doctrines.
9-3 REVOLUTIONARY INFLU"ENCES IN EUROPE
Whether or not the civilization of medieval Europe might in due time have
given rise to mathematical activity will never be known. But dramatic changes,
largely initiated by non-European forces, drastically altered the Christian
world. The earliest influence tending to transform thought and life in medieval
Europe may be credited to the Arabs. While the Church was gradually civi
lizing the European barbarians and establishing the Christian way of life, the
Arabs, perhaps more ruthless in proselytizing and certainly more dynamic and
aggressive. succeeded in establishing their own civilization and culture in
southern Europe, North Africa, and the Near East. Though fanatic in the
advancement of their own religion, once their empire was stabilized, the Arabs
displayed great tolerance toward alien ideas and learning, readily absorbed the
mathematics and science of the Greeks and Hindus, and built cultural centers
in Spain and the Near East. They translated the Greek works into Arabic and
added commentaries and contributions of their own to mathematics, astron
omy, medicine, optics, meteorology, and science in general.
By about 1 1 00 A.D. Europeans were trading freely with Arabs. The
Crusades, which attempted to wrest Palestine from the Arabs, brought further
contacts between Christians and Moslems. Through these channels the Euro
peans became aware of the Greek works and Arab additions. They were so
fascinated by this material that they aroused themselves to acquire it. Wealthy
merchants, princes, and popes sent agents to the Arab centers to purchase
manuscripts. Many Europeans went to live in Spain and learned Arabic in
order to read the works and translate them into Latin. Others were assisted by
Jewish and Arab scholars in making the translations. Plato, Aristotle, Eltclid,
Ptolemy, and the Greek literary works were avidly grasped.
In the fifteenth century Italy made new contacts with the Greek heritage.
Ambassadors from Constantinople, the capital of the Eastern Roman empire,
which still possessed the largest collection of ancient manuscripts, came to
Italy several times in the first half of the fifteenth century, largely to seek
help against the Turks. The Italians learned about the Greek works and
like the Europeans of three centuries earlier, sought eagerly to possess them.
In addition, some Greek scholars discouraged by the poverty in Eastern
Europe and Alexandria migrated to Italy. When the Turks finally captured
Constantinople in 1453, a Rood of these men bringing their manuscripts with
them came to Italy.
By financing the geographical explorations of the fifteenth and sixteenth
centuries, which were intended to discover new trade rOlltes, the merchants
REVOLUTIONARY INFLUENCES IN EUROPE 201
affected the life of Europe. The discovery of America and of a route to China
around Africa resulted in acquainting Europe with strange lands, beliefs, cus
toms, religions. and ethical doctrines. Catholics met Mohammedans, Chinese,
and the American Indian. To the broadening influence of trade itself was
added knowledge which conflicted sharply with the doctrines and way of life
hitherto accepted in Europe. Questioning of the accepted doctrines and values
ensued.
The merchant class and the large classes of artisans and free laborers
introduced new interests. Employers and employees sought material gain, and
so looked for commodities, machinery, and natural phenomena which might
be employed to advantage. The rulers of the Italian cities and states also
spurred on these interests. They coveted power and magnificence and, to
acquire the necessary wealth, favored trade, industries, and inventions. The
cities competed to surpass one another in skills, devices, and quality of mer
chandise. These groups, though selfishly motivated, were nevertheless effective
in orienting the civilization toward the physical world and in fostering the
accumulation of empirical knowledge.
The Protestant Revolution, or the Reformation as it is called, also upset
the old culture in Europe. We are not concerned here with justification of
the break from the Church. But Luther fanned the fire of discontent which
had spread throughout Europe. Disputes about the nature of the sacrament,
the validity of the control of the Church by Rome, and the meaning of
passages of the Scriptures raised doubts in many people, who were thus em
boldened to turn to other sources of knowledge, notably the physical world
itself.
Several discoveries and inventions of the late medieval period had effects
far greater than one might at first expect. In the twelfth century the Euro
peans learned from the Chinese about the compass. The introduction of the
compass was important because it was an immense aid to navigators on long
sea voyages. The explorers who dared the Atlantic might not have been
willing to do so without it.
The introduction of gunpowder in the thirteenth century produced as its
most obvious effects changes in methods of warfare and the design of forti
fications. It also introduced a new physical problem, the motion of projectiles.
An indirect result was the granting of more power to the common man because
with a musket he could be effective in warfare. Previously only those who
could afford expensive armor, that is, the wealthy nobles, could wield military
power.
The invention of printing (about 1450) was immensely important in help
ing to spread Greek knowledge across Europe. Another invention, paper made
of cotton and later of rags, which replaced costly parchment, also helped to
make books plentiful and cheap. Many editions and translations of Greek
works were printed in the century following these inventions. They helped
202 THE AVVAKENING OF EUROPE
to bridge the gulf between the learned and the untutored just at the time when
great numbers were seeking to obtain knowledge.
Advances in the subject of optics had a vast effect on future scientific
activity. The first was the discovery made in the thirteenth century that lenses
can be used to magnify objects and thus aid in the examination of materials and
natural phenomena. Lens grinders began to produce spectacles. Early in the
seventeenth century, two of them discovered that a pair of lenses held at some
distance from each other could be used to make distant objects seem close.
Thus the telescope became available and was immediately applied to astronomy
with results we shall describe later. At about the same time, it was found that
a combination of lenses would do even better than a single lens to magnify
nearby objects, and the microscope was invented. The investigation of the
biological world and the revelation of hitherto unsuspected small-scale phe
nomena soon followed.
9-4 NEW DOCTRINES OF THE RENAISSANCE
It was to be expected that the insular world of medieval Europe accustomed
for centuries to one rigid, dogmatic system of thought would be shocked and
aroused by the series of events we have just described. The European world
was in revolt. As John Donne put it, "All in pieces, all coherence gone."
Europe revolted against scholastic domination of thought, rigid authority, and
restrictions on the physical life. It revolted against the Scriptures as the source
of all knowledge and the authority for all assertions. It revolted against en
forced conformity to the established canons of conduct.
A leading figure in the revolt from the old modes of thought is Leonardo
da Vinci ( 1452-15 1 9 ) . Because he saw how most scholars accepted as author
itative all that they read, he distrusted the men who took their learning only
from books and professed their knowledge so dogmatically. He describes
them as puffed up and pompous, strutting about, and adorned only by the
labors of others whom they merely repeated. These were only the reciters
and trumpeters of other people's learning. Leonardo determined to learn for
himself and made exhaustive studies of plants, animals, the human body, light,
the principles of mechanical devices, rocks, the flight of birds, and hundreds
of other subjects. Although he is most often remembered as one of the great
masters of painting, he also was a psychologist, linguist, botanist, zoologist,
anatomist, geologist, musician, sculptor, architect, and engineer.
Many scholars turned to exhaustive studies of the Greek authors, to trans
lations, and to compilations. They gave to these works the same infinitely
detailed and critical attention that they and others had formerly given to
biblical documents. The writings of Luca Pacioli ( 1445-1 514) show this
tendency. He was a monk, who in 1499 published Summa de Arithmetica,
Geometric., Proportione et Proportionalita. As a full, almost encyclopedic,
NEW DOCTRINES OF THE RENAISSANCE 203
account of the mathematical knowledge available to Europe by 1500, it was
enormously helpful.
More interesting as a transitional figure is Jerome Cardan whom we met
in Chapter 5. He wrote a great number of works which exhibit a critical
attirude only in the sense that he traced the origins of stories, miracles, and
"facts" to the authorities. However, he accepted freely any number of medie
val superstitions, legends, accounts of supernatural events, pseudo-sciences, and
even magical medical trearments. He believed in the significance of dreams,
ghosts, portents, palmistry, and astrology, which to him were sciences. He
also wrote volumes on moral aphorisms and on the varieties of beings and
bodies which fill the universe. Among these were spirits which took the form
of sylphs, salamanders, gnomes, and ondines. Communion with these spirits
was the highest aim in life.
Cardan's writings in the above fields were compilations; much of the
material, incidentally, he stOle from Leonardo da Vinci, who was a friend of
Cardan's father. In his mathematical and scientific work, however, he shows
the new influences. His still famous ATs Magna ( 1 545) , which contains a full
account of the algebraic methods known to the Arabs, also contains results due
to himself and his contemporaries. He is the first European mathematician of
consequence. Some indication of what was new in his work was given in
Chapter 5.
Pacioli and Cardan are mathematical figures in the movement commonly
known as humanism. The humanists, and we speak now of those active in
all fields, have been criticized because they idolized the past too much and
looked backward rather than forward. They slavishly accepted the Greek
works and pored over them, even undertaking extensive philological studies
to determine the meanings of dubious words. To their credit may be noted
that they prepared the atmosphere for the revival of reason, spread the Greek
ideas through Europe, secularized education, and stressed the individual, ex
perience, and the natural world.
The period devoted to the collection and study of the classics was followed
by one in which intellectuals groped for positive doctrines and methods to
replace or at least alter the medieval culture. We cannot trace in detail the
oscillations of thought, the mixture of medieval fantasy and rational specula
tions, the commingling of fine observations with outmoded principles, all of
which one finds especially in the sixteenth century. Many European thinkers
finally broke away from the endless rationalizing on the basis of dogmatic
principles which were vague in meaning and unrelated to experienct, and chose
human inquiry rather than divine authority.
It was from the Greek works that the leaders in this intellectual revitaliza
tion of Europe derived the principles of a new approach to man and the
universe. They learned that man could enjoy a physical life and find pleasure
in food, sports, and the development of his own body. Beauty was not a snare,
204 THE AVVAKENING OF EUROPE
and pleasure not a sin. Man, the unworthy creature, who had been commanded
to regard himself as a sinner, to spend his life in abstinence, penance and ab
jecmess, and to prepare for death, the only real event of life, could find dignity
in his own being, and demand a full life on this earth as his birthright. In place
of sin, death, and judgment, men should seek beauty, pleasure, and joy. The
Renaissance world began to see man as the goal of God rather than God as
the goal of man.
The human spirit was emancipated and inspired to refashion its ideals of
existence. Perhaps the most important decision was to tum to nature herself
as the source of knowledge. "Back to nature" became the new cry. Europeans
turned to nature's laws instead of divine pronouncements gleaned from the
Scriptures, to the universe of God instead of God. Man himself was included
in the study of nature.
Leonardo is a representative figure in this shift to nature as the prime
focus. He almost boasts that he is not a man of letters and that he chose to
learn from experience. His observations and inventions recorded in his note
books give evidence of his extensive and detailed physical studies. He says,
"If you do not rest on the good foundation of nature, you will labor with
little honor and less profit." Sciences which arise in thought and end in
thought do not give truths because no experience enters into these purely
mental reflections, and without experience no thing is sure.
A new school of biologists arose, of whom Andreas Vesalius ( 1514-64)
was the leader. His On the Structure of the Human Body ( 1 543) may be
regarded as the beginning of modern anatomy. Although this work is based
on Galen, he corrected many of Galen's errors and added new observations.
Vesalius asserted that the true Bible is the human body, and he dissected
corpses to learn the human structure. William Harvey ( 1 578-1657), the
famous seventeenth-century doctor, voices the spirit of Vesalius in the preface
to his book On the Movement of the Heart and the Blood: "I profess to learn
and to teach anatomy, not from books, but from dissections; not from the
positions of philosophers, but from the fabric of nature." Harvey also fol
lowed Galen but, like Vesalius, added new material derived from his own ob
servations and thought. Andrew Cesalpinus ( 1520-1603 ), the botanist, clearly
advocated starting from observation and then proceeding through careful
differentiation of the species observed to inductive truths.
We shall see in the next chapter how the artists, too, turned to the study
of nature and to new goals in painting which obliged them to study anatomy,
perspective, light, and mechanics. Regard for the primacy of observation
forced Johannes Kepler to devise revolutionary doctrines in astronomy. In
deed, experience became the source of all basic scientific laws and, in this
respect, usurped the role of mind.
The second guiding principle adopted by the Europeans of the Renaissance
was to let reason be the judge of what to accept. Revelation, faith, and author-
NEW DOCTRINES OF THE RENAISSANCE 205
ity were to be subordinated as support for assertions about man and the uni
verse, and reason was to be applied freely to all problems man sought to solve.
Although the Church itself had used reason to erect its own theology, it had
said that some matters were beyond reason. Moreover, the results obtained by
reasoning were not put forth to be scrutinized rationally but rather to be
accepted. In the Renaissance, mind replaced faith as the sovereign authority,
and man was encouraged to apply it to the problems besetting his age.
The new impulse to study nature and the decision to apply reason instead
of. relying upon authority were forces which might in themselves have led to
mathematical activity. But the Europeans also had the Greek works. From
the Greeks the Europeans learned that nature is mathematically designed, and
that this design is harmonious, aesthetically pleasing, and the inner truth about
nature. Nature is not only rational, simple, and orderly but it acts in accord
ance with inexorable and immutable laws.
Almost from the beginning of the period in which Greek works began to
be known in Europe, one finds leading thinkers impressed with the importance
of the mathematical study of nature. In the thirteenth century, Roger Bacon
believed that the laws of nature are but the laws of geometry. Mathematical
truths are identical with things as they are in nature. Moreover mathematics
is basic to the other sciences because it takes cognizance of quantity. Leonardo,
too,-although his knowledge of Greek works was rather limited and his
appreciation of what mathematical proof means almost nil-had caught the
new spirit. He says that only by holding fast to mathematics can the mind
safely penetrate to the essence of nature. "No human inquiry can be called
true science unless it proceeds through mathematical demonstrations." He
also says, "The man who discredits the supreme certainty of mathematics is
feeding on confusion and can never silence the contradictions of sophistical
sciences, which lead to eternal quackery." Leonardo was not a mathematician,
and his understanding of the principles of mechanics, the study of bodies at
rest and in motion, was intuitive and but a dim foreshadowing of the work of
Galileo and Newton, but he had prophetic vision. He says in one of his
notebooks, "Mechanics is the paradise of the mathematical sciences because
in it we come to the fruits of mathematics." Leonardo does stress the role
of theory in science and says, "Theory is the general; experiments are the
soldiers." However he did not appreciate the precise role of theory or foresee
what later became the true method of science. He, in fact, lacked method
ology. Copernicus and Kepler, whom we shall study in more detail later,
were also convinced that the world is mathematically and harmoniously
designed, and this belief sustained them in their scientific endeavors.
Galileo speaks of mathematics as the language in which God wrote the
great book-the universe-and unless one knows this language, it is impossible
to comprehend a single word. Rene Descartes, father of coordinate geometry,
was convinced that nature is but a vast geometrical system. He says that he
206 THE AWAKENING OF EUROPE
"neither admits nor hopes for any principles in Physics other than those
which are in Geometry or in abstract Mathematics, because thus all the phe
nomena of nature are explained, and some demonstrations of them can be
given." Certainly by 1600 the conviction that mathematics is the key to
nature's behavior had taken firm hold and stimulated the great scientific work
which was to follow.
To the intellectuals of the Renaissance mathematics appealed for stiII an
other reason. The Renaissance, as we have seen, was a period in which medieval
civilization and culture were challenged and new influences, information, and
revolutionary movements were sweeping Europe. These men sought new and
sound bases for the erection of knowledge, and mathematics offered such a
foundation. Mathematics remained the one accepted body of truths amid
crumbling philosophical systems, disputed theological beliefs, and changing
ethical values. Mathematical knowledge was certain knowledge and offered a
secure foothold in a morass. The search for truth was redirected toward
mathematics.
9-5 THE RELIGIOUS MOTIVATION IN THE STUDY OF NATURE
The decisions to study nature, to apply reason, and to seek the mathematical
design of natu;"e led to a revival of mathematical activity and to the emergence
of great mathematicians. But the thinking of these men took a turn which is
of interest because it shows one of the strong motivations for mathematical
activity over a couple of centuries and because it played a role in the subse
quent cultural history.
The mathematicians and scientists of the Renaissance were brought up in
a religious world which stressed the universe as the handiwork of God. The
scientists whom we shall meet shortly, Copernicus, Brahe, Kepler, Pascal,
Galileo, Descartes, Newton, and Leibniz, accepted this doctrine. These men
were in fact orthodox Christians. Copernicus was a member of the Church.
Kepler studied for the ministry although he did not take orders. Newton was
deeply religious and, when late in life he felt too exhausted to pursue creative
scientific work, turned to religious studies.
However, in the sixteenth century the new goal in the intellectual world
became to study nature through mathematics and indeed to uncover the mathe
matical design of nature. Now Catholic teachings had by no means included
this last principle, which is Greek. How then was the attempt to understand
God's universe to be reconciled with the search for the mathematical laws of
nature? The answer was to add a new doctrine, namely, that God had designed
the universe mathematicaI1y. Thus the Catholic doctrine postulating the
supreme importance of seeking to understand God and his creations took the
form of a search for God's mathematical design of nature. Indeed the work of
the sixteenth, seventeenth, and even some eighteenth-century mathematicians
RELIGIOUS MOTIVATION IN THE STUDY OF NATURE 207
was a religious quest, motivated by religious beliefs, and justified in their
minds because their work served this larger purpose. The search for the
mathematical laws of nature was an act of devotion. It was the study of the
ways and nature of God which would reveal the glory and grandeur of his
handiwork. The Renaissance scientist was a theologian studying nature instead
of the Bible. Copernicus, Kepler, and Descartes speak repeatedly of the
harmony which God imparted to the universe through his mathematical design.
Mathematical knowledge, being in itself truth about the universe, is as sac
rosanct as any line of the Scriptures. Galileo says, "Nor does God less
admirably discover Himself to us in Nature's actions than in the Scripture's
sacred dictions." Man could not hope to perceive the divine plan as clearly as
God himself understood it, but man could with humility and modesty seek to
at least approach the mind of God.
One can go further and assert that these men were sure of the existence of
mathematical laws underlying natural phenomena and persisted in the search
for them because they were convinced a priori that God had incorporated
them into the construction of the universe. Each discovery of a law of nature
was hailed as evidence testifying more to God's brilliance than to the ingenuity
of the investigator. Kepler in particular wrote paeans to God on the occasion
of each discovery. The beliefs and attitudes of the mathematicians and scien
tists exemplify the larger cultural phenomenon which swept Renaissance
Europe. The Greek works impinged on a deeply devour Christian world, and
the intellectual leaders born in one and attracted by the other fused the
doctrines of both.
EXERCISES
1. In view of what we know about Greek and medieval attitudes toward the
physical world and mathematical activities in these two cultures, would you draw
any conclusion about the connection between interest in the physical world and
the pursuit of mathematics?
2. What events and influences led to a revival of interest in mathematics?
3. How did Renaissance scientists and mathematicians reconcile the Greek doctrine
that the world is mathematically designed and the Christian doctrine that the
universe is the creation of God?
Topics for Further Investigation
1. The rise of algebra in the sixteenth century.
2. Hindu and Arab mathematics.
3 . The life and work of Roger Bacon.
4. The life and work of Jerome Cardan.
S. The life and work of Leonardo da Vinci.
208 THE AWAKENING OF EUROPE
Recommended Reading
BALL, W. W. ROUSE: A Short Account of the History of Mathematics, 4th ed.,
Chaps. 6 to 12, Dover Publications, Inc., New York, 1960.
CAJOR., FLORIAN: A History of Mathematics, 2nd ed., pp. 83-129, The Macmillan
Co., New York, 1938.
CARDAN, JEROME: The Book of My Life, E. P. Dutton and Co., New York, 1930.
CROMBIE, A. C.: Augustine to Gatileo, Chaps. 1 to 5, Falcon Press, London, 1952.
Also published in paperback under the title Medieval and Early Modern Science,
2 vols., Doubleday and Co. Anchor Books, New York, 1959.
CROMBIE, A. C.: Robert Grosseteste and the Origins of Experimental Science, Ox
ford University Press, London, 1953.
DAMPIER-WHETHAM, WILLIAM C. D.: A History of Science, pp. 65-138, Cambridge
University Press, London, 1929.
DA VINCI, LEONARDO: Philosophical Diary, Philosophical Library, Inc., New York,
1959.
EASTON, STEWART C.: Roger Bacon and His Search for a Universal Science, Colum
bia University Press, New York, 1952.
HOFMANN, JOSEPH E.: The History of Mathematics, Chaps. 3 and 4, The Philosophi
cal Library, New York, 1957.
M ACCURDY, EDWARD: The Notebooks of Leonardo da Vinci, George Braziller,
New York, 1954.
ORE, OYSTEIN: Cardano, The Gambling Scholar, Princeton University Press, Prince
ton, 1953.
RANDALL, JOHN HERMAN, JR.: The Making of the Modern Mind, rev. ed., Chaps.
I through 9, Houghton Mifflin Co., Boston, 1940.
RUSSELL, BERTRAND: A History of Western Philosophy, pp. 324-545, Simon and
Schuster, New York, 1945.
SMITH, DAVID EUGENE: History of Mathematics, Vol. I, Chaps. 5 through 8, Dover
Publications, Inc., New York, 1958.
VALLENTIN, ANTONINA: Leonardo da Vinci, The Viking Press, New York, 1938.
CHAPTER 10*
MATHEMATICS AND PAINTING IN THE RENAISSANCE
Mighty is geometry; joined with art, resistless.
EURIPIDES
10-1 INTRODUCTION
The new currents of thought in the European Renaissance, the search for new
truths to replace the discredited ones, the tum to the study of nature to obtain
reliable facts, and the revived Greek conviction that the essence of nature's
behavior should be sought in mathematical laws, bore fruit first in the field of
art rather than science. While philosophers and scientists sought to unearth
basic facts which might somehow be incorporated into their yet to be formu
lated new scientific merhod, and while mathematicians were still digesting the
Greek works and awaiting inspiration for new themes, the artists, particularly
the painters, reacted far more quickly and revolutionized the art of painting.
That the painters turned to mathematics to formulate their new style of
painting is a little surprising, but the phenomenon has an explanation. The
painters of the fourteenth, fifteenth, and sixteenth centuries were the architects
and engineers of their time. They were also the sculptors, inventors, gold
smiths, and stonecutters. They designed and built churches, hospitals, palaces,
cloisters, bridges, dams, fortresses, canals, town walls, and weapons. Thus
Leonardo da Vinci, in offering his services to Lodovico Sforza, ruler of Milan,
promises to serve as engineer, constructor of military works, and designer of
war machines, as well as architect, sculptor, and painter. The artist was even
expected to predict the motion of cannon balls, by no means a simple problem
for the mathematics of those times. In view of these manifold activities the
painter necessarily had to be something of a scientist.
Further, the Renaissance painter, unlike the builder of Gothic cathedrals,
was influenced by the current doctrines which proclaimed that he learn truths
from nature and that the essence of natural phenomena is best expressed
through mathematics. Again, in comparison with his predecessors, he had the
advantage of gleaning some mathematical knowledge from the newly recovered
Greek works that were exciting the Europeans. The Renaissance painters went
209
210 MATHEMATICS AND PAINTING IN THE RENAISSANCE
so far in assimilating this knowledge and in applying mathematics to painting
that they produced the first really new mathematics in Europe. In the fifteenth
century they were the most accomplished and also the most original mathe
maticians.
10-2 GROPINGS TOWARD A SCIENTIFIC SYSTEM OF PERSPECTIVE
Before we examine just how Renaissance painters employed mathematics and
thereby revolutionized the art of painting, let us see what had been going on
in this field. Rather early in the medieval period painting became an extensive
activity. Kings, princes, and church leaders commissioned works of art to
enhance buildings. The system which the medieval painters used until about
1 300 was conceptual. Their objective was to portray and embellish the central
themes in the Christian drama. Since the intent was to stir up religious feelings
rather than to present real scenes, people and objects were drawn in accordance
with conventions which had acquired symbolic meaning. Thus people were
placed in unnatural, stylized positions; the general impression was one of
flatness; and the entire painting had a two-dimensional effect. The backgrounds
were usually solid gold to suggest that the action or people existed in some
supra-earthIy region.
Examples of this style of representation are abundant. A classic example
of the late medieval period is found in Simone Martini's ( 1 285-1344) "Majesty"
(Fig. 1 0- 1 ) . Clearly this is no real scene. The background is blue. Despite the
assemblage the scene looks flat; the throne especially lacks depth. There is
hardly the suggestion of a floor on which the figures stand, and these appear
lifeless and unrelated to one another. Moreover, sizes are not important. This
painting also illustrates another conceptual device used in medieval painting,
known as terraced perspective. To show a group of people arranged in depth,
those farther back are placed somewhat above those in front.
Toward the end of the thirteenth century, the painters began to be influ
enced by the Renaissance. Since the preoccupation with religious themes still
existed and paintings. in fact, continued to be commissioned mainly by church
officials, the same subjects appear but in more realistic settings. The painters
had turned to the observation of nature and saw a real world, physical beings,
earth, sea, and air. Their paintings reveal this interest in natural scenes by
reflecting their efforts to render space, depth, mass, volume, and other visual
effects largely through the use of lines, surfaces, and other geometric forms.
To achieve naturalism they also tried to render emotions and to depict drapery
folding around parts of the body as drapery actually does. People began to
look like real individuals instead of types. Mysticism gradually gave way to
realism and art became more and more secular.
Cimabue (ca. 1 300), Cavallini (ca. 12 50-1 3 30), Duccio ( 1 2 5 5-1 3 1 8), and
Giotto ( 1 266-1 3 3 7 ) were the leaders of the new movement to inject realism
GROPINGS TOWARD A SYSTEM OF PERSPECTIVE 211
Fig. 10-1.
Simone Martini: Majesty. Pallazzo Communale. Siena.
into painting and to incorporate the beauty of nature. Giono, in particular,
is often called the father of modem painting. In the works of the men cited
and in those of their immediate successors, we can readily observe the search
for an optical system of perspective.
Duccio's "Last Supper" (Fig. 10-2) shows what could be a real scene and
offers an ambitious attempt at depth. The receding wall and ceiling lines
create this effect. Moreover, pairs of lines which are parallel in the actual
scene and symmetrically placed with respect to the center are drawn so as
to meet on a vertical line through the center of the painting. This scheme
is referred to as vertical perspective and was developed further by other
painters.
On the whole the picture is not too successful. The table seems to slant
toward the front. The objects on the table are too much in the foreground
and appear to be on the point of sliding off or toppling over. The table and
the room are not seen from the same point of view. The various parts lack
proportion. The failure of the painting to depict depth properly causes one
to look from side to side instead of into the painting. An interesting feature
Fig. 10-2. Ouccio: Last Supper. Opera del Ouomo, Siena.
�, . "
Fig. 10-3. Giotto : Birth and Naming of St. John the Baptist. Church of Santa Croce,
Florence.
REALISM LEADS TO MATHEMATICS 213
characteristic of the period is the setting in a partially boxed-in room. The
artists were beginning to treat nature, but for the moment limited themselves
to scenes which had both interior and exterior components. They were already
looking into space and were about to venture into the wide world.
Giotto painted with the definite goal of reproducing visual perceptions
and spatial relations, and his paintings tend to produce the effect of photo
graphic copies. His figures possess . mass, volume, and vitality, are grouped
appealingly, and are interrelated. His "Birth and Naming of St. John the
Baptist" (Fig. 10-3) is typical. The partially boxed-in interior is again evident
as is the use of lines and surfaces. The side walls are drawn small or fore
shortened to suggest depth. The ground plane is a clear surface. Although
Giotto's paintings are not visually correct and although he introduced no new
principles, his results are far better than those of his predecessors. He chose
homelike scenes, gave human feelings to his figures, and distributed them in
space. He catches shades of emotions and expresses them through the features
and postures of the bodies. There is no mysticism nor ecstatic piety; "real"
angels, Christ, and disciples stand before us. He was aware of the progress he
had made and he delighted in showing his skill.
A step forward in the achievement of realism was made by Ambrogio
Lorenzetti (fourteenth century) . His outdoor panoramas are the best of this
period. However, from the standpoint of the significant development which
was to follow, his "Presentation in the Temple" (Fig. 10-4) is more worthy
of attention. There is a definite foreground or horizontal plane as opposed to
the background or vertical plane. The lines on the floor clearly recede and
meet in one point. Other pairs of receding parallel lines meet in respective
points of a vertical line. Also significant is the gradual decrease (foreshorten
ing) in the size of the floor blocks to suggest distance. But the floor and the
rest of the painting are not unified.
These few samples of fourteenth-century Renaissance painting show the
increasing efforts to achieve naturalism, real scenes, and three-dimensionality.
The innovators were groping for an effective technique but did not succeed.
Visualization and sheer artistic skill were not enough.
10-3 REALISM LEADS TO MATHEMATICS
There was rather little progress in the second half of the fourteenth century
because the Black Death seriously disturbed the life of Europe and decimated
the population. The fifteenth century witnessed, as we noted in the preceding
chapter, a new flood of Greek works to Italy, a new series of translations, and
enormous support for artists. The Greek ideals became better known and
were discussed enthusiastically in Italy. Secularization was hastened, and the
artist acquired a heightened interest in humanity and in the study of nature,
and a zeal for science.
214 MATHEMATICS AND PAINTING IN THE RENAISSANCE
Fig. 10-4.
Ambrogio Lorenzetti: Presentation in the Temple. Uffizi, Florence.
To achieve an accurate delineation of actual objects and a system of
painting which would yield sound portraiture, painters studied nudes, the body
in various postures, anatomy, expression, light, and color. The Madonna and
Son were portrayed as human beings suffering human emotions, and Church
history was enacted by real people. Religious themes became predominantly
a conventional or habitual outlet for the depiction of the real world. Later,
instead of humanizing religious themes, the artists turned to glorifying man
and nature. The ascetic, mystical, and devotional attitudes were dropped
THE BASIC IDEA OF MATHEMATICAL PERSPECTIVE 215
entirely. Still later pagan subjects were adopted. The glory and gladness of
nature, the delight in physical existence, the beauty of earth, sea, and air were
the new values. Painting became entirely secular.
In their striving for realism the artists went one step further and decided
that their function was to imitate nature, to depict what they saw as realis
rically as they could. Nature was to be the authority for what appeared on
canvas, and painting was to be the science of reproducing nature accurately.
The objective of painting, says Leonardo da Vinci, is to reproduce nature and
the merit of a painting lies in the exactness of the reproduction. Even a purely
imagined scene must appear to the spectator as if it existed exactly as pictured.
Painting was to be a veridical reproduction of reality.
But how was the reproduction to be achieved? Here, too, the Renaissance
artist adopted a Greek ideal. By the fifteenth century he had become thor
oughly familiar and imbued with the Greek doctrine that mathematics is the
essence of the real world. Hence to penetrate to the real substance of the
theme he sought to display on canvas, the Renaissance artist believed that he
must reduce it to its mathematical content. To capture the essence of forms,
the organization of objects in space, and the structure of space the artist
decided that he must find the underlying mathematical laws.
But realistic painting includes more than the mathematical properties of
the objects being portrayed. The eye sees the painting, and this must create
on the eye the same impression as the scene itself. Also, since vision and the
light which carries the scene to the eye are involved, these too must be
analyzed. But the study of light also led to mathematics. From Greek times on,
as we have already seen in earlier chapters, light had been shown to be subject
to mathematical laws. Indeed, the few mathematical laws of light were about
the only precise knowledge about the phenomenon which the Greek and
Renaissance worlds possessed, because the nature of light itself was a mystery.
And so, to study the impress of scene and painting on the eye, the artists were
once again led to mathematics.
Thus, although the artists made extensive and intensive physical studies
of light and shade, color, the chemistry of pigments, the laws of movement
and balance, the eye, anatomy, and the effect of distance on sight, they were
chiefly dominated by the new thought that mathematics must be used to
achieve realism in painting and that geometry is the key to the solution of this
problem. Thereupon they created and perfected a totally new mathematical
system of perspective which enabled them to "place reality on their canvases."
10-4 THE BASIC IDEA OF MATHEMATICAL PERSPECTIVE
The mathematical system of perspective which the Renaissance painters created
and which is known as the system of focused perspective was founded about
1425 by the architect and sculptor Brunelleschi ( 1 l77-1446). His ideas were
furthered and written down by the architect and painter Leone Battista
216 MATHEMATICS AND PAINTING IN THE RENAISSANCE
Alberti (\404-1472) . It is not Alberti's artistic work which entitles him to
fame but his technical knowledge. He studied architecture, painting, perspec
tive, and sculpture, wrote several books explaining theoretical matters to
artists, and exercised enormous influence. In his Della Pittura ( 1435) Alberti
says that learning is essential to the artist. The arts are learned by reason
and method; they are mastered by practice. He says further that the first
necessity of a painter is to know geometry and that painting by incorporating
and revealing the mathematical structure of nature can even improve on
nature.
The mathematical scheme was developed and perfected by Paolo Uccello
( 1 397-1475), Piero della Francesca ( 1416--1492 ) , and Leonardo da Vinci
( 1452-1519). The system these men and others created and which Leonardo
called the rudder and guide rope of painting has been used since the Renais
sance by all artists who seek exact depiction of reality, and is taught in art
schools today.
In their study of light, vision, and the representation of objects on canvas,
these artists discovered the following facts. Suppose that a person looks at a
real scene from a fixed position. Of course, he sees with both eyes, bur each
eye sees the same scene from a slightly different position. Although in ordinary
vision we need both sensations to give us some perception and measure of
depth, this perception is really not very good. Experience teaches us how to
interpret the combined sensations, as Leonardo points out in his Treatise on
Painting. The Renaissance artists decided to concentrate on what one eye
sees and to compensate for the deficiency by shading, shadows where pertinent,
and by what is known as aerial perspective, that is the gradual diminution of
the intensity of colors with distance.·
Let us imagine that lines of light are drawn from one eye to various points
on the objects in the scene. This collection of lines is called a projection. Let
us imagine next, as did Alberti, Leonardo, and the German artist Albrecht
Durer ( 147 1-1528), that a glass screen is interposed between the eye and the
scene itself. Thus when one looks out of a window at a scene outside, the
window serves as the glass screen. The lines of the projection will pierce the
glass screen, and we may imagine a dot placed on the screen where each line
pierces it. The figure formed by these dots on the screen is called a section.
The most important fact which the Renaissance artists discovered is that this
section makes the same impression on the eye as does the scene itself, for all
that the eye sees is light traveling along a straight line from each point on
the object to the eye, and if the light emanates from points on the glass screen
hut travels along the very same lines, it should still create the same impression .
• The difference between a drawing made according to the laws of perspective and
a three-dimensional picture is clear when one views a stereoscopic drawing with
both eyes through colored glasses.
THE BASIC IDEA OF MATHEMATICAL PERSPECTlVE 217
Fig.10-5.
Albrecht Durer: Designer of the Sitting Man.
Fig. 10-6.
Albrecht Durer: Designer of the Lying Woman.
218 MATHEl\lATICS AND PAINTING IN THE RENAISSANCE
Fig. 10-7.
Albrecht DUrer: Designer of the Lute.
Hence this section, which is two-dimensional, is what the artist must place on
the canvas to create the correct impression on the eye. DUrer used the word
"perspective" because the Latin verb from which it is derived means "to see
through."
Before we investigate just how the painter is to put this section on canvas,
let us study the idea of projection and section. Fortunately some woodcuts
made by DUrer, who learned the mathematical system of perspective in Italy
and then returned to Germany to teach it to his countrymen, are very helpful.
The woodcuts are in DUrer's text Underweysung der MesSlmg mit dem Zyrkel
lind Rycbtscbeyed ( 1 5 2 5 ) . The first of these, "The Designer of the Sitting
""an" (Fig. 10-5), shows an artist looking through a glass screen; he holds his
eye at a fixed position, and marks on the screen the point at which a line of light
from his eye to some point on the man's body pierces the screen.
The second woodcut, "The Designer of the Lying Woman" (Fig. 10-6),
shows the artist again holding his eye at a fixed position and noting on paper
the points where the lines of light from his eye to the woman pierce the screen.
To facilitate the process of reproducing the correct location of the dots on
the paper, he has divided screen and paper into little squares.
MATHEMATICAL THEOREMS ON PERSPECTIVE DRAWING 219
The third woodcut, "The Designer o f the Lute" (Fig. 10-7 ) , delineates on
the screen the section which the eye would see if it viewed the lute from the
point on the wall where the rope is attached.
These woodcuts, then. illustrate what the artists meant by a section on a
glass screen. Of course, a section depends upon the position of the glass screen
as wen as on the position of the observer. But this implies no more than that
there can be many different paintings of the same scene. Thus, for example,
two paintings can be the same except for size, and size is determined by the
distance between glass screen and eye. Two paintings may differ in that one
shows a frontal view and the other represents the same scene viewed somewhat
from the side. The difference is due to a change in the observer's position.
10-5 SOME MATHEMATICAL THEOREMS ON PERSPECTIVE DRAWING
Let us accept, then, the principle that the canvas must contain the same section
that a glass screen placed between the eye of the painter and the actual scene
would contain. Since the artist cannot look through his canvas at the actual
scene and may even be painting an imaginary scene, he must have theorems
which tell him how to place his objects on the canvas so that the painting will,
in effect, contain the section made by a glass screen.
Gluss screen
Fig. 10-8.
The image of a line horizontal
and parallel to the screen is
horizontal .
Suppose then that the eye at E (Fig. 10-8) looks at the horizontal line GH
and that GH is parallel to a vertical glass screen. The lines from E to the
points of GH lie in one plane, namely the plane determined by the point E
and the line GH, for a point and a line determine a plane. This plane will cut
the screen in a line. G'H', because two planes which meet at all meet in a line.
It is apparent that the line G'H' must also be horizontal, but we can prove this
fact and so be certain. We can imagine a vertical plane through GH. Since
GH is parallel to the screen and the latter is also vertical, the two planes
must be parallel. The plane determined by E and GH cuts these parallel
planes, and a plane which intersects two parallel planes intersects them in
parallel lines. Hence G'H' is parallel to GH, and since GH is horizontal, so
220 MATHEMATICS AND PAINTING IN THE RENAISSANCE
is G'H'. Bur GH was any horizontal line parallel to the screen. Hence the
image on the screen of any horizontal line parallel to the screen or picture
plane must be horiwntal. Thus in a painting which is to contain what this
glass screen contains, the line G'H' must be drawn horizontally.
We can present practically the same argument to show that the image of
any vertical line, which is automatically parallel to the vertical screen, must
appear on the screen as a vertical line. Thus all vertical lines must be drawn
vertically .
Now let us consider a somewhat more complicated situation. Suppose
that AB and CD (Fig. 10-9) are two parallel, horizontal lines in an actual
scene. Moreover, assume that these lines are perpendicular to the screen. The
eye is at E. If we now imagine that lines go from E to each point of AB, these
lines, that is the projection, will lie in one plane for the point E, and the line
AB will determine this plane by virtue of the theorem of solid geometry
already mentioned. Similarly, E and the line CD determine another plane.
The screen cuts the two planes we have just described, The sections must lie
on the screen, and our problem is to determine where they should lie.
A--/
C
Fig. 10-9.
The images of two horizontal parallel lines which are perpen
dicular to the screen meet at a point on the screen.
Of course, the intersection of two planes is a line, and so the section cor
responding to AB and that corresponding to CD will be lines, A'B' and C'D',
respectively. Moreover, as the eye at E looks farther and farther out along
the parallels AB and CD, the lines of sight will become more and more
horizontal. As the eye follows AB and CD to infinity, so to speak, the lines
from E tend to merge into one horizontal line which will be parallel to AB
and CD. This line from E will pierce the screen at some point, say 0', and
this point corresponds to the imaginary point 0 where AB and CD seem to
MATHEMATICAL THEOREMS ON PERSPECTIVE DRAWING 221
meet at infinity. Of course, AB and CD are parallel and do not meet, but it is
convenient to think of them as meeting at a point at infinity. Indeed, the eye
gets the impression that they do meet. Then the line EO' will be perpendicular
to the screen because it is parallel to AB and CD and these two lines are
perpendicular to the screen. The point 0' corresponds to the imagined meet
ing point at infinity of AB and CD, but because this point does not actually
exist, 0' is called the principal vanishing point. It vanishes in the sense that it
does not correspond to any actual point on AB or CD, whereas other points
on A'B' or C'D' do correspond to actual points on AB or CD, respectively.
Now the lines AB and CD extend out to infinity to the hypothetical
meeting point 0; that is, ABO and CDO are lines in the real scene. The sec
tions of these lines, A'B'O' and C'D'O', must therefore meet at 0'. What we
have shown then is that A'B' and C'D' must be placed on the screen so that
they meet at 0', and 0' is the foot of the perpendicular extending from the
eye to the screen. Let us now note that AB and CD are any horizontal lines
perpendicular to the screen. Hence all horizontal lines which are perpendicu
lar to the screen must be drawn so as to go through 0', the principal vanish
ing point, which is the foot of the perpendicular from the eye to the screen.
We may draw another important conclusion from the preceding situation.
The distances AC and BD are equal, for they are the distances between
parallel lines. However, the corresponding images A'C' and B'D' are not
equal because the lines A'B' and C'D' converge to 0'. Moreover, B'D' will
be shorter than A'C' because it is closer to 0'. But B'D' corresponds to the
actual distance BD which is farther from the screen than A C is. Hence
lengths which are farther from the screen must be drawn shorter than equal
lengths closer to the screen. This fact is often described by the statement
that, to obtain proper perspective in a painting, lengths farther away from
the observer must be foreshortened.
We shall establish one more theorem about perspective drawing. Let us
now suppose that JK (Fig. 10-10) is a horizontal line which makes an angle
of 45" with the screen. Assume that the eye at E looks out along the line JK
toward infinity. Then the line from the eye to the point at infinity on JK
will be parallel to JK. Since JK is horizontal, the new line, EL in Fig. 10-10,
will also be horizontal. It will pierce the screen at some point, say D" and will
also make an angle of 45" with the screen. The triangle D,EO' is a right
triangle because EO' is perpendicular to the screen. In view of the acute
angles of 45", O'D, EO'. Then the point D, is as far from 0' as E is. The
�
projection from E to the various points of JK cuts the screen in some line,
J'K', say. As the eye continues to follow JK toward infinity, the projection
cuts the screen in points lying on an extension of J'K'. and we have already
established that when the eye looks toward infinity on JK, the projection cuts
the screen at D,. Hence J'K' must go through D,. We now have another im
portant result. The image of any horizontal line which makes an angle of 45"
222 MATHEMATICS AND PAINTING IN THE RENAISSANCE
E
K
Fig. 10-10.
The image of a horizontal line which makes a 45°·angle with the screen
goes through a diagonal vanishing point.
with the screen must go through the point DI which lies on the screen, on
the same level as E, but is as far to the right of the principal vanishing point
as E is from the principal vanishing point. The point D, is called a diagonal
vanishing point.
Had we considered instead of JK lines which make an angle of 1 3 5 ° with
the screen, we would have found that their images must go through a point
D, which lies as far to the left of 0' as E is from 0'. The point D, is also
called a diagonal vanishing point.
We see, then, that the points 0', D1, and D2 correspond to points at in
finity in the actual scene. As a matter of fact, all points on the horizontal
line D20'D1 correspond to points at infinity in the actual scene, and this line
is called the vanishing line. It is the image of what one might call the
horizon in the actual scene, that is, the points at infinity toward which the
eye gazes when it looks in a horizontal direction.
The above theorems hardly begin to illustrate what one must know and
apply to draw actual scenes realistically. The treatment of curves is especially
difficult. For example, actual circles and spheres cannot, in general, be drawn
as circles unless their centers happen to lie on the perpendicular from the eye
to the screen. In all other cases, they must be drawn as ellipses or as arcs of
parabolas or hyperbolas, depending upon their position relative to the observer.
This fact becomes clear if one considers that the lines from the eye to each
point on the edge of the circle or sphere, the projection, in other words, form
RENAISSANCE PAINTING: MATHEMATICAL PERSPECfIVE 223
a cone and that the section of this cone on the screen will be one of the conic
sections discussed in Chapter 6. We shall not investigate the more complicated
theorems because to do so would require a course in the subject and because
the detailed theorems are of interest only for the specific purpose of learning
to paint realistically. We may have seen enough of the basic principles to
appreciate that the problem of painting realistically is handled by the applica
tion of a thoroughly mathematical system.
We know that the construction of a painting in accordance with the
focused scheme presupposes a definite fixed position of the painter in relation
to the scene. To view properly a painting so constructed, the observer should
place himself in precisely the position the painter used in planning the painting.
Otherwise the observer will get a distorted view. Strictly speaking, paintings
In museums should be hung so that the observer can conveniently take that
position.
1� RENAISSANCE PAINTINGS EMPLOYING MATHEMATICAL PERSPECTIVE
Renaissance painters achieved their goal of devising a mathematical system
which permitted the realistic representation of actual scenes and joyously
hastened to employ it. Realistic paintings constructed in accordance with
the focused scheme of perspective begin to appear about 1430.
The artist who contributed key principles of mathematically determined
perspective, including new methods of construction. and who was the best
mathematician of his times is Piero della Francesca. This highly intellectual
painter with a passion for geometry planned all his works mathematically to
the last detail. Each scene to be painted was a mathematical problem. The
placement of each figure was calculated to ensure its correctness in relation to
other figures and to the painting as a whole. He loved geometrical forms so
much that he used them for hats, parts of the body, and other details in his
paintings. Piero practically identified painting and perspective. His De pros
petth'Q pingendi, a treatise on painting and perspective in which he uses
Euclid's deductive method, presents perspective as a science and provides
sample constructions illustrating how perspective problems are to be handled.
Though incidental to our purposes, it is worth noting that Piero painted the
first Renaissance portraits of real people, the Duke and Duchess of Urbina,
Federigo de Montefeltro and his wife Battista Sforza.
There are numerous examples which illustrate Piero's excellent perspec
tive. His "Flagellation" (Fig. 10-1 1 ) is one of the best. As in all of his paint
ings a geometric framework underlies the design. The principal vanishing
point is chosen to be near the figure of Christ. This device of placing the
principal vanishing point within the most important area in the painting is
deliberate because the eye tends to focus on that vanishing point. All objects
are carefully foreshortened; this is especially noticeable in the marble blocks on
224 MATHEMATICS AND PAINTING IN THE RENAISSANCE
Fig. 10-11.
Piero della Francesca: The Flagellation. Ducal Palace. Urbino.
Fig. 10-12.
Piero della Francesca: Architectural View of a City. Kaiser Friedrich Museum. Berl in.
RENAISSANCE PAINTING: MATHEMATICAL PERSPECTIVE 225
Fig. 10-13.
Leonardo d a Vinci : Study for the Adoration o f the Magi. Uffizi. Florence.
the floor and in the beams. The immense labor which went into the calculation
of these sizes is indicated by a drawing in the book referred to above wherein
he explains a similar construction.
Piero achieves unity of the various parts by means of the system of
perspective. All parts are mathematically tied together to produce this syn
thesis. Indeed, it was somewhat because of this effect that the Renaissance
painters valued the system and were excited about it. The example shown
here should be compared with the foun.enth-century works (Section 10-2),
where unity is lacking. The entire layout of Piero's painting is so carefully
planned that movement is sacrificed to the unity of design.
To illustrate the power of perspective Piero painted several scenes of cities.
His "Architectural View of a City" (Fig. 10-12) gives a striking illusion of
depth. These examples of Piero's paintings show his obsession for perspective
and his great technique.
Leonardo da Vinci's work provides excellent examples of paintings em
bodying mathematical perspective. Leonardo prepared for painting by deep
and extensive studies in anatomy, perspective, geometry, physics, and chem
istry. In his Treatise on Painting, a scientific treatise on painting and perspec
tive, Leonardo gives his views. He opens with the statement, "Let no one who
is not a mathematician read my works." Painting, he says, is a science which
should be founded on the study of nature and, like all sciences, must also be
based on mathematics. He scorns those who think they can ignore theory and
oy mere practice pro-iuce art: "Practice must be founded on sound theory."
226 MATHEMATICS AND PAINTING IN THE RENAISSANCE
Fig. 10-14.
Leonardo da Vinci: Adoration of the Magi. Uffizi. Florence.
Fig. 10-15.
Leonardo da Vinci: Annunciation. Uffizi. Florence.
RENAISSANCE PAINTING: MATHEMATICAL PERSPt:CTlVE 227
Painting, which he regarded as superior to architecture, music, and poetry.
is a science because it deals with the geometry of surfaces.
The detailed mathematical studies which Leonardo undertook in prepara
tion for his paintings are illustrated by one of several sketches he made for
his "Adoration of the Magi" (Fig. 10- 1 3 ) . The painting itself, which was
never completed, is shown in Fig. 10-14. His "Last Supper" is another excellent
example of mathematical perspective, but is so well known that we shall
reproduce instead "The Annunciation" (Fig. 10-1 5 ) . Although the action
takes place in the foreground and the chief figures are far apart, they and the
distant scene in the rear are all brought together by the perspective structure.
Raphael ( 1483-1520) supplies many superb paintings which exhibit ex
cellent perspective. In his "School of Athens" (Fig. 10-16) he boldly tackles
an enormous scene encompassing a vast number of people within a magnifi
cent architectural setting. The portrayal of depth, the harmonious organiza
tion, coherence, and exactness of proportions achieved despite the difficulty
of the undertaking are extraordinary. This picture, especially, shows how
perspective unifies a composition and ties figures at the sides to the central
theme.
The history of this painting is of interest. Pope Julius II ( 1443-1 5 1 3) was
impressed with ancient learning and regarded Christianity as the climax of
Fig. 10-16.
Raphael : School of Athens. Vatican.
228 MATHEMATlCS AND PAINTING IN THE RENAISSANCE
Flg. 1G-17.
Raphael: Fire in the 8orgo. Vatican.
Jewish religious thought and Greek philosophy. He wished to have his idea
embodied in paintings and commissioned both Michelangelo and Raphael to
develop this theme. Michelangelo treated it in his frescoes on the ceiling of
the Sistine Chapel, where he shows the human race led to Christ through a
long line of Jewish prophets and pagan sibyls. Raphael executed the same
theme in a somewhat different manner. In four frescoes which cover the
walls of the Pope's principal official room, the Camera della Segnatura, he
teaches that the human soul is to aspire to God through each of its faculties:
reason, the artistic capacity, the sense of order and good government. and
the religious spirit. "The School of Athens" glorifies reason and naturally
exhibits the people who excelled in the intellectual sphere. Plato and Aristotle
are the central figures. Plato points upward to the eternal ideas and Aristotle
down to the earth as the field of experience. At Plato's left is Socrates. In
the left foreground Pythagoras writes in a book. The right foreground shows
the bald-headed Euclid; Archimedes stoops to demonstrate a theorem; Ptolem)'
holds up a sphere. All the way to the right is Raphael himself.
Raphael offers so many examples of excellent perspective that it is difficult
to limit oneself to one or two representative samples. His "The Fire in the
OTHER VALUES OF MATHEMATICAL PERSPECTIVE 229
Borgo" «Fig. 10-- 1 7) shows exquisite depth, perfect handling of figures in
various positions, the proper foreshortening, and again the unification of a
scene in which many actions take place.
10-7 OTHER VALUES or MATHEMATICAL PERSPECTIVE
We could offer countless examples illustrating the application of the mathemat
ical system of perspective by the Renaissance masters.' All painters of this
period in which western European art reached one of its pinnacles employed it
and employed it well. Many, among them Uccello and Piero, were obsessed
by it and painted scenes viewed from unusual positions just to solve the
mathematical problems involved. The essential difference between the art of
the Renaissance and that of the Middle Ages is the introduction of the third
dimension, and Renaissance painting is characterized by the importance at
tached to realism, to the realistic rendering of space, distance, and forms,
achieved by means of the mathematical system of perspective. Through it,
the process of seeing was rationalized; the extended world was brought under
control; and the rational interests of the painters were satisfied.
In the history of culture the accomplishments of the Renaissance artists
have broad significance. Their apparent goal was to gaze at nature and to
depict what they saw on canvas, but their true, more profound objective
was to uncover the very secrets of nature. The Renaissance artist was a
scientist, and painting was a science not merely in the sense that it had a
highly technical and even mathematical content, but because it was inspired
by the ultimate goal of science, understanding nature. Art and science are
never separated in the thinking and work of Ghiberti, Alberti and Leonardo,
for example. Leonardo's Paragone, A Comparison of the Arts ( Treatise on
Painting) contains a chapter on "Painting and Science" in which he asserts
that painting seeks the truths of nature. The artist of that period regarded
himself as the servant of science. These men who explored and represented
nature with methods peculiar to their art were motivated precisely by the
spirit and objectives of the scientists who studied astronomy, light, motion,
and other phenomena. They were in fact the forerunners in spirit and goals
of the great physical scientists of modern times and they revealed truth in
a form which means more to many people than the deep and intricate analyses
of modern mathematical physics. That mathematics proved to be the founda
tion of painting and thereby enabled painting to reveal the structure of nature
was no more than fitting, for the Greeks had already shown that mathematics
was the essence of design, and later scientists were to confirm this fact in ever
more striking fashion.
• The same theme is treated in the author's Mathematics in Western Culture. Other
examples can be found there.
230 MATHEMATICS AND PAINTING IN THE RF.NAISSANCl-:
The works of the Renaissance artists are hung in art museums. They
could, with as much justification, be hung in science museums. The lover of
Renaissance art is consciously or unconsciously a lover of science and mathe
matics.
EXERCISES
1. Relate the "back to nature" movement of the Renaissance to the development of
a mathematical system of perspective.
2. Distinguish between conceptual and optical systems of perspective.
3. Which artists did most to create a mathematical system of perspective?
4. What is the principle of projection and section in the theory of perspective?
5. Draw the rear wall, and the visible portions of the side walls, ceiling, and floor
of a room as seen by an observer in the room whose eye is looking directly at
the rear wall.
6. Add to the drawing of the preceding exercise a square table, two of whose edges
are parallel to the rear wall.
7. Draw a cube positioned in such a way that one edge is closest to you and that
neighboring edges make angles of 45° and 1350, respectively, with the canvas.
Go as far as you can with the theorems at your disposal.
8. State three theorems of the geometry of perspective drawing.
Topics for Further Investigation
I. The influence of mathematics on Renaissance painting.
2. Theories of the artists on human proportions. Use the reference to Panofsky's
Meaning in the Visual Arts.
3. Vision and painting. Use the reference to Helmholtz.
Recommended Reading
BLUNT, ANTHONY: Artistic Theory in Italy, Oxford University Press, London, 1940.
BUN 1M, MIRIAM: Space ill Medieval Painting and the Forerunners of Perspective,
Columbia University Press, New York, 1940.
CLARK, KENNETH: Piero della Francesca, Oxford University Press, New York, 1951.
COLE, REX V.: Perspective, Seeley, Service and Co., Ltd., London, 1927.
COOLIDGE, JULIAN L.: Mathematics of Great Amateurs, Dover Publications, Inc.,
New York, 1963.
DA VINCI, LEONARDO: Treatise on Painting, Princeton University Press, Princeton,
1956.
FRY, ROGER: Visioll alld DesiJ.{n, pp. 1 1 2-168, Penguin Books Ltd., Baltimore, 1937.
OTHER VALUES OF MATHEMATICAL PERSPECTIVE 231
HELMHOLTZ, HERMAN VON: Popular Lectures on Scientific Subjects, pp. 250-286,
Dover Publications, Inc., New York, 1962.
IVINS, WM. M., JR.: Art and Geometry, Dover Publications, Inc., New York, 1964.
JOHNSON , MARTIN: Art and Scientific Thought, Pan Four, Faber and Faber, Ltd.,
London, 1944.
KLINE, MORRIS: Mathematics in Western Culture, Chap. to, Oxford University
Press, New York, 1953.
LAWSON, PHILIP J.: Practical Perspective Drawing, McGraw-Hill Book Co., Inc.,
New York, 1943.
PANOFSKY, ERWIN: "Durer as a Mathematician," pp. 603--621 of James R. Newman:
The World of Mathematics, Simon and Schuster, New York, 1956.
PANOFSKY, ERWIN: Meaning in the Visual Arts, Chap. 6, Doubleday Anchor Books,
New York, 1955.
POPE-HENNESSY, JOHN: The Complete Work of Paolo Uccello, Phaidon Press,
London, 1950.
PORTER, A. T.: The Principles of Perspective, University of London Press Ltd.,
London, 192 7.
VASARI, GIORGIO: Lives of the Most Famous Painters, Sculptors and Architects, E. P.
Dunon, New York, 1927, and many other editions.
CHAPTER 1 1
PROJECTIVE GEOMETRY
The gods did not reveal all things to men at the start; but as time goes on,
by searching, they discover more and more.
XENOPHANES
11-1 THE PROBLEM SUGGESTED BY PROJECTION AND SECTION
The origins of mathematical ideas are far more novel and surprising than is
commonly believed. Practical and scientific problems no doubt most fre
quently suggest new areas of exploration. However, not only are there other
sources, hut these often give rise to major branches of mathematics, some of
which become valuable tools in scientific and practical endeavors. The ques
tions raised by the painters during their work on the mathematics of perspec
tive caused them and. later, professional mathematicians to develop the subject
known as projective geometry. This subject, the most original creation of
the seventeenth century, is now one of the principal branches of mathematics.
Let us see just what problems led to this development. The basic mathe
matical concepts in the system of focused perspective are projection and sec
tion. A projection we may recall, is a set of lines of light from the eye to the
points of an object or scene; a section is the pattern formed by the intersection
of these lines with a glass screen placed between the eye and the object viewed.
A ___
C
Fig. 11-l.
The section of the projection
of a horizontal square.
232
PROBLEM SUGGESTED BY PRO]ECfION AND SECfION 233
Let us consider, as an example, the section o f the projection o f a square.
Suppose that the square is horizontal (Fig. I I-I) and is viewed from a point
somewhere above the level of the plane in which it lies. Furthermore, let us
suppose that a vertical glass screen is placed paraIlel to the front and back edges
of the square. We know from our work on perspective that the section on the
screen of the two sides of the square which are perpendicular to the glass
screen will consist of two line segments which tend to meet in the principal
vanishing point. That is, the extensions of A'B' and C'D' meet at 0'. Since the
sides AC and BD of the square are paraIle! to the screen, they give rise to the
paraIlel sections A'C' and B'D'.
The section A'B'D'C' is not a square because the sides A'B' and C'D' are
not paraIle!. Nor is it a rectangle. The angles, then, of the section do not
equal the corresponding angles of the original figure. The size of the section
clearly depends upon where the glass screen is placed. Hence neither the
lengths of the sides of the section nor the area equals the corresponding
quantities in the original figure. In the language of Euclidean geometry, we
may say that the section is neither congruent, similar, nor equivalent to the
original figure. But the section does create the same impression on the eye as
the original figure does. Hence it should possess some properties in common
with the origina!. The question then becomes, What geometrical properties
do the section and original figure have in common?
It is a natural step from this question to the next one. If two observers
view the same scene from different positions, two different projections are
formed (Fig. 1 1-2). If a section of each of these projections is made, then,
in view of the fact that the sections are determined by and suggest the same
scene, the sections should possess common geometrical properties. What are
they?
D C
Glass screen
Glass screen E"
Fig. 11-2.
E' The different sections of two different projections of a square.
2 J4 PROJECTIVE GEOMETRY
These questions, which were originally raised by the artists, could occur
to anybody who thinks about what he sees. Suppose that an observer looks
at a rectangular picture frame from various positions. The figure he actually
perceives varies with his position. Thus he would be led to ask the same
question that we raised above, What properties do these various shapes have
in common? Again, as a man walks along a street and near a street lamp, his
shadow changes in size and shape. The projection here consists of the lines
of light from the street lamp, which takes the place of the eye, extending to
the outlines of the man's body and then continuing to the ground. The section
made by the plane of the ground is the shadow, although in this case the lamp
and the section are on opposite sides of the actual object. The mathematical
question raised by this example is, What properties do the various shadows
have in common?
Another familiar, contemporary example illustrating projection and sec
tion is the photograph. In this case, the eye is the lens of the camera, and
rays of light proceed from the scene through the lens to the film. The section
is made by the film. We know that different sections can be obtained by
placing the camera closer or farther from the scene being photographed or
by tilting the camera. Yet all photographs of the same scene should contain
some common geometrical properties.
The srudy of properties common to a figure and a section of a projection
of that figure, or to two sections of the same projection, or to two sections
of two different projections of the same figure has led to new concepts and
theorems which today comprise a whole new branch of geometry, namely
projective geometry. We shall attempt in this chapter to gain some under
standing of the nature of this subject.
1 1-2 THE WORK OF DESARGUES
If we study any section of the projection of a figure and consider its relation
to the original figure, then a few facts are readily noted. Mathematically the
eye is a point. and this point and any line in the actual figure determine a plane,
which is the projection determined by the line. The glass screen which cuts
the projection is also a plane, and since two nonparallel planes meet in a line,
the section is a line. Hence corresponding to a line in the actual scene there is
a line in the section. We may therefore say that the property of linearity is
common to an actual line and any section of a projection of that line. Similarly,
it is easy to visualize that two intersecting lines of the actual figure will gen
erate two intersecting lines in the section. This, then, is another minor mathe
matical property which is common to actual object and section. It follows
that a triangle will give rise to a triangular section, although the shape of the
triangle in the section will not necessarily be the same as that of the original
triangle. Likewise a quadrilateral will correspond to a quadrilateral.
THE ·WORK OF DESARGUES 235
But the discovery of these few properties which a figure and its section
or two sections of the same projection have in common hardly elates one nor
does it give us any significant answers to the question of what properties
figures and sections or various sections have in common. The first man to
explore the problem and come up with nontrivial answers was Girard
Desargues ( 1593-1662 ). Desargues was not a professional mathematician; he
was a self-educated architect and engineer. His motive in tackling the subject
was to help his colleagues. He believed that he could compile in compact
form the many theorems on perspective that were useful to architects, engi
neers, painters, and stone-cutters. He even invented a special terminology
which he thought would be more comprehensible to craftsmen and artists than
the usual mathematical language. Of his motives Desargues writes:
I freely confess that I never had taste for study or research either in physics or
geometry except in JO far as they could serve as a means of arriving at some sort
of knowledge of the proximate causes . . . for the good and convenience of life,
in maintaining health, in the practice of some art, . . . having observed that a
good part of the arts is based on geometry, among others the cutting of stone in
architecture, that of sundials, that of perspective in particular.
Desargues began by organizing numerous theorems and published one book
on the construction of sundials and another on the application of his own
geometric theories to stone-cutting and masonry. He lectured in Paris about
1626, and wrote a pamphlet on perspective ten years later. Desargues' chief
contribution, a book on projective geometry, appeared in 1639.
The basic theorem of projective geometry, a theorem now fundamental in
the entire field of mathematics, was formulated and proved by Desargues and
is named after him. It illustrates how mathematicians responded to the ques
tions raised by perspective.
Suppose the eye at point 0 (Fig. 1 1-3) looks at a triangle ABC. The
lines from 0 to the various points on the sides of the triangle constitute, as
we know, a projection. A section of this projection will then contain a
triangle, A'B'C', where A ' corresponds to A, B' to B, and C' to C. Alterna
tively, we may regard both triangles as sections of a projection of a third
triangle. The two triangles, ABC and A'We', are said to be perspective from
the point O. Desargues' theorem states an important geometrical fact which
relates triangles ABC and A'B'C'. If we prolong the corresponding sides AC
and A'e', they will meet in a point P; the sides AB and A'B' extended will
meet in a point Q; and the extensions of BC and We' will meet in a point R.
Then, the theorem asserts, P, Q, and R will lie on a straight line. In more
compact language, the theorem says:
If two triangles in different planes are perspective from a point, the three pairs
of corresponding sides meet in three points which lie on one straight line.
236 PROJECTIVE GEOMETRY
The proof of this theorem is simple. The lines AC and A'C' lie in one plane
because OAA' and OCC' are two intersecting lines and, as such, determine a
plane. Then AC and A 'C' will meet in a point because any two lines in one
pbne meet in a point.' Let us denote this point by P. The same argument
shows that AB and A'B' meet in a point Q, and that BC and B'C' meet in a
point R.
\
\
\
\
, \
I \
I \
I \
\
0 ...;:=---
A
Fig. 11-3.
Desargues' theorem.
We now wish to show that P, Q, and R lie on one line. Now P, Q, and R
lie in the plane of triangle ABC because P lies on AC, Q on AB, and R on
Be. Likewise P, Q, and R lie in the plane of triangle A'B'C'. Now the
points common to two planes must lie on one line, the line of intersection of
the two planes. Hence, P, Q, and R lie on one line.
The reader may be troubled about the assertion in Desargues' theorem
that each pair of corresponding sides of the two triangles must meet in a
point. He may ask, If these sides happen to be parallel, does the theorem
fail' Desargues had taken account of this possibility. We observed in the
preceding chapter that a set of lines which are parallel in a particular scene
being painted may have to be drawn on the canvas so as to meet in a point.
In this case, there is a point in the section which does not correspond to any
point in the scene itself. This breakdown of the correspondence between
points in the scene and points in the section can be repaired by agreeing that
any set of parallel lines is to be regarded as having a point in common. Where
is this point? The answer is that it cannot be visualized. although the student
is often advised to think of it as being at infinity, a bit of advice which es-
• We shall neglect for the moment the special case of twO parallel lines.
THE WORK OF DESARGUES 237
sentially amounts to answering a question b y not answering it. However,
whether or not one can visualize the point common to parallel lines, a point
distinct from the usual, finitely located points of the lines. it is convenient to
say that they have a point in common. In addition. it is agreed that two or
more parallel lines are to have just one point in common as do nonparallel lines.
Hence we say of any two lines in projective geometry ·that they meet in one
and only one point. This agreement is further recommended by the argument
that projective geometry is concerned with problems which arise from the
phenomenon of vision, and we never see parallel lines, as the familiar example
of apparently converging railroad tracks illustrates.
One more agreement must be made about these new points introduced in
projective geometry. We just agreed that any set of parallel lines has one
point in common. Since there are many sets of parallel lines in one plane,
each set having its own direction, there are many such new points in the
plane of projective geometry. It is agreed that all these new points lie on one
new line, sometimes called the line at infinity.
Let us now turn back to Desargues' theorem. If each of the three pairs
of corresponding sides of the triangles presented in Desargues' theorem con
sisted of parallel lines, it would follow from our agreements that the two
lines of each pair intersect in one point and that the three points of inter
section lie on one line, the line at infinity. These conventions or agreements
about points and a line at infinity obviate the necessity for making special
statements when parallel lines happen to be involved in the theorems. This
is as it should be, because the property of parallelism plays no role in pro
jective geometry as opposed to Euclidean geometry. The reader who balks
at accepting the agreements about parallel lines may nevertheless accept the
theorems of projective geometry, with the mental reservation that they fail
to state the truth when parallel lines are involved.
Desargues discovered another fundamental property which is common to
a figure and to a section of a projection of that figure. Let us consider the
figure consisting of the line I on which any four points are selected and
designated A, B, C, and D (Fig. 1 1-4). We may form a projection of this
figure from an arbitrary point 0 and cut this projection with the usual glass
Fig. 1 1-4.
A section of a projection of
four points on a line.
238 PROJECfIVE GF.OMETRY
screen to obtain a section. The original figure and the point 0 determine a
plane. Then the section consists of the line I' on which A' corresponds to
A , B' to B, and so on. Since length in the figure differs from length in a section
or, to use more technical phraseology, length is not invariant under projection
and section, we should not expect that A'B' would equal AB, or that any
segment on I would equal the corresponding segment on 1'. One might next
consider the ratio CAICB and venture that perhaps this ratio would equal the
corresponding ratio C'A' je'H', This conjecture is not correct. However,
one can prove the surprising fact, namely that
CA/CB C'A'/C'B'
-
DA/DB D'A'/D'B'
Thus this ratio of ratios, or cross ratio as it is called, is a projective invariant.
This is a very surprising fact. It does not matter where the points A , B, C,
and D lie on the line I or which points are labeled A , B, C, or D. The cross
ratio of the lengths they determine and the cross ratio of the corresponding
lengths in the section will be the same.
Incidentally the fact that the cross ratio of four points on a line is the
same in the section as in the original figure permits us to check the correctness
of a painting executed in accordance with the system of focused perspective.
If four points A', B', e', D' in the painting correspond to four points A, H, C,
D which lie on one line in the original scene, then the cross ratio of the first
set must equal the cross ratio of the second set. This fact, however, is not roo
useful in constructing the painting itself.
EXERCISES
I. What fact or facts may you assert about:
a) a section of a projection of an equilateral triangle;
b ) a section of a projection of a square?
2 . Why should one expect that a figure and a section of a projection of that figure
should possess some geometrical properties in common?
B
c
A' A Fig. 11-5
3. Figure 1 1-5 shows two triangles lying in the same plane, the plane of the paper.
Moreover, these two triangles are perspective from the point O.
a) What is the difference between this figure and the one considered in the text?
THE WORK OF PASCAL 239
b ) Verify b y actually drawing lines that the pairs of corresponding sides o f the
two triangles in Fig. 1 1-5 meet in three points which lie on one straight line.
c) What generalization of Desargues' theorem is suggested by the result in (b)?
2 3
•
A B c D Fig. 11-6
4. Given the points and lengths in Fig. 1 1--6, what is the cross ratio of the lengths
determined by A, B, C, and D?
5. What is the meaning of the statement that a geometrical property is invariant
under projection and section?
A c B D Fig. 1 1-7
6. In projective geometry, lengths are sometimes regarded as directed. For the
positions shown in Fig. 1 1-7, CA would be taken to be negative, whereas CB is
positive because it is in the opposite direction. Then the cross ratio
CAICB
DAIDB
in the present case is negative. When the cross ratio is -1, the four points A, B,
C, D are said to form a har11lonic set. Suppose now that D is moved indefinitely
far to the right and C is moved so as to keep the cross ratio I Can you discover
-
.
any special property of the point C in relation to A and B?
11-3 THE WORK OF PASCAL
In the domain of projective geometry, Desargues' ideas were further advanced
by Blaise Pascal ( 1623-1662 ) . This man of many contradictory qualities, beset
by deep emotional conflicts, is commended to us also by his superb original
work in other branches of mathematics, physics, literature, and theology. His
father, a judge and tax commissioner, recognized that his son was bright and
guided his education. He decided that Blaise should not tackle mathematics
until he was 16 years old, but somehow the boy got started on his own and
learned a good deal quickly.
When Pascal was still a youth in Paris his father tOok him to the weekly
sessions of a group of noted intellectuals, Roberval, Mersenne, Mydorge, and
others. There Blaise met Desargues and as a result became interested in study
ing properties of geometric figures which remain invariant under projection
and section. At the age of 16 he proved a famous theorem, still called Pascal's
theorem, which we shall examine shortly. He then wrote the Essay 011 Conic
Sections, which contains many original results. Mathematics became his great
240 PROJECTIVE GEOMETRY
passion. To aid his father he conceived the idea of having a machine perform
arithmetic operations, and he constructed the first successful computing ma
chine. He also was one of the notable precursors of Newton and Leibniz in
the creation of the calculus and, together with Pierre de Fermat, another great
French mathematician whom we shall meet later, founded the theory of
probability . •
Pascal made some famous physical experiments which confirmed the
discovery by Evangelista Torricelli, a pupil of Galileo, that the air presses
down upon us, or that the air has weight, and he also clarified and furthered
the study of pressure in liquids, a study which, in technical parlance, is known
as hydrostatics.
It was clear to Pascal that the data of experience are the starting points of
knowledge and he respected and superbly exercised the power of reason. His
Spirit of Geometry, an essay on method and rules of thought, may well be
classed with Descartes' Discourse 011 Method, another landmark on the role
of reason. But as he grew older, he became more and more dissatisfied with
the limited results attained by reason. About ten years before he died, he
began to find emptiness in the knowledge of nature and acquired some distaste
for it. "Don't overrate science," he cautioned. He became convinced that the
truths of mathematics were not broad enough to encompass all of man's world.
He would frequently say that all the sciences could not comfort one in days
of affliction, but that the doctrines of Christian truth would comfort one at all
times both in affliction and in one's ignorance of these sciences. Famous are
his epigrams, "The heart has its reasons which the reason knows nothing of"
and "Nothing that has to do with faith can be the concern of reason." More
and more he turned to religion. He had been brought up as a Catholic, but
he would not accept the strict dogmatic theology of the powerful Jesuits. He
became a Jansenist and his Provincial Letters, one of his famous literary works.
is filled with anti-Jesuitical polemics. In his Penst!es, another literary classic,
he penned many more of his thoughts on religion. The conflict between
science and faith ended with a victory for religion. Ironically, Pascal, the de
fender of faith, helped immensely to found the ensuing Age of Reason.
Typical of theorems in projective geometry is the one conceived and
proved by Pascal. This theorem, like Desargues', states the property of a
geometrical figure which is common to the figure and to any section of any
projection of that figure; that is, it states a property of a geometrical figure
which is invariant under projection and section.
Pascal has this to say: Draw any six-sided polygon (hexagon) inscribed
in a circle and letter the vertices A , B, C, D, E, F (Fig. 1 1-8). Prolong a pair
of opposite sides, AB and DE say, until they meet in a point, P. Extend an
other pair of opposite sides, AF and CD say, until they meet in a point, Q.
• See Chapter 23.
THE WORK OF PASCAL 241
Finally, prolong the third pair until they meet in a point, R. Then, Pascal
asserts, P, Q, and R will always lie on one straight line. The mathematician,
with his usual passion for brevity, says:
If a hexagon is inscribed in a circle, the pairs of opposite sides intersect in three
points which lie on one straight line.
We shall not give the proof of this theorem because it would require more
time than we should devote to the subject. The statement, however, offers
another illustration of the type of theorem investigated in projective geom
etry.
p Il
9
- -- //
o
'-
� , --.....
_-
, /
, /
, /
, /
,
,
/
/
,\ C
Fig. 11-8.
Pascal's theorem. B
F E
c
Fig. 11-9.
A section of a projection of a
hexagon inscribed in a circle. B
As stated, Pascal's theorem seems to have little to do with properties com
mon to all sections of a projection. However, let us visualize (Fig. 1 1-9) a
projection of the figure involved in Pascal's theorem and a section of this
projection. The projection of the circle is a cone, and a section of this cone
will not necessarily be a circle, but, as we know from the work of the Greeks,
it may be an ellipse, a parabola, or a hyperbola, that is, a conic section. To
each side of the hexagon inscribed in the circle there corresponds a side of the
hexagon inscribed in this conic section, and to each intersection of lines in
the original figure there belongs an intersection of the corresponding lines in the
section. Finally, since the points P, Q, R lie on one line in the original figure,
the corresponding points will be on one line in the section. Hence Pascal's
theorem states a property of a circle which continues to hold in any section
of any projection of that circle.
242 PROJECTIVE GEOMETRY
EXERCISES
1. Draw a circle, choose any six points on it as the vertices of a hexagon, find the
points of intersection of the three pairs of opposjte sides, and see whether they
lie on a straight line.
2. Draw any two straight lines. Choose three points on one and label them A, B, C.
Choose three points on the second line and label them A', B', C'. Find the point
of intersection of AB' and A'B; do the same for AC' and A'C, and for BC' and
B'e. Do you observe any interesting fact about these three points of intersection?
11-4 THE PRINCIPLE OF DUALITY
It would be pleasant to relate that the innovations of Desargues and Pascal
were immediately appreciated by their fellow mathematicians and that the
potentialities in their methods and ideas were eagerly seized upon and further
developed. Actually this pleasure is denied to us. Perhaps Desargues' novel
terminology batRed his contemporaries just as many people today are batRed
and repulsed by the language of mathematics. At any rate, except for Des
cartes, Pascal, and Fermat. Desargues' colleagues exhibited the usual reactions
to radical ideas: they dismissed them, called Desargues crazy, and forgot pro
jective geometry. Desargues himself became discouraged and returned to the
practice of architecture and engineering. Every printed copy of Desargues'
book, originally published in 1639, was lost. Pascal's work on conics and his
other studies on projective geometry, though published in 1640, also remained
unknown until almost 1 800. Fortunately, a pupil of Desargues, Philippe de
La Hire, made a manuscript copy of Desargues' book which was accidentally
picked up in a bookshop by the nineteenth-century geometer Michel Chasles,
and thus the world finally learned the full extent of Desargues' major work.
Apart from some results which La Hire used and which were incorrectly
credited to him for 1 50 years, Desargues' and Pascal's discoveries had to be
remade one by one by the nineteenth-century geometers.
Another reason for the neglect of projective geometry during the seven
teenth and eighteenth centuries was that analytic geometry, created by Desar
gues' contemporaries, Descartes and Fermat (see Chapter 1 2 ) , and the calculus,
developed chiefly by Newton and Leibniz during the latter half of the seven
teenth century (see Chapters 16 and 1 7 ) , proved to be so useful in the rapidly
expanding branches of physical science that mathematicians concentrated on
these subjects.
The study of projective geometry was revived through a series of accidents
and events almost as striking as those which had first stimulated interest in this
discipline. The problem of designing fortifications attracted the geometrical
talents of Gaspard Monge ( 1 746-1 8 1 8 ) , the inventor of descriptive geometry.
It is relevant that this subject, .though distinct from projective geometry, uses
projection and section. Monge was a most inspiring teacher, and there gathered
THE PRINClPLE OF DUALITY 243
about him a host of bright pupils, among them Charles J. Brianchon ( 1 785-
1 864), L. N. M. Carnot ( 1753-1823 ) , and Jean Victor Poncelet ( 1788-1867).
These men were so impressed by Monge's geometry that they sought to show
that geometric methods could accomplish as much and more than the algebraic
or analytical methods of treating geometry introduced by Descartes. Carnot
in particular wished "to free geometry from the hieroglyphics of analysis." As
if to take revenge on Descartes whose creation had caused the abandonment
of pure geometry, the early nineteenth-century geometers made it their ob
jective to outdo Descartes.
The revival of projective geometry was launched dramatically by Poncelet.
While serving as an officer in Napoleon's army, he was captured during the
campaign in Russia and spent the year 1 8 1 3-1814 in a Russian prison. There
Poncelet reco'nstructed without the aid of any books all he had learned from
Monge and Carnot and then proceeded to create new results in projective
geometry. He was the first mathematician to appreciate fully that this subject
was indeed a totally new branch of mathematics, and he consciously sought
properties of geometrical figures which were common to all sections of any
projection of a given figure. A group of French and, later, a group of German
mathematicians continued Poncelet's work and developed intensively the sub
ject of projective geometry.
B c D
L
(a)
d
a
Fig. 11-10. c
(a) A set of points on a line; b
(b) a set of lines on a point. (h)
The many accomplishments of this period were capped by the discovery
of one of the most beautiful principles in all mathematics-the principle of
duality. It is true in projective geometry, as in Euclidean geometry, that any
two points determine one line or, as we now prefer to put it, any two points
lie on one line. But in projective geometry it is also true that any two lines
determine, or lie on, one point. (The reader who has refused to accept the
convention that parallel lines in Euclid's sense are also to be regarded as having
a point in common will have to forego the next few paragraphs and pay for
his stubbornness.) It will be noted that the second italicized statement can be
obtained from the first one by merely interchanging the words point and line.
We say in projective geometry that we have dualized the original statement
or that one is the dual of the other. If we are discussing a set of points on a
line and interchange "point" and "line," we obtain the phrase a set of lines on
a point. Figure 1 1-10 illustrates the two dual statements.
244 PROJECfIVE GEOMETRY
A triangle consists of three points not all on the same line and the lines
joining them. We could speak of three lines 110t all on the same point and the
points joining them. We usually do not speak of a point as joining two lines;
rather, we refer to such a point as the point of intersection of the lines. But
the meaning is clear either way. The figure described by the rephrased or
dualized Statement is again a triangle. Because the dual figure of the triangle
is a triangle, the triangle is called self-dual.
In projective geometry the quadrilateral is defined as a figure consistng
of four lines and the six points in which the lines join in pairs. This use of
the term "quadrilateral" differs slightly from the one common in Euclidean
geometry. A picture of a quadrilateral is shown in Fig. I I-I I (a). We can
equally well speak of a figure consisting of four points and the six lines which
join the points in pairs (Fig. I I- l i b ) . This new figure is called a quadrangle.
Hence quadrilateral and quadrangle are dual figures.
Fig. 11-11.
(a) A quadrilatera l ;
(a) (b)
(b) a quadrangle.
We seem to be able to take the statement describing any figure and by
dualizing the statement obtain a new figure. Let us try next something more
ambitious. We shall dualize Desargues' theorem. We shall consider the case
where the two triangles and the point 0 from which the triangles are per
spective all lie in one plane, and see what results when we interchange point
and line. We shall use the fact already noted that the dual of a triangle is a
triangle.
Desargues' Theorem Dual of Desargues' Theorem
If we have two triangles If we have two triangles
such that lines joining such that points which join
corresponding vertices corresponding sides
lie on one point, 0, lie on one line, 0,
then corresponding sides then corresponding vertices
join in three points are joined by three lines
which lie on one straight line. which lie on one point.
If we examine the new statement we see that it is actually the converse of
Desargues' theorem; that is, the hypothesis and the conclusion in Desargues'
theorem are now interchanged. Hence by interchanging point and line we
THE PRINCIPLE OF DUALITY 245
have discovered a possible theorem. It would be too much to ask that the
proof of the new statement should be obtainable from the proof of the
original theorem by interchanging point and line. Although It IS too much
to ask, the gods have been generous beyond our merits, for the new proof
can indeed be obtained in this way.
The principle of duality, as thus far described, tells us how to obtain a
new statement or theorem from a given one involving points and lines. But
projective geometry also deals with curves. How should one dualize state
ments describing curves? The clue lies in the fact that a curve is after all but
a collection of points satisfying some condition. For example, the circle is the
set of all points at a fixed distance from a given point. The principle of duality
suggests, then, that the figure dual to a given curve might be a collection of
lines satisfying the condition dual to the condition defining the given curve.
(However the definition given for the circle is not in the form which can be
dualized.) This collection of lines may also be called a curve, for a collection
of lines suggests a curve as well as does a collection of points (Fig. 1 1-12). It
is called a line curve.
Fig. 11-12. Fig. 11-13.
(a) A point curve; (b) a line curve. Brianchon's theorem, the dual of
Pascal's theorem.
For conic sections, the figure dual to a point conic, that is, a conic re
garded as a collection of points, turns out to be the collection of tangents to
that point conic. Thus if the conic section is a circle, the dual figure is the
collection of tangents to that circle. This collection of tangents suggests the
circle as well as does the usual collection of points, and we shall call this
collection of tangents the line circle.
,Ve have dualized statements about simple figures with suggestive results.
Let us now see whether the application of the principle of duality to
theorems on curves is equally productive. As a test we shall dualize Pascal's
theorem. Figure 1 1-1J illustrates the content of the dual statement.
246 PROJECTIVE GEOMETRY
Pascal's theorem Dual of Pascafs theorem
If we take six points, A, B, C, D, E, If we take six lines a, h, c, d, e,
and F on the point circle, then and f on the line circle, then
the lines which join A and B and the points which join a and b and
D and E join in the point Pj d and e are joined by the line p;
the lines which join B and C and E the points which join h and c and e
and F join in the point Qj and f are joined by the line qj
the lines which join C and D and F the points which join c and d and f
and A join in the point R. and a are joined by the line T.
The three points P, Q, and R lie on The three lines p, q, and T lie on
one line, 1. one point, L.
Geometrically the dual statement has the following meaning: Since the
line circle is the collection of tangents to the point circle, the six lines on
the line circle are any six tangents to the point circle, and these six tangents
form a hexagon circumscribed about the point circle. Hence the dual state
ment tells us that if we circumscribe a hexagon about a point circle, the lines
joining opposite vertices of the hexagon, lines p, q, and r in the dual statement,
meet in one point. This dual statement is indeed a theorem of projective
geometry. It is called Brianchon's theorem after the man who discovered it
by applying the principle of duality to Pascal's theorem.
The principle of duality in projective geometry says that we can inter
change point and line in a theorem about figures lying in one plane and obtain
a meaningful statement. Moreover-although nothing said so far justifies the
assertion-the new, or dual, statement will itself be a theorem; that is, it can
be proved. However, it is possible to show by one proof that every rephrasing
of a theorem of projective geometry in accordance with the principle of
duality must lead to a theorem. The principle of duality is a remarkable
property of projective geometry. It reveals the symmetry in the roles which
point and line play in the structure of that geometry, and this symmetry in
turn reveals that line and point are equally fundamental concepts.
The principle of duality also gives us some insight into the process of
creating mathematics. Whereas the discovery of this principle as well as of
theorems such as Desargues' and Pascal's calls for imagination and genius, the
discovery of new theorems by means of the principle is an almost mechanical
procedure.
EXERCISES
1. Given the figure consisting of four points no three of which are on the same
line, what is the dual figure?
2. State the principle of duality.
3. Given the figure consisting of four points all on one line, what is the dual figure?
PROJECTIVE AND EUCLIDEAN GEOMETRIES: RELATIONSHIP 247
4. Given the figure consisting of three points on one line. a founh point not on
that line, and the lines joining any two of these points. what is the dual figure?
5. In what way is the principle of duality a means of discovering new theorems?
11-5 THE RELATIONSHIP BETWEEN PROJECTIVE AND EUCLIDEAN GEOMETRIES
Projective geometry offers many more exciting concepts than we can hope
to survey. Let us see rather what the broader features of the subject are. The
basic concept is projection and section and the main goal is to find properties
of geometric figures which hold for any section of any projection of those
figures. A careful examination of the properties which prove to be invariant
under projection and section shows that these properties deal with the col
linearity of points, that is, with points lying on the same line; with the con
currence of lines, that is with a set of lines meeting in one point; with cross
ratio; and with the fundamental roles of point and line as exhibited by the
principle of duality. On the other hand, Euclidean geometry, which, of course,
was well known to the nineteenth-century projective geometers, deals, for
example, with the equality of lengths, angles, and areas. A comparison of these
two classes of properties suggests that the projective properties are simpler than
those treated in Euclidean geometry. One might say that projective geometry
deals with the very formation of the geometrical figures whose congruence,
similarity, and equivalence (equal area) are discussed in Euclidean geometry.
With hindsight to aid us, we can see that there should be a geometry
more fundamental than Euclidean geometry. Anybody first perceives the
position in space of trees, houses, roads, and other objects and only then thinks
of distances and sizes. When traveling, one must first choose a particular road
to follow before being concerned with how far to move along the road. That
is. position and relative position precede distance in importance both practically
and logically.
Hence one might suspect that logically projective geometry is the more
fundamental and encompassing subject and that Euclidean geometry is in
some sense a specialization. This conjecture is coqect. The clue to the rela
tionship between the two geometries may be obtained by again examining pro
jection and section. Consider a geometric figure, a rectangle say (Fig. 1 1-14) .
Let a projection of this figure be formed from an arbitrary point 0,
and then let a section ()f this projection be made by a plane parallel to the
plane of the rectangle. By applying some theorems of Euclidean geometry one
can prove that the section will be a rectangle similar to the original one. Hence
the relationship of similarity is a special type of projective relationship in
which the plane of the section and the plane of the original figure are parallel.
If, now, the point ° moves indefinitely far to the left, the lines of the
projection come closer and closer to being parallel to each other. When the
center of the projection is the "point at infinity"( ! ) , these lines are parallel;
248 PROJECTIVE GEOMETRY
C
C
C'
o �' A'
A
o�
Fig. 11-14. Fig. 11-15.
Similar figures related by projection and Congruent figures related by projection
section. and section.
then the section made by a plane parallel to the rectangle is a rectangle con
gruent to the original one (Fig. 1 1- 1 5 ) . This last type of projection, called
parallel projection, thus yields congruent sections. In other words, from the
standpoint of projective geometry, the relationships of congruence and sim
ilarity which are so intensively studied in Euclidean geometry can be studied
through projection and section for special projections. We see therefore that
Euclidean geometry is not only a logical subdivision of projective geometry,
but we can now look upon it in a new light, namely, as a study of properties
of geometric figures which are invariant under special projections.
Although projective geometry was initiated by Desargues for the very
practical purpose of extending and systematizing theorems which might help
the artists, it is not highly significant from the standpoint of applications to art
or science. The subject has been developed and cultivated by mathematicians
who sought and found pleasure in its ideas. The Renaissance artists and
geometers opened up new themes which the Greek world did not grasp, the
investigation of the propenies of intersections of lines, cross ratio, duality,
projection and section, and, above all. the theme of properties invariant under
projection and section. Projective geometry is now a vast branch of mathe
matics because it does offer latitude to intuition, new methods of proof, ele
gant results, and aesthetically satisfying ideas. This subject born of art makes
its primary contribution to mathematics as an art.
EXERCISES
1. What major mathematical problem was suggested by the artists' use of projection
and section?
2. How would you distinguish projective geometry from Euclidean geometry with
respect to the propenies of geometrical figures?
3. Write a shan essay on how the rise of realistic painting in the Renaissance
stimulated a new mathematical development.
PROJECfIVE AND EUCLmEAN GEOMETRIES: RELATIONSHIP 249
Recommended Reading
BELL, E. T.: Men of Mathematics, Chaps. 5 and 1 3, Simon and Schuster, New
York, 1937.
IVINS, WM. M., JR.: Art and Geometry, Dover Publications, Inc., New York, 1964.
KLINE, MORRIS: "Projective Geometry," an article in James R. Newman: The
World of Mathematics, pp. 622-641, Simon and Schuster, New York, 1956.
MORTIMER, ERNEST: Blaise Pascal: The Life and Work of a Realist, Harper and
Bros., New York, 1959.
SAWYER, W. W.: Prelude to Mathematics, Chap. 10, Pelican Books Ltd., England,
1955.
YOUNG, JACOB W. A.: Monographs on Topics of Modern Mathematics, Chap. 2,
Dover Publications, Inc., New York, 1955.
YOUNG, JOHN W.: Projective Geometry, The Open Court Publishing Co., Chicago,
1930.
CHAPTER 12
COORDINATE GEOMETRY
In order to seek truth it is necessary once in the course of our life to doubt
as far as possible all things.
DESCARTES
12-1 DESCARTES AND FERMAT
Doubts as to the soundness of the knowledge and outlook possessed by medi
eval Europe had already been raised during the Renaissance. The revived
Greek knowledge, great explorations, new inventions, the rise of an artisan
class with problems of its own which could not be answered by purposive or
teleological explanations of natural phenomena, and the advocacy of experi
ence as the source of all knowledge, all tended to undermine the old founda
tions. No one appreciated more the need for the reconstruction of knowledge
than did Rene Descartes ( 1 596-1650) .
Born to moderately wealthy parents in La Haye, France, Descartes re
ceived an excellent formal and traditional education at the Jesuit College of
La Fleche. But while still at school, he had already become critical of the
truths which so many of his contemporaries and teachers professed so con
fidently and he began to question the kind of knowledge that was being im
parted to him. Of the traditional studies, he said, eloquence has incomparable
force and beauty, and poetry has its ravishing graces and delights. However
these attainments he judged to be gifts of nature rather than the fruits of
study. He respected theology because it pointed out the path to heaven and
he, too, aspired to heaven, but "being given assuredly to understand that the
way is not less open to the most ignorant than to the most learned, and that
the revealed truths which lead to heaven are above our comprehension," he
did not presume to subject these truths to his impotent reason. Philosophy,
he agreed, "affords the means of discoursing with an appearance of truth on all
matters, and commands the admiration of the more simple," but, though
cultivated for ages by the most distinguished men, it had not produced doc
trines which were beyond dispute. Law, medicine, and other professions
secure riches and honor for their practitioners, but since these subjects borrow
250
DESCARTES AND FERMAT 251
principles from philosophy, they could not be solid structures, and fortunately
he was not obliged to pursue them to better his fortune. Logic he also de
precated because its syllogisms and the majority of its other rules are of use
only in the communication of what one already knows or in speaking without
judgment about things of which one is ignorant. It does not in itself proffer
knowledge. Treatises on morals contain useful precepts and exhortations to be
virtuous but no evidence that these ace founded on truths.
Because he had a critical mind and because he lived at a time when the
world outlook which had dominated Europe for a thousand years was being
vigorously challenged, Descartes could not be satisfied with the tenets so
forcibly and dogmatically pronounced by his teachers and other leaders. He
felt all the more justified in his doubts when he realized that he was in one of
the most celebrated schools of Europe and that he was not an inferior student.
At the end of his course of study he concluded that all his education had
advanced him only to the point of discovering man's ignorance.
At the age of 20, after having graduated from the University of Poi tiers,
where he studied law, Descartes decided to learn some things that were not
in books. He began by living a gay life in Paris, after which he retired for
a period of reflection in a quiet corner of that city. To see the world he
joined an army, participated in military campaigns, and traveled. Finally he
decided to settle down.
Because Descartes thought he could more easily find peace and seclusion
in the saner atmosphere of Holland, he secured in 1628 a house in Amsterdam.
There he devoted himself over a period of twenty years to critical and pro
found thinking about the nature of truth, the existence of God, and the
physical structure of the universe. There he created his best works. As he
continued to write. he and his audience became more and more impressed
with the greatness of his work. Lucid thoughts set forth in literary classics
which revealed the clarity, precision, and effectiveness of the French language
made Descartes famous and his philosophy popular.
His retirement from the world was broken by an invitation to serve as a
tutor to Queen Christina of Sweden. Reluctant as he was to leave his com
fortable home, he could not resist the attraction of royalty and so moved to
Stockholm. The queen preferred to begin her day at 5 a.m. by studying in
an icy library, and her tutor was obliged to meet her at that hour. This
regimen was too much for the frail Descartes. His flesh was weak, and his
spirit unwilling. He.caught cold and died in the year 1650.
From the profound thinking and writings carried on during the years in
Holland there emerged new foundations of knowledge. Descartes is the
acknowledged father of modern mathematics and philosophy, and he founded
a new cosmology which dominated the seventeenth century until it was ulti
mately displaced by the work of Galileo and Newton. Convinced that the
knowledge he had acquired in school was either unteliable or worthless,
252 COORDINATE GEOMETRY
Descartes swept away all opinions, prejudices, dogmas, pronouncements of
authorities, and, so he believed, preconceived notions. He began reconstruction
by seeking a new method of obtaining sure and reliable knowledge. The
answer, he says, came to him in a dream while he was on one of his campaigns.
The "long chains of simple and easy reasonings by which geometers are
accustomed to reach the conclusions of their most difficult demonstrations"
led him to the conclusion that "all things to the knowledge of which man is
competent are naturally connected in the same way." He decided, then, that
a sound body of philosophy could be deduced only by the methods of the
geometers, for only they had been able to reason clearly and unimpeachably
and to arrive at universally accepted truths. Having concluded that mathe
matics "is a more powerful instrument of knowledge than any other that has
been bequeathed to us by human agency," he sought to distill from a study
of the subject some general principles which would provide a method of
obtaining exact knowledge in all fields, a method which he called a "universal
mathematic."
Following the pattern of mathematics which builds on axioms, he decided
that he would accept nothing as true which was not so clear and distinct
to his mind as to exclude all doubt. He would begin, in other words, with
unquestionable, self-evident truths. The next principle of his method was to
break down larger problems into smaller ones; he would proceed from the
simple to the complex. Then he would write out the steps of his reasoning
and review them so thoroughly that nothing would be inadvertently assumed
or necessary arguments omitted. These four principles are the core of his
method.
However, he first had to find those simple, clear, and distinct truths which
would play the part in his philosophy that axioms play in mathematics proper.
And here Descartes took a backward step. Whereas his age was turning to
experience as the reliable source of knowledge, Descartes looked into his mind.
After much critical refle ction he decided that he was sure of the following
truths: (a) I think, therefore I am. (b) Each phenomenon must have a cause.
(c) An effect cannot be greater than the cause. (d) The mind has within it
the ideas of perfection, space, time, and motion.
He then proceeded to reason on the basis of these axioms. The full story
of his search for method and of the application of the method to actual prob
lems of philosophy is presented in his famous Discourse 071 Method ( 1 637 ) .
I n later writings h e continued to follow the procedure outlined in this work
and thereby founded the first great modern system of philosophy. What is
relevant at the moment is that the truths of mathematics and mathematical
method served as a beacon to a great thinker lost in the intellectual storms of
the seventeenth century and enabled him to develop a philosophy which was
more rational, less mystical, and less bound to theology than the systems
produced by all his European predecessors.
THE NEED FOR NEW METHODS IN GEOMETRY 253
To show what his new method could accomplish in fields outside of phi
losophy, Descartes applied it to geometry and published these results in his
Geometry, an appendix to his Discourse on Method. But before we examine
how Descartes revolutionized method in geometry, we must note another
great seventeenth-century thinker, Pierre de Fermat, who, equally concerned
with improving geometrical methods, independently arrived at the same broad
idea.
In contrast to Descartes' adventurous, romantic, and purposive life Fennat's
was highly conventional. He was born in 1601 to a French leather merchant.
After studying law at Toulouse, he earned his living as a lawyer and served
as King's councillor for the parliament of Toulouse, a position much like that
of a modern district attorney. Fermat's home life was also quite ordinary.
He married and brought up five ·children. His evenings he devoted to study.
Whereas Descartes cared little for knowledge as such or for the beauty and
harmony in mathematics and the arts but sought truths and useful knowledge,
Fermat was faithful to the Greek ideals of speculative knowledge and intel
lectual pleasures. He was a student of Greek literature, wrote poetry, joined
in the solution of the scientific problems of his day, and above all regaled him
self in all branches of mathematics. Despite the brief amount of time he could
devote to study, he made fundamental contributions to algebra, the calculus,
the mathematical theory of probability, coordinate geometry, and the theory
of numbers. Fermat's mathematical achievements entitle him to the honor of
being considered one of the best mathematicians the world has had.
12-2 THE NEED FDR NEW METHODS IN GEOMETRY
Descartes and Fermat developed a new approach to geometry, specifically, an
algebraic method of representing and analyzing curves. Why were the mathe
maticians of the seventeenth century so much concerned with ways of work
ing with curves? The general reason is that the rise of science and the vast
expansion of commercial and industrial activities had raised problems involving
curves. Let us see what these problems were.
The heliocentric theory of Copernicus and Kepler, which we shall ex
amine in Chapter I S, won gradual acceptance during the seventeenth century,
and scientists and mathematicians, at least, began to apply it intensively to the
purely scientific problem of understanding the heavenly motions and to the
more practical concerns, such as navigation, wherein astronomical knowledge
was essential. Now the heliocentric theory called for the use of ellipses and,
to some extent, of parabolas and hyperbolas. Many new facts about these
curves were needed.
In the seventeenth century the idea that a ship's longitude could be de
termined most easily and accurately by means of a clock was actively pursued.
The precise details of this method of determining longitude do not matter at
254 COORDINATE GEOMETRY
the moment, but it is relevant that there were no clocks at that time which
could be carried conveniently aboard ships. The adaptation of a spring and
a pendulum to a clock was initiated by several scientists, notably Galileo
Galilei, Robert Hooke, and Christian Huygens. The motion of the bob of a
pendulum and of objects suspended from springs (see Chapter 18) is studied
by the use of curves.
The very motion of ships at sea raised problems involving curves. Al
though the ideal path of a ship on a spherical surface is a great circle, the
actual path cannot always be that since ships must obviously detour around
land. Moreover, on maps the ideal and actual paths have to be represented
by even more complicated curves whose shapes depend on the method of
projection used to draw the flat map.
The increased interest in light raised numerous problems involving curves.
When light travels long distances through the earth's atmosphere, it is grad
ually bent or refracted and hence follows a curved path (see Chapter 1 and
Section 6 of Chapter 7 ) . Since observations of the positions of heavenly bodies
may be in error because the path of light is curved, it is obviously necessary to
know something about these curved paths to correct our observations. A
knowledge of curves was needed to design lenses used in telescopes and micro
scopes. Both of these instruments had been invented in the early part of the
seventeenth century and attracted considerable attention. Lenses for spectacles
had by this time been in use for 300 years, but improvement in design was a
constant problem. Both Descartes and Fermat were very much interested in
optics, and Descartes in particular did a great deal of work on the design of
lenses. A good part of his findings are contained in the essay Dioptrics which,
incidentally, appeared with the Geometry as one of three appendices to his
Discourse on Method. Fermat, too, was a notable contributor to optics; his
principle of least time, which we described earlier (Chapter 7 ) , still stands as
a basic postulate in that subject.
Another class of problems calling for the study of curves was presented
by the increasing use of cannons. The balls or shells shot from cannons are
called projectiles, and a number of questions were raised concerning their
motion. What paths or curves do projectiles follow? How do the paths de
pend upon the angle at which the cannon is inclined? What is the range or
horizontal distance traveled by the projectile) And how is the path affected
by the initial velocity imparted to the ball?
The motion of projectiles was but one of a wider class of problems in
volving motion. As we shall note more fully in a later chapter, the motion
of objects on and near the surface of the earth became an active concern in
the early seventeenth century because the heliocentric theory raised basic
problems in this domain. Since all such motions take place along straight-line
or curved paths, these more fundamental problems of motion also led to the
study of curves.
THE NEED FOR NEW METHODS IN GEOMETRY 255
Of course, this study was not new to mathematics. The Greeks had
studied extensively the line, circle, and conic sections and had deduced hun
dreds of theorems about these curves. These contributions were known to
seventeenth-century Europe. Why, then, did Fermat and Descartes decide
that mathematics needed new methods of working with curves? The reason
was stated by Descartes. He complained that Greek geometry was so much
tied to figures "that it can exercise the understanding only on condition of
greatly fatiguing the imagination." Descartes also deplored that the methods
of Euclidean geometry were exceedingly diverse and specialized and did not
allow for general applicability. Each theorem required a new type of proof,
and much imagination, effort, and ingenuity had to be expended to find such
proofs. The Greeks, with ample time at their disposal and no concern for
immediate application, were not troubled by the lack of general procedures.
However, these conditions no longer obtained in the seventeenth century.
J\1oreover, new curves were needed for the applications which were of im
portance to the seventeenth century, and the Greek geometrical methods did
not seem to be effective in these cases.
There is another limitation inherent in Greek geometry which the seven
teenth century could no longer tolerate. One might indeed determine by
some geometrical argument what type of curve a projectile shot from a can
non follows and one might prove some geometrical facts about this curve, but
geometry could never answer such questions as how high the projectile would
go or how far from the starting point it would land. The seventeenth century
sought quantitative or numerical information because such data are paramount
in practical applications.
It was clear ro Descartes and Fermat that entirely new methods of work
ing with curves were needed. Descartes, impatient with the methods of the
Greeks and their disinterest in application, says,
I have resolved to quit only abstract geometry, that is to say, the consideration
of questions which serve only to exercise the mind, and this, in order to study
another kind of geometry, which has for its object the explanation of the
phenomena of nature.
Both Descartes and Fermat, working independently of each other, saw
clearly the potentialities of algebra for the representation and study of curves.
This realization was not entirely a bolt from the blue. A great deal of progress
had been made in algebra during the latter half of the sixteenth and the early
part of the seventeenth century, much of it contributed by Descartes and
Fermat themselves. Cardan, Tartaglia, Vieta, Descartes, and Fermat had ex
tended the theory of the solution of equations (d. Chapter 5), had introduced
symbolism, and had established a number of algebraic theorems and methods.
What impressed Descartes especially was that algebra enables man to reason
efficiently. It mechanizes thought, and hence produces almost automatically
256 COORDINATE GEOMETRY
results that may otherwise be difficult to establish. This value of algebra has
already been pointed out earlier in this book, but historically it was Descartes
who clearly perceived and called attention to this feature. Whereas geometry
contained the truth about the universe, algebra offered the science of method.
It is, incidentally, somewhat paradoxical that great thinkers should be enamored
of ideas which mechanize thought. Of course, their goal is to get at more
difficult problems, as indeed they do.
EXERCISES
I . How did mathematics help Descanes build his philosophy?
2. What were the four steps of mathematical method emphasized by Descartes?
3. Why did Descartes and Fennat seek a new method of deriving properties of
curves?
4. What scientific prob1ems of the seventeenth century required more know1edge
about curves?
12-3 THE CONCEPTS OF EQUATION AND CURVE
We can understand more readily what Descartes and Fermat accomplished if
we examine a somewhat modernized version of their idea. In his general study
of methodology, Descartes had decided to solve problems by proceeding from
the simple to the complex. Now the simplest figure in geometry is the straight
line, and so Descartes sought to analyze curves by working with straight lines.
He observed, first of all, that if one introduces (Fig. 12-1) a horizontal line,
OX, then the shape of a curve C could be studied by observing how vertical
line segments such as O'PI> 02P2, O,P" . . . changed in length.
y
P, C
PI P2
P3
/
-
/" r---
5
3 X 12-1
0 QI Q2 Q3 Q,
Fig.
The next step was to express this information in arithmetical terms. The
position of 0" for example, could be specified by stating its distance from
a fixed point 0, called the origin, on the horizontal line. The length O,P,
could certainly be specified by a number. Thus the position of P, would be
determined by two numbers, the length 00, and the length O,P,. The length
OQ" which is 3 in Fig. 12-1, is called the abscissa of PI> and the length P,O"
THE CONCEPTS OF EQUATION AND CURVE 257
which is 5 in Fig. 12-1, is called the ordinate. The line OX is called the X-axis,
and the line OY, which is perpendicular to OX and which shows the direction
in which Q,P, is taken, is called the Y-axis.
Stated in more general terms, Descartes' and Fermat's first step was to
describe the position of any point P on a curve by two numbers, an abscissa
and an ordinate. The first expresses the distance or length from 0 along the
X-axis to the point Q directly below P, and the second denotes the distance
or length from Q to P along a line perpendicular to the X-axis or parallel to
the Y-axis. The pair of numbers is called the coordinates of P and is written
thus: (3, 5 ) .
y
R
5
_ _3-..,, ...; --,- _ X
3
-+ -", o+- ,- _
-4 -4
Fig. 12-2.
Plotting points on a
s T rectangular Cartesian
coordinate system.
To distinguish points which are reached by proceeding along the X-axis
to the right of 0 from those which are reached by proceeding to the left of
0, distances to the left are represented by negative numbers. Thus to arrive
at the point R of Fig. 12-2 one proceeds 3 units to the left along the X-axis
and 5 units upward in the direction of the Y-axis. The coordinates of R are
therefore - 3 and 5 and are represented as ( 3, 5 ) . The distinction between
-
upward and downward is also made by using positive and negative numbers,
and as a consequence the coordinates of S in Fig. 12-2 are - 3 and -4, and the
coordinates of T are 3 and -4.
Thus far, then, Descartes and Fermat had a simple scheme for representing
the position of any point in the plane by means of numbers, these numbers
being distances from two arbitrarily chosen but fixed axes. To each point
there corresponds a pair of numbers, and to each pair of numbers a unique
point. The system using axes and coordinates to represent points is called a
rectangular Cartesian coordinate system.
To represent a curve such as C of Fig. 12-1, one could list the coordinates
of the many points on the curve, that is, the coordinates of Ph P2, Pa, . . .
But such a representation would hardly be convenient, for each curve consists
of an infinite number of points. Descartes and Fermat had a better idea. First
of all they introduced the lette" x and y to stand for the coordinates of any
258 CooRDINATF. GEOMETRY
one of the points on the curve. When x and y have specific numerical values,
for example, 2 and 3, respectively, then they refer, of course, to a definite
point. Otherwise they represent an arbitrary point. The use of x and y here
is analogous to using the words "man" and "woman" to represent any man or
woman in the United States, whereas John and Mary would describe a par
ticular couple.
y
c
y
--�o�--�X---L--- X
Fig. 12-3
Now, if one looks at the curve C of Fig. 12-3 and observes the abscissas
and ordinates of the points on this curve, he notices that as the abscissas in
crease <as one looks from left to right), the corresponding ordinates at first
increase and then decrease. The behavior of the ordinates changes as the
abscissas change. Might it not be possible to describe the relationship between
these abscissas and ordinates by specifying how large any ordinate is when
the corresponding abscissa is named? This description should hold for all
points on the curve and apply to that curve and no other. The answer is yes,
and the general description turns out to be an algebraic equation involving
x and y, the coordinates of an arbitrary point. This statement is a bit vaguej
let us see, therefore what it means in concrete examples.
Suppose, first, that the curve happens to be a straight line inclined 45' to
the horizontal.' To describe the line algebraically, we introduce a horizontal
line which passes through any point 0 of the given line (Fig. 1 2-4) and con
sider this horizontal line as the X-axis of our coordinate system. The Y-axis
is then a vertical line through O. Consider any point P on the line. The
coordinates of this general point P are the x and y shown in the figure. Now
Euclidean geometry tells us that the triangle OOP is an isosceles right triangle.
Hence 00 = OP or x = y. Therefore the line OP appears to be characterized
by the algebraic equation
y = x (I)
because for any point P on the line, the coordinates are such that y = x.
• In coordinate geometry and in higher mathematics, in general, the word "curve"
includes straight lines.
THE CONCEPTS OF EQUATION AND CURVE 259
--�'-4--+-- x
P'
Fig. 12-4. Fig. 12-5.
A straight line on a rectangular Cartesian A straight line of slope 3 on a rectangular
coordinate system. Cartesian coordinate system.
We should note that this equation also describes points, such as P', which
lie to the left of the vertical axis. Thus, for example, the abscissa of P' may
be -4; now angle Q'OP' is also 45', and hence the ordinate of P' must also be
-4. Since abscissa and ordinate arc equal, x y also holds for P'. Then the
=
all-important fact about the line P'P is that it may be described algebraically
by the equation y x. In other words, we may say that the coordinates of any
=
point on the line satisfy the equation y x. On the other hand, points not on
=
that line, such as R, will have ordinates unequal to the abscissas because. while
R has the <arne abscissa as P, the ordinate of R is larger than the ordinate of P,
and hence for R, y does not equal x.
Let us consider a second example. One may describe the line of Fig. 1 2-4
by saying that it rises one unit for each unit of horizontal distance or, in the
customary expression, that it has slope 1 . We now consider a line which rises
more steeply, for example, one which rises 3 units for each horizontal distance
of 1 unit (Fig. 12-5 ) . Again let P be any point on the line. The coordinates
of this general point P are then (x, y ) . Then from the similar triangles OQ'P'
and OQP we may argue that
y 1
=
-
_ .
x 1
Hence
y = lx (2)
is the equation of the line.
The nature of equations such as y x and y 3x requirp.s some attention.
= =
In elementary algebra we also treat equations, for example, :Ii' - 5x + 6 = 0
260 COORDINATE GEOMETRY
or 2x + 3 = 7. In these latter equations, however, x represents some definite
but unknown quantity and our aim is to find the value or values of x. On
the other hand, when we represent a curve by an equation such as y 3x, =
we are not seeking to determine unknowns. In fact, x and y are not unknown.
They represent the coordinates of any point on the line. Thus x = 3 and y = 9
are one pair of values satisfying the equation; x = 4 and y = 1 2 constitute
another such pair; there are millions of others. The end product of the process
of finding the equation of a curve is then an equation involving x and y which
states the relationship between x and y peculiar to all points on the curve. Of
course, if one wishes to find the coordinates of a particular point on the curve,
he can, provided he knows the abscissa of the point, substitute this number for
x in the equation and now solve for the ordinate. Thus, if the given line has
the equation y = 3x and we wish to determine the ordinate of the point whose
abscissa is 21, we substitute 2i for x and immediately find that the ordinate of
this point is 7 i.
Let us cons:der next another example which will enlarge our understanding
of these new equations. Suppose we are given two straight lines as shown in
Fig. 1 2-6. The line 01' is the one we have just discussed and its equation is
y = 3x. The line 0'1" is supposed to be parallel to 01' and 2 units above it;
that is, 1'1" is 2. What is the equation of the line 0'1''1
To answer this question, we again seek the relation between x and y which
holds for any point on this line. Now l' and 1" have the same x-value or
abscissa, namely OQ. But the ordinato of 1" is larger than that of l' by the
amount pP'. The distance PP' is given to be 2. Since the two lines are parallel,
the vertical distance between a point on 01' and a point on 0'1" will always be
2. Hence, whereas the ordinate of each point on the line 01' is always 3 times
the abscissa, the ordinate of each point on 0'1" will be 3 times the abscissa
plus 2. That is, the equation of O'P' is
y = 3x + 2 . (3 )
We should note that although the straight line 0'1" is identical except
for position with the straight line 01', its equation is different. The differ
ence results from the fact that 01' passes through the origin 0 of the system
of coordinates, wheieas Q'P' does not. Hence the very same curve may be
represented by a different equation if its position with respect to the coordinate
axes is changed.
Why do we bother with different equations for the same curve' If a
straight line having a slope of 3 can always be placed on a coordinate system
so that its equation is y = 3x, why do we have to consider the more compli
cated form y 3x + 2 ? The answer is that if one wishes to work with two
=
straight lines simultaneously and keep these lines in the same relative positions
which they may happen to have in some physical application, one cannot assign
to them an identical position on the set of axes.
THE CONCEPTS OF EQUATION AND CURVE 261
p y
--;,--o-:--'---1t"--- X
Fig. 12-6. Fig. 12-7.
Two parallel lines of slope 3 on a A line with negative slope.
rectangular Cartesian coordinate system.
While we are discussing equations of straight lines, we wish to note one
more case which will be of interest later. Let us compare the line OP of Fig. 1 2-7
with the line OP of Fig. 12-5. The line in Fig. 1 2-7 may be said to "fall"
as one views it from left to right, whereas that in Fig. 12-5 rises. Alternatively,
we may say that the ordinates in Fig. 1 2-7 decrease as the abscissas increase.
What is the equation of the line OP of Fig. 1 2-7? Suppose that the coordinates
of some point P' on the line are ( - 1, l ) . The point P is an arbitrary point of
the line OP, so let us denote its coordinates by x and y. Again we have the
similar triangles OQP and OQ'P'. Insofar as mere lengths, without regard to
sign, are concerned, we can say that
!= 1
x
However, we know that for the line OP, the ordinate of any point is always
opposite in sign to the abscissa of that point. That is, when y is a positive
number, x is a negative one and conversely. Hence, in the present case, not
y = lx, but
y = - lx (4)
is the equation of the line OP. Thus the fact that OP falls to the right is
reflected in the negative sign of the coefficient of x. To distinguish, with
respect to slope, lines which fall to the right from those which rise to the
262 COORDINATE GEOMETRY
right, we say that the former have negative slope. Thus the slope of OP
is 3 .
-
To illustrate Descartes' and Fermat's idea once more we shall seek the
equation of a circle. Suppose that we choose our axes so that the origin is
at the center of the circle (Fig. 1 2-8). Now the circle is defined to be the
collection of all points which are at the same distance from one point, called
the center. Let us assume that this distance is 5 units. (The quantity 5 is, of
course, the length of the radius.) Since each point on the circle is described by
a pair of coordinates, the problem of finding the algebraic equation of this
circle can be solved by answering the following question. What property or
relationship do the coordinates of points on the circle possess which dis
tinguishes them from those of other points? If we consider a typical point P
on the circle and let x and y represent its two coordinates, then we see that the
lengths x, y, and 5 form a right triangle. According to the Pythagorean theo
rem of Euclidean geometry, the square of OP must equal the square of OQ plus
the square of QP; hence
x2 + y2 = 5 2. (5)
This same statement also holds for points on the circle such as P', for even
though the coordinates of P' are negative, their squares are positive and so
satisfy equation (5). Equation ( 5 ) is the algebraic representation of the circle.
It says in words that the square of the abscissa of any point on the curve plus
the square of the ordinate equals the square of the radius.
y
-+- - o "---c ---1 -
- - �- L- -
'f Q X
P'
Fig. 12-8.
A circle on a rectangular
Cartesian coordinate
system.
To decide algebraically whether any point belongs to the circle repre
sented by equation ( 5 ) , we have to test whether the coordinates of the point
satisfy the equation. Thus the point whose abscissa is 3 and whose ordinate is
4 belongs to the circle under discussion because, substituting 3 for x and 4 for
y in equation ( 5 ) , we see that the resulting left side equals the right side, that
is, 32 + 42 5 2 . As another example, let us consider the point whose abscissa
=
THE CONCEPTS OF EQUATION AND CURVE 263
is 2 and whose ordinate is y'IT. Again, substituting 2 for x and y'IT for y in
equation ( 5 ) , we find that
2 ' + (vIzl)' = 25,
becallse the square of y'IT is 2 1 . Thus the point whose coordinates are
(2, y'IT) lies on the circle.
EXERCISES
l . What is meant by the coordinates of a point?
2. State in your own words what the equation of a curve is.
3. Find the equation of the straight line which
a) rises 2 units for each horizontal distance of one unit and passes through the
origin of the coordinate system chosen;
b) makes an angle of 300 with the X-axis and passes through the origin [Sug
gestion: tan 30· = 1/v'l.l;
c) falls 4 units for each unit of horizontal distance traversed and passes through
the origin;
d) passes through the origin and has slope 4;
e) passes through the origin and has slope -4.
4. Find the coordinates of one point which lies on the curve whose equation is
a) , + 2y = 7 1 ; b) " + y' = 36.
5 . Does the point whose coordinates are ( - 3, 5 ) lie on the curve whose equation
is X2 + 2y2 = 59?
6. Detennine whether the point whose coordinates are (3, -2) lies on the curve
whose equation is X2 + y2 = 4x + 1 .
7 . Describe the curve whose equation is
a) y = 3, + 7, b) " + y' = 49, c) " + y' = 20,
d) , + 2y = 6, e) y' = 20 - x'.
8. Would latitude and longitude serve as a coordinate system for points on the
surface of the earth?
9. Can a curve have more than one equation? If so, how is this possible?
to. Can you say anything about the slope of the line whose equation is y = mx + 2 ?
I I. What are the coordinates of the point of intersection of the line y = 3x + 7 and
the Y-axis?
12. What are the coordinates of the point of intersection of the line y = 3x + b and
the Y-axis?
13. What is the slope of the line whose equation is y = 111X + b, and what are the
coordinates of the point where the line cuts the Y-axis?
264 COORDINATE GEOMETRY
14. The equation of a circle with center at the origin and radius 1 is
.' + y' = 1 . <a)
The equation of a circle with center at the origin and radius 2 is
.' + y' = 4. <b)
If we subtract equation (a) from equation (b), we obtain the result 0 = 3. What
is wrong?
y
p
y - 3
F y
- � ---Y
(0, 3) � - -
3
--------��o����x�--�- x
3 3 Fig. 12-9.
A parabola on a rectangular
Q d D Cartesian coordinate system.
12-4 THE PARABOLA
The curve most widely used, next to the straight line and circle, is the
parabola. Let us see how this curve is represented algebraically. Recalling the
definition of the parabola given in Chapter 6, we start with a fixed line d,
called the directrix, and a fixed point F (Fig. 12-9), called the focus. We now
consider all points each of which is equidistant from d and F. This set of points
is called a parabola. Thus if P is a typical point on the parabola determined by
d and F, then the distance from P to F must equal the distance from P to ':,
that is
PF = PD. (6)
To obtain the equation of this curve, we first introduce a set of coordinate
axes. We know from the discussion of the straight line that the same curve
may have different equations, depending upon how one chooses the axes in
relation to the curve. Mathematicians have learned by experience that a simple
equation results if the axes are chosen in the following way (see Fig. 1 2-9) .
Let the Y-axis be the line through F and perpendicular to d. The X-axis,
which is, of course, perpendicular to the Y-axis, is drawn halfway between
F and d.
Since the point F and the line d are fixed, the distance from F to d, namely
FQ, is fixed. Let us suppose that this distance is 6 units. Then the distance
OF is J units, and the distance OQ is also J units because the X-axis is halfway
THE PARABOLA 265
between F and d. Then the coordinates of F are (0, 3). We now wish to
express equation (6) in algebraic tenns. Let P be any point on the parabola.
Then its coordinates are ( x, y ) . We see that PF is the hypotenuse of a right
triangle whose sides are x and y - 3. Hence, by the Pythagorean theorem,
PF = vx' + (y 3) ' .
The perpendicular distance from P to d is y + 3. Thus, in algebraic terms,
equation (6) states that
yx' + (y - 3) ' = y + 3. (7)
Equation (7) is the equation of the parabola.
Next we shall perform some algebraic manipulations to simplify the form
of this equation. We begin by squaring both sides of the equation, an
operation which amounts to multiplying equals by equals. Squaring the left
side removes the radical, and squaring the right side yields y ' + 6y + 9.
Hence we now have
x' + (y - 3) ' = y' + 6y + 9.
We now write out in full the square called for on the left side and obtain
x' + y ' - 6y + 9 = y' + 6y + 9.
Subtracting y ' and 9 from both sides yields
x' - 6y = 6y.
We add 6y to both sides and obtain x' � 12y or, as we prefer to write it,
12y = x' .
Dividing both sides by 12, we obtain
1 ,
- (8)
Y T2 X '
_
This equation is much simpler than (7) and yet expresses the same fact.
We might note incidentally that the number 1 2 in the denominator is
twice the distance from F to d. Had we called this distance a, the resulting
equation would have read
-
I ,
Y = x . (9)
2a
In Fig. 1 2-9 we drew a curve which resembles a parabola, but we did not
know at the time its position in relation to the axes we chose. To obtain a
266 COORDINATE GEOMETRY
y
(-5, p,)
Fig. 12-10
-------"�o:i--"""-- X
y
y
d d
3 F
� ::::-
----::;; � X
--+-'�+-��---- X
(3, 0)
--- ---
o
(0, - 3) F
Fig. 12-11. Fig. 12-12.
A parabola with focus below the A parabola with focus to the right of the
directrix. directrix.
rough idea of this posltlon, let us substitute into equation (8) any pOSItive
value of x, say 5. Then y �H. This tells us that (5, H) is a point on the
curve (Fig. 1 2-10). But if we substitute - 5 for x, we obtain the same result
for y . Thus ( 5, H ) is another point on the curve. These two points are
-
symmetrically situated with respect to the Y-axis. Moreover, no matter which
abscissa and ordinate we calculate, the negative of that abscissa will always
yield the same ordinate because the abscissa has to be squared. This means that
to each point on the curve to the right of the Y-axis there corresponds a point
symmetrically situated to the left of the Y-axis. Hence, once we have deter
mined the shape of the curve to the right of the Y-axis, we automatically
know what the curve looks like to the left.
If we are interested only in the shape of the curve, it is now sufficient to
observe that as x increases from zero to any value, x2 increases, and r/12
also increases. This means that the points on the curve move out and
up from the origin. Of course, there are many curves that meet this descrip
tion. To obtain a more precise picture of the curve we need to calculate a few
sets of coordinates. Thus, when x J, Y � .,;, so that ( 3 , U are the coor
�
dinates of a point on the curve. We should note, too, that the curve lies
entirely above the X-axis except, of course, for the point at the origin. because
for every value of x, y is zero or positive.
THE PARABOLA 267
In work involving parabolas and their equations it is sometimes convenient
to consider one whose focus lies below the directrix. If we now choose the
axes as shown in Fig. 12-1 1, what is the equation of the parabola? We could,
of course, obtain the answer by going through steps analogous to those con
tained in equations (6) through (8). However, there is no need to do so since
Figs. 12-9 and 12-1 1 differ only in the sign of the y-values: whereas the y
values in the former are positive, the y-values in the latter must be negative.
Hence the equation of the parabola in Fig. 12-11 is
y = _n-X2 . (10)
Let us consider Fig. 12-11, but suppose now that distances downward on
the Y-axis are chosen to be positive. How does this change affect the equation
of the parabola? The suggested situation does not really differ from that shown
in Fig. 1 2-9. If we imagine this whole figure rotated about the X-axis through
1 80°, that is through half a rotation, we obtain the situation we have just
proposed. Since the position of the curve in relation to the X- and Y-axes is
exactly the same in Fig. 12-11 as in Fig. 12-9, the parabola has the equation
y = n-X2. ( 1 1)
Let us consider one more variation. Suppose that directrix and focus
happen to lie as in Fig. 12-12. We choose the X- and Y-axes as shown, with
the Y-axis halfway between focus and directrix. The upward direction on the
Y-axis is positive. What is the equation of the parabola determined by this
choice of focus, directrix, and axes? To answer this question we have but to
compare Figs. 1 2-9 and 12-12. The X-axis in Fig. 12-9 plays the role of the
Y-axis in Fig. 12-12, and the Y-axis in Fig. 1 2-9 plays the role of the X-axis
in Fig. 12-12; in other words, the roles of abscissa and ordinate are exchanged.
Hence the equation of the parabola in Fig. 12-12 is
( 1 2)
These last few equations illustrate some of the various forms which the
equation of the parabola can take. Which of these one should use is a matter
of convenience in application, as we shall see in later chapters.
EXERCISES
1. Determine the shapes of the following parabolas by choosing a number of values
of x, calculating the corresponding values of y, and plotting the points whose
coordinates are thereby determined.
a) y = 3x2 b) Y = J\x2 c) y = _ lx2 d) x = Zy2
e) x = !y2 f) Zx = y2
268 COORDINATE GEOMETRY
y y
d
___ � -1 o 3 _ X
F '-
� ,"", -1
Fig. 12-14
y
Fig. 12-13
Fig. 12-15
2. Compare the curves of Exercises l (a) and l (b).
3. For the parabola shown in Fig. 12-13, the distance from focus to directrix is 6.
What is the equation of the parabola?
4. If the equation of a parabola is y = ir, what are the coordinates of the focus?
Describe the position of the directrix.
5. The following exercises specify the position of focus and directrix of a parabola
in relation to a coordinate system. Find the equation of the parabola.
a) Focus (0, 4); directrix parallel to, and 4 units below, the X-axis.
b) Focus (0, 6); directrix parallel to, and 6 units below, the X-axis.
c) Focus (0, - 5 ) ; directrix parallel to, and 5 units above, the X-axis.
d ) Focus (4, 0); directrix parallel to, and 4 units to the left of, the Y-axis.
6. Suppose that the designer of a bridge has decided to use the parabolic cable
y = x2 for the range x = -5 to x = 5 (Fig. 12-14). The roadbed of the bridge is
to be the X-axis. Compute the lengths of straight wire needed to suspend the
roadbed from the cable at X = I, 2, 3, 4, and 5.
7. Suppose that the designer of a bridge has decided to use the parabolic cable
y = x2 for the range x = -5 to x = 5 (Fig. 12 1 5 ). The roadbed is to have the
-
shape y = -ToX2. Compute the length of straight wire needed to suspend the
roadbed from the cable at X = I, 2, 3, 4, and 5.
FINDING A CURVE FROM ITS EQUATION 269
12-5 FINDING A CURVE FROM ITS EQUATION
The great merit of the idea conceived by Fermat and Descartes is that it
permits us to represent a curve algebraically and, as we shan see later, learn
much about the curve by working with the equation. But another value of
their idea, hardly secondary in importance, is that any equation in x and y
determines a curve. Hence by merely writing down any equation we please
and by finding out what curve belongs to the equation we can discover many
new curves. Although we shall not, at this time, make any sensational discov
eries, let us see how the process of determining the curve of an equation can
be carried out.
Suppose we consider the equation
y = x2 - 6x (Il)
and try to determine the curve which has this equation. A direct method
would be to calculate coordinates of points on the curve and then plot these
points. Thus, when x = 2, Y = -8, and so (2, - 8 ) are the coordinates of a
point on the curve. By calculating many such sets of coordinates and by
plotting the corresponding points one can determine the shape of the curve.
However, one often learns more by applying a little algebra.
If equation ( 1 3 ) had been y = x2, we would know at once by comparing
with equation (9) that it is a parabola with distance from focus to directrix
equal to t. Let us see whether a little algebraic juggling with equation ( 1 3 )
might bring it into a form which will permit u s to identify the curve. By
adding 9 to both sides of equation ( 1 3 ) we obtain
y + 9 = x2 - 6x + 9.
Y Y'
Now our knowledge of algebra tells us
that the right side is (x - 3 )2. Hence we p
have ��--+on-�-----r- x
y + 9 = (x - 3) 2 . (14)
Suppose we introduce new letters x' and 9
y ' such that
x' = x - 3 and y' = y + 9. ( 1 5)
-- -4 3 0� -- --
� '
-- -- � � -- -- x,
Then substitution in equation ( 14) yields
Fig. 12-16
y' = x,2. ( 1 6)
The curve corresponding to this equation is the parabola shown in
Fig. 1 2-16, where it is graphed with respect to the X'- and Y'-axes. Of course,
we wish to find the curve corresponding to equation ( 1 3 ) , not that described
by ( 16). However, equations ( 1 5 ) provide the necessary connection. The
270 COORDINATE GEOMETRY
equation x' = x - 3, or x = 3 + x', tells us that the abscissas x of points be
longing to ( 1 3 ) should be 3 more than the abscissas x' of points belonging to
(16). How can we increase by 3 the abscissas of each point in Fig. 12-16?
The answer is simple. We draw a new Y-axis 3 units to the left of the
Y'-axis. Then the x-value of a typical point such as P is 3 units more than
its x'-value. We now use the second equation in ( 15), that is, y' = y + 9,
or y y' - 9. This equation says that the y-values of points should be 9
=
units less than the y'-values. How can we reduce by 9 the ordinates of the
curve in Fig. 12-16? We have already indicated the essential trick. We intro
duce a new X-axis 9 units above the X'-axis. Now the y-value of P is 9 units
less than the y'-value. Hence with respect to the X- and Y-axes, the points of
the parabola y' = x" have the correct x- and y-values called for by equation
( 1 3 ) . Since we did not change the curve in any way by introducing the X
and Y-axes, but merely changed the axes, the curve of equation ( 1 3 ) is a
parabola placed with respect to the X- and Y-axes as shown in Fig. 12-16.
We were a bit lucky in analyzing equation ( 1 3 ) because the introduction
of new coordinates in equation ( 1 5 ) reduced equation ( I 3 ) to equation ( 1 6)
whose curve we already knew. If this change to a familiar form is not
possible, then the initial equation may indeed represent some new curve,
and by analyzing the equation we might get to know the properties of this
new curve. The results of such studies would be a further addition to the
stock of knowledge about equations and their corresponding curves, a type
of knowledge which the professional mathematician builds up for his work
just as a writer may build up a bigger and bigger vocabulary. Thus through
the notion of equation and curve, Fermat and Descartes opened up to
mathematicians a vast variety of new curves.
EXERCISES
1. For equation ( 1 3 ) of the text calculate the coordinates of a number of points on
the curve and plot the points. Then sketch in the curve. Does your graph look
like the one in Fig. 12-16?
2. Determine the curve whose equation is y x2 - lOx.
=
3. Determine the curve whose equation is y = _x2 + 6x. [Suggestion: Note that
the given equation is the same as -y = x2 - 6x and use the results obtained for
equation ( 1 3 ) of the text.]
4. Sketch the curve whose equation is y = -x2 + 6x by finding and plotting the
coordinates of a number of points on the curve.
S. Knowing the curve which corresponds to y = x2 - 6x, can you determine the
curve which corresponds to y = x2 - 6x + 9?
6. What does one mean by the statement that a curve can be associated with any
equation in x and y?
THE ELLIPSE 271
7. Sketch the curves whose equations are given below.
I
a) y = x3 b) y = x3 + 9 c) y =
x
Does the sketch of part (c) suggest one of the conic sections?
12-6 THE ELLIPSE
Another very widely used curve is the ellipse. Let us review the definition
given in Chapter 6. We start with two fixed points, F and F', called foci, and
a constant quantity which is greater than the distance FF'. We now consider
all points, the sum of whose distances from F and F' is the constant quantity.
This set of points is called an ellipse. To be more concrete, suppose that the
distance FF' is 6 and that the constant quantity is 10. If P is a point such that
PF + PF' is 10, then P is a point on the ellipse.
y
p
F' 3 F
1---;���===3����X
0 \
x
Fig.12-17.
An ellipse on a rectangular
Cartesian coordinate system.
From the standpoint of coordinate geometry, the first thing of interest
about the ellipse is its equation. Let us see whether we can find it. As in
the case of the parabola, experience has taught mathematicians that the result
ing equation will be simplest if the line FF' (Fig. 12-17) is chosen as the
X-axis and the Y-axis is chosen to be the line perpendicular to the X-axis and
halfway between F and F'. Let us consider the ellipse for which the length FF'
is 6 units. Then the coordinates of F are (3, 0) and those of F' are ( - 3, 0).
Now let P be any point on the curve and let us denote its coordinates by
(x, y ) . If the constant quantity which determines the ellipse is 10, then the
condition which any point P on the ellipse satisfies is
PF + PF' = 10. (17)
We wish to express this condition algebraically. The procedure IS straight-
272 COORDlNATE GEOMETRY
forward. The distance PF is the hypotenuse of the right triangle PQF whose
arms are x - 3 and y . Hence PF � V(x 3 ) ' + y '. The distance PF' is the
hypotenuse of the right triangle PQF' whose arms are x + 3 and y. Hence
PF' � V(x + 3 ) ' + y ' . Thus equation ( 1 7 ) amounts to
v(x - 3) 2 + y 2 + v(x + 3 ) 2 + y 2 = 10. (18)
We are now in the same position that we were in when we arrived at
equation (7) for the parabola. We could maintain that ( 1 8) is the equation
of the ellipse, for it is indeed the condition which the coordinates (x, ,Y) of
any point on the ellipse satisfy. However, as for the parabola, a little algebra
applied to ( 1 8) will simplify the equation. We shall not carry out the algebraic
steps explicitly because they afC uninteresting, and it is not important for us
to acquire great facility in algebra. The result is
1 6x2 + 25y 2 = 400 . (19) y
We know what an ellipse looks like, But • •
( - a, b) (a, b)
we do not know how our ellipse lies in relation
to the axes chosen in Fig. 1 2-17. An analysis
of equation ( 19) will supply the answer. First --------�o�-- x
of all let us note that if (a, b) should happen to
be the coordinates of a point which satisfy ( - a, - b) (a, -b)
•
equation ( 19), that is, if •
1002 + 25b2 = 400, (20) Fig. 12-18
then the sets of coordinates ( -a, b), (a, -b), and (-a, -b) will also satisfy
the equation, because the substitution of any one of these latter three pairs of
coordinates will yield the same equation as (20). Figure 12-18 shows where
the various points (a, b), (-a, b), (a, -b), and ( -a, -b) lie in relation to
the axes. We see, for example, that (a, b) and ( -a, b) are symmetrically
placed with respect to the Y-axis. What we have learned so far is that if the
ellipse contains a point (a, b) which lies in the first quadrant, it contains the
point ( -a, b) which is symmetrically situated with respect to the Y-axis; it
contains the point (a, - b ) which is symmetrically situated with respect to the
X-axis; and it contains the point ( -a, -b) which is symmetric to (-a, b)
with respect to the X-axis. Hence, if we can determine which points lie on
the ellipse in the first quadrant, we can, by symmetry, decide what the ellipse
looks like in the other three quadrants.
The shape of the ellipse in the first quadrant is easily determined. We have
but to calculate the coordinates of a number of points in the first quadrant and
plot the points carefully with respect to the coordinate axes. By symmetry
we obtain the shape of the curve in the other three quadrants. The final graph
is that shown in Fig. 12-17.
THE EQUATIONS OF SURFACES 273
We could investigate the equations of other curves and the curves cor
responding to other equations. Bur what we have done should make the
primary idea clear. To each curve there corresponds an equation which
describes that curve. The equation depends upon how we choose the axes,
but once this choice is made, the equation is unique. Conversely, given an
equation involving x and y, we can find the curve which this equation de
scribes, namely the collection of points whose coordinates satisfy the equation.
EXERCISES
I . For equation ( 19) of the text, calculate the coordinates of the points whose
abscissas are 0, 1, 2, 3, 4, 5. Plot the points.
2. Sketch the ellipse whose equation is 9x2 + 16y2 144.
=
3. Calculate the length of the X-axis which is contained within the ellipse repre
sented by equation ( 19). Does this length have any relation to any of the quan
tities which determine the ellipse? This length is called the major axis of the
ellipse.
4. Suppose that the constant quantity which defines an ellipse, 1 0 in the example
discussed in the text, is retained, but the distance between the foci F and F' is 0.
What changes must one make in equation ( I S ) ? Can you now simplify the equa
tion and recognize the curve that it represents?
5. Kepler's first law of planetary motion says that the path of each planet is an
ellipse with the sun at one focus. Let us suppose that equation ( 19) of the text is
the equation of some planet's path and that the sun is at F. What is the planet's
distance from the sun when it crosses the positive X-axis and what is the distance
when it crosses the negative X-axis?
* 12-7 THE EQUATIONS OF SURFACES
The mathematician has but to get hold of an idea, and he will develop it for
all that it is worth. It had already occurred to Descartes that the idea of
equations for curves might be extended to finding equations for surfaces. This
possibility was soon explored.
A curve can lie in one plane, but surfaces such as a sphere or the ellipsoid
(of which the surface of the earth and the surface of a football are examples) ,
do not lie in one plane. They exist i n three-dimensional space. To pursue the
idea of finding equations for surfaces, we must first introduce coordinates for
points in space. This is readily done. One introduces three mutually perpen
dicular lines (Fig. 12-19) as axes instead of the two lines used for points in the
plane. These are called the coordinate axes. The X- and Y-axes determine a
plane called the XY-plane. Similarly, the X- and Z-axes determine the XZ
plane, and the Y- and Z-axes define the YZ-plane.
The location of a point P in space is described by three numbers. For
example, the point P of Fig. 12-19 is described by ( 3, 4, 5 ) . The number 5
274 COORDINATE GEOMETRY
z
z
I
I
I
�� ___ I_ �
2- ____ y I
x
/
/
+klm
x /
Y
Q
y
Fig. 12-19. Fig. 12-20.
A three-dimensional rectangular A sphere on a three·dimensional
Cartesian coordinate system rectangular Cartesian coordinate system.
describes the perpendicular distance of P above the XY-plane, while 3 and 4
are the x- and y-coordinates of Q, the foot of the perpendicular from P to
this plane. Alternatively one can say that if one proceeds a distance of 3 units
along the X-axis, then a distance of 4 units along a parallel to the Y-axis, and
finally travels upward a distance of 5 units along the perpendicular to the
XY-plane, he will arrive at the point P. To represent all points in space, we
must, as in the two-dimensional system, use negative numbers also. Thus points
below the XY-plane have negative third coordinates.
Let us now consider the problem of finding the equation of a surface.
We shall use the sphere as an example. A surface, like a curve, has some
defining property which states just which points belong to it. By definition,
the sphere is the set of all points in space at a given distance, the radius, from
a fixed point called the center. To be concrete let us suppose that all points
of our sphere are 5 units from the center and that the sphere is located
so that its center is at the origin of our three-dimensional coordinate system
(Fig. 12-20). The general point on a surface is represented by threo letters,
x, y, z. Thus the coordinates of the general point P are ( x, y, z). Let us now
express algebraically the fact that the distance of any point (x, y, z) on the
sphere is 5 units from the origin. The lengths x, y, and z are shown in
Fig. 1 2-20. Now x and y are the arms of a right triangle (lying in the XY
plane) whose hypotenuse is OQ. Then by the Pythagorean theorem,
x2 + y2 = OQ2. (21)
Further OQ and z are the arms of the right triangle OQP, whose hypotenuse
FOUR-DIMENSIONAL GEOMETRY 275
is OP or 5 units. Hence
OQ2 + Z2 = 25. (22)
But OQ' has the value given by equation ( 2 1 ) . If we substitute this value in
equation (22), we obtain
2 2 2
x + y + Z = 25. (23)
This is the equation of a sphere in the sense that the left side equals the
right side when and only when the coordinates of a point on the sphere are
substituted for x, y, and z.
We could readily obtain the equations of a few other surfaces, such as
plane, paraboloid, and ellipsoid. But we shall not do so because we shall
not utilize three-dimensional coordinate geometry and the procedure involved
is a more or less apparent extension of a familiar concept.
EXERCISES
1. Plot the points whose coordinates are given below.
a) (1 , 2, 3) b) (1, 2 , - 3 ) c) ( I 2, 3)
-
, d) (I, -2, 3)
2. What geometrical figure does an equation in x, y, and z represent?
9.
3 . Describe the surface whose equation is x2 + y 2 + Z2 4
=
4. We know that an equation such as x + y = 5 represents a straight line. What
does the equation x + y + z = 5 represent?
* 12-8 FOUR-DIMENSIONAL GEOMETRY
Our experience is limited to figures lying in a plane and in space. But our
intellects are not. The idea of a four-dimensional world and of figures in it
had tantalized mathematicians such as Pascal at a time when coordinate
geometry was still being fashioned. During the next 200 years the subject was
mentioned occasionally, but it was not taken seriously until some startling
developments (which we shall consider in Chapter 20) caused a number of
mathematicians. notably Bernhard Riemann, to investigate it. Four-dimensional
geometry proved to be more than a speculation, for some of the deepest
developments in modern science, notably the theory of relativity, use this
concept. Let us see how coordinate geometry can be employed to portray a
four-dimensional world.
We have seen that the equation of a circle of radius 5 in a two-dimensional
coordinate system is
2
x + y 2 = 25, (24)
and we have just seen that the equation of a sphere of radius 5 in a three-
276 COORDINATE GEOMETRY
dimensional coordinate system is
(25)
Even idle speculation would suggest that we at least write down the equation
(26)
and consider what meaning it might have. It would seem reasonable to
interpret x, y, Z, and w as the coordinates of a point in four-dimensional space
and, by analogy with equations (24) and (25), to interpret equation (26) as
the equation of a hypersphere in four-dimensional space. This is exactly what
mathematics does. However, mathematics does not suppose that there is
anywhere a real four-dimensional space in which four mutually perpendicular
axes can be set up. Nor do mathematicians claim that they, wise and farsighted
as they think they are, can visualize figures in a four-dimensional space. It
follows that no one else can either.
Four-dimensional geometry is entirely a creation of the mind; it is a
geometry without pictures. One speaks of the coordinates (x, y, z, w) as
representing a point, and one uses the term hypersphere as though it were a
real geometrical figure corresponding to equation (26), but these geometrical
terms are merely a convenience and a carry-over from two- and three
dimensional geometry. The words are suggestive but not descriptive of
actual figures.
Suppose it is agreed then that four-dimensional geometry is indeed a mental
creation. Is it of any value? There is excellent reason to study this "geometry".
and there is excellent use for it. We can understand these facts better if we
backtrack a bir. Consider the equation x'- + y 2 = 25. It represents, as we know,
a circle. But where is this curve that knows no end, this "arc unbroken", the
cherished figure of the Greeks? Every geometrical fact we know or can
establish about the geometrical circle has its algebraic equivalent and can be
derived algebraically from the equation of the circle. Hence the equations of
the circle and of other curves in plane, or two-dimensional, geometry are a
complete substitute for their geometrical counterparts, and we could. if we
wished to, eliminate geometry altogether. In four-dimensional "geometry"
we have only the equations, but we talk about them as though they represented
figures in a four-dimensional world. The properties of these figures are com
pletely specified by the equations. What is lacking is the possibility of actually
constructing such figures.
So far then we have tried to see what meaning this four-dimensional
geometry has. Now we wish to know how it is used. One application is made
in studying physical events wherein time plays a role. Consider for example,
the motion of a planet. The location of a planet is described by three coordin
ates x, y, and z. But the instant at which the planet occupies that location is
SUMMARY 277
also important. An eclipse of the sun, for example, occurs because planet and
sun are in certain positions at the same instant. Hence the full description of
the position of a heavenly body requires four coordinates, the fourth being a
value of time. The path of a celestial object is described by equations involving
four letters, usually x, y , z, and t.
But is there any value to thinking geometrically about equations involving
four letters' There is. Let us consider the usual sphere for a moment. Some
curves on this sphere, a circle of latitude, for example, lie in one plane, and
hence only two-dimensional geometry is needed to visualize some curves which
lie on a three-dimensional sphere. Similarly, the path on which a planet moves
in the four dimensions of space and time may be a curve which can exist in
three-dimensional space. This curve may be part of a "geometrical structure"
which lies in four-dimensional space, just as the circle of latitude is part of a
sphere and yet the curve can be visualized. This visualization aids the under
standing. This same visualization might be possible for the paths of other
planets and one can consequently better understand their motions. Yet the
proper interrelationship of these several paths can be represented only in four
dimensional space just as the relationship of the circles of latitude to one an
other can be represented only in three-dimensional space. We see therefore
that it is helpful to think in terms of geometrical figures lying in a four
dimensional space.
This brief presentation of four-dimensional geometry may give some
further indication of the direction in which scientific thought has been moving
with the aid of mathematics. Copernicus asked the world to accept a theory of
planetary motions which violated some sense impressions for the sake of a better
mathematical account. The utilization of a four-dimensional geometry which
has no sensuous or visual content means complete reliance upon the mind.
12-9 SUMMARY
From the purely mathematical standpoint coordinate geomery offers a brand
new thought, the representation of geometrical figures by equations. It also
offers, as Descartes and Fermat had expected, a new mathematical method of
deriving properties of figures from equations. For example, the fact that a
curve is symmetric with respect to some line is readily seen from the equation,
as we observed in the case of parabola and ellipse.
But Descartes' and Fermat's union of algebra and geometry means far more
than a new mathematical method of working with curves. The forms of all
physical objects which are studied for any reason whatsoever are, at least when
idealized, curves and surfaces. The fusilage of an airplane, the wings of an
airplane, the hull of a boat, and the shape of a projectile are surfaces. The paths
of all moving objects, a ball thrown by a child, an electron expelled from an
atom, a ship on the ocean, a plane in the air, the planets in the heavens, and
278 COORDINATE GEOMETRY
the tracks of light, are curves. These surfaces and curves can be represented by
equations, and the shapes or motions studied by applying algebra to these
equations. In other words, Descartes and Fermat made possible the algebraic
representation and the study by algebraic means of the various objects and
paths of interest to scientists. In addition, algebra supplies quantitative knowl
edge. This method of working with curves and surfaces is so basic in science
that Descartes and Fermat may very weIl be caIled the founders of mathemat
ical physics. Part of Descartes' greatness and perhaps the largest part of his
contribution was his vision of what his method accomplished; he said he had
"reduced physics to mathematics." The investigation of nature which Renais
sance Europe had determined to undertake was enormously expedited as we
shall soon see. The story of coordinate geometry illustrates how an interest in
geometric method became immensely valuable for science and engineering.
EXERCISES
1 . In what sense does a four-dimensional geometry exist?
2. What geometrical language would be appropriate to describe the figure whose
equation is x + y + z + w = 5?
3. Did Descartes and Fermat introduce a new method for working with curves?
If so, describe it.
4. Does coordinate geometry replace Euclidean geometry?
REVIEW EXERCISES
1. What is the x-coordinate of any point on the Y-axis?
2. What is the y-coordinate of any point on the X-axis?
3. If two points lie on a line parallel to the Y-axis, what can you say about their
x-coordinates?
4. Write the equation of a line whose slope is 4 and which
a) passes through the origin,
b) cuts the Y-axis at the point (0, 2 ) .
5. Write the equation o f a line whose slope is - 4 and which
a) passes through the origin,
b) cuts the Y-axis at the point (0, 2 ) ,
c) cuts the Y-axis at the point (0, - 2 ) .
6 . Show that the point whose coordinates are ( 2 . V2i) lies on the circle
x' + y' = 2 5.
7. Sketch the curves whose equations are given below.
a) y = ix' b) y = 8x' c) y = -tx'
SUMMARY 279
8. Sketch the curves whose equations are given below.
a) y = x2 + 6x b) Y = _x2 -6x
c) y = x2 + 6x + 9, by relating the curve to the one in part (a).
9. Plot the graph of 16x' + 25y' = 400 in the first quadrant by solving for y and
then making a table of values.
10. Plot the entire graph of the equation x2 - y2 = 4. Can you identify the curve?
1 1 . One can regard the quadratic equation x2 - 6x 0 as a special case of
=
y = x2 - 6x,
the special nature being that the values of x which satisfy the first equation are
those which correspond to y = 0 in the second one. Can you then suggest a
graphical method of solving the quadratic equation?
Topics for Further Investigation
1. The life and work of Rene Descartes.
2. The life and work of Pierre de Fermat.
3. Four-dimensional geometry.
Recommended Reading
Asson, E. A.: Flatland, A Romance of Many Dimensions, Dover Publications, Inc.,
New York, 1952.
DESCARTES, RENE: Discourse on Method, Penguin Books Ltd., Harmondsworth,
England, 1960 (also many other editions).
DESCARTES, RENE: La Geo1nitrie (the original French and an English translation) ,
Dover Publications, Inc., New York, 1954.
HALDANE, ELIZABETH S.: Descartes, His Life and Times, J. Murray, London, 1905.
MANNlNG, H. A.: The Fourth Dimension Simply Explained, Dover Publications, Inc.,
New York, 1960.
SAWYER, W. W.: Mathematicians" Delight, Chap. 9, Penguin Books Ltd., Har
mondsworth, England, 1943.
ScOTI, J. F.: The Scientific Work of Rene Descartes, Taylor and Francis, Ltd.,
London, 1952.
WHITEHEAD, ALFRED N.: An Introduction to Mathematics, Chaps. 9 and 10, Holt,
Rinehart and Winston, Inc., New York, 1939 (also in paperback).
CHAPTER 13
THE SIMPLEST FORM U LAS I N ACTION
When you can measure what you are talking about and express it in
numbers, you know something about it.
LORD KELVIN
13-1 MASTERY OF NATURE
We have already mentioned not only a revival of interest in the study of
nature but a decided effort on the part of manufacturers, artisans, and engi
neers to utilize materials effectively and to lighten the burden of labor. This
more practical interest in exploiting knowledge of materials and of natural
phenomena in behalf of economic and social needs was an incentive for scien
tific activity which was adjoined to the older goal so strongly pursued by the
Greeks, namely, the understanding of nature. The new motivation for scien
tific work, mastery of nature for the welfare of man, was proclaimed and
advocated by such prominent and respected thinkers as Francis Bacon ( 1 562-
1626) and Rene Descartes. Bacon criticizes the Greeks. He says that the
interrogation of nature should be pursued not to delight scholars but to serve
man. It is to relieve suffering, to better the mode of life, and to increase
happiness. Let us put nature to use. Knowledge should bear fruit in works;
science should be applied to industry. In Bacon's words, let us ascend to knowl
edge and descend to work. Man should reconstitute his knowledge to apply it
to the relief of man's estate. "The true and lawful goal of science is to endow
human life with new powers and inventions." Bacon foresaw that science could
provide man with "infinite commodities" and minister to the conveniences
and comfort of man.
Descartes, too, is explicit about employing science for practical ends. He
says, "It is possible to attain knowledge which is very useful in life, and instead
of that speculative philosophy which is taught in the schools, we may find a
practical philosophy by means of which, knowing the force and action of fire,
water, air, the stars, heavens and all other bodies that environ us, as distinctly
as we know the different crafts of our artisans, we can in the same way
employ them in all those uses to which they are adapted, and thus render
ourselves the masters and possessors of nature."
280
THE SEARCH FOR SCIENTIFIC METHOD 281
The founder of modern chemistry, Robert Boyle, expressed the same
thought: "The good of mankind may be much increased by the naturalist's in
sight into the trades." The mathematician and philosopher Leibniz, about
whom we shall learn more later, proposed in 1669 the organization of a society
devoted to making inventions in mechanics and discoveries in chemistry and
physiology which would be useful to people. He, toO, wanted to put knowl
edge to use. He called the universities monkish and said that they were ab
sorbed in trifles. They possessed learning but no judgment. Instead he urged the
pursuit of real knowledge, mathematics, physics, geography, chemistry, anat
omy, botany, zoology, and history. To Leibniz the skills of the artisan and
the practical man were more valuable than the learned subtleties of professional
scholars.
One should not conclude that science and mathematics became concerned
exclu,ively with the solution of problems facing society. It is true that the
scientists of the seventeenth century worked on many specific practical prob
lems, the invention of a clock, the improvement in methods of determining
longitude, the design of better lenses, and so on. And they focused even their
more general theoretical effort on those fields of pure science-astronomy,
motion, and optics-in which practical problems predominated or whose in
vestigation gave promise of solving practical problems. But the desire to
understand nature's ways was by no means lost; it remained the outstanding
motivation for the truly great scientists and mathematicians.
13-2 THE SEARCH FOR SCIENTIFIC METHOD
We have tried to point out thus far that the scientific needs anJ interests of
seventeenth-century Europe were great. But need and interest do not in them
selves produce results. A need for money and an interest in money do not
provide money. The question still remains, How did the European scientists
go about solving scientific problems? How does one come l.0 grips with nature
either to understand or subjugate her' One might be tempted to guess that the
Europeans found the proper scientific method in the revived Greek literature.
But this was not the case.
We shall review a few principles of Greek science and late medieval
science, such as it was, to appreciate the changes made in the seventeenth
century. First of all most Greeks and medieval thinkers believed that the
basic truths exist within the human rriind. They are already implanted at birth
and are called upon when desired, or they are so clearly truths that when pro
posed, the mind immediately rec'>gnizes them as such. Thus the axioms of
Euclidean geometry were accepted by the Greeks as self-evident truths. The
medievalists added revelation from God as another source of truth, but again
a source communicated to man's mind. The task of science. then. was to
determine the implications of these principles by reasoning.
282 THE SIMPLEST FORMULAS IN ACTION
To this source of knowledge, Aristotle and his followers added observation
and induction on the basis of observations. Although Aristotle, Galen, the
celebrated physician, and astronomers such as Hipparchus and Ptolemy cer
tainly made observations, inductive conclusions did not play a great role. Also,
observational results were more likely to be forced to fit a preconceived notion
than allowed to suggest some new conclusion. For example, the principle that
heavenly motions must somehow consist of circular paths because only circular
motion is complete and perfect dominated all Greek and medieval astronomy.
Another methodological principle employed by the Greeks was classifica
tion, an approach stressed by Aristotle and taken over by his medieval fol
lowers. Thus one observed varieties of animals, flowers, fruits, and humans and
classified them according to genus and species. This method is, of course, still
used in biology and has some general applicability. It at least reduces the
variety of organisms to a few major types and permits systematic study of
whole classes in one swoop. It is relevant that Aristotle himself was a physician.
The Aristotelians pursued another scientific doctrine which is best de
scribed by the key words "qualitative study of nature." They believed that all
phenomena could be explained in terms of the acquisition or loss of basic sub
stances. Thus they and the Platonists believed that heat, coldness, wetness,
and dryness were basic substances, and these substances, combined in different
proportions, produced other substances. Heat and dryness produced fire; heat
and wetness produced air; coldness and wetness produced waterj and coldness
and dryness produced earth. The hardness or softness, coarseness or fineness
of various substances was accordingly determined by the relative abundance
of the four basic elements in them. Solids, fluids, and gases were also distin
guished by the possession of special substances. Thus a fluid such as mercury
possessed some quality, fluidity, which gold did not have. To change mercury
into gold meant that one had to take away the fluidity and substitute a new
quality which supplied rigidity. Today we recognize that solidity, fluidity, and
gaseousness are states of the same matter. However, explanation in terms of
special substances was employed right up to modern times. Early chemists, for
example Robert Boyle, ascribed the fact that substances such as sulphur were
easily set afire to the presence of a special substance called phlogiston. Until
the nineteenth century heat was considered to be a substance called caloric
which bodies lost or gained as they lost or gained heat. Electricity in the
eighteenth century was conceived of as a fluid which flowed through metals.
Aristotelian and medieval science also emphasized another objective for
science, namely explanation. To explain meant to give the cause of a phenom
enon. However, there were four distinct types of causes, each important in
its own way. Suppose that an architect builds a church. The material cause
of the church is the brick, stone, and mortar of which the church is constructed.
The formal cause is the design which the architect has in mind. Then there
is the effective cause, that is, the actual building process. The fourth type,
THE SEARCH FOR SCIENTIFIC METHOD 283
called final cause, is the purpose which the entire project serves. I n the present
example, the purpose might be to provide a house of worship or to glorify
God. Of these four types the final cause was considered most important be
cause it supplied the meaning people usually seek. Thus when we ask why
some one was killed and are told that the killer sought revenge, we are satisfied.
An entirely different explanation might be furnished in terms of the physical
and physiological processes which took place. But such an explanation is
usually not of as much interest.
In medieval thought, the final cause dominated. Rain falls to water the
crops and supply drinking water. Plants grow to supply food for man. Balls
fall to earth because all objects seek their natural place, and the natural place of
heavy objects is the center of the universe which is the center of the earth.
By the sixteenth century many scholars realized that science could not be
advanced by such means. They recognized that new principles and entirely
new methods were needed, but did not have a clear conception of what these
should be. Prior to Galileo's work, one idea emerged distinctly from the writ
ings of Aristotle's critics, namely the need for systematic experimentation.
Francis Bacon issued the manifesto for the experimental method. He attacked
preconceived philosophical systems, barren speculations, and idle displays of
learning. Scientific work, he said, should not become entangled in a search for
final causes which belonged to philosophy. In his Advancement of Learning
( 1605) and in his Novum Organum ( 1 620) he points out the feebleness of
efforts and the paucity of results in past studies of nature. Man, he observes,
has put very little thought and labor into science. Let us come to grips with
nature. Let us not have desultory and haphazard experimentation, but let it be
thorough and directed. He then makes the acute and most important statement
that the only hope for progress lies in a change of method for science. All
knowledge begins with observations. But it must proceed by gradual and suc
cessive inductions rather than by hasty generalizations. He contrasts the antici
pation of nature with the interpretation of nature. The one skims; the other
is orderly. We gain our ends only if we start with correct laws of nature. He
criticizes the then current notions of substance, quality, action, being, heaviness,
lightness, density, rareness, moistness, dryness, generation, corruption, attrac
tion, and repulsion. The Aristotelian emphasis on form, he says, is fantastic
and ill defined. Man masters nature by understanding her.
Bacon's stress on experimentation and induction from experimental results
did reflect what was beginning to take place in Europe. The work of the
biologists Vesalius, Cesalpinus, and Harvey was mentioned in Chapter 9.
Famous for his systematic experimentation is W,illiam Gilbert ( 1 540--160 3 ) ,
physician to Queen Elizabeth. I n his D e Magnete ( 1600) h e presents the de
tails and results of his clear and fruitful work on magnetism, a phenomenon
about which practically nothing was known. Gilbert states explicitly that we
must start from experiments. Kepler's regard for observational facts has al-
284 THE SIMPLEST FORMULAS IN ACTION
ready been mentioned. Galileo, too, performed some key experiments on mo
tion; about his results in this area we shall say more later. Moreover, he and
his pupils, notably Evangelista Torricelli ( 1 608-1647) , having convinced them
selves that air has weight, proceeded to carry out relevant experiments. Tor
ricelli also investigated the flow of water through nozzles. Blaise Pascal and
Robert Boyle ( \ 627-1691) worked on the pressure of fluids. Boyle and the
French priest Edme Mariotte ( \ 620-1684) studied gases such as air. Otto von
Guericke ( 1 602-1686) invented the air pump and used it to demonstrate the
pressure of air. Rene Descartes experimented in chemistry, biology, and optics.
Robert Hooke was a famous experimenter, whose work on springs we shall
discuss later, and Christian Huygens ( 1 629-1695) obtained distinguished results
from his experiments with the pendulum. Newton's work on light was one of
the greatest experimental achievements of the seventeenth century.
It is also true that the artists, engineers, and craftsmen, concerned with the
practical problems of their trades or professions, did not wait for new scien
tific methods to gain further knowledge of nature. They investigated mechan
ical forces, the design of lenses, the chemistry of paints, the motion of cannon
balls, and other phenomena and discovered new facts. To this class belongs the
self-educated sixteenth-century mathematician Tartaglia, who worked on pro
jectile motion and arrived at results which contradicted Aristotelian physics.
The Dutch engineer Simon Stevin ( \ 548-1620) learned about the pressure
exerted by water on the walls of canals, and made precise observations of the
nature of stable and unstable equilibrium of bodies. He also studied the motion
of bodies on slopes. It was spectacle makers who, without discovering a single
law of optics, nevertheless invented the telescope and microscope. Many of
these men sought not ultimate meanings but common useful knowledge.
There is no doubt that experimentation and practical investigations by tech
nicians and engineers did produce new facts and even opened up new lines of
inquiry, but the rise of experimentation was not the reason that science sud
denly blossomed in the seventeenth century. The value and import of seven
teenth-century experimentation has been vastly overrated. Modern science
owes its origins and present flourishing state to a new scientific method which
was fashioned almost entirely by Galileo Galilei. Galileo's method is doubly
important to us because, as we shall see, it assigned a major role to mathematics.
13-3 THE SCIENTIFIC METHOD OF GALILEO
Galileo, born in Pisa in 1 564, entered the university of his native city to study
medicine. He also took private lessons in mathematics and was so strongly
attracted to the subject that he decided to make mathematics his profession. At
the age of 2 3 when his application for a teaching position at the University of
Bologna was rejected because he did not seem worthy of an appointment, he
accepted a professorship of mathematics at Pisa. Galileo was one of the men
THE SCIENTIFIC METHOD OF GALILEO 285
who attacked Aristotelian science, and he did not hesitate to express his views
even though these criticisms alienated his colleagues. He had also begun to
write important mathematical papers which aroused jealousy in the less com
petent. Galileo was made to feel uncomfortable, and left in 1592 to accept the
position of professor of mathematics at the University of Padua. After 1 8 years
at Padua he was invited to Florence by the Grand Duke Cosimo II. He ap
pointed Galileo "Chief Mathematician" of his court, gave him a home and
handsome salary, and protected him from the Jesuits who had gained domina
tion of the Papacy and who had already threatened Galileo because of his
defense of Copernican theory. In Florence Gali!eo had leisure to pursue his
studies and to write. There he spent 23 years. In gratitude Galileo named the
satellites of Jupiter, which he discovered in the first year of his service under
Cosimo de'Medici, the Medicean stars.
After his condemnation by the Roman Inquisition in 1633 he was forbidden
to publish any more. But he undertook to write up his years of thought and
work on phenomena of motion and the strength of materials. The manuscript,
entitled Discourses and Mathematical Demonstrations Concerning Two New
Sciences (also referred to as Dialogues Concerning Two New Sciences) , was
secretly transported to Holland and published there in 1638. Galileo defended
his actions with the words that he had never "declined in piety and reverence
for the Church and my own conscience." He died in 1642.
Galileo began his investigation of the methodology of science by asking
what is fundamental about the world of phenomena perceived by the senses,
a question also considered by Descartes. Both agreed, as some philosophers had
asserted earlier, that color, tastes, smells, sounds, and the various sensations of
heat, hardness, and softness of objects are not distinct physical substances, but
are effects which physically existing properties produce in human beings.
What then does exist outside of man and is independent of man? The exten
sion of objects, their shapes and sizes. and their motion are real and external to
human perception. G.lileo says,
If ears, tongues, and noses were removed, I am of the opinion that shape, quan
tity [size], and motion would remain, but there would be an end of smells,
tastes, and sounds, which, ab:;tractedly from the living creature, I take to be
mere words.
Descartes' famous words in this connection are. "Give me extension and motion
and I will construct the universe." The idea advocated by these two men is
known as the doctrine of primary and secondary qualities. The primary quali
ties exist in the physical world, and their effects on the sense organs of human
beings produce the secondary qualities.
Thus in one sweeping blow Descartes and Galileo stripped away a thou
sand phenomena and qualities to concentrate on matter and motion. But this
was only the first step in the new approach to nature which Galileo was fash-
286 THE SIMPLEST FORMULAS IN ACTION
ioning. His next thought, one also voiced by Descartes and even by Aristotle,
was that any branch of science should be patterned on the model of mathemat
ics. This implies two essential steps. Mathematics starts with axioms, that is,
clear, self-evident truths. From these it proceeds by deductive reasoning to
establish new truths. So any branch of science should start with axioms or first
principles and proceed deductively
Galileo departs radically from the Greeks, medievalists, and even Descartes
in the method of obtaining these first principles. As noted earlier, the pre
Galileans believed that the mind supplies the basic principles. These men, we
might say, first decided how the world should function and then fitted what
they saw into their preconceived principles. Galileo decided that in physics as
opposed to mathematics basic principles must come from experience and ex
perimentation; they will be correct if attention is paid to what nature says
rather than what the mind prefers. He openly criticized scientists and philos
ophers who accepted principles which conformed to their preconceived ideas
of how nature should and must behave. He said that nature did not first make
men's brains and then arrange the worli:l so that it would be acceptable to
human intellects. To the medievalists who kept repeating Aristotle and debat
ing the meaning of his works, Galileo addressed the criticism that knowledge
comes from observation and not from books. It was useless to debate about
Aristotle. Those who did he called paper scientists who fancied that science
was to be studied like the A eneid or the Odyssey or by collation of texts.
"When we have the decrees of nature, authority goes for nothing; . . ." Of
course some Renaissance thinkers and Galileo's contemporary, Francis Bacon,
had also arrived at the conclusion that experimentation was necessary. With
respect to this particular plank of his new method Galileo was not ahead of all
others. Yet even a modernist as great as Descartes did not grant the wisdom
of Galileo's reliance upon experimentation. The facts supplied by the senses,
he said, can only lead to delusion. Reason penetrates such delusions. Particular
phenomena of nature can be deduced from, and understood in terms of, the
innate general principles. In much of his scientific work, Descartes did
experiment and require that theory lit facts, but in his philosophy he was still
tied to truths of the mind.
The phenomena one observes are so numerous, so varied, so unlike each
other that one can well despair of finding any principles at all in nature.
Galileo decided that he must penetrate to the core of a phenomenon and
begin there. He says in his Two New Sciences that it is impossible to treat
the infinite variety of weights, shapes, and velocities. But he had observed
that different objects fall with more nearly equal speeds in air than in water.
Hence the thinner the medium, the smaller the difference in speed of fall
among bodies. "Having observed this I came to the conclusion that in a
medium totally devoid of resistance all bodies would fall with the same speed."
What Galileo was doing here was to strip away the incidental or minor effects
THE SCIENTIFIC METHOD OF GALILEO 287
in the effort to get at the essential or major one. He then imagined what would
happen if all resistance were removed, that is, if bodies fell in a vacuum, and
he obtained the principle that in a vacuum all bodies fall according to the
same law. Thus Galileo did not just experiment and infer from experiments.
He tried to discard the relatively unimportant and nonessential, and here he
showed genius, for, as any card player knows, to recognize what to discard is
wisdom. In other words, he idealized. He did just what the mathematician
does in studying real figures. The mathematician strips away molecular struc
ture. color, and thickness of lines. to get at some basic properties, and he con
centrates on these. So did Galileo penetrate to basic physical principles.
Of course, actual bodies do fall in resisting media. What could Galileo
say about such motions? His answer was
. . . hence, in order to handle this matter in a scientific way, it is necessary to cut
loose from these difficulties (air resistance, friction, etc.) and having discovered
and demonstrated the theorems, in the case of no resistance, to use them and apply
them with such limitations as experience will teach.
Thus far Galileo had formulated a number of methodological principles,
many of which were suggested by the pattern mathematics employed in
algebra and in geometry. His next principle was to apply mathematics itself.
Galileo proposed to seek for science axioms and theorems of a special kind.
Unlike the Aristotelians and the medieval scientists, who had fastened upon
the notion of fundamental qualities, studied the acquisition and loss of these
qualities, or debated their meaning, Galileo proposed to seek quantitative
axioms. The change is most important, and we shall see its full significance in
several succeeding chapters. But an elementary example may help at the
moment to demonstrate some of its implications. The Aristotelians said that a
ball falls because it has weight and it falls to the earth because it, like every
object, seeks its natural place, and the natural place of heavy bodies is the
center of the earth. The natural place of a light body, such as fire, is in the
heavens, and hence fire rises. These principles are qualitative. By contrast let
us consider the statement that the speed (in feet per second) with which a
ball falls is 32 times the number of seconds it has been falling. This statement
can be expressed more briefly in symbols. If we denote by v the speed of the
body and by t the number of seconds it has been falling, then the above
assertion amounts to v = 32t. This simple statement illustrates many impor
tant ideas. But the relevant one at the moment is that it is primarily quantita
tive. It tells us the speed that a ball will acquire in a given number of seconds.
In two seconds its speed will be 64 feet per second; in 3 seconds, 96 feet per
second; and so on. In the expression v� 32t, the letters v and t stand for
many values. We can substitute for t any number we please and calculate the
corresponding value of v. Technically, v and t are called variables, and the
relation v� 32t is called a formula.
288 THE SIMPLEST FORMULAS IN ACTION
Galileo intended to adopt such formulas as his axioms, and he expected,
by mathematical means, to deduce from them new formulas which would
serve as theorems. Since formulas give quantitative knowledge, we can perhaps
begin to comprehend the meaning of the statement that Galileo sought quan
titative knowledge. Moreover we see that mathematics was to be the essential
medium in his scientific reasoning.
The decision to seek quantitative knowledge expressed in formulas en
gendered another decision which was also radical, although at first contact it
hardly reveals its full significance. As pointed out earlier in this chapter, the
Aristotelians believed that one of the tasks of science was to explain why
things happened, and explanation meant unearthing the causes of a phenom
enon. The statement that a body falls because it has weight gives the effective
cause of the fall, and the statement that it seeks its natural place gives the final
cause. But the quantitative statement v = 32t, for whatever it may be worth,
does not explain why a ball falls. It tells only how speed changes with time.
In other words, formulas do not explain; they describe. And the knowledge
of nature Galileo sought was descriptive. He says, for example, in his Two
New Sciences that he will investigate and demonstrate some of the properties
of motion without regard to what the causes might be. Positive scientific in
quiries were to be separated from questions of ultimate causation.
First reactions to this tl:ought of Galileo are likely to be negative. De
scriptions of phenomena in terms of formulas hardly seem to be more than
a first step. It would appear that the Aristotolians had really grasped the true
function of science, namely, to explain why phenomena happened. Even
Descartes protested Galileo's decision to seek descriptive formulas. He said,
"Everything that Galileo says about bodies falling in empty space is built with
out foundation: he ought first to have determined the nature of weight." Fur
ther, said Descartes, Galileo should reflect about ultimate reasons. But we shall
see more clearly in the space of a few chapters that Galileo's decision to aim
for description was the most profound and the most fruitful thought that any
one has had about scientific methodology. We merely wish to recapitulate here
that the scientific knowledge which Galileo envisioned was to consist of a
series of mathematical formulas deduced from a few fundamental ones.
Since the laws Galileo proposed to find were to be quantitative, they
obviously had to relate measures, sizes, or amounts of some physical quantities,
just as v = 32t relates measures of speed and time. Here, too, Galileo made a
fundamental contribution. Whereas the Aristotelians had talked in terms of
qualities such as earthiness, fluidity, rigidity, essences, natural places, natural
and violent motion, potentiality, actuality, and purpose, Galileo not only in
troduced an entirely new set of concepts but chose concepts which were
measurable so that their measures could be related by formulas. Some of his
concepts, such as distance, time, speed, acceleration, force, mass, and weight,
are, of course, familiar to us and so the choice does not surprise us. But to
Galileo's contemporaries these choices and in particular their adoption as funda-
THE SCIENTIFIC METHOD OF GALILEO 289
mental concepts, were startling. However, these very ones did prove to be
most instrumental in the task of understanding and mastering nature.
We have described the essential features of Galileo's program. Some of
his ideas had been espoused by others. Some were entirely original with him.
But what establishes Galileo's greatness in the invention of this methodology
is that he saw clearly what was wrong or deficient in the scientific efforts of
his age, completely shed the older ways, and formulated the new steps-almost
in so many words. Moreover, he applied his method to problems of motion
and in this work not only managed to provide a lucid example of the procedure
but succeeded in obtaining brilliant results. He showed, in other words, that it
worked. Galileo was fully conscious of what he had accomplished. He says
toward the end of his Two New Sciences, "So that we may say the door is now
opened, for the first time, to a new method fraught with numerous and wonder
ful results which in future years will command the attention of other minds."
But others were also aware of Galileo's greatness. The seventeenth-century
philosopher Thomas Hobbes said of Galileo, "He has been the first to open to
us the door to the whole realm of Physics . . .
"
Since we are interested in the role of mathematics in the modern world.
it may be worth while to emphasize one point. The scholars who fashioned
modern science, Descartes, Galileo, and Newton, approached the study of
nature as mathematicians. They proposed to find broad, profound, but also
simple and clear mathematical principles either through intuition or through
crucial observation and experiments and then expected to deduce new laws
from these principles, entirely in the manner in which mathematics proper
had constructed its geometry and algebra. Mathematical deduction was to
take up the major share of scientific activity. Galileo says he valued a scientific
principle, whether or not obtained by experimentation, far more because of
the abundance of theorems which he could deduce from it than because of the
knowledge afforded by the principle itself.
What these great thinkers envisioned did in fact prove to be the profitable
course. For the next two centuries, scientists formulated precise and sweeping
mathematical laws of nature on the basis of slim, almost trivial, observations
and experiments. The greatest progress in the seventeenth and eighteenth cen
turies occurred in mechanics and in astronomy, and in both these fields experi
mental results were hardly startling and certainly not decisive. The significant
contribution, as we shall see, was the creation of vast branches of mathematical
theory.
The expectations of these scientists, seemingly rash, can be explained. These
men were convinced that nature is mathematically designed and therefore saw
no reason why they could not proceed. in scientific matters as mathematics had
proceeded in the study of numbers and geometric figures. As Randall says in
his Making of the Modern Mind, "Science was born of a faith in the mathe
matical interpretation of Nature, held long before it had been empirically
verified."
290 THE SIMPLEST FORMULAS IN ACTION
EXERCISES
1. What properties of physical objects did Descartes and Galileo regard as funda
mental and real?
2. What is the distinction between a qualitative and a quantitative study of nature?
3. Describe the essential principles in GaIileo's plan of scientific activity and contrast
them with those of his predecessors.
4. Contrast the Greek objectives in the study of nature with those advocated by
Bacon and Descartes.
5. How does mathematics enter into Galileo's scientific method?
13-4 FUNCTIONS AND FORMULAS
We intend to pursue the seventeenth-century developments initiated by Galileo
and to pay particular attention to the role of mathematics in Galileo's method.
Let us recall just what Galileo set out to do. He proposed to find fundamental
quantitative physical principles or laws and to apply mathematical reasoning
to these quantitative statements in order to deduce new physical laws. These
physical laws would then provide the answers to a variety of scientific and
practical problems. To express the physical principles in the manner he re
garded as significant, Galileo introduced a new mathematical concept, the
extremely important concept of a function. For the next two centuries mathe
maticians devoted themselves to the construction of functions and to the study
of their properties. But the purely mathematical aspect of these creations is in
itself rather barren. It is merely the sketch of a picture. And the picture in
the present case is precisely the physical world which Galileo set out to in
vestigate. Hence, as we study functions, we shall also study the situations
which gave rise to them and the good that was accomplished with them. In
fact, it is artificial to separate the physical thinking from the accompanying
mathematics, for the two were developed as one. The leading mathematicians
of the seventeenth and eighteenth centuries were also the leading scientists.
And the accomplishments of these two centuries were a triumph of mathe
matics and science conjoined.
Before proceeding with Galileo's work, we shall familiarize ourselves with
the notion of a function. Let us consider the situation in which a ball is
dropped from some point above the ground and let us suppose that we wish
to describe the distance the ball falls with increasing time. (Why we should
seek such a description and what we can do with it are questions we shall
answer later.) It is understood that the distance is measured downward from
the point at which the ball begins to drop, and the time of fall is measured
from the instant the ball begins to fall. Then the correct description, which
we can accept for the moment as a fact, says that the distance the ball falls,
measured in feet, is 16 times the square of the number of seconds it falls. The
italicized statement is an example of a function. As such it is important in two
FUNCTIONS AND FORMULAS 291
respects. First of all, it deals with varying quantities or vll7iables. The number
of seconds that the ball falls increases from zero to larger and larger values.
The distance that the ball falls also increases from zero to larger and larger
values. Secondly, the statement specifies exactly the relationship between the
variables time and distance. What is characteristic of functions, then, is thar
they are precise statements of relationships among variables.
We know that verbal statements are clumsy to work with. Our experience
with algebra teaches us that we can be more effective by introducing symbols.
Let us then introduce the symbol t to stand for any number of seconds that
the ball has been falling and the symbol d to stand for the distance that the ball
falls in t seconds. In these symbols the italicized statement above says that
d = 1 6t2•
The algebraic expression of a functional relationship is called a formula.
Several facts about formulas are important for their proper understanding and
use. In the present case, for example, we must be sure to note that the letters
d and t represent not just one particular value of distance and time but whole
ranges of values. Thus, if the ball falls for 5 seconds, the variable t can repre
sent any number from 0 to 5. The variable d can represent any distance which
the ball may have fallen during the 5 seconds. Of course, the values of d are
not independent of the values of t. In fact, the whole point of the formula
is to tell us precisely what d is for a given t. Thus when t is 2, for example,
d is 16 · 2', or 64. That is, by substituting a particular value of t in the formula,
we can calculate the distance d that an object has fallen in that number of
seconds chosen for t. The values of d depend upon the values of t and, for
this reason, t is called the independent vll7iable and d, the dependent variable.
One also says that the formula expresses d as a function of t. Since we can
calculate d for millions of values of t, the formula is indeed a compact repre
sentation of millions of bits of information.
Suppose that the dropped ball falls for just 5 seconds. It then hits the
ground and remains at rest. However, the formula d = 1 6t2 does not "stop"
at the end of 5 seconds. We could substitute 6 for t and find that d is 16 · 36,
or 576. Likewise, we could substitute 7, or 9!, or even -2 for t and in each
case calculate the correponding value of d. Thus, the mathematical formula
has meaning for all positive and negative values of t. However, if the ball
falls for only 5 seconds, the formula represents the physical situation only for
values of t from 0 to 5. In other words, the mathematical formula is more
extensive than the physical situation.
We used the letters d and t to represent the variables distance and time.
We could have used y and x, in which case the very same formula would read
y = 16x2•
292 THE SIMPLEST FORMULAS IN ACTION
The letters d and t happen to be better because they suggest the physical
meaning. But nothing would be altered mathematically if we used y and x.
Discussion of a formula and of its physical significance is often aided by
utilizing the ideas of coordinate geometry. We can think of d = 16t' as the
equation of a curve. Since the choice of particular letters does not have any
mathematical significance, we can introduce axes d and t (Fig. 1 3-\). The
curve corresponding to d = 16t2 consists of those points whose abscissa and
ordinate or whose t- and d-coordinates satisfy the equation. Thus, since for
t = 1, we have d = 16, the point whose abscissa is 1 and whose ordinate is
16 lies on the curve. (For convenience we use a smaller unit on the d-axis.)
d
150
120
00
(2,64)
60
Fig. 13-1.
The graph of d = 16t'.
The curve need not be and is not here a picture of the physical motion.
Nevertheless, it does show that compared with t, d increases very rapidly
as t increases beyond the value O. Moreover, the curve reveals that the formula
has mathematical meaning for all positive and negative values of t, whereas
only that part of the curve which extends from t = 0 to t = 5 represents the
physical situation.
EXERCISES
1. What is a function?
2. Distinguish between a function and a formula.
3. "Cook up" some mathematical formulas of your own, whether or not they have
any physical significance.
4. For the formula v = 32t, calculate the value of v when t is 0, 3, 7, 4t, and -6.
5. ]n the formula A = '1Tr, which quantity would you regard as the independent
variable and which as the dependent one?
6. For the formula d= 16t2, calculate the value of d when t= 2!. -4, and 7.
7. For the formula v = 32t, calculate the value of t when v = 64, 80. 128.
8. For the formula d= 16t2, calculate the value of t when d= 144. Are both
answers physically significant?
FORMULAS FOR MOTION OF DROPPED OBJECTS 293
9. For any given temperature, the relationship between readings in the fahrenheit
scale and the centigrade scale is
F=!C + 32.
What is F when C=O? What is F when C= 100? Do these values of F and C
have special physical significance?
13-5 THE FORMULAS DESCRIBING THE MOTION OF DROPPED OBJECTS
Galileo not only formulated the general program for science, but he put it into
effect. And here, too, he showed immense wisdom. He did not try, as had
scientists and philosophers before him, to tackle the whole universe or to em
brace man and nature in one theory. He decided to concentrate on a few
classes of phenomena, and principally on motions near the surface of the earth.
Galileo possessed the restraint which proves the master.
As we noted earlier in this chapter, Galileo thought as a mathematician,
and he began his work by idealizing the problem he set out to solve. He, too,
considered the motion of a ball, say rolling along the ground, and he asked,
What if air resistance, friction between ball and ground, and any other hinder
ing forces were not present' What would the ball do once it were set into
motion? He concluded that the ball will continue indefinitely to move at a
constant speed in a straight line. In more general terms, if no force acts on a
body and the body is at rest, it will remain at rest; if no force acts and the
body is in motion, it will continue to move at a constant speed in a straight line.
This fundamental principle of motion or axiom of physics is now known as
Newton's first law of motion. We should note that it contains two important
assertions. The first, that a body in motion will continue to move in a straight
line, is no innovation. It says that straight-line motion is the natural motion of
bodies, the motion they will pursue unless they are forced to deviate from such
a path. Bur the second assertion, that the body will continue to move at a con
stant speed indefinitely, is a radical departure from Aristotle, for Galileo was
saying that no force is needed to keep the body going once it is set into motion.
Bur what if a force is applied to a body? Galileo answered that if the body
is at rest, the force will set it in motion and change its speed from zero to some
nonzero quantity. If the body is already in motion, the force will change the
speed, the direction of the motion, or both. Thus, an object set into motion
along a rough surface encounters friction. Friction is a force, and its effect
is to reduce the speed of the object. Galileo's second principle, then, states that
force produces change in speed or direction. In other words, Galileo said that
force causes acceleration.
Let us consider an object which moves along a straight line, but which is
being accelerated. The acceleration is a gain or loss in speed per unit time.
Thus, if an object has been moving at a speed of 30 feet per second and if in
294 THE SIMPLEST FORMULAS IN ACfIQN
one second its speed increases to 40 feet per second, then the acceleration is
10 feet per second for that one second, or 10 feet per second per second or, in
scientific shorthand, 10 ft/sec2• If the increase in speed had been 10 feet per
second over two seconds, then the acceleration would have been 5 ft/sec2.
Now, as the Greeks had observed, a body which is dropped falls with
increasing speed; its motion is accelerated. Since falling bodies possess accelera
tion, that is, they do not move at a constant speed, it follows that some force
must be causing the change in speed. By Galileo's time the concept of gravity
had become more or less accepted. The earth exerts a force on any object and
this force, if not offset by some other force, gives the object an acceleration.
The surprising fact which Galileo discovered is that if one neglects air resis
tance, then an object falls to earth with a constant acceleration, and, moreover,
this constant is the same for all bodies, namely 32 ft/sec2• Thus if we let a
stand for acceleration, the third fundamental law of motion, an axiom of
physics, states that for all bodies falling to earth·
a = 32. (1)
We now have a few fundamental principles about motion. Let us see
next whether, in accordance with Galileo's plan, mathematical reasoning can
lead to new information. Let us consider the motion of an object which is
dropped, that is, whose initial speed is zero. Galileo's third principle says that
the object gains speed each second at the rate of 32 ft/sec. Hence, at the end
of one second its speed is 3 2 ft/sec. At the end of two seconds its speed is
2 times 32 ft/sec or 64 ft/sec. At the end of t seconds its speed is t times
3 2 ft/sec.or 32t ft/sec. If we let v denote the speed at the end of t seconds,
then
v = 321. (2)
We now have a formula which tells us the precise speed which a dropped
body acquires in t seconds. It can, of course, be used to calculate v for any
given value of t. Thus at the end of 6 seconds the speed of the body is
192 ft/sec.
Formula (2) is of some interest but hardly a surprise. Let us see whether
we can obtain more significant conclusions by the further application of
mathematics. We wish to determine the distance which a dropped body falls
in t seconds. To be specific, let us consider for the moment that t = 6. Now
the speed at the end of 6 seconds is 192 ft/sec. To obtain the distance traveled
in 6 seconds, one is tempted to multiply 192 by 6, that is, the speed by the
time. However, the object did not travel at 192 ft/sec throughout the 6
seconds. In fact, it started with zero speed and only gradually increased its
• This axiom applies only to objects near the surface of the earth. We shall say
more about it in Chapter 15.
FORMULAS FOR MOTION OF DROPPED OB]ECfS 295
speed to 192. Which speed should we use to compute the distance traveled?
Presumably, the average speed.
A reasonable guess would be that the average speed is the arithmetic
average of the initial and final speeds, that is, (0 + 192)/2 or 96 ft/sec. Let us
see whether we can establish the correctness of this guess.
Suppose we calculate the speeds at the instants t 0, 1 , 2, 3, 4, 5, and 6.
=
If we substitute these values of t in (2) we obtain the speeds
0, 32, 64, 96, 1 28, 1 60, 192.
If we take the average of these seven speeds, that is,
0+ 32 + 64+ 96+ 1 28 + 1 60+ 192
7
we obtain 96 ft/sec. Of course, this calculation does not prove that the
average speed is 96 because the object falls with varying speed even during
the first second, the second second, and so forth. We might therefore average
the speeds attained by the object after each half-second of fall, that is, the
speeds at t = 0, t, I, t, 2, . . . . , 6. These speeds are
0, 16, 32, 48, 64, 80, 96, 1 1 2, 1 28, 144, 1 60, 1 76, 192. (3)
If we average these speeds we again get 96 ft/sec. This calculation is no more
of a proof that 96 is correct than the preceding one because even in each half
second the object falls with a varying speed. Vet we continue to get the
average of 96. Let us see whether we can find the reason that 96 results each
time.
We note that 96 is the speed the object attains when t = 3, because if we
substitute 3 for t in ( 2 ) we get v = 96. Now it seems significant that the
average speed occurs halfway through the time interval from ° to 6. This
probably is the case because the speed at each instant before t = 3 and the
speed at some corresponding instant after t = 3 average to 96. Indeed, if we
examine the speeds in ( 3 ) we see that ° and 192, 16 and 176, J2 and 160, and
so on, each averages to 96. In other words, if for each instant before t = 3,
for example, t = It, and for the instant as far beyond t = 3, in our example
t = 4t, the average speed is 96, then the average for all the instants will be 96.
Just to check once more, at t = Ii the speed is 48 and at t = 4t the speed is 144.
The average of 48 and 144 is 96.
This argument can be generalized. Let h be any interval of time. Then
3 - h is some instant before 3 seconds, and, by formula (2), the speed of the
falling body at the instant t = 3 - h is 32 ( J - h) = 96 - 32h. At the instant
t = 3 + h, which is h seconds after 3 seconds, the speed is J2 ( J + h) or
96 + 32h. We see that the speed at t = 3 - h is 32h less than 96, and the speed
296 THE SIMPLEST FORMULAS IN ACTION
at t = 3 + h is 32h more than 96. Hence the average speed for these two
instants is 96 because
96 - 32k + 96 + 32k
96.
2
Since the object falls for as many instants during the interval from t = 0 to
t = 3 as it does from t = 3 to t = 6, and since the pairing of instants equally
far from t = 3 produces the average of 96, the average speed over the entire
interval of 6 seconds is 96 ft/sec. It is important to note that this average
speed is attained after one-half of the time of travel.
If, instead of 6 seconds, we had used the general value of t seconds, then
our conclusion would read that the average speed is attained after t/2 seconds.
Since v = 32t, the average speed in the interval from 0 to t is given by
average speed = 1 6t.
The argument used to derive the average speed utilizes formula (2) and
this is correct only when (1) holds. One should not, then, use the conclusion
that the average speed is the speed attained at t/2 seconds in other kinds of
motion. However. the argument does hold when the acceleration is constant,
even if that constant is not 32.
Now that we have the average speed of an object which is dropped and
falls for t seconds, we can calculate the distance fallen. The average speed is
that constant speed with which the object could have fallen to cover the same
distance in t seconds. Since this average is a constant speed, we have but to
multiply it by the time of travel to obtain the distance fallen. Then since
161 . t = 1 6t', if we let d represent the distance fallen in t seconds, we have
the new result:
d = 16t'. (4)
Formula (4) says, for example, that in 3 seconds the object falls 1 6 · 3', or 144
feet. With a little mathematics we have derived an important law of falling
bodies. It tells us the distance which any body that is dropped and freely
falling travels in t seconds.
We can derive a few significant consequences of formulas (2) and (4) by
the application of simple algebra. Dividing both sides of formula (4) by 1 6 and
taking the square root of both sides of the resulting equation, we obtain
t=±�.
This result tells us the time required for a dropped body to fall d feet. Of
FORMULAS FOR MOTION OF DROPPED OBjECfS 297
course, of the two roots (one positive and one negative), only the positive
value possesses physical significance because we are dealing with a physical
situation in which t-ime is positive and measured from the instant the body
begins to fall. Hence we shall forget about the negative root and consider that
t= �. (5)
If we now wished to calculate how long it takes an object to fall 1000 feet, we
would substitute this value in formula (5) and calculate t.
From formula (5) one can draw a most significant conclusion. The for
mula does not tell us the name of the President of the United States, but this
is not so surprising. However, it is surprising that the formula does not involve
the weight or any other property of the falling body. This means that all
bodies take the same time to fall a given distance, provided, of course that air
resistance is neglected. A feather and a piece of lead take the same time to
fall a given distance in a vacuum. This is the les:;on which Galileo is supposed
to have learned by dropping various objects from the leaning tower of Pisa.
Many people still hesitate to accept this conclusion because they observe bodies
falling in air, and the resistance encountered by feathers is Quite different from
that offered to lead. Undoubtedly it was this difference gleaned from actual ob
servations which led the Aristotelians to the conclusion that heavier bodies
fall faster.
Fonnula (5) was derived by merely rearranging, so to speak, formula (4).
But Galileo's plan envisaged also combining existing formulas to obtain new
knowledge. To illustrate this process, suppose one takes the value of t given
by (5) and substitutes it in the formula v = 32t. This yields
v = 32 (d.
'\f16
Now the square root of a fraction is equal to the square root of the numerator
divided by the square root of the denominator. Hence
Vd
v = 32 - = 8Vd. (6)
4
The new formula enables us to calculate the speed which a dropped body will
acquire in falling d feet. While this information. is implicit in fonnulas (4)
and ( 5 ) , which yield (6), we now see clearly something we might not have
appreciated before. We should note that formula (6) says that the speed
increases as the square root of d. The predecessors of Galileo believed that
the speed increased directly with distance.
298 THE SIMPLEST FORMULAS IN ACfION
EXERCISES
1. Was Aristotle wrong in asserting that, to keep an object moving at a constant
speed in a real medium, a force must constantly be applied?
2. Suppose that gravity does not exist and a man steps off the roof of a building.
What would his subsequent motion be? What would it be in the presence of
gravity?
3. An automobile travels at the speed of 10 mi/hr for 59 min and at a speed of
50 mi/hr for 1 min. What is its average speed?
4. What is the speed of an object 4 sec after it is dropped? What is its average
speed during the 4 sec? At what instant does the object actually possess this
average speed?
5. Distinguish between speed and acceleration.
6. We may regard the formula v= 32t as an equation in v and t, and we may
therefore plot t-values as abscissas and v-values as ordinates. Draw the curve of
the formula v = 32t.
7. Using the formula d = 16t2, calculate how far a body will drop in 5 sec, 6! sec,
10 sec. Is the drop the same from one second to another?
8. Apply the instructions of Exercise 6 to the formula d = 16t2• What is the name
of the resulting curve?
9. Graph the curve of d= 16t2• but let the downward direction of the vertical
axis, that is the d-axis, be positive.
10. A window washer at the 50th floor of a skyscraper (500 ft above the street)
steps back to observe the results of his work. Describe mathematically his subse
quent behavior.
1 1 . Using the fonnula v = 8#, calculate the speed with which an object dropped
from the top of the Empire State Building (about 1000 ft above street level) hits
the ground.
12. If the relation between speed and distance were v = 8d instead of v=8Vd,
what difference would there be in the behavior of falling bodies?
13. Suppose that we are considering the motion of an object which is dropped
from a point near the surface of the moon. On the moon all objects also fall to
the surface with a constant acceleration, and the value of this acceleration is
5.3 ft/sec'. What change would you make in formulas ( 2 ) and (4) to have
them represent speed acquired and distance traveled for objects falling to the
moon's surface? Incidentally, the moon has no atmosphere, and hence air re
sistance can surely be neglected.
14. Suppose that an object is dropped and falls with a constant acceleration a. What
would you propose as formulas for speed and distance faIlen in time t?
15. Show first that formula (6) implies d= v'/64. Now suppose a dropped object
acquires a speed of 88 ft/sec. What distance must it fall to acquire this speed?
16. Suppose an object is traveling along a straight line with a speed of 88 ft/sec and
then starts to lose speed, that is decelerates, at the constant rate of 32 ft/sec2•
What distance must it travel for its speed to become zero? [Suggestion: The
FORMULAS FOR MOTION OF OBJECTS THROWN DOWNWARD 299
distance it must travel to reach zero speed is the same as the distance it would
travel if it starred with zero speed and accelerated at 32 ft/sec2 to attain a speed
of 88 ft/sec.]
17. As a direct generalization of the thought in Exercise 16, we may state that if
an object is traveling in a straight line at a speed of v ft/sec and then loses speed
at the rate of 32 ft/sec2, the distance d it travels before attaining zero speed is
d=v/64. Suppose the deceleration is 11 ft/sec2. What formula gives the dis
tance the object travels before attaining zero speed?
18. Using the result of Exercise 17, answer the following question. An automobile
is traveling at 60 mijhr (or 88 ft/sec) , and the brakes are applied. The action of
the brakes decelerates the automobile at the rate of 11 ft/sec2. How far will the
automobile travel before stopping? The answer gives the minimum distance in
which one can, even under most favorable road conditions, stop a car traveling
at 60 mi/hr. However, it takes about 1 sec before a person who decides to apply
the brakes actually does so. What distance will the automobile travel in that
time?
19. A man drops a stone into a well and listens for the sound of the splash. He
finds that 6! sec elapse from the instant the stone is dropped until he hears the
sound. How far below is the surface of the water? Assume that sound travels at
1152 ft/sec.
13-6 THE FORMULAS DESCRIBING THE MOTION OF OBJ ECTS THROWN DOWNWARD
Thus far we have seen how simple formulas describe the motion of a body
which is dropped. By employing slightly more complicated fonnulas Galileo
was able to tackle further phenomena of motion. Suppose that instead of being
dropped a ball is thrown downward. Now the ball does not start its motion
with zero speed but with whatever speed the hand imparts to it. The problem
we shall look into is, What is the subsequent motion of the ball? To be specific,
suppose the hand imparts to the ball a speed of 96 ft/sec. Neglecting for the
moment the action of the force of gravity, we can say that the ball will con
tinue to travel downward in a straight line with a speed of 96 ft/sec. The basis
for this assertion is, of course, the first law of motion. We know, however, that
gravity will also act on the ball and give it a speed of 32t ftlsec in t seconds.
Since both speeds will operate simultaneously to make the ball move down
ward, the total speed, v, of the ball is represented by the formula:
v = 96 + 32/. (7)
Let us compare this formula with v = 32/. We see that the term 96 in
formula (7) represents the speed given to the ball by the hand. Both formulas
are said to be of the first degree in t because the independent variable, t, ap
pears only to the first power. That is, the formulas contain 32t as opposed to
32t', or 32t', or some other power of t. First-degree formulas are often called
linear functions because the curve representing each is a straight line.
300 THE SIMPLEST FORMULAS IN ACTION
We can also obtain the formula for the distance, d, which the ball will
fall in 1 seconds. If there were no gravity, the ball would fall a distance of 961
feet in 1 seconds because it would have the constant speed of 96 ft/sec im
parted by the hand. But during the same 1 seconds, the force of gravity will
exert an additional downward pull which, according to formula (4), will cause
the ball to fall 16/2 feet. Since both forces, the hand and gravity, cause the
ball to fall downward, the total distance, d, traveled in 1 seconds is
d = 961 + 1 6t2 • (8)
Comparison of formula (8) with formula (4) shows that formula (8) contains
a new term, 96t, which represents the contribution made by the action of the
hand to the distance the ball falls. Formulas (4) and (8), incidentally, are of
the second degree in 1 because the independent variable 1 occurs to the second
power. Second-degree functions are also called quadratic functions.
EXERCISES
1. If a ball, instead of being merely dropped, is thrown downward with a speed of
128 ft/sec, will its speed be greater in t seconds? Will the distance fallen in t
seconds be greater?
2. Write the formula representing the speed acquired and distance traveled in
seconds by a ball which is thrown downward with a speed of 128 ft/sec.
3. Suppose a ball is thrown downward with a speed of 96 ft/sec. What are the
speed and distance traveled after 3 sec? after 4i sec?
4. Graph formula (8) by plotting points whose coordinates satisfy the equation.
What is the name of the resulting curve?
5. Graph formula (8) by applying the method of change of coordinates presented
in Chapter 12.
13-7 FORMULAS FOR THE MOTION OF BODIES PROJECTED UPWARD
A more interesting phenomenon both physically and mathematically is the
motion of a ball thrown straight up into the air. Suppose, for example, that the
ball is thrown upward with a speed of 96 ft/sec, and let us again consider the
questions, What are the speed and distance traveled after 1 seconds of motion?
If gravity is neglected, then the action of the hand will cause the ball to start
upward with a speed of 96 ft/sec and, according to the first law of motion, it
should continue to travel upward at that speed indefinitely. However, we
know that the downward pull of gravity causes the ball to acquire in t seconds
a downward speed of 32t ft/sec. Since the hand gives the ball an upward speed
of 96 and gravity gives it a downward speed of 32/, the net speed, v, of the ball
at the end of 1 secrmds is
v = 96 - 3 2/. (9)
FORMULAS FOR MOTION OF BODIES PROJECTED UPWARD 301
The minus sign in formula (9) takes care of the fact that the speed resulting
from the action of gravity reduces the speed imparted by the hand. Formula
(9) should be compared with formula (7).
Let us turn to the second question: How far does the ball travel? Since the
ball travels upward to some maximum height and then falls down, we shall
instead ask the more pertinent question, What height above the ground does
the ball possess at any time t1 If there were no gravity, the ball would move
upward at the constant velocity of 96 ft/sec. Hence in t seconds it would
travel upward 96t feet. However, we know that a ball moving above the
surface of the earth for t seconds will experience a downward pull of gravity
amounting to a distance of 1 6t' feet in t seconds. Hence the net height, d,
reached by the ball is
d = 96t - 16t'. (10)
As in formula (9), the minus sign here represents the fact that the action of
gravity offsets the action of the hand.
We can now answer some questions about the motion of the ball. We
know from experience that the ball will rise to some height and then fall back
to the ground. How high will it go? We would expect the ball to continue
to rise until its upward speed, which is continually decreasing, becomes zero.
This fact can be put to use through formula (9). We now ask the question,
What is t when v = O? Suppose we denote by t, this particular unknown value
of t. Then we may say an the basis of formula (9) that
o = 96 - 32t,.
To find t, we have but to solve this simple equation. Clearly t, = 3.
We have determined the time it takes the ball to reach its maximum height
but not the height itself. However, formula ( 10) gives us the height at any
time t. Suppose then that we substitute the value t, = 3 in ( 1 0) and calculate
d" the height of the ball above the ground at this instant. Substitution of the
quantity 3 for t in ( 10) yields
d, = 96 ' 3 - 1 6 ' 3 2 = 144.
Thus the maximum height above the ground to which the ball will rise is
144 feet.
We know that the ball will fall to the ground after reaching this maximum
height. Will it take as much time to reach the ground from its maximum
height as it did to travel from the ground to the maximum height' Those
people who have lots of confidence in their intuition should answer this ques
tion before we settle it by mathematical reasoning.
To obtain an answer we must proceed somewhat indirectly. Our informa
tion about the motion is contained in formulas (9) and ( 1 0). Of these two
302 THE SIMPLEST FORMULAS IN ACfION
formulas, ( 10) offers some prospect of being useful because it relates time
and height reached by the ball. There is one bit of information which might
be used in connection with formula ( 10), namely, that when the ball reaches
the ground, the height of the ball above the ground is zero. Let us therefore
find the time at which the ball reaches the ground and then see what it suggesrs.
We denote by I, the value of 1 at which the ball reaches the ground.
Hence, by formula ( 1 0 ) ,
o = 961, - 161�. (II)
Our problem now is to determine the value o f I, which, according to ( I I ) ,
satisfies a second-degree equation. This equation is easily solved. We apply
the distributive axiom to justify writing
0 = 1 61,(6 - I,) . ( 1 2)
We are seeking the value or values of I, for which the right side of ( \ 2 )
equals the left side. Clearly when I, 0 , one factor o n the right side is zero,
=
and hence the product is zero. Likewise, when t2 = 6, the product is zero.
Hence, there are two values of I, namely 0 and 6, when d, the height above the
ground, is zero.
Why two values? Mathematically the two values result from the fact that
we are solving a quadratic or second-degree equation. Physically the two
values are readily understandable. The value I, = 0 is the value of 1 at the
instant at which the ball is about to start out; I, = 6 is the value of 1 at the
instant at which the ball hits the ground after traveling up and down. With
respect to the problem in hand, the second value is the interesting one because
it tells us that 6 seconds elapse during the upward and downward travel of
the ball. Since we found earlier that the ball requires 3 seconds to reach its
highest position, it is evident that only 3 seconds are required for the ball to
return to the ground. Hence it takes exactly the same time for the ball to go
up as it does to come down.
We can now ask mathematics to answer another question for us. What
speed does the ball possess when it strikes the ground? Is this speed the same,
more, or less than the 96 fr/sec with which it was thrown up? The answer
can be obtained at once. Formula (9) gives the speed of the ball at any instant
of its flight. The ball strikes the ground at the instant 12 = 6. If we substitute
6 for 1 in (9), we find that V2, the speed at the instant the ball strikes the
ground, is
v, = 96 - 3 2 ' 6 = -96.
Thus mathematics tells us that the speed is 96 ft/sec, the very same speed with
which it was projected upward. Obligingly, mathematics also tells us, through
the minus sign, that the speed is in the opposite direction to that of the upward
throw.
FORMULAS FOR MOTION OF BODIES PROjECfED UPWARD 303
EXERCISES
1. For a ball thrown upward with a speed of 128 ft/sec, what formula describes the
relationship between the subsequent height above the ground and the time of
travel?
2. If a ball is thrown upward with a speed of 160 ft/sec, then the formula relating
its subsequent height above the ground and the time of travel is d = 160t - 16t2,
and the speed of the ball is given by the formula
v = 160 - 321.
a) How high is the ball after 4 sec?
b) What is its speed at the end of the fourth second?
c) How high will the ball go?
3. If the height of a ball thrown upward is given by the formula d = 1441 - 161',
what is d when t = 9? Interpret the result physically.
4. If the height above the ground of a ball is representable by the formula
d = 192t - 16t2, then its height after 4 sec is 5 1 2 ft, and its height after 8 sec is
also 512 ft. Verify these heights and account for the fact that the height is the
same after 4 additional seconds.
5. If a gun capable of firing a bullet at the speed of 1000 ft/sec is fired straight
upward, how high will the bullet go?
6. Suppose that a ball is dropped from the top of a building 100 ft high. Let d
represent the height of the ball above the ground and t the time of travel meas
ured from the instant the ball is dropped. Write a formula representing the
motion in terms of d and t.
7. Since man seems to be preparing for experiences on the moon, it may be well
to consider the following question. Suppose that a ball is thrown up from the
surface of the moon with a speed of 96 ft/sec. How high will it go and how
long will it take to reach its maximum height? Remember that on the moon the
value of 5.3 ft/sec2 corresponds to the acceleration of 32 ft/sec2 which holds on
the earth.
8. Suppose that a bullet shot straight up into the air returns to the ground 60 sec
later. What was the initial speed? [Suggestion: Use formula ( 10). However,
the initial speed, which was 96 in formula ( 10), is now unknown.]
9. A rocket is shot straight up into the air to a height of 50 mi at which point its
velocity is 300 mi/hr. Its fuel is now exhausted, and hence the rocket receives
no further acceleration from this source. Write a formula which describes the
subsequent motion of the rocket. [Suggestion: You can choose the 50-mi point
as the origin for height and zero time as the instant when the rocket is at that
height.]
10. A ball is thrown up into the air from the roof of a building with a speed of
96 ft/sec. Write a formula for its subsequent height above the roof as a function
of time.
11. Using the data of Exercise 10 and the additional fact that the roof is 1 12 ft above
the ground, find the time when the ball reaches the ground.
304 THE SIMPLEST FORMULAS IN ACfION
REVIEW EXERCISES
In all of the exercises below the units are feet and seconds.
1. An object which is dropped near the surface of the earth acquires a speed in
ft/sec of v = 32t in t sec. Calculate v when t is
a) 7 b) zt d) <Ii e) 9.
2. An object falls and acquires the speed v= 32t in t sec. How long does it take to
acquire a speed of
a) IZ8 b) 160 c) 400 d) 16?
3. If an object acquires speed according to the formula v= 32t, what is its average
speed during
a) the first 5 sec of fall,
b) the first 8 sec of fall?
4. Suppose an object acquires speed according to the formula v = 32t. and we
wished to compute the average speed during the eighth second of fall. Could
we use the argument in the text (properly modified) to average the speed at
t = 7 and the speed at t = 8?
5. Suppose an object is dropped and falls with a constant acceleration of g ft/sec2
instead of 32 ft/sec2• What formula relates the speed and time of fall?
6. Suppose an object is dropped and falls with a constant acceleration of g ft/sec2•
Then in t sec it acquires a speed of v = gt ft/sec. One might convince himself
that the argument given in the text where the acceleration is 32 ft/sec2 and the
average speed proves to be 16t carries over to the case where the acceleration
is g. Suppose the argument does carry over.
a) What is the average speed for t sec of fall?
b ) What is the distance fallen in t sec?
7. If an object is dropped near the surface of the earth, it falls d = 16t2 feet in
t sec. Calculate d when t is
a) 4 b) 7 c) 3t e) 5t.
8. If an object falls according to the formula d= 16t2, how much is t when d is
a) 64 b) 96 c) 144 d) ZOO e) 169'
9. If an object is thrown downward from a point near the surface of the earth
with an initial speed of 64 ft/sec, the speed it acquires in t sec is given by the
formula v = 64 + 32t. Calculate the speed when t is
a) 3 b) 3! c) 5 d) 5t e) 7.
10. If an object acquires speed according to the formula v = 64 + 32t, how long
does it take to acquire a speed in ft/sec of
a) 96 b) 100 c) 300 d) 1 50?
1 1. Suppose the constant acceleration of a falling object is g ft/sec2 and the object
is thrown downward at speed of 100 ft/sec. Guess the formulas which represent
a) the speed acquired in t sec,
b) the distance fallen in t sec.
FORMULAS FOR MOTION OF BODIES PROjECfED UPWARD 305
12. Suppose an object falls d ft in t sec where d = 128t + 16t'. How long does it
take to fall
a) 320 f[ b) 768 f[ c) 304 ft d) 1 5 6 f[?
1 3 . Suppose the constant acceleration of a faIling object is g ft/sec2 and the object
is thrown upward with an initial speed of 128 ft/sec. Guess the formulas which
represent
a) the speed acquired in t sec,
b) the height above the ground in t sec.
14. If the height above the ground of an object IS given by the formula
d = 96t - 16t2, how high is the object when t is
a) 3 b) 2! c) 5 d) 5!?
15. The height above the ground of an object is given by d = 96t - 16:2. Calculate
the height when t = 7. What is the physical meaning of the result?
16. If an object is thrown up from the roof of a building 200 ft high and the height
of the object above the roof is given by d = 96t - 16t2, what is the height of
the object when t = 7 ? What is the physical meaning of the result?
17. If we had reason to believe that the acceleration of any object which is dropped
near the surface of any planet, the sun, or the moon is a constant, could we carry
over the mathematics of this chapter to motions of objects near the surface of
these bodies?
Topics for Further Investigation
1. Galileo's scientific work.
2. Huygens' scientific work.
3. The importance of experimental work versus that of mathematical deduction
from basic principles in seventeenth-century science.
4. The scientific ideas espoused by Francis Bacon.
Recommended Reading
BELL, A. E.: Christian Huygens and the Development of Science in the Seventeenth
Century, Edward Arnold and Co., London, 1947.
BONNER, FRANCIS T. and MELBA PHILLIPS: Principles of Physical Science, pp. 37-65,
Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957.
BURTT, E. A.: The Metaphysical Foundations of Modern Physical Science, 2nd ed.,
Chaps. 1 through 6, Routledge and Kegan Paul Ltd., London, 1932.
BUITERFIELD, HERBERT: The Origins of Modern Science, Chaps. 4 through 7, The
Macmillan Co., New York, 1951.
COHEN, I. BERNARD: The Birth of a New Physics, Chap. 5, Doubleday and Co.,
Anchor Books, New York, 1960.
306 THE SIMPLEST FORMULAS IN ACfION
CROMBIE, A. C.: Augustine to Galileo, Chap. 6, Falcon Press Ltd., London, 1952.
Also in paperback under the title: Medieval and Early Modern Science, 2 vols.,
Doubleday and Co., Anchor Books, New York, 1959.
DAMPIER-WHETHAM, WM. C. D.: A History of Science and Its Relations with
Philosophy and Religion, Chap. 3, Cambridge University Press, London, 1929.
FARRINGTON, BENJAMIN: Francis Bacon: Philosopher of Industrial Science, Henry
Schuman, Inc., New York, 1949.
GALILEI, G ALILEO: Dialogues Concerning Two New Sciences, pp. 147-2 33, Dover
Publications, Inc., New York, 1952.
HOLTON, GERALD and DUANE H. D. ROLLER: Foundations of Modern Physical
Science, Chaps. 1, 2, and 1 3 through 15. Addison-Wesley Publishing Co., Inc.,
Reading, Mass., 1958.
KLINE, MORRIS: Mathematics and the Physical World, Chaps. 12 and 13, T. Y. Crowell
Co., New York, 1959. Also in paperback, Doubleday and Co., New York, 1963.
MOODY, ERNEST A.: "Galileo and Avempace: Dynamics of the Leaning Tower
Experiment," an essay in PHILIP P. WIENER and AARON N OLAND : Roots of Scien
tific Thought, Basic Books, Inc., New York, 1957.
RANDALL, JOHN HERMAN, JR.: Making of the Modern Mind, rev. ed., Chaps. 9 and
10, Houghton Mifflin Co., Boston, 1940.
SAWYER, W. W.: Mathematician's Delight, Chaps. 8 and 9, Penguin Books Ltd.,
Harmondswonh, England, 1943.
SMITH, PRESERVED: A History of Modern Culture, Vol. I, Chap. 6, Henry Holt &
Co., New York, 1930.
STRONG, EDWARD W.: Procedures and Metaphysics, University of California Press,
Berkeley, 1936.
TAYLOR, HENRY OSBORN: Thought and Expression in the Sixteenth Century, 2nd cd.,
Vol. II, Chaps. 30 rhrough 35, The Macmillan Co., New York, 1930.
TAYLOR, LLOYD WM.: Physics, The Pioneer Science, Chaps. 3 through 7, Dover
Publications, Inc., New York, 1959.
WHITEHEAD, ALFRED N.: Introduction to Mathematics, Chaps. 2 through 4, Holt,
Rinehart and Winston, Inc., New York, 1939.
WHITEHEAD, ALFRED N.: Science and the Modern World, Chap. 3, Cambridge Uni
versity Press, London, 1926.
WOLF, ABRAHAM: A History of Science, Technology and Philosophy in the 16th
and 17th Centuries, 2nd ed., Chap. 3, George AIlen and Unwin Ltd., London,
1950. Also in paperback.
CHAPTER 14
PARAMETRIC EQUATIONS AND CURVI LINEAR MOTION
I now propose to set forth those properties which belong to a body whose
motion is compounded of two other motions, namely, one uniform and
one naturally accelerated; these properties, well worth knowing, I propose
to demonstrate in a rigorous manner.
GALILEO
14-1 INTRODUCTION
We saw in the preceding chapter that simple functions can be used to express
physical principles and that by applying algebra to the formulas which express
the functions symbolically, we can obtain new physical knowledge. To some
extent, then, we have come to recognize the broader significance and useful
ness of functions and mathematical processes for science in general. However,
we have hardly penetrated as yet the mathematical domain of functions nor
have we learned enough applications to sense its real power.
In this chapter we shall extend slightly the use of functions. In Chapter
1 3, we represented the acceleration and speed attained and distance traveled
by a falling body by using one formula for each physical quantity. We were
enabled thereby to study motion along straight-line paths. We shall now ex
amine motion along curved paths, for example, the motion of an object
dropped from a moving plane, or the motion of a projectile shot out from a
cannon. It was again Galileo who perceived the basic principle underlying the
phenomenon of curvilinear motion. He presented the concept and its mathe
matical treatment to the world in the Dialogues Concerning Two New
Sciences, the very same book in which he treated motion in a straight line.
Galileo's purpose in investigating curvilinear motion was to study the behavior
of cannon balls, or projectiles in general. The cannon, introduced in the
fourteenth century, had undergone such improvement by Galileo's time that
it could fire a projectile over several miles. However, the theory of projectile
motion was not we11 understood before Galileo's work because mathematicians
and physicists had attempted to apply Aristotle's laws of motion, and these
were nOt correct.
307
308 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
The problems that Galileo treated, e.g., the motion of cannon balls, un
fortunately did not lose their importance in the succeeding centurie�. In fact,
they have become even more common and more complicated in our times, since
such phenomena as the motion of bombs dropped from moving airplanes, the
trajectories of death-dealing projectiles capable of traveling thousands of miles,
and similar problems of modern "civilization," also fall within the puissance of
Galileo's method. However, the value of this phase of Galileo's work is not
limited to meting out death and destruction. Aside from using his results as
an illustration of the power of mathematics, we shall see in the space of one
chapter how an extension of Galileo's ideas on projectile motion led, in the
hands of Newton, to the greatest advance in science which our civilization
has achieved.
() 3 X
14-2 THE CONCEPT OF PARAMETRIC EQUATIONS
/
//
Let us suppose that a stone is thrown oUt hori /
zontally from the top 0 of a cliff (Fig. 14-1). I
I
We know from physical experience that the stone I 9
will travel out and down and follow the curved I
path OAB. If we introduce a set of coordinate
axes on which the positive direction of the Y-axis
is downward, then we know from our work in
coordinate geometry that this curve can be repre A
sented by an equation. Let us suppose, for defi
niteness, that this equation is y = x". Then it
y B
follows from equation (9) of Chapter 1 2 that the
path is part of a parabola opening downward. Fig. 14-l.
The path of a stone thrown
We shall call y = x', which is, of course, also the out horizontally from the top
fMmula that tells us how y changes when x o of a cliff.
changes, the direct relationship between x and y .
As the stone travels out and down, the horizontal distance and the down
ward distance which it travels from the point 0 keep changing with time.
Thus, at the point A, the horizontal distance traveled may be 3, and since
y = x', the vertical distance must be 9. At the point B, the horizontal distance
may be 4, in which case the vertical distance must be 16.
The direct relationship between x and y is frequently useful, but it does
not involve the time that the object is in motion. We may wish instead to
utilize the equation which gives the relationship between horizontal distance
traveled and time and the equarion which relates vertical distance and time.
Let us mppose for t:,e moment that the stone travels straight out at the rate
of J ft/sec. Then the relationship between horizontal distance and time is
x = Jt. Since y = x', then in terms of t, y = (Jt)' or y = 9t'.
THE CONCEPT OF PARAMETRIC EQUATIONS 309
The two equations:
x = 3t, y = 9t2, .(1)
are called the parametric equations of the curve OAB. They describe the
curve OAB just as well as does the single equation y = x', provided that we
understand how parametric equations are to be used. For each value of t,
equations ( 1 ) yield a value of x and a value of y. These values of x and y
which belong to the sa.me value of t are the coordinates of one point on the
curve OAB. Thus for t = 1, x = 3, and y = 9. Then (3, 9) are the coordinates
of a point on the curve, namely, the point A, which we discussed earlier. For
t = �, x = 4 and y = 16, and (4, 16) are the coordinates of the point B.
We may also say that the two formulas x = 3t and y = 9t' are equivalent
to the single formula y = x'. Whether we speak of equations of curves or
formulas is really immaterial. The word formula emphasizes the idea of change
because formulas are relationships among variables, and we often like to think
of what happens to one variable as another, related variable changes. On the
other hand, when a curve is given in its entirety, the concept of change may
not be relevant, and then we speak of the equation of the curve.
If the two formulas in ( 1 ) are entirely equivalent to the single formula
y = x', why do we bother with two formulas instead of one? There are two
reasons: ( 1 ) When one argues from physical principles, it is often easier to
arrive at the parametric representation of a given phenomenon, and (2) it is
easier to study the phenomenon by working with p·arametric equations. We
shall recognize the utility of parametric representations as we study the next
few sections.
There is one more mathematical detail. Suppose that we find the para
metric formulas describing a motion and we wish to determine the direct
relationship between x and y. Can we do this? Yes indeed. For example, if
x = 3t and y = 4t2 are the parametric formulas, we can solve the first one for
t and obtain t = xj3. We substitute this value of t in y = 4t' and obtain
Y=4 G) ', or or
4x'
y = -- .
9
This is the direct relationship between x and y.
EXERCISES
1. If the parametric formulas representing a phenomenon are x = 2t and y = 3t,
what is the direct relationship between x and y? What curve represents the
parametric formulas or the cirect relationship?
2. If the parametric formulas are x = 4t and y = 5t2, what is the direct relationship
between x and y? What curve represents the direct re1ationship?
3 10 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
3. Suppose the parametric formulas are x = 2t and y = lOt + 4t2• What is the direct
relationship between x and y and what curve describes it?
4. Suppose x = 3t and y = (!)t are the parametric equations of a curve. Sketch the
the curve by using the parametric equations only.
14-3 THE MOTION OF A PROJECTILE DROPPED FROM AN AIRPLANE
Let us see now how parametric formulas arise in the study of physical phe
nomena and how they can be useful in deducing new information about the
phenomena. Suppose a bomb is released from an airplane which is flying
horizontally at 60 miles per hour (an untealistic figure used for computational
convenience) . If there were no gravity, the bomb would continue to move
forward alongside the airplane at the rate of 60 miles per hour. This fact
seems surprising, but it is a consequence of the first law of motion, which states
that if an object is in motion and no force is applied to alter that motion, then
the object will continue to move indefinitely at the speed it already has. Since
the bomb has been moving with the airplane, it already possesses a horizontal
speed of 60 miles per hour. We have assumed that no forces are acting on
the bomb and hence it will continue to move forward at that speed. There
are more familiar analogous situations which may make the truth of what was
just said a little more acceptable. Suppose that a person rides in an automobile
which is moving at the rate of 60 miles per hour and the driver suddenly
applies the brakes. The automobile's motion is then checked, but the passenger's
motion is not, and he continues to move forward at 60 miles per hour, at least
until he hits the windshield.
Let us return to the motion of the bomb released from the plane. We had
assumed that gravity was not acting. But it does act and it pulls the bomb
downward at the same time as the bomb moves forward so that the bomb
follows a curved path. Here Galileo made a discovery applying to projectile
motion, namely, that one could study its horizontal and vertical motions as
though they were occurring separately, and that the position of the bomb at
any time could be determined by finding how far it had traveled horizontally
and vertically. This idea was new and radical in Galileo's time. Aristotle had
argued that one motion would interfere with the other, and that only one could
operate at any given time. Thus he would have said that the violent motion
imparted to the bomb by the airplane would prevail until the acting force was
used up, and then the natural motion downward would take over and cause
the bomb to fall straight down.
Let us apply Galileo's way of analyzing the motion. The bomb moves
horizontally at the constant speed of 60 miles per hour, or 88 feet per second.
If we measure time from the instant the bomb is released from the plane, and
if we measure horizontal distance from the point at which it is released, then
MOTION OF PRO]ECfILE DROPPED FROM AN AIRPLANE 311
the horizontal distance x covered by the bomb in t seconds is given by the
formula
x = 88t. (2)
This formula describes the horizontal motion.
According to Galileo, the vertical motion downward takes place as though
it were independent of the horizontal motion. But the vertical motion is due
to gravity only, and we know that an object which falls straight down under
the action of gravity and starts with zero speed falls 161' feet in t seconds.
Hence, if we let y represent the distance downward from the point at which
the bomb is released, then
y = 1 6t2• (3)
Formulas (2) and ( 3 ) together describe the entire motion. We observe
that x and y, the horizontal and vertical distances traveled, are given in terms
of a third variable, t. In fact, they are the parametric formulas for the morion
in question. To draw the graph of the path described hy the bomb, we may
=
adopt either of two methods. We can choose various values of t, say
t 0, 1, 2, 3, and so on, and calculate the values of x and y for each value of t.
Thus when t = I , x = 88 and y = 16; then (88, 16) are the coordinates of one
point on the curve. Calculating many such sets of coordinates will give us
some idea of the shape of the curve
(Fig. 14-2 ) .
Or we may proceed b y the second
method. that is, determine the direct
relationship berween x and y . Solving
equation ( 2 ) for t, we have t = x/88.
Substituting this expression for t in
( 3 ) yields
y = 16 (;S
or
Fig. 14-2.
x2 Path of a bomb released from an airplane
y = _. (4) flying horizontally.
484
From formula (9) of Chapter 12 we know that the curve is a parabola. We
have thus called upon our knowledge of curve and equation to determine that
the curve is a parabola. If we had not been familiar with the curve of equation
(4), we would have had to analyze the equation or plot points whose coordi
nates satisfy (4) and thereby determine the curve. In other words, we would
have been faced with a problem of coordinate geometry.
312 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
We should note that only part of the parabola is of physical interest. The
full parabola extends to the right and left of the Y-axis. However, only the
part to the right, that is, the half corresponding tt) positive x-values, repre
sents the motion of the bomb. And of this right-hand half, which mathe
matically extends downward indefinitely, only an arc is of physical interest,
namely the arc from 0 to the ground.
We have learned so far that the path of the bomb released from an airplane
traveling horizontally is an arc of a parabola. Let us now see whether we can
use mathematics to derive more information about the motion of bombs or, in
general, about objects which move outward and downward. Suppose that two
airplanes flying horizontally at speeds of 60 and 1 20 miles per hour, respec
tively, release bombs from the same point at the same instant of time. Which
of these bombs would reach the ground sooner? The reader might try to
answer this question by using his intuition before resorting to the use of
mathematics.
Both bombs must fall the same vertical distance to reach the ground. The
vertical motion is independent of the horizontal motion and is governed by
formula ( 3 ) . Hence this formula applies to both bombs. When they reach the
ground, the value of y will be the same for both bombs. It follows that the
value of t will also be the same for both. That is, both bombs will reach the
ground at the same time .
.'I C
Fig. 14-3.
Paths of two bombs released
� il!lll il!lll il!lll � W!I! W!l!*
iI!Ill iI!Ill il!lll il!lll W!i W!I! rl>TIffl iiiii from two airplanes with different
y horiZontal speeds.
How docs the difference in the speeds of the two airplanes affect the
motion? The plane flying at 60 miles per hour gives its bomb a horizontal
speed of 60 miles per hour, or 88 feet per second, and the plane flying at 1 20
miles per hour imparts to its bomb a speed of 1 20 miles per hour, or 176 feet
per second. Hence the bombs move with different horizontal speeds, and in the
same time, t, the second one will travel farther horizontally. Thus OeD, the
path of the second bomb, will be a wider parabola than OAB, the path traveled
by the first bomb (Fig. 14--3 ) .
MOTION OF PROJECTILES LAUNCHED BY CANNONS 313
EXERCISES
1. Suppose that there were no force of gravity, and that an object is released from
an airplane fiying horizontally at the rate of 100 mi/hr. Describe the subsequent
motion of the object.
2. One object is dropped from an airplane flying horizontally at the rate of 100
mi/hr and another from a plane flying horizontally at 200 mi/hr. Both planes
are at the same altitude. Compare the times required for the two objects to reach
the ground. What principle is involved?
3. From a cliff 500 ft high a stone is thrown horizontally with a speed of 100 ft/sec.
How long does it take the stone to reach the ground below? What horizontal
distance has the stone traveled by the time it strikes the ground?
4. A gun installed in a plane which is flying in a horizontal line at a speed of 2000
ft/sec fires a bullet in the direction of the plane's motion at the initial speed of
1000 ft/sec. What is the horizontal speed of the bullet relative to the gra:und?
5. A bullet fired horizontally hits a point on a wall 300 ft away. The point is 1 ft
below the lovel at which the bullet is fired. What is the horizontal speed of the
bullet?
6. A plane is traveling in a horizontal line at a speed of 300 ft/sec and at an altitude
of 1 mi. Where (at what horizontal distance from the target) should the gunner
release a bomb to hit a given point on the ground?
7. Suppose that a plane flying in a horizontal lir.e at the rate of 200 ft/sec releases a
bomb and continues to fly horizontally at the same rate. Where is the plane in
relation to the bomb when the bomb strikes the ground?
14-4 THE MOTION OF PROJECTILES LAUNCHED BY CANNONS
A slight extension of the mathematics just introduced to treat the motion of
bombs dropped from airplanes will enable us to handle the motion of projec
tiles shot out from cannons inclined at some angle to the ground. It was thi;
latter problem which Galileo investigated in the seventeenth century. We shall
see how neady mathematic, answers a variety of problems raised by such
motions.
Suppose that a cannon inclined at an angle of 3�" to the ground fires a
shell with a velo�ity· of 1 000 ft/sec (Fig. 14-4). What is the subsequent
motion of the shell? We know from intuition or experience with balls thrown
at a similar angle of elevation that the shell will travel out and up along some
cl.rved path and will then return to the ground. This qualitative knowledge
is not, of course, sufficienr to answer significant questions about the motion.
The initial velocity of the shell is in the direction which makes an angle
of 3�" to the ground. To treat the motion of the shell, it is mathematically
... The terms speed and velocity are often used interchangeably. However, the word
velocity implies that direction as well as magnitude of the speed are under discussion.
314 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
y
Fig. 14-4.
Shell fired with an initial velocity of
1000 ft/sec from a cannon inclined
at an angle of 30° to the ground.
simpler to consider its horizontal and vertical motions separately, that is, to
obtain the parametric formulas. For this purpose we must know the horizontal
and vertical velocities of the shell.
Suppose that the shell travels for one second in the direction OR in which
it is fired. How far will it travel horizontally and vertically in that second?
Let us drop a perpendicular from R onto the X-axis and from R onto the
Y-axis. Thus we determine the lengths OP and OQ, respectively. The length
OP is the horizontal distance which the shell travels in one second and the
length OQ is the corresponding vertical distance. Since OP and OQ are
distances traveled in one second, they also represent the horizontal and vertical
velocities.
The horizontal and vertical velocities are then OP and OQ. What are
their magnitudes? We see from Fig. 14-4 that
OP
cos 30° =
1000
or
OP = 1000 cos 30° = 1000(0.8660) 866 ft/sec.
Similarly,
PR
sin 300 =
1000
or
PR = 1000 sin 30° = 1000(0.5000) = 500 ft/sec.
S;nce PR = OQ, the horizontal and vertical velocities of the shell are 866 ft/sec
and 500 ft/sec, respectively.
We now utilize the physical fact that the horizontal and vertical motions
can be treated independently. Let us begin with the horizontal motion. The
shell has an initial horizontal velocity of 866 ft/sec, and no force acts to ac
celerate or decelerate the horizontal motion. Hence the shell will continue to
move horizontally at a constant speed of 866 ft/sec, and the horizontal distance
x traveled in time t is given by
x = 866/. (5)
MOTION OF PROJECTILES LAUNCHED BY CANNONS 315
Next we consider the vertical motion of the shell. Gravity gives the shell
a constant acceleration downward of 32 ft/sec'. Since the upward direction
has been chosen to be positive, the downward acceleration must be written
a = - 32. (6)
The downward velocity acquired in time 1 is - 32t. However, the shell has an
initial upward velocity of 500 ft/sec which, by the first law of motion, would
continue indefinitely, were it not affected by gravity. The net velocity v is
then (compare Section 1 3-7)
v = - 32t + 500. (7)
To obtain the distance traveled upward in any time t, we use the same
reasoning as in the preceding chapter. If only the velocity of 500 ft/sec were
acting, the distance traveled upward in t seconds would be 5001. But in that
time gravity pulls the shell downward a distance of 161'. Hence y, the net
height above the ground, is
y = - 1 6t2 + 500t. (8)
Fonnulas ( 5 ) and (8) give the horizontal and vertical distances from the
starting point O. We note that once more motion is represented by parametric
formulas.
Several questions about the motion arise in practice. The first is, What
path does the shell take? We may save ourselves the work of plotting the
curve by determining the direct relationship between x and y, provided we
recognize the curve of the resulting equation. Let us try this. Solving ( 5 ) for
t yields t = x/866. We substitute this value of 1 in (8) and obtain
y = _16 � ( )
866
2
+ 500 �
866
( )
or
x2 250
Y = -
46,87 2 + 433 x. (9)
In Section 12-5 we discussed equations of the form (9)-albeit by means of
numerically simpler examples. We could have proved the quite general state
ment that an equation of the form
y = _ax' + bx, (10)
where a and b are any positive numbers, represents a parabola which opens
downward and passes through the origin. (See Exercise 3 of Section 12-5,
which is a special case.) Hence with just a little more work in coordinate
316 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
geometry, we could have proved what we shall now state without proof,
namely, that equation (9) describes a parabola. Thus the parabola, as Galileo
readily established, appears once more as the path of a projectile.
What is the range of the shell? That is, how far from the starting point
will the projectile strike the ground again? The answer is important because
it tells us whether a given target on the ground can be reached. Unfortunately
neither formula (5) nor formula (8) answers this question directly. However,
when the shell reaches the ground, the value of y in (8) should be zero. Let us
determine, then, the value of t, say t" when y = O. From (8)
0 = - 1 6tI + 500t,. (I I)
Equation ( I I ), which is of the second degree 10 t" is rather easy to solve.
Applying the distributive axiom, we may write
0 = t,(- I6t, + 5(0) . ( 1 2)
The right side of ( 1 2 ) equals zero when either factor is zero, that IS, when
t, = 0 and when
- 1 6t, + 500 = O.
The second alternative leads to
(Il)
The first value, t l = 0, corresponds to the instant when the shell first starts
its flight. Then the second value, 125/4, must be the time when the shell
returns to the ground.
To determine the range, one more step is necessary. Formula (5) tells us
how far the shell travels horizontally in any time t. Since the shell travels
125/4 seconds by the time it reaches the ground, we have but to substitute
this value of t in (5) to get the range. If XI denotes the range, then (see
Fig. 14-5)
X, = 866 � = 27,063 feet. (14)
We might also like to know how high the shell will go in its flight and how
long it takes to reach that height. These questions are readily answered. At
the highest point in its flight, the vertical velocity is zero, else the shell would
continue to rise. Formula (7) gives us the vertical velocity at any time t. Let
us ask for the value of t, say t2 , when v = O. Then
0 = - 32t2 + 500
or
t2 = W = .if'- . ( 1 5)
MOTION OF PROJECTILES LAUNCHED BY CANNONS 317
y
A
Y2 = 3906
O
� == � == �B
( == == == == B X Fig. 14-5.
Path of a shell shot from a cannon.
XI =- 27,063
Hence it takes 1 2 5/8 seconds for the shen to reach the highest point. Now
formula (8) tells us how high the shell is at any time t. Let us therefore
find the height, Y 2, when t = 1 25/8. We substitute 1 2 5/8 for t in (8) and
obtain
Y, = - 16 (.1f') ' + 500 (.q.»
or
Y' = 3906 feet. ( 1 6)
Hence the shell reaches a maximum height of 3906 feet above the ground.
Another interesting question is whether the shell takes as long to travel
from the cannon to its maximum height as it does to return from the latter
position to the ground. Or, to fit the situation shown in Fig. 1 4--5, we may
restate the question and ask, Does it take as long for the shen to travel from
o to A as from A to B? We considered an analogous problem in the preceding
chapter while discussing the motion of an object thrown straight up into the
air and found that the two time intervals were equal. What does intuition
suggest as the answer in the present case?
We can show at once that the time required to get from 0 to A is the
same as the time required to travel from A to B. Equation ( 1 3 ) supplies the
time it takes the shen to reach B, that is, to travel the path OAB. Equation
( 1 5 ) gives the time required to travel the path OA. We see at once that the
value of t, is twice the value t,. Hence the time of travi:! along path AB
must equal the time of travel along path OA.
EXERCISES
I. Suppose that a shell is fired in a direction making an angle of 400 with the ground
and with a velocity of 300 ft/sec. What are the horizontal and vertical velocities
of the shell? What are the parametric equations describing the motion?
2 . Suppose that the parametric formulas for the motion of a projectile are x = 20t
and y = - 16t2 + 30t. What is the direct relationship between x and y? What is
the nature of the curve represented by the direct relationship between x and y?
3. Suppose that the parametric formulas for the motion of a projectile are x = 3t
and y -16t2 + St. Working with these formulas, plot a few points of the path.
=
4. Find the range of the projectile whose motion is described in Exercise 2 .
318 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
5. Find the maximum height of the projectile whose motion is described in Exer
cise 2.
6. What velocity does the shell whose motion is treated in Section 14--4 have on
striking the ground? How does this terminal velocity compare with the initial
velocity?
* 14-5 THE MOTION OF PROJECTILES FIRED AT AN ARBITRARY ANGLE
In the preceding section we saw how we could study the motion of a shell
fired from a cannon which is inclined at an angle of 30° to the ground. The
initial velocity of the shell in this direction was 1000 ft/sec. Since the angle of
fire and initial velocity given in our example are representative values, the
example teaches a great deal about the phenomenon of projectile motion.
However, suppose that we sought to answer such questions as: What is the
effect of the initial velocity on the range and on the maximum height attained
by the projectile? What is the effect of the angle of fire on the range and on
the maximum height of the projectile? At what angle should one fire a pro
jectile to hit a given target? One could repeat the procedures pursued in the
preceding section, using different initial velocities and angles of fire, and thus
perhaps obtain answers to some of these questions. But the work would be
considerable and still leave us with the problem of trying to infer a general
conclusion from a number of special cases. The mathematician would not
proceed in this way. He would suppose that the initial velocity is an arbitrary
value, V, and that the angle of fire is an arbitrary angle, A, and then study the
motion with these arbitrary values V and A. He might thereby obtain con
clusions about all such motions because his results would hold for any initial
velocity and any angle of fire.
y
Q ---- ,R
I
I
v Fig. 14-6.
Shell fired with an initial velocity of
" V ft/sec from a cannon inclined at
�Of-��p�----� X an angle A to the ground.
Let us pursue this general investigation of projectile motion. Suppose that
a shell is fired from a cannon which is inclined at an angle A to the ground
(Fig. 14-6), and that the initial velocity of the shell is V ft/sec. What is the
subsequent motion of the shell?
We shall follow the procedure of the preceding section. The first major
point to remember is the physical principle that the horizontal and vertical
MOTION OF PRO}ECfILES FIRED AT ARBITRARY ANGLE 3 19
motions of the projectile can be studied as though the motions were taking
place independently. Hence let us find the initial horizontal and vertical
velocities of the shell. By the very same argument that we used in Section
14-4, we know that the components of the velocity OR in the horizontal and
vertical direction (Fig. 14-6) are obtained by dropping a perpendicular from
R onto the X- and Y-axes, respectively. Thus OP is the horizontal velocity,
and OQ is the vertical velocity. Now
OP
cos A =
OR
Hence
OP = OR cos A
or
OP = V cos A . (17)
Since
PR
sin A - ,
OR
and since OQ = PR, we have
00 = OR sin A
or
00 = V sin A. (IS)
Formulas ( 1 7) and ( I S) give us the initial horizontal and vertical velocities,
respectively. We must now see what happens when the shell is in motion. The
horizontal motion is uniform; i.e., no force acts to speed it up or slow it down.
The shell will therefore continue to travel indefinitely in the horizontal direc
tion at the velocity given by ( 1 7 ) . Let us use Vx to indicate the velocity in the
X-direction. Then at any time t,
Vx = V cos A. (19)
Since the velocity in the horizontal direction is constant, the distance traveled
is velocity multiplied by time. Then
x = ( V cos A) t.
This expression is best written as
x = Vt cos A, (20)
so that there is no confusion about the fact that the quantity whose cosine is
to be taken is A, whereas if we had written V cos At, one might think that
320 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
the quantity is At. Formula (20) is a generalization of formula (5), for the
866 in (5) is just 1000 cos 30°.
To obtain the vertical velocity of the shell we must take into aCCount the
fact that gravity does produce a vertical acceleration which affects the vertical
velocity. This vertical acceleration is 3 2 ft/sec' and is downward. Since we
have chosen the upward direction as positive, we have
a = - 32.
Because the acceleration is constant, the downward velocity gained by the
shell in t seconds is - J2t. However the shell has an upward initial velocity of
V sin A. Hence the net vertical velocity v, is
v, = - 32/ + V sin A. (21)
Next we use an old argument to determine the vertical height attained by
the shell in t seconds. If only the velocity V sin A were acting, then in t
seconds the shell would reach the height ( V sin A ) t or Vt sin A. However,
in these t seconds gravity pulls the shell downward a distance of 161'. The
net height, y, of the shell therefore is
y = - 1 612 + Vl sin A. (22)
Formulas ( 19) through (22) supply the general equations of projectile
motion. We are now in a position to answer with respect to the arbitrary
values V and A the same questions that were discussed in the preceding section
for a specific numerical example. Thus, by solving (20) for t and substituting
this result in (22), we could find the direct relationship between x and y . We
could then see that for any fixed value of V and any fixed value of A, the
path is a parabola. Similarly, we could obtain general expressions for the
maximum height reached by the shell, the time required to reach that height,
and, say, the time required for the shell to return to the ground. In other
words, we could reproduce for any V and A the results derived in the pre
ceding section for special values of V and A.
Let us turn instead to answering questions which we could not treat be
fore. Let us study the effect of the initial velocity, V, and angle of fire, A,
on the range of the cannon. To do this we must first determine the general
expression for the range. The method is the same as the one used in the pre
ceding section.
We begin with the physical fact that when the shell strikes the ground, the
y-value of its position is zero. Hence let t, be the value of 1 when y = o.
Setting y = 0 in ( 2 2 ) , we have
0 = - 161; + VI, sin A .
MOTION OF PROjECfILES FIRED AT ARBITRARY ANGLE 321
We may apply the distributive axiom to write
0 = /,(- 1 611 + V sin A) .
Now the right side is zero when 11 = 0 and when
- 1 6/, + V sin A = O. (23)
Sin"e I I 0 corresponds physically to the instant when the shell starts its
=
motion, it follows that the value given by ( 2 3 ) is the value of I at the instant
the shell strikes the ground. If we solve ( 2 3 ) for t" we find
V sin A
tl = (24)
16
--_ .
To determine the shell's range, that is, its horizontal distance from the starring
point at the instant when tl has the value just found, we use formula (20).
Thus, if X , is the value of X when t has the value t " then
V sin A
x, -,-
V cos A - -
16
=
or the range is
V'
Xl = - sin A cos A . (25)
16
Formula (25) answers one question immediately. If the angle A is held
fixed, then the range depends upon V2. If V is increased, then Xl increases,
and indeed XI increases rapidly because it depends upon V2 rather than upon
just V. Also, if we wished to attain a given range with a given angle of fire,
that is, if X, and A were specified, we could use (25) to calculate the necessary
initial velocity, V.
The more practical problem is to study the dependence of range upon
angle of fire, for it is easier to change the angle of fire of a cannon than it is
to change the initial velocity of the shells. Let us suppose, then, that the
initial velocity V is fixed and ask the question, What is the maximum range
that can be ob,.ined by varying angle A ? Since V is fixed, our question, in
view of (25), amounts to: For what value of A is the product sin A cos A a
maximam? A little mathematics provides the answer.
'Ve know from our work on the trigonometric ratios that sin A and cos A
are certain ratios of sides of a right triangle. The size of the right triangle
. used to determine sin A and cos A J"es not matter because all possible right
triangles containing a definite angle A are similar and therefore the ratio of
two particular sides in any one of these triangles is always the same.· Hence
• The reader can review this point in Chapter 7, where it was first made.
322 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
let us choose a right triangle whose hypotenuse AB (Fig. 14-7) is the diameter
of a definite circle. The vertex of the right angle of this right triangle must
lie on the circle because a theorem of plane geometry states that the vertices of
all right triangles with vertex C and hypotenuse AB must lie on a circle with
AB as diameter. Let us denote the sides of the right triangle ABC by a, b,
and c. We draw the perpendicular CD and denote it by h. Then, using the
right triangle A DC, we obtain
. h
Sin A = _ . c
b
b
From the right triangle ABC we have
A I""----c;:---!,.l-� B
b
cos A = - .
c
So far, then, we have found that
. h b h Fig. 14-7
sm A cos A = - . - = (26)
b e e
We may vary angle A and continue to regard it as an angle of a right
triangle with hypotenuse AB because, as we noted above, the size of the right
triangle is immaterial. Of course, variations in the angle A will produce
changes in the position of C, but for the reason given earlier, C will continue
to be a point on the circle with diameter AB. Hence we can consider all
possible acute angles A by considering all triangles ABC with AB fixed and
C varying on the circle. Let us now look at (26) again. Since the quantity c
is fixed, sin A cos A will be a maximum when h is a maximum, and h is great
est when it is the radius of the circle. But when h is the radius, C is directly
above the center of the circle. Then A C = BC. In this case angle A = 45"
because the right triangle ABC is isosceles. Hence
sin A cos A
is a maximum when angle A = 45°.
We may now return to formula ( 2 5 ) . We have found that when V is
fixed, the maximum range, that is, the maximum possible value of Xl, is ob
tained when A = 45". Since for A = 45", sin A = cos A = y'f/2, the maxi
mum range is given by the formula
v2 V2 V2 v2
maxImum Xl = _ . _ . _ -
16 2 2 32
This famous result was first proved by Galileo.
SUMMARY 323
EXERCISES
1 . Formulas (20) and (22) give the parametric representation of the motion of
a projectile. Find the direct relationship between x and y .
2. Derive the formula in terms of V and A for the time it takes a projectile to reach
its maximum height.
3. What is the formula in terms of V and A for the maximum height reached by a
projectile?
4. Show generally that it takes as long for a projectile to return from its highest
position to the ground as it does to travel from the cannon to the highest position.
5. What is the range of a projectile which is fired with an initial velocity of
2000 fe/sec and at an angle of 40° to the ground?
6. What is the maximum range of a projectile fired with an initial velocity of
800 ft/sec?
7. How long does it take for a projectile fired with an initial velocity of 2000 ft/sec
to reach a target located at maximum range?
8. Show that the highest point which a gun can reach for all possible angles of fire
but with the same initial velocity is attained by firing straight up.
14-6 SUMMARY
In this chapter we have seen how the simultaneous application of two simple
formulas, the parametric formulas, makes it possible to represent an entire class
of curves which phy'ically happen to be the paths of projectiles. The value
of the parametric formulas is, then, that they enable us to answer readily a
variety of questions about projectile motion. To appreciate how much mathe
matics accomplishes in this area, one might consider how he would proceed
experimentally to find, for example, the dependence of the range of a projectile
on the angle of fire. One would have to fire at least dozens of projectiles at
different angles, making certain that other factors, such as the velocity with
which the projectiles are fired, the shape of the projectiles, and the state of the
atmosphere, are constant, and accurately measure the range and angle each
time. With all these precautions taken and the information secured, the ex
perimenter might obtain some limited results. He might, for example, learn
that as the angle of fire increases in 1 °-degree steps from 1 ° to 40°, the range
increases steadily, but he might miss the all-important fact that above 45" the
range decreases. The dependence of range upon velocity would still be en
tirely unknown and require further experimentation. But we have seen how a
little mathematics costing only pencil and paper can supply the full story of
the dependence of range upon angle of fire and initial velocity.
Thus the present chapter, too, illustrates how the cotr.bination of mathe
matics and simple physical axioms, such as the fact that the acceleration of
bodies near the surface of the earth is 3 2 ft/sec', the first law of motion, and
the independence of the horizontal and vertical motions, permits us to deduce
324 PARAMETRIC EQUATIONS AND CURVILINEAR MOTION
a vast amount of knowledge about our physical world. The knowledge re
ferred to at the moment concerns projectile motion under idealized conditions;
that is, the resistance of air is neglected; the earth is assumed to be flat over the
short distances which the projectiles cover; and the projectiles are limited to
travel near the surface of the earth. One might regard the whole story as of
minor interest because it deals with just one phenomenon and one which seems
limited to bombs and guns. However, the study of this phenomenon has
proved to be of immeasurable scientific importance. First of all, the deductions
made from the physical axioms mentioned above can be checked experimen
tally. If the deductions agree with experience we have some reason to believe
that the axioms are correct. We must remember in this connection that phys
ical axioms are generalizations from limited experience and that our confidence
in them depends upon how well they continue to lead to new physical facts.
Secondly, the study of motions near the surface of the earth, projectile motion
in particular, led to the most important advance in science since 1600, namely
Newtonian mechanics. The step to the broad science of mathematical mechan
ics will be taken in the next chapter.
REVIEW EXERCISES
I. Sketch the curves whose parametric equations are given below.
a) x = 31, Y = 71 b) x = 31, Y = 512
c) x = 312, Y = 51 d) x = 31 + 7, y = 51 + 9
e) x = 5 cos 8, y = 5 sin 8 f) x = 21, Y = 512 + 3t
2. Find the direct equations in parts (a), (b), (c), and (d) of Exercise l.
3. Suppose a bomb is released from an airplane which is flying horizontally at a
speed of 240 mph.
a) Write the parametric equations of motion of the bomb.
b ) If the airplane is one mile above the ground, how long will it take the bomb
to strike the ground?
c) Suppose the airplane flies at 300 mph instead of 240. How long will it take
the bomb to strike the ground?
d) Suppose the airplane is 2 mi above the ground and flies at 240 mph. How long
will it take the bomb to reach the ground?
e) How far from the point on the ground directly below the point at which the
bomb is released does the bomb strike the ground?
4. Suppose a shell is shot from a cannon inclined at an angle of 450 with the ground
and the shell is given an initial velocity of 2000 ft/sec.
a ) What are the horizontal and vertical velocities of the shell?
b) Write the parametric equations of the motion of the shell.
c) Find the range of the shell.
d) What is the maximum height above the ground attained by the shell during
its flight?
SUMMARY 325
Recommended Reading
GALlLEI. GALILEO: Dialogues Concerning Two New Sciences, pp. 234 through 282,
Dover Publications, Inc., New York, 1952.
HOLTON, GERALD and DUAN E H. D. ROLLER: Foundations of Modern Physical
Science, Chap. 3, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1958.
KLINE, MORRIS: Mathematics and the Physical World, Chap. 14, T. Y. Crowell Co.,
New York, 1959. Also in paperback, Doubleday and Co., New York, 1963.
CHAPTER 15
THE APPLICATION OF FORMULAS TO GRAVITATION
. . . from motion's simple laws
Could trace the secret hand of Providence
Wide-working through this universal frame.
JAMES THOMSON in his
Memorial Poem to Newton
15-1 THE REVOLUTION IN ASTRONOMY
While Galileo was fashioning the new science of motion, Johannes Kepler was
making dramatic contributions to one of the most far-reaching developments
in the history of Western civilization. This development was begun by
Nicolaus Copernicus and its essence was a radically new mathematical theory
of planetary motions.
Up to the sixteenth century the only sound and useful astronomical theory
was the geocentric system of Hipparchus and Ptolemy which we examined
in Chapter 8. This was the theory accepted by professional astronomers and
applied to calendar reckoning and navigation. It was, however, a rather
sophisticated creation in that its strength lay entirely in the mathematical
effectiveness of the scheme. The deferents and epicycles had no physical sig
nificance in themselves nor did the theory give any physical or intuitive rea
sons that the planets should move on epicycles attached to deferents.
The author of the next great celestial drama was Nicolaus Copernicus, who
lived about 1400 years after Ptolemy. Copernicus was born in Poland in 1473
and, after studying mathematics and science at the University of Cracow, de
cided to go to Italy, the center of the revived Greek learning. At the Uni
versity of Bologna, which he entered in 1 497, he studied astronomy. Then for
ten years he studied medicine and law and secured a doctor's degree in both
fields. He also became learned in Greek and mathematics. In 1 500 Copernicus
was appointed a canon of the Cathedral of Frauenberg in East Prussia, but he
did not assume his duties until 1 5 1 2 when he had finished his studies in Italy.
The job, which entailed mainly the management of estates owned by the
Cathedral, left Copernicus with plenty of time to make astronomical observa-
326
THE REVOLUTION IN ASTRONOMY 327
tions and to think about the relevant theory. After years of reflection and
observation Copernicus finally evolved a new theory of planetary motions
which he incorporated in a classic work, On the l<ev% ,tions of the Heavenly
Spheres. This appeared in 1 543, the year in which Copernicus died.
As we have already noted, when Copernicus began to think about astron
omy, the Ptolemaic theory was the only sound and effective system in exis
tence. This theory had become somewhat more complicated during the inter
vening centuries in that more epicycles had been added to those introduced by
Ptolemy, to make the theory fit the increased amount of observational data
gathered largely by the Arabs. In Copernicus' time the theory required a total
of 77 circles to describe the motion of the sun, moon, and the five planets
known then.
Copernicus had studied the Greek works and had become convinced that
the universe was mathematically and harmoniously designed. Harmony de
manded a more pleasing theory than the complicated extensions of Ptolemaic
theory. Copernicus read that some Greek authors, notably Aristarchus, had
suggested the possibility that the sun might be stationary and that the earth
revolved about the sun and rotated on its axis at the same time. He decided to
explore this possibility. He was in a sense overimpressed with Greek thought,
for he, too, believed that the motions of heavenly bodies must be circular or,
at worst, a combination of circular motions since circular motion was natural
motion. Moreover, he also accepted the belief that each planet must move at
a constant speed on its epicycle, and that the center of each epicycle must
move at a constant speed on the circle which carried it. Such principles were
axiomatic for him. Copernicus even adds an argument which shows the some
what mystic character of sixteenth-century thinking. He says that a variable
speed could be caused only by a variable power; but God, the cause of all
motions, is constant.
The upshot of such reasoning was that Copernicus used the scheme of
deferent and epicycles to describe the motions of the heavenly bodies, with,
however, the all important difference that the sun was at the center of each
deferent, while the earth itself became a planet moving about the sun and
rotating on its axis. Nevertheless, he achieved considerable simplification. He
was able to reduce the total number of circles, deferents and epicycles, to 34
instead of the 77 required under the geocentric view.
However, the remarkable simplification was achieved by Johannes Kepler,
one of the most intriguing figures in the history of science. In a life beset by
many personal misfortunes and hardships occasioned by social and political
events, Kepler had the good fortune to become in 1 600 an assistant to the
famous astronomer Tycho Brahe. Brahe was then engaged in making exten
sive new observations, the first such major undertaking since Greek times.
These observations, together with others which Kepler made himself, were
invaluable to him in his later work. When Brahe died in 1601 Kepler suc-
328 THE APPLICATION O F FORMULAS T O GRAVITATION
ceeded him as Imperial Mathematician to the Emperor Rudolph II of Austria,
King of Bohemia.
Kepler's scientific reasoning is fascinating. Like Copernicus, he was a
mystic and, like Copernicus, he believed that the world was designed by God
in accordance with some simple and beautiful mathematical plan. This belief
dominated all his thinking. But Kepler also had qualities which we now asso
ciate with scientists. He could be coldly rational. His fertile imagination trig
gered the conception of new theoretical systems. But he knew that theories
must fit observations and, in his later years, saw even more clearly that em
pirical data may indeed suggest the fundamental principles of science. Coper
nicus, too, wanted his theory to fit observational data; yet he held to the
heliocentric view, although the differences between theoretical predictions and
astronomical data were greater than might be accounted for by experimental
errors alone. Kepler, on the other hand, sacrificed his most beloved theories
when he saw that they did not fit observational data, and it was precisely this
incredible persistence in refusing to tolerate discrepancies which any other
scientist of his day would have disregarded that led him to espouse radical
ideas. He also had the humility, patience, and energy to perform extraordinary
labor which mark great men.
In his book 0" the Motion of the Planet Mars, published in 1609, Kepler
announced the first two of his three famous laws of planetary motion. The
first of these is especially remarkable, for Kepler broke with the tradition held
for 2000 years that circles or spheres must be used to describe heavenly mo
tions. Instead of resorting to deferent and several epicycles, which both Ptol
emy and Copernicus had used to describe the motion of any one planet, Kepler
found that a single ellipse would do. His first law states that each planet moves
on an ellipse and that the sun is at one ( common) focus of each of these
elliptical paths (Fig. 1 5-1 ) . The other focus of each ellipse is merely a
mathematical point at which nothing physical exists.
Kepler's first law utilizes a geometrical figure which had been introduced
and studied by the Greeks of the classical period. Had the ellipse and its
properties not yet been known, and had Kepler been faced with the double
Q'
s.
Phmet
(
s
•
)
P'
Fig. 15-1. Fig. 15-2.
Each planet moves in an ellipse about the Kepler's law of equal areas.
sun.
THE REVOLUTION IN ASTRONOMY 329
problem of abstracting the proper path from a multitude of data and con
ceiving the ellipse, he might possibly have ended up in an impasse. By working
out the properties of this curve Euclid, Apollonius, and Archimedes determined
the course of our civilization just as decisively as if they had stood at Kepler's
side.
Kepler's first law is of immense value in comprehending readily the paths
of the planets. But astronomy must go much further if it is to be interesting
in itself and useful. It must tell us how to predict the positions of the planets.
If one finds by observation that a planet is at a particular position, P say in
Fig. 1 5-2, he might like to know when it might be at some other position, a
solstice or an equinox, for example. What is needed is the velocity with which
the planets move along their respective paths.
Here, too, Kepler made a radical step. Copernicus, as we noted earlier,
and the Greeks had always used constant velocities. A planet moved along
its epicycle so as to cover equal arcs in equal times, and the center of each
epicycle also moved at a constant velocity on another epicycle or on a deferent.
But Kepler's observations told him that a planet moving on its ellipse does
not move at a constant speed. Kepler searched hard and long for the correct
law of velocities and found it. What he discovered was that if a planet moves
from P to Q (Fig. 1 5-2) in, say one month, then it will also move from P' to Q'
in one month, provided that the area PSQ equals the area P'SQ'. Since P is
nearer the sun than P' is, the arc PQ must be larger than the arc P'Q' if the
areas PSQ and P'SQ' are equal. Hence the planets do not move at a constant
velocity. In fact, they move faster when closer to the sun.
Kepler was overjoyed to discover this second law. Although it is not so
simple to apply as a law of constant velocity, it nonetheless confirmed his
fundamental belief that God had used mathematical principles to design the
universe. God had chosen to be just a little more subtle, but a mathematical
law clearly determined how fast the planets moved.
Another major problem remained open. What law described the distances
of the planets from the sun? The problem was now complicated by the fact
that a planet's distance from the sun was not constant but varied from a least
to a greatest value (see Fig. 1 5-2 ). Hence Kepler searched for a new principle
which would take this fact into account. Now he believed that nature was
not only mathematically but harmoniously designed and he took this word
"harmony" very literally. Thus he believed that there was a music of the
spheres which produced a harmonious tonal effect, not one given off in actual
sounds but discernible by some translation of the facts about planetary motions
into musical notes. He followed this lead and after an amazing combination of
mathematical and musical arguments, arrived at the law that if T is the period
of revolution of any planet and D is its mean distance from the sun, then
T' = kD',
3 30 THE APPLICATION OF FORMULAS TO GRAVITATION
where k is a constant which is the same for all the planets. This statement is
Kepler's third law of planetary motion and the one which he triumphantly
announced in his book The Harmony of the World ( 1619).
15-2 THE OBJECTIONS TO A HELIOCENTRIC THEORY
The work of Copernicus and Kepler is by far the most dramatic, startling, and
influential development in the formation of modern culture. The first sur
prising feature, and one which in itself makes their work astounding, is the
sharp break from existing thought. Copernicus and Kepler were educated in a
milieu which accepted the geocentric theory of Ptolemy as almost unquestion
able truth. Moreover, both were scientifically cautious. Nor did either really
have at his disposal any unusual observations which conflicted sharply with
Ptolemy's theory. Copernicus, as a matter of fact, was not a great observer
and did not seem to mind leaving his work somewhat at odds with observations.
Kepler did have access to more numerous and more reliable data and showed
greater tenacity in making the theory fit the data, but there was nothing in
these observations which suggested that an entirely new theory must be in
troduced.
'''hile casting aside the weight of centuries, Copernicus and Kepler could
give only token rebuttals to the numerous scientific arguments against a mov
ing earth which Ptolemy had advanced against Aristarchus. How could the
heavy earth be put into motion? That the other planets were in motion even
according to Ptolemaic theory was explained by the doctrine that these were
made of special light matter and therefore easily moved. About the best
answer Copernicus could give was that it was natural for a sphere to move.
Another scientific argument against the earth's rotation maintained that rotation
would cause objects to fly off into space just as an object on a rotating platform
will fly off. Copernicus had no answer to this argument. To the further ob
jection that a rotating earth should itself fly apart, Copernicus replied weakly
that since the earth's motion was natural, it could not destroy the body. Then
he countered by asking why the sky did not fall apart under the very rapid
daily motion which the geocentric theory called for. Vet another objection
declared that if the earth rotated from west to east, an object thrown up into
the air should fall back to the west of its original position because the earth
moved on while the object was in the air. If, moreover, the earth revolved
about the sun, then, since the velocity of an object is proportional to its weight,
or so at least Greek and Renaissance physics maintained, lighter objects on the
earth should be left behind. Even the air should be left behind. To the last
argument Copernicus replied that air is earthy and so moves in sympathy with
the earth.
These scientific objections to a moving earth were weighty ones and could
not be dismissed as the stubborness of doubters who refused to see the truth.
THE ARGUMENTS FOR THE HELIOCENTRIC THEORY 331
The substance of the matter is that a rotating and revolving earth did not fit in
with the physical theory of motion due to Aristotle and common in Coper
nicus' and Kepler's time.
Another class of scientific arguments against a heliocentric theory came
from astronomy proper. The most serious one stemmed from the fact that
the heliocentric theory regarded the stars as fixed. In six months the earth
changes its position in space by 1 86 million miles. Hence if one notes the
direction of a particular star at one time and again six months later, he should
observe a difference in direction. But this difference was not observed in
Copernicus' and Kepler's time. Copernicus answered that the stars are so far
away that the difference in direction was too small to be observed. However,
his explanation did not satisfy the critics, who countered that if the stars were
that distant, then they should not be clearly observable. In this installce,
Copernicus' answer was correct. The change in direction over a six-month
period for the nearest star is an angle of 0.76", and this was first detected by
the mathematician Bessel in 1838 who, of course, by that time had a good tele
scope at his disposal.
A further, powerful argument against a moving earth contended that we
do not feel any motion despite the fact that the earth is presumably moving
around the sun at 1 8 miles per second and that a person on the equator is
rotating at the rate of about 0.3 miles per second. Our senses, on the contrary,
tell us that the sun is moving in the sky. Of course, there are counterargu
ments which would mean more to us today because we have the experience
of traveling at high speeds. A pe,"on traveling in an airplane at 400 miles per
hour does not feel the motion. But to the people of Copernicus' time the
argument that we do not feel ourselves moving at the very high speeds called
for by the new astronomy was convincing.
15-3 THE ARGUMENTS FOR THE HELIOCENTRIC THEORY
In view of the numerous and sound arguments against the heliocentric theory
and the challenge it posed to the prevailing religious thinking of the times,
what made Copernicus and Kepler take up this long-discarded thought and
pursue it so courageously? For what most other men would call a mes' of pot
tage, they broke with established physics, philosophy, religion, and common
sense.
Both were convinced that the universe is mathematically designed and
hence that a true pattern of the motions was inherent. Moreover. this design
was instituted by God, and God would surely have used a simple and har
monious pattern. But Ptolemaic theory had become so encumbered in the
sixteenth century that it was no longer simple or beautiful. Hence Copernicus
and Kepler believed, when each found a more harmonious and simpler theory,
that their work was indeed a description of the divine order of things.
332 THE APPLICATION OF FORMULAS TO GRAVITATION
There are many passages in Copernicus' On the Revolutions of the Heav
enly Spheres and in Kepler's writings which bear unmistakable testimony to
this motivation as the central force in the search for a new theory and to their
conviction that they had found the right one when it proved to be simpler.
Copernicus says of his theory:
We find therefore, under this orderly arrangement, a wonderful symmetry in
the universe, and a definite relation of harmony in the motion and magnitude
of the orbs, Of a kind that it is not possible to obtajn in any other way.
Kepler remarks of his later work wherein he had already introduced the el
liptical theory of motion, "I have attested it as true in my deepest soul and 1
contemplate its beauty with incredible and ravishing delight." The work of
Copernicus and Kepler is the work of men searching the universe for the
harmony which their religious convictions assured them must exist and which
must be describable mathematically and simply because God had so designed
the universe. What distinguishes their religious convictions from those of their
contemporaries is that they did not tie themselves to literal interpretations of
the Holy Writings. They searched for the word of God in the heavens.
The core of the argument which Copernicus and Kepler presented for the
heliocentric theory was its mathematical simplicity. Their philosophical and
religious convictions assured them that the world is mathematically and simply
designed; accordingly the fact that a heliocentric view was mathematically
simpler than the geocentric one determined their position. The mathematical
simplicity of the new view was, in fact, the sole argument they could advance.
Only persons possessed of the unshakable conviction that mathematics is the
essence of the design of the universe and that the omnipotent mathematician
would necessarily prefer simplicity would dare to advance such a radical theory
and would have had the courage to defend it against the opposition it was sure
to and did encounter in those times.
The new theory appealed to astronomers, geographers, and navigators
because it simplified their theoretical and arithmetical work. Hence many
of these men adopted the new view just as a mathematical convenience, even
though they were not convinced of its truth. While this feature of the new
theory carried little weight with Copernicus and Kepler, it nonetheless had the
effect of making more and more people think in terms of a heliocentric view,
and, since one tends to accept as true what is familiar, there is no doubt that
this practical aspect did, in the long run, help to gain adherents for the theory.
Support for the new theory came from an unexpected development. Early
in the seventeenth century the telescope was invented. and Galileo, upon hear
ing of this invention, built one himself. He then proceeded to make observa
tions of the heavens which startled his age. He detected four moons of
Jupiter (we now can observe twelve ) , and this discovery shoy/ed that a
moving planet can have satellites. Hence it was likely that the earth, toO,
THE ARGUMENTS FOR THE HELIOCENTRIC THEORY 333
could be in motion and yet have a satellite, our moon. Galileo saw irregular
surfaces and mountains on the moon, spots on the sun, and a bulge around
the equator of Saturn (which we now call the rings of Saturn). Here was
further evidence that the planets were like the earth and certainly not perfect
bodies composed of some special ethereal substance, as Greek and medieval
thinkers had believed. The Milky Way, which had hitherto appeared to be
just a broad band of light, could be seen with the telescope to be composed of
thousands of separate stars, each of which gave off light. Thus, there were
other suns and presumably other planetary systems suspended in the heavens.
Moreover, the heavens clearly contained more than seven moving bodies, a
number which had been accepted as sacrosanct. Copernicus had predicted
that if human sight could be enhanced, then man would be able to observe
phases of Venus and Mercury, that is, to observe that more or less of each
planet's hemisphere facing the earth is lit up by the sun, just as the naked eye
can discern the phases of the moon. Galileo did discover the phases of Venus.
All of Galileo's observations were made with a telescope of such limited
power that, as has been said, it is remarkable he could find Jupiter, let alone
the moons of Jupiter. Many of his discoveries were in direct support of a
heliocentric theory; others served primarily to challenge current beliefs and
to at least prepare some minds for a more objective examination of the new
theory. Galileo, himself, though he lectured on Ptolemaic theory until 1605,
had been converted to Copernicanism by a work of Kepler. In 1 6 1 1 he openly
declared for Copernicanism. His own observations convinced him that the
Copernican system was correct, and in the classic Dialogue on the Great
World Systems he defended it strongly. By the middle of the seventeenth
century the scientific world was willing to proceed on a heliocentric basis.
One word of caution regarding the work of Copernicus and Kepler: these
men believed that the heliocentric theory was true for the reasons already
cited. This is not the view we hold today. If the criterion is truth, then
heliocentric theory is not to be preferred to Ptolemaic theory. Scientific
theories, we now believe, are the work of man. The mind supplies the patterns
which organize observations. We may indeed prefer the heliocentric theory
because it is simpler and agrees better with observations, but we do not regard
it as the last word. Another theory, which still may not be the truth, may be
conceived and produce even better results. As a matter of fact, one was-the
theory of relativity. We shall not anticipate too much. The evolution of the
concept of truth as it applies to mathematics and mathematical theories of
science will be a continuing concern.
EXERCISES
1. What is a geocentric astronomical system? a heliocentric astronomical system?
2. Did Copernicus break completely with Greek astronomy?
3. Is the sun at the center of the Keplerian system?
334 THE APPLICATION O F FORMULAS TO GRAVITATION
4. What innovations did Kepler introduce into the Copernican system?
5. To reconstruct Kepler's improvement on Copernicus, suppose that a planet P
moves once around its epicycle while the center of the epicycle moves once
completely around the sun. What path does the planet seem to follow in relation
to the sun? Would it be simpler to accept this single path as opposed to the
combination of epicycle and deferent?
6. What scientific objections were there to an earth in motion?
7. Why did Copernicus and Kepler advocate the new heliocentric theory?
8. Why do you accept the heliocentric theory?
9. State Kepler's first law of planetary motion.
10. State Kepler's second law of planetary motion.
1 1. If we take the earth's average distance from the sun, 93,000,000 mi, as the unit of
distance and the earth's period of revolution, 1 year, as the unit of time, then
Kepler's third law says that T'l= D3. If the average distance of Neptune from
the sun is 2,797,000,000 mi, how long does it take the planet to complete one
revolution around the sun?
15-4 THE PROBLEM OF RELATING EARTHLY AND HEAVENLY MOTIONS
In view of the fact that Galileo had discovered the laws which underlie ter
restrial motions and Kepler had discovered the basic laws of planetary motion
one would expect that the scientists of the seventeenth century would have
regarded the theory of motion to be complete. But to scientists who seek the
ultimate design of our universe, the two accomplishments we have just de
scribed immediately suggested more profound problems. A comparison of
these two classes of laws, namely Galileo's for terrestrial motions and Kepler's
for heavenly motions, revealed several basic differences. In the first place,
Galileo had started with clear physical principles, such as the first law of
motion and the constant downward acceleration of objects moving near the
surface of the earth, and had deduced the formulas which describe straight
line and curvilinear motions. Kepler's three laws, though they fitted observa
tions within the limits of observational errors, did not rest on physical prin
ciples. They were merely accurate mathematical descriptions of collections of
data. Moreover, the three laws were logically independent of one another.
Secondly, for terrestrial motions, the parabola was found to be the basic path
of curvilinear motion, whereas for planetary motion, the ellipse was the basic
path.
This comparison raised several questions. Could one establish any logical
relationship among the Keplerian laws or were they really independent? What
physical principles determined planetary motions? The mathematical laws,
accurate and succinct as they were, presented after all only a rather bleak
account, without giving any insight into, or rationale for, the motions. And
why should parabolic paths prevail on earth and elliptical paths in the heavens?
RELATING EARTHLY AND HEAVENLY MOTIONS 335
The overriding question, however, which bothered the leading scientists
of the latter half of the seventeenth century was: Could one establish a con
nection between the laws of terrestrial motion and the laws of planetary
motion? Perhaps the very ,.me physical principles which Galileo had used
to deduce the paths of objects moving near the earth could lead to the laws
describing the motion of the planets. In this event, the fWO classes of laws
would be united; the Keplerian laws would be related to each other by being
deduced from a common basis; and the physical reasons for planetary motion
would be revealed.
The thought that all the phenomena of motion should follow from one
set of physical principles might seem grandiose and inordinate to reasonable
people, but it occurred very naturally to the religious mathematicians of the
seventeenth century. God had designed the universe, and it was to be ex
pected that all phenomena of nature would follo'v one master plan. One mind
designing a universe would almost surely have employed one set of basic
principles to govern as many related phenomena as possible. Since the scien
tists of the seventeenth century were engaged in the quest for God's design
of nature, it seemed very reasonable to them that they should seek the unity
underlying the diverse earthly and heavenly motions. As phrased by Newton,
this goal was
to derive two or three general principles of motion from phenomena, and after
wards to tell us how the properties and actions of all corporeal things follow
from these manifest principles. . .
A less cogent but to mathematicians nonetheless significant indication of the
existence of some unity was f:lrnished by the fact that parabola and ellipse
were both conic sections. The common mathematical origin of these curves
warranted some belief that parabolic and elliptical motions were but special
cases of some fundamental principle of motion.
In the seventeenth century, there were other less weighty but perhaps
more pressing reasons to pursue the study of motion beyond the stage
reached by Galileo and Kepler. Another open question was how to relate
heavenly and earthly motions in a more limited but practical connection. This
was the problem described earlier of determining the longitude of a ship at sea.
Although navigators had used the stars, sun, and moon to determine the loca
tions of their ships, the positions of these celestial bodies at various times of
the year had yet to be related more precisely to the longitudes of points on the
earth. In the seventeenth century it seemed that the moon would be most
suitable for the determination of longitude because its closeness permitted
accurate observation of its position from points on the earth. Hence, more
precise information about the motion of the moon around the earth was needed.
This became a major scientific problem of the age.
336 THE APPLICATION OF FORMULAS TO GRAVITATION
15-5 A SKETCH OF NEWTON'S LIFE
Any great advance in mathematics and science is almost always the work of
many men contributed bit by bit over hundreds of years. Then one man smart
enough to distinguish the worthy ideas of his predecessors from the welter of
suggestions and results and imaginative and audacious enough to fit the sig
nificant ideas into a master plan makes the culminating and definitive step. In
the problem of unifying all the phenomena of motion, the decisive step was
made by Isaac Newton.
He was born in 1642, premature and weak. His mother was already
widowed and so preoccupied with running the family farm that she could
pay no attention to the boy. The elementary education Newton received in
local schools of a small English town could hardly have given him much of a
start, and in his youth Newton showed no promise. His family sent him to
Cambridge University, where he entered Trinity College in 1 66 1 . Here, at
last, Newton got the opportunity to study the works of Copernicus, Kepler,
and Galileo, and here he had at least one good teacher, the distinguished mathe
matician Isaac Barrow. His university work was not outstanding and he had,
in fact, such difficulties with geometry that he almost changed his course of
study from science to law. However, Barrow did recognize that Newton had
ability.
Newton finished his undergraduate work; at that point an outbreak of the
plague in the area around London led to the closing of the university. He,
therefore, spent the years 1665 and 1666 in the quiet of the family home at
Woolsthotpe. During this period Newton initiated his great work in me
chanics, mathematics, and optics. He realized that the law of gravitation.
which we shall examine shortly, was the key to an embracing science of
mechanics; he obtained a general method for treating the problems of the
calculus (see Chapters 1 6 and 1 7 ) ; and through experiments he made the epochal
discovery that white light such as sunlight is really composed of all colors
from violet to red. "All this," Newton said later in life, "was in the two
plague years of 1665 and 1666, for in those days I was in the prime of my
age for invention, and minded mathematics and philosophy [science] more
than at any other time since."
Newton returned to Cambridge in 1667 and was elected a Fellow of
Trinity College. In 1 669, Isaac Barrow resigned his professorship of mathe
matics to devote himself to theology, and Newton was appointed in Barrow's
place. Newton apparently was not a successful teacher, for few students at
tended his lectures; nor did anyone comment on the originality of the material
he presented.
In 1684 his friend Edmond Halley, the astronomer of Halley's comet fame,
urged him to publish his work on gravitation and even assisted him editorially
and financially. Thus in 1687 the classic of science, the Mathematical Principles
of Natural Philosophy, often briefly referred to as the Principia or the Prin-
NEWTON'S KEY IDEA 337
ciples, appeared. This book received much acclaim and, aside from three Latin
editions, appeared in many languages. One popularization was entitled New
tonial1ism for Ladies. The Principia is written in the deductive manner of
Euclid; that is, it contains definitions, axioms, and hundreds of theorems and
corollaries. Its conciseness makes it difficult reading. To excuse this aspect,
Newton told a friend that he had made the Principia difficult on purpose "to
avoid being baited by little smatterers in mathematics." He thereby hoped to
avoid the criticisms heaped on his earlier papers on light.
After about thirty years of creative activity which included some work in
chemistry, Newton became depressed and suffered a nervous breakdown. He
left Cambridge University to become Warden of the British Mint in 1696 and
thereafter confined his scientific activities to the investigation of an occasional
problem. He did, however, devote himself to theological studies, which he
regarded as more fundamental than science and mathematics because the latter
disciplines concerned only the physical world. In fact, had Newton been born
two hundred years earlier he would almost surely have become a theologian.
An example of his theological writing is The Chronology of the Ancient Kings
Amended, in which he sought to determine the dates of Biblical events by
utilizing astronomical facts mentioned in connection with these events.
During his last years and posthumously he was honored in many ways.
He was President of the Royal Society of London from 1703 to his death; he
was knighted in 1705; and he was buried in Westminster Abbey.
15-6 NEWTON'S KEY IDEA
In his philosophy and method of science, Newton followed Galileo. He, too,
believed that the universe was mathematically designed by God and that mathe
matics and science should strive to uncover that glorious design. Like Galileo,
he was convinced that fundamental physical principles should be quantitative
statements about the real qualities of the world, space, time, mass, weight, and
force. From these principles and with the axioms and theorems of mathematics,
it should be possible to deduce the laws of nature. Newton expressed this phi
losophy in the preface to his Principles:
. . . fOT the whole burden of philosophy [science] seems to consist in this
from the phenomena of motion to investigate the fOTces of nature, and from
these forces to demonstrate the other phenomena . . .
By investigating the forces of nature, he meant to arrive at the basic laws gov
erning the operation of these forces and to deduce the consequences.
The first problem, then, in executing such a program is to discover the
fundamental principles. Like Galileo, Newton insisted on obtaining these by
direct study of the physical world rather than by searching one's mind for
hypotheses that seemed to be reasonable or by accepting Biblical passages.
338 THE APPLICATION O F FORMULAS TO GRAVITATION
Here, too, Newton is explicit. In another of his famous books, Opticks, first
published in 1 704, he says:
Thus analysis consists in making observations and experiments and in drawing
general conclusions by induction, and admitting of no objections against the
conclusions, but such as are taken fTom experiments or other certain truths.
What Newton sought to emphasize and what required emphasis in his time is
that generalizations must be based on some experimental Of observational
grounds, and that no hypothesis can be tolerated which is contrary to a single
bit of physical evidence. Further, deductions made from the basic principles
must also be ia accord with physical evidence, for only by continued agree
ment between deductively established conclusions and experimental tests can
one acquire confidence that the original generalizations afe correct.
With such principles of scientific method clearly in mind, Newton turned
to the problem of finding the physical principles which would lead to a
unifying theory of earthly and celestial motions. He was, of course, familiar
with the principles unearthed by GaIileo. But these were presumably not
enough. It was clear from the first law of motion that the planets must be
acted on by a force which pulls them toward the sun, for if no force were
acting, each planet would move in a straight line. The idea of a force which
constantly pulls each planet toward the sun had occurred to many men,
Kepler, the famous experimental physicist, Robert Hooke, the physicist and
renowned architect, Christopher Wren, Halley and others, even before New
ton set to work. It had also been conjectured that this force exerted on a dis
tant planet must be weaker than that exerted on a nearer one and, in fact, that
this force must decrease as the square of the distance between sun and planet
increased. But, prior to Newton's work, none of these thoughts about a
gravitational force advanced beyond speculation.
Newton adopted these ideas. However, in his attempt to tie in the action
of the gravitational force with motions on the earth, a line of thinking oc
curred to him which was highly imaginative and certainly original in his time
but which is now an almost daily experience. He considered the problem of
what happens when a projectile is shot out horizontally from the top of a
mountain. As Newton knew and as we know from our study of Galileo's
work, the projectile follows a parabolic path to earth (see Chapter 14,
Fig. 14-3 ) . If the horizontal speed of the projectile is increased, then the path
is wider but remains parabolic. However, Galileo had assumed that the earth
was Rat and that the projectiles were given moderate initial horizontal speeds
such as a cannon might impart to shells. Newton then asked himself what
would happen if the sphericity of the earth were taken into account, and if
the horizontal speeds of t�.e projectiles were gradually increased. If the
sphericity of the earth is taken into account, then projectiles with small hori
zontal speeds will follow the paths PA and PB of Fig. 1 5-3. As the speed is
' 3 39
NEWTON S KEY IDEA
increased somewhat, the projectile might take a path such as PC. Suppose now
that the speed is increased still more. Would the projectile fall off into space?
Not necessarily. As the projectile travels into space, it is pulled toward the
earth. But the pull of . spherical earth is directed toward the center, and hence
the projectile, subjected to this continuous pull toward the center, need not
fall off into space. It might, in fact, continue to circle the earth indefinitely if
the earth pulled it in just enough so that it would not wander out into space
and yet not fall to earth.
And so Newton concluded in his Principles:
And after the same manner that a projectile, by the force of gravity, may be
made to revolve in an orbit, and go round the whole earth, the moon also, either
by the force of gravity, if it is endowed with gravity, or by any other force,
that impels it toward the earth, may be continually drawn aside towards the
earth, out of the rectilinear [straight-line] way which by its innate force [inertia]
it would pursue; and would be made to revolve in the orbit which it now
describes; nor could the moon without some such force be retained in its orbit.
If this force were too small, it would not sufficiently turn the moon out of a
rectilinear course; if it were too great, it would turn it too much, and draw the
moon from its orbit toward the earth. It is necessary that the force be of a just
quantity, and it belongs to the mathematicians to find the force that may serve
exactly to retain a body in a given orbit with a given velocity; .
p
Fig. 15-3.
Projectiles shot out horizontally from the top
of a mountain with increasing horizontal
velocities.
This argument which showed how the motion of the moon around the earth
could be related to m�tions occurring on earth was immediately extended to
the motions of the planets about the sun. The planets, set into motion some
how, are attracted by the sun and are presumably pulled in just enough to keep
them from flying off into space or from crashing into the sun.
Thus Newton had some reason to suppose that the same force which
pulled projectiles to earth caused the moon to revolve around the earth and
the planets to revolve around the sun. He had now to determine precisely how
strong the force of gravitation is, that is, how it depends upon the bodies
involved and upon the distances between the bodies.
340 THE APPLICATION OF FORMULAS TO GRAVITATION
15-7 MASS AND WEIGHT
Before we can understand Newton's law of gravitation we must distinguish
two properties of matter, mass and weight. Newton's first law of motion says
that if no force is applied, bodies continue at the speed they already have.
Stated otherwise, the law says that bodies have inertia; they persist in the
motion they already have unless compelled to do otherwise by the application
of force. This inertia or resistance of matter to change in speed is called
inertial mass or just mass.
Do all objects have the same mass? Not at all. Since mass exhibits itself
in an object's resistance to change in speed, we can appeal to experience to
see that different objects may possess different masses. Suppose, for example,
that a small and a large ball of lead are at rest on the ground and one wishes to
start them moving. Experience tells us that we must exert more force to get
the larger ball rolling than to get the smaller one rolling with the same speed.
Since more force is required in the first case, the larger ball must possess more
mass. Or we can imagine the force that might be required to stop these balls
if they were rolling toward us at the same speed. Again, more force would be
required to stop the larger one. Thus the masses of objects are not the same.
We shall not present physical methods of measuring mass. It suffices to
know that by adopting a unit of mass, just as we adopt a unit of length, we can
compare all other masses with this unit and so determine exactly how much
mass there is in any individual piece of matter. Mass is measured in pounds or
grams, the pound being approximately 454 grams.
Bodies falling to earth possess acceleration. Hence some force must be
acting to produce this change in speed. The force, as Galileo and others
realized, is the pull of the earth or the force of gravity. We feel this pull when
we hold an object in our hands. This particular force applied to an object is
called the weight of the object. Hence weight and mass are by no means the
same. Mass is inertia or resistance fO change in speed, and weight is a force
exerted by the earth.
However, there is a remarkable, experimentally determined relationship
between the mass and weight of an object, namely, near the surface of the
earth the weight is always 32 times the mass; in symbols:
w = 32m. (I)
The quantity 32 is precisely the acceleration which all bodies falling to earth
possess. Thus equation ( I ) says that the force, w, which the earth exerts on
a mass, m, is the acceleration with which it causes the mass to fall times rhe
mass. When the mass m is measured in pounds, the weight w is measured in
poundals. Thus a mass of one pound has a weight of 32 poundals. For con
venience, 32 poundals are called one pound of weight. Of course, a pound of
mass and a pound of weight are not the same physical quantities; and it is con-
THE LAW OF GRAVITATION 341
fusing at times to use the same unit for both mass and weight. Yet, as we shall
see in a moment, this confusion is not roo serious. (If the mass is measured in
grams, the quantity which replaces 32 is 980, and the units are centimeters per
sec'. Then instead of ( I ) we have
w = 98Om,
and the weight w is measured in dynes.)
Because weight and mass are so intimately related, we do not trouble in
ordinary life to distinguish between the two properties. Large masses have
large weights, and so, even in those instances where we are actually concerned
with the mass of an object, we often tend to think of weight. For example, if
one were to try to start an automobile rolling by pushing it, he would have to
exert considerable force. The average person relates this fact to the great
weight of the automobile. However, weight plays no role here because the
force of gravity acts downward and has no effect on motion along the ground.
The forceful push is required because the mass resists change in speed. Hence,
it is the mass of the automobile rather than the weight which calls for the
exertion of great force.
EXERCISES
1. Why do people usually fail to distinguish between mass and weight?
2. Let us assume that the relationship between weight and mass on the moon has the
same mathematical form as on the earth; that is, the weight is a constant times
the mass. The acceleration which the moon gives to all bodies falling toward its
surface is 5.3 ft/sec2. If a man weighs 160 lb on the earth, what will he weigh on
the moon?
3. If the acceleration which the sun imparts to bodies falling toward its surface is
27 times that imparted by the earth, what would a man whose weight on the eanh
is 160 lb weigh on the sun?
15-8 THE LAW OF GRAVITATION
Newton adopted the conjecture already made by his contemporaries, namely,
that the force of attraction, F, between any two bodies of masses m and M,
respectively, separated by a distance r is given by the formula
mM
F = G -
2 • (2)
r
In this formula, G is a constant; i.e., it is the same number, no matter what
m, M, and r may be. The numerical value of this constant, which we shall
determine later, depends upon the units used for mass, force, and distance.
342 THE APPLICATION OF FORMULAS TO GRAVITATION
From the mathematical standpoint, formula (2) represents a new type of
functional relationship. The quantity F is the dependent variable which
depends upon three independent variables, m, M, and r, the quantity G being
a constant. If we give values to m, M, and T, then the value of F is determined.
Such a function is. of course, more complicated than, say the formula
d = 96t - 16t2, which contains just one independent variable. t. and one
dependent variable, d. When one is working in a situation in which m and M
are fixed and only r can vary, then F is a function of just one independent
variable. For example, if G were 1, m were 2, and M were 3. then the rela
tionship between F and r would be F = 6/T'. This formula expresses the
dependence of the gravitational force between two fixed masses on the distance
between them.
Newton had yet to show that formula ( 2 ) was the correct quantitative
expression for this force. To apply formula ( 2 ) and to work with forces in
general, Newton adopted a second quantitative physical principle which
proved to be just as important as his law of gravitation. As we noted in the
preceding section, near the surface of the earth the force of gravity of the earth
gives to objects an acceleration of 32 ft/sec', and the force is 32 times the
mass of the object. Newton generalized this relationship and affirmed that
whenever any force acts on an object it gives that object an acceleration.
Moreover, the relationship of force, mass of the object, and acceleration im
parted to the object, is
F = mao ( 3)
In this formula F is the amount of force applied to the object of mass m, and
a is the amount of acceleration imparted to the object. In the special case
where F is the weight w, the value of a is 32 ft/sec'. Formula ( 3 ) is known as
Newton's second law of motion. It applies to any force, whether or not it be
the force of gravity. As in the case of formula ( 1 ) , if m is measured in pounds
and a in feet and seconds, then F is measured in poundals. Thus a force of 32
poundals gives a mass of I pound an acceleration of J 2 ft/sec'. The unit,
pound, is also used for forces with the understanding that one pound of force
equals 32 poundals.
Let us see how Newton tested his law of gravitation. We shall write (2)
in the slightly different form
GM ·
F= m --
(4)
r'
If we compare ( 3 ) and (4), we observe that the quantity GM/r' in (4) plays
the role of a in ( J ) ; that is, the law of gravitation can be viewed as stating that
the gravitational force F gives a mass m the acceleration GM/r'. In symbols,
(5)
FURTHER DISCUSSION OF MASS AND WEIGHT 343
Now let M be the mass of the earth and let m be the mass of a small body near
the surface of the earth. Then there is the question of what r in ( 5 ) represents.
I t is supposed to represent the distance between the two masses. Shall we take
it then to be the distance from the mass m to the surface of the earth or to
some point in the interior? If the two masses were separated by millions of
miles, as are the earth and sun, one might idealize each mass and regard it as
concentrated at one point because the size of each mass is small compared with
the distance between them. But for objects near the surface of the earth, the
value of r depends heavily upon what point in or on the surface of the earth is
chosen as the position of the earth. Newton conjectured (and later proved)
that for purposes of gravitational attraction the mass of the earth could be
regarded as though it were concentrated at the earth's center. Hence, with
respect to the earth's gravitational acceleration acting on a mass m near the
surface of the earth, r in ( 5 ) can be taken to be 4000 miles or 2 1,120,000 feet.
This value of r is essentially the same for all objects near the surface of the
earth. Moreover, the mass M of the earth is constant, and so is G. Hence, for
all objects near the surface of the earth, the entire right side of ( 5 ) is constant.
Consequently, the acceleration which gravity imparts to all objects near the
surface of the earth is constant. This is precisely what Galileo had found and,
in fact, he had determined that the constant is 32 ft/sec'. Thus Newton's law
of gravitation met its first test, for it yielded as a special case a well established
fact.
EXERCISES
1. Suppose that the gravitational force varies with the distance between two definite
masses according to the formula F 6 /r2. Show graphically how F varies with r.
=
2. Knowing that the acceleration of objects near the surface of the earth is 32 ft/sec2,
use fonnula ( 5 ) to calculate the acceleration which the earth exerts on objects
1000 mi above the surface of the eanh.
3. Suppose that an object falls to earth from a point 1000 mi above the surface.
May we use the formula d= 16t2 to compute the time it takes to fall this distance?
4. What is the mass of an object which weighs 150 Ib? (One pound of weight is 32
poundals.)
5. How much force is required to give an automobile weighing 3000 Ib an accelera
tion of 12 ft/sec2?
15-9 FURTHER DISCUSSION OF MASS AND WEIGHT
With some support for the law of gravitation we can now, following Newton's
example, adopt it as an axiom of physics and see what conclusions we may
draw from this axiom and the other axioms of physics and mathematics. The
law itself states that the force of gravitation F between any two masses m and
344 THE APPLICATION OF FORMULAS TO GRAVITATION
M is given by the formula
(6)
where 7 is the distance between the masses. Formula (6) leads immediately
to a better understanding of the relationship between weight and mass and to
an extension of the concept of weight. Let M be the mass of the earth and let
m be the mass of some other object. Since F is the force with which the earth
attracts this object, we can regard F as the weight of the object, for this at
tractive force is what we have meant by weight. However, we now see
that the force or weight depends upon the distance 7 between the two masses.
Hence, the weight of an object is not really a fixed number but varies with the
distance of the object from the earth or, more precisely, from the center of
the earth (see Section 1 5-8). If an object of mass m is at the surface of the
earth, its weight is given by
Mm ,
F, = G (7)
(4000 . 5280) 2
but the same object taken 1000 miles above the surface of the earth will have
the weight:
Mm
F2 - G (8)
(5000 ' 5280) 2
The value F2 is considerably less than F, because the denominator in the
second expression is much larger. We see, then, that the farther an object of
mass m is from the surface of the earth, the less is its weight. On the other
hand, the mass of the object, that is its resistance to change in speed, is the
same at all locations. Thus we can see more clearly that the weight and mass of
an object are quite different properties.
The concept of weight can, and in the present scientific era must, be ex
tended still further. So far we have considered the weight of an object to be
the force with which the earth attracts the object. But now let us imagine that
the object were taken to the moon and, for simpliciry, let us suppose that no
matter other than the moon and the object exist in space. May we speak of
the weight of the object on the moon? The law of gravitation applies to moon
and object, and so the moon will attract the object. This attractive force will
be the weight of the object on the moon. To calculate this weight, we have
but to let M in (6) be the mass of the moon, m, the mass of the object, and 7,
the radius of the moon. We know that the radius of the moon is 1080 miles or
1080 · 5280 feet. The mass of the moon can be determined by methods similar
to those used later (Section 15-10) to compute the masses of the earth and
FURTHER DISCUSSION OF MASS AND WEIGHT 345
sun. The result of the ca\culation, which we may accept for present purposes,
is that the weight of an object on the moon is i its weight on earth.
We can extend the notion of weight still further. Suppose that an object
is in space somewhere between earth and moon. According to the law of
gravitation, the earth ar:rracts the object, and so does the moon. Since these
attractions oppose each other, we may regard the weight of the object as the
net am·action. If we now think of the object as moving from the earth to the
moon, then the ar:rracrive force of the earth decreases while that of the moon
increases. At the outset, the eanh's force is stronger, but at some point in the
path to the moon the two forces will be equal and oppositely directed so that
the net weight of the object will be zero. This point is located at a distance of
about 24,000 miles from the moon along the line from the earth to the moon.
All of the above considerations about weight are now no longer purely aca
demic flights of fancy but are important factors in the process of determining
the paths of rockets which are sent out to strike the moon.
EXERCISES
1. Suppose that a person weighs 150 lb at the surface of the earth where, of course,
his distance from the center of the eanh is 4000 mi. What would this person
weigh at a point 4000 mi above the surface of the earth?
2. How does the law of gravitation enable us to further differentiate between the
mass and weight of an object?
3. Suppose that of two objects on the earth, one has twice the mass of the other.
Show that the force with which the earth attracts the first one is twice the force
with which the earth attracts the second.
4. What would a man whose present weight is 150 lb weigh if the eanh's mass
were one-tenth of what it is?
5. Suppose that the earth's mass were twice as large as it is. What change would
there be in the acceleration of falling bodies? Would a body which is dropped
from a height of 1000 ft reach the ground sooner than it now does?
6. It is stated in the text that all bodies near the surface of the earth fall with the
same acceleration. Suppose that an object is several thousand miles from the
surface of the earth. How would the acceleration of its fall to the earth compare
with the acceleration of a body near the surface?
7. Suppose that the mass of the moon were the same as the mass of the earth.
The radius of the moon is about ! the radius of the earth. What would a man
who weighs 150 lb on the earth weigh on the moon?
8. The earth's attractive force acts quite differently on objects in the interior of
the earth than on objects outside the earth. In the former case the force is given
by the formula F=-=G111MT/R3, where 1n is the mass of the object, M the mass of
the earth, R the rauiu� of the earth, and r the distance of the object from the
center of the earth. Compare the variation of this attractive force as r varies,
with the force given by formula (6).
346 THE APPLICATION OF FORMULAS TO GRAVITATION
9. Suppose that the law of gravitation were F = GmM/r instead of formula (6).
Compare the variation of weight with distance from the center of the earth
according to this formula with the variation of weight according to (6).
10. Consider all objects at a distance of 5000 mi from the center of the earth. Is
the ratio of weight to mass the same for all these objects?
15-10 SOME DEDUCTIONS FROM THE LAW OF GRAVITATION
The essence of the scientific method created by Galileo and Newton is to
establish basic quantitative physical principles and to apply mathematical reason
ing to these principles. The law of gravitation and the first and second laws
of motion are such physical principles. We shall see now that Newton was
able to make some remarkable deductions from these principles.
The law of gravitation contains the constant G. Many calculations based
on the law of gravitation require that one know G. In principle, this quantity
is easily measured. One has but to take two known masses, place them a
measured distance apart, and measure the force with which the two masses
attract each other. Then, since
mM
F= G- '
,
- (9)
T
we see that every quantity in (9) is given except G, so that we have a simple
algebraic equation for G. The actual experiments which have been made to
measure G are a little more complicated because the force F is small for
ordinary masses. However, the experiments have been performed, and the
value of G turns out to be 1 .07/10'. The notation 10' is scientific shorthand
for the product in which 1 0 occurs as a factor 9 times, i.e., one billion. This
value of G presupposes that masses are measured in pounds, distances in feet,
and forces in poundals (practical English system ) . (In the centimer-gram-sec
ond (cgs) system of unit;, G is 6.67/10'.)
With the value of G known, it is a simple matter to calculate the mass of
the earth. We may recall that formula ( 5 ) , which is an immediate conse
quence of the law of gravitation and the second law of motion, states that the
acceleration which the earth imparts to any other mass is
( 1 0)
where r is the distance between the two masses. We have also learned that
when r is 4000 miles or 2 1 , 1 2 0,000 feet, then Q = 32. Let us substitute these
values and the value of G in ( 1 0) . Then
1 .07 . M
32 = - -:-
-- -;;
---, (1 1)
10' (21,1 20,000) '
SOME DEDUCTIONS FROM THE LAW OF GRAVITATION 347
and we have obtained a simple equation for the unknown M. To shorren
the somewhat complicated arithmetic, let us approximate and write 2 1,1 20,000
as 2 1 ,000,000 or as 2 1 · 1 0'. Then by a theorem on exponents (see Section 5-3)
(21 ' 1 0 6) 2 = (2 1 ) 2 . (106) 2 = 441 . 10 1 2.
Substituting the value just obtained in ( I I ) yields
1 .07 M
32 ' ( 1 2)
= 10. 441 . 101 2
The factors 10· and 1012 can be combined, for the first factor means
1 0 · 1 0 · 10 · . " wherein 10 occurs 9 times, and the second factor means that
10 occurs 1 2 times. Then in the product of these two factors 1 0 occurs 2 1
times. Hence
- .07M
1- -=
32 = (13)
441 . 102 1
Multiplying both sides o f this equation by 441 . 1021 and dividing both sides
by 1.07, we obtain
3 2 ' 441 . 1021
M =
1.07
or
M = 1 3.1 . 1024 pounds. (14 )
Since there are 2000 pounds in one ton, we may divide the right side by 2000
and write
M = 6.5 . 1021 tons. ( 1 5)
Hence some simple algebra applied to formula ( 10) was all that was needed
to calculate the mass of the earth. Let us note clearly that this quantity is not
the weight of the earth. Technically the earth has no weight since weight is,
by definition, the force which the earth exerts on other masses. However, a
mass of 6.5 · 1021 tons would weigh the same amount of tons, and so one can
get some idea of the earth's mass.
From the knowledge just obtained we can deduce some information abour
the interior of the earth. The earth is approximately spherical in shape, and
since the volume, V, of a sphere is 47rfJ/3, where r is the radius, we can compute
the volume of the earth. Thus
348 THE APPLICATION OF FORMULAS TO GRAVITATION
We shall approximate 2 1 1 2 by 2 1 · 10' and use the value of 3 . 14 for 1r. Then
v = ! ( 3 . 1 4) (2 1 . 10 2) 3 . 1 0 ' 2 = �(3. 1 4) (2 1) 3 . 10 6 . 1 OI 2.
Since (t) ( 3 . 14) ( 2 1 ) 3 is about 39,000, we have
V = 39,000 · 106 • 1OI2 = 39 · 10 2 1 = 3.9 · 1 0 22 cubic feet.
We next divide the mass of the earth, in pounds, by the volume to find the
mass per cubic foot. Thus
M 1 3 . 1 . 1 02 41 3 10 . 1 0 22 1 3 10
- = -- = 3 36. ( 1 6)
V 3.9 · 1 0 22 3.9 · 1 0 22 3.9
The mass per cubic foot of water is 62.5 pounds. We see then that the mass
per cubic foot of earth is about 5.5 times the mass per cubic foot of water.
This figure of 5.5, incidentally, is the density of the earth.
Examination of the earth's surface shows that it consists mostly of water
and sand. Since the quantity of rock visible on the surface does not account
for the ratio 5.5, the conclusion follows that the interior of the earth must
contain heavy minerals.
Only a little more work is required to compute the mass of the sun. We
shall again begin with the law of gravitation and the second law of motion.
The two masses involved now are the mass of the sun, S, and the mass of the
earth, E. Then the law of gravitation states that the force with which the
sun attracts the earth is
SE
F = G. , (17)
T
where r is the distance from the earth to the sun. According to Newton's
second law of motion, the force which the sun exerts on the earth gives the
earth an acceleration a such that
F = Ea. ( 1 8)
Since the forces in ( 1 7) and ( 1 8) are the same, we may equate the right sides.
Then
SE
Ea G.· (19)
T
=
We may next divide both sides of ( 19) by E and obtain
(20)
SOME DEDUCTIONS FROM THE LAW OF GRAVITATION 349
In this last equation we know G and r. If we knew a, the acceleration of the
earth, we could calculate S. Let us see what we can do about calculating a.
The acceleration which the sun imparts to the earth causes the earth to
depart from a straight-line path, which it might otherwise pursue, and "fall"
toward the sun just enough to keep it on its elliptical path. (The acceleration
which the earth imparts to the moon has the same effect on the lunar orbit.)
d
r
H
r
s Fig. 15-4.
The sun's pull on the earth causes
the earth to "fall" the distance QR
in t seconds.
We shall suppose, for the sake of simplicity, that the path of the earth is
circular. Let us imagine that the earth is at the point P (Fig. 1 5-4) in its
path around the sun. If there were no gravitational force, the earth would
shoot straight out along the tangent at P into space in accordance with the
first law of motion. Let us suppose that in time t, the earth would have
reached the point O. The distance traveled would be the velocity of the
earth in its path around the sun, v say, multiplied by I. Hence PO = vi.
However, during that time t, the sun pulls the earth in a distance OR or d .
Since SPO is a right triangle,
(r + d)' = r' + (VI) ' .
Squaring r + d and substituting the result, we obtain
r' + 2dr + d' = r' + v't'.
We subtract r' from both sides of this equation and find that
2dr + d' = v't'.
Applying the distributive axiom on the left side permits us to write
( �)
2d r + = v't'. (21)
Now d is the distance thar the earth falls in time I . Let us suppose that it
falls with constant acceleration. (We shall soon let I become very small so
that the acceleration can well be taken as constant.) If a body falls a distance
350 THE APPLICATION OF FORMULAS TO GRAVITATION
d with constant acceleration a, then we know from our work in Chapter 13
(Section 1 3-5, Exercise 14) that
or
2d = at'. (22)
Let us substitute this value of 2d in ( 2 1 ) . Then
( �) = v't'.
at' r +
We now divide both sides of this equation by t' and obtain
(23)
Thus far t was arbitrarily chosen, and d was the distance the earth fell toward
the sun in time t. Our result so far, then, is valid for any value of t. If we
now let t become smaller and smaller, d will also decrease. When t = 0, it
follows from (22) that d = O. In this case, (23) becomes
aT = v'
or
v'
a= (24)
r
This result states that the acceleration which the sun imparts to the earth at
each point P of the earth's path is the square of the earth's velocity divided
by the distance of the earth from the sun. This acceleration is called centripetal
(i.e., center-seeking) acceleration, because it causes the earth to mOve toward
the center of its path.
We now have the quantity a which we needed in (20) . Substitution of
(24) in (20) yields
We may multiply both sides by r and divide both sides by G to obtain
v r '
s=_· (25)
G
Every term on the right side of this equation is known. The distance r is
93,000,000 miles or 4.9 · 10" feet. The velocity, v, of the earth is the circum-
SOME DEDUCTIONS FROM THE LAW OF GRAVITATION 351
ference o f the earth's path divided by the number of seconds in one year:
2rl.9 . 10 1 1 3 0.8 . 10 1 1
V= ."..., = :.::.. -,
= - -, - "':- ''7
:'::" "":':' ' = 9.8 ' 104 ft/sec.
365 ' 24 ' 60 ' 60 3 . 1 5 ' 10
Hence
v2 = (9.8 ' 104) 2 = (9.8) 2 . lO B = 96 ' lO B. (26)
In Section 1 5-10 we learned that G � 1.07/ 10'. Thus, using these values of
r, V, and G in (25), we have
96 ' lO B . 4.9 . 10 1 1 96 ' lO B . 4.9 . 101 1 . 10"
s =
1 .07/10" 1 .07
or
s = 440 ' 10 2 B = 4.40 ' 10 3 0 (27)
Hence the mass of the sun is 4.40 · 10'10 pounds. Since the earth's mass was
previously found to be 1 . 3 1 . 10" pounds, we see that the mass of the sun is
3.36 · 10' or 3 36,000 times the mass of the earth.
We can determine the mlSS per cubic foot of the sun in the same manner as
we calculated the mass per cubic foot of the earth. The mass of the sun is
now known, and the radius, computed in Chapter 7, is 432,000 miles, or
2.28 · 10' feet. We shall not reproduce the calculations, but state the result:
the mass per cubic foot proves to be 90 pounds. Since a cubic foot of water
has a mass of 62.5 pounds, we see that the mass per cubic foot of the sun
is about It that of water; that is, the density of the sun is about I i.
The examples given in this section further illustrate how mathematical
reasoning can be applied to physical laws (in our case, to the second law
of motion and the law of gravitation) in order to deduce fundamental knowl
edge about the universe. We did, of course, also use some experimentally
obtained facts such as the value of G and the acceleration of bodies near the
earth's surface. However, mathematics has been the main tool, and it obtains
for us such remarkable information as the mass of the earth and the mass of
the sun.
EXERCISES
1. Suppose an object moves in a circle at a constant speed. Is the motion subject
to an acceleration?
2. Does the formula for the acceleration of the earth given in (24) depend upon
the law of gravitation?
3. If you whirl an object on a string of radius 5 ft, at the rate of 50 ft/sec, what is
the centripetal acceleration acting on the object? What force exerts this cen
tripetal acceleration?
352 THE APPLICATION OF FORMULAS TO GRAVITATION
4. Use fonnula (24) with the understanding that v is the velocity of the moon and
T is the distance of the moon from the earth, to calculate the acceleration of the
moon. (The period of the moon's path around the earth is 27! days, and the
distance of the moon from the earth is 240,000 mi.)
5. Using the figures in the text for the mass and radius of the sun, calculate the
ratio of the mass to the volume of the sun.
* 15-11 THE ROTATION OF THE EARTH
We have repeatedly used the quantity 3 2 ft/sec2 as the acceleration which the
earth gives to objects near its surface. This figure is perfectly satisfactory for
most purposes, but it is not strictly accurate even for motions riear the earth's
surface. Actually, the acceleration of falling bodies decreases from
32.257 ft/sec2 at either pole to 32.089 ftlsec' at the equator.· The discovery
of this decrease was at first not surprising to the seventeenth-century scientists.
Newton had already proved that the earth is not strictly spherical, but has the
shape of a somewhat flattened sphere (Fig. 1 5-5 ) ; that is, for example, the
lengths OA, OB, OC, and OD are not equal, but are successively larger. Since
the general formula for the acceleration due to gravity [see ( 5 ) 1 is GMlr',
where G and M are fixed and r is the distance from the center of the earth,
this acceleration is less at C, say, than at B because r is larger at C than at B.
Hence we should expect the acceleration due to
gravity to decrease as the location varies from A
to D. Now the values of G and M were known.
Moreover, Newton and Huygens had computed
lengths such as OA, OB, and so forth, and there
fore were able to determine what the acceleration II"----] D
should be at points such as A, B, C, and D. These
calculations based on the expression GMIr', call
for only a small percentage of the decrease actu-
ally measured. Thus precise measurement re
Fig. 15-5.
vealed a discrepancy between the acceleration The spheroidal shape of the
predicted by the law of gravitation and the actual earth.
acceleration of falling bodies. This discrepancy
required explanation.
The problem was solved by Huygens. Objects on the surface of the earth
would fly off into space if the earth did not pull them toward the center, just
as an object whirled at the end of a string would fly off into space if the hand
• The numerical values can, in principle, be obtained by measuring the accelerations
with which bodies near the surface fall to earth. However, a more accurate method
utilizes the formula for the period of a pendulum.
THE ROTATION OF THE EARTH 353
at the center did not exert an inward pull. Thus the earth's gravitational force
has two effects. Even if the earth did not rotate, it would pull all objects
toward the center, simply because the earth's mass attracts the object. But
since the earth does rotate, it must also exert an inward pull so that objects do
not fly off into space but remain on or near the surface of the earth. This
latter effect is a centripetal force. ]n a sense, the twO effects of the earth's
gravitational force, that is, the force which causes objects to fall to the surface,
or weight, and the centripetal force, are of the same nature. The centripetal
force also pulls objects toward the earth's center, but pulls teem in just enough
to keep them on a circular course. The weight, on the other hand, pulls objects
toward the earth from the circular course to which they are kept by the
centripetal force.
Let us express quantitatively what we have just described. By Newton's
second law, the centripetal force must produce an acceleration (centripetal
acceleration) on the object. Now formula (24) gives the centrip·otal accelera
tion which the sun exerts on the earth. However, this formula is really quite
general, that is, if we replace sun and earth in the argument which led to (24)
by the earth and an object on or near the earth's surface, then the argument
still holds, provided that v is the velocity of the object and r is its distance
from the center of the circle in which the object rotates. Newton's second
law then tells us that the centripetal force must be the mass of the object
times the centripetal acceleration, that is 'I1l'lfl./r.
The force with which the earth pulls an object straight down, i.e., the
weight, equals the mass of the object times the acceleration of its fall. It is
this acceleration which we measure when we observe the fall of objects and
which varies from pole to equator. We shall now denote it by g. Then the
weight is mg.
According to Huygens the gravitational force which the earth exerts on
objects supplies both the centripetal force and the weight. The centripetal
force must be directed toward the center of the circle of latitude on which
the object rotates. However, the weight is directed toward the center of the
earth. Hence we cannot write a simpJe formula which expresses precisely how
the earth's gravitational force is apportioned to provide the centripetal force
and the weight at any latitude. However at the extreme cases of latitude,
that is, at the equator, and at the poles, the apportionme'lt is simple. At the
equator the centripetal force must be directed toward the center of the earth.
If we denote the radius of the earth by R, then at the equator
GMm mv2
R2 = T + mg. (28)
At the North Pole, for example, an object does not travel in a circle as the earth
354 THE APPLICATION O F FORMULAS TO GRAVITATION
rotates, and so no centripetal force is required to keep it rotating with the
earth. Hence at that location,
GMm
R' = mg. (29)
Clearly g is larger in (29) than in (28).
What can we say about the apportionment at intermediate latitudes?
Formula (28) is no longer correct because the circle on which an object
rotates does not have radius R, but has a smallf:r radius. Also the velocity, v,
of the object depends on its latitude. Moreover, as we have already noted,
the direction of the centripetal furce required to keep the object rotating
with the earth must be directed toward the center of the latitude circle. The
effect of all these factors is to decrease the cent! ipetal force required to keep
the object rotating with the earth as the latitude increases, and this f('rce is zero
at the poles. Hence more and more of the gravitational force is applied to the
weight of the object, the quantity 11Zg, and since m is constant, g increases
from the equator to the poles. Almost the full increase in g is due to the
rotation of the earth, the balance being due to the shape of the earth. We can
turn our argument around. We observe that g increases from the equator to
the poles. This increase can be explained by assuming that the earth rotates.
Hence we have reason to believe that the earth rotates.
The numerical value of g, that is, the acceleration of falling bodies, has,
of course, been of importance for centuries. But it has additional importance
today. Let us consider a satellite which circles the earth once every hour. The
circular paths of satellites do have the center of the earth as their center. Al
though the satellite moves at a height of a few hundred miles above the surface
of the earth, we shall ignore this distance and suppose that it travels very near
the surface. What is significant is that the satellite covers 25,000 miles per
hour. Hence the centripetal force required to keep it in its path is considerably
greater than that required to keep an object which travels 2 5,000 miles in 24
hours from flying off into space. We see this fact �rom the middle term in
(28) which tells us that the centripetal force increases with the square of the
velocity. Thus a great deal of the earth's gravitational force must be expended
in centripetal force. In fact, since the satellite does not fall to earth, the value
of g, that is, the acceleration with which it should fall to earth, must be zero.
In other words, the full gravitational force of the earth is expended in keeping
the satellite on its circular path around the earth, and the satellite neither flies
off into space nor falls to earth.
But the weight of any object is the product of its mass and the acceleration,
g, with which gravity makes it fall to earth. Since for the satellite, g = 0, it
follows that the satellite has no weight. Objects contained in the satellite
would also be weightless and so would not experience any earthward pull.
GRAVITATION AND THE KEPLERIAN LAWS 355
In view of the importance which satellites are likely to have in future
scientific investigations, it is desirable to know the velocity which a satellite
must possess if it is to stay in orbit at some desired distance from the center
of the earth. This velocity is readily calculated from (28). Since the satellite
does not fall to earth, the value of g for it must be zero. Then
GMm mv'
� = T
If we divide both sides by m and multiply both sides by T, we obtain
GM
v2 = -_ . (30)
T
We know G and M, the mass of the earth. When T, the distance of the
satellite from the center of the earth, is chosen, then we know all the quantities
on the right side of (30). The quantity GM can be calculated once and for
all. Thus
1 .0
GM = : ( 1 3 . 1 ) 1 0 2 4 = 1 4 ' l OIS
10
The value of r must be in feet. We can now calculate v.
EXERCISES
I. Since the weight of an object is mg, how does a person's weight change as he
travels from the North Pole to the equator?
2. Suppose that a satellite stays close to the earth's surface. How fast would it have
to travel to stay on its circular path and not fall to earth? [Suggestion: Use
formula (30).]
3. The moon is a satellite of the earth. Since the moon stays on its path and does
not fall to the earth, we may conclude that the earth's entire gravitational force
acts as centripetal force on the moon. Using the assumption that the moon's path
is a circle and that it is 240,000 mi from the earth, calculate the velocity of the
moon. [Suggestion: Use (30).]
4. Using the result of Exercise 3, calculate the time it takes the moon to make one
complete revolution around the earth.
5. Calculate the speed required to maintain a satellite in an orbit 500 mi above the
surface of the earth.
* 15-12 GRAVITATION AND THE KEPLERIAN LAWS
Thus far in this chapter we examined the evidence which convinced Newton
that the law of gravitation was correct, and we have seen how it can be applied
to answer a variety of questions about objects and motions on the earth and in
356 THE APPLICATION OF FORMULAS T O GRAVITATION
the heavens. We should now recall that one of the major problems challenging
seventeenth-century scientists was the question whether the same physical
principles could account for terrestrial and celestial motions. Since the law of
gravitation when applied to bodies falling near the earth's surface reduces to
fall with constant acceleration (see Section 15-8), Newton's principles cer
tainly encompassed earthly motions. As to heavenly motions, the three famous
laws of Kepler, which he had inferred from observations, were seemingly
independent of the law of gravitation. The truly great triumph of Newton
was his demonstration that all three Keplerian laws were mathematical con
sequences of the law of gravitation and the two laws of motion.
We shall illustrate what Newton did by showing how the third Keplerian
law can be deduced from the basic laws just mentioned. However, we shall
simplify Newton's work and suppose that the path of a planet around the
sun is circular, whereas the true path, as Kepler proved, is an ellipse.
Let m be the mass of any planet, M the mass of the sun, and r the distance
between them. Then the law of gravitation says that the force F exerted by
the sun on the planet is
GmM .
F-
- (31)
r2
We also know that the sun's force causes any planet to depart from straight
line motion and "fall" toward the sun with some acceleration. This accelera
tion, a, is none other than the centripetal acceleration given by formula (24),
that is vir. The derivation of (24) dealt with the sun and earth, but it applies
to any planet, provided that v is the velocity of the planet and r is its distance
from the sun. We may also assert, by the second law of motion, that the
centripetal force F with which the sun attracts that planet is
", 2
F= m-' (3 Z)
r
The velocity v of any planet is the circumference of its path divided by the
time T of revolution around the sun; that is, v = ZrrrIT. Hence, from (32),
(33)
Now formulas ( J I ) and ( J J ) yield two different expressions for the force
with which the sun attracts any one planet.· Hence we may equate these
• In the light of Section 15-11 we can say that the gravitational force equals the
centripetal force because the sun does not cause any planet to fall toward the sun
from the circular path.
GRAVITATION AND THE KEPLERIAN LAWS 357
two expressions and obtain
Dividing both sides of this equation by m eliminates that quantity. Multiply
ing both sides by 1", we obtain
If we now multiply both sides of this last equation by r'IGM, we find
(34)
The quantity +rr'/GM is the same no matter what planet is being con
sidered, because G is a constant, M is the mass of the sun, and 411"2 is a constant.
Hence formula (34) says that 1" is the product of some constant, say K, and
r'; in symbols,
T2 = Kr3. (35)
Thus the square of the time of revolution of any planet is a constant (i.e., the
same for all planets) times the cube of that planet's distance from the sun.
Formula ( 3 5 ) is, then, Kepler's third law of planetary motion. We have
derived it from the two laws of motion and the law of gravitation by a purely
mathematical argument.
As we remarked earlier, Newton demonstrated that all three of Kepler's
laws, which the latter had obtained only after years of observation and trial
and error, were mathematical consequences of the laws of motion and gravi
tation. Hence the laws of planetary motion, which prior to Newton's work
seemed to have no relationship to earthly motions, were shown to follow from
the same basic principles as did the laws of earthly motions. In this sense,
Newton "explained" the laws of planetary motion. These facts were as much
a consequence of basic physical laws as the straight-line motion of objects
falling to earth from rest or of projectiles following parabolic paths. Newton's
original conjecture that the parabolic motion of projectiles should be intimately
related to the elliptical motion of the planets was gloriously established.
Further, since the Keplerian laws agree with observations, their derivation
from the law of gravitation constituted superb evidence for the correctness of
that law.
The few deductions from the laws of motion and gravitation which we
have presented are just a sample of what Newton and his colleagues were able
to accomplish. Newton applied the law of gravitation to explain a phenomenon
358 THE APPLICATION O F FORMULAS TO GRAV,IATION
which heretofore had not been understood, namely the tides in the oceans.
He showed that these were due to the gravitational forces exerted by the moon
and, to a lesser extent, the sun on large bodies of water. From data collected
on the height of lunar tides, that is, tides due to the moon, Newton calculated
the moon's mass. Newton and Huygens calculated the bulge of the earth
around the equator. Newton and others showed that the paths of comets are
in conformity with the law of gravitation. Hence the comets, too, were recog
nized as lawful members of our solar system and ceased to be viewed as ac
cidental occurrences or visitations from God intended to wreak destruction
upon us. Newton then showed that the attraction of the moon and the sun on
the earth's equatorial bulge cause the axis of the earth to describe a cone over
a period of 26,000 years instead of always pointing to the same star in the sky.
This motion of the earth's axis causes a slight change each year in the time of
the spring and fall equinoxes, a fact which had been observed by Hipparchus
1 800 years earlier. Thus Newton explained the precession of the equinoxes.
Finally Newton solved a number of problems involving the motion of the
moon. The plane in which the moon moves is inclined somewhat to the plane
in which the earth moves. He was able to show that this phenomenon follows
from the interaction of the sun, earth, and moon under the law of gravitation.
As the moon travels around the earth, it cuts the plane of the earth's motion
around the sun. The points in which it intersects are called the nodes. The
nodes change in position, and this variation (regression of the nodes) also
proved to be a consequence of the gravitational effect of the sun and earth
on the moon. As the moon moves around the earth in an almost elliptical path,
the point farthest from the earth, called the apogee, shifts about 2 0 per revolu
tion. This effect. Newton showed, was due to the sun's attraction. Newton
and his immediate successors deduced so many and such weighty consequences
about the motions of the planets. the comets, the moon, and the sea, that their
accomplishments were viewed as "the explication of the System of the World."
Today we have almost daily evidenc" that Newton had found sound
physical principles which govern the operation of the universe. By applying
just those principles man can now create satellites which circle the earth. In
fact, Newton's suggestion that projectiles shot out horizontally and with large
velocities from the top of a mountain would circle the earth is, in essence, the
one used to launch satellites. Strictly speaking, scientists do not operate from
mountain tops because accessible peaks are not high enough to ensure that the
satellite will clear other mountains, and because the air resistance at such al
titudes is still considerable. Instead rockets project the satellite upward to a high
altitude where the air resistance is negligible; there a mechanism turns the satel
lite to a horizontal direction and another rocket gives it a horizontal velocity.
Then the satellite follows an elliptical path.
Newton went further in his speculations and conjectured that the planets
must have been shot from the sun at some angle and, upon reaching their
IMPLICATIONS OF THE THEORY OF GRAVITATION 359
present distances. must have retained enough "horizontal" velocity to start
moving in their elliptical paths around the sun. This conjecture is still the
accepted theory of the origin of our solar system.
EXERCISES
I. What reason would there be for calling Newton's law of gravitation a universal
law?
2. In what sense did Newton incorporate the Keplerian laws in his science of
motion?
3. What suppon did the heliocentric theory receive from Newton's work on
gravitation?
4. What support did Newton's principles derive from the heliocentric theory?
* 15-13 IMPLICATIONS OF THE THEORY OF GRAVITATION
The work on gravitation presented mankind with a new world order, a uni
verse controlled throughout by a few universal mathematical laws which in
turn were derived from a common set of mathematically expressible physical
principles. Here was a majestic scheme which embraced the fall of a stone,
the tides of the oceans, the moon, the planets, the comets which seemed to
sweep defiantly through the orderly system of planets, and the most distant
stars. This view of the universe came to a world seeking to secure a new
approach to truth and a body of sound truths which were to replace the al
ready discredited doctrines of medieval culture. Thus it was bound to give
rise to revolutionary systems of thought in almost all intellectual spheres. And
it did. But, for the moment, we wish to confine ourselves to the implications
and consequences of the theory of gravitation for mathematics proper.
Newton's work followed and considerably broadened the plan laid down
by Galileo, who proposed to find basic quantitative physical principles and
to deduce from them the description of physical phenomena. Galileo had
discovered and utilized such axioms as the first law of motion, the constant
acceleration of bodies moving near the surface of the earth, and the inde
pendence of the horizontal and vertical motions of projectiles. His results
were confined to terrestrial motions. Newton added to the axioms the second
law of motion and replaced the principle of constant acceleration of falling
bodies by the more general law of gravitation. He then found that the result
ing set of principles enabled him to deduce the description of all motions of
matter on earth and in the heavens. Thus the scientific method of Galileo and
Newton involves mathematics not only in the expression of axioms and the
laws which are deduced but also in the deductive process itself. Indeed, mathe
matics offered not merely the vehicle for scientific expression but the most
powerful tool for the real work of science, that is the acquisition of knowledge
360 THE APPLICATION OF FORMULAS TO GRAVITATION
about the physical world and the organization of that knowledge in coherent
systems. From the time of Newton, these roles of mathematics have been un
questionably accepted and utilized. Hence, as the success of Newtonian
mechanics spurred efforts in other physical domains, mathematics was con
fronted with new challenges and received new suggestions for the creation of
concepts and methods which in turn gave greater power to science. This
interaction of mathematics and science has grown immensely since its begin
ning in rhe seventeenth century and has become the outstanding feature of the
intellectual life of our own century.
The most surprising development of the theory of gravitation and one
which established a new and unanticipated role for mathematics took place
after Newton had deduced a number of conclusions about our solar system.
Galileo and Newton had set about finding quantitative laws that related matter,
space, time, forces, and other physical properties, but had wisely decided not
to look into causal relationships; that is, they had deliberately avoided such
questions as why bodies fall to earth or why planets move around the sun. In
other words, they had concentrated on description. Nevertheless, they did
utilize the force of gravitation, a concept which had been vaguely suggested
even before Galileo's time-for example, by Copernicus and Kepler. Since the
force of gravitation now assumed central importance, it was natural to ask,
What is the mechanism that enables the earth to attract objects and the sun to
attract planets? The heightened emphasis on this universal force could not
but push such questions to the fore. The properties ascribed to the force of
gravitation were indeed remarkable. It acted over distances of inches and
millions of miles. It acted instantaneously and through empty space. Nor
could the action of the force be suspended or blocked. Even when the moon
was between the earth and the sun, the sun continued to attract the earth.
Although he tried to provide some physical explanation for the action of
gravity, Newton did not succeed, and he concluded, "I have not been able to
deduce from phenomena the cause of the properties of gravity and I frame
no hypotheses." In spite of his ignorance of the workings of gravitation,
Newton insisted on adopting the laws of motion and gravitation. He says,
But to derive two or three general principles of motion from phenomena, and
afterwards to tell us how the properties and actions of all corporeal things
follow from those manifest principles, 'Would be a very great thing though the
causes of those principles were not yet discovered: and therefore 1 scruple not
to propose the principles of motion above mentioned, they being of very general
extent, and leave their causes to be found out.
Concerning his work in his Principles, he says,
But our purpose is only to trace out the quantity and properties of this force
[gravitation] from the phenomena, and to apply what we discover in S011le
simple cases as principles, by which, in a mathematical way, we may estimate
IMPLICATIONS OF THE THEORY OF GRAVITATION 361
the effects thereof in more involved cases; for it would be endless and impossible
to bring every particular to direct and immediate observation. We said, in a
mathematical way [note Newton's emphasis on the mathematics], to avoid all
questions about the nature or quality of this force, 'Which 'We 'Would not be
understood to determine by any hypothesis; . . .
Newton was indeed troubled that he could give no explanation. But all he
could do to justify the introduction of this force is summed up at the end of
his Principles,
And to us it is enough that gravity does really exist, and act according to the
la'Ws 'We have explained, and abundantly serves to account for all the motions of
the celestial bodies, and of our sea.
Contrary to popular belief, no one ever discovered gravitation, for the
physical reality of this force has never been demonstrated. However, the
mathematical deductit)ns from the quantitative law proved so effective that
the phenomenon has been accepted as an integral part of physical science.
What science has done, then, in effect is to sacrifice physical intelligibility
for the sake of mathematical description and mathematical prediction. This
basic concept of physical science is a complete mystery, and all we know
about it is a mathematical law describing the action of a force as though it
were real. We see therefore that the best knowledge we have of a funda
mental and universal phenomenon is a mathematical law and its consequences.
And it has become more and more true since Newton's days that our best
knowledge of the physical world is mathematical knowledge.
REVIEW EXERCISES
1. '\tYrite as a decimal:
1 2 10'
a) c) d) JO'
J03 J08 e) 106
2. Express each of the following numbers as a number between 1 and 10 multi
plied or divided by a power of 10:
a) 58,000 b) 58,790 c) 63.4 ' J03 d ) 46.75 e) 0.05 f) 0.0074
3. Express each of the following quantities as a number between 1 and 10 multiplied
or divided by a power of 10:
5 · J03 · I I · J O' 6 ' 10' 9' J04 . 1 2 ' JO'
a) b) c)
3 · 10' 3 ' 10" 5 ' 1 0 3 ' JO" 5 ' I O'
4 . J03 . 3 . JO' 10' 10 1 2 . 3 . 108
d) e) f)
5 . JO' . I I . 106 J · IO' · 1 2 · J03 5 · 10'9
362 THE APPLICATION OF FORMULAS TO GRAVITATION
4. The frequency at which a frequency-modulation ( FM ) station broadcasts is 91
million cycles per second. Write the frequency as a number between 1 and 10
multiplied by a power of 10.
5. The mass of the earth is 13.1 ' 1()24 lb. A gram of mass is 0.002205 lb. Find the
mass of the earth in grams.
6. The mass of the sun is 4.40 ' l()3o lb. Use the data of Exercise 5 to compute the
mass of the sun in grams.
In the foI1owing exercises you may use the fact that when M is the mass of the
earth, GM = 32 · (4000)2( 5280)2.
7. Calculate the acceleration which gravity imparts to an object
a) 2000 mi above the surface of the earth,
b ) 10,000 mi above the surface of the earth.
8. Suppose a man weighs 200 lb at the surface of the earth. Calculate his weight
when he is
a) 2000 mi above the surface of the earth,
b) 10,000 mi above the surface of the eanh.
9. What does a man who weighs 200 lb on the earth weigh on the moon if the
weight there is due only to the attraction of the moon. The acceleration which
the moon impans to objects near its surface is 5.3 ft/sec2•
Topics for Further Investigation
1. The astronomical work of Copernicus. The books by Armitage, Dreyer. Koyre,
Kuhn, Wolf. and any number of others listed in the Recommended Reading
would be fine source material.
2. The astronomical work of Kepler. The books by Caspar, Dreyer, Koyre. Kuhn
and Wolf listed in the Recommended Reading would be fine source material.
3. Show how the history of the heliocentric theory exemplifies the influence of
mathematics on western European culture. The books by Kline in the Recom-
.
mended Reading will provide material.
Recommended Reading
ARMITAGE, ANGUS: Sun, Stand Thou Still, Henry Schuman, New York, 1947. Also
in paperback under the tirIe The World of Copernicus.
ARMITAGE, ANGUS: Copernicus, W. W. Nonon and Co., New York, 1938.
BAUMGARDT, CAROLA: Johannes Kepler, Life and Letters, Victor Gollancz Ltd.,
London, 1952.
BELL, E. T.: Men of Mathematjcs, Chaps. 6, 9, 10, and 1 1, Simon and Schuster, New
York, 1937.
BONNER, FRANCIS T. and M ELBA PHILLIPS; Principles of Physical Science, Chaps. 1
and 4, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957.
IMPLICATIONS OF THE THEORY OF GRAVITATION 363
BURTT, E. A.: The Metaphysical Foundations of Modern Physical Science, rev. ed.,
Chap. 2 and pp. 202-262, Routledge and Kegan Paul Ltd., London, 1932.
BUITERFlELD, HERBERT: The Origins of Modern Science, Chaps. 2 and 8, The Mac
millan Co., New York, 1951.
CASPAR, MAX: Johannes Kepler, Abelard-Schuman, New York, 1960.
COHEN, I. BERNARD: The Birth of a New Physics, Chap. 7, Doubleday and Co.,
Anchor Books, New York, 1960.
DAMPIER-WHETHAM, WM. C. D.: A History of Science and Its Relations with
Philosophy and Religion, pp. 160-195, Cambridge University Press, London,
1929.
DE SANTILLANA, GIORGIO: The Crime of Galileo, University of Chicago Press,
Chicago, 1955.
DRAKE, STILLMAN: Discoveries and Opinions of Galileo, Doubleday & Co., Anchor
Books, New York, 1957.
DREYER, j. L. E.: A History of Astronomy From Thales to Kepler, 2nd ed., Dover
Publications, Inc., New York, 1953.
DREYER, J. L. E.: Tycho Brahe, A Picture of Scientific Ufe and Work in the Sixteenth
Century, Dover Publications, Inc., New York, 1963.
GADE, JOHN A.: The Life and Times of Tycho Brahe, Princeton University Press,
Princeton, 1947.
GALILEI, GALILEO : Dialogue on the Great World Systems, The University of
Chicago Press, Chicago, 1953. Other editions of this work, originally published
in 1632, �lso exist.
HALL, A. R.: The Scientific Revolution, Chap. 9, Longmans, Green and Co., Inc.,
New York, 1954.
HOLTON, GERALD and DUANE H. D. ROLLER: Foundations of Modern Physical Sci
ence, Chaps. 4, 5, 8 through 12, Addison-Wesley Publishing Co., Inc., Reading,
Mass., 1958.
JEANS, SIR JAMES: The Growth of Physical Science, 2nd ed., Chap. 6, Cambridge
University Press, London, 1951.
JONES, SIR HAROLD SPENCER: "john Couch Adams and the Discovery of Neptune,"
in JAMES R. NEW MA N: The World of Mathematics, Vol. II, pp. 820-839, Simon
and Schuster, Inc., New York, 1956.
KLINE, MORRIS: Mathematics: A Cultural Approach, Chapter 12, Addison-Wesley
Publishing Co., Inc., Reading, Mass., 1962.
KLINE, MORRIS: Mathematics in Western Culture, Chap. 9, Oxford University Press,
N.Y., 1953. Also in paperback.
KOYRE, ALEXANDRE: From the Closed World to the Infinite Universe, Chaps. 1
through 4, The johns Hopkins Press, Baltimore, 1957.
KUHN , THOMAS S.: The Copernican Revolution, Harvard University Press, Cam
bridge, 1957.
MASON, S. F.: A History of the Sciences, Chaps. 17 and 25, Routledge and Kegan
Paul Ltd., London, 1953.
MORE, LOUIS T.: Isaac Newton, Dover Publications, Inc., New York, 1962.
364 THE APPLICATION OF FORMULAS TO GRAVITATION
NEWMAN, JAMES R.: The World of Mathematics, Vol. I, pp. 254-285, Simon and
Schuster, Inc., New York, 1956.
SMITH , PRESERVED: A History of Modern Culture, Vol I, Chap. 2 and Vol. II, Chap.
2, Holt, Rinehart and Winston, Inc., New York, 1934.
SULLIVAN, JOHN WM. N.: Isaac Newton, The Macmillan Co., New York, 1938.
TAYLOR, LLOYD WM.: Physics, The Pioneer Science, Chaps. 9, 10, and 13, Dover
Publications, Inc., New York, 1959.
WIGHTMAN, WM. P. D.: The Growth of Scientific Ideas, Chaps. 8, 10, and 1 1, Yale
University Press, New Haven, 1951.
WOLF, ABRAHAM: A History of Science, Technology and Philosophy in the Six
teenth and Seventeenth Centuries, 2nd ed., Chaps. 2, 3, 6, and 7, George Allen
and Unwin Ltd., London, 1950. Also in paperback.
CHAPTER 1 6 *
THE DI FFERENTIAL CALCULUS
No nature except an extraordinary one could ever easily formulate a theory.
PLATO
16-1 INTRODUCTION
The mathematical ideas explored in the preceding chapters, arithmetic, algebra,
Euclidean geometry, trigonometry, coordinate geometry, and the various types
of functions, comprise a considerable amount of mathematics. Of course, the
development of each of these ideas is far more extensive than we have indicated
or than school courses usually cover. But the seventeenth century, which in
spired and initiated the modern scientific movement, provided the problems
and suggestions for new branches of mathematics which dwarf in extent, depth,
and power the mathematics we have examined thus far. The most significant
mathematical creation of that century and the one which proved to be most
fruitful for the modern development of mathematics and scienc |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.