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Full text of "A History of Mathematics"

A History 


Carl R Boyer 



1. Syracuse 




2. Crotona 



(Euclid, Heron, Ptolemy, Pappus 
Menelaus and others) 

3. Elea 



(Plato, Theaetetus) 

4. Rome 




5. Tarentum 




6. Cyrene 




7. Elis 




8. Athens 




9. Stagira 




10. Abdera 



(Theodorus, Eratosthenes) 

11. Byzantium 




12. Chalcedon 



(Parmenides, Zeno) 

13. Nicaea 




14. Cyzicus 




15. Pergamum 




16. Chios 




17. Samos 




18. Smyrna 




19. Miletus 



(Eudemus, Geminus) 

20. Cnidus 




21. Rhodes 



(Pythagoras, Conon, Aristarchus) 

22. Perga 




23. Chalcis 




24. Gerasa 




25. Alexandria 




26. Syene 



(Pythagoras, Archytas, Philolaus 


A History of Mathematics 

Carl B. Boyer 

Professor of Mathematics 
Brooklyn College 

A History 

of Mathematics 

JOHN WILEY& SONS, INC. New York London Sydney 

| ACCESSiqdk'o. 


Aj nr ■■■■" 

f 29 MAR 1979 "7 

' N 


Copyright © 1968 by John Wiley & Sons, Inc. 

All rights reserved. No part of this book may be 
reproduced by any means, nor transmitted, nor 
translated into a machine language without the 
written permission of the publisher. 

Library of Congress Catalog Card Number: 68-1 650i 
Printed in the United States of America 

To the Memory of My Parents 

Howard Franklin Boyer 


Rebecca Catherine (Eisenhart) Boyer 


Numerous histories of mathematics have appeared during this century, 
many of them in the English language. Some are very recent, such as J. F. 
Scott's A History of Mathematics 1 ; a new entry in the field therefore should 
have characteristics not already present in the available books. Actually, 
few of the histories at hand are textbooks, at least not in the American sense 
of the word, and Scott's History is not one of them. It appeared, therefore, 
that there was room for a new book — one that would meet more satis- 
factorily my own preferences and possibly those of others. 

The two-volume History of Mathematics by David Eugene Smith 2 was 
indeed written "for the purpose of supplying teachers and students with a 
usable textbook on the history of elementary mathematics," but it covers too 
wide an area on too low a mathematical level for most modern college 
courses, and it is lacking in problems of varied types. Florian Cajori's 
History of Mathematics 2, still is a very helpful reference work; but it is not 
adapted to classroom use, nor is E. T. Bell's admirable The Development of 
Mathematics. 4 The most successful and appropriate textbook today appears 
to be Howard Eves, An Introduction to the History of Mathematics, 5 which 
I have used with considerable satisfaction in at least a dozen classes since it 
first appeared in 1953. I have occasionally departed from the arrangement 
of topics in the book in striving toward a heightened sense of historical- 
mindedness and have supplemented the material by further reference to the 
contributions of the eighteenth and nineteenth centuries especially by the 
use of D. J. Struik, A Concise History of Mathematics. 6 

The reader of this book, whether a layman, a student, or a teacher of a 
course in the history of mathematics, will find that the level of mathematical 
background that is presupposed is approximately that of a college junior or 
senior, but the material can be perused profitably also by readers with either 
stronger or weaker mathematical preparation. Each chapter ends with a set 
of exercises that are graded roughly into three categories. Essay questions 
that are intended to indicate the reader's ability to organize and put into his 
own words the material discussed in the chapter are listed first. Then follow 
relatively easy exercises that require the proofs of some of the theorems 
mentioned in the chapter or their application to varied situations. Finally, 

1 London: Taylor and Francis, 1958. 

2 Boston: Ginn and Company, 1923-1925. 

3 New York: Macmillan, 1931, 2nd edition. 

4 New York: McGraw-Hill, 1945, 2nd edition. 

5 New York: Holt, Rinehart and Winston, 1964, revised edition. 

6 New York: Dover Publications, 1967, 3rd edition. 


there are a few starred exercises, which are either more difficult or require 
specialized methods that may not be familiar to all students or all readers. 
The exercises do not in any way form part of the general exposition and 
can be disregarded by the reader without loss of continuity. 

Here and there in the text are references to footnotes, generally biblio- 
graphical, and following each chapter there is a list of suggested readings. 
Included are some references to the vast periodical literature in the field, 
for it is not too early for students at this level to be introduced to the wealth 
of material available in good libraries. Smaller college libraries may not be 
able to provide all of these sources, but it is well for a student to be aware of 
the larger realms of scholarship beyond the confines of his own campus. 
There are references also to works in foreign languages, despite the fact that 
some students, hopefully not many, may be unable to read any of these. 
Besides providing important additional sources for those who have a reading 
knowledge of a foreign language, the inclusion of references in other lan- 
guages may help to break down the linguistic provincialism which, ostrich- 
like, takes refuge in the mistaken impression that everything worthwhile 
appeared in, or has been translated into, the English language. 

The present work differs from the most successful presently available 
textbook in a stricter adherence to the chronological arrangement and a 
stronger emphasis on historical elements. There is always the temptation in a 
class in history of mathematics to assume that the fundamental purpose of 
the course is to teach mathematics. A departure from mathematical standards 
is then a mortal sin, whereas an error in history is venial. I have striven to 
avoid such an attitude, and the purpose of the book is to present the history 
of mathematics with fidelity, not only to mathematical structure and exacti- 
tude, but also to historical perspective and detail. It would be folly, in a book 
of this scope, to expect that every date, as well as every decimal point, is 
correct. It is hoped, however, that such inadvertencies as may survive beyond 
the stage of page proof will not do violence to the sense of history, broadly 
understood, or to a sound view of mathematical concepts. It cannot be too 
strongly emphasized that this single volume in no way purports to present 
the history of mathematics in its entirety. Such an enterprise would call for 
the concerted effort of a team, similar to that which produced the fourth 
volume of Cantor's Vorlesungen uber Geschichte der Mathematik in 1908 
and brought the story down to 1799. In a work of modest scope the author 
must exercise judgment in the selection of the materials to be included, 
reluctantly restraining the temptation to cite the work of every productive 
mathematician ; it will be an exceptional reader who will not note here what 
he regards as unconscionable omissions. In particular, the last chapter 
attempts merely to point out a few of the salient characteristics of the 
twentieth century. In the field of the history of mathematics perhaps nothing 


is more to be desired than that there should appear a latter-day Felix Klein 
who would complete for our century the type of project Klein essayed for 
for the nineteenth century, but did not live to finish. 

A published work is to some extent like an iceberg, for what is visible 
constitutes only a small fraction of the whole. No book appears until the 
author has lavished time on it unstintingly and unless he has received en- 
couragement and support from others too numerous to be named individ- 
ually. Indebtedness in my case begins with the many eager students to whom 
I have taught the history of mathematics, primarily at Brooklyn College, but 
also at Yeshiva University, the University of Michigan, the University of 
California (Berkeley), and the University of Kansas. At the University of 
Michigan, chiefly through the encouragement of Professor Phillip S. Jones, 
and at Brooklyn College through the assistance of Dean Walter H. Mais 
and Professors Samuel Borofsky and James Singer, I have on occasion 
enjoyed a reduction in teaching load in order to work on the manuscript of 
this book. Friends and colleagues in the field of the history of mathematics, 
including Professor Dirk J. Struik of the Massachusetts Institute of Tech- 
nology, Professor Kenneth O. May at the University of Toronto, Professor 
Howard Eves of the University of Maine, and Professor Morris Kline at 
New York University, have made many helpful suggestions in the prepara- 
tion of the book, and these have been greatly appreciated. Materials in the 
books and articles of others have been expropriated freely, with little 
acknowledgment beyond a cold bibliographical reference, and I take this 
opportunity to express to these authors my warmest gratitude. Libraries 
and publishers have been very helpful in providing information and illustra- 
tions needed in the text; in particular it has been a pleasure to have worked 
with the staff of John Wiley and Sons. The typing of the final copy, as well 
as of much of the difficult preliminary manuscript, was done cheerfully and 
with painstaking care by Mrs. Hazel Stanley of Lawrence, Kansas. Finally, 
I must express deep gratitude to a very understanding wife, Dr. Marjorie N. 
Boyer, for her patience in tolerating disruptions occasioned by the develop- 
ment of yet another book within the family. 

Brooklyn, New York Carl B. Boyer 

January 1968 


Chapter I. Primitive Origins 1 

1 The concept of number. 2 Early number bases. 3 Number language and the 
origin of counting. 4 Origin of geometry. 

Chapter II. Egypt g 

1 Early records. 2 Hieroglyphic notation. 3 Ahmes papyrus. 4 Unit fractions. 5 
Arithmetic operations. 6 Algebraic problems. 7 Geometrical problems. 
8 A trigonometric ratio. 9 Moscow papyrus. 10 Mathematical weaknesses. 

Chapter III. Mesopotamia 26 

1 Cuneiform records. 2 Positional numeration. 3 Sexagesimal fractions. 4 Funda- 
mental operations. 5 Algebraic problems. 6 Quadratic equations. 7 Cubic 
equations. 8 Pythagorean triads. 9 Polygonal areas. 10 Geometry as applied 
arithmetic. 11 Mathematical weaknesses. 

Chapter IV. Ionia and the Pythagoreans 48 

1 Greek origins. 2 Thales of Miletus. 3 Pythagoras of Samos. 4 The Pythagorean 
pentagram. 5 Number mysticism. 6 Arithmetic and cosmology. 7 Figurate 
numbers. 8 Proportions. 9 Attic numeration. 10 Ionian numeration. 11 
Arithmetic and logistic. 

Chapter V. The Heroic Age 69 

1 Centers of activity. 2 Anaxagoras of Clazomenae. 3 Three famous problems. 

4 Quadrature of lunes. 5 Continued proportions. 6 Hippias of Ellis. 7 
Philolaus and Archytas of Tarentum. 8 Duplication of the cube. 9 In- 
commensurability. 10 The golden section. 11 Paradoxes ofZeno. 12 Deduc- 
tive reasoning. 13 Geometrical algebra. 14 Democritus of Abdera. 

Chapter VI. The Age of Plato and Aristotle 91 

1 The seven liberal arts. 2 Socrates. 3 Platonic solids. 4 Theodorus of Cyrene. 

5 Platonic arithmetic and geometry. 6 Origin of analysis. 7 Eudoxus of 
Cnidus. 8 Method of exhaustion. 9 Mathematical astronomy. 10 Menaech- 
mus. 11 Duplication of the cube. 12 Dinostratus and the squaring of the 
circle. 13 Autolycus of Pitane. 14 Aristotle. 15 End of the Hellenic period. 

Chapter VII. Euclid of Alexandria 111 

1 Author of the Elements. 2 Other works. 3 Purpose of the Elements. 4 Definitions 
and postulates. 5 Scope of Book I. 6 Geometrical algebra. 7 Books III and 
IV. 8 Theory of proportion. 9 Theory of numbers. 10 Prime and perfect 
numbers. 11 Incommensurability. 12 Solid geometry. 13 Apocrypha. 14 
Influence of the Elements. 


Chapter VIII. Archimedes of Syracuse 134 

1 The siege of Syracuse. 2 Law of the lever. 3 The hydrostatic principle. 4 The 
Sand- Reckoner. 5 Measurement of the circle. 6 Angle trisection. 7 Area of a 
parabolic segment. 8 Volume of a paraboloidal segment. 9 Segment of a 
sphere. 10 On the Sphere and Cylinder. 11 Book of Lemmas. 12 Semiregular 
solids and trigonometry. 13 The Method. 14 Volume of a sphere. 15 Recovery 
of the Method. 

Chapter IX. Apollonius of Perga 157 

1 Lost works. 2 Restorations of lost works. 3 The problem of Apollonius. 
4 Cycles and epicycles. 5 The Conies. 6 Names of the conic sections. 7 The 
double-napped cone. 8 Fundamental properties. 9 Conjugate diameters. 
10 Tangents and harmonic division. 11 The three-and-four-line locus. 

12 Intersecting conies. 13 Maxima and minima, tangents and normals. 
14 Similar conies. 15 Foci of conies. 16 Use of coordinates. 

Chapter X. Greek Trigonometry and Mensuration 176 

1 Early trigonometry. 2 Aristarchus of Samos. 3 Eratosthenes of Cyrene. 4 
Hipparchus of Nicaea. 5 Menelaus of Alexandria. 6 Ptolemy's Almagest. 

7 The 360 degree circle. 8 Construction of tables. 9 Ptolemaic astronomy. 
10 Other works by Ptolemy. 11 Optics and astrology. 12 Heron of Alex- 
andria. 13 Principle of least distance. 14 Decline of Greek mathematics. 

Chapter XI. Revival and Decline of Greek Mathematics 196 

1 Applied mathematics. 2 Diophantus of Alexandria. 3 Nicomachus of Gerasa. 

4 The Arithmetica of Diophantus. 5 Diophantine problems. 6 The place of 
Diophantus in algebra. 7 Pappus of Alexandria. 8The Collection. 9 Theorems 
of Pappus. 10 The Pappus problem. 11 The Treasury of Analysis. 12 The 
Pappus-Guldin theorems. 13 Proclus of Alexandria. 14 Boethius. 15 End of 
the Alexandrian period. 16 The Greek Anthology. 17 Byzantine mathemati- 
cians of the sixth century. 

Chapter XII. China and India 217 

1 The oldest documents. 2 The Nine Chapters. 3 Magic squares. 4 Rod numerals. 

5 The abacus and decimal fractions. 6 Values of pi. 7 Algebra and Horner's 
method. 8 Thirteenth-century mathematicians. 9 The arithmetic triangle. 

10 Early mathematics in India. 11 The Sulvasutras. 12 The Siddhantas. 

13 Aryabhata. 14 Hindu numerals. 15 The symbol for zero. 16 Hindu 
trigonometry. 17 Hindu multiplication. 18 Long division. 19 Brahmagupta. 
20 Brahmagupta's formula. 21 Indeterminate equations. 22 Bhaskara. 
23 The Lilavati. 24 Ramanujan. 

Chapter XIII. The Arabic Hegemony 249 

1 Arabic conquests. 2 The House of Wisdom. 3 Al-jabr. 4 Quadratic equations. 
5 The father of algebra. 6 Geometric foundation. 7 Algebraic problems. 

8 A problem from Heron. 9 Abd al-Hamid ibn-Turk. 10 Thabit ibn-Qurra. 

11 Arabic numerals. 12 Arabic trigonometry. 13 AbuT-Wefa and al- 
Karkhi. 14 Al-Biruni and Alhazen. 15 Omar Khayyam. 16 The parallel 
postulate. 17 Nasir Eddin. 18 Al-Kashi. 


Chapter XIV. Europe in the Middle Ages 272 

1 From Asia to Europe. 2 Byzantine mathematics. 3 The Dark Ages. 4 Alcuin 
and Gerbert. 5 The century of translation. 6 The spread of Hindu-Arabic 
numerals. 7 The Liber abaci. 8 The Fibonacci sequence. 9 A solution of a 
cubic equation. 10 Theory of numbers and geometry. 11 Jordanus Nemor- 
arius. 12 Campanus of Novara. 13 Learning in the thirteenth century. 
14 Medieval kinematics. 15 Thomas Bradwardine. 16 Nicole Oresme. 17 
The latitude of forms. 18 Infinite series. 19 Decline of medieval learning. 

Chapter XV. The Renaissance 297 

1 Humanism. 2 Nicholas of Cusa. 3 Regiomontanus. 4 Application of algebra 
to geometry. 5 A transitional figure. 6 Nicolas Chuquet's Triparty. 7 Luca 
Pacioli's Summa. 8 Leonardo da Vinci. 9 Germanic algebras. 10 Cardan's 
Ars magna. 11 Solution of the cubic equation. 12 Ferrari's solution of the 
quartic equation. 13 Irreducible cubics and complex numbers. 14 Robert 
Recorde. 15 Nicholas Copernicus. 16 Georg Joachim Rheticus. 17 Pierre de 
la Ramee. 18 Bombelli's Algebra. 19 Johannes Werner. 20 Theory of 
perspective. 21 Cartography. 

Chapter XVI. Prelude to Modern Mathematics 333 

1 Francois Viete. 2 Concept of a parameter. 3 The analytic art. 4 Relations 
between roots and coefficients. 5 Thomas Harriot and William Oughtred. 
6 Horner's method again. 7 Trigonometry and prosthaphaeresis. 8 Trigono- 
metric solution of equations. 9 John Napier. 10 Invention of logarithms. 
11 Henry Briggs. 12 Jobst Biirgi. 13 Applied mathematics and decimal 
fractions. 14 Algebraic notations. 15 Galileo Galilei. 16 Values of pi. 
17 Reconstruction of Apollonius' On Tangencies. 18 Infinitesimal analysis. 
19 Johannes Kepler. 20 Galileo's Two New Sciences. 21 Galileo and the 
infinite. 22 Bonaver.tura Cavalieri. 23 The spiral and the parabola. 

Chapter XVII. The Time of Fermat and Descartes 367 

* 1 Leading mathematicians of the time. 2 The Discours de la methode. 3 Invention 
of analytic geometry. 4 Arithmetization of geometry. 5 Geometrical algebra. 
6 Classification of curves. 7 Rectification of curves. 8 Identification of conies. 
9 Normals and tangents. 10 Descartes' geometrical concepts. 11 Fermat's 
loci. 12 Higher-dimensional analytic geometry. 13 Fermat's differentiations. 
14 Fermat's integrations. 15 Gregory of St. Vincent. 16 Theory of numbers. 
17 Theorems of Fermat. 18 Gilles Persone de Roberval. 19 Evangelista 
Torricelli. 20 New curves. 21 Girard Desargues. 22 Projective geometry. 
23 Blaise Pascal. 24 Probability. 25 The cycloid. 

Chapter XVIII. A Transitional Period 404 

1 Philippe de Lahire. 2 Georg Mohr. 3 Pietro Mengoli. 4 Frans van Schooten. 
5 Jan de Witt. 6 Johann Hudde. 7 Rene Francois de Sluse. 8 The pendulum 
clock. 9 Involutes and evolutes. 10 John Wallis. 11 On Conic Sections. 
12 Arithmetica infinitorum. 13 Christopher Wren. 14 Wallis' formulas. 
15 James Gregory. 16 Gregory's series. 17 Nicolaus Mercator and William 
Brouncker. 18 Barrow's method of tangents. 


Chapter XIX. Newton and Leibniz 429 

1 Newton's early work. 2 The binomial theorem. 3 Infinite series. 4 The Method 
of Fluxions. 5 The Principia. 6 Leibniz and the harmonic triangle. 7 The 
differential triangle and infinite series. 8 The differential calculus. 9 Deter- 
minants, notations, and imaginary numbers. 10 The algebra of logic. 1 1 The 
inverse square law. 12 Theorems on conies. 13 Optics and curves. 14 Polar 
and other coordinates. 15 Newton's method and Newton's parallelogram. 
16 The Arithmetica universalis. 17 Later years. 

Chapter XX. The Bernoulli Era 455 

1 The Bernoulli family. 2 The logarithmic spiral. 3 Probability and infinite series. 
4 L'Hospital's rule. 5 Exponential calculus. 6 Logarithms of negative 
numbers. 7 Petersburg paradox. 8 Abraham de Moivre. 9 De Moivre's 
theorem. 10 Roger Cotes. 11 James Stirling. 12 Colin Maclaurin. 13 Taylor's 
series. 14 The Analyst controversy. 15 Cramer's rule. 16 Tschirnhaus 
transformations. 17 Solid analytic geometry. 18 Michel Rolle and Pierre 
Varignon. 19 Mathematics in Italy. 20 The parallel postulate. 21 Divergent 

Chapter XXI. The Age of Euler 481 

1 Life of Euler. 2 Logarithms of negative numbers. 3 Foundation of analysis. 
4 Infinite series. 5 Convergent and divergent series. 6 Life of d'Alembert. 
7 The Euler identities. 8 D'Alembert and limits. 9 Differential equations. 
10 The Clairauts. 11 The Riccatis. 12 Probability. 13 Theory of numbers. 
14 Textbooks. 15 Synthetic geometry. 16 Solid analytic geometry. 17 
Lambert and the parallel postulate. 18 Bezout and elimination. 

Chapter XXII. Mathematicians of the French Revolution 510 

1 The age of revolutions. 2 Leading mathematicians. 3 Publications before 1789. 
4 Lagrange and determinants. 5 Committee on Weights and Measures. 

6 Condorcet on education. 7 Monge as administrator and teacher. 8 
Descriptive geometry and analytic geometry. 9 Textbooks. 10 Lacroix on 
analytic geometry. 11 The Organizer of Victory. 12 Metaphysics of the 
calculus and geometry. 13 Geometrie de position. 14 Transversals. 15 
Legendre's Geometry. 16 Elliptic integrals. 17 Theory of numbers. 18 
Theory of functions. 19 Calculus of variations. 20 Lagrange multipliers. 
21 Laplace and probability. 22 Celestial mechanics and operators. 23 
Political changes. 

Chapter XXIII. The Time of Gauss and Cauchy 544 

1 Early discoveries by Gauss. 2 Graphical representation of complex numbers. 
3 The fundamental theorem of algebra. 4 The algebra of congruences. 5 
Reciprocity and frequency of primes. 6 Constructible regular polygons. 

7 Astronomy and least squares. 8 Elliptic functions. 9 Abel's life and work. 
10 Theory of determinants. 11 Jacobians. 12 Mathematical journals. 13 
Complex variables. 14 Foundations of the calculus. 15 Bernhard Bolzano. 
16 Tests for convergence. 17 Geometry. 18 Applied mathematics. 


Chapter XXIV. The Heroic Age in Geometry 572 

1 Theorems of Brianchon and Feuerbach. 2 Inversive geometry. 3 Poncelet's 
projective geometry. 4 Pliicker's abridged notation. 5 Homogeneous 
coordinates. 6 Line coordinates and duality. 7 Revival of British mathe- 
matics. 8 Cayley's ^-dimensional geometry. 9 Geometry in Germany. 
10 Lobachevsky and Ostrogradsky. 11 Non-Euclidean geometry. 12 The 
Bolyais. 13 Riemannian geometry. 14 Spaces of higher dimension. 15 
Klein's Erlanger Programm. 16 Klein's hyperbolic model. 

Chapter XXV. The Arithmetization of Analysis 598 

1 Fourier series. 2 Analytic number theory. 3 Transcendental numbers. 4 Un- 
easiness in analysis. 5 The Bolzano-Weierstrass theorem. 6 Definition of 
real number. 7 Weierstrassian analysis. 8 The Dedekind "cut". 9 The limit 
concept. 10 Gudermann's influence. 11 Cantor's early life. 12 The "power" 
of infinite sets. 13 Properties of infinite sets. 14 Transfinite arithmetic. 
15 Kronecker's criticism of Cantor's work. 

Chapter XXVI. The Rise of Abstract Algebra 620 

1 The Golden Age in mathematics. 2 Mathematics at Cambridge. 3 Peacock, the 
"Euclid of algebra." 4 Hamilton's quaternions. 5 Grassmann and Gibbs. 
6 Cayley's matrices. 7 Sylvester's algebra. 8 Invariants of quadratic forms. 
9 Boole's analysis of logic. 10 Boolean algebra. 11 De Morgan and the 
Peirces. 12 The tragic life of Galois. 13 Galois theory. 14 Field theory. 15 
Frege's definition of cardinal number. 16 Peano's axioms. 

Chapter XXVII. Aspects of the Twentieth Century 649 

1 The nature of mathematics. 2 Poincare's theory of functions. 3 Applied 
mathematics and topology. 4 Hilbert's problems. 5 Godel's theorem. 6 
Transcendental numbers. 7 Foundations of geometry. 8 Abstract spaces. 
9 The foundations of mathematics. 10 Intuitionism, formalism, and 
logicism. 11 Measure and integration. 12 Point set topology. 13 Increasing 
abstraction in algebra. 14 Probability. 15 High-speed computers. 16 
Mathematical structure. 17 Bourbaki and the "New Mathematics." 

General Bibliography 679 

Appendix: Chronological Table 683 

Index 697 

A History of Mathematics 


Primitive Origins 

Did you bring me a man who cannot number his 

From the Book of the Dead 

Mathematicians of the twentieth century carry on a highly sophisticated 
intellectual activity which is not easily defined ; but much of the subject that 
today is known as mathematics is an outgrowth of thought that originally 
centered in the concepts of number, magnitude, and form. Old-fashioned 
definitions of mathematics as a "science of number and magnitude" are no 
longer valid, but they do suggest the origins of the branches of mathematics. 
Primitive notions related to the concepts of number, magnitude, and form 
can be traced back to the earliest days of the human race, and adumbrations 
of mathematical notions can be found in forms of life that may have ante- 
dated mankind by many millions of years. Darwin in Descent of Man (1871) 
noted that certain of the higher animals possess such abilities as memory 
and imagination, and today it is even clearer that the abilities to distinguish 
number, size, order, and form — rudiments of a mathematical sense — are not 
exclusively the property of mankind. Experiments with crows, for example, 
have shown that at least certain birds can distinguish between sets containing 
up to four elements. 1 An awareness of differences in patterns found in their 
environment is clearly present in many lower forms of life, and this is akin 
to the mathematician's concern for form and relationship. 

At one time mathematics was thought to be directly concerned with the 
world of our sense experience, and it was only in the nineteenth century that 
pure mathematics freed itself from limitations suggested by observations of 
nature. It is clear that originally mathematics arose as a part of the everyday 
life of man, and if there is validity in the biological principle of the "survival 
of the fittest," the persistence of the human race probably is not unrelated 
to the development in man of mathematical concepts. At first the primitive 
notions of number, magnitude, and form may have been related to contrasts 

1 See Levi Conant, The Number Concept. Its Origin and Development (1923). Cf. H. Kalmus, 
"Animals as Mathematicians," Nature, 202 (1964), 1156-1160. 


India Iran Hoopta* Syria Egypt Asia minor Greece Italy Spain 



Chronological scheme representing the extent of some ancient and medieval civilizations. 
(Reproduced, with permission, from O. Neugebauer, The Exact Sciences in Antiquity.) 

rather than likenesses — the difference between one wolf and many, the 
inequality in size of a minnow and a whale, the unlikeness of the roundness of 
the moon and the straightness of a pine tree. Gradually there must have 
arisen, out of the welter of chaotic experiences, the realization that there are 


samenesses ; and from this awareness of similarities in number and form both 

science and mathematics were born. The differences themselves seem to point 

to likenesses, for the contrast between one wolf and many, between one 

sheep and a herd, between one tree and a forest, suggests that one wolf, one 

sheep, and one tree have something in common— their uniqueness. In the 

same way it would be noticed that certain other groups, such as pairs, can 

be put into one-to-one correspondence. The hands can be matched against 

the feet, the eyes, the ears, or the nostrils. This recognition of an abstract 

property that certain groups hold in common, and which we call number, 

represents a long step toward modern mathematics. It is unlikely to have 

been the discovery of any one individual or of any single tribe; it was more 

probably a gradual awareness which may have developed as early in man's 

cultural development as his use of fire, possibly some 300,000 years ago That 

the development of the number concept was a long and gradual process is 

suggested by the fact that some languages, including Greek, have preserved 

in their grammar a tripartite distinction between one and two and more than 

two, whereas most languages today make only the dual distinction in 

"number" between singular and plural. Evidently our very early ancestors 

at first counted only to two, any set beyond this level being stigmatized as 

"many." Even today many primitive peoples still count objects by arranging 

them into bundles of two each. 

The awareness of number ultimately became sufficiently extended and 
vivid so that a need was felt to express the property in some way, presumably 
at first in sign language only. The fingers on a hand can be readily used to 
indicate a set of two or three or four or five objects, the number one generally 
not being recognized at first as a true "number." By the use of the fingers on 
both hands, collections containing up to ten elements could be represented; 
by combining fingers and toes, one could mount as high as twenty. When 
the human digits were inadequate, heaps of stones could be used to represent 
a correspondence with the elements of another set. Where primitive man 
used such a scheme of representation, he often piled the stones in groups of 
five, for he had become familiar with quintuples through observation of the 
human hand and foot. As Aristotle had noted long ago, the widespread use 
today of the decimal system is but the result of the anatomical accident that 
most of us are born with ten fingers and ten toes. From the mathematical 
point of view it is somewhat inconvenient that Cro-Magnon man and his 
descendants did not have either four or six fingers on a hand. 

Although historically finger counting, or the practice of counting by fives 
and tens, seems to have come later than countercasting by twos and threes, 
the quinary and decimal systems almost invariably displaced the binary and 
ternary schemes. A study of several hundred tribes among the American 


Indians, for example, showed that almost one third used a decimal base and 
about another third had adopted a quinary or a quinary-decimal system ; 
fewer than a third had a binary scheme, and those using a ternary system 
constituted less than 1 per cent of the group. The vigesimal system, with 
twenty as a base, occurred in about 10 per cent of the tribes. 2 

Groups of stones are too ephemeral for preservation of information; 
hence prehistoric man sometimes made a number record by cutting notches 
in a stick or a piece of bone. Few of these records remain today, but in Czecho- 
slovakia a bone from a young wolf was found which is deeply incised with 
fifty-five notches. These are arranged in two series, with twenty-five in the 
first and thirty in the second ; within each series the notches are arranged in 
groups of five. Such archaeological discoveries provide evidence that the 
idea of number is far older than such technological advances as the use of 
metals or of wheeled vehicles. It antedates civilization and writing, in the 
usual sense of the word, for artifacts with numerical significance, such as the 
bone described above, have survived from a period of some 30,000 years ago. 
Additional evidence concerning man's early ideas on number can be found 
in our language today. It appears that our words "eleven" and "twelve" 
originally meant "one over" and "two over," indicating the early dominance 
of the decimal concept. However, it has been suggested that perhaps the 
Indo-Germanic word for eight was derived from a dual form for four, and 
that the Latin novem for nine may be related to novus (new) in the sense that 
it was the beginning of a new sequence. Possibly such words can be inter- 
preted as suggesting the persistence for some time of a quaternary or an 
octonary scale, just as the French quatre-vingt of today appears to be a 
remnant of a vigesimal system. 

Man differs from other animals most strikingly in his language, the 
development of which was essential to the rise of abstract mathematical 
thinking ; yet words expressing numerical ideas were slow in arising. Number 
signs probably preceded number words, for it is easier to cut notches in a 
stick than it is to establish a well-modulated phrase to identify a number. 
Had the problem of language not been so difficult, rivals to the decimal 
system might have made greater headway. The base five, for example, was 
one of the earliest to leave behind some tangible written evidence ; but by the 
time that language became formalized, ten had gained the upper hand. The 
modern languages of today are built almost without exception around the 
base ten, so that the number thirteen, for example, is not described as three 
and five and five, but as three and ten. The tardiness in the development of 

2 W. C. Eels, "Number Systems of North American Indians," American Mathematical 
Monthly, 20 (1913), 293. Cf. also D. J. Struik, "Stone Age Mathematics," Scientific American, 179 
(December 1948), 44-49. 


language to cover abstractions such as number is seen also in the fact that 
primitive numerical verbal expressions invariably refer to specific concrete 
collections — such as "two fishes" or "two clubs" — and later some such 
phrase would be adopted conventionally to indicate all sets of two objects. 
The tendency for language to develop from the concrete to the abstract is 
seen in many of our present-day measures of length. The height of a horse is 
measured in "hands," and the words "foot" and "ell" (or elbow) have 
similarly been derived from parts of the body. 

The thousands of years required for man to separate out the abstract 
concepts from repeated concrete situations testify to the difficulties that 
must have been, experienced in laying even a very primitive basis for mathe- 
matics. Moreover, there are a great many unanswered questions relating to 
the origins of mathematics. It usually is assumed that the subject arose in 
answer to man's practical needs, but anthropological studies suggest the 
possibility of an alternative origin. It has been suggested 3 that the art of 
counting arose in connection with primitive religious ritual and that the 
ordinal aspect preceded the quantitative concept. In ceremonial rites depict- 
ing creation myths it was necessary to call the participants onto the scene in 
a specific order, and perhaps counting was invented to take care of this 
problem. If theories of the ritual origin of counting are correct, the concept 
of the ordinal number may have preceded that of the cardinal number. 
Moreover, such an origin would tend to point to the possibility that counting 
stemmed from a unique origin, spreading subsequently to other portions of 
the earth. This view, although far from established, would be in harmony 
with the ritual division of the integers into odd and even, the former being 
regarded as male, the latter as female. Such distinctions were known to 
civilizations in all corners of the earth, and myths regarding the male and 
female numbers have been remarkably persistent. 

The concept of whole number is one of the oldest in mathematics, and its 
origin is shrouded in the mists of prehistoric antiquity. The notion of a 
rational fraction, however, developed relatively late and was not in general 
closely related to man's systems for the integers. Among primitive tribes 
there seems to have been virtually no need for fractions. For quantitative 
needs the practical man can choose units that are sufficiently small to obviate 
the necessity of using fractions. Hence there was no orderly advance from 
binary to quinary to decimal fractions, and decimals were essentially the 
product of the modern age in mathematics, rather than of the ancient period. 

Statements about the origins of mathematics, whether of arithmetic or 4 
geometry, are of necessity hazardous, for the beginnings of the subject are 

3 See A. Seidenberg, "The Ritual Origin of Counting," Archive for History of Exact Sciences, 2 
(1962), 1-40. 


older than the art of writing. It is only during the last half-dozen millennia, 
in a career that may have spanned thousands of millennia, that man has been 
able to put his records and thoughts in written form. For data about the 
prehistoric age we must depend on interpretations based on the few surviving 
artifacts, on evidence provided by current anthropology, and on a conjectural 
backward extrapolation from surviving documents. Herodotus and Aristotle 
were unwilling to hazard placing origins earlier than the Egyptian civiliza- 
tion, but it is clear that the geometry they had in mind had roots of greater 
antiquity. Herodotus held that geometry had originated in Egypt, for he 
believed that the subject had arisen there from the practical need for re- 
surveying after the annual flooding of the river valley. Aristotle argued that it 
was the existence of a priestly leisure class in Egypt that had prompted the 
pursuit of geometry. We can look upon the views of Herodotus and Aristotle 
as representing two opposing theories of the beginnings of mathematics, one 
holding to an origin in practical necessity, the other to an origin in priestly 
leisure and ritual. The fact that the Egyptian geometers sometimes were 
referred to as "rope-stretchers" (or surveyors) can be used in support of 
either theory, for the ropes undoubtedly were used both in laying out temples 
and in realigning the obliterated boundaries. We cannot confidently contra- 
dict either Herodotus or Aristotle on the motive leading to mathematics, 
but it is clear that both men underestimated the age of the subject. Neolithic 
man may have had little leisure and little need for surveying, yet his drawings 
and designs suggest a concern for spatial relationships that paved the way 
for geometry. Pottery, weaving, and basketry show instances of congruence 
and symmetry, which are in essence parts of elementary geometry. Moreover, 
simple sequences in design, such as that in Fig. 1.1, suggest a sort of applied 

FIG. 1.1 

group theory, as well as propositions in geometry and arithmetic. The design 
makes it immediately obvious that the areas of triangles are to each other 
as squares on a side, or, through counting, that the sums of consecutive odd 
numbers, beginning from unity, are perfect squares. For the prehistoric 
period there are no documents, hence it is impossible to trace the evolution 


of mathematics from a specific design to a familiar theorem. But ideas are 
like hardy spores, and sometimes the presumed origin of a concept may be 
only the reappearance of a much more ancient idea that had lain dormant. 
The concern of prehistoric man for spatial designs and relationships may 
have stemmed from his aesthetic feeling and the enjoyment of beauty of 
form, motives that often actuate the mathematician of today. We would like 
to think that at least some of the early geometers pursued their work for the 
sheer joy of doing mathematics, rather than as a practical aid in mensuration ; 
but there are other alternatives. One of these is that geometry, like counting, 
had an origin in primitive ritualistic practice. The earliest geometrical results 
found in India constituted what were called the Sulvasutras, or "rules of the 
cord." These were simple relationships that apparently were applied in the 
construction of altars and temples. It is commonly thought that the geo- 
metrical motivation of the "rope-stretchers" in Egypt was more practical 
than that of their counterparts in India ; but it has been suggested 4 that both 
Indian and Egyptian geometry may derive from a common source — a proto- 
geometry that is related to primitive rites in somewhat the same way in which 
science developed from mythology and philosophy from theology. We must 
bear in mind that the theory of the origin of geometry in a secularization of 
ritualistic practice is by no means established. The development of geometry 
may just as well have been stimulated by the practical needs of construction 
and surveying or by an aesthetic feeling for design and order. We can make 
conjectures about what led men of the Stone Age to count, to measure, and 
to draw. That the beginnings of mathematics are older than the oldest 
civilizations is clear. To go further and categorically identify a specific origin 
in space or time, however, is to mistake conjecture for history. It is best to 
suspend judgment on this matter and to move on to the safer ground of the 
history of mathematics as found in the written documents that have come 
down to us. 


Conant, Levi, The Number Concept. Its Origin and Development (New York : Macmillan, 

Eels, W. G, "Number Systems of North American Indians," American Mathematical 

Monthly, 20 (1913), 293. 
Kalmus, H., "Animals as Mathematicians," Nature, 202 (1964), 1156-1160. 
Menninger, Karl, Zahlwort und Ziffer: Eine Kulturgeschichte der Zahlen, 2nd ed. 

(Gottingen : Vandenhoeck & Ruprecht, 1957-1958, 2 vols.). 

A. Seidenberg, "The Ritual Origin of Geometry," Archive for History of Exact Sciences, 1 
(1962), 488-527. 


Seidenberg, A., "The Ritual Origin of Geometry," Archive for History of Exact Sciences, 

1 (1962), 488-527. 

Seidenberg, A., "The Ritual Origin of Counting," Archive for History of Exact Sciences, 

2 (1962), 1-40. 

Smeltzer, Donald, Man and Number (New York : Emerson Books, 1958). 

Smith, D. E., History of Mathematics (Boston : Ginn, 1923-1925, 2 vols. ; paperback ed., 

New York : Dover, 1958). 
Smith, D. E., and Jekuthiel Ginsburg, Numbers and Numerals (Washington, D.C.: 

National Council of Teachers of Mathematics, 1958). 
Struik, D. J., "Stone Age Mathematics," Scientific American, 179 (December 1948), 



1. Describe the type of evidence on which an account of prehistoric mathematics is based, 
citing some specific instances. 

2. What evidence, if any, is there that mathematics began with the advent of man? Do you 
think that mathematics antedates man? 

3. List evidences from language for the use at some time of bases other than ten. 

4. What are the advantages and disadvantages of the bases two, three, four, five, ten, twenty, 
and sixty? Do you think that these influenced early man in his choice of a base? 

5. If you had to choose a number base, which would it be? Why? 

6. Which do you think came first, number names or number symbols? Why? 

7. Why are there few traces of scales from six to nine? 

8. What do you think were the first plane and solid geometric figures to be consciously and 
systematically studied? Why? 

9. Which do you think was more influential in the rise of early geometry, an interest in 
astronomy or a need for surveying? Explain. 

10. Which of the following time divisions was prehistoric man likely to notice : the year, the 
month, the week, the day, the hour? Explain. 



Sesostris . . . made a division of the soil of Egypt among 
the inhabitants . . . If the river carried away any portion 
of a man's lot, ... the king sent persons to examine, and 
determine by measurement the exact extent of the 
loss . . . From this practice, I think, geometry first came 
to be known in Egypt, whence it passed into Greece. 


It is customary to divide the past of mankind into eras and periods, with 
particular reference to cultural levels and characteristics. Such divisions are 
helpful, although we should always bear in mind that they are only a frame- 
work arbitrarily superimposed for our convenience and that the separations 
in time they suggest are not unbridged gulfs. The Stone Age, a long period 
preceding the use of metals, did not come to an abrupt end. In fact, the type 
of culture that it represented terminated much later in Europe than in certain 
parts of Asia and Africa. The rise of civilizations characterized by the use of 
metals took place at first in river valleys, such as those in Egypt, Mesopotamia, 
India, and China ; hence we shall refer to the earlier portion of the historical 
period as the "potamic stage." Chronological records of the civilizations in 
the valleys of the Indus and Yangtze rivers are quite unreliable, but fairly 
dependable information is available about the peoples living along the Nile 
and in the "fertile crescent" of the Tigris and Euphrates rivers. Before the 
end of the fourth millennium B.C. a primitive form of writing was in use in 
both the Mesopotamian and Nile valleys. There the early pictographic 
records, through a steady conventionalizing process, evolved into a linear 
order of simpler symbols. In Mesopotamia, where clay was abundant, 
wedge-shaped marks were impressed with a stylus upon soft tablets which 
then were baked hard in ovens or by the heat of the sun. This type of writing 
is known as cuneiform (from the Latin word cuneus or wedge) because of the 
shape of the individual impressions. The meaning to be transmitted in cunei- 
form was determined by the patterns or arrangements of the wedge-shaped 
impressions. Cuneiform documents had a high degree of permanence ; hence 
many thousands of such tablets have survived from antiquity, many of them 



I I innn 
I I I- R 


! ! ML7&*=>2£^7 < 


-«&<=>-<s&-| I ! ID 

i^zpk^) © 1 1 1 in< 

I I innn m 
i i i rW 

i inn-esa 
1 1 nn^r7 5 

Reproduction (fop) of a portion of the Moscow Papyrus showing the problem on the 
volume of a frustum of a square pyramid, together with hieroglyphic transcription (below). 

dating back some 4000 years. Of course, only a small fraction of these touch 
on themes related to mathematics. Moreover, until about a century ago the 
message of the cuneiform tablets remained muted because the script had not 
been deciphered. In the 1870s significant progress in the reading of cuneiform 
writing was made when it was discovered that the Behistun Cliff carried a 
trilingual account of the victory of Darius over Cambyses, the inscriptions 
being in Persian, Elamitic, and Babylonian. Knowledge of Persian consequently 
supplied a key to the reading of Assyrian, a language closely related to the 
older Babylonian. Even after this important discovery, decipherment and 
analysis of tablets with mathematical content proceeded slowly, and it was 
not until the second quarter of the twentieth century that awareness of 
Mesopotamian mathematical contributions became appreciable, largely 
through the pioneer work of Fr. Thureau-Dangin in France and Otto 
Neugebauer in Germany and America. : 

1 See, for example, O. Neugebauer, Vorgriechische Mathematik (Berlin: Springer, 1934). For a 
more general account in English see his The Exact Sciences in Antiquity (1957). 


Egyptian written records meanwhile had fared better than Babylonian in 2 
one respect. The trilingual Rosetta Stone, playing a role similar to that of the 
Behistun Cliff, had been discovered in 1799 by the Napoleonic expedition. 
This large tablet, found at Rosetta, an ancient harbor near Alexandria, con- 
tained a message in three scripts: Greek, Demotic, and Hieroglyphic. 
Knowing Greek, Champollion in France and Thomas Young in England 
made rapid progress in deciphering the Egyptian hieroglyphics (that is, 
"sacred carvings"). Inscriptions on tombs and monuments in Egypt now 
could be read, although such ceremonial documents are not the best source 
of information concerning mathematical ideas. Egyptian hieroglyphic 
numeration was easily disclosed. The system, at least as old as the pyramids, 
dating some 5000 years ago, was based, as we might expect, on the ten-scale. 
By the use of a simple iterative scheme and of distinctive symbols for each 
of the first half-dozen powers often, numbers over a million were carved on 
stone, wood, and other materials. A single vertical stroke represented a unit, 
an inverted wicket or heel bone was used for 10, a snare somewhat resembling 
a capital letter C stood for 100, a lotus flower for 1000, a bent finger for 
10,000, a burbot fish resembling a polywog for 100,000, and a kneeling figure 
(perhaps God of the Unending) for 1,000,000. Through repetition of these 
symbols the number 12,345, for example, would appear as 


Sometimes the smaller digits were placed on the left, and sometimes the digits 
were arranged vertically. The symbols themselves occasionally were reversed 
in orientation, so that the snare might be convex toward either the right or 
the left. 

Egyptian inscriptions indicate familiarity with large numbers at an early 
date. A museum at Oxford has a royal mace more than 5000 years old on 
which a record of 120,000 prisoners and 1,422,000 captive goats appears. 2 
These figures may have been exaggerated, but from other considerations it 
is nevertheless clear that the Egyptians were commendably accurate in 
counting and measuring. The pyramids exhibit such a high degree of precision 
in construction and orientation that ill-founded legends have grown up 
around them. The suggestion, for example, that the ratio of the perimeter of 
the base of the Great Pyramid (of Khufu or Cheops) to the height was 
consciously set at 2n is clearly inconsistent with what we know of the 
geometry of the Egyptians. 3 Nevertheless, the pyramids and passages within 
them were so precisely oriented that attempts are made to determine their 
age from the known rate of change of the position of the polestar. 

2 J. E. Quibell, Hierakonpolis (London: B. Quaritch, 1900). See especially Plate 26B. 

3 NoelF. Wheeler, "Pyramids and Their Purpose," Antiquity, 9(1935), 5-21,161-189,292-304. 


The Egyptians early had become interested in astronomy and had observed 
that the annual flooding of the Nile took place shortly after Sirius, the dog- 
star, rose in the east just before the sun. By noticing that these heliacal risings 
of Sirius, the harbinger of the flood, were separated by 365 days, the Egyptians 
established a good solar calendar made up of twelve months of thirty days 
each and five extra feast days. But this civil year was too short by a quarter 
of a day, hence the seasons advanced about one day every four years until, 
after a cycle of about 1460 years, the seasons again were in tune with the 
calendar. Inasmuch as it is known through the Roman scholar Censorinus, 
author of De die natale (a.d. 238), that the calendar was in line with the 
seasons in a.d. 139, it has been suggested through extrapolation backward 
that the calendar was instituted in the year 4241, just three cycles earlier. 
More precise calculations (based on the fact that the year is not quite 365^ 
days long) have modified the date to 4228, but other scholars feel that the 
backward extrapolation beyond two cycles is unwarranted and suggest 
instead an origin around 2773 B.C. 

There is a limit to the extent of mathematical information that can be 
inferred from tombstones and calendars, and our picture of Egyptian 
contributions would be sketchy in the extreme if we had to depend on 
ceremonial and astronomical material only. Mathematics is far more than 
counting and measuring, the aspects generally featured in hieroglyphic 
inscriptions. Fortunately we have other sources of information. There are a 
number of Egyptian papyri that somehow have survived the ravages of time 
over some three and a half millennia. The most extensive one of a mathe- 
matical nature is a papyrus roll about 1 foot high and some 18 feet long which 
now is in the British Museum (except for a few fragments in the Brooklyn 
Museum). It had been bought in 1858 in a Nile resort town by a Scottish 
antiquary, Henry Rhind ; hence it often is known as the Rhind Papyrus or, 
less frequently, as the Ahmes Papyrus in honor of the scribe by whose hand 
it had been copied in about 1650 b.c. 4 The scribe tells us that the material is 
derived from a prototype from the Middle Kingdom of about 2000 to 
1800 B.C., and it is possible that some of this knowledge may have been 
handed down from Imhotep, the almost legendary architect and physician 
to the Pharaoh Zoser, who supervised the building of his pyramid about 
5000 years ago. In any case, Egyptian mathematics seems to have stagnated 
for some 2000 years after a rather auspicious beginning. 

4 There are two good English editions, one edited by T. E. Peet and published in London in 
1923, the other by A. B. Chace et al. and published in two volumes at Oberlin, Ohio, in 1927-1929. 
Volume I of the latter contains an extensive general account of Egyptian mathematics by 
R. C. Archibald, a translation with commentary of the Ahmes Papyrus, and a very extensive 
bibliography of articles on Egyptian mathematics. 


The numerals and other material in the Rhind Papyrus are not written 
in the hieroglyphic forms described above, but in a more cursive script, 
better adapted to the use of pen and ink on prepared papyrus leaves and 
known as hieratic ("sacred," to distinguish it from the still later demotic or 
popular script). Numeration remains decimal, but the tedious repetitive 
principle of hieroglyphic numeration has been replaced by the introduction 
of ciphers or special signs to represent digits and multiples of powers often. 
Four, for example, usually is no longer represented by four vertical strokes, 
but by a horizontal bar; and seven is not written as seven strokes, but as a 
single cipher \ resembling a sickle. In hieroglyphic the number twenty-eight 
had appeared as nn||! !, but in hieratic it is simply = A. Note that the cipher = 
for the smaller digit eight (or two fours) appears on the left rather than on 
the right. The principle of cipherization, introduced by the Egyptians some 
4000 years ago and used in the Rhind Papyrus, represented an important 
contribution to numeration, and it is one of the factors that makes our own 
system in use today the effective instrument that it is. 

Men of the Stone Age had no use for fractions, but with the advent of more 
advanced cultures during the Bronze Age the need for the fraction concept 
and for fractional notations seems to have arisen. Egyptian hieroglyphic 
inscriptions have a special notation for unit fractions — that is, fractions with 
unit numerators. The reciprocal of any integer was indicated simply by 
placing over the notation for the integer an elongated oval sign. The fraction 

| thus appeared as ijn, and ^ was written as nn. In the hieratic notation, 

appearing in papyri, the elongated oval is replaced by a dot, which is placed 
over the cipher for the corresponding integer (or over the right-hand cipher 
in the case of the reciprocal of a multidigit number). In the Ahmes Papyrus, 
for example, the fraction £ appears as ==, and ^o is written as x. Such unit 
fractions were freely handled in Ahmes' day, but the general fraction seems 
to have been an enigma to the Egyptians. They felt comfortable with the 
fraction §, for which they had a special hieratic sign ?■; occasionally they 
used special signs for fractions of the form n/(n + 1), the complements of the 
unit fractions. To the fraction § the Egyptians assigned a special role in 
arithmetic processes, so that in finding one third of a number they first found 
two thirds of it and subsequently took half of the result ! They knew and 
used the fact that two thirds of the unit fraction 1/p is the sum of the two 
unit fractions l/2p and l/6p; they were also aware that double the unit 
fraction l/2p is the unit fraction 1/p. However, it looks as though, apart from 
the fraction f , the Egyptians regarded the general proper rational fraction of 
the form m/n not as an elementary "thing," but as part of an uncompleted 
process. Where today we think of f as a single irreducible fraction, Egyptian 


scribes thought of it as reducible to the sum of the three unit fractions i and 
| and T5. To facilitate the reduction of "mixed" proper fractions to the sum 
of unit fractions, the Rhind Papyrus opens with a table expressing 2/n as a 
sum of unit fractions for all odd values of n from 5 to 101. The equivalent of 
| is given as ^ and ^ ; T 2 T is written as £ and ^ ; and ^ is expressed as y\j and 
^. The last item in the table decomposes 5 t§t i nto tot and jm and 303 and 
5^6. It is not clear why one form of decomposition was preferred to another 
of the indefinitely many that are possible. At one time it was suggested that 
some of the items in the 2/n table were found by using the equivalent of the 

2 11 


n + I n(n + 1) 

or from 


p - q p. P - 


Yet neither of these procedures yields the combination for ^ that appears in 
the table. Recently it has been suggested 6 that the choice in most cases was 
dictated by the Egyptian preference for fractions derived from the "natural" 
fractions \ and y and f by successive halving. Thus if one wishes to express 
^ as a sum of unit fractions, he might well begin by taking half of ys and 
then seeing if to the result, ^, he can add a unit fraction to form ^; or he 
could use the known relationship 

2 1 _ _i_ J_ 

3 f 2p 6p 

to reach the same result tt = to + To- ° ne problem in the Rhind Papyrus 
specifically mentions the second method for finding two thirds of £ and asserts 
that one proceeds likewise for other fractions. Passages such as this indicate 
that the Egyptians had some appreciation of general rules and methods above 
and beyond the specific case at hand, and this represents an important step 
in the development of mathematics. For the decomposition of f the halving 

5 A list of fractional decompositions of 2/n from n = 5 to n = 101 is given in B. L. van der 
Waerden, Science Awakening (1961) and in Kurt Vogel, Vorgriechische Mathematik, Vol. 1, 
Vorgeschichte und Agypten (ca. 1958). A clear-cut explanation of Egyptian fractions appears also 
in O. Neugebauer, The Exact Sciences in Antiquity. All three works give excellent accounts of 
Egyptian mathematics. 

6 See Neugebauer, Exact Sciences in Antiquity, pp. 74 ff. 


procedure is not appropriate ; but by beginning with a third of j one finds 
the decomposition given by Ahmes, f = ^ + yj. In the case off one applies 
the halving procedure twice to j to reach the result j = ? + j§; successive 
halving yields also the Ahmes decomposition ts = ^ + tj + xc?- The 
Egyptian obsession with halving and taking a third is seen in the last entry 
in the table 2/n for n = 101, for it is not at all clear to us why the decomposition 
2/n = 1/n + l/2« + l/3w + 1/2 - 3 • n is better than 1/n + 1/n. Perhaps one 
of the objects of the 2/n decomposition was to arrive at unit fractions smaller 
than 1/n. 

The 2/n table in the Ahmes Papyrus is followed by a short n/10 table for 
n from 1 to 9, the fractions again being expressed in terms of the favorites — 
unit fractions and the fraction f . The fraction yg, f° r example, is broken 
into yo an d i an d §• Ahmes had begun his work with the assurance that it 
would provide a "complete and thorough study of all things . . . and the 
knowledge of all secrets," and therefore the main portion of the material, 
following the 2/n and n/10 tables, consists of eighty-four widely assorted 
problems. The first six of these require the division of one or two or six or 
seven or eight or nine loaves of bread among ten men, and the scribe makes use 
of the n/10 table that he has just given. In the first problem the scribe goes to 
considerable trouble to show that it is correct to give to each of the ten men 
one tenth of a loaf! If one man receives yg loaf, two men will receive fg or 5 
and four men will receive § of a loaf or 3 + yj of a loaf. Hence eight men will 
receive f + fs of a loaf or f + tV + To of a loaf, and eight men plus two men 
will receive f-r-y + yo + yo, ora whole loaf. Ahmes seems to have had a 
kind of equivalent to our least common multiple which enabled him to com- 
plete the proof. In the division of seven loaves among ten men, the scribe 
might have chosen \ + 5 of a loaf for each, but the predilection for § led 
him instead to § and 3% of a loaf for each. 7 

The fundamental arithmetic operation in Egypt was addition, and our 
operations of multiplication and division were performed in Ahmes' day 
through successive doubling or "duplation." Our own word "multiplication" 
or manifold is, in fact, suggestive of the Egyptian process. A multiplication of, 
say, 69 by 19 would be performed by adding 69 to itself to obtain 138, then 
adding this to itself to reach 276, applying duplation again to get 552, and 
once more to obtain 1 104, which is, of course, sixteen times 69. Inasmuch as 
19 = 16 + 2 + 1, the result of multiplying 69 by 19 is 1104 + 138 + 69— 
that is, 131 1. Occasionally a multiplication by ten also was used, for this was 
a natural concomitant of the decimal hieroglyphic notation. Multiplication 
of combinations of unit fractions was also a part of Egyptian arithmetic. 

7 For further details see R. J. Gillings, "Problems 1 to 6 of the Rhind Mathematical Papyrus," 
The Mathematics Teacher, 55 (1962), 61-69. 


Problem 13 in the Ahmes Papyrus, for example, asks for the product of 
T5 + Til and 1 + j + i; the result is correctly found to be £. For division 
the duplation process is reversed, and the divisor is successively doubled 
instead of the multiplicand. That the Egyptians had developed a high degree 
of artistry in applying the duplation process and the unit fraction concept is 
apparent from the calculations in the problems of Ahmes. Problem 70 calls 
for the quotient when 100 is divided by 7 + \ + \ + £; the result, 12 + f + 
T2 + lie. is obtained as follows. Doubling the divisor successively, we first 
obtain 15 + \ + \, then 31 + \, and finally 63, which is eight times the 
divisor. Moreover, two thirds of the divisor is known to be 5 + \. Hence 
the divisor when multiplied by 8 + 4 + f will total 99f , which is £ short of 
the product 100 that is desired. Here a clever adjustment was made. Inasmuch 
as eight times the divisor is 63, it follows that the divisor when multiplied by 
^3 will produce £. From the 2/n table one knows that ^ is ^ + T26> hence 
the desired quotient is 12 + f + ij + jh- Incidentally, this procedure 
makes use of a commutative principle in multiplication, with which the 
Egyptians evidently we're familiar. 

Many of Ahmes' problems show a knowledge of manipulations of propor- 
tions equivalent to the "rule of three." Problem 72 calls for the number of 
loaves of bread of "strength" 45 which are equivalent to 100 loaves of 
"strength" 10, and the solution is given as 100/10 x 45 or 450 loaves. In 
bread and beer problems the "strength" or pesu is the reciprocal of the 
grain density, being the quotient of the number of loaves or units of volume 
divided by the amount of grain. Bread and beer problems are numerous in 
the Ahmes Papyrus. Problem 63, for example, requires the division of 700 
loaves of bread among four recipients if the amounts they are to receive are 
in the continued proportion \:\:\:\- The solution is found by taking the 
ratio of 700 to the sum of the fractions in the proportion. In this case the 
quotient of 700 divided by if is found by multiplying 700 by the reciprocal 
of the divisor, which is \ + -fe. The result is 400; by taking f and | and ^ 
and 5 of this, the required shares of bread are found. 

The Egyptian problems so far described are best classified as arithmetic, 
but there are others that fall into a class to which the term algebraic is appro- 
priately applied. These do not concern specific concrete objects, such as 
bread and beer, nor do they call for operations on known numbers. Instead 
they require the equivalent of solutions of linear equations of the form 
x + ax = b or x + ax + bx = c, where a and b and c are known and x is 
unknown. The unknown is referred to as "aha" or heap. Problem 24, for 
instance, calls for the value of heap if heap and a seventh of heap is 19. The 
solution given by Ahmes is not that of modern textbooks, but is character- 
istic of a procedure now known as the "method of false position" or the 


"rule of false." A specific value, most likely a false one, is assumed for heap, 
and the operations indicated on the left-hand side of the equality sign are 
performed on this assumed number. The result of these operations then is 
compared with the result desired, and by the use of proportions the correct 
answer is found. In problem 24 the tentative value of the unknown is taken 
as 7, so that x + jx is 8, instead of the desired answer, which was 19. Inasmuch 
as 8(2 + i + i) = 19, one must multiply 7 by 2 + £ + i to obtain the correct 
heap ; Ahmes found the answer to be 16 + \ + |. Ahmes then "checked" his 
result by showing that if to 16 + j + £ one adds a seventh of this (which is 
2 + i + i), one does indeed obtain 19. Here we see another significant step 
in the development of mathematics, for the check is a simple instance of a 
proof. Although the method of false position was generally used by Ahmes, 
there is one problem (Problem 30) in which x + fx + \x + jx = 37 is 
solved by factoring the left-hand side of the equation and dividing 37 by 
1 + f + i + t. the result being 16 + ^ + ^ + jrg- 

Many of the "aha" calculations in the Rhind Papyrus evidently are practice 
exercises for young students. Although a large proportion of them are of a 
practical nature, in some places the scribe seems to have had puzzles or 
mathematical recreations in mind. Thus Problem 79 cites only "seven houses, 
49 cats, 343 mice, 2401 ears of spelt, 16807 hekats." It is presumed that the 
scribe was dealing with a problem, perhaps quite well known, in which in 
each of seven houses there are seven cats each of which eats seven mice, each 
of which would have eaten seven ears of grain, each of which would have 
produced seven measures of grain. The problem evidently called not for the 
practical answer, which would be the number of measures of grain that were 
saved, but for the impractical sum of the numbers of houses, cats, mice, ears 
of spelt, and measures of grain. This bit of fun in the Ahmes Papyrus seems 
to be a forerunner of our familiar nursery rhyme : 

As I was going to St. Ives, 
I met a man with seven wives ; 
Every wife had seven sacks, 
Every sack had seven cats, 
Every cat had seven kits. 
Kits, cats, sacks, and wives, 
How many were going to St. Ives? 

The Greek historian Herodotus tells us that the obliteration of boundaries 
in the overflow of the Nile emphasized the need for surveyors. The accom- 
plishments of the "rope-stretchers" of Egypt evidently were admired by 
Democritus, an accomplished mathematician and one of the founders of an 
atomic theory, and today their achievements seem to be overvalued, in part 
as a result of the admirable accuracy of construction of the pyramids. It often 


is said that the ancient Egyptians were familiar with the Pythagorean theorem, 
but there is no hint of this in the papyri that have come down to us. There 
are nevertheless some geometrical problems in the Ahmes Papyrus. Problem 
51 of Ahmes shows that the area of an isosceles triangle was found by taking 
half of what we would call the base and multiplying this by the altitude. 
Ahmes justified his method of finding the area by suggesting that the isosceles 
triangle can be thought of as two right triangles, one of which can be shifted 
in position, so that together the two triangles form a rectangle. The isosceles 
trapezoid is similarly handled in Problem 52, in which the larger base of a 
trapezoid is 6, the smaller base is 4, and the distance between them is 20. 
Taking half the sum of the bases, "so as to make a rectangle," Ahmes multi- 
plied this by 20 to find the area. In transformations such as these, in which 
isosceles triangles and trapezoids are converted into rectangles, we see the 
beginnings of a theory of congruence and of the idea of proof in geometry, 
but the Egyptians did not carry their work further. A serious deficiency in 
their geometry was the lack of a clear-cut distinction between relationships 
that are exact and those that are approximations only. A surviving deed from 
Edfu, dating from a period some 1500 years after Ahmes, gives examples of 
triangles, trapezoids, rectangles, and more general quadrilaterals ; the rule 
for finding the area of the general quadrilateral is to take the product of the 
arithmetic means of the opposite sides. Inaccurate though the rule is, the 
author of the deed deduced from it a corollary—that the area of a triangle is 
half the sum of two sides multiplied by half the third side. This is a striking 
instance of the search for relationships among geometric figures, as well as 
an early use of the zero concept as a replacement for a magnitude in geometry. 
The Egyptian rule for finding the area of a circle has long been regarded 
as one of the outstanding achievements of the time. In Problem 50 the scribe 
Ahmes assumed that the area of a circular field with a diameter of nine units 
is the same as the area of a square with a side of eight units. If we compare 
this assumption with the modern formula A = nr 2 , we find the Egyptian 
rule to be equivalent to giving n a value of about 3g, a commendably close 
approximation; but here again we miss any hint that Ahmes was aware 
that the areas of his circle and square were not exactly equal. It is possible 
that Problem 48 gives a hint to the way in which the Egyptians were led to 
their area of the circle. In this problem the scribe formed an octagon from a 
square of side nine units by trisecting the sides and cutting off the four corner 
isosceles triangles, each having an area of 4\ units. The area of the octagon, 
which does not differ greatly from that of a circle inscribed within the square, 
is sixty-three units, which is not far removed from the area of a square with 
eight units on a side. That the number 4(8/9) 2 did indeed play a role com- 
parable to our constant n seems to be confirmed by the Egyptian rule for 
the circumference of a circle, according to which the ratio of the area of a 


circle to the circumference is the same as the ratio of the area of the circum- 
scribed square to its perimeter. This observation represents a geometrical 
relationship of far greater precision and mathematical significance than the 
relatively good approximation for n. Degree of accuracy in approximation is, 
after all, not a good measure of either mathematical or architectural achieve- 
ment, and we should not overemphasize this aspect of Egyptian work. 
Recognition by the Egyptians of interrelationships among geometrical 
figures, on the other hand, has too often been overlooked, and yet it is here 
that they came closest in attitude to their successors, the Greeks. No theorem 
or formal proof is known in Egyptian mathematics, but some of the geometric 
comparisons made in the Nile Valley, such as those on the perimeters and 
areas of circles and squares, are among the first exact statements in history 
concerning curvilinear figures. 

Problem 56 of the Rhind Papyrus is of special interest in that it contains 
rudiments of trigonometry and a theory of similar triangles. In the construc- 
tion of the pyramids it had been essential to maintain a uniform slope for the 
faces, and it may have been this concern that led the Egyptians to introduce 
a concept equivalent to the cotangent of an angle. In modern technology it is 
customary to measure the steepness of a straight line through the ratio of the 
"rise" to the "run." In Egypt it was customary to use the reciprocal of this 
ratio. There the word "seqt" meant the horizontal departure of an oblique 
line from the vertical axis for every unit change in the height. The seqt thus 
corresponded, except for the units of measurement, to the batter used today 
by architects to describe the inward slope of a masonry wall or pier. The 
vertical unit of length was the cubit ; but in measuring the horizontal distance, 
the unit used was the "hand," of which there were seven in a cubit. Hence 
the seqt of the face of a pyramid was the ratio of run to rise, the former 
measured in hands, the latter in cubits. In Problem 56 one is asked to find 
the seqt of a pyramid that is 250 ells or cubits high and has a square base 
360 ells on a side. The scribe first divided 360 by 2 and then divided the result 
by 250, obtaining i + i + To in ells. Multiplying the result by 7, he gave 
the seqt as 5yj in hands per ell. In other pyramid problems in the Ahmes 
Papyrus the seqt turns out to be 5|, agreeing somewhat better with that of 
the great Cheops Pyramid, 440 ells wide and 280 high, the seqt being 5^ hands 
per ell. 

There are many stories about presumed geometrical relationships among 
dimensions in the Great Pyramid, some of which are patently false. For 
instance, the story that the perimeter of the base was intended to be precisely 
equal to the circumference of a circle of which the radius is the height of the 
pyramid is not in agreement with the work of Ahmes. The ratio of perimeter 
to height is indeed very close to ^, which is just twice the value of ^ often 


used today for n ; but we must recall that the Ahmes value for n is about 
3£, not 3j. That Ahmes' value was used also by other Egyptians is confirmed 
in a papyrus roll from the twelfth dynasty (the Kahun Papyrus, now in 
London) in which the volume of a cylinder is found by multiplying the height 
by the area of the base, the base being determined according to Ahmes' rule. 

Much of our information about Egyptian mathematics has been derived 
from the Rhind or Ahmes Papyrus, the most extensive mathematical docu- 
ment from ancient Egypt ; but there are other sources as well. 8 Besides the 
Kahun Papyrus, already mentioned, there is a Berlin Papyrus of the same 
period, two wooden tablets from Akhmim (Cairo) of about 2000 B.C., a leather 
roll containing lists of unit fractions and dating from the later Hyksos period, 
and an important papyrus, known as the Golenischev or Moscow Papyrus, 
purchased in Egypt in 1893. The Moscow Papyrus is about as long as the 
Rhind Papyrus — about 18 feet — but it is only one-fourth as wide, the width 
being about 3 inches. It was written, less carefully than the work of Ahmes, 
by an unknown scribe of the twelfth dynasty (ca. 1890 B.C.). It contains 
twenty-five examples, mostly from practical life and not differing greatly 
from those of Ahmes, except for two that have special significance. Associated 
with Problem 14 in the Moscow Papyrus is a figure that looks like an isosceles 
trapezoid (see Fig. 2.1), but the calculations associated with it indicate that 


FIG. 2.1 

a frustum of a square pyramid is intended. Above and below the figure are 
signs for two and four respectively, and within the figure are the hieratic 
symbols for six and fifty-six. The directions alongside make it clear that the 
problem calls for the volume of a frustum of a square pyramid six units high 
if the edges of the upper and lower bases are two and four units respectively. 
The scribe directs one to square the numbers two and four and to add to 
the sum of these squares the product of two and four, the result being twenty- 
eight. This result is then multiplied by a third of six ; and the scribe concludes 
with the words, "See, it is 56 ; you have found it correctly." That is, the volume 

8 A good account of these appears in the work of Archibald cited in footnote 4. 



of the frustum has been calculated in accordance with the modern formula 
V = h(a 2 + ab + b 2 )/3, where h is the altitude and a and b are the sides of 
the square bases. Nowhere is this formula written out, but in substance it 
evidently was known to the Egyptians. If. as in the deed from Edfu, one takes 
b = 0, the formula reduces to the familiar formula, one-third the base times 
the altitude, for the volume of a pyramid. How these results were arrived at 
by the Egyptians is not known. An empirical origin for the rule on volume 
of a pyramid seems to be a possibility, but not for the volume of the frustum. 
For the latter a theoretical basis seems more likely ; and it has been suggested 
that the Egyptians may have proceeded here as they did in the cases of the 
isosceles triangle and the isosceles trapezoid — they may in thought have 
broken the frustum into parallelepipeds, prisms, and pyramids. 9 Upon 
replacing the pyramids and prisms by equal rectangular blocks, a plausible 
grouping of the blocks leads to the Egyptian formula. One could, for example, 
have begun with a pyramid having a square base and with the vertex directly 
over one of the base vertices. An obvious decomposition of the frustum would 
be to break it into four parts as in Fig. 2.2 — a rectangular parallelepiped 

FIG. 2 2 

having a volume b 2 h, two triangular prisms, each with a volume of b{u — b)h/2, 
and a pyramid of volume (a — b) 2 h/3. The prisms can be combined into a 
rectangular parallelepiped with dimensions b and a - h and /t; and the 
pyramid can be thought of as a rectangular parallelepiped with dimensions 
ci — b and a - b and &/3, Upon cutting up the tallest parallelepipeds so that 
all altitudes are /r/3. one can easily arrange the slabs so as to form three 

* Van der Wacrden, Science Awakening, p. 35. C(. R. J. GilJings. "The Volume of a Truncated 
Pyramid in Ancient Egypt," Mathematics Teacher, 57 (1%4), 552-555. 


layers, each of altitude /i/3, and having cross-sectional areas of a 2 and ah and 
b 2 respectively. 

Problem 10 in the Moscow Papyrus presents a more difficult question of 
interpretation than does Problem 14. Here the scribe asks for the surface 
area of what looks like a basket with a diameter of 4j. He proceeds as 
though he were using the equivalent of a formula S = (1 - i) 2 (2x) • x, where 
x is 4j, obtaining an answer of 32 units. Inasmuch as (1 - ^) 2 is the Egyptian 
approximation for Jt/4, the answer 32 would correspond to the surface of a 
hemisphere of diameter 4^ ; and this was the interpretation given to the 
problem in 1930. 10 Such a result, antedating the oldest known calculation of 
a hemispherical surface by some 1500 years, would have been amazing, and 
it seems, in fact, to have been too good to be true. Later analysis 1 x indicates 
that the "basket" may have been a roof — somewhat like that of a quonset 
hut in the shape of a half cylinder of diameter A\ and length 4|. The calcula- 
tion in this case calls for nothing beyond knowledge of the length of a semi- 
circle ; and the obscurity of the text makes it admissible to offer still more 
primitive interpretations, including the possibility that the calculation is only 
a rough estimate of the area of a domelike barn roof. In any case, we seem 
to have here an early estimation of a curvilinear surface area. 

1 For many years it had been assumed that the Greeks had learned the rudi- 
ments of geometry from the Egyptians, and Aristotle argued that geometry 
had arisen in the Nile Valley because the priests there had the leisure to 
develop theoretical knowledge. That the Greeks did borrow some elementary 
mathematics from Egypt is probable, for the use of unit fractions persisted 
in Greece and Rome well into the Medieval period, but evidently they exag- 
gerated the extent of their indebtedness. The knowledge indicated in extant 
Egyptian papyri is mostly of a practical nature, and calculation was the chief 
element in the questions. Where some theoretical elements appear to enter, 
the purpose may have been to facilitate technique rather than understanding. 
Even the once-vaunted Egyptian geometry turns out to have been mainly a 
branch of applied arithmetic. Where elementary congruence relations enter, 
the motive seems to be to provide mensurational devices rather than to gain 
insight. The rules of calculation seldom are motivated, and they concern 
specific concrete cases only. The Ahmes and Moscow papyri, our two chief 
sources of information, may have been only manuals intended for students, 

10 See W. W. Struve, "Mathematischer Papyrus des Staatlichen Museums der Schonen 
Kiinste in Moskau," Quellen und Studien zur Geschkhte der Mathematik, Part A, Quellen, I ( 1 930). 

" See van der Waerden, Science Awakening, p. 34. Cf., however, R. J. Gillings, "The Area of 
the Curved Surface of a Hemisphere in Ancient Egypt," The Australian Journal of Science, 30 
(1967), 113-116, in which the author concludes that the scribe of the Moscow Papyrus was 
indeed dealing correctly, in Problem 10, with the curved surface of a hemisphere. 


but they nevertheless indicate the direction and tendencies in Egyptian 
mathematical instruction; further evidence provided by inscriptions on 
monuments, fragments of other mathematical papyri, and documents from 
related scientific fields serves to confirm the general impression. It is true 
that our two chief mathematical papyri are from a relatively early period, a 
thousand years before the rise of Greek mathematics, but Egyptian mathe- 
matics seems to have remained remarkably uniform throughout its long 
history. It was at all stages built around the operation of addition, a dis- 
advantage that gave to Egyptian computation a peculiar primitivity com- 
bined with occasionally astonishing complexity. The fertile Nile Valley has 
been described as the world's largest oasis in the world's largest desert. 
Watered by one of the most gentlemanly of rivers and geographically shielded 
to a great extent from foreign invasion, it was a haven for peace-loving 
people who pursued, to a large extent, a calm and unchallenged way of life. 
Love of the beneficent gods, respect for tradition, and preoccupation with 
death and the needs of the dead, all encouraged a high degree of stagnation. 
Geometry may have been a gift of the Nile, as Herodotus believed, but the 
Egyptians did little with the gift. The mathematics of Ahmes was that of his 
ancestors and of his descendants. For more progressive mathematical 
achievements one must look to the more turbulent river valley known as 


Chace, A. B., L. S. Bull, H. P. Manning, and R. C. Archibald, eds., The Rhind Mathemat- 
ical Papyrus (Oberlin, Ohio, 1927-1929, 2 vols.). This contains a comprehensive 
bibliography of works on Egyptian mathematics published in the interval from 
1706 through 1927, as well as an extensive general account of Egyptian mathematics. 

Gillings, R. J., "Problems 1 to 6 of the Rhind Mathematical Papyrus," The Mathematics 
Teacher, 55 (1962), 61-69. Continuations are found in later volumes of the journal. 

Guggenbuhl, Laura, "Mathematics in Ancient Egypt : A Checklist," The Mathematics 
Teacher, 58 (1965), 630-634. 

Neugebauer, O., Die Grundlagen der dgyptischen Bruchrechnung (Berlin : Springer, 1926). 

Neugebauer, O., The Exact Sciences in Antiquity, 2nd ed. (Providence, R. I.: Brown 
University Press, 1957; paperback ed., New York ; Harper Torchbook). 

Parker, R. A., The Calendars of Ancient Egypt (Chicago: University of Chicago Press, 

Struve, W. W., "Mathematischer Papyrus des Staatlichen Museums der Schonen 
Kiinste in Moskau," Quellen and Studien zur Geschichte der Mathematik, Part A, 
Quellen, 1 (1930). 

Van der Waerden, B. L., "Die Entstehungsgeschichte der agyptischen Bruchrechnung," 
Quellen und Studien zur Geschichte der Mathematik, Part B, Studien, IV (1937-1938), 


Van der Waerden, B. L., Science Awakening, trans, by Arnold Dresden (New York : 
Oxford University Press, 1961 ; paperback ed., New York: Wiley, 1963). 

Vogel, Kurt., Vorgriechische Mathematik, Vol. I, Vorgeschichte und Agypten (paperback 
ed., Hannover: Hermann Schroedel, ca. 1958). 

Wheeler, Noel F., "Pyramids and Their Purpose," Antiquity, 9 (1935), 5-21, 161-189, 


1. Describe the evidence on which our estimate of Egyptian mathematics is based. Do you 
think that this is likely to be altered by the discovery of new documents? Explain. 

2. Do you think that astronomy was a more significant factor than surveying in the rise of 
Egyptian mathematics? Explain. 

3. What does the word "geometry" mean etymologically? Is the use of the word justifiable in 
the light of the historical origin of the subject? Explain. 

4. What do you regard as the three chief shortcomings in Egyptian mathematics? Explain 
why you regard these as the most significant. 

5. What do you regard as the three chief contributions of Egypt to the development of mathe- 
matics? Explain why you regard them as important. 

6. Write the number 7654 in Egyptian hieroglyphic form. How does this differ from the way 
in which Ahmes would write this number? 

7. Express jfj as a sum of two unequal unit fractions, and write these in Egyptian hieroglyphic 
notation. How does the hieratic form differ from this? 

8. Solve by the method of false position the equation x + jx = 16. (This is Problem 25 in the 
Ahmes papyrus.) 

9. Solve the following problem from the Ahmes Papyrus (Problem 40): Divide 100 loaves 
among five men so that the shares are in arithmetic progression and so that one seventh of 
the sum of the three largest shares is equal to the sum of the two smallest. 

10. Solve in the Egyptian manner the simultaneous equations x 2 + y 2 = 100, y = 3x/4, taken 
from a Berlin papyrus from ancient Egypt. (Use the method of "double false," starting from 
an assumed value of x, finding the corresponding value of y from the second equation, and 
adjusting the values so that they satisfy the first equation.) 

11. Through duplation and mediation (that is, successive doubling and halving) find 101 -h 16, 
expressing the result in Egyptian hieroglyphic form. 

12. Derive the Egyptian formula for volume of a frustum of a square pyramid algebraically 
from the known formula for volume of a pyramid, using proportions established in elemen- 
tary geometry. Do you believe that the Egyptians could have derived their formula in this 
way? Explain. 

13. To what extent is it fair to say that the Egyptians knew the area of the circle? Explain clearly. 

14. Why do you think that the Egyptians preferred the decomposition A = to + 30 to the 
alternative fV = tV + A^ 

15. Show that if n is a multiple of three, 2/n can be broken into the sum of two unit fractions, 
one of which is half of 1/n. 

16. Show that if n is a multiple of five, 2/n can be broken into the sum of two unit fractions, one 
of which is a third of 1/n. 

17. Justify the method of solution used by Ahmes in his Problem 63. (See text.) 

18. Justify the assumption made by Ahmes that the ratio of the area of a circle to its circum- 
ference is the same as the ratio of the area of the circumscribed square to the perimeter of 


this square. 
19. Iftheseqtofapyramidis5palms(orhands)and 1 finger per cubit, and if the side of its base is 
140 cubits, what is its altitude? (This is Problem 57 in the Ahmes Papyrus.) There are five 
fingers in a palm. 
*20. Using the Egyptian method of division, solve the following problem (Problem 31) from the 
Ahmes Papyrus : A quantity and its two thirds and its half and its one seventh together 
make 33. Find the quantity. [The answer given is 14 + I + ^g + wr + T54 + tss + 675 + 




How much is one god beyond the other god? 

An Old Babylonian astronomical text 

The fourth millennium before our era was a period of remarkable cultural 
development, bringing with it the use of writing, of the wheel, and of metals. 
As in Egypt during the first dynasty, which began toward the end of this 
wonderful millennium, so also in the Mesopotamian valley there was at the 
time a high order of civilization. There the Sumerians had built homes and 
temples decorated with artistic pottery and mosaics in geometrical patterns. 
Powerful rulers united the local principates into an empire which completed 
vast public works, such as a system of canals to irrigate the land and to 
control flooding. The Biblical account of the Noachian flood had an earlier 
counterpart in the legend concerning the Sumerian hero Utnapischtum and 
the flooding of the region between the Tigris and Euphrates rivers, where the 
overflow of the rivers was not predictable, as was the inundation of the Nile 
Valley. The Bible tells us that Abraham came from the city of Ur, a Sumerian 
settlement where the Euphrates emptied into the Persian Gulf, for at that 
time the two rivers did not join, as they now do, before reaching the Gulf. 
The cuneiform pattern of writing that the Sumerians had developed during 
the fourth millennium, long before the days of Abraham, may have been the 
earliest form of written communication, for it probably antedates the 
Egyptian hieroglyphic, which may have been a derivative. Although they 
have nothing in common, it is an interesting Coincidence that the origins 
of writing and of wheeled vehicles are roughly coeval. 

The Mesopotamian civilizations of antiquity often are referred to as 
Babylonian, although such a designation is not strictly correct. The city 
of Babylon was not at first, nor was it always at later periods, the center of 
the culture associated with the two rivers, but convention has sanctioned 
the informal use of the name "Babylonian" for the region during the interval 
from about 2000 to roughly 600 B.C. When in 538 B.C. Babylon fell to Cyrus 
of Persia, the city was spared, but the Babylonian empire had come to an 
end. "Babylonian" mathematics, however, continued through the Seleucid 



period in Syria almost to the dawn of Christianity. Occasionally the area 
between the rivers is known also as Chaldea, because the Chaldeans (or 
Kaldis), originally from southern Mesopotamia, were for a time dominant, 
chiefly during the late seventh century B.C., throughout the region between 
the rivers. Then, as today, the Land of the Two Rivers was open to invasions 
from many directions, making of the Fertile Crescent a battlefield with 
frequently changing hegemony. One of the most significant of the invasions 
was that by the Semitic Akkadians under Sargon I (ca. 2276-2221 B.C.) 
or Sargon the Great. He established an empire that extended from the 
Persian Gulf in the south to the Black Sea in the north, and from the steppes 
of Persia on the east to the Mediterranean Sea on the west. Under Sargon 
there was begun a gradual absorption by the invaders of the indigenous 
Sumerian culture, including the cuneiform script. Later invasions and revolts 
brought varying racial strains — Ammorites, Kassites, Elamites, Hittites, 
Assyrians, Medes, Persians, and others — to political power at one time or 
another in the valley, but there remained in the area a sufficiently high degree 
of cultural unity to justify referring to the civilization simply as Mesopota- 
mian. In particular, the use of cuneiform script formed a strong bond. Laws, 
tax accounts, stories, school lessons, personal letters — these and many other 
records were impressed on soft clay tablets with a stylus, and the tablets 
then were baked in the hot sun or in ovens. Such written documents, for- 
tunately, were far less vulnerable to the ravages of time than were Egyptian 
papyri ; hence there is available today a much larger body of evidence about 
Mesopotamian than about Nilotic mathematics. From one locality alone, 
the site of ancient Nippur, we have some 50,000 tablets. The university 
libraries at Columbia, Pennsylvania, and Yale, among others, have large 
collections of ancient tablets from Mesopotamia, some of them mathematical. 
Despite the availability of documents, however, it was the Egyptian hiero- 
glyphic rather than the Babylonian cuneiform that first was deciphered in 
modern times. Some progress in the reading of Babylonian script had been 
made early in the nineteenth century by Grotefend, but it was only during 
the second quarter of the twentieth century that substantial accounts of 
Mesopotamian mathematics began to appear in histories of antiquity. 1 

The early use of writing in Mesopotamia is attested by hundreds of clay 
tablets found in Uruk and dating from about 5000 years ago. By this time 
picture writing had reached the point where conventionalized stylized forms 
were used for many things : ~ for water, O for eye, and combinations of 

1 See especially O. Neugebauer, The Exact Sciences in Antiquity (1957) and B. L. van der 
Waerden, Science Awakening (1961). Cf. also O. Neugebauer and A. Sachs, Mathematical 
Cuneiform Texts (American Oriental Series, Vol. 29, 1945). A good secondary account and 
further references will be found in R. C. Archibald, Outline of the History of Mathematics (Ameri- 
can Mathematical Monthly, 56 (1949), No. 1, supp.). 


these to indicate weeping. Gradually the number of signs became smaller, 
so that of some 2000 Sumerian signs originally used only a third remained 
by the time of the Akkadian conquest. Primitive drawings gave way to 
combinations of wedges: water became ^ and eye £T>-. At first the scribe 
wrote from top to bottom in columns from right to left ; later, for convenience, 
the table was rotated counter clockwise through 90°, and the scribe wrote 
from left to right in horizontal rows from top to bottom. The stylus, which 
formerly had been a triangular prism, was replaced by a right circular 
cylinder — or, rather, two cylinders of unequal radius. During the earlier 
days of the Sumerian civilization, the end of the stylus was pressed into the 
clay vertically to represent ten units and obliquely to represent a unit, using 
the smaller stylus; similarly, an oblique impression with the larger stylus 
indicated sixty units and a vertical impression indicated 3600 units. Combina- 
tions of these were used to represent intermediate numbers. 

As the Akkadians adopted the Sumerian form of writing, lexicons were 
compiled giving equivalents in the two tongues, and forms of words and 
numerals became less varied. Thousands of tablets from about the time of 
the Hammurabi dynasty (ca. 1800-1600 B.C.) illustrate a number system that 
had become well established. The decimal system, common to most civiliza- 
tions, both ancient and modern, had been submerged in Mesopotamia 
under a notation that made fundamental the base sixty. Much has been 
written about the motives behind this change; it has been suggested that 
astronomical considerations may have been instrumental or that the 
sexagesimal scheme may have been the natural combination of two earlier 
schemes, one decimal and the other using the base six. It appears more 
likely, however, that the base sixty was consciously adopted and legalized 
in the interests of metrology, for a magnitude of sixty units can be subdivided 
easily into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, 
twentieths, and thirtieths, thus affording ten possible subdivisions. Whatever 
the origin, the sexagesimal system of numeration has enjoyed a remarkably 
long life, for remnants survive, unfortunately for consistency, even to this 
day in units of time and angle measure, despite the fundamentally decimal 
form of our society. 

Babylonian cuneiform numeration, for smaller whole numbers, proceeded 
along the same lines as did the Egyptian hieroglyphic, with repetitions of 
the symbols for units and tens. Where the Egyptian architect, carving on 

stone might write fifty-nine as n n n n ft \\\, the Mesopotamian scribe could simi- 
larly represent the same number on a clay tablet through fourteen wedge- 
shaped marks — five broad sideways wedges or "angle-brackets," each 
representing ten units, and nine thin vertical wedges, each standing for a 


unit, all juxtaposed in a neat group as <|f$. Beyond the number fifty-nine, 
however, the Egyptian and Babylonian systems differed markedly. Perhaps 
it was the inflexibility of the Mesopotamian writing materials, possibly it 
was a flash of imaginative insight that made the Babylonians aware that 
their two symbols for units and tens sufficed for the representation of any 
integer, however large, without excessive repetitiveness. This was made pos- 
sible through their invention, some 4000 years ago, of the positional nota- 
tion — the same principle that accounts for the effectiveness of our present 
numeral forms. That is, the ancient Babylonians saw that their symbols could 
do double, triple, quadruple, or any degree of duty simply by being assigned 
values that depend on their relative positions in the representation of a 
number. The wedges in the cuneiform symbol for fifty-nine are tightly 
grouped together so as to form almost the equivalent of a single cipher. 
Appropriate spacing between groups of wedges can establish positions, read 
from right to left, that correspond to ascending powers of the base ; each 
group then has a "local value" that depends on its position. Our number 222 
makes use of the same cipher three times, but with a different meaning each 
time. Once it represents two units, then it means two tens, and finally it 
stands for two hundreds (that is, twice the square of the base ten). In a 
precisely analogous way the Babylonians made multiple use of such a 
symbol as TT. When they wrote TT TT TT, clearly separating the three groups of 
two wedges each, they understood the right-hand group to mean two units, 
the next group to mean twice their base, sixty, and the left-hand group to 
signify twice the square of their base. This numeral therefore denoted 
2(60) 2 + 2(60) + 2 (or 7322 in our notation). 

There is a wealth of primary material concerning Mesopotamian mathe- 
matics, but oddly enough most of it comes from two periods widely separated 
in time. There is an abundance of tablets from the first few hundred years 
of the second millennium B.C. (the Old Babylonian age), and there are many 
also from the last few centuries of the first millennium B.C. (the Seleucid 
period). Most of the important contributions to mathematics will be found 
to go back to the earlier period, but there is one contribution not in evidence 
until almost 300 B.C. The Babylonians seem at first to have had no clear 
way in which to indicate an "empty" position — that is, they did not have a 
zero symbol, although they sometimes left a space where a zero was intended. 
This meant that their forms for the numbers 122 and 7202 looked very 
much alike, for TT TT might mean either 2(60) + 2 or 2(60) 2 + 2. Context in 
many cases could be relied on to relieve some of the ambiguity ; but the lack 
of a zero symbol, such as enables us to distinguish at a glance between 22 
and 202, must have been quite inconvenient. By about the time of the con- 
quest by Alexander the Great, however, a special sign, consisting of two 
small wedges placed obliquely, was invented to serve as a placeholder where 


a numeral was missing. From that time on, as long as cuneiform was used, 
the number TT -> TT, or 2(60) 2 + 0(60) + 2, was readily distinguishable from 
TT TT, or 2(60) + 2. 

The Babylonian zero symbol apparently did not end all ambiguity, for 
the sign seems to have been used for intermediate empty positions only. 
There are no extant tablets in which the zero sign appears in a terminal 
position. This means that the Babylonians in antiquity never achieved an 
absolute positional system. Position was relative only; hence the symbol 
TTTT could represent 2(60) + 2 or 2(60) 2 + 2(60) or 2(60) 3 + 2(60) 2 or any 
one of indefinitely many other numbers in which two successive positions 
are involved. 

3 Had Mesopotamian mathematics, like that of the Nile Valley, been based 
on the addition of integers and unit fractions, the invention of the positional 
notation would not have been of great significance at the time. It is not much 
more difficult to write 98,765 in hieroglyphic notation than in cuneiform, 
and the latter is definitely more difficult to write than the same number in 
hieratic script. The secret of the clear superiority of Babylonian mathematics 
over that of the Egyptians undoubtedly lies in the fact that those who lived 
"between the two rivers" took the most felicitous step of extending the 
principle of position to cover fractions as well as whole numbers. That is, 
the notation TTTT was used not only for 2(60) + 2, but also for 2 + 2(60) _1 
or for 2(60) _1 + 2(60) ~ 2 or for other fractional forms involving two suc- 
cessive positions. This meant that the Babylonians had at their command the 
computational power that the modern decimal fractional notation affords 
us today. For the Babylonian scholar, as for the modern engineer, the addition 
or the multiplication of 23.45 and 9.876 was essentially no more difficult 
than was the addition or multiplication of the whole numbers 2345 and 
9876 ; and the Mesopotamians were quick to exploit this important discovery. 
An Old Babylonian tablet from the Yale Collection (No. 7289) includes the 
calculation of the square root of two to three sexagesimal places, the answer 
being written "1«^<#T<. In modern characters this number can be appro- 
priately written as 1 ; 24,5 1,10, where a semicolon is used to separate the 
integral and fractional parts and a comma is used as a separatrix for the 
sexagesimal positions. This form will generally be used throughout this 
chapter to designate numbers in sexagesimal notation. This Babylonian 
value for y/l is equal to approximately 1.414222, differing by about 0.000008 
from the true value. Accuracy in approximations was relatively easy for the 
Babylonians to achieve with their fractional notation, the best that any 
civilization afforded until the time of the Renaissance. 

4 The effectiveness of Babylonian computation did not result from their 
system of numeration alone. Mesopotamian mathematicians were skillful 


in developing algorithmic procedures, among which was a square-root 
process often ascribed to later men. It sometimes is attributed to the Greek 
scholar Archytas (428-365 B.C.) or to Heron of Alexandria (ca. 100) ; occasion- 
ally one finds it called Newton's algorithm. This Babylonian procedure is as 
simple as it is effective. Let x = N /a be the root desired and let a 1 be a first 
approximation to this root; let a second approximation b x be found from 
the equation b t = aja x . If a^ is too small, then b x is too large, and vice versa. 
Hence the arithmetic mean a 2 = ^^ + b x ) is a plausible next approximation. 
Inasmuch as a 2 always is too large, the next approximation b 2 = a/a 2 will 
be too small, and one takes the arithmetic mean a 3 = j(a 2 + b 2 ) to obtain 
a still better result ; the procedure can be continued indefinitely. The value of 
y/2 on Yale table 7289 will be found to be that of a 3 , where a t = 1 ;30. In 
the Babylonian square-root algorithm one finds an iterative procedure that 
could have put the mathematicians of the time in touch with infinite processes, 
but scholars of the time did not pursue the implications of such problems. 
The algorithm just described is equivalent to a two-term approximation 
to the bin omial series, a case with which the Babylonians were familiar. If 
sja 2 + b is desired, the approximation a x — a leads to b^ = (a 2 + b)/a and 
a 2 = (a l + bJ/2 = a + b/(2a), which is in agreement with the first two 
terms in the expansion of (a 2 + b) 1 ' 2 and provides an approximation found 
in Old Babylonian texts. Despite the efficacy of their rule for square roots, 
the Mesopotamian scribes seem to have imitated the modern applied 
mathematician in having frequent recourse to the ubiquitous tables that 
were available. In fact, a substantial proportion of the cuneiform tablets 
that have been unearthed are "table texts," including multiplication tables, 
tables of reciprocals, and tables of squares and cubes and of square and cube 
roots written, of course, in cuneiform sexagesimals. One of these, for example, 
carries the equivalents of the entries shown in the table below. The product 


















of elements in the same line is in all cases 60, the Babylonian number base, 
and the table apparently was thought of as a table of reciprocals. The sixth 
line, for example, denotes that the reciprocal of 8 is 7/60 + 30/(60) 2 . It will 
be noted that the reciprocals of 7 and 1 1 are missing from the table, because 
the reciprocals of such "irregular" numbers are nonterminating sexagesimals, 


just as in our decimal system the reciprocals of 3, 6, 7, and 9 are infinite when 
expanded decimally. Again the Babylonians were faced by the problem of 
infinity, but they did not consider it systematically. At one point, however, 
a Mesopotamian scribe seems to give upper and lower bounds for the recipro- 
cal of the irregular number 7, placing it between 0;8,34,16,59 and 0;8,34,18. 
With their penchant for multipositional computations, it is tantalizing not 
to find among them a recognition of the simple three-place periodicity in the 
sexagesimal representation of j, a discovery that could have provoked 
considerations of infinite series. 

It is clear that the fundamental arithmetic operations were handled by 
the Babylonians in a manner not unlike that which would be employed 
today, and with comparable facility. Division was not carried out by the 
clumsy duplication method of the Egyptians, but through an easy multiplica- 
tion of the dividend by the reciprocal of the divisor, using the appropriate 
items in the table texts. Just as today the quotient of 34 divided by 5 is easily 
found by multiplying 34 by 2 and shifting the decimal point, so in antiquity 
the same division problem was carried out by finding the product of 34 by 12 
and shifting one sexagesimal place to obtain 6f§. Tables of reciprocals in 
general furnished reciprocals of "regular" integers only — that is, those that 
can be written as products of twos, threes, and fives — although there are a 
few exceptions. One table text includes the approximations jg = ; 1,1,1 and 
i _ ; o,59,0,59. Here we have sexagesimal analogues of our decimal expres- 
sions 5- = .111 and yi = .0909, unit fractions in which the denominator is 
one more or one less than the base ; but it appears again that the Babylonians 
did not notice, or at least did not regard as significant, the infinite periodic 
expansions in this connection. 2 

One finds among the Old Babylonian tablets some table texts containing 
successive powers of a given number, analogous to our modern tables of 
logarithms, or, more properly speaking, of antilogarithms. Exponential (or 
logarithmic) tables have been found in which the first ten powers are listed 
for the bases 9 and 16 and 1,40 and 3,45 (all perfect squares). The question 
raised in a problem text, to what power must a certain number be raised in 
order to yield a given number, is equivalent to our question, what is the 
logarithm of the given number in a system with the certain number as base. 
The chief differences between the ancient tables and our own, apart from 
matters of language and notation, are that no single number was systemati- 
cally used as a base in varied connections and that the gaps between entries 
in the ancient tables are far larger than in our tables. Then, too, their "log- 
arithm tables" were not used for general purposes of calculation, but rather 
to solve certain very specific questions. 

2 In addition to the references cited in footnote 1, see also Kurt Vogel, Vorgriechische Mathe- 
matik, Vol. II, Die Mathematik der Babylonier (1959). 


Despite the large gaps in their exponential tables, Babylonian mathe- 
maticians did not hesitate to interpolate by proportional parts to approxi- 
mate intermediate values. Linear interpolation seems to have been a common- 
place procedure in ancient Mesopotamia, and the positional notation lent 
itself conveniently to the rule of three. A clear instance of the practical use 
of interpolation within exponential tables is seen in a problem text that asks 
how long it will take money to double at 20 per cent annually ; the answer 
given is 3; 47, 13,20. It seems to be quite clear that the scribe used linear 
interpolation between the values for (1 ; 12) 3 and (1 ; 12) 4 , following the com- 
pound interest formula a = P(l + r) n , where > is 20 per cent or £§, and 
reading values from an exponential table with powers of 1 ; 12. 

One table for which the Babylonians found considerable use is not 
generally included in handbooks of today. This is a tabulation of the values 
of n 3 + n 2 for integral values of n, a table essential in Babylonian algebra ; 
this subject reached a considerably higher level in Mesopotamia than in 
Egypt. Many problem texts from the Old Babylonian period show that the 
solution of the complete three-term quadratic equation afforded the Babylon- 
ians no serious difficulty, for flexible algebraic operations had been developed. 
They could transpose terms in an equation by adding equals to equals, and 
they could multiply both sides by like quantities to remove fractions or to 
eliminate factors. By adding Aab to (a - b) 2 they could obtain (a + b) 2 , for 
they were familiar with many simple forms of factoring. They did not use 
letters for unknown quantities, for the alphabet had not yet been invented, 
but words such as "length," "breadth," "area," and "volume" served 
effectively in this capacity. That these words may well have been used in a 
very abstract sense is suggested by the fact that the Babylonians had no 
qualms about adding a "length" to an "area" or an "area" to a "volume." 
Such problems, if taken literally, could have had no practical basis in 

Egyptian algebra had been much concerned with linear equations, but 
the Babylonians evidently found these too elementary for much attention. 
In one problem the weight x of a stone is called for if (x + x/7) + y^x + x/7) 
is one mina ; the answer is simply given as 48 ; 7,30 gin, where 60 gin make a 
mina. In another problem in an Old Babylonian text we find two simultane- 
ous linear equations in two unknown quantities, called respectively the 
"first silver ring" and the "second silver ring." If we call these x and y in 
our notation, the equations are x/7 + v/11 = 1 and 6x/7 = 10y/ll. The 
answer is expressed laconically in terms of the rule 

n n l y 7 l 

x/7 = 1 and — = 

7 + 11 72 11 7+11 72 


In another pair of equations part of the method of solution is included in the 
text. Here \ width + length = 7 hands, and length + width = 10 hands. 
The solution is first found by replacing each "hand" by 5 "fingers" and then 
noticing that a width of 20 fingers and a length of 30 fingers will satisfy 
both equations. Following this, however, the solution is found by an alterna- 
tive method equivalent to an elimination through combination. Expressing 
all dimensions in terms of hands, and letting the length and width be x and y 
respectively, the equations become y + 4x = 28 and x + y = 10. Subtracting 
the second equation from the first, one has the result 3x = 18; hence x = 6 
hands or 30 fingers and y = 20 fingers. 

The solution of a three-term quadratic equation seems to have exceeded 
by far the algebraic capabilities of the Egyptians, but Neugebauer in 1930 
disclosed that such equations had been handled effectively by the Babylonians 
in some of the oldest problem texts. For instance, one problem calls for the 
side of a square if the area less the side is 14,30. The solution of this problem, 
equivalent to solving x 2 - x = 870, is expressed as follows : 

Take half of 1, which is 0;30, and multiply 0;30 by 0;30, which is 0; 15; add 
this to 14,30 to get 14,30; 15. This is the square of 29; 30. Now add 0;30 to 
29 ;30, and the result is 30, the side of the square. 
The Babylonian solution is, of course, exactly equivalent to the formula 
x = ^/(p/2) 2 + q + p/2 for a root of the equation x 2 - px = q— the 
quadratic formula that is familiar to schoolboys of today. In another text 
the equation llx 2 + 7x = 6; 15 was reduced by the Babylonians to the 
standard type x 2 + px = q by first multiplying through by 11 to obtain 
(llx) 2 + 7(1 lx) = 1,8 ;45. This is a quadratic in normal form in the unknown 
quantity y = llx, and the solution for y is easily obtained by the familiar 
rule y = J (p/2) 2 + q - p/2, from which the value of x is then determined. 
This solution is remarkable as an instance of the use of algebraic transforma- 

Until modern times there was no thought of solving a quadratic equation 
of the form x 2 + px + q = 0, where p and q are positive, for the equation 
has no positive root. Consequently, quadratic equations in ancient and 
Medieval times — and even in the early modern period — were classified 
under three types : 

(1) x 2 + px = q 

(2) x 2 = px + q 

(3) x 2 + q = px 

All three types are found in Old Babylonian texts of some 4000 years ago. 


The first two types are illustrated by the problems given above; the third 
type appears frequently in problem texts, where it is treated as equivalent to 
the simultaneous system x + y = p, xy = q. So numerous are problems in 
which one is asked to find two numbers when given their product and either 
their sum or their difference that these seem to have constituted for the 
ancients, both Babylonian and Greek, a sort of "normal form" to which 
quadratics were reduced. Then by transforming the simultaneous equations 
xy = a an d x + y = b into the pair of linear equations x + y = b and 
x + y = ^/F + 4a, the values of x and y are found through an addition 
and a subtraction. A Yale cuneiform tablet, for example, asks for the solution 
of the system x + y = 6; 30 and xy = 7; 30. The instructions of the scribe 
are essentially as follows. First find 

x + y 
—^= 3;15 

and then find 


x + y 

x + y 


= 10:33,45 


- xy = 3;3,45 




x + y 


- xy = 1 ;45 

x + y\ x - y\ 

1 + hr 2 =3;15 + 1;45 

= 3; 15 - 1;45 

From the last two equations it is obvious that x = 5 and y = \{. Because 
the quantities x and y enter symmetrically in the given conditional equations, 
it is possible to interpret the values of x and y as the two roots of the quadratic 
equation x 2 + 7; 30 = 6;30x. Another Babylonian text calls for a number 
which when added to its reciprocal becomes 2; 0,0,33,20. This leads to a 
quadratic of type 3, and again we have two solutions, 1 ; 0,45 and ; 59,1 5,33,20. 


The Babylonian reduction of a quadratic equation of the form ax 2 + bx = c 
to the normal form y 2 + by = ac through the substitution y = ax shows 
the extraordinary degree of flexibility in Mesopotamian algebra. This 
facility, coupled with the place-value idea in computation, accounts in large 
measure for the superiority of the Babylonians in mathematics. There is no 
record in Egypt of the solution of a cubic equation, but among the Babylon- 
ians there are many instances of this. Pure cubics, such as x i = 0;7,30, were 
solved by direct reference to tables of cubes and cube roots, where the 
solution x = ; 30 was read off. Linear interpolation within the tables was 
used to find approximations for values not listed in the tables. Mixed cubics 
in the standard form x 3 + x 2 = a were solved similarly by reference to the 
available tables which listed values of the combination n 3 + n 2 for integral 
values of n from 1 to 30. With the help of these tables they read off easily 
that the solution, for example, of x 3 + x 2 = 4,12 is equal to 6. For 
still more general cases of equations of third degree, such as 144x 3 + 12x 2 = 
21, the Babylonians used their method of substitution. Multiplying both 
sides by 12 and using y = 12x, the equation becomes y 3 + y 2 = 4,12, from 
which y is found to be equal to 6, hence x is just \ or 0;30. Cubics 
of the form ax 3 + bx 2 = c are reducible to the Babylonian normal form by 
multiplying through by a 2 /b 3 to obtain {ax/b} 3 + (ax/b) 2 = ca 2 /b 3 , a cubic 
of standard type in the unknown quantity ax/b. Reading off from the tables 
the value of this unknown quantity, the value of x is determined. Whether or 
not the Babylonians were able to reduce the general four-term cubic, 
ax 3 + bx 2 + ex = d, to their normal form is not known. That it is not too 
unlikely that they could reduce it is indicated by the fact that a solution of a 
quadratic suffices to carry the four-term equation to the three-term form 
px 3 + qx 2 = r, from which, as we have seen, the normal form is readily 
obtained. There is, however, no evidence now available that would suggest 
that the Mesopotamian mathematicians actually carried out such a reduction 
of the general cubic equation. 

The solution of quadratic and cubic equations in Mesopotamia is a 
remarkable achievement to be admired not so much for the high level of 
technical skill as for the maturity and flexibility of the algebraic concepts 
that are involved. With modern symbolism it is a simple matter to see that 
(ax) 3 + (ax) 2 = b is essentially the same type of equation as y 3 + y 2 = b ; 
but to recognize this without our notation is an achievement of far greater 
significance for the development of mathematics than even the vaunted 
positional principle in arithmetic that we owe to the same civilization. 
Babylonian algebra had reached such an extraordinary level of abstraction 
that the equations ax 4 + bx 2 = c and ax 8 + bx A = c were recognized as 
nothing worse than quadratic equations in disguise — that is, quadratics in 
x 2 and x 4 . 


The algebraic achievements of the Babylonians are admirable, but the 8 
motives behind this work are not easy to understand. It commonly has been 
supposed that virtually all pre-Hellenic science and mathematics were purely 
utilitarian ; but what sort of real-life situation in ancient Babylon could 
possibly lead to problems involving the sum of a number and its reciprocal 
or a difference between an area and a length? If utility was the motive, then 
the cult of immediacy was less strong than it is now, for direct connections 
between purpose and practice in Babylonian mathematics are far from 
apparent. That there may well have been toleration for, if not encouragement 
of, mathematics for its own sake is suggested by a tablet (No. 322) in the 
Plimpton Collection at Columbia University. 3 The tablet dates from the 
Old Babylonian period (ca. 1900 to 1600 B.C.), and the tabulations it contains 
could easily be mistaken for a record of business accounts. Analysis, however, 
shows that it has deep mathematical significance in the theory of numbers 
and that it was perhaps related to a kind of proto trigonometry. Plimpton 322 
was part of a larger tablet, as is illustrated by the break along the left-hand 
edge, and the remaining portion contains four columns of numbers arranged 
in fifteen horizontal rows. The right-hand column contains the digits from 
one to fifteen, and its purpose evidently was simply to identify in order the 
items in the other three columns, arranged as follows. 

1,59,0,15 1,59 2,49 1 

1,56,56,58,14,50,6,15 56,7 1,20,25 2 

1,55,7,41,15,33,45 1,16,41 1,50,49 3 

1,53,10,29,32,52,16 3,31,49 5,91 4 

1,48,54,1,40 1,5 1,37 5 

1,47,6,41,40 5,19 8,1 6 

1,43,11,56,28,26,40 38,11 59,1 7 

1,41,33,59,3,45 13,19 20^49 8 

1,38,33,36,36 8,1 12,49 9 

1,35,10,2,28,27,24,26,40 1,22,41 2,16,1 10 

1 > 33 >45 45,0 1,15,0 11 

1,29,21,54,2,15 27,59 48,49 12 

1,27,0,3,45 2,41 4,49 13 

1,25,48,51,35,6,40 29,31 53,49 14 

1,23,13,46,40 56 1,46 15 

The tablet is not in such excellent condition that all the numbers can still 
be read, but the clearly discernible pattern of construction in the table made 
it possible to determine from context the few items that were missing because 

3 Further description of this table will be found in Neugebauer, Exact Sciences in Antiquity, 
pp. 36-40. A good account of it appears also in Howard Eves, An Introduction to the History 
oj Mathematics (1964), pp. 35-37. A scholarly interpretation of the possible motivation behind 
the table text is given by D. J. de Solla Price, "The Babylonian 'Pythagorean Triangle' Tablet " 
Centaurus, 10 (1964), 219-231. 



of small fractures. To understand what the entries in the table probably 
meant to the Babylonians, consider the right triangle ABC (Fig. 3.1). If the 
numbers in the second and third columns (from left to right) are thought of 
as the sides a and c respectively of the right triangle, then the first or left-hand 
column contains in each case the square of the ratio of c to b. The left-hand 
column therefore is a short table of values of sec 2 A, but we must not assume 
that the Babylonians were familiar with our secant concept. Neither the 
Egyptians nor the Babylonians introduced a measure of angles in the modern 
sense. Nevertheless, the rows of numbers in Plimpton 322 are not arranged 
in haphazard fashion, as a superficial glance might imply. If the first comma 

6 C 

FIG. 3.1 

in column one (on the left) is replaced by a semicolon, it is obvious that the 
numbers in this column decrease steadily from top to bottom. Moreover, 
the first number is quite close to sec 2 45°, and the last number in the column 
is approximately sec' 2 31°, with the intervening numbers close to the values 
of sec 2 A as A decreases by degrees from 45° to 31°. This arrangement ob- 
viously is not the result of chance alone. Not only was the arrangement 
carefully thought out, but the dimensions of the triangle were also derived 
according to a rule. Those who constructed the table evidently began with 
two regular sexagesimal integers, which we shall call p and q, with p > q, 
and then formed the triple of numbers p 2 - q 2 and 2pq and p 2 + q 2 . The 
three integers thus obtained are easily seen to form a Pythagorean triple in 
which the square of the largest is equal to the sum of the squares of the other 



two. Hence these numbers can be used as the dimensions of the right triangle 
ABC, with a = p 2 - q 2 and b = 2pq and c = p 3 + </ 2 , Restricting themselves 
to values of p less than 60 and to corresponding values of q such that 
\ < p/q < \ + ^fl — that is, to right triangles for which a < b -the Babylon- 
ians presumably found that there were just 38 possible pairs or values of 
p and q satisfying the conditions, and for these they apparently formed the 
38 corresponding Pythagorean triples. Only the first 15, arranged in descend- 
ing order for the ratio {p 2 + q 2 )/2pq, are included in the table on the tablet, 
but it is likely that the scribe had intended to continue the table on the other 
side of the tablet. It has been suggested also 4 that the portion of Plimpton 322 
that has been broken off' from the left side contained four additional columns 
in which were tabulated the values of p and q and 2pq and what we should 
now call tan 2 A. 

The Plimpton tablet 322 might give the impression that it is an exercise 
in the theory of numbers, but it is likely that this aspect of the subject was 
merely ancillary to the problem of measuring the areas of squares on the 
sides of a right triangle. The Babylonians disliked working with the recipro- 
cals of irregular numbers, for these could not be expressed exactly in finite 

Plimpton 322. 

"See the explanation given by Price (whose suggestion we have here been following) in the 
article cited in footnote 3, 


sexagesimal fractions. Hence they were interested in values of p and q that 
should give them regular integers for the sides of right triangles of varying 
shape, from the isosceles right triangle down to one with a small value for 
the ratio a/b. For example, the numbers in the first row are found by starting 
with p = 12 and q = 5, with the corresponding values a = 119 and b = 120 
and c = 169. The values of a and c are precisely those in the second and third 
positions from the left in the first row on the Plimpton tablet; the ratio 
c 2 /b 2 = 28561/14400 is the number 1; 59,0,15 that appears in the first 
position in this row. 5 The same relationship is found in the other fourteen 
rows; the Babylonians carried out the work so accurately that the ratio 
c 2 /b 2 in the tenth row is expressed as a fraction with eight sexagesimal 
places, equivalent to about fourteen decimal places in our notation. 

So much of Babylonian mathematics is bound up with tables of reciprocals 
that it is not surprising to find that the items in Plimpton 322 are related to 
reciprocal relationships. If a = 1, then 1 = (c + b)(c - b), so that c + b 
and c - b are reciprocals. If one starts with c + b = n, where n is any regular 
sexagesimal, then c - b = 1/n ; hence a = 1 and b = ^n - l/ri) and 
and c = j(n + 1/n) are a Pythagorean fraction triple which can easily be 
converted to a Pythagorean integer triple by multiplying each of the three 
by In. All triples in the Plimpton tablet are easily calculated by this device. 

The account of Babylonian algebra that we have given is representative 
of their work, but it is not intended to be exhaustive. There are in the Babylon- 
ian tablets many other things, although none so striking as those in the 
Plimpton tablet 322. For instance, in one tablet the geometrical progression 
1 + 2 + 2 2 + • • ■ + 2 9 is summed and in another the sum of the series of 
squares l 2 + 2 2 + 3 2 + • • • + 10 2 is found. One wonders if the Babylonians 
knew the general formulas for the sum of a geometrical progression and 
the sum of the first n perfect squares. It is quite possible that they did, and it 
has been conjectured that they were aware that the sum of the first n perfect 
cubes is equal to the square of the sum of the first n integers. 6 Nevertheless, 
it must be borne in mind that tablets from Mesopotamia resemble Egyptian 
papyri in that only specific cases are given, with no general formulations. 

A few years ago it used to be held that the Babylonians were better in 
algebra than were the Egyptians, but that they had contributed less to 
geometry. The first half of this statement is clearly substantiated by what 
we have learned above ; attempts to bolster the second half of the comparison 

5 Vogel, in Vorgriechische Mathematik, II, 37-41, interprets this number, and also the others 
in this column, as a 2 /fc 2 rather than as c 2 /b 2 — that is, as tan 2 A rather than sec 2 A. The difference 
between these functions is always one, and the unit wedges in the left-hand column in Plimpton 
322 have in most cases been broken away; but careful inspection of this edge seems to substan- 
tiate the interpretation of the column as squares of secants rather than of tangents. 

6 See Archibald, Outline of the History of Mathematics, p. 11. 


generally are limited to the measure of the circle or to the volume of the 
frustrum of a pyramid. 7 In the Mesopotamian valley the area of a circle was 
generally found by taking three times the square of the radius, and in accuracy 
this falls considerably below the Egyptian measure. However, the counting 
of decimal places in the approximations for n is scarcely an appropriate 
measure of the geometrical stature of a civilization, and a recent discovery 
has effectively nullified even this weak argument. In 1936 a group of mathe- 
matical tables were unearthed at Susa, a couple of hundred miles from 
Babylon, and these include significant geometrical results. True to the 
Mesopotamian penchant for making tables and lists, one tablet in the Susa 
group compares the areas and the squares of the sides of the regular polygons 
of three, four, five, six, and seven sides. The ratio of the area of the pentagon, 
for example, to the square on the side of the pentagon is given as 1 ;40, a 
value that is correct to two significant figures. For the hexagon and heptagon 
the ratios are expressed as 2;37,30 and 3; 41 respectively. In the same tablet 
the scribe gives ; 57,36 as the ratio of the perimeter of the regular hexagon 
to the circumference of the circumscribed circle; and from this we can 
readily conclude 8 that the Babylonian scribe had adopted 3; 7,30 or 3|, as 
an approximation for n. This is at least as good as the value adopted in 
Egypt. Moreover, we see it in a more sophisticated context than in Egypt, 
for the tablet from Susa is a good example of the systematic comparison of 
geometric figures. One is almost tempted to see in it the genuine origin of 
geometry, but it is important to note that it was not so much the geometrical 
context that interested the Babylonians as the numerical approximations 
that they used in mensuration. Geometry for them was not a mathematical 
discipline in our sense, but a sort of applied algebra or arithmetic in which 
numbers are attached to figures. 

There is some disagreement as to whether or not the Babylonians were 
familiar with the concept of similar figures, although this appears to be quite 
likely. The similarity of all circles seems to have been taken for granted in 
Mesopotamia, as it had been in Egypt, and the many problems on triangle 
measure in cuneiform tablets seem to imply a concept of similarity. A tablet 
in the Baghdad Museum has a right triangle ABC (Fig. 3.2) with sides a = 60 
and b = 45 and c = 75, and it is subdivided into four smaller right triangles 
ACD, CDE, DEF, and EFB. The areas of these four triangles are then given 
as 8,6 and 5,11; 2,24 and 3,19; 3,56,9,36 and 5,53 ;53,39,50,24 respectively. 
From these values the scribe computed the length of AD as 27, apparently 
using a sort of "similarity formula" equivalent to our theorem that areas 
of similar figures are to each other as squares on corresponding sides. The 

7 See, for example, George Sarton, A History of Science, Vol. I (Cambridge, Mass. : Harvard 
University Press, 1952), pp. 73-74. 

8 See Neugebauer, Exact Sciences in Antiquity (2), p. 47. 



FIG. 3.2 

lengths of CD and BD are found to be 36 and 48 respectively, and through 
an application of the "similarity formula" to triangles BCD and DCE the 
length of CE is found 9 to be 21 ; 36. The text breaks off in the middle of the 
calculation of DE. 

1 Measurement was the keynote of algebraic geometry in the Mesopotamian 
valley, but a major flaw, as in Egyptian geometry, was that the distinction 
between exact and approximate measures was not made clear. The area of a 
quadrilateral was found by taking the product of the arithmetic means of 
the pairs of opposite sides, with no warning that this is in most cases only a 
crude approximation. Again, the volume of a frustum of a cone or pyramid 
sometimes was found by taking the arithmetic mean of the upper and lower 
bases and multiplying by the height; sometimes, for a frustum of a square 
pyramid with areas a 2 and b 2 for the lower and upper bases, the formula 

was applied. However, for the latter the Babylonians used also a rule equiva- 
lent to r , , \ 2 w . \ 2 - 

a formula that is correct and reduces to the one known to the Egyptians. 

It is not known whether Egyptian and Babylonian results were always 
independently discovered, but in any case the latter were definitely more 
extensive than the former, both in geometry and algebra. The Pythagorean 
theorem, for example, does not appear in any form in surviving documents 
from Egypt, but tablets even from the Old Babylonian period show that in 
Mesopotamia the theorem was widely used. A cuneiform text from the Yale 

9 See Vogel, Vorgriechische Mathematik, II, 78-79. 


Collection, for example, contains a diagram of a square and its diagonals 
in which the number 30 is written along one side and the numbers 42 ; 25,35 
and 1 ;24,51,10 appear along a diagonal. The last number obviously is the 
ratio of the lengths of the diagonal and a side, and this is so accurately 
expressed that it agrees with y/l to within about a millionth. The accuracy 
of the result was made possible by knowledge of the Pythagorean theorem. 
Sometimes, in less precise computations, the Babylonians used 1; 25 as a 
rough-and-ready approximation to this ratio. Of more significance than the 
precision of the values, however, is the implication that the diagonal of any 
square could be found by multiplying the side by ^fl. Thus there seems to 
have been some awareness of general principles, despite the fact that these 
are exclusively expressed in special cases. 

Babylonian recognition of the Pythagorean theorem was by no means 
limited to the case of a right isosceles triangle. In one Old Babylonian problem 
text a ladder or beam of length ; 30 stands against a wall ; the question is, 
how far will the lower end move out from the wall if the upper end slips down 
a distance of 0;6 units? The answer is correctly found by use of the Pythag- 
orean theorem. Fifteen hundred years later similar problems, some with new 
twists, were still being solved in the Mesopotamian valley. A Seleucid tablet, 
for example, proposes the following problem. A reed stands against a wall. 
If the top slides down three units when the lower end slides away nine units, 
how long is the reed? The answer is given correctly as fifteen units. 

Ancient cuneiform problem texts provide a wealth of exercises in what we 
might call geometry, but which the Babylonians probably thought of as 
applied arithmetic. A typical inheritance problem calls for the partition of a 
right triangular property among six brothers. The area is given as 11,22,30 
and one of the sides is 6,30 ; the dividing lines are to be equidistant and parallel 
to the other side of the triangle. One is asked to find the difference in the 
allotments. Another text gives the bases of an isosceles trapezoid as 50 and 
40 units and the length of the sides as 30 ; the altitude and area are required. 10 

The ancient Babylonians were aware of other important geometrical 
relationships. Like the Egyptians, they knew that the altitude in an isosceles 
triangle bisects the base. Hence, given the length of a chord in a circle of 
known radius, they were able to find the apothem. Unlike the Egyptians, 
they were familiar with the fact than an angle inscribed in a semicircle is a 
right angle, a proposition generally known as the Theorem of Thales, despite 
the fact that Thales lived well over a millennium after the Babylonians had 
begun to use it. This misnaming of a well-known theorem in geometry is 
symptomatic of the difficulty in assessing the influence of pre-Hellenic 
mathematics on later cultures. Cuneiform tablets had a permanence that 

10 These and other problems are found in van der Waerden, Science Awakening, pp. 76-77. 


could not be matched by documents from other civilizations, for papyrus 
and parchment do not so easily survive the ravages of time. Moreover, 
cuneiform texts continued to be recorded down to the dawn of the Christian 
era ; but were they read by neighboring civilizations, especially the Greeks? 
The center of mathematical development was shifting from the Mesopota- 
mian valley to the Greek world half a dozen centuries before the beginning of 
our era, but reconstructions of early Greek mathematics are rendered 
hazardous by the fact that there are virtually no extant mathematical docu- 
ments from the pre-Hellenistic period. It is important, therefore, to keep 
in mind the general characteristics of Egyptian and Babylonian mathematics 
so as to be able to make at least plausible conjectures concerning analogies 
that may be apparent between pre-Hellenic contributions and the activities 
and attitudes of later peoples. 

11 A number of deficiencies in pre-Hellenic mathematics are quite obvious. 
Extant papyri and tablets contain specific cases and problems only, with no 
general formulations, and one may question whether these early civilizations 
really appreciated the unifying principles that are at the core of mathematics. 
Further study is somewhat reassuring, for the hundreds of problems of 
similar types in cuneiform tablets seem to be exercises that schoolboys were 
expected to work out in accordance with certain recognized methods or 
rules. That there are no surviving statements of these rules does not necessarily 
mean that the generality of the rules or principles was missing in ancient 
thought. Were a rule not there in essence, the similarity of the problems 
would be difficult to explain. Such large collections of similar problems could 
not have been the result of chance. 

More serious, perhaps, than the lack of explicit statements of rules is the 
absence of clear-cut distinctions between exact and approximate results. 
The omission in the tables of cases involving irregular sexagesimals seems 
to imply some recognition of such distinctions, but neither the Egyptians 
nor the Babylonians seem to have raised the question of when the area of a 
quadrilateral (or of a circle) is found exactly and when only approximately. 
Questions about the solvability or unsolvability of a problem do not seem 
to have been raised ; nor was there any investigation into the nature of proof. 
The word "proof" means various things at different levels and ages ; hence it 
is hazardous to assert categorically that pre-Hellenic peoples had no concept 
of proof, nor any feeling of the need for proof. There are hints that these 
people occasionally were aware that certain area and volume methods could 
be justified through a reduction to simpler area and volume problems. 
Moreover, pre-Hellenic scribes not infrequently checked or "proved" their 
divisions by multiplication; occasionally they verified the procedure in a 
problem through a substitution that verified the correctness of the answer. 


Nevertheless, there are no explicit statements from the pre-Hellenic period 
that would indicate a felt need for proofs or a concern for questions of logical 
principles. The lack of such statements often has led to judgments that pre- 
Hellenic civilizations had no true mathematics, despite the obviously high 
level of technical facility. 

Critics also point to what they regard as an absence of abstraction in 
Egyptian and Babylonian mathematics. The language of the documents does 
seem always to remain close to concrete cases, as we have seen ; but this, 
too, can be misleading. In Mesopotamian problems the words "length" and 
"width" should perhaps be interpreted much as we interpret the letters 
x and v, for the writers of cuneiform tablets may well have moved on from 
specific instances to general abstractions. How else does one explain the addi- 
tion of a length to an area? In Egypt also the use of the word for quantity is not 
incompatible with an abstract interpretation such as we read into it today. 

Evaluations of pre-Hellenic civilizations frequently point to the fact that 
there was no clearly discernible intellectual activity of a characteristically 
unified sort comparable to that which later carried the label "mathematics" ; 
but here, too, it is easy to be excessively dogmatic. It may be true that geometry 
had not yet been crystallized out of a crude matrix of space experience that 
included all sorts of things that could be measured ; but it is difficult not to see 
in Babylonian and Egyptian concern with number and its applications some- 
thing very close to what usually, in ages since, has been known as algebra. 

Pre-Hellenic cultures have been stigmatized also as entirely utilitarian, 
with little or no interest in mathematics for its own sake. Here, too, a matter 
of judgment, rather than of incontrovertible evidence, is involved. Then, as 
now, the vast majority of mankind were preoccupied with immediate 
problems of survival. Leisure was far scarcer than it is now, but even under 
this handicap there were in Egypt and Babylonia problems that have the 
earmarks of recreational mathematics. If a problem calls for a sum of cats 
and measures of grain, or of a length and an area, one cannot deny to the 
perpetrator either a modicum of levity or a feeling for abstraction. Of course 
much of pre-Hellenic mathematics was practical, but surely not all of it. 
The truth probably lies somewhere between extremes recently published by 
two historians of mathematics. One of them 11 claims that Babylonian 
mathematics was directed solely toward practical ends ; the other has upheld 
the diametrically opposite view that "Sumerian mathematics was not used 
for the solution of problems in practical life, but only for enjoyment or for 
exultation of the spirit." 12 A cautious reader may safely assume that neither 

"M. Cipolla, Storia delta matematica dai primordia a Leibniz (Mozara: Societa editrice 
siciliana, 1949), p. 23. 

12 Quoted from Ettore Bortolotti by Ettore Carruccio in his Mathematics and Logic in 
History and in Contemporary Thought (Chicago : Aldine, 1964), p. 15. 


of these extreme positions can be held with impunity. In the practice of 
computation, which stretched over a couple of millennia, the schools of 
scribes used plenty of exercise material, often, perhaps, just as good clean 


Archibald, R. C, Outline of the History of Mathematics, 6th ed. (Herbert Ellsworth 
Slaught Memorial Papers, No. 2, Buffalo, N.Y. : The Mathematical Association of 
America, 1949). 

Bruins, E. M., and M. Rutten, Textes mathematiques de Suse (Pans, 1961). 

Eves, Howard, An Introduction to the History of Mathematics, 2nd ed. (New York : 

Holt, 1964). 
Kugler, F. X., Sternkunst und Sterndienst in Babel (Miinster in Westphalia : Aschendoett, 

1907-1935, 2 vols, and 3 supps.). 
Neugebauer, O., The Exact Sciences in Antiquity, 2nd ed. (Providence R. I. : Brown 

University Press, 1957 ; paperback ed., New York : Harper). 
Neubebauer, O., Mathematische Keilschift-Texte (Berlin: Springer, 1935-1937, 3 vols.). 

This is Vol. II of Quellen und Studien zur Geschichte der Mathematik, Astronomie 

und Physik, Part A, Quellen. See also numerous articles by Neugebauer and others 

in Quellen und Studien, Part B, Studien, I-IV (1928-1938). 
Neugebauer, O., Vorgriechische Mathematik (Berlin: Springer, 1934). 
Neugebauer, O., and A. Sachs, Mathematical Cuneiform Texts (New Haven, Conn. : 

Yale University Press, 1945). 
Thureau-Dangin, F., Textes mathematiques Babyloniens (Leiden : Brill, 1938). 
Van der Waerden, B. L., Science Awakening, trans, by Arnold Dresden (New York : 

Oxford University Press, 1961 ; paperback ed., New York : Wiley, 1963). 
Vogel, Kurt, Vorgriechische Mathematik, Vol. II, Die Mathematik der Babylonier 

(paperback ed., Hannover : Hermann Schroedel, ca. 1959). 


1. What do you regard as the four most significant contributions of the Mesopotamians to 
mathematics? Justify your answer. 

2. What do you regard as the four chief weaknesses of Mesopotamian mathematics? Justify 
your answer. 

3. Compare, as to significance and possible influence on later civilizations, the geometry and 
trigonometry of the Babylonians with that of the Egyptians. 

4. Describe the relative advantages and disadvantages of the number notations of the Babylon- 
ians and the Egyptians. 

5. Write the number 10,000 in Babylonian notation. 
6 Write the number 0.0862 in Babylonian notation. 

7. Use the Babylonian algorithm for square roots to find the square root of two to half a dozen 
decimal places and compare with the Babylonian value 1 ;24,51,10. 

8. Verify that if (c/a) 2 is 1 ;33,45 and b = 45 and c = 1 , 1 5, then a, b, c form a Pythagorean triad. 


9. Verify that the parameters p = 9 and q = 4 lead to the values in line 5 of the Plimpton 
Tablet 322. 

10. Show that ifp and q are positive numbers such that p 2 - q 2 < 2p</,then 1 < p/q < 1 + ^/l. 

11. How closely does the Babylonian approximation 3;41 agree with the correct value for the 
ratio of the area of the regular heptagon to the square of a side? 

12. The Babylonians estimated the ratio of the area of a regular hexagon to the square of one 
side as 2; 37,30. How does this compare with the correct ratio? 

13. Solve the following Old Babylonian problem: The area of two squares together is 1000, 
and the side of one square is 1 less than two thirds of the side of the other square. Find the 
sides of the two squares. 

14. Find as a sexagesimal fraction to the nearest minute the ratio of the area of a regular pentagon 
to the square of a side, and compare your answer with the value 1 ;40 given by the Babylon- 

15. Solve the following Old Babylonian problem: One side of a right triangular property is 
50 units long. Parallel to the other side and 20 units from the other side a line is drawn 
cutting off a right trapezoidal area of 5,20 units. Find the lengths of the parallel sides of this 

16. Verify the result of an Old Babylonian computation in which the area of an isosceles trape- 
zoid whose sides are 30 units and whose bases are 14 and 50 is given as 12,48. 

17. Solve the following Old Babylonian problem: Ten brothers receive 1;40 minas of silver, 
and brother over brother received a constant difference. If the eighth brother received 
6 shekels, find how much each earned. (There are 60 shekels in a mina.) 

18. Find the length of the ladder in the problem described in the text. 

19. Solve the problem of the six brothers described in the text. 

20. An Old Babylonian tablet unearthed at Susa asks for the radius of the circle circumscribing 
a triangle whose sides are 50, 50, and 60. Solve this problem. 

21. Show that the sexagesimal representation of j has a three-place periodicity. How many 
places are there in the periodicity of the decimal representation? 

*22. Another tablet from Susa calls for the sides x and y of a rectangle if xy = 20,0 and x 3 d = 
14,48,53,20, where d is the length of a diagonal. Solve this problem. 

Chapter IV 

Ionia and 

the Pythagoreans 

To Thales ... the primary question was not What do we 
know, but How do we know it. 


The intellectual activity of the Potamic civilizations in Egypt and Mesopo- 
tamia had lost its verve well before the Christian era; but as learning in 
the river valleys was declining, and as bronze was giving way to iron in 
weaponry, vigorous new cultures were springing up all along the shores of 
the Mediterranean Sea. To indicate this change in the centers of civilization, 
the interval from roughly 800 B.C. to a.d. 800 sometimes is known as the 
Thalassic Age (that is, the "sea" age). There was, of course, no sharp disrup- 
tion to mark the transition in intellectual leadership from the valleys of the 
Nile, Tigris, and Euphrates rivers to the shores of the Mediterranean, for 
time and history flow continuously, and changing conditions are associated 
with antecedent causes. Egyptian and Babylonian scholars continued to 
produce papyrus and cuneiform texts for many centuries after 800 B.C. ; but 
a new civilization meanwhile was rapidly preparing to take over scholarly 
hegemony, not only around the Mediterranean but, ultimately, in the chief 
river valleys as well. To indicate the source of the new inspiration, the first 
portion of the Thalassic Age is labeled the Hellenic era, so that the older 
cultures are consequently known as pre-Hellenic. 

The Greeks of today still call themselves Hellenes, continuing a name used 
by their early forebears who settled along the coasts of the Mediterranean 
Sea. Greek history is traceable back into the second millennium B.C. when, 
as unlettered invaders, they pressed down relentlessly from the north. They 
brought with them no mathematical or literary tradition ; they seem to have 
been very eager to learn, however, and it did not take them long to improve 
on what they were taught. For example, they took over, perhaps from the 
Phoenicians, an existing alphabet, consisting only of consonants, and to it they 
added vowels. The alphabet seems to have originated between the Babylonian 
and Egyptian worlds, possibly in the region of the Sinai Peninsula, through a 
process of drastic reduction in the number of cuneiform or hieratic symbols. 



This alphabet found its way to the new colonies — Greek, Roman, and 
Carthaginian — through the activities of traders. It is presumed that some 
rudiments of computation traveled along the same routes, but the more 
esoteric portions of priestly mathematics may have remained undiffused. 
Before long, however, Greek traders, businessmen, and scholars made their 
way to the centers of learning in Egypt and Babylonia. There they made 
contact with pre-Hellenic mathematics; but they were not willing merely 
to receive the long-established traditions, for they made the subject so 
thoroughly their own that it shortly took a drastically different form. 

The first Olympic Games were held in 776 B.C., and by that time a wonderful 
Greek literature already had developed, evidenced by the works of Homer 
and Hesiod. Of Greek mathematics at the time we know nothing. Presum- 
ably it lagged behind the development of literary forms, for the latter lend 
themselves more readily to continuity of oral transmission. It was to be 
almost another two centuries before there was any word, even indirectly, 
concerning Greek mathematics. Then, during the sixth century B.C., there 
appeared two men, Thales and Pythagoras, who seem to have played in 
mathematics a role similar to that of Homer and Hesiod in literature. Most 
of what is reported in this chapter centers on Thales and Pythagoras, but a 
note of warning is in order. Homer and Hesiod are somewhat shadowy 
figures, but at least we have a consistent tradition attributing to them certain 
literary masterpieces which, first transmitted orally from generation to 
generation, ultimately were copied down and preserved for posterity. Thales 
and Pythagoras also are somewhat indistinct figures, historically, although 
less so than Homer and Hesiod ; but as far as their scholarly work is con- 
cerned, the parallel with Homer and Hesiod ceases. No mathematical master- 
piece from either one has survived, nor is it even established that either Thales 
or Pythagoras ever composed such a work. What they may have done must 
be reconstructed on the basis of a none too trustworthy tradition that grew 
up around these two early mathematicians. Certain key phrases are attributed 
to them — such as "Know thyself," in the case of Thales, and "All is number," 
in the case of Pythagoras — but not much more of a specific nature. Neverthe- 
less, the earliest Greek accounts of the history of mathematics, which no 
longer survive, ascribed to Thales and Pythagoras a number of very definite 
discoveries in mathematics. We outline these contributions in this chapter, 
but the reader should understand that it is largely persistent tradition, rather 
than any extant historical document, on which the account is based. 

The Greek world for many centuries had its center between the Aegean and 
Ionian Seas, but Hellenic civilization was far from localized there. Greek 
settlements by about 600 B.C. were to be found scattered along the borders 
of most of the Black Sea and the Mediterranean Sea, and it was on these 
outskirts that a new surge in mathematics developed. In this respect the 


sea-bordering colonists, especially in Ionia, had two advantages : they had 
the bold and imaginative spirit typical of pioneers, and they were in closer 
proximity to the two chief river valleys from which knowledge could be 
derived. Thales of Miletus (ca. 624^548 B.C.) and Pythagoras of Samos 
(ca. 580-500 B.C.) had in addition a further advantage : they were in a position 
to travel to centers of ancient learning and there acquire firsthand information 
on astronomy and mathematics. In Egypt they are said to have learned 
geometry ; in Babylon, under the enlightened Chaldean ruler Nebuchadnez- 
zar, Thales probably came in touch with astronomical tables and instruments. 
Tradition has it that in 585 B.C. Thales amazed his countrymen by predicting 
the solar eclipse of that year. The historicity of this tradition is very much 
open to question, especially because an eclipse of the sun is visible over only 
a very small portion of the earth's surface, and it does not seem likely that 
there were in Babylon tables of solar eclipses that would have enabled Thales 
to make such a prediction. It is quite likely, on the other hand, that the 
gnomon or sundial entered Greece from Babylon, and perhaps the water 
clock came from Egypt. The Greeks were far from hesitant in taking over 
elements of foreign cultures, else they would never have learned so quickly 
how to advance beyond their predecessors ; but everything they touched, they 

What is really known about the life and work of Thales is very little indeed. 
His birth and death are estimated from the fact that the eclipse of 585 B.C. 
probably occurred when he was in his prime, say about forty, and that he was 
said to have been seventy-eight when he died. However, serious doubts about 
the authenticity of the eclipse story make such extrapolations hazardous, 
and they shake our confidence concerning the discoveries fathered upon 
Thales. Ancient opinion is unanimous in regarding Thales as an unusually 
clever man and the first philosopher — by general agreement the first of the 
Seven Wise Men. He was regarded as "a pupil of the Egyptians and the 
Chaldeans," an assumption that appears plausible. The proposition now 
known as the Theorem of Thales — that an angle inscribed in a semicircle 
is a right angle — may well have been learned by Thales during his travels to 
Babylon. However, tradition goes further and attributes to him some sort of 
demonstration of the theorem. For this reason Thales frequently has been 
hailed as the first true mathematician — as the originator of the deductive 
organization of geometry. This report — or legend — was embellished by 
adding to this theorem four others that Thales is said to have proved : 

1. A circle is bisected by a diameter. 

2. The base angles of an isosceles triangle are equal. 

3. The pairs of vertical angles formed by two intersecting lines are equal. 


4. If two triangles are such that two angles and a side of one are equal 
respectively to two angles and a side of the other, then the triangles are 

There is no document from antiquity that can be pointed to as evidence of 
this achievement, and yet the tradition has been persistent. About the nearest 
one can come to reliable evidence on this point is derived from a source a 
thousand years after the time of Thales. A student of Aristotle by the name of 
Eudemus of Rhodes (fl. ca. 320 B.C.) wrote a history of mathematics. This 
has been lost, but before it disappeared, someone had summarized at least 
part of the history. The original of this summary also has been lost, but during 
the fifth century of our era information from the summary was incorporated 
by the Neoplatonic philosopher Proclus (410-485) in the early pages of his 
Commentary on the First Book of Euclid's Elements. Following introductory 
remarks on the origin of geometry in Egypt, the Commentary of Proclus 
reports that Thales 

. . . first went to Egypt and thence introduced this study into Greece. He discovered 
many propositions himself, and instructed his successors in the principles underlying 
many others, his method of attack being in some cases more general, in others more 
empirical. ' 

It is largely upon this quotation at third hand that designations of Thales as 
the first mathematician hinge. Proclus later in his Commentary, again 
depending on Eudemus, attributes to Thales the four theorems mentioned 
above. There are other scattered references to Thales in ancient sources, but 
most of these describe his more practical activities. Diogenes Laertius, 
followed by Pliny and Plutarch, reported that he measured the heights of the 
pyramids in Egypt by observing the lengths of their shadows at the moment 
when the shadow of a vertical stick is equal to its height. 2 Herodotus, the 
historian, recounts the story of Thales' prediction of a solar eclipse ; the 
philosopher Aristotle reports that Thales made a fortune by "cornering" 
the olive presses during a year in which the olive crop promised to be abun- 
dant. Still other legends picture Thales as a salt merchant, as a stargazer, 
as a defender of celibacy, or as a farsighted statesman. Such reports, however, 
provide no further evidence concerning the important question of whether or 
not Thales actually arranged a number of geometrical theorems in a deductive 
sequence. The tale that he calculated the distance of a ship at sea through the 
proportionality of sides of similar triangles is inconclusive, for the principles 
behind such a calculation had long been known in Egypt and Mesopotamia. 
Such stories do not establish the bold conjecture that Thales created 

1 The translation is taken from T. L. Heath, History of Greek Mathematics (1921), I, 128. Cf. 
Ivor Thomas, ed.. Selections Illustrating the History of Greek Mathematics (1939-1941), I, 147. 

2 For a full account see Heath, op. cit., 1, 128-140. 


demonstrative geometry ; but in any case Thales is the first man in history to 
whom specific mathematical discoveries have been attributed. 3 We know now 
that a large body of mathematical material was familiar to the Babylonians a 
millennium before the time of Thales, and yet among the Greeks it was under- 
stood that Thales had made definite advances. It would appear reasonable to 
suppose, in the light of Proclus' statements, that Thales contributed some- 
thing in the way of rational organization. That it was the Greeks who added 
the element of logical structure to geometry is virtually universally admitted 
today, but the big question remains whether this crucial step was taken by 
Thales or by others later — perhaps as much as two centuries later. On this 
point we must suspend final judgment until there is additional evidence on 
the development of Greek mathematics. 

Pythagoras is scarcely less controversial a figure than Thales, for he has 
been more thoroughly enmeshed in legend and apotheosis. Thales had been 
a man of practical affairs, but Pythagoras was a prophet and a mystic, born 
at Samos, one of the Dodecanese islands not far from Miletus, the birthplace 
of Thales. Although some accounts picture Pythagoras as having studied 
under Thales, this is rendered unlikely by the half-century difference in their 
ages. Some similarity in their interests can readily be accounted for by the 
fact that Pythagoras also traveled to Egypt and Babylon — possibly even to 
India. During his peregrinations he evidently absorbed not only mathemat- 
ical and astronomical information, but also much religious lore. Pythagoras 
was, incidentally, virtually a contemporary of Buddha, of Confucius, and of 
Lao-Tze, so that the century was a critical time in the development of religion 
as well as of mathematics. When he returned to the Greek world, Pythagoras 
settled at Croton on the southeastern coast of what is now Italy, but at 
that time was known as Magna Graecia. There he established a secret society 
which somewhat resembled an Orphic cult except for its mathematical and 
philosophical basis. 

That Pythagoras remains a very obscure figure is due in part to the loss of 

documents from that age. Several biographies of Pythagoras were written 

in antiquity, including one by Aristotle, but these have not survived. A 

further difficulty in identifying clearly the figure of Pythagoras lies in the fact 

that the order he established was communal as well as secret. Knowledge and 

property were held in common, hence attribution of discoveries was not to be 

made to a specific member of the school. It is best, consequently, not to 

speak of the work of Pythagoras, but rather of the contributions of the 

Pythagoreans, although in antiquity it was customary to give all credit to the 


3 B. L. van der Waerden, in Science Awakening, p. 80, accepts the conjecture that Thales used 
deduction ; O. Neugebauer. in Exact Sciences in Antiquity, pp. 142, 143, 148, rejects it. 


The Pythagorean school of thought was politically conservative and with a 
strict code of conduct. Vegetarianism was enjoined upon the members, 
apparently because Pythagoreanism accepted the doctrine of metempsy- 
chosis, or the transmigration of souls, with the resulting concern lest an 
animal to be slaughtered might be the new abode of a friend who had died. 
Among other taboos of the school was the eating of beans (more properly 
lentils). Perhaps the most striking characteristic of the Pythagorean order 
was the confidence it maintained in the pursuit of philosophical and math- 
ematical studies as a moral basis for the conduct of life. The very words 
"philosophy" (or "love of wisdom") and "mathematics" (or "that which is 
learned") are supposed to have been coined by Pythagoras himself to describe 
his intellectual activities. He is said to have given two categories of lectures, 
one for members of the school or order only, and the other for those in the 
larger community. It is presumed that it was in the lectures of the first category 
that Pythagoras presented whatever contributions to mathematics he may 
have made. Having described, in the quotation above, the work in geometry 
done by Thales, Proclus went on to say : 

Pythagoras, who came after him, transformed this science into a liberal form of 
education, examining its principles from the beginning and probing the theorems in an 
immaterial and intellectual manner. He discovered the theory of proportionals and the 
construction of the cosmic figures. 4 

Even if we do not accept this statement at its face value, it is evident that the 
Pythagoreans played an important role — possibly the crucial role — in the 
history of mathematics. In Egypt and Mesopotamia the elements of arith- 
metic and geometry were primarily exercises in the application of numerical 
procedures to specific problems, whether concerned with beer or pyramids or 
the inheritance of land. There had been little in the way of intellectual struc- 
ture, and perhaps nothing resembling philosophical discussion of principles. 
Thales is generally regarded as having made a beginning in this direction, 
although tradition supports the view of Eudemus and Proclus that the new 
emphasis in mathematics was due primarily to the Pythagoreans. With them 
mathematics was more closely related to a love of wisdom than to the exigen- 
cies of practical life ; and it has had this tendency ever since. How far the 
Pythagoreans went in this direction is not at all clear, and at least one eminent 
scholar 5 regards all reports of important mathematical contributions by 
Pythagoras as unhistorical. It is indeed difficult to separate history and legend 
concerning the man, for he meant so many things to the populace — the 
philosopher, the astronomer, the mathematician, the abhorrer of beans, the 

4 See Ivor Thomas, op. cit., I, 149. Cf. also Heath, op. cit., I, 141, and van der Waerden op cit 
p. 90. 

-"' See Neugebauer, op. cit., p. 148. 


saint, the prophet, the performer of miracles, the magician, the charlatan. 
That he was one of the most influential figures in history is difficult to deny, 
for his followers, whether deluded or inspired, spread their beliefs over most 
of the Greek world. The Pythagorean purification of the soul was accom- 
plished in part through a strict physical regimen and in part through cultist 
rites reminiscent of worshippers of Orpheus and Dionysus ; but the harmo- 
nies and mysteries of philosophy and mathematics also were essential parts 
in the rituals. Never before or since has mathematics played so large a role 
in life and religion as it did among the Pythagoreans. If, then, it is impossible 
to ascribe specific discoveries to Pythagoras himself, or even collectively to 
the Pythagoreans, it is nevertheless important to understand the type of 
activity with which, according to tradition, the school was associated. 

The motto of the Pythagorean school is said to have been "All is number." 
Recalling that the Babylonians had attached numerical measures to things 
around them, from the motions of the heavens to the values of their slaves, 
we may perceive in the Pythagorean motto a strong Mesopotamian affinity. 
The very theorem to which the name of Pythagoras still clings quite likely 
was derived from the Babylonians. It has been suggested, as justification for 
calling it the Theorem of Pythagoras, that the Pythagoreans first provided a 
demonstration ; but this conjecture cannot be verified. Legends that Pythag- 
oras sacrified an ox (a hundred oxen, according to some versions) upon 
discovering the theorem — or its proof — are implausible in view of the 
vegetarian rules of the school. Moreover, they are repeated, with equal 
incredibility, in connection with several other theorems. It is reasonable to 
assume that the earliest members of the Pythagorean school were familiar 
with geometrical properties known to the Babylonians; but when the 
Eudemus-Proclus summary ascribes to them the construction of the "cosmic 
figures" (that is, the regular solids), there is room for doubt. The cube, the 
octahedron, and the dodecahedron could perhaps have been observed in 
crystals, such as those of pyrite (iron disulphide) ; but a scholium in Elements 
XIII reports that the Pythagoreans knew only three of the regular polyhedra : 
the tetrahedron, the cube, and the dodecahedron. Familiarity with the 
last figure is rendered plausible by the discovery near Padua of an Etruscan 
dodecahedron of stone dating from before 500 B.C. It is not improbable, 
therefore, that even if the Pythagoreans did not know of the octahedron and 
the icosahedron, they knew of some of the properties of the regular pentagon. 
The figure of a five-pointed star (which is formed by drawing the five diagonals 
of a pentagonal face of a regular dodecahedron) is said to have been the 
special symbol of the Pythagorean school. The star pentagon had appeared 
earlier in Babylonian art, and it is possible that here, too, we find a connecting 
link between pre-Hellenic and Pythagorean mathematics. 


One of the tantalizing questions in Pythagorean geometry concerns the 
construction of a pentagram or star pentagon. If we begin with a regular 
polygon ABCDE (Fig. 4.1) and draw the five diagonals, these diagonals 
intersect in points A'B'C'D'E' which form another regular pentagon. Noting 
that the triangle BCD', for example, is similar to the isosceles triangle BCE 
and noting also the many pairs of congruent triangles in the diagram, it is 
not difficult to see that the diagonal points A'B'C'D'E' divide the diagonals 
in a striking manner. In each case a diagonal point divides a diagonal into 
two unequal segments such that the ratio of the whole diagonal is to the larger 

FIG. 4.1 

segment as this segment is to the smaller segment. This subdivision of a 
diagonal is the well-known "golden section" of a line segment, but this name 
was not used until a couple of thousand years later— just about the time when 
Kepler wrote lyrically : 

Geometry has two great treasures : one is the Theorem of Pythagoras ; the other, the 
division of a line into extreme and mean ratio. The first we may compare to a measure 
of gold ; the second we may name a precious jewel. 

To the ancient Greeks this type of subdivision soon became so familiar 
that no need was felt for a special descriptive name ; hence the longer designa- 
tion "the division of a segment in mean and extreme ratio" generally was 
replaced by the simple words "the section." 

One of the important properties of "the section" is that it is, so to speak, 
self-propagating. If a point P x divides a segment RS (Fig. 4.2) in mean and 
extreme ratio, with RP t the longer segment, and if on this larger segment we 
mark off a point P 2 such that RP 2 = P X S, then segment RP t will in turn be 
subdivided in mean and extreme ratio at point P 2 . Again, upon marking off 

I I l l i 

^3 Pi Pi 

FIG. 4.2 



on RP 2 point P 3 such that RP 3 = P 2 P X , segment RP 2 will be divided in mean 
and extreme ratio at F 3 . This iterative procedure can be carried out as many 
times as desired, the result being an ever smaller segment RP„ divided in 
mean and extreme ratio by point P n+1 . Whether or not the earlier Pythagor- 
eans noticed this unending process or drew significant conclusions from it 
is not known. Even the more fundamental question of whether or not the 
Pythagoreans of about 500 B.C. could divide a given segment into mean and 
extreme ratio cannot be answered with certainty, although the probability 
that they could and did seems to be high. The construction required is equiva- 
lent to the solution of a quadratic equation. To show this, let RS = a and 
RP t = x in Fig. 4.2. Then, by the property of the golden section, 
a:x = x:(a - x), and upon multiplying means and extremes we have the 
equation x 2 = a 2 - ax. This is a quadratic equation of type 1 described in 
Chapter 3, and Pythagoras could have learned from the Babylonians how to 
solve this equation algebraically. However, if a is a rational number, then 
there is no rational number x satisfying the equation. Did Pythagoras realize 
this? It seems unlikely. Perhaps instead of the Babylonian algebraic type of 
solution, the Pythagoreans may have adopted a geometrical procedure 
similar to that found in Euclid's Elements II. 11 and VI. 30. To divide a line 
segment AB in mean and extreme ratio, Euclid first constructed on the seg- 
ment AB the square ABCD (Fig. 4.3). Then he bisected AC at point £, drew 


FIG. 4.3 

line segment EB, and extended line CEA to F so that EF = EB. When the 
square AFGH is completed, point H will be the point desired, for one can 
readily show that AB.AH = AH.HB. Knowing what solution, if any, the 
earlier Pythagoreans used for the golden section would go far toward 
clarifying the problem of the level and characteristics of pre-Socratic math- 
ematics. If Pythagorean mathematics began under a Babylonian aegis, with 


strong faith that all is number, how (and when) did it happen that this gave 
way to the familiar emphasis on pure geometry that is so firmly enshrined in 
the classical treatises? 

,~ It has been customary to hold that most of the material in the first two 
books of the Elements was due to the Pythagoreans. This would presuppose a 
high level of achievement, implying a fairly rapid development of the subject 
after the days of Thales and Pythagoras. This view requires faith in what has 
been called the "Greek miracle," by which relatively unlettered newcomers 
on the Mediterranean scene mastered the material inherited from their 
neighbors and rapidly rose to new heights, establishing on the way the 
essential deductive pattern of theorems. In recent years serious doubt has 
been cast on the traditional view by those who call attention to relatively 
primitive concepts in Pythagorean arithmetic. If, for example, the leading 
Pythagorean mathematician of the early fourth century B.C., Archytas of 
Tarentum (428-365 B.C.), could assert that not geometry, but arithmetic 
alone, could provide satisfactory proofs, 6 there would appear to be little 
ground for placing the rise of the axiomatic method in geometry among the 
Pythagoreans of a century or two before this time. On the other hand, it may 
be argued that Archytas represented only one point of view, insisting on an 
orthodox Pythagorean numerology that others had abandoned or modified. 
Certainly there had been shifting attitudes in Pythagorean astronomy, and 
we can assume that there were comparable modifications in mathematics. 

Number mysticism was not original with the Pythagoreans. The number 
seven, for example, had been singled out for special awe, presumably on 
account of the seven wandering stars or planets from which the week (hence 
our names for the days of the week) is derived. The Pythagoreans were not the 
only people who fancied that the odd numbers had male attributes and the 
even female— with the related (and not unprejudiced) assumption, found as 
late as Shakespeare, that "there is divinity in odd numbers." Many early 
civilizations shared various aspects of numerology, but the Pythagoreans 
carried number worship to its extreme, basing their philosophy and their 
way of life upon it. The number one, they argued, is the generator of numbers 
and the number of reason ; the number two is the first even or female number, 
the number of opinion ; three is the first true male number, the number of 
harmony, being composed of unity and diversity; four is the number of 
justice or retribution, indicating the squaring of accounts ; five is the number 
of marriage, the union of the first true male and female numbers ; and six is 
the number of creation. Each number in turn had its peculiar attributes. 
The holiest of all was the number ten or the tetractys, for it represented the 

6 Neugebauer, Exact Sciences in Antiquity, p. 148. 


number of the universe, including the sum of all the possible geometric 
dimensions. A single point is the generator of dimensions, two points 
determine a line of dimension one, three points (not on a line) determine a 
triangle with area of dimension two, and four points (not in a plane) determine 
a tetrahedron with volume of dimension three; the sum of the numbers 
representing all dimensions therefore is the revered number ten. It is a 
tribute to the abstraction of Pythagorean mathematics that the veneration 
of the number ten evidently was not dictated by anatomy of the human hand 
or foot. 

In Mesopotamia geometry had been not much more than number applied 
to spatial extension ; it appears that at first it may have been much the same 
among the Pythagoreans — but with a modification. Number in Egypt had 
been the domain of the natural numbers and the unit fractions ; among the 
Babylonians it had been the field of all rational fractions. In Greece the word 
number was used only for the integers. A fraction was not looked upon as a 
single entity, but as a ratio or relationship between two whole numbers. 
(Greek mathematics in its earlier stages frequently came closer to the 
"modern" mathematics of today than to the ordinary arithmetic of a genera- 
tion ago.) As Euclid later expressed it (Elements V. 3), "A ratio is a kind of 
relation in respect of size of two magnitudes of the same kind." Such a view, 
focusing attention on the connection between pairs of numbers, tends to 
sharpen the theoretical or rational aspects of the number concept and to 
deemphasize the role of number as a tool in computation or approximation 
in mensuration. Arithmetic now could be thought of as an intellectual disci- 
pline as well as a technique, and a transition to such an outlook seems to have 
been nurtured in the Pythagorean school. If tradition is to be trusted, the 
Pythagoreans not only established arithmetic as a branch of philosophy ; they 
seem to have made it the basis of a unification of all aspects of the world 
about them. Through patterns of points, or unextended units, they associated 
number with geometrical extension ; this in turn led them to an arithmetic 
of the heavens. Philolaus (died ca. 390 B.C.), a later Pythagorean who shared 
the veneration of the tetractys or decad, wrote that it was "great, all-powerful 
and all-producing, the beginning and the guide of the divine as of the ter- 
restrial life." 7 This view of the number ten as the perfect number, the symbol 
of health and harmony, seems to have provided the inspiration for the earliest 
nongeocentric astronomical system. Philolaus postulated that at the center 
of the universe there was a central fire about which the earth and the seven 

7 For an especially extensive account of Pythagoreanism see Eduard Zeller, A History of 
Greek Philosophy from the Earliest Period to the Time of Socrates (1881), I, 306-533. On the 
tetractys see especially pp. 428 ff. A longer description of the role of the tetractys is given on 
pp. 180-188 of Thomas Taylor, The Theoretic Arithmetic of the Pythagoreans (Los Angeles, 
1934), but this book must be read with circumspection. 



planets (including the sun and the moon) revolved uniformly. Inasmuch as 
this brought to only nine the number of heavenly bodies (other than the 
sphere of fixed stars), the Philolaic system assumed the existence of a tenth 
body — a "counterearth" collinear with the earth and the central fire — 
having the same period as the earth in its daily revolution about the central 
fire. The sun revolved about the fire once a year, and the fixed stars were 
stationary. The earth in its motion maintained the same uninhabited face 
toward the central fire, hence neither the fire nor the counterearth ever was 
seen. The postulate of uniform circular motion that the Pythagoreans 
adopted was to dominate astronomical thought for more than 2000 years. 
Copernicus, almost 2000 years later, accepted this assumption without ques- 
tion, and it was to the Pythagoreans that Copernicus referred to show that his 
doctrine of a moving earth was not so new or revolutionary. 

The thoroughness with which the Pythagoreans wove number into their 1 
thought is well illustrated by their concern for figurate numbers. Although 
no triangle can be formed by fewer than three points, it is possible to have 
triangles of a larger number of points, such as six, ten, or fifteen (see Fig. 4.4). 

FIG. 4.4 

Numbers such as three, six, ten, and fifteen or, in general, numbers given by 
the formula 

n(n + 1) 

iV = l + 2+3 + --- + n = —— — - 

were called triangular; and the triangular pattern for the number ten, the 
holy tetractys, vied with the pentagon for veneration in Pythagorean number 
theory. There were, of course, indefinitely many other categories of privileged 
numbers. Successive square numbers are formed from the sequence 
1 + 3 + 5 + 7 + --- + (2n — 1), where each odd number in turn was looked 
upon as a pattern of dots resembling a gnomon (the Babylonian shadow 
clock) placed around two sides of the preceding square pattern of dots 
(see Fig. 4.4). Hence the word gnomon (related to the word for knowing) came 


to be attached to the odd numbers themselves. The sequence of even numbers, 
2 + 4 + 6 + • • • + In = n(n + 1), produces what the Greeks called "oblong 
numbers," each of which is double a triangular number. Pentagonal patterns 
of points illustrated the pentagonal numbers given by the sequence 

n(3n - 1) 
N=l + 4 + 7 + --- + (3n-2) = 

and hexagonal numbers were derived from the sequence 

1 + 5 + 9 + • • • + (4w - 3) = In 2 - n 

In similar manner polygonal numbers of all orders are designated; the 
process, of course, is easily extended to three-dimensional space, where one 
deals with polyhedral numbers. Emboldened by such views, Philolaus is 
reported to have maintained that 

All things which can be known have number; for it is not possible that without 
number anything can be either conceived or known. 

The dictum of Philolaus seems to have been a tenet of the Pythagorean 
school, hence stories arose about the discovery by Pythagoras of some simple 
laws of music. Pythagoras is reputed to have noticed that when the lengths of 
vibrating strings are expressible as ratios of simple whole numbers, such as 
two to three (for the fifth) or as three to four (for the fourth), the tones will be 
harmonious. If, in other words, a string sounds the note C when plucked, 
then a similar string twice as long will sound the note C an octave below ; 
and tones between these two notes are emitted by strings whose lengths are 
given by intermediate ratios: 16:9 for D, 8:5 for E, 3:2 for F, 4:3 for G, 
6:5 for A, and 16:15 for B, in ascending order. Here we have perhaps the 
earliest quantitative laws of acoustics — possibly the oldest of all quantitative 
physical laws. So boldly imaginative were the early Pythagoreans that they 
extrapolated hastily to conclude that the heavenly bodies in their motions 
similarly emitted harmonious tones, the "harmony of the spheres." Pythagor- 
ean science, like Pythagorean mathematics, seems to have been an odd 
congeries of sober thought and fanciful speculation. The doctrine of a 
spherical earth often is ascribed to Pythagoras, but it is not known whether 
this conclusion 8 was based on observation (perhaps of new constellations as 
Pythagoras traveled southward) or on imagination. The very idea that the 
universe is a "cosmos," or a harmoniously ordered whole, seems to be a 
related Pythagorean contribution — one which at the time had little basis in 
direct observation but which has been enormously fruitful in the development 

8 The tradition that attributes the spherical-earth concept to the Pythagoreans has been 
questioned. See W. A. Heidel, The Frame of the Ancient Greek Maps with a Discussion of the 
Sphericity of the Earth (New York: Amer. Geog. Soc, 1937). 


of astronomy. As we smile at ancient number fancies, we should at the 
same time be aware of the impulse these gave to the development of both 
mathematics and science. The Pythagoreans were among the earliest people 
to believe that the operations of nature could be understood through math- 

Proclus, quoting perhaps from Eudemus, ascribed to Pythagoras two 
specific mathematical discoveries: (1) the construction of the regular solids 
and (2) the theory of proportionals. Although there is question about the 
extent to which this is to be taken literally, there is every likelihood that the 
statement correctly reflects the direction of Pythagorean thought. The theory 
of proportions clearly fits into the pattern of early Greek mathematical 
interests, and it is not difficult to find a likely source of inspiration. It is 
reported that Pythagoras learned in Mesopotamia of three means — the 
arithmetic, the geometric, and the subcontrary (later called the harmonic) — 
and of the "golden proportion" relating two of these : the first of two numbers 
is to their arithmetic mean as their harmonic mean is to the second of the 
numbers. This relationship is the essence of the Babylonian square-root 
algorithm, hence the report is at least plausible. At some stage, however, 
the Pythagoreans generalized this work by adding seven new means to 
make ten in all. If b is the mean of a and c, where a < c, then the three quan- 
tities are related according to one of the following ten equations : 

b — a a 





c - 

-a a 
-b b 


b - 
c - 

-a a 
- b c 



c - 

-a c 
- b a 



c - 

-a b 
- b a 


b — a c 

c - b b 


c — a c 
b — a a 


c — a c 
c — b a 


c — a b 
b — a a 


c — a b 
c — b a 

The first three equations are, of course, the equations for the arithmetic, 
the geometric, and the harmonic means respectively. 

It is difficult to assign a date to the Pythagorean study of means, and similar 
problems arise with respect to the classification of numbers. The study of 
proportions or the equality of ratios presumably formed at first a part of 
Pythagorean arithmetic or theory of numbers. Later the quantities a, b, 


and c entering in such proportions were more likely to be regarded as geomet- 
rical magnitudes ; but the period in which the change took place is not clear. 
In addition to the polygonal numbers mentioned above and the distinction 
between odd and even, the Pythagoreans at some stage spoke of odd-odd 
and even-odd numbers, according as the number in question was the product 
of two odd numbers or of an odd and an even number, so that sometimes the 
name even number was reserved for integral powers of two. By the time of 
Philolaus the distinction between prime and composite numbers seems to 
have become important. Speusippus, nephew of Plato and his successor as 
head of the Academy, asserted that ten was "perfect" for the Pythagoreans 
because, among other things, it is the smallest integer n for which there are 
just as many primes between one and n as nonprimes. (Occasionally prime 
numbers were called linear inasmuch as they usually are represented by dots 
in one dimension only.) Neo-Pythagoreans sometimes excluded two from 
the list of primes on the ground that one and two are not true numbers, but 
the generators of the odd and even numbers. The primacy of the odd numbers 
was assumed to be established by the fact that odd + odd is even, whereas 
even + even remains even. 

To the Pythagoreans has been attributed the rule for Pythagorean triads 
given by (m 2 - l)/2, m, (m 2 + l)/2, where m is an odd integer; but inasmuch 
as this rule is so closely related to the Babylonian examples, it is perhaps not 
an independent discovery. Also ascribed to the Pythagoreans, with doubt 
as to the period in question, are the definitions of perfect, abundant, and 
deficient numbers according as the sum of the proper divisors of the number 
is equal to, greater than, or less than the number itself. According to this 
definition, six is the smallest perfect number, with twenty-eight next. That 
this view probably was a later development in Pythagorean thought is 
suggested by the early veneration of ten rather than six. Hence the related 
doctrine of "amicable" numbers also is likely to have been a later notion. 
Two integers a and b are said to be "amicable" if a is the sum of the proper 
divisors of b and if b is the sum of the proper divisors of a. The smallest such 
pair are the integers 220 and 284. 

The picture of Pythagorean mathematics that has been presented is 
based largely on reports of commentators who lived many centuries later 
and who were, almost without exception, interested in philosophical aspects 
of thought. Although it appears plausible to assume, with the commentators, 
that it was the Pythagoreans who were largely responsible for the abstract 
and intellectual view that fashioned mathematics into a liberal discipline, 
the level of sophistication during the sixth and fifth centuries B.C. may not 
have been as high as that attributed to them by tradition. It must have been 
all too tempting to later devotees of a philosophical school, such as the 


Pythagorean, to exaggerate the accomplishments of the founder and of the 
early members of the sect. It is highly probable that elements of primitivity 
were present during the early stages of Pythagoreanism, but went unreported. 
It is obvious also that the type of attitude toward mathematics represented 
by the Pythagoreans almost certainly was atypical of Greek thought as a 
whole. The Hellenes were celebrated as shrewd traders and businessmen, 
and there must have been a lower level of arithmetic or computation that 
satisfied the needs of the vast majority of Greek citizens. Number activities 
of this type would have been beneath the notice of philosophers, and recorded 
accounts of practical arithmetic were unlikely to find their way into libraries 
of scholars. If, then, there are not even fragments surviving of the more 
sophisticated Pythagorean works, it is clear that it would be unreasonable 
to expect manuals of trade mathematics to survive the ravages of time. 
Hence it is not possible to tell at this distance how the ordinary processes of 
arithmetic were carried out in Greece 2500 years ago. About the best one can 
do is to describe the systems of numeration that appear to have been in use. 
In general there seem to have been two chief systems of numeration in 
Greece : one, probably the earlier, is known as the Attic (or Herodianic) nota- 
tion ; the other is called the Ionian (or alphabetic) system. Both systems are, 
for integers, based on the ten-scale, but the former is the more primitive, 
being based on a simple iterative scheme found in the earlier Egyptian 
hieroglyphic numeration and in the later Roman numerals. In the Attic 
system the numbers from one to four were represented by repeated vertical 
strokes. For the number five a new symbol — the first letter n (or T) of the 
word for five, pente — was adopted. (Only capital letters were used at the time, 
both in literary works and in mathematics, lower-case letters being an 
invention of the later ancient or early Medieval period.) For numbers from 
six through nine, the Attic system combined the symbol T with unit strokes, 
so that eight, for example, was written as Pin. For positive integral powers of 
the base (ten), the initial letters of the corresponding number words were 
adopted— a for deka (ten), h for hekaton (hundred), x for khilioi (thousand), 
and m for myrioi (ten thousand). Except for the forms of the symbols, the 
Attic system is much like the Roman ; but it had one advantage. Where the 
Latin world adopted -distinctive symbols for 50 and 500, the Greeks wrote 
these numbers by combining letters for 5, 10, and 100, using P (or 5 times 10) 
for 50, and P (or 5 times 100) for 500. In the same way they wrote P 1 for 5000 
and f 51 for 50000. In Attic script the number 45,678, for example, would appear 

MM mm pnrpi Hf 3 ^ Pm 

The Attic system of notation (known also as Herodianic inasmuch as it 10 
was described in a fragment attributed to Herodian, a grammarian of the 


second century) appears in inscriptions at various dates from 454 to 95 B.C. f 
but by the early Alexandrian Age, at about the time of Ptolemy Philadelphus, 
it was being displaced by the Ionian or alphabetic numerals. Similar alphabe- 
tic schemes were used at one time or another by various Semitic peoples, 
including the Hebrews, Syrians, Arameans, and Arabs — as well as by other 
cultures, such as the Gothic — but these would seem to have been borrowed 
from the Greek notation. The Ionian system probably was used as early as 
the fifth century B.C. and perhaps as early as the eighth century B.C. One reason 
for placing the origin of the notation relatively early is that the scheme called 
for twenty-seven letters of the alphabet — nine for the integers less than 10, 
nine for multiples of 10 that are less than 100, and nine for multiples of 100 
that are less than 1000. The classical Greek alphabet contains only twenty- 
four letters ; hence use was made of an older alphabet that included three 
additional archaic letters— F (vau or digamma or stigma), 1 (koppa), 
and A (sampi)— to establish the following association of letters and numbers : 


i 2 3 4 5 6 7 8 9 10 20 30 40 50 
son^px T T$X*SIA 

60 70 80 90 100 200 300 400 500 600 700 800 900 
Since the three archaic letters occupy the positions in the numeral scheme 
that they held in the older alphabet, it has been suggested that the Ionian 
system was introduced before the abandonment of the three letters — say in 
the eighth century B.C. ; this view becomes less convincing when we consider 
the long time interval between the presumed introduction and the ultimate 
triumph of the system in the third century B.C. 10 The obvious advantage in 
conciseness of the alphabetic system might have been expected to find a 
readier adoption for the system than the indicated delay of half a millennium. 
The cipherization in the Ionian notation bears to the Attic numeration 
essentially the same relationship as did the Egyptian hieratic to the more 
cumbersome hieroglyphic, where the superiority of the cursive script had 
been clear to scribes. 

After the introduction of small letters in Greece, the association of letters 
and numbers appeared as follows : 

1 2 3 4 5 6 7 8 9 10 20 30 40 50 

60 70 80 90 100 200 300 400 500 600 700 800 900 

9 See Heath, op. cit, I. 30. See also James Gow, A Short History of Greek Mathematics (Cam- 
bridge, 1884). 

10 For further discussion and references see C. B. Boyer, "Fundamental Steps in the Develop- 
ment of Numeration," Isis 35 (1944), 153-168. 


Since these forms are more familiar today, we shall use them here. For the 
first nine multiples of a thousand, the Ionian system adopted the first nine 
letters of the alphabet, a partial use of the positional principle ; but for added 
clarity these letters were preceded by a stroke or accent : 

,a ,P ,y ,5 , e ,«■ Z ,r\ ,0 
1000 2000 3000 4000 5000 6000 7000 8000 9000 

Within this system any number less than 10,000 was easily written with only 
four characters. The number 8888, for example, would appear as ^conrj or 
as rjconri, the accent sometimes being omitted when the context was clear. 
The use of the same letters for thousands as for units should have suggested 
to the Greeks the full-fledged positional scheme in decimal arithmetic, but 
they do not seem to have appreciated the advantages of such a move. That 
they had such a principle more or less in mind is evident not only in the 
repeated use of the letters a through 9 for units and thousands, but also in 
the fact that the symbols are arranged in order of magnitude, from the smallest 
on the right to the largest on the left. At 10,000, which for the Greeks was 
the beginning of a new count or category (much as we separate thousands 
from lower powers by a comma), the Ionian Greek notation adopted a 
multiplicative principle. A symbol for an integer from 1 to 9999, when placed 
above the letter M, or after it, separated from the rest of the number by a dot, 
indicated the product of the integer and the number 10,000— the Greek 
myriad. Thus the number 88888888 would appear as M^cony ■ r\(o%r\. Where 
still larger numbers are called for, the same principle could be applied to the 
double myriad, 100000000 or 10 8 . 

Early Greek notations for integers were not excessively awkward, and they 
served their purposes effectively. It was in the use effractions that the systems 
were weak. Like the Egyptians, the Greeks were tempted to use unit fractions, 
and for these they had a simple representation. They wrote down the denom- 
inator and then simply followed this with a diacritical mark or accent to 
distinguish it from the corresponding integer. Thus 3^ would appear as 15'. 
This could, of course, be confused with the number 30£, but context or the 
use of words could be assumed to make the situation clear. In later centuries 
general common fractions and sexagesimal fractions were in use ; these will 
be discussed later in connection with the work of Archimedes, Ptolemy, and 
Diophantus, for there are extant documents which, while not actually 
dating from the time of these men, are copies of works written by them — a 
situation strikingly different from that concerning mathematicians of the 
Hellenic period. 

The history of mathematics during the time of Thales and the Pythagoreans 11 
necessarily depends, to an undesirable degree, on conjecture and inference, 


since documents from the period are entirely missing. In this respect there is 
far more uncertainty about Greek mathematics from 600 to 450 B.C. than 
about Babylonian algebra or Egyptian geometry from about 1700 B.C. Not 
even mathematical artifacts have survived from the early days of Greece. 
It is evident that some form of counting board or abacus was used in calcula- 
tion, but the nature and operation of the device must be inferred from the 
Roman abacus and from some casual references in Greek authors. Herodotus, 
writing in the early fifth century B.C., says that in counting with pebbles, as in 
writing, the Greek hand moved from left to right, the Egyptian from right to 
left. A vase from a somewhat later period pictures a collector of tribute with 
a counting board which was used not only for integral decimal multiples of 
the drachma, but for nondecimal fractional subdivisions. Beginning on the 
left, the columns designate myriads, thousands, hundreds, and tens of 
drachmas, respectively, the symbols being in Herodianic notation. Then, 
following the units column for drachmas, there are columns for obols (six 
obols = one drachma), for half the obol, and for the quarter obol. Here we 
see how ancient civilizations avoided an excessive use of fractions : they 
simply subdivided units of length, weight, and money so effectively that 
they could calculate in terms of integral multiples of the subdivisions. This 
undoubtedly is the explanation for the popularity in antiquity of duodecimal 
and sexagesimal subdivisions, for the decimal system here is at a severe 
disadvantage. Decimal fractions were rarely used, either by the Greeks or 
by other Western peoples, before the period of the Renaissance. The abacus 
can be readily adapted to any system of numeration or to any combination 
of systems; it is likely that the widespread use of the abacus accounts at least 
in part for the amazingly late development of a consistent positional system 
of notation for integers and fractions. In this respect the Pythagorean Age 
contributed little if anything. The point of view of the Pythagoreans seems 
to have been so overwhelmingly philosophical and abstract that technical 
details in computation were of little concern to them. Such techniques were 
relegated to a separate discipline, called logistic. This dealt with the number- 
ing of things, rather than with the essence and properties of number as such, 
matters of concern in arithmetic. That is, the ancient Greeks made a clear 
distinction between mere calculation on the one hand and what today is 
known in America as theory of numbers (and in England as the higher 
arithmetic) on the other. Whether or not such a sharp distinction was a 
disadvantage to the historical development of mathematics may be a moot 
point, but it is not easy to deny to the early Ionian and Pythagorean math- 
ematicians the primary role in establishing mathematics as a rational and 
liberal discipline. It is for this reason that Thales often is called the first 
mathematician and that Pythagoras is known as the father of mathematics. 
The extent to which we accept such ascriptions literally, in view of the absence 


of supporting documentary evidence, will depend on our confidence in 
tradition. It is obvious that tradition can be quite inaccurate, but it seldom is 
entirely misdirected. 


Allman, G. J., Greek Geometry from Thales to Euclid (Dublin : Dublin University Press, 

Clagett, Marshall, Greek Science in Antiquity (New York : Abelard-Schuman, ca. 1955 ; 

2nd paperback ed., New York : Collier, 1966). 
Dantzig, Tobias, The Bequest of the Greeks (New York : Scribner's, 1955). 
Freeman, Kathleen, Ancilla to the Pre-Socratic Philosophers (Cambridge, Mass.: 

Harvard University Press, 1948). 
Gow, James, A Short History of Greek Mathematics (reprint, New York : Hafner, 1923). 
Heath, T. L., A History of Greek Mathematics (Oxford : Clarendon, 1921, 2 vols.). 
Heath, T. L., Manual of Greek Mathematics (New York : Oxford University Press, 1931 ; 

paperback ed., New York : Dover, 1963). 
Loria, Gino, Historie des sciences mathematiques dans Vantiquite hellenique (Paris: 

Gauthier-Villars, 1929). 
Michel, Paul-Henri, De Pythagore a Euclide (Paris, 1950). 
Neugebauer, O., The Exact Sciences in Antiquity, 2nd ed. (Providence, R.I. : Brown 

University Press, 1957; paperback ed., New York : Harper). 
Tannery, Paul, La geometrie grecque, comment son histoire nous est parvenue et ce que 

nous en savons (Paris, 1887). 
Thomas, Ivor, ed., Selections Illustrating the History of Greek Mathematics (2 vols., 

Cambridge, Mass. : Harvard University Press, 1939-1941). 
Van der Waerden, B. L., Science Awakening, trans, by Arnold Dresden (New York : 

Oxford University Press, 1961 ; paperback ed., New York: Wiley, 1963). 
Zeller, Eduard : A History of Greek Philosophy from the Earliest Period to the Time 

of Socrates, trans, by S. F. Alleyne (London: Longmans, Green, 1881, 2 vols.). 


1. Prove two theorems attributed to Thales and tell, with reasons, whether or not you think 
he may have used similar reasoning. 

2. Prove the Pythagorean theorem. Do you think Pythagoras used your method? Explain. 

3. Theon of Smyrna, a Neoplatonist and Neo-Pythagorean of the second century, is said to 
have found that the sum of two consecutive triangular numbers is a square number. Prove 
this theorem. 

4. What are the first four heptagonal numbers (corresponding to regular polygons of seven 

5. Write the numbers 3456 and 4567 and their sum in the early Greek Attic notation and in the 
Ionian or alphabetic system. 


6. Prove that if three numbers, a, b, c are in arithmetic progression in that order, and if ,4, B, C 
are their reciprocals respectively, B is the harmonic mean of A and C. 

7. Philolaus called the cube a "geometrical harmony," because of the number of its faces, 
vertices, and edges. Justify his designation in the light of Pythagorean theory of proportions. 

8. Show that 1184 and 1210 are amicable numbers. 

9. Show, in the manner of the Pythagoreans, that an oblong number is the sum of two equal 
triangular numbers. 

10. Prove carefully that the diagonals of a regular pentagon divide each other in mean and 
extreme ratio. 

11. Using straightedge and compasses only, construct a regular pentagon, given the side of the 

12. Using straightedge and compasses only, construct a regular pentagon, given a diagonal of 
the pentagon. 

13. In a given circle construct a regular pentagon, using straightedge and compasses only. 

14. All polygonal numbers are of the form P m = an 2 + bn, where m is the number of sides and n 
is the order. Using this fact, find a and b for octagonal numbers (m = 8) and verify geometric- 
ally for n — 3. 

15. Find the fifth pentagonal number and the sixth hexagonal number. 

16. Is 4567 a heptagonal number? Justify your answer. 

17. Show that, if a > b > c, the three equations 

a — b a a — b a a — b a 

b — c a' b — c b' b — c c 

define b respectively as the arithmetic, the geometric, and the harmonic mean of a and c. 
*18. All polyhedral numbers are of the form P„ = an 3 + bn 2 + en, where m is the number of 

faces and n is the order. Use this fact to find a and b and c for tetrahedral numbers (m = 4) 

and verify geometrically for n = 4. 
*19. Polyhedral numbers are found by adding successive polygonal numbers of the same kind. 

Show how to generalize this procedure to define polytopal numbers in n-dimensional space 

and find three nontrivial polytopal numbers. 


The Heroic Age 

I would rather discover one cause than gain the king- 
dom of Persia. 


Accounts of the origins of Greek mathematics center on the so-called Ionian 1 
and Pythagorean schools and the chief representative of each — Thales and 
Pythagoras — although reconstructions of their thought rest on fragmentary 
reports and traditions built up during later centuries. To a certain extent 
this situation prevails throughout the fifth century B.C. There are virtually 
no extant mathematical or scientific documents until the days of Plato in 
the fourth century b.c. Nevertheless, during the last half of the fifth century 
there circulated persistent and consistent reports concerning a handful of 
mathematicians who evidently were intensely concerned with problems that 
formed the basis for most of the later developments in geometry. We shall 
therefore refer to this period as the "Heroic Age of Mathematics," for seldom 
either before or since have men with so little to work with tackled mathe- 
matical problems of such fundamental significance. No longer was mathe- 
matical activity centered almost entirely in two regions nearly at opposite 
ends of the Greek world ; it flourished all about the Mediterranean. In what 
is now southern Italy there were Archytas of Tarentum (born ca. 428 B.C.) 
and Hippasus of Metapontum (fl. ca. 400 B.C.); at Abdera in Thrace we find 
Democritus (born ca. 460 B.C.) ; nearer the center of the Greek world, on 
the Attic peninsula, there was Hippias of Ellis (born ca. 460 B.C.); and at 
nearby Athens there lived at various times during the critical last half of the 
fifth century B.C. three scholars from other regions : Hippocrates of Chios 
(fl. ca. 430 B.C.), Anaxagoras of Clazomenae (t428 b.c), and Zeno of Elea 
(fl. ca. 450 B.C.). Through the work of these seven men we shall describe the 
fundamental changes in mathematics that took place a little before the year 
400 b.c. 

The fifth century b.c. was a crucial period in the history of Western 2 
civilization, for it opened with the defeat of the Persian invaders and closed 
with the surrender of Athens to Sparta. Between these two events lay the 



great Age of Pericles, with its accomplishments in literature and art. The 
prosperity and intellectual atmosphere of Athens during the century attracted 
scholars from all parts of the Greek world, and a synthesis of diverse aspects 
was achieved. From Ionia came men, such as Anaxagoras, with a practical 
turn of mind ; from southern Italy came others, such as Zeno, with stronger 
metaphysical inclinations. Democritus of Abdera espoused a materialistic 
view of the world, while Pythagoras in Italy held idealistic attitudes in science 
and philosophy. At Athens one found eager devotees of old and new branches 
of learning, from cosmology to ethics. There was a bold spirit of free inquiry 
that sometimes came into conflict with established mores. In particular, 
Anaxagoras was imprisoned at Athens for impiety in asserting that the sun 
was not a deity, but a huge red-hot stone as big as the whole Peloponnessus, 
and that the moon was an inhabited earth that borrowed its light from the 
sun. He well represents the spirit of rational inquiry, for he regarded as the 
aim of his life the study of the nature of the universe— a purposefulness that 
he derived from the Ionian tradition of which Thales had been a founder. 
The intellectual enthusiasm of Anaxagoras was shared with his countrymen 
through the first scientific best-seller— a book On Nature which could be 
bought in Athens for only a drachma. Anaxagoras was a teacher of Pericles, 
who saw to it that his mentor ultimately was released from prison. Socrates 
was at first attracted to the scientific ideas of Anaxagoras, but the gadfly of 
Athens found the naturalistic Ionian view less satisfying than the search for 
ethical verities. 

Greek science had been rooted in a highly intellectual curiosity which 
often is contrasted with the utilitarian immediacy of pre-Hellenic thought ; 
Anaxagoras clearly represented the typical Greek motive — the desire to 
know. In mathematics also the Greek attitude differed sharply from that of 
the earlier potamic cultures. The contrast was clear in the contributions 
generally attributed to Thales and Pythagoras, and it continues to show 
through in the more reliable reports on what went on in Athens during the 
Heroic Age. Anaxagoras was primarily a natural philosopher rather than a 
mathematician, but his inquiring mind led him to share in the pursuit of 
mathematical problems. We are told by Plutarch that while Anaxagoras was 
in prison he occupied himself in an attempt to square the circle. Here we have 
the first mention of a problem that was to fascinate mathematicians for more 
than 2000 years. 1 There are no further details concerning the origin of 
the problem or the rules governing it. At a later date it came to be under- 
stood that the required square, exactly equal in area to the circle, was to be 

1 See E. W. Hobson, Squaring the Circle (ca. 1913), p. 14. This work has been reprinted several 
times. The accuracy of Plutarch's statement in this connection has been questioned recently. 
On the work of Anaxagoras see D. E. Gershenson and D. A. Greenberg, Anaxagoras and the 
Birth of Physics (New York: Blaisdell, 1964). 


constructed by the use of compasses and straightedge alone. Here we see a type 
of mathematics that is quite unlike that of the Egyptians and Babylonians. 
It is not the practical application of a science of number to a facet of life 
experience, but a theoretical question involving a nice distinction between 
accuracy in approximation and exactitude in thought. The mathematical 
problem that Anaxagoras here considered was no more the concern of the 
technologist than were those he raised in science concerning the ultimate 
structure of matter. In the Greek world mathematics was more closely 
related to philosophy than to practical affairs, and this kinship has persisted 
to the present day. 

Anaxagoras died in 428 B.C., the year that Archytas was born, just one 
year before Plato's birth and one year after Pericles' death. It is said that 
Pericles died of the plague that carried off perhaps a quarter of the Athenian 
population, and the deep impression that this catastrophe created is perhaps 
the origin of a second famous mathematical problem. It is reported that a 
delegation had been sent to the oracle of Apollo at Delos to inquire how the 
plague could be averted, and the oracle had replied that the cubical altar 
to Apollo must be doubled. The Athenians are said to have dutifully doubled 
the dimensions of the altar, but this was of no avail in curbing the plague. 
The altar had, of course, been increased eightfold in volume, rather than 
twofold. Here, according to the legend, was the origin of the "duplication 
of the cube" problem, one that henceforth was usually referred to as the 
"Delian problem" — given the edge of a cube, construct with compasses 
and straightedge alone the edge of a second cube having double the volume 
of the first. At about the same time there circulated in Athens still a third 
celebrated problem — given an arbitrary angle, construct by means of 
compasses and straightedge alone an angle one-third as large as the given 
angle. These three problems — the squaring of the circle, the duplication of 
the cube, and the trisection of the angle — have since been known as the "three 
famous (or classical) problems" of antiquity. More than 2200 years later 
it was to be proved that all three of the problems were unsolvable by means 
of straightedge and compasses alone. Nevertheless, the better part of Greek 
mathematics, and of much later mathematical thought, was suggested by 
efforts to achieve the impossible — or, failing this, to modify the rules. The 
Heroic Age failed in its immediate objective, under the rules, but the efforts 
were crowned with brilliant success in other respects. 

Somewhat younger than Anaxagoras, and coming originally from about 
the same part of the Greek world, was Hippocrates of Chios. He should not 
be confused with his still more celebrated contemporary, the physician 
Hippocrates of Cos. Both Cos and Chios are islands in the Dodecanese 
group ; but Hippocrates of Chios in about 430 B.C. left his native land for 


Athens in his capacity as a merchant. Aristotle reports that Hippocrates 
was less shrewd than Thales and that he lost his money in Byzantium through 
fraud ; others say that he was beset by pirates. In any case, the incident was 
never regretted by the victim, for he counted this his good fortune in that 
as a consequence he turned to the study of geometry, in which he achieved 
remarkable success— a story typical of the Heroic Age. Proclus wrote that 
Hippocrates composed an "Elements of Geometry," anticipating by more 
than a century the better-known Elements of Euclid. However, the textbook 
of Hippocrates — as well as another reported to have been written by Leon, 
a later associate of the Platonic school— has been lost, although it was known 
to Aristotle. In fact, no mathematical treatise from the fifth century has 
survived ; but we do have a fragment concerning Hippocrates which Simpli- 
cius (fl. ca. 520) claims to have copied literally from the History of Mathematics 
(now lost) by Eudemus. This brief statement, the nearest thing we have to an 
original source on the mathematics of the time, describes a portion of the 
work of Hippocrates dealing with the quadrature of lunes. A lune is a figure 
bounded by two circular arcs of unequal radii ; the problem of the quadrature 
of lunes undoubtedly arose from that of squaring the circle. The Eudemian 
fragment attributes to Hippocrates the following theorem : 

Similar segments of circles are in the same ratio as the squares on their bases. 

The Eudemian account reports that Hippocrates demonstrated this by first 
showing that the areas of two circles are to each other as the squares on 
their diameters. Here Hippocrates adopted the language and concept of 
proportion which played so large a role in Pythagorean thought. In fact, it 
is thought by some that Hippocrates became a Pythagorean. The Pythagor- 
ean school in Croton had been suppressed (possibly because of its secrecy, 
perhaps because of its conservative political tendencies), but the scattering 
of its adherents throughout the Greek world served only to broaden the 
influence of the school. This influence undoubtedly was felt, directly or 
indirectly, by Hippocrates. 

The theorem of Hippocrates on the areas of circles seems to be the earliest 
precise statement on curvilinear mensuration in the Greek world. Eudemus 
believed that Hippocrates gave a proof of the theorem, but a rigorous 
demonstration at that time (say about 430 B.C.) would appear to be unlikely. 
The theory of proportions at that stage probably was established for com- 
mensurable magnitudes only. The proof as given in Euclid XII. 2 comes from 
Eudoxus, a man who lived halfway between Hippocrates and Euclid. How- 
ever, just as much of the material in the first two books of Euclid seems to 
stem from the Pythagoreans, so it would appear reasonable to assume that 
the formulations, at least, of much of Books III and IV of the Elements 
came from the work of Hippocrates. Moreover, if Hippocrates did give a 


demonstration of his theorem on the areas of circles, he may have been 
responsible for the introduction into mathematics of the indirect method of 
proof. That is, the ratio of the areas of two circles is equal to the ratio of the 
squares on the diameters or it is not. By a reductio ad absurdum from the 
second of the two possibilities, the proof of the only alternative is established. 
From his theorem on the areas of circles Hippocrates readily found the 
first rigorous quadrature of a curvilinear area in the history of mathematics. 
He began with a semicircle circumscribed about an isosceles right triangle, 
and on the base (hypotenuse) he constructed a segment similar to the circular 
segments on the sides of the right triangle (Fig. 5.1). Because the segments 

are to each other as squares on their bases, and from the Pythagorean 
theorem as applied to the right triangle, the sum of the two small circular 
segments is equal to the larger circular segment. Hence the difference 
between the semicircle on AC and the segment ADCE equals triangle ABC. 
Therefore the lune ABCD is precisely equal to triangle ABC; and since 
triangle ABC is equal to the square on half of AC, the quadrature of the 
lune has been found. 2 

Eudemus describes also an Hippocratean lune quadrature based on an 
isosceles trapezoid ABCD inscribed in a circle so that the square on the 
longest side (base) AD is equal to the sum of the squares on the three equal 
shorter sides AB and BC and CD (Fig. 5.2). Then if on side AD one constructs 
a circular segment AEDF similar to those on the three equal sides, lune 
ABCDE is equal to trapezoid ABCDF. 

That we are on relatively firm ground historically in describing the 
quadrature of lunes by Hippocrates, is indicated by the fact that scholars 
other than Simplicius also refer to this work. Simplicius lived in the sixth 
century, but he depended not only on Eudemus (fl. ca. 320 B.C.) but also on 
Alexander of Aphrodisias (fl. ca. a.d. 200), one of the chief commentators on 

2 An excellent account of Hippocrates' quadratures is found in B. L. van der Waerden, Science 
Awakening (1961), pp. 131 ff. 



Aristotle. Alexander describes two quadratures other than those given above. 
(1) If on the hypotenuse and sides of an isosceles right triangle one constructs 
semicircles (Fig. 5.3), then the lunes created on the smaller sides together 
equal the triangle. (2) If on a diameter of a semicircle one constructs an 
isosceles trapezoid with three equal sides (Fig. 5.4), and if on the three equal 

sides semicircles are constructed, then the trapezoid is equal in area to the 
sum of four curvilinear areas : the three equal lunes and a semicircle on one 
of the equal sides of the trapezoid. From the second of these quadratures it 
would follow that if the lunes can be squared, the semicircle— hence the 
circle— can also be squared. This conclusion seems to have encouraged 
Hippocrates, as well as his contemporaries and early successors, to hope 
that ultimately the circle would be squared. 

The Hippocratean quadratures are significant not so much as attempts at 
circle-squaring as indications of the level of mathematics at the time. They 
show that Athenian mathematicians were adept at handling transformations 
of areas and proportions. In particular, there was evidently no difficulty in 
converting a rectangle of sides a and b into a square. This required finding the 
mean proportional or geometric mean between a and b. That is, if a :x = x :b, 
geometers of the day easily constructed the line x. It was natural, therefore, 
that geometers should seek to generalize the problem by inserting two means 


between two given magnitudes a and b. That is, given two line segments a 
and b, they hoped to construct two other segments x and y such that 
a:x = x:y = y.b. Hippocrates is said to have recognized that this problem 
is equivalent to that of duplicating the cube ; for if b = 2a, the continued 
proportions, upon the elimination of y, lead to the conclusion that x 3 = 2a 3 . 
There are three views on what Hippocrates deduced from his quadrature 
of lunes. Some have accused him of believing that he could square all lunes, 
hence also the circle ; others think that he knew the limitations of his work, 
concerned as it was with some types of lunes only. At least one scholar has 
held that Hippocrates knew he had not squared the circle but tried to deceive 
his countrymen into thinking that he had succeeded. 3 There are other 
questions, too, concerning Hippocrates' contributions, for to him has been 
ascribed, with some uncertainty, the first use of letters in geometric figures. 
It is interesting to note that whereas he advanced two of the three famous 
problems, he seems to have made no progress in the trisection of the angle, 
a problem studied somewhat later by Hippias of Ellis. 

Toward the end of the fifth century B.C. there flourished at Athens a group 
of professional teachers quite unlike the Pythagoreans. Disciples of Pythag- 
oras had been forbidden to accept payment for sharing their knowledge with 
others. The Sophists, however, openly supported themselves by tutoring 
fellow citizens — not only in honest intellectual endeavor, but also in the art 
of "making the worse appear the better." To a certain extent the accusation 
of shallowness directed against the Sophists was warranted ; but this should 
not conceal the fact that Sophists usually were very widely informed in many 
fields and that some of them made real contributions to learning. Among 
these was Hippias, a native of Ellis who was active at Athens in the second 
half of the fifth century B.C. He is one of the earliest mathematicians of whom 
we have firsthand information, for we learn much about him from Plato's 
dialogues. We read, for example, that Hippias boasted that he had made 
more money than any two other Sophists. He is said to have written much, 
from mathematics to oratory, but none of his work has survived. He had a 
remarkable memory, he boasted immense learning, and he was skilled in 
handicrafts. To this Hippias (there were many others in Greece who bore the 
same name) we apparently owe the introduction into mathematics of the 
first curve beyond the circle and the straight line. Proclus and other com- 
mentators ascribe to him the curve since known as the trisectrix or quadratrix 
of Hippias. 4 This is drawn as follows : In the square ABCD (Fig. 5.5) let side 

3 See Bjornbo's article "Hippocrates" in Pauly-Wissowa, Real-Enzyklopiidie der klassischen 
Altertumswissenschaft, Vol. VIII, p. 1796. 

4 An excellent account of this is found in Kathleen Freeman, The Pre-Socratic Philosophers. 
A Companion to Diels, Fragmente der Vorsokratiker (1949), pp. 381-391. See also the article on 
Hippias in Pauly-Wissowa, op. cit., VIII, 1707 ff. 



AB move down uniformly from its present position until it coincides with 
DC and let this motion take place in exactly the same time that side DA 
rotates clockwise from its present position until it coincides with DC. If 
the positions of the two moving lines at any given time are given by A'B' 
and DA" respectively and if P is the point of intersection of A'B' and DA", 
the locus of P during the motions will be the trisectrix of Hippias — curve 
APQ in the figure. Given this curve, the trisection of an angle is carried out 
with ease. For example, if PDC is the angle to be trisected, one simply 
trisects segments B'C and AD at points R, S, T, and U. If lines TR and US 
cut the trisectrix in V and W respectively, lines VD and WD will, by the 
property of the trisectrix, divide angle PDC in three equal parts. 

The curve of Hippias generally is known as the quadratrix, since it can 
be used to square the circle. Whether or not Hippias himself was aware of 
this application cannot now be determined. It has been conjectured that 
Hippias knew of this method of quadrature but that he was unable to justify 
it. Since the quadrature through Hippias' curve was specifically given later 
by Dinostratus, we shall describe this work in the next chapter. 

Hippias lived at least as late as Socrates (t399 B.C.), and from the pen of 
Plato we have an unflattering account of him as a typical Sophist— vain, 
boastful, and acquisitive. Socrates is reported to have described Hippias as 
handsome and learned, but boastful and shallow. Plato's dialogue on 
Hippias satirizes his show of knowledge, and Xenophon's Memorabilia 
includes an unflattering account of Hippias as one who regarded himself an 
expert in everything from history and literature to handicrafts and science. 
In judging such accounts, however, we must remember that Plato and 
Xenophon were uncompromisingly opposed to the Sophists in general. It 
is well to bear in mind also that both Protagoras, the "founding father 
of the Sophists," and Socrates, the archopponent of the movement, were 


antagonistic to mathematics and the sciences. With respect to character, Plato 
contrasts Hippias with Socrates, but one can bring out much the same 
contrast by comparing Hippias with another contemporary — the Pythagor- 
ean mathematician Archytas of Tarentum. 

Pythagoras is said to have retired to Metapontum toward the end of his 
life and to have died there about 500 B.C. Tradition holds that he left no 
written works, but his ideas were carried on by a large number of eager 
disciples. The center at Croton was abandoned when a rival political group 
from Sybaris surprised and murdered many of the leaders, but those who 
escaped the massacre carried the doctrines of the school to other parts of the 
Greek world. Among those who received instruction from the refugees was 
Philolaus of Tarentum, and he is said to have written the first account of 
Pythagoreanism — permission having been granted, so the story goes, to 
repair his damaged fortunes. Apparently it was this book from which Plato 
derived his knowledge of the Pythagorean order. The number fanaticism that 
was so characteristic of the brotherhood evidently was shared by Philolaus, 
and it was from his account that much of the mystical lore concerning the 
tetractys was derived, as well as knowledge of the Pythagorean cosmology. 
The Philolaean cosmic scheme is said to have been modified by two later 
Pythagoreans, Ecphantus and Hicetas, who abandoned the central fire 
and counterearth and explained day and night by placing a rotating earth 
at the center of the universe. The extremes of Philolaean number worship 
also seem to have undergone some modification, more especially at the hands 
of Archytas, a student of Philolaus at Tarentum. 

The Pythagorean sect had exerted a strong intellectual influence through- 
out Magna Graecia, with political overtones that may be described as a sort 
of "reactionary international," or perhaps better as a cross between Orphism 
and Freemasonry. At Croton political aspects were especially noticeable, but 
at outlying Pythagorean centers, such as Tarentum, the impact was primarily 
intellectual. Archytas believed firmly in the efficacy of number ; his rule of 
the city, which allotted him autocratic powers, was just and restrained, for 
he regarded reason as a force working toward social amelioration. For many 
years in succession he was elected general, and he was never defeated ; yet 
he was kind and a lover of children, for whom he is reported to have invented 
"Archytas' rattle." Possibly also the mechanical dove, which he is said to 
have fashioned of wood, was built to amuse the young folk. 

Archytas continued the Pythagorean tradition in placing arithmetic above 
geometry, but his enthusiasm for number had less of the religious and mystical 
admixture found earlier in Philolaus. He wrote on the application of the 
arithmetic, geometric, and subcontrary means to music, and it was probably 
either Philolaus or Archytas who was responsible for changing the name of 


the last one to "harmonic mean." Among his statements in this connection 
was the observation that between two whole numbers in the ratio n:(n + 1) 
there could be no integer that is a geometric mean. Archytas gave more 
attention to music than had his predecessors, and he felt that this subject 
should play a greater role than literature in the education of children. Among 
his conjectures was one that attributed differences in pitch to varying rates 
of motion resulting from the flow causing the sound. Archytas seems to 
have paid considerable attention to the role of mathematics in the curriculum, 
and to him has been ascribed the designation of the four branches in the 
mathematical quadrivium — arithmetic (or numbers at rest), geometry (or 
magnitudes at rest), music (or numbers in motion), and astronomy (or 
magnitudes in motion). These subjects, together with the trivium consisting 
of grammar, rhetoric, and dialectics (which Aristotle traced back to Zeno), 
later constituted the seven liberal arts; hence the prominent role that 
mathematics has played in education is in no small measure due to Archytas. 

8 It is likely that Archytas had access to an earlier treatise on the elements 
of mathematics, and the iterative square-root process often known by the 
name of Archytas had been used long before in Mesopotamia. Nevertheless, 
Archytas was himself a contributor of original mathematical results. The 
most striking contribution was a three-dimensional solution of the Delian 
problem which may be most easily described, somewhat anachronistical^, 
in the modern language of analytic geometry. Let a be the edge of the cube 
to be doubled, and let the point (a, 0, 0) be the center of three mutually 
perpendicular circles of radius a and each lying in a plane perpendicular 
to a coordinate axis. Through the circle perpendicular to the x-axis construct 
a right circular cone with vertex (0, 0, 0); through the circle in the xy-plane 
pass a right circular cylinder ; and let the circle in the xz-plane be revolved 
about the z-axis to generate a torus. The equations of these three surfaces 
are respectively x 2 = y 2 + z 2 and lax = x 2 + y 2 and (x 2 + y 2 + z 2 ) 2 = 
4a 2 (x 2 + y 2 ). These three surfaces intersect in a point whose x-coordinate is 
crfl; hence the length of this line segment is the edge of the cube desired. 
The achievement of Archytas is the more impressive when we recall that 
his solution was worked out synthetically without the aid of coordinates. 
Nevertheless, the most important contribution of Archytas to mathematics 
may have been his intervention with the tyrant Dionysius to save the life 
of his friend, Plato. The latter remained to the end of his life deeply com- 
mitted to the Pythagorean veneration of number and geometry, and the 
supremacy of Athens in the mathematical world of the fourth century B.C. 
resulted primarily from the enthusiasm of Plato, the "maker of mathemati- 
cians." However, before taking up the role of Plato it is necessary to discuss 
the work of an earlier Pythagorean— an apostate by the name of Hippasus. 


Hippasus of Metapontum (or Croton), roughly contemporaneous with 
Philolaus, is reported to have been originally a Pythagorean but to have 
been expelled from the brotherhood. One account has it that the Pythagor- 
eans erected a tombstone to him, as though he were dead; another story 
reports that his apostasy was punished by death at sea in a shipwreck. The 
exact cause of the break is unknown, in part because of the rule of secrecy, 
but there are three suggested possibilities. According to one, Hippasus was 
expelled for political insubordination, having headed a democratic move- 
ment against the conservative Pythagorean rule. A second tradition attributes 
the expulsion to disclosures concerning the geometry of the pentagon or 
the dodecahedron — perhaps a construction of one of the figures. A third 
explanation holds that the expulsion was coupled with the disclosure of a 
mathematical discovery of devastating significance for Pythagorean philo- 
sophy — the existence of incommensurable magnitudes. 

It had been a fundamental tenet of Pythagoreanism that the essence of 
all things, in geometry as well as in the practical and theoretical affairs of 
man, are explainable in terms of arithmos, or intrinsic properties of whole 
numbers or their ratios. The dialogues of Plato show, however, that the 
Greek mathematical community had been stunned by a disclosure that 
virtually demolished the basis for the Pythagorean faith in whole numbers. 
This was the discovery that within geometry itself the whole numbers and 
their ratios are inadequate to account for even simple fundamental properties. 
They do not suffice, for example, to compare the diagonal of a square or a 
cube or a pentagon with its side. The line segments are incommensurable, 
no matter how small a unit of measure is chosen. Just when and how the 
discovery was made is not known, but much ink has been spilled in support 
of one hypothesis or another. Earlier arguments in favor of a Hindu origin 
of the discovery 5 lack foundation, and there seems to be little chance that 
Pythagoras himself was aware of the problem of incommensurability. The 
most plausible suggestion is that the discovery was made by the later Pythag- 
oreans at some time before 410 B.C. 6 Some would attribute it specifically to 
Hippasus of Metapontum during the earlier portion of the last quarter of 
the fifth century B.C., 7 while others place it about another half a century later. 

5 See Heinrich Vogt, "Haben die alten Inder den Pythagoreischen Lehrsatz und das Irra- 
tionale gekannt?," Bibliotheca Mathematica (3), 7 (1906-1907), 6-23; also Leopold von 
Schroeder, Pythagoras und die Inder (Leipzig. 1 884). 

6 See especially Heinrich Vogt, "Die Entdeckungsgeschichte des Irrationalen nach Plato 
und anderen Quellen des 4. Jahrhunderts," Bibliotheca Mathematica (3), 10 (1910), 97-155, 
and the same author's paper, "Zur Entdeckungsgeschichte des Irrationalen," Bibliotheca 
Mathematica (3), 14 (1914), 9-29. Cf. Heath, History of Greek Mathematics (1921), 1, 157. 

7 See Kurt von Fritz, "The Discovery of Incommensurability by Hippasus of Metapontum," 
Annals of Mathematics (2), 46 (1945), 242-264. 


The circumstances surrounding the earliest recognition of incommensur- 
able line segments are as uncertain as is the time of the discovery. Ordinarily 
it is assumed that the recognition came in connection with the application 
of the Pythagorean theorem to the isosceles right triangle. Aristotle refers 
to a proof of the incommensurability of the diagonal of a square with respect 
to a side, indicating that it was based on the distinction between odd and 
even. 8 Such a proof is easy to construct. Let d and s be the diagonal and side 
of a square, and assume that they are commensurable — that is, that the 
radio d/s is rational and equal to p/q, where p and q are integers with no 
common factor. Now, from the Pythagorean theorem it is known that 
d 2 = s 2 + s 2 ; hence (d/s) 2 = p 2 /q 2 = 2, or p 2 = 2q 2 . Therefore p 2 must be 
even ; hence p must be even. Consequently q must be odd. Letting p = 2r 
and substituting in the equation p 2 = 2q 2 , we have 4r 2 = 2q 2 , or q 2 = 2r 2 . 
Then q 2 must be even ; hence q must be even. However, q was shown above 
to be odd, and an integer cannot be both odd and even. It follows therefore, 
by the indirect method, that the assumption that d and s are commensurable 
must be false. 

10 In this proof the degree of abstraction is so high that the possibility that 
it was the basis for the original discovery of incommensurability has been 
questioned. There are, however, other ways in which the discovery could 
have come about. Among these is the simple observation that when the five 
diagonals of a regular pentagon are drawn, these diagonals form a smaller 
regular pentagon (Fig. 5.6), and the diagonals of the second pentagon in 

FIG. 5.6 

turn form a third regular pentagon, which is still smaller. This process can 
be continued indefinitely, resulting in pentagons that are as small as desired 
and leading to the conclusion that the ratio of a diagonal to a side in a regular 
pentagon is not rational. The irrationality of this ratio is, in fact, a consequence 

8 See H. G. Zeuthen, "Sur l'origine historique de la connaissance des quantites irrationelles," 
Oversigt over del Kongelige Danske Videnskabernes Selskabs. Forhandlinger, 1915, pp. 333-362. 



of the argument presented in connection with Fig. 4.2 in which the golden 
section was shown to repeat itself over and over again. Was it perhaps this 
property that led to the disclosure, possibly by Hippasus, of incommensur- 
ability? There is no surviving document to resolve the question, but the 
suggestion is at least a plausible one. In this case, it would not have been 
y/2 but ^/5 that first disclosed the existence of incommensurable magnitudes, 
for the solution of the equation a:x = x:{a — x) leads to (^/S — l)/2 as the 
ratio of the side of a regular pentagon to a diagonal. The ratio of the diagonal 
of a cube to an edge is y/3, and here, too, the spectre of the incommensurable 
rears its ugly head. 

A geometric proof somewhat analogous to that for the ratio of the diagonal 
of a pentagon to its side can be provided also for the ratio of the diagonal of a 
square to its side. If in the square ABCD (Fig. 5.7) one lays off on the diagonal 

FIG. 5.7 

AC the segment AP = AB and at P erects the perpendicular PQ, the ratio 
of CQ to PC will be the same as the ratio of A C to AB. Again, if on CQ one 
lays off QR = QP and constructs RS perpendicular to CR, the ratio of 
hypotenuse to side again will be what it was before. This process, too, can 
be continued indefinitely, thus affording a proof that no unit of length, 
however small, can be found so that the hypotenuse and a side will be com- 

The Pythagorean doctrine that "Numbers constitute the entire heaven" 
was now faced with a very serious problem indeed ; but it was not the only 
one, for the school was confronted also by arguments propounded by the 
neighboring Eleatics, a rival philosophical movement. Ionian philosophers 
of Asia Minor had sought to identify a first principle for all things. Thales 
had thought to find this in water, but others preferred to think of air or fire 



as the basic element. The Pythagoreans had taken a more abstract direction, 
postulating that number in all its plurality was the basic stuff behind phenom- 
ena; this numerical atomism, beautifully illustrated in the geometry of 
figurate numbers, had come under attack by the followers of Parmenides 
of Elea (fi. ca. 450 B.C.). The fundamental tenet of the Eleatics was the unity 
and permanence of being, a view that contrasted with the Pythagorean ideas 
of multiplicity and change. Of Parmenides' disciples the best known was 
Zeno the Eleatic (fi. ca. 450 B.C.) who propounded arguments to prove the 
inconsistency in the concepts of multiplicity and divisibility. The method 
Zeno adopted was dialectical, anticipating Socrates in this indirect mode of 
argument: starting from his opponent's premises, he reduced these to an 

The Pythagoreans had assumed that space and time can be thought of 
as consisting of points and instants ; but space and time have also a property, 
more easily intuited than defined, known as "continuity." The ultimate 
elements making up a plurality were assumed on the one hand to have the 
characteristics of the geometrical unit — the point — and on the other to have 
certain characteristics of the numerical units or numbers. Aristotle described 
a Pythagorean point as "unity having position" or as "unity considered in 
space." It has been suggested 9 that it was against such a view that Zeno 
propounded his paradoxes, of which those on motion are cited most fre- 
quently. As they have come down to us, through Aristotle and others, four 
of them seem to have caused the most trouble: (1) the Dichotomy, (2) the 
Achilles, (3) the Arrow, and (4) the Stade. The first argues that before a moving 
object can travel a given distance, it must first travel half this distance ; but 
before it can cover this, it must travel the first quarter of the distance ; and 
before this, the first eighth, and so on through an infinite number of sub- 
divisions. The runner wishing to get started, must make an infinite number of 
contacts in a finite time ; but it is impossible to exhaust an infinite collection, 
hence the beginning of motion is impossible. The second of the paradoxes 
is similar to the first except that the infinite subdivision is progressive rather 
than regressive. Here Achilles is racing against a tortoise that has been given 
a headstart, and it is argued that Achilles, no matter how swiftly he may run, 
can never overtake the tortoise, no matter how slow it may be. By the time 
that Achilles will have reached the initial position of the tortoise, the latter 
will have advanced some short distance; and by the time that Achilles will 
have covered this distance, the tortoise will have advanced somewhat 
farther; and so the process continues indefinitely, with the result that the 
swift Achilles can never overtake the slow tortoise. 

9 See Paul Tannery, La geometrie grecque (1887), pp. 217-261. For a different view see B. L. 
van der Waerden, "Zenon und die Grundlagenkrise der griechischen Mathematik," Mathe- 
matische Annalen, 117 (1940), 141-161. 



The Dichotomy and the Achilles argue that motion is impossible under the 
assumption of the infinite subdivisibility of space and time ; the Arrow and 
the Stade, on the other hand, argue that motion is equally impossible if 
one makes the opposite assumption — that the subdivisibility of space and 
time terminates in indivisibles. In the Arrow Zeno argues that an object in 
flight always occupies a space equal to itself; but that which always occupies 
a space equal to itself is not in motion. Hence the flying arrow is at rest at 
all times, so that its motion is an illusion. 

Most controversial of the paradoxes on motion, and most awkward to 
describe, is the Stade (or Stadium), but the argument can be phrased somewhat 
as follows. Let A x , A 2 , A 3 , A 4 be bodies of equal size that are stationary; 
let B^ , B 2 , B 3 , B A be bodies, of the same size as the ^4's, that are moving to 
the right so that each B passes each A in an instant — the smallest possible 
interval of time. Let C t , C 2 , C 3 , C 4 also be of equal size with the A's and 
B's and let them move uniformly to the left with respect to the A's so that 
each C passes each A in an instant of time. Let us assume that at a given time 
the bodies occupy the following relative positions : 

A x 

A 2 


A A 


B 2 

B 3 

B 4 


C 2 

c 3 

C 4 

Then after the lapse of a single instant — that is, after an indivisible sub- 
division of time — the positions will be as follows : 


A 2 

A 3 



B 2 

B 3 

B A 


C 2 

c 3 

C 4 


It is clear, then, that C t will have passed two of the B's; hence the instant 
cannot be the minimum time interval, for we can take as a new and smaller 
unit the time it takes C l to pass one of the B's. 

The arguments of Zeno 10 seem to have had a profound influence on the 
development of Greek mathematics, comparable to that of the discovery of 
the incommensurable, with which it may have been related. Originally, in 
Pythagorean circles, magnitudes were represented by pebbles or calculi, 
from which our word calculation comes, but by the time of Euclid there is 
a complete change in point of view. Magnitudes are not in general associated 
with numbers or pebbles, but with line segments. In the Elements even the 
integers themselves are represented by segments of lines. The realm of 
number continued to have the property of discreteness, but the world of 
continuous magnitudes (and this included most of pre-Hellenic and Pythag- 
orean mathematics) was a thing apart from number and had to be treated 
through geometrical method. It seemed to be geometry rather than number 
that ruled the world. This was perhaps the most far-reaching conclusion of 
the Heroic Age, and it is not unlikely that this was due in large measure to 
Zeno of Elea and Hippasus of Metapontum. 

12 It has generally been held that the deductive element had been introduced 
into mathematics by Thales, but recently it has been argued against this 
thesis that the mathematics of the sixth and fifth centuries B.C. was too 
primitive to countenance such a contribution. Those who hold to this thesis 
sometimes refer to the arguments of Zeno and Hippasus as possible inspira- 
tion for the deductive approach. Certainly the doubts and problems raised 
in this connection would have been a fertile field for the growth of deduction ; 
and it would not be unreasonable to regard the end of the fifth century B.C. 
as a terminus ante quern for the rational deductive form with which we have 
become so familiar. It may be well to indicate at this point, therefore, that 
there are several conjectures as to the causes leading to the conversion of 
the mathematical prescriptions of pre-Hellenic peoples into the deductive 
structure appearing in Greece. Some have suggested 11 that Thales in his 
travels had noted discrepancies in pre-Hellenic mathematics — such as the 
Egyptian and Babylonian rules for the area of a circle — and that he and his 
early successors therefore saw the need for a strict rational method. Others, 
more conservative, would place the deductive form much later — perhaps 
even as late as the early fourth century, following the discovery of the 

10 The bibliography on the paradoxes is enormous. Among the most informative historical 
treatments is that by Florian Cajori, "History of Zeno's Arguments on Motion," American 
Mathematical Monthly, 22 (1915), 1-6, 39-47, 77-82, 109-115, 145-149, 179-186, 215-220, 
253-258, 292-297. For sources, see Zeno of Elea (text, translation, and notes by H. D. P. Lee; 

' ' See van der Waerden, Science Awakening (1961), p. 89. 



incommensurable. 12 Other suggestions find the cause outside mathematics. 
One, for example, sees in the sociopolitical development of the Greek city- 
states the rise of dialectics and a consequent requirement of a rational basis 
for mathematics and other studies ; another somewhat similar suggestion is 
that deduction may have come out of logic in attempts to convince an oppo- 
nent of a conclusion by looking for premises from which the conclusion 
necessarily follows. * 3 

Whether deduction came into mathematics in the sixth century b.c. or 
the fourth and whether incommensurability was discovered before or after 
400 b.c, there can be no doubt that Greek mathematics had undergone 
drastic changes by the time of Plato. The dichotomy between number and 
continuous magnitude required a new approach to the Babylonian algebra 
that the Pythagoreans had inherited. The old problems in which, given the 
sum and the product of the sides of a rectangle, the dimensions were required, 
had to be dealt with differently from the numerical algorithms of the Babylon- 
ians. A "geometrical algebra" had to take the place of the older "arithmetical 
algebra," and in this new algebra there could be no adding of lines to areas 
or of areas to volumes. From now on there had to be a strict homogeneity of 
terms in equations, and the Mesopotamian normal forms, xy = A,x ± y - b, 
were to be interpreted geometrically. The obvious conclusion, which the 
reader can arrive at by eliminating y, is that one must construct on a given 
line b a rectangle whose unknown width x must be such that the area of 
the rectangle exceeds the given area A by the square x 2 or (in the case of the 
minus sign) falls short of the area A by the square x 2 (Fig. 5.8). In this way the 


FIG. 5.8 

Greeks built up the solution of quadratic equations by their process known 
as "the application of areas," a portion of geometrical algebra that is fully 
covered by Euclid's Elements. Moreover, the uneasiness resulting from 
incommensurable magnitudes led to an avoidance of ratios, insofar as 

12 Neugebauer, The Exact Sciences in Antiquity, pp. 148-149. 

13 See Arpad Szabo, "Anfange des euklidischen Axiomensystems," Archive for History of 
Exact Sciences, 1 (1960), 37-106. 



possible, in elementary mathematics. The linear equation ax = be, for 
example, was looked upon as an equality of the areas ax and be, rather than 
as a proportion — an equality between the two ratios a:b and c:x. Conse- 
quently, in constructing the fourth proportion x in this case, it was usual to 
construct a rectangle OCDB with sides b = OB and c = OC (Fig. 5.9) and 


FIG. 5.9 

then along OC to lay off OA = a. One completes rectangle OAEB and draws 
the diagonal OE cutting CD in P. It is now clear that CP is the desired line x, 
for rectangle OARS is equal in area to rectangle OCDB. Not until Book V 
of the Elements did Euclid take up the difficult matter of proportionality. 

Greek geometrical algebra strikes the modern reader as excessively 
artificial and difficult ; to those who used it and became adept at handling 
its operations, however, it probably appeared to be a convenient tool. The 
distributive law a(b + c + d) = ab + ac + ad undoubtedly was far more 
obvious to a Greek scholar than to the beginning student of algebra today, 
for the former could easily picture the areas of the rectangles in this theorem, 
which simply says that the rectangle on a and the sum of segments b, c, d is 
equal to the sum of the rectangles on a and each of the lines b, c, d taken 
separately (Fig. 5.10). Again, the identity (a + b) 2 = a 2 + lab + b 2 becomes 
obvious from a diagram that shows the three squares and the two equal 
rectangles in the identity (Fig. S.ll^andadifferenceoftwosquaresa 2 - b 2 = 







6 2 

FIG. 5.10 

FIG. 5.11 



(a + b){a - b) can be pictured in a similar fashion (Fig. 5.12). Sums, differ- 
ences, products, and quotients of line segments can easily be constructed 
with straightedge and compasses. Square roots also afford no difficulty in 
geometric algebra. If one wishes to find a line x such that x 2 = ab, one simply 
follows the procedure found in elementary geometry textbooks today. One 
lays off on a straight line the segment ABC, where AB = a and BC = b 



FIG. 5.12 

(Fig. 5.13). With AC as diameter one constructs a semicircle (with center O) 
and at B erects the perpendicular BP, which is the segment x desired. It is 
interesting that here too the proof as given by Euclid, probably following the 
earlier avoidance of ratios, makes use of areas rather than proportions. 
If in our figure we let PO = AO = CO = r and BO = s, Euclid would 

say essentially that x 

r - s z = (r - s){r + s) = ab. 

The Heroic Age in mathematics produced half a dozen great figures, and 
among them must be included a man who is better known as a chemical 
philosopher. Democritus of Abdera (ca. 460 B.c.-ca. 370 B.C.) is today 
celebrated as a proponent of a materialistic atomic doctrine, but in his time 
he had acquired also a reputation as a geometer. He is reported to have 
traveled more widely than anyone of his day — to Athens, Egypt and Mesopo- 
tamia, and possibly India — acquiring what learning he could ; but his own 
achievements in mathematics were such that he boasted that not even the 



"rope-stretchers" in Egypt excelled him. He wrote a number of mathematical 
works, not one of which is extant today, but we have the titles of a few : On 
Numbers, On Geometry, On Tangencies, On Mappings, and On Irrationals. 
So great was his fame that in later centuries many treatises in chemistry and 
mathematics were unwarrantedly attributed to him. In particular, early 
alchemical works by a pseudo-Democritus are not to be ascribed to our 
Abderite; but other books, On the Pythagoreans, On the World Order, 
and On Ethics, may have been genuine. His scientific material was said to 
be clear, but clothed in a literary style ; Cicero wrote of Democritus that he 
had rhythm that made him more poetical than the poets. Yet of the mass of 
writings thought to have been by Democritus, nothing beyond a few words 
has survived. 

The key to the mathematics of Democritus is without doubt to be found 
in his physical doctrine of atomism. All phenomena were to be explained, 
he argued, in terms of indefinitely small and infinitely varied (in size and 
shape) impenetrably hard atoms moving about ceaselessly in empty space. 
The creation of our world — and of innumerable others also — was the result 
of an ordering or coagulation of atoms into groups having certain similarities. 
This was not a new theory, for it had been proposed earlier by Leucippus ; 
therefore the opponents of Democritus (and there were many of these) 
accused him of plagiarism from others, including Anaxagoras and Pythagoras. 
The physical atomism of Leucippus and Democritus may indeed have been 
suggested by the geometrical atomism of the Pythagoreans, and it is not 
surprising that the mathematical problems with which Democritus was 
chiefly concerned were those that demand some sort of infinitesimal approach. 
The Egyptians, for example, were aware that the volume of a pyramid is 
one-third the product of the base and the altitude, but a proof of this fact 
almost certainly was beyond their capabilities, for it requires a point of view 
equivalent to the calculus. Archimedes later wrote that this result was due 
to Democritus, but that the latter did not prove it rigorously. This creates a 
puzzle, for if Democritus added anything to the Egyptian knowledge here, 
it must have been some sort of demonstration, albeit inadequate. Perhaps 
Democritus showed that a triangular prism can be divided into three tri- 
angular pyramids which, two by two, are equal in height and area of the base, 
and then deduced, from the assumption that pyramids of the same height 
and equal bases are equal, the familiar Egyptian theorem. 

This assumption can be justified only by the application of infinitesimal 
techniques. If, for example, one thinks of two pyramids of equal bases and 
the same height as composed of indefinitely many infinitely thin equal 
cross sections in one-to-one correspondence (a device usually known as 
Cavalieri's principle in deference to the seventeenth-century geometer), 
the assumption appears to be justified. Such a fuzzy geometrical atomism 


might have been at the base of Democritus' thought, although this has not 
been established. In any case, following the paradoxes of Zeno and the 
awareness of incommensurables, such arguments based on an infinity of 
infinitesimals were not acceptable. Archimedes consequently could well hold 
that Democritus had not given a rigorous proof. The same judgment would 
be true with respect to the theorem, also attributed by Archimedes to 
Democritus, that the volume of a cone is one-third the volume of the circum- 
scribing cylinder. This result probably was looked upon by Democritus as a 
corollary to the theorem on the pyramid, for the cone is essentially a pyramid 
whose base is a regular polygon of infinitely many sides. 

Democritean geometrical atomism was immediately confronted by certain 
problems. If the pyramid or the cone, for example, is made up of infinitely 
many infinitely thin triangular or circular sections parallel to the base, a 
consideration of any two adjacent laminae creates a paradox. If the adjacent 
sections are equal in area, then, since all sections are equal, the totality will 
be a prism or a cylinder, and not a pyramid or a cone. If, on the other hand, 
adjacent sections are unequal, the totality will be a step pyramid or a step 
cone, and not the smooth-surfaced figure one has in mind. This problem is 
not unlike the difficulties with the incommensurable and with the paradoxes 
of motion. Perhaps, in his On the Irrational, Democritus analysed the 
difficulties here encountered, but there is no way of knowing what direction 
his attempts may have taken. His extreme unpopularity in the two dominant 
philosophical schools of the next century, those of Plato and Aristotle, may 
have encouraged the disregard of Democritean ideas. Nevertheless, the 
chief mathematical legacy of the Heroic Age can be summed up in six 
problems : the squaring of the circle, the duplication of the cube, the trisection 
of the angle, the ratio of incommensurable magnitudes, the paradoxes on 
motion, and the validity of infinitesimal methods. To some extent these can 
be associated, although not exclusively, with men considered in this chapter : 
Hippocrates, Archytas, Hippias, Hippasus, Zeno, and Democritus. Other 
ages were to produce a comparable array of talent, but perhaps never again 
was any age to make so bold an attack on so many fundamental mathematical 
problems with such inadequate methodological resources. It is for this 
reason that we have called the period from Anaxagoras to Archytas the 
Heroic Age. 


Allman, G. J., Greek Geometry from Thales to Euclid (Dublin : Dublin University Press, 

1 889). 
Cajori, Florian, "History of Zeno's Arguments on Motion," American Mathematical 

Monthly, 22 (1915), 1-6, 39-47, 77-82, 109-115, 145-149, 179-186, 215-220, 

253-258, 292-297. 


Freeman, Kathleen, The Pre-Socratic Philosophers, 2nd ed. (Oxford : Black well, 1949). 
Gow, James, A Short History of Greek Mathematics (reprint, New York : Hafner, 1923). 
Heath, T. L., History of Greek Mathematics (New York : Oxford University Press, 

1921, 2 vols.). 
Hobson, E. W., Squaring the Circle (Cambridge, ca. 1913). 
Lee, H. D. P., ed., Zeno of Elea (Cambridge: Cambridge University Press, 1936). 
Michel, Paul-Henri, De Pythagore a Euclide (Paris: Societe d'Edition "Les Belles 

Lettres," 1950). 
Neugebauer, O., The Exact Sciences in Antiquity, 2nd ed., (Providence R.I. : Brown 

University Press, 1957 ; paperback, New York : Harper). 
Szabo, Arpad, "The Transformation of Mathematics into Deductive Science and the 

Beginnings of its Foundation on Definitions and Axioms." Scripta Mathematica, 

27(1964), 27^8, 113-139. 
Tannery, Paul, La geometrie grecque, comment son histoire nous est parvenue et ce que 

nous en savons (Paris, 1887). 
Thomas, Ivor, ed., Selections Illustrating the History of Greek Mathematics (Cambridge, 

Mass.: Harvard University Press, 1939-1941, 2 vols). 
Van der Waerden, B. L., Science Awakening (trans, by Arnold Dresden, New York : 

Oxford University Press, 1961 ; paperback ed., New York, Wiley, 1963). 
Von Fritz, Kurt, "The Discovery of Incommensurability by Hippasus of Metapontum," 

Annals of Mathematics (2), 46 (1945), 242-264. 


1. Justify the two quadratures attributed by Alexander of Aphrodisias to Hippocrates. 

2. Draw an angle of 60° and use Hippias' trisectrix to divide the angle into seven equal parts. 

3. Prove carefully that the segments into which the diagonals of a regular pentagon divide 
each other are incommensurable with respect to the diagonals. 

4. Which do you believe was discovered first, the irrationality of J 2 or of ^/Sl Justify your 
answer in terms of historical evidence. 

5. Using ruler and compasses only, construct the line x if ax = b 2 , where a and b are any 
given line segments. 

6. Given line segments a and b and using compasses and straightedge only, construct x and y 
if x + y = a and xy = b 2 . 

7. Given line segments a, b, and c, construct x and y if x - y = a and xy = be. 

8. Solve the equation x 2 + ax = b 2 by constructing a line segment satisfying the given condi- 

9. Given a unit line segment (of length 1), construct a line segment of length ^3 + f . 

10. Show that in polar coordinates the equation of Hippias' trisectrix is nr sin 9 = 2a0. Sketch 
the main branch of this curve for -tc/2 ■< 9 < 3ji/2 and tell why Hippias did not draw this 
complete branch. 

1 1. Are the diagonals of a regular hexagon incommensurable with respect to a side? Explain 
fully, indicating whether your conclusion could have been reached in antiquity. 

*12. Carry out the steps needed to show that Archytas' construction duplicates the cube. 


The Age of Plato and 

Willingly would I burn to death like Phaeton, were this 
the price for reaching the sun and learning its shape, its 
size, and its substance. 


The Heroic Age lay largely in the fifth century B.C., and from this period 
little remains in the way of direct evidence about mathematical developments. 
The histories of Herodotus and Thucydides and the plays of Aeschylus, 
Euripides, and Aristophanes have in some measure survived, but scarcely 
a line is extant of what was written by mathematicians of the time. Firsthand 
mathematical sources from the fourth century B.C. are almost as scarce, but 
this inadequacy is made up for in large measure by accounts written by 
philosophers who were au courant with the mathematics of their day. We 
have most of what Plato wrote and about half of the work of Aristotle ; 
with the writings of these intellectual leaders of the fourth century B.C. as a 
guide, we can give a far more dependable account of what happened in their 
day than we could about the Heroic Age. 

We included Archytas among the mathematicians of the Heroic Age, but 
in a sense he really is a transition figure in mathematics during Plato's time. 
Archytas was among the last of the Pythagoreans, both literally and figura- 
tively. He could still believe that number was all-important in life and in 
mathematics, but the wave of the future was to elevate geometry to the 
ascendancy, largely because of the problem of incommensurability. On the 
other hand, Archytas is reported to have established the quadrivium — 
arithmetic, geometry, music, and astronomy — as the core of a liberal educa- 
tion, and here his views were to dominate much of pedagogical thought to our 
day. The seven liberal arts, which remained a shiboleth for almost two 
millennia, were made up of Archytas' quadrivium and the trivium of gram- 
mar, rhetoric, and Zeno's dialectic. Consequently, one may with some justice 
hold that the mathematicians of the Heroic Age were responsible for much of 



Plato and Aristotle in Raphael's "School of Athens." 


the direction in Western educational traditions, especially as transmitted 
through the philosophers of the fourth century B.C. 1 

The fourth century B.C. had opened with the death of Socrates, a scholar 
who adopted the dialectic method of Zeno and repudiated the Pythagorean- 
ism of Archytas. Socrates admitted that in his youth he had been attracted by 
such questions as why the sum 2 + 2 was the same as the product 2 x 2, as 
well as by the natural philosophy of Anaxagoras ; but upon realizing that 
neither mathematics nor science could satisfy his desire to know the essence 
of things, he gave himself up to his characteristic search for the good. 

In the Phaedo of Plato, the dialogue in which the last hours of Socrates are 
so beautifully described, we see how deep metaphysical doubts precluded a 
Socratic concern with either mathematics or natural science. 

I cannot satisfy myself that, when one is added to one, the one to which the addition 
is made becomes two, or that the two units added together make two by reason of the 
addition. I cannot understand how when separated from the other, each of them was one 
and not two, and now, when they are brought together, the mere juxtaposition or 
meeting of them should be the cause of their becoming two. 2 

Hence the influence of Socrates in the development of mathematics was 
negligible, if not actually negative. This makes it all the more surprising that 
it was his student and admirer, Plato, who became the mathematical inspira- 
tion of the fourth century b.c. We shall concentrate in this chapter on the 
mathematical achievements of half a dozen men who lived between the 
death of Socrates in 399 b.c. and the death of Aristotle in 322 B.C. The six 
men whose work we shall describe (in addition to that of Plato and Aristotle) 
are Theodorus of Cyrene (fl. ca. 390 B.C.), Theaetetus (T368 B.C.), Eudoxus of 
Cnidus (t ca. 355 B.C.), Menaechmus (fl. ca. 350 B.C.) and his brother Dino- 
stratus (fl. ca. 350 B.C.), and Autolycus of Pitane (fl. ca. 330 B.C.). 

The six mathematicians were not scattered throughout the Greek world, 
as had been those in the fifth century B.C. ; they were associated more or less 
closely with the Academy of Plato at Athens. Although Plato himself made 
no outstanding specific contribution to technical mathematical results, he 
was the center of the mathematical activity of the time and guided and 
inspired its development. Over the doors of his school was inscribed the 
motto, "Let no one ignorant of geometry enter here" ; his enthusiasm for the 

1 The firm establishment of this particular group of seven liberal arts was, however, achieved 
only in the fourth century of our era, and the recognized division of these into the trivium and 
quadrivium became traditional only with the Carolingian renaissance. Marshall Clagett, in 
Greek Science in Antiquity, 2nd ed. New York: Collier, 1966, p. 185, writes that the use of the 
Latin term quadrivium seems to stem from Boethius (ca. 480-524). 

2 Dialogues of Plato (1875), I, 476-477. 


subject led him to become known not as a mathematician, but as "the maker 
of mathematicians." It is clear that Plato's high regard for mathematics did 
not come from Socrates; in fact, the earlier Platonic dialogues seldom refer 
to mathematics. The one who converted Plato to a mathematical outlook 
undoubtedly was Archytas, a friend whom he visited in Sicily in 388 B.C. 
Perhaps it was there that he learned of the five regular solids, which were 



associated with the four elements of Empedocles in a cosmic scheme that 
fascinated men for centuries. Possibly it was the Pythagorean regard for the 
dodecahedron that led Plato to look on this, the fifth and last, regular solid as 
a symbol of the universe. Plato put his ideas on the regular solids into a 
dialogue entitled the Timaeus, presumably named for a Pythagorean who 
serves as the chief interlocutor. It is not known whether Timaeus of Locri 
really existed or whether Plato invented him as a character through whom to 
express the Pythagorean views that still were strong in what is now Southern 
Italy. The regular polyhedra have often been called "cosmic bodies" or 
"Platonic solids" because of the way in which Plato in the Timaeus applied 
them to the explanation of scientific phenomena. Although this dialogue, 
probably written when Plato was near seventy, provides the earliest definite 
evidence for the association of the four elements with the regular solids, 
much of this fantasy may be due to the Pythagoreans. Proclus attributes the 
construction of the cosmic figures to Pythagoras ; but the scholiast Suidas 
reported that Plato's friend Theaetetus, born about 414 B.C. and the son of one 
of the richest patricians in Attica, first wrote on them. A scholium (of un- 
certain date) to Book XIII of Euclid's Elements reports that only three of 
the five solids were due to the Pythagoreans, and that it was through 
Theaetetus that the octahedron and icosahedron became known. It seems 
likely that in any case Theaetetus made one of the most extensive studies of 
the five regular solids, and to him probably is due the theorem that there are 
five and only five regular polyhedra. Perhaps he is responsible also for the 
calculations in the Elements of the ratios of the edges of the regular solids to 
the radius of the circumscribed sphere. 


Theaetetus was a young Athenian who died in 369 B.C. from a combination 
of wounds received in battle and of dysentery, and the Platonic dialogue 
bearing his name was a commemorative tribute by Plato to his friend. In the 
dialogue, purporting to take place some thirty years earlier, Theaetetus 
discusses with Socrates and Theodorus the nature of incommensurable 
magnitudes. It has been assumed that this discussion took somewhat the 
form that we find in the opening of Book X of the Elements. Here distinctions 
are made not only between commensurable and incommensurable magni- 
tudes, but also between those that while incommensurable in length are, or 
are not, commensurable in square. Surds such as ^3 and ^/s are incom- 
mensurable in length, but they are commensurable in squa re, for th eir 
squares have the ratio 3 to 5. The magnitudes ^/l + ^/3 and y/l + N /5, on 
the other hand, are incommensurable both in length and in square. 

The dialogue that Plato composed in memory of his friend, Theaetetus, 
contains information on another mathematician whom Plato admired and 
who contributed to the early development of the theory of incommensurable 
magnitudes. Reporting on the then recent discovery of what we call the 
irrationality of ,/2, Plato in the Theaetetus says that his teacher, Theodorus 
of Cyrene — of whom Theaetetus also was a pupil — was the first to prove the 
irrationality of the square roots of the nonsquare integers from 3 to 17 
inclusive. It is not known how he did this, nor why he stopped with N /l7. 
The proof in any case could have been constructed along the lines of that for 
~Jl as given by Aristotle and interpolated in later versions of Book X of the 
Elements. References in ancient historical works indicate that Theodorus 
made discoveries in elementary geometry that later were incorporated in 
Euclid's Elements ; but the works of Theodorus are lost. 

Plato is important in the history of mathematics largely for his role as 
inspirer and director of others, and perhaps to him is due the sharp distinction 
in ancient Greece between arithmetic (in the sense of the theory of numbers) 
and logistic (the technique of computation). Plato regarded logistic as 
appropriate for the businessman and for the man of war, who "must learn 
the art of numbers or he will not know how to array his troops." The philos- 
opher, on the other hand, must be an arithmetician "because he has to arise 
out of the sea of change and lay hold of true being." Moreover, Plato says 
in the Republic, "arithmetic has a very great and elevating effect, compelling 
the mind to reason about abstract number." So elevating are Plato's thoughts 
concerning number that they reach the realm of mysticism and apparent 
fantasy. In the last book of the Republic he refers to a number that he calls 
"the lord of better and worse births." There has been much speculation 


concerning this "Platonic number," and one theory is that it is the number 
60 4 = 12,960,000 — important in Babylonian numerology and possibly 
transmitted to Plato through the Pythagoreans. In the Laws the number of 
citizens in the ideal state is given as 5040 (that is, 7 • 6 ■ 5 • 4 • 3 • 2 • 1). This 
sometimes is referred to as the Platonic nuptial number, and various theories 
have been advanced to suggest what Plato had in mind. 

As in arithmetic Plato saw a gulf separating the theoretical and computa- 
tional aspects, so also in geometry he espoused the cause of pure mathematics 
as against the materialistic views of the artisan or technician. Plutarch, in his 
Life of Marcellus, speaks of Plato's indignation at the use of mechanical 
contrivances in geometry. Apparently Plato regarded such use as "the mere 
corruption and annihilation of the one good of geometry, which was thus 
shamefully turning its back upon the unembodied objects of pure intel- 
ligence." Plato may consequently have been largely responsible for the 
prevalent restriction in Greek geometrical constructions to those that can be 
effected by straightedge and compasses alone. The reason for the limitation 
is not likely to have been the simplicity of the instruments used in constructing 
lines and circles, but rather the symmetry of the configurations. Any one of the 
infinitely many diameters of a circle is a line of symmetry of the figure ; any 
point on an infinitely extended straight line can be thought of as a center of 
symmetry, just as any line perpendicular to the given line is a line with respect 
to which the given line is symmetric. Platonic|philosophy, with its,apotheosiza- 
tion of ideas, would quite naturally find a favored role for the line and the 
circle among geometrical figures. In a somewhat similar manner Plato 
glorified the triangle. The faces of the five regular solids in Plato's view were 
not simple triangles, squares, and pentagons. Each of the four faces of the 
tetrahedron, for example, is made up of six smaller right triangles formed by 
altitudes of the equilateral triangular faces. The regular tetrahedron he 
therefore thought of as made up of twenty-four scalene right triangles in 
which the hypotenuse is double one side; the regular octahedron contains 
8 x 6 or 48 such triangles, and the icosahedron is made up of 20 x 6 or 120 
triangles. In a similar way the hexahedron (or cube) is constructed of twenty- 
four isosceles right triangles, for each of the six square faces contains four 
right triangles when the diagonals of the squares are drawn. 

To the dodecahedron Plato had assigned a special role as representative 
of the universe, cryptically saying that "God used it for the whole" (Timaeus 
55C). 3 Plato looked upon the dodecahedron as composed of 360 scalene 
right triangles, for when the five diagonals and five medians are drawn in 
each of the pentagonal faces, each of the twelve faces will contain thirty 
right triangles. The association of the first four regular solids with the 

3 References here and elsewhere, unless otherwise noted, are to the dialogues of Plato and 
are from Plato, Dialogues, trans, by Benjamin Jowett (Oxford, 1871, 4 vols.). 


traditional four universal elements provided Plato in the Timaeus with a 
beautifully unified theory of matter according to which everything was con- 
structed of ideal right triangles. The whole of physiology, as well as the 
sciences of inert matter, is based in the Timaeus on these triangles. Normal 
growth of the body, for example, is explained as follows : 

When the frame of the whole creature is young and the triangles of its constituent 
bodies are still as it were fresh from the workshop, their joints are firmly locked together. 
. . . Accordingly, since any triangles composing the meat and drink ... are older and 
weaker than its own, with its new-made triangles, it gets the better of them and cuts them 
up, and so causes the animal to wax large. 

In old age, on the other hand, the triangles of the body are so loosened by use 
that "they can no longer cut up into their own likeness the triangles of the 
nourishment as they enter, but are themselves easily divided by the intruders 
from without," and the creature wastes away. 4 

Pythagoras is reputed to have established mathematics as a liberal subject, 
but Plato was influential in making the subject an essential part of the cur- 
riculum for the education of statesmen. Influenced perhaps by Archytas, 
Plato would add to the original subjects in the quadrivium a new subject, 
stereometry, for he believed that solid geometry had not been sufficiently 
emphasized. Plato also discussed the foundations of mathematics, clarified 
some of the definitions, and reorganized the assumptions. He emphasized 
that the reasoning used in geometry does not refer to the visible figures that 
are drawn but to the absolute ideas that they represent. The Pythagoreans 
had defined a point as "unity having position," but Plato would rather think 
of it as the beginning of a line. The definition of a line as "bread thless length" 
seems to have originated in the school of Plato, as well as the idea that a line 
"lies evenly with the points on it." In arithmetic Plato emphasized not only 
the distinction between odd and even numbers, but also the categories 
"even times even," "odd times even," and "odd times odd." Although we 
are told that Plato added to the axioms of mathematics, we do not have an 
account of his premises. 

Few specific mathematical contributions are attributed to Plato. A formula 
for Pythagorean triples— (2n) 2 + (n 2 - l) 2 = (n 2 + l) 2 , where n is any 
natural number — bears Plato's name, but this is merely a slightly modified 
version of a result known to the Babylonians and the Pythagoreans. Perhaps 
more genuinely significant is the ascription to Plato of the so-called analytic 
method. In demonstrative mathematics one begins with what is given, either 
generally in the axioms and postulates or more specifically in the problems 
at hand. Proceeding step by step, one then arrives at the statement that was 

4 Timaeus 81B-81D. Translation is from F. M. Cornford, Plato's Cosmology (1937), p. 329. 


to have been proven. Plato seems to have pointed out that often it is pedago- 
gically convenient, when a chain of reasoning from premises to conclusion is 
not obvious, to reverse the process. One might begin with the proposition 
that is to be proved and from it deduce a conclusion that is known to hold. 
If, then, one can reverse the steps in this chain of reasoning, the result is a 
legitimate proof of the proposition. It is unlikely that Plato was the first to 
note the efficacy in the analytic point of view, for any preliminary investiga- 
tion of a problem is tantamount to this. What Plato is likely to have done is 
to formalize this procedure, or perhaps to give it a name. 

The role of Plato in the history of mathematics is still bitterly disputed. 
Some 5 regarded him as an exceptionally profound and incisive thinker; 
others picture him as a mathematical pied piper who lured men away from 
problems concerning the world's work and encouraged them in idle specula- 
tion. 6 In any case, few would deny that Plato had a tremendous effect on the 
development of mathematics. The Platonic Academy in Athens became the 
mathematical center of the world, and it was from this school that the leading 
teachers and research workers came during the middle of the fourth century 
B.C. Of these the greatest was Eudoxus of Cnidus (4087-355? B.C.), a man who 
was at one time a pupil of Plato and who became the most renowned math- 
ematician and astronomer of his day. 

We sometimes read of the "Platonic reform" in mathematics, and al- 
though the phrase tends to exaggerate the changes taking place, the work 
of Eudoxus was so significant that the word "reform" is not inappropriate. 
In Plato's youth the discovery of the incommensurable had caused a veritable 
logical scandal, for it had raised havoc with theorems involving proportions. 
Two quantities, such as the diagonal and side of a square, are incommensur- 
able when they do not have a ratio such as a (whole) number has to a (whole) 
number. How, then, is one to compare ratios of incommensurable mag- 
nitudes? If Hippocrates really did prove that the areas of circles are to each 
other as squares on their diameters, he must have had some way of handling 
proportions or the equality of ratios. We do not know how he proceeded, or 
whether to some extent he anticipated Eudoxus, who gave a new and generally 
accepted definition of equal ratios. Apparently the Greeks had made use of 
the idea that four quantities are in proportion, a : b = c : d, if the two ratios 
a:b and c:d have the same mutual subtraction. That is, the smaller in each 
ratio can be laid off on the larger the same integral number of times, and the 
remainder in each case can be laid off on the smaller the same integral number 
of times, and the new remainder can be laid off on the former remainder the 

5 See, for example, Francois Lasserre, The Birth of Mathematics in the Age of Plato (1964). 
"Lancelot Hogben, Science for the Citizen (New York: 1938). p. 64. Cf. George Sarton. 
A History of Science (Cambridge, Mass.: Harvard University Press, 1952), Vol. 1, pp. 431 ff. 


same integral number of times, and so on. Such a definition would be awk- 
ward to use, and it was a brilliant achievement of Eudoxus to discover the 
theory of proportion used in Book V of Euclid's Elements. The word ratio 
denoted essentially an undefined concept in Greek mathematics, for Euclid's 
"definition" of ratio as a kind of relation in size between two magnitudes of 
the same type is quite inadequate. More significant is Euclid's statement that 
magnitudes are said to have a ratio to one another if a multiple of either can 
be found to exceed the other. This is essentially a statement of the so-called 
"Axiom of Archimedes" — a property that Archimedes himself attributed to 
Eudoxus. The Eudoxian concept of ratio consequently excludes zero and 
clarifies what is meant by magnitudes of the same kind. A line segment, for 
example, is not to be compared, in terms of ratio, with an area ; nor is an area 
to be compared with a volume. 

Following these preliminary remarks on ratios, Euclid gives in Definition 5 
of Book V the celebrated formulation by Eudoxus : 

Magnitudes are said to be in the same ratio, the first to the second and the third to 
the fourth, when, if any equimultiples whatever be taken of the first and the third, and 
any equimultiples whatever of the second and fourth, the former equimultiples alike 
exceed, are alike equal to, or are alike less than, the latter equimultiples taken in corres- 
ponding order. 7 

That is, a/b = c/d if, and only if, given integers m and n, whenever ma < nb, 
then mc < nd; or if ma = nb, then mc = nd, or if ma > nb, then mc > nd. 

The Eudoxian definition of equality of ratios is not unlike the process of 
cross-multiplication that is used today for fractions — a/b = c/d according as 
ad = be — a process equivalent to a reduction to a common denominator. 
To show that § is equal to f , for example, we multiply 3 and 6 by 4, to obtain 
12 and 24, and we multiply 4 and 8 by 3, obtaining the same pair of numbers 
12 and 24. We could have used 7 and 13 as our two multipliers, obtaining 
the pair 21 and 42 in the first case and 52 and 104 in the second ; and as 21 is 
less than 52, so is 42 less than 104. (We have here interchanged the second and 
third terms in Eudoxus' definition to conform to the common operations 
as usually used today, but similar relationships hold in either case.) Our 
arithmetical example does not do justice to the subtlety and efficacy of 
Eudoxus' thought, for the application here appears to be trivial. To gain a 
heightened appreciation of his definition it would be well to replace a, b, c, d by 
surds or, better still, to let a and b be spheres and c and d cubes on the radii 
of the spheres. Here a cross-multiplication becomes meaningless, and the 
applicability of Eudoxus' definition is far from obvious. In fact, it will be 
noted that, strictly speaking, the definition is not far removed from the 

7 The Thirteen Books of Euclid's Elements, ed. by T. L. Heath (Cambridge, 1908, 3 vols.), 
II, 114. 


nineteenth-century definitions of real number, for it separates the class of 
rational numbers m/n into two categories, according as ma < nb or ma > nb. 
Because there are infinitely many rational numbers, the Greeks by implica- 
tion were faced by the concept that they wished to avoid — that of an infinite 
set — but at least it was now possible to give satisfactory proofs of theorems 
involving proportions. 

8 A crisis resulting from the incommensurable had been successfully met, 
thanks to the imagination of Eudoxus ; but there remained another unsolved 
problem— the comparison of curved and straight-line configurations. Here, 
too, it seems to have been Eudoxus who supplied the key. Earlier math- 
ematicians seem to have suggested that one try inscribing and circumscribing 
rectilinear figures in and about the curved figure and continue to multiply 
indefinitely the number of sides; but they did not know how to clinch the 
argument, for the concept of a limit was unknown at the time. According to 
Archimedes, it was Eudoxus who provided the lemma that now bears 
Archimedes' name — sometimes known as the axiom of continuity— which 
served as the basis for the method of exhaustion, the Greek equivalent of the 
integral calculus. The lemma or axiom states that, given two magnitudes 
having a ratio (that is, neither being zero), one can find a multiple of either 
one which will exceed the other. This statement excluded a fuzzy argument 
about indivisible line segments, or fixed infinitesimals, that was sometimes 
maintained in Greek thought. It also excluded the comparison of the so-called 
angle of contingency or "horn angle" (formed by a curve C and its tangent 
Tat a point P on C) with ordinary rectilinear angles. The horn angle seemed 
to be a magnitude different from zero, yet it does not satisfy the axiom of 
Eudoxus with respect to the measures of rectilinear angles. 

From the axiom of Eudoxus (or Archimedes) it is an easy step, by a 
reductio ad absurdum, to prove a proposition that formed the basis of the 
Greek method of exhaustion : 

If from any magnitude there be subtracted a part not less than its half, and if from the 
remainder one again subtracts not less than its half, and if this process of subtraction is 
continued, ultimately there will remain a magnitude less than any preassigned magnitude 
of the same kind. 8 

This proposition, which we shall refer to as the "exhaustion property," 
is equivalent to the modern statement that if M is a given magnitude, e is a 
preassigned magnitude of the same kind, and r is a ratio such that \ < r < 1, 
then we can find a positive integer N such that M(\ — rf < e for all positive 
integers n > N. That is, the exhaustion property is equivalent to the modern 

8 See Elements of Euclid (ed. by T. L. Heath, reprinted, New York : Dover, 3 vols., 1956), III, 14. 
The axiom is, of course, still legitimate if half is changed to third or quarter or other proper part. 


statement that lim M(l - rf = 0. Moreover, the Greeks made use of this 


property to prove theorems about the areas and volumes of curvilinear 
figures. In particular, Archimedes ascribed to Eudoxus the earliest satisfac- 
tory proof that the volume of the cone is one-third the volume of the cylinder 
having the same base and altitude, a statement that would seem to indicate 
that the method of exhaustion was derived by Eudoxus. If so, then it is to 
Eudoxus (rather than to Hippocrates) that we probably owe the Euclidean 
proofs of theorems concerning areas of circles and volumes of spheres. 
Facile earlier suggestions had been made that the area of a circle could be 
exhausted by inscribing in it a regular polygon and then increasing the 
number of sides indefinitely, but the Eudoxian method of exhaustion first 
made such a procedure rigorous. (It should be noted that the phrase "method 
of exhaustion" was not used by the ancient Greeks, being a modern inven- 
tion ; but the phrase has become so well established in the history of math- 
ematics that we shall continue to make use of it.) As an illustration of the 
way in which Eudoxus probably carried out the method, we give here, in 
somewhat modernized notation, the proof that areas of circles are to each 
other as squares on their diameters. The proof, as it is given in Euclid, 
Elements XII. 2, is probably that of Eudoxus. 

Let the circles be c and C, with diameters d and D and areas a and A. 
It is to be proven that a/ A = d 2 /D 2 . The proof is complete if we proceed 
indirectly and disprove the only other possibilities, namely, a/ A < d 2 /D 2 and 
a/A > d 2 /D 2 . Hence we first assume that a/A > d 2 /D 2 . Then there is a 
magnitude a' < a such that a'/A = d 2 /D 2 . Let a - a' be a preassigned 
magnitude e > 0. Within the circles c and C inscribe regular polygons of 
areas p n and P H , having the same number of sides n, and consider the inter- 
mediate areas outside the polygons but inside the circles (Fig. 6.1). If the 

FIG. 6.1 

number of sides should be doubled, it is obvious that from these intermediate 
areas we would be subtracting more than the half. Consequently, by the 
exhaustion property, the intermediate areas can be reduced through succes- 
sive doubling of the number of sides (that is, by letting n increase) until 


a- p„< e. Then, since a - a' = e, we have p„ > a'. Now, from earlier 
theorems it is known that pJP n = d 2 /D 2 and since it was assumed that 
a' I A = d 2 /D 2 , we have pJP n = a' /A. Hence if p„ > a', as we have shown, we 
must conclude that P„ > A. Inasmuch as P„ is the area of a polygon inscribed 
within the circle of area A, it is obvious that P„ cannot be greater than A. 
Since a false conclusion implies a false premise, we have disproved the 
possibility that a/A > d 2 /D 2 . In an analogous manner we can disprove the 
possibility that a/ A < d 2 /D 2 , thereby establishing the theorem that areas of 
circles are to each other as squares on their diameters. 

The property that we have just demonstrated appears to have been the 
first precise theorem concerning the magnitudes of curvilinear figures; 
it marks Eudoxus as the apparent originator of the integral calculus, the 
greatest contribution to mathematics made by associates of the Platonic 
Academy. Eudoxus, moreover, was by no means a mathematician only, and 
in the history of science he is known as the father of scientific astronomy. 
Plato is said to have proposed to his associates that they attempt to give a 
geometrical representation of the movements of the sun, the moon, and the 
five known planets. It evidently was tacitly assumed that the movements were 
to be compounded of uniform circular motions. Despite such a restriction, 
Eudoxus was able to give for each of the seven heavenly bodies a satisfactory 
representation through a composite of concentric spheres with centers at 
the earth and with varying radii, each sphere revolving uniformly about an 
axis fixed with respect to the surface of the next larger sphere. For each planet, 
then, Eudoxus gave a system known to his successors as "homocentric 
spheres" ; these geometrical schemes were combined by Aristotle into the 
well-known Peripatetic cosmology of crystalline spheres that dominated 
thought for almost 2000 years. 

Eudoxus was without doubt the most capable mathematician of the Hel- 
lenic Age, but all of his works have been lost. 9 It is possible that the Aristotel- 
ian estimate for the circumference of the earth— about 400,000 stades, or 
40,000 miles — is due to Eudoxus, for Archimedes reported that Eudoxus had 
calculated that the diameter of the sun was nine times that of the earth. In 
his astronomical scheme Eudoxus had seen that by a combination of circular 
motions he could describe the motions of the planets in looped orbits along 
a curve known as the hippopede or horse fetter. This curve, resembling a 
figure eight on a sphere, is obtained as the intersection of a sphere and a 
cylinder tangent internally to the sphere— one of the few new curves that the 
Greeks recognized. At the time there were only two means of defining curves : 

9 For an extensive and authoritative account of what Eudoxus probably did, see O. Becker, 
"Eudoxus-Studien," Quellen und Studien zur Geschichte der Mathematik, Part B, II (1933), 
311-333, 369-387; III (1936), 236-244, 370-410. 


(1) through combinations of uniform motions and (2) as the intersections 
of familiar geometric surfaces. The hippopede of Eudoxus is a good example 
of a curve that is derivable in either of these two ways. Proclus, who wrote 
some 800 years after the time of Eudoxus, reported that Eudoxus had added 
many general theorems in geometry and had applied the Platonic method of 
analysis to the study of the section (probably the golden section) ; but the 
two chief claims to fame of Eudoxus remain the theory of proportions and the 
method of exhaustion. 

Eudoxus is to be remembered in the history of mathematics not only for 1 
his own work, but also through that of his pupils. In Greece there was a strong 
thread of continuity of tradition from teacher to student. Thus Plato learned 
from Archytas, Theodorus, and Theaetetus ; the Platonic influence in turn 
was passed on through Eudoxus to the brothers Menaechmus and Dinostra- 
tus, both of whom achieved eminence in mathematics. We saw that Hippo- 
crates of Chios had shown that the duplication of the cube could be achieved 
provided that one could find, and was permitted to use, curves with the 
properties expressed in the continued proportion a/x = x/y = y/2a; we 
noted also that the Greeks had only two approaches to the discovery of new 
curves. It was consequently a signal achievement on the part of Menaechmus 
when he disclosed that curves having the desired property were near at hand. 
In fact, there was a family of appropriate curves obtainable from a single 
source — the cutting of a right circular cone by a plane perpendicular to an 
element of the cone. That is, Menaechmus is reputed to have discovered the 
curves that were later known as the ellipse, the parabola, and the hyperbola. 

Of all the curves, other than circles and straight lines, that are apparent to 
the eye in everyday experience, the ellipse should be the most obvious, for it 
is present by implication whenever a circle is viewed obliquely or whenever 
one saws diagonally through a cylindrical log. Yet the first discovery of the 
ellipse seems to have been made by Menaechmus as a mere by-product in a 
search in which it was the parabola and hyperbola which proffered the 
properties needed in the solution of the Delian problem. Beginning with a 
single-napped right circular cone having a right angle at the vertex (that is, 
a generating angle of 45°), Menaechmus found that when the cone is cut by a 
plane perpendicular to an element, the curve of intersection is such that, in 
terms of modern analytic geometry, its equation can be written in the form 
y 2 = Ix, where / is a constant depending on the distance of the cutting plane 
from the vertex. We do not know how Menaechmus derived this property, 
but it depends only on theorems from elementary geometry. Let the cone be 
ABC and let it be cut in the curve EDG by a plane perpendicular to the element 
ADC of the cone (Fig. 6.2). Then through P, any point on the curve, pass a 
horizontal plane cutting the cone in the circle PVR, and let Q be the other 



FIG. 6.2 

point of intersection of the curve (parabola) and the circle. From the sym- 
metries involved it follows that line PQ 1 RV at O. Hence OP is the mean 
proportional between RO and OV. Moreover, from the similarity of triangles 
OVD and BCA it follows that OV/DO = BC/AB, and from the similarity 
of triangles R'DA and ABC it follows that R'D/AR' = BC/AB. If OP <= y and 
OD = x are coordinates of point P, we have y 2 = RO ■ OV, or, on substituting 

BC BC AR' ■ BC 2 

Inasmuch as segments -4/t', BC, and AB are the same for all points P on the 
curve EQDPG, we can write the equation of the curve, a "section of a right- 
angled cone," as y 1 = Ix, where / is a constant, later to be known as the 
latus rectum of the curve. In an analogous way we can derive an equation of 
the form v 2 = foe - b 2 x 2 /a 2 for a "section of an acute-angled cone" and an 
equation of the form y 1 = ix + b 2 x 2 /a 2 for a "section of an obtuse-angled 
cone," where a and b are constants and the cutting plane is perpendicular 
to an element of the acute-angled or obtuse-angled right circular cone. 

Menaechmus apparently derived these properties of the conic sections, 
and others as well. Since this material has a strong resemblance to the use of 
coordinates, as illustrated above, it has sometimes been maintained that 
Menaechmus had analytic geometry. 10 Such a judgment is warranted only in 
part, for certainly Menaechmus was unaware that any equation in two 
unknown quantities determines a curve. In fact, the general concept of an 
equation in unknown quantities was alien to Greek thought. It was short- 

10 See J. L. Coolidge, A History of Geometrical Methods (1940), pp. U7-1 19, and H. G. Zeulhen. 
"Sur 1' usage des coordonnees dans l'antiquite," Kongeiige Danske Videxsk&eHtes Setskahs, 
rorhamtlinger. Overset, 1888, pp. 127-144. 



comings in algebraic notations that, more than anything else, operated 
against the Greek achievement of a full-fledged coordinate geometry. 

Menaechmus had no way of foreseeing the hosts of beautiful properties 
that the future was to disclose. He had hit upon the conies in a successful 
search for curves with the properties appropriate to the duplication of the 
cube. In terms of modern notation the solution is easily achieved. By shifting 
the cutting plane (Fig. 6.2), we can find a parabola with any latus rectum. 
If, then, we wish to duplicate a cube of edge a, we locate on a right-angled 
cone two parabolas, one with latus rectum a and another with latus rectum 
2a. If, then, we place these with vertices at the origin and with axes along the 
y- and x-axes respectively, the point of intersection of the two curves will 
have coordinates (x, y) satisfying the continued proportion a/x = x/y = y/2a 
(Fig. 6.3}— that is, x = a*/l, y = a^4. The x-coordinate therefore is the 
edge of the cube sought. 

y* = 2ax 

FIG. 6.3 

It is probable that Menaechmus knew that the duplication could be 
achieved also by the use of a rectangular hyperbola and a parabola. If the 
parabola with equation y 2 = (a/2)x and the hyperbola xy = a 2 are placed 
on a common coordinate system, the point of intersection will have co- 
ordinates x = a*/l, y = a/j/l, the x-coordinate being the side of the cube 
desired. Menaechmus probably was acquainted with many of the now 
familiar properties of the conic sections, including the asymptotes of the 
hyperbola which would have permitted him to operate with the equivalents 
of the modern equations that we used above. Proclus reported that Menaech- 
mus was one of those who "made the whole of geometry more perfect" ; but 
we know little concerning his actual work. We do know that Menaechmus 
taught Alexander the Great, and legend attributes to Menaechmus the 
celebrated comment, when his royal pupil asked for a shortcut to geometry : 
"O King, for travelling over the country there are royal roads and roads for 
common citizens; but in geometry there is one road for all." Among the 
chief authorities for attributing to Menaechmus the discovery of conic 




sections is a letter from Eratosthenes to King Ptolemy Euergetes, quoted 
some 700 years later by Eutocius, in which several duplications of the cube 
are mentioned. Among them is one by Archytas' unwieldy construction and 
another by "cutting the cone in the triads of Menaechmus." 

12 Dinostratus, brother of Menaechmus, was also a mathematician, and 
where one of the brothers "solved" the duplication of the cube, the other 
"solved" the squaring of the circle. The quadrature became a simple matter 
once a striking property of the end point Q of the trisectrix of Hippias had 
been noted, apparently by Dinostratus. If the equation of the trisectrix 
(Fig. 6.4) is nr sin 9 — 2a9, where a is the side of the square ABCD associated 
with the curve, the limiting value of r as 9 tends toward zero is 2a/%. This is 

obvious to one who has had calculus and recalls that lim sin 9/9 = 1 for 

radian measure. The proof as given by Pappus, and probably due to 

Dinostratus, is based only on considerations from elementary geometry. 

The theorem of Dinostratus states that side a is the mean proportional 

between the segment DQ and the arc of the quarter circle AC — that is, 

AC/AB = AB/DQ. Using a typically Greek indirect proof, we establish the 

theorem by demolishing the alternatives. Hence assume first that AC/AB = 

AB/DR where DR > DQ. Then let the circle with center D and radius DR 

intersect the trisectrix at S and side AD of the square at T. From S drop the 

perpendicular SU to side CD. Inasmuch as it was known to Dinostratus 

that corresponding arcs of circles are to each other as the radii, we have 

AC/AB = fk/DR ; and since by hypothesis AC/AB = AB/DR, it follows 

that TR = AB. But from the definitional property of the trisectrix it is 

known that TR/SR = AB/SU. Hence, since TR = AB, it must follow that 

SR = SU, which obviously is false, since the perpendicular is shorter than 

any other line or curve from point S to line DC. Hence the fourth term DR 

in the proportion AC/AB = AB/DR cannot be greater than DQ. In a similar 

manner we can prove that this fourth proportional cannot be less than DQ ; 

hence Dinostratus' theorem is established— that is, AC/AB = AB/DQ. 


Given the intersection point Q of the trisectrix with DC, we then have a 
proportion involving three straight-line segments and the circular arc AC. 
Hence by a simple geometric construction of the fourth term in a proportion, 
a line segment b equal in length to AC can be easily drawn. Upon drawing 
a rectangle with 2b as one side and a as the other, we have a rectangle exactly 
equal in area to the area of the circle with radius a ; a square equal to the 
rectangle is easily constructed by taking as the side of the square the geometric 
mean of the sides of the rectangle. Inasmuch as Dinostratus showed that the 
trisectrix of Hippias serves to square the circle, the curve more commonly 
came to be known as the quadratrix. It was, of course, always clear to the 
Greeks that the use of the curve in the trisection and quadrature problems 
violated the rules of the game— that circles and straight lines only were 
permitted. The "solutions" of Hippias and Dinostratus, as their authors 
realized, were sophistic ; hence the search for further solutions, canonical or 
illegitimate, continued, with the result that several new curves were discovered 
by Greek geometers. 

A few years after Dinostratus and Menaechmus there nourished a math- 1 3 
ematician who has the distinction of having written the oldest surviving 
Greek mathematical treatise. We have described rather fully the work of 
earlier Hellenic mathematicians, but it must be borne in mind that the 
accounts have been based not on original works, but on later summaries, 
commentaries, or descriptions. Occasionally a commentator appears to be 
copying from an original work extant at the time, as when Simplicius in the 
sixth century of our era is describing the quadrature of lunes by Hippocrates. 
But not until we come to Autolycus of Pitane, a contemporary of Aristotle, 
do we find a Greek author one of whose works has survived. One reason for 
the survival of this little treatise, On the Moving Sphere, is that it formed part 
of a collection, known as the "Little Astronomy," widely used by ancient 
astronomers. On the Moving Sphere is not a profound, and probably not a very 
original work, for it includes little beyond elementary theorems on the geom- 
etry of the sphere that would be needed in astronomy. Its chief significance 
lies in the fact that it indicates that Greek geometry evidently had reached 
the form that we regard as typical of the classical age. Theorems are clearly 
enunciated and proved. Moreover, the author uses without proof or indica- 
tion of source other theorems that he regards as well known ; we conclude, 
therefore, that there was in Greece in his day, about 320 B.C., a thoroughly 
established textbook tradition in geometry. 

Autolycus was a contemporary of Aristotle — the most widely learned 1 4 
scholar of all times, whose death is usually taken to mark the end of the 
first great period, the Hellenic Age, in the history of Greek civilization. 


Aristotle, like Eudoxus, was a student of Plato and, like Menaechmus, 
a tutor of Alexander the Great. Aristotle was primarily a philosopher and 
biologist, but he was thoroughly au courant with the activities of the math- 
ematicians. He may have taken a role in one of the leading controversies of 
the day, for to him was ascribed a treatise On Indivisible Lines. Modern 
scholarship questions the authenticity of this work, but in any case it probably 
was the result of discussions carried on in the Aristotelian Lyceum. The thesis 
of the treatise is that the doctrine of indivisibles espoused by Xenocrates, a 
successor of Plato as head of the Academy, is untenable. The indivisible, or 
fixed infinitesimal of length or area or volume, has fascinated men of many 
ages; Xenocrates thought that this notion would resolve the paradoxes, 
such as those of Zeno, that plagued mathematical and philosophical thought. 
Aristotle, too, devoted much attention to the paradoxes of Zeno, but he 
sought to refute them on the basis of common sense. Inasmuch as he hesitated 
to follow Platonic mathematicians into the abstractions and technicalities 
of the day, Aristotle made no lasting contribution to the subject. He is said 
to have written a biography of Pythagoras, although this is lost; and 
Eudemus, one of his students, wrote a history of geometry, also lost. More- 
over, through his foundation of logic and through his frequent allusion to 
mathematical concepts and theorems in his voluminous works, 11 Aristotle 
can be regarded as having contributed to the development of mathematics. 
The Aristotelian discussion of the potentially and actually infinite in arith- 
metic and geometry influenced many later writers on the foundations of 
mathematics ; but Aristotle's statement that the mathematicians "do not need 
the infinite or use it" should be compared with the assertions of our day that 
the infinite is the mathematician's paradise. Of more positive significance 
are Aristotle's analysis of the roles of definitions and hypotheses in math- 

15 In 323 B.C. Alexander the Great suddenly died, and his empire fell apart. 
His generals divided the territory over which the young conqueror had ruled ; 
Ptolemy took Egypt, Seleucus and Lysimachus vied for Syria and the East, 
and Antigonus and Cassander each for a while ruled Macedon. At Athens, 
where Aristotle had been regarded as a foreigner, the philosopher found 
himself unpopular, now that his powerful soldier-student was dead. He left 
Athens and died the following year. Throughout the Greek world the old 
order was changing, politically and culturally. Under Alexander there had 
been a gradual blending of Hellenic and Oriental customs and learning, so 
that it was more appropriate to speak of the newer civilization as Hellenistic, 
rather than Hellenic. Moreover, the new city of Alexandria, established by the 

1 ' See T. L. Heath, Mathematics in Aristotle (1949). 


world conqueror, now took the place of Athens as the center of the math- 
ematical world. In the history of civilization it is therefore customary to 
distinguish two periods in the Greek world, with the almost simultaneous 
deaths of Aristotle and Alexander (as well as that of Demosthenes) as a 
convenient dividing line. The earlier portion is known as the Hellenic Age, 
the later as the Hellenistic or Alexandrian Age; in the next few chapters we 
describe the mathematics of the first century of the new era, often known as 
the Golden Age of Greek mathematics. 


Becker, O., "Eudoxus-Studien," Quellen und Studien zur Geschichte der Mathematik, 

Part B, Studien, II (1933), 311-333, 369-387 ; III (1936), 236-244, 370-410. 
Brumbaugh, R. S., Plato's Mathematical Imagination (Bloomington, Ind.: Indiana 

University Press, 1954). 
Coolidge, J. L., A History of the Conic Sections and the Quadric Surfaces (Oxford • 

Clarendon, 1945). 
Coolidge, J. L., A History of Geometrical Methods (Oxford : Clarendon, 1940 ; paperback 

ed., New York : Dover, 1963). 
Cornford, F. M., Plato's Cosmology, The Timaeus of Plato translated with a running 

commentary (London : Routledge and Kegan Paul, 1937). 
Gorland, Albert, Aristoteles und die Mathematik (Marburg, 1899). 
Heath, T. L., History of Greek Mathematics (Oxford, 1921, 2 vols.). 
Heath, T. L., Mathematics in Aristotle (Oxford, 1949). 
Heiberg, J. L., "Mathematisches zu Aristoteles," Abhandlungen zur Geschichte der 

Mathematischen Wissenschaften, 18 (1904), 1-49. 
Lasserre, Francois, The Birth of Mathematics in the Age of Plato, trans, by Helen 

Mortimer (London: Hutchinson, 1964). 
Loria, Gino, Historie des sciences mathematiques dans I'antiquite hellenique (Paris: 

Gauthier-Villars, 1929). 
Michel, Paul-Henri, De Pythagore a Euclide (Paris: Societe d'Edition "Les Belles 

Lettres," 1950). 
Plato, Dialogues, trans, by Benjamin Jowett (Oxford, 1871, 4 vols.). 
Solmsen, Friedrich, "Platos Einfluss auf die Bildung der mathematischen Methode," 

Quellen und Studien zur Geschichte der Mathematik, Part B, Studien, I (1929-1931) 

Toeplitz, Otto, "Das Verhaltnis von Mathematik und Ideenlehre bei Plato," Quellen 

und Studien zur Geschichte der Mathematik, Part B, Studien, I (1931), 3-33. 
Wedberg, Anders, Plato's Philosophy of Mathematics (Stockholm : Almquist & Wiksell 

Zeuthen, H. G., Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886). 



1. It is believed that Theaetetus found for the regular solids the ratio of the edge to the radius 
of the circumscribed sphere. Do this for three of the regular solids. (See Book XIII of Euclid's 

2. Prove the theorem, probably due to Theaetetus, that there are not more than five regular 
solids. (See Book XIII of Euclid's Elements.) 

3. Plato in the Theaetetus says that Theodorus proved ^3 irrational. Give a careful proof of 
this theorem. 

4. Find the angles of the 360 scalene right triangles that Plato indicated on the surface of the 

5. Complete the other half of the proof by exhaustion (see text) that the areas of circles are to 
each other as squares on their radii. (Use circumscribed polygons.) 

6. Describe a method by which Eudoxus could have measured the circumference of the earth. 

7. Using the method suggested in connection with the work of Menaechmus, prove that the 
section of a cylinder is an ellipse. This was proved by Serenus, who probably lived in the 
fourth century of our era. 

8. In his theory of the rainbow Aristotle used a locus commonly attributed to Apollomus, a 
later mathematician : the locus of all points P such that the distances of P from two fixed 
points ?! and P 2 are in a fixed ratio different from one. Identify the locus. 

*9. Prove that a Menaechmean section (perpendicular to an element) of an acute-angled cone 
is an ellipse. 
*10. Complete the proof by Dinostratus (see text) by showing that the assumption DR < DQ leads 

to an absurdity. 


Euclid of Alexandria 

Ptolemy once asked Euclid whether there was any 
shorter way to a knowledge of geometry than by a 
study of the Elements, whereupon Euclid answered 
that there was no royal road to geometry. 

Proclus Diadochus 

The death of Alexander the Great had led to internecine strife among the 
generals in the Greek army ; but by 306 B.C. control of the Egyptian portion 
of the empire was firmly in the hands of Ptolemy I, and this enlightened 
ruler was able to turn his attention to constructive efforts. Among his early 
acts was the establishment at Alexandria of a school or institute, known as 
the Museum, second to none in its day. As teachers at the school he called a 
band of leading scholars, among whom was the author of the most fabulously 
successful mathematics textbook ever written — the Elements (Stoichia) of 
Euclid. Considering the fame of the author and of his best seller, remarkably 
little is known of Euclid's life. So obscure was his life that no birthplace is 
associated with his name. Although editions of the Elements often bore the 
identification of the author as Euclid of Megara and a portrait of Euclid of 
Megara often appears in histories of mathematics, this is a case of mistaken 
identity. * The real Euclid of Megara was a student of Socrates and, although 
concerned with logic, was no more attracted to mathematics than was his 
teacher. Our Euclid, by contrast, is known as Euclid of Alexandria, for he was 
called there to teach mathematics. From the nature of his work it is presumed 
that he had studied with students of Plato, if not at the Academy itself. 
Legends associated with Euclid picture him as a kindly and gentle old man. 
The tale related above in connection with a request of Alexander the Great 
for an easy introduction to geometry is repeated in the case of Ptolemy, 
whom Euclid is reported to have assured that "there is no royal road to 
geometry." Evidently Euclid did not stress the practical aspects of his subject, 
for there is a tale told of him that when one of his students asked of what use 
was the study of geometry, Euclid asked his slave to give the student three- 
pence, "since he must needs make gain of what he learns." 

* See, for example, the title-page on p. 298 below. 111 


Euclid and the Elements are often regarded as synonymous; in reality the 
man was the author of about a dozen treatises covering widely varying topics, 
from optics, astronomy, music, and mechanics to a book on the conic sections. 
With the exception of the Sphere of Autolycus, surviving works by Euclid are 
the oldest Greek mathematical treatises extant ; yet of what Euclid wrote 
more than half has been lost, including some of his more important com- 
positions, such as a treatise on conies. Euclid regarded Aristaeus, a con- 
temporary geometer, as deserving great credit for having written an earlier 
treatise on Solid Loci (the Greek name for the conic sections, stemming 
presumably from the stereometric definition of the curves in the work of 
Menaechmus). The treatises on conies by Aristaeus and Euclid have both 
been lost, probably irretrievably, perhaps because they were soon superseded 
by the more extensive work on conies by Apollonius to be described below. 
Among Euclid's lost works are also one on Surface Loci, another on Pseudaria 
(or fallacies), and a third on Porisms. It is not even clear from ancient refer- 
ences what material these contained. The first one, for example, might have 
concerned the surfaces known to the ancients — the sphere, cone, cylinder, 
tore, ellipsoid of revolution, paraboloid of revolution, and hyperboloid of 
revolution of two sheets — or perhaps curves lying on these surfaces. As far 
as we know, the Greeks did not study any surface other than that of a solid 
of revolution. 

The loss of the Euclidean Porisms is particularly tantalizing, for it may have 
represented an ancient approximation to an analytic geometry. Pappus later 
reported that a porism is intermediate between a theorem, in which some- 
thing is proposed for demonstration, and a problem, in which something is 
proposed for construction. Others have described a porism as a proposition 
in which one determines a relationship between known and variable or 
undetermined quantities, perhaps the closest approach in antiquity to the 
concept of function. If a porism was, as has been thought, a sort of verbal 
equation of a curve, Euclid's book on Porisms may have differed from our 
analytic geometry largely in the lack of algebraic symbols and techniques. 
The nineteenth-century historian of geometry, Michel Chasles, suggested as 
a typical Euclidean porism the determination of the locus of a point for 
which the sum of the squares of its distances from two fixed points is a 

Five works by Euclid have survived to our day: the Elements, the Data, 
the Division of Figures, the Phaenomena, and the Optics. The last-mentioned 
is of interest as an early work on perspective, or the geometry of direct vision. 
The ancients had divided the study of optical phenomena into three parts : 
(1) optics (the geometry of direct vision), (2) catoptrics (the geometry of 
reflected rays), and (3) dioptrics (the geometry of refracted rays). A Catoptrica 


sometimes ascribed to Euclid is of doubtful authenticity, being perhaps by 
Theon of Alexandria who lived some six centuries later. Euclid's Optics 1 is 
noteworthy for its espousal of an "emission" theory of vision according to 
which the eye sends out rays that travel to the object, in contrast to a rival 
Aristotelian doctrine in which an activity in a medium travels in a straight 
line from the object to the eye. It should be noted that the mathematics of 
perspective (as opposed to the physical description) is the same no matter 
which of the two theories is adopted. Among the theorems found in Euclid's 
Optics is one widely used in antiquity — tan a/tan fi < a/p if < a < (i < n/2. 
One object of the Optics was to combat an Epicurean insistence that an 
object was just as large as it looked, with no allowance to be made for the 
foreshortening suggested by perspective. 

Euclid's Phaenomena is much like the Sphere of Autolycus — that is, a work 
on spherical geometry of use to astronomers. A comparison of the two works 
indicates that both authors drew heavily on a textbook tradition that was we.ll 
known to their generation. It is quite possible that much the same was true of 
Euclid's Elements, but in this case there is no contemporary work extant with 
which it can be compared. 

The Euclidean Division of Figures is significant in that it is a work that 
would have been lost, had it not been for the learning of Arabic scholars. 
It has not survived in the original Greek ; but before the disappearance of 
the Greek versions, an Arabic translation had been made (omitting some 
of the original proofs "because the demonstrations are easy"), which in turn 
was later translated into Latin, and ultimately into current modern 
languages. 2 This is not atypical of other ancient works. The Division of Figures 
includes a collection of thirty-six propositions concerning the division of 
plane configurations. For example, Proposition 1 calls for the construction 
of a straight line that shall be parallel to the base of a triangle and shall 
divide the triangle into two equal areas. Proposition 4 requires a bisection 
of a trapezoid abqd (Fig. 7.1) by a line parallel to the bases; the 

FIG. 7.1 

' See M. R. Cohen and I. E. Drabkin : A Source Book in Greek Science (1948), pp. 257 ff. 
2 An English version entitled Euclid's Book on Divisions of Figures was edited by R. C. 
Archibald (1915). 



required line zi is found by determining z such that ze 2 = ^eb 2 + ea 2 ). Other 
propositions call for the division of a parallelogram into two equal parts by 
a line drawn through a given point on one of the sides (Proposition 6) or 
through a given point outside the parallelogram (Proposition 10). The final 
proposition asks for the division of a quadrilateral in a given ratio by a 
line through a point on one of the sides of the quadrilateral. Somewhat 
similar in nature and purpose to the Division of Figures is Euclid's Data, a 
work that has come down to us through both the Greek and the Arabic. 
It seems to have been composed for use at the university of Alexandria, 
serving as a companion volume to the first six books of the Elements in much 
the way that a manual of tables supplements a textbook. It was to be useful 
as a guide to the analysis of problems in geometry in order to discover proofs. 
It opens with fifteen definitions concerning magnitudes and loci. The body 
of the text comprises ninety-five statements concerning the implications of 
conditions and magnitudes that may be given in a problem. The first two 
state that if two magnitudes a and b are given, their ratio is given, and that 
if one magnitude is given and also its ratio to a second, the second magnitude 
is given. There are about two dozen similar statements, serving as algebraic 
rules or formulas. Then follow simple geometrical rules concerning parallel 
lines and proportional magnitudes, reminding the student of the implications 
of the data given in a problem, such as the advice that when two line segments 
have a given ratio, then one knows the ratio of the areas of similar rectilinear 
figures constructed on these segments. Some of the statements are geometrical 
equivalents of the solution of quadratic equations. For example, we are told 
that if a given (rectangular) area AB is laid off along a line segment of given 
length AC (Fig. 7.2) and if the area BC by which the area AB falls short of 

the entire rectangle AD is given, the dimensions of the rectangle BC are 
known. The truth of this statement is easily demonstrated by modern algebra. 
Let the length of AC be a, the area of AB be b 2 , and the ratio of FC to CD be 
c : d. Then if FC = x and CD = y, we have x/y = c/d and (a — x)y = b 2 . 
Eliminating y we have (a — x)dx = b 2 c or dx 2 — adx + b 2 c = 0, from which 

x = a/2 ± ^/(a/2) 2 — b 2 c/d. The geometric solution given by Euclid is 
equivalent to this, except that the negative sign before the radical is used. 
Statements 84 and 85 in the Data are geometrical replacements of the familiar 


Babylonian algebraic solutions of the systems xy = a 2 , x ± y = b, which 
again are the equivalents of solutions of simultaneous equations. The last 
few statements in the Data concern relationships between linear and angular 
measures in a given circle. 

The university at Alexandria evidently was not unlike modern institutions 3 
of higher learning. Some of the faculty probably excelled in research, others 
were better fitted to be administrators, and still others were noted for teaching 
ability. It would appear, from the reports we have, that Euclid very definitely 
fitted into the last category. There is no new discovery attributed to him, but 
he was noted for expository skill. This is the key to the success of his greatest 
work, the Elements. It was frankly a textbook and by no means the first one. 
We know of at least three earlier such elements, including that by Hippocrates 
of Chios ; but there is no trace of these, nor of other potential rivals from 
ancient times. The Elements of Euclid so far outdistanced competitors that 
it alone survived. The Elements was not, as is sometimes thought, a com- 
pendium of all geometrical knowledge ; it was instead an introductory text- 
book covering all elementary mathematics — that is, arithmetic (in the sense 
of the English "higher arithmetic" or the American "theory of numbers"), 
synthetic geometry (of points, lines, planes, circles, and spheres), and algebra 
(not in the modern symbolic sense, but an equivalent in geometrical garb). 
It will be noted that the art of calculation is not included, for this was not a 
part of university instruction ; nor was the study of the conies or higher 
plane curves part of the book, for these formed a part of more advanced 
mathematics. Proclus described the Elements as bearing to the rest of 
mathematics the same sort of relation as that which the letters of the alphabet 
have in relation to language. Were the Elements intended as an exhaustive 
store of information, the author probably would have included references 
to other authors, statements of recent research, and informal explanations. 
As it is, the Elements is austerely limited to the business in hand — the exposi- 
tion in logical order of the fundamentals of elementary mathematics. Occa- 
sionally, however, later writers interpolated into the text explanatory scholia, 
and such additions were copied by later scribes as part of the original text. 
Some of these appear in every one of the manuscripts now extant. Euclid 
himself made no claim to originality, and it is clear that he drew heavily 
from the works of his predecessors. It is believed that the arrangement is his 
own, and presumably some of the proofs were supplied by him ; but beyond 
this it is difficult to estimate the degree of originality that is to be found in 
this, the most renowned mathematical work in history. 

The Elements is divided into thirteen books or chapters, of which the first 4 
half dozen are on elementary plane geometry, the next three on the theory 


of numbers, Book X on incommensurables, and the last three chiefly on solid 
geometry. There is no introduction or preamble to the work, and the first 
book opens abruptly with a list of twenty-three definitions. The weakness 
here is that some of the definitions do not define, inasmuch as there is no 
prior set of undefined elements in terms of which to define the others. Thus 
to say, as does Euclid, that "a point is that which has no part," or that 
"a line is breadthless length," or that "a surface is that which has length and 
breadth only," is scarcely to define these entities, for a definition must be 
expressed in terms of things that precede, and are better known than the things 
defined. Objections can easily be raised on the score of logical circularity to 
other so-called "definitions" of Euclid, such as "The extremities of a line are 
points," or "A straight line is a line which lies evenly with the points on itself," 
or "The extremities of a surface are lines," all of which may have been due 
to Plato. The Euclidean definition of a plane angle as "the inclination to one 
another of two lines in a plane which meet one another and do not lie in a 
straight line" is vitiated by the fact that "inclination" has not been previously 
defined and is not better known than the word "angle." 

Following the definitions, Euclid lists five postulates and five common 
notions. Aristotle had made a sharp distinction between axioms (or common 
notions) and postulates ; the former, he said, must be convincing in them- 
selves — truths common to all studies— but the latter are less obvious and 
do not presuppose the assent of the learner, for they pertain only to the 
subject at hand. Some later writers distinguished between the two types of 
assumptions by applying the word axiom to something known or accepted 
as obvious, while the word postulate referred to something to be "demanded." 
We do not know whether Euclid subscribed to either of these views, or even 
whether he distinguished between two types of assumptions. Surviving 
manuscripts are not in agreement here, and in some cases the ten assumptions 
appear together in a single category. Modern mathematicians see no essential 
difference between an axiom and a postulate. In most manuscripts of the 
Elements we find the following ten assumptions: 3 

Postulates. Let the following be postulated : 

1. To draw a straight line from any point to any point. 

2. To produce a finite straight line continuously in a straight line. 

3. To describe a circle with any center and radius. 

4. That all right angles are equal. 

5. That, if a straight line falling on two straight lines makes the interior 
angles on the same side less than two right angles, the two straight lines, if 

3 See The Thirteen Books of Euclid's Elements, translated and edited by T. L. Heath (1956, 
3 vols.). 


produced indefinitely, meet on that side on which the angles are less than the 
two right angles. 

Common notions : 

1. Things which are equal to the same thing are also equal to one another. 

2. If equals be added to equals, the wholes are equal. 

3. If equals be subtracted from equals, the remainders are equal. 

4. Things which coincide with one another are equal to one another. 

5. The whole is greater than the part. 

Aristotle had written that "other things being equal, that proof is the better 
which proceeds from the fewer postulates," and Euclid evidently subscribed 
to this principle. For example, Postulate 3 is interpreted in the very limited 
literal sense, sometimes described as the use of Euclidean (collapsible) com- 
passes, whose legs maintain a constant opening so long as the point stands 
on the paper, but fall back upon each other when they are lifted. That is, the 
postulate is not interpreted to permit the use of a pair of dividers to lay off 
a distance equal to one line segment upon a noncontiguous longer line seg- 
ment, starting from an end point. It is proved in the first three propositions 
of Book I that the latter construction is always possible, even under the 
strict interpretation of Postulate 3. The first proposition justifies the con- 
struction of an equilateral triangle ABC on a given line segment AB by 
constructing through B a circle with a center at A and another circle through 
A with center at B, and letting C be the point of intersection of the two circles. 
(That they do intersect is tacitly assumed.) Proposition 2 then builds on 
Proposition 1 by showing that from any point A as extremity (Fig. 7.3) one 

FIG. 7.3 


can lay off a straight line segment equal to a given line segment EC. First 
Euclid draws AB, and on this he constructs the equilateral triangle ABD, 
extending the sides DA and DB to E and F respectively. With B as center 
describe the circle through C, intersecting BF in G; then with D as center 
draw a circle through G, intersecting DE in H. Line AH is then easily shown 
to be the line required. Finally, in Proposition 3, Euclid makes use of 
Proposition 2 to show that, given any two unequal straight lines, one can 
cut off from the greater a segment equal to the smaller. 

In the first three propositions Euclid went to great pains to show that a 
very restricted interpretation of Postulate 3 nevertheless implies the free use 
of compasses as is usually done in laying off distances in elementary geometry. 
Nevertheless, by modern standards of rigor the Euclidean assumptions are 
woefully inadequate, and in his proofs Euclid often makes use of tacit postu- 
lates. In the first proposition of the Elements, for example, he assumes without 
proof that the two circles will intersect in a point. For this and similar situa- 
tions it is necessary to add to the postulates one equivalent to a principle of 
continuity. Moreover, Postulates 1 and 2 as they were expressed by Euclid 
guarantee neither the uniqueness of the straight line through two non- 
coincident points nor even its infinitude ; they simply assert that there is at 
least one and that it has no termini, yet in his proofs Euclid freely made use 
of the uniqueness and infinitude. It is, of course, easy to criticize the work of 
a man in the light of later developments and to forget that "sufficient unto 
the day is the rigor thereof." In its time the Elements evidently was the most 
tightly reasoned logical development of elementary mathematics that had 
ever been put together, and two thousand years were to pass before a 
more careful presentation occurred. During this long interval most mathe- 
maticians regarded the treatment as logically satisfying and pedagogically 

Most of the propositions in Book I of the Elements are well known to 
anyone who has had a high school course in geometry. Included are the 
familiar theorems on congruence of triangles (but without an axiom justifying 
the method of superposition), on simple constructions by straightedge and 
compasses, on inequalities concerning angles and sides of a triangle, on 
properties of parallel lines (leading to the fact that the sum of the angles of 
a triangle is equal to two right angles), and on parallelograms (including the 
construction of a parallelogram having given angles and equal in area to a 
given triangle or to a given rectilinear figure). The book closes (in Proposi- 
tions 47 and 48) with the proof of the Pythagorean theorem and its converse. 
The proof of the theorem as given by Euclid was not that usually given in 
textbooks of today, in which simple proportions are applied to the sides of 
similar triangles formed by dropping an altitude upon the hypotenuse. It 



has been suggested that Euclid avoided such a proof because of difficulties 
involved in commensurability. Only in Book V did Euclid turn to the well- 
founded theory of proportions, and up to that point the use of proportion- 
alities is avoided as far as possible. For the Pythagorean theorem Euclid used 
instead the beautiful proof with a figure sometimes described as a windmill 
or as the peacock's tail or as the bride's chair (Fig. 7.4). The proof is accom- 
plished by showing that the square on AC is equal to twice the triangle FAB 

FIG. 7.4 

or to twice the triangle CAD or to the rectangle AL, and that the square on 
BC is equal to twice the triangle ABK or to twice the triangle BCE or to the 
rectangle BL. Hence the sum of the squares is equal to the sum of the rect- 
angles, that is, to the square on AB. It has been assumed that this proof was 
original with Euclid, and many conjectures have been made as to the possible 
form of earlier proofs. Since the days of Euclid many alternative proofs have 
been proposed. 

It is to Euclid's credit that the Pythagorean theorem is immediately 
followed by a proof of the converse : If in a triangle the square on one of the 
sides is equal to the sum of the squares on the other two sides, the angle 
between these other two sides is a right angle. Not infrequently in modern 
textbooks the exercises following the proof of the Pythagorean theorem are 
such that they require not the theorem itself, but the still unproved converse. 
There may be many a minor flaw in the Elements, but the book had all the 
major logical virtues. 



Book II of the Elements is a short one, containing only fourteen proposi- 
tions, not one of which plays any role in modern textbooks ; yet in Euclid's 
day this book was of great significance. This sharp discrepancy between 
ancient and modern views is easily explained — today we have symbolic 
algebra and trigonometry that have replaced the geometrical equivalents 
from Greece. For instance, Proposition I of Book II states that "If there be 

rb Oewpi}fia. rrjs vvfufy^s. 
(With apologies to La Vie Parisienne.) 

The "Bride's Chair" diagram of Euclid's Elements I. 47 in a World War I setting. [The 
Mathematical Gazette, 11 (1922-1923), 364.] 



two straight lines, and one of them be cut into any number of segments 
whatever, the rectangle contained by the two straight lines is equal to the 
rectangles contained by the uncut straight line and each of the segments." 
This theorem, which asserts (Fig. 7.5) that AD(AP + PR + RB) = AD ■ AP + 

AD ■ PR + AD ■ RB, is nothing more than a geometrical statement of one 
of the fundamental laws of arithmetic known today as the distributive law : 
a(b + c + d) = ab + ac + ad. In later books of the Elements (V and VII) we 
find demonstrations of the commutative and associative laws for multiplica- 
tion. Whereas in our time magnitudes are represented by letters that are 
understood to be numbers (either known or unknown) on which we operate 
with the algorithmic rules of algebra, in Euclid's day magnitudes were 
pictured as line segments satisfying the axioms and theorems of geometry. 
It is sometimes asserted that the Greeks had no algebra, but this is patently 
false. They had Book II of the Elements, which is a geometrical algebra that 
served much the same purpose as does our symbolic algebra. There can be 
little doubt that modern algebra greatly facilitates the manipulation of rela- 
tionships among magnitudes. But it is undoubtedly also true that a Greek 
geometer versed in the fourteen theorems of Euclid's "algebra" was far more 
adept in applying these theorems to practical mensuration than is an ex- 
perienced geometer of today. Ancient geometrical algebra was not an ideal 
tool, but it was far from ineffective. Euclid's statement (Proposition 4), "If a 
straight line be cut at random, the square on the whole is equal to the squares 
on the segments and twice the rectangle contained by the segments," is a 
verbose way of saying that (a + b) 2 = a 2 + lab + b 2 , but its visual appeal 
to an Alexandrian schoolboy must have been far more vivid than its modern 
algebraic counterpart can ever be. True, the proof in the Elements occupies 
about a page and a half; but how many high school students of today could 
give a careful proof of the algebraic rule they apply so unhesitatingly? The 
same holds true for Elements II. 5, which contains what we should regard as an 

impractical circumlocution for a 2 

(a + b)(a - b): 

If a straight line be cut into equal and unequal segments, the rectangle con- 
tained by the unequal segments of the whole, together with the square on the 
straight line between the points of section, is equal to the square on the half. 



The diagram that Euclid uses in this connection played a key role in Greek 
algebra ; hence we reproduce it 4 with further explanation. If in the diagram 
(Fig. 7.6) we let AC = CB = a, and CD = b, the theorem asserts that 







E G 

FIG. 7.6 


(a + b)(a - b) + b 2 = a 2 . The geometrical verification of this statement is 
not difficult. However, the significance of the diagram lies not so much in 
the proof of the theorem as in the use to which similar diagrams were put 
by Greek geometrical algebraists. The pride of the modern schoolboy in 
algebra is the solution of the quadratic equation (which he may or may not 
be able to justify), and a diagram similar to Fig. 7.6 was the Greek schoolboy's 
geometrical equivalent. If the Greek scholar were required to construct a 
line x having the property expressed by ax - x 2 = b 2 , where a and b are 
line segments with a>2b, he would drawline^B=a andbisectitat C.Then at 
C he would erect a perpendicular CP equal in length to b ; with P as center 
and radius a/2 he would draw a circle cutting AB in point D. Then on AB 
he would construct rectangle ABMK of width BM = BD and complete the 
square BDHM. This square is the area x 2 having the property specified in the 
quadratic equation. As the Greeks expressed it, we have applied to the seg- 
ment AB (= a) a rectangle AH (= ax - x 2 ) which is equal to a given square 
(b 2 ) and falls short (of AM) by a square DM. The demonstration of this is 
provided by the proposition cited above (II. 5) in which it is clear that the 
rectangle ADHK equals the concave polygon CBFGHL — that is, it differs 
from (a/ 2) 2 by the square LHGE, the side of which by construction is 
CD = V(a/2) 2 - b 2 . 

In an exactly analogous manner the quadratic equation ax + x 2 = b 2 is 
solved through the use of II. 6 : 

If a straight line be bisected and a straight line be added to it in a straight 
line, the rectangle contained by the whole (with the added straight line) and 
the added straight line together with the square on the half is equal to the 
square on the straight line made up of the half and the added straight line. 

4 Throughout this chapter the translations and most of the diagrams are from the Thirteen 
Books of Euclid's Elements as edited by T. L. Heath. 



This time we "apply to a given straight line (AB = a) a rectangle 
(AM = ax + x 2 ) which shall be equal to a given square (b 2 ) and shall 
exceed ( AH) by a square figure" (Fig. 7.7). In this case the distance 
CD = ^/(a/2) 2 + b 2 ; since from the proposition it is known that rectangle 
AM (= ax + x 2 ) plus square LG [=(a/2) 2 ] is equal to square CF 
[= (a/2) 2 + b 2 ], it follows that the condition ax + x 2 = b 2 is satisfied. 

The next few propositions of Book II are variations of the geometric 
algebra that we have illustrated, with II. 11 being an important special case 
of II. 6. Here Euclid solves the equation ax + x 2 = a 2 by drawing a square 
ABCD with side a, bisecting side AD at E, drawing EB, extending side DA 
to F such that EF = EB, and completing the square AFGH (Fig. 7.8). Then 


K L 








H G 

E G 

FIG. 7.7 


FIG. 7.8 

on extending GH to intersect DC in K, we shall have applied to segment 
AD a rectangle FK (= ax + x 2 ) equal to a given square AC (= a 2 ) and 
exceeding by a square (x 2 ). 

The figure used by Euclid in Elements II. 11, and again in VI. 30 (our 
Fig. 7.8), is the basis for a diagram that appears today in many geometry 
books to illustrate the iterative property of the golden section. To the gnomon 
BCDFGH (Fig. 7.8) we add point Lto complete the rectangle CDFL (Fig. 7.9), 
and within the smaller rectangle LBGH, which is similar to the larger 
rectangle LCDF, we construct, by making GO = GL, the gnomon LBMNOG 
similar to gnomon BCDFGH. Now within the rectangle BHOP, which is 
similar to the larger rectangles CDFL and LBHG, we construct the gnomon 
PBHQRN similar to the gnomons BCDFGH and LBMNOG. Continuing 
indefinitely in this manner, we have an unending sequence of nested similar 
rectangles tending toward a limiting point Z. It turns out that Z, which is 
easily seen to be the point of intersection of lines FB and DL, is also the 
pole of a logarithmic spiral tangent to the sides of the rectangles at points 

C, A, G,P,M,Q, Other striking properties can be found in this fascinating 

diagram. 5 

5 See, for example, H. S. M. Coxeter, "The Golden Section, Phyllotaxis, and Wythoff s 
Game," Scripta Mathematics 19(1953), 135-143. 


D A 

^ ^ ■ 



"^ ^ 


\ </^ 




/^ x 


/ ^-^ 


/ ^ 


/ ^^ 


/ \ 


/ " 

I M 

Q o/ 











' \— 



FIG. 7.9 

Propositions 12 and 13 of Book II are of interest because they adumbrate 
the concern with trigonometry that was shortly to blossom in Greece. These 
propositions will be recognized by the reader as geometric formulations — 
first for the obtuse angle and then for the acute angle — of what later became 
known as the law of cosines for plane triangles : 

Proposition 12 

In obtuse-angled triangles the square on the side subtending the obtuse 
angle is greater than the squares on the sides containing the obtuse angle by 
twice the rectangle contained by one of the sides about the obtuse angle, 
namely that on which the perpendicular falls, and the straight line cut off 
outside by the perpendicular toward the obtuse angle. 

Proposition 13 

In acute-angled triangles the square on the side subtending the acute 
angle is less than the squares on the sides containing the acute angle by twice 
the rectangle contained by one of the sides about the acute angle, namely 
that on which the perpendicular falls, and the straight line cut off within by 
the perpendicular toward the acute angle. 

The proofs of Propositions 12 and 13 are analogous to those used today in 
trigonometry through double application of the Pythagorean theorem. 

It generally has been supposed that the contents of the first two books of 
the Elements are largely the work of the Pythagoreans. Books III and IV, on 
the other hand, deal with the geometry of the circle, and here the material 
is presumed to have been drawn largely from Hippocrates of Chios. The two 
books are not unlike the theorems on circles contained in textbooks of today. 


The first proposition of Book III, for example, calls for the construction of 
the center of a circle; and the last, Proposition 37, is the familiar statement 
that if from a point outside a circle a tangent and a secant are drawn, the 
square on the tangent is equal to the rectangle on the whole secant and the 
external segment. Book IV contains sixteen propositions, largely familiar 
to modern students, concerning figures inscribed in, or circumscribed about, 
a circle. Theorems on the measure of angles are reserved until after a theory 
of proportions has been established. 

Of the thirteen books of the Elements those most admired have been the 
fifth and the tenth — the one on the general theory of proportion and the 
other on the classification of incommensurables. The discovery of the in- 
commensurable had threatened a logical crisis which cast doubt on proofs 
appealing to proportionality, but the crisis had been successfully averted 
through the principles enunciated by Eudoxus. Nevertheless, Greek mathe- 
maticians tended to avoid proportions. We have seen that Euclid put off 
their use as long as possible, and such a relationship among lengths as 
x :a = b :c would be thought of as an equality of the areas ex = ab. Sooner 
or later, however, proportions are needed, and so Euclid tackled the problem 
in Book V of the Elements. Some commentators have gone so far as to suggest 
that the whole book, consisting of twenty-five propositions, was the work of 
Eudoxus, but this seems to be unlikely. Some of the definitions — such as that 
of a ratio — are so vague as to be useless. Definition 4, however, is essentially 
the axiom of Eudoxus and Archimedes : "Magnitudes are said to have a 
ratio to one another which are capable, when multiplied, of exceeding one 
another." Definition 5, the equality of ratios, is precisely that given earlier in 
connection with Eudoxus' definition of proportionality. 

To the casual reader Book V might appear as superfluous as Book II, for 
both have now been displaced by corresponding rules in symbolic algebra. 
A more careful reader interested in axiomatics will see that Book V deals 
with topics of fundamental importance in all mathematics. It opens with 
propositions that are equivalent to such things as the left-hand and right- 
hand distributive laws for multiplication over addition, the left-hand distri- 
butive law for multiplication over subtraction, and the associative law for 
multiplication (ab)c = a(bc). Then follow rules for "greater than" and 
"less than" and the well-known properties of proportions. It often is 
asserted that Greek geometrical algebra could not rise above the second 
degree in plane geometry, nor above the third degree in solid geometry, but 
this is not really the case. The general theory of proportions would permit 
work with products of any number of dimensions, for an equation of the 
form x 4 = abed is equivalent to one involving products of ratios of lines such 
as x/a ■ x/b = c/x ■ d/x. 


Having developed the theory of proportions in Book V, Euclid exploited 
it in Book VI by proving theorems concerning ratios and proportions related 
to similar triangles, parallelograms, and other polygons. Noteworthy is 
Proposition 31, a generalization of the Pythagorean theorem: "In right- 
angled triangles the figure on the side subtending the right angle is equal to 
the similar and similarly described figures on the sides containing the right 
angle." Proclus credits this extension to Euclid himself. Book VI contains (in 
Propositions 28 and 29) also a generalization of the method of application of 
areas, for the sound basis for proportion given in Book V enabled tfie author 
now to make free use of the concept of similarity. The rectangles of Book II 
are now replaced by parallelograms, and it is required to apply to a given 
straight line a parallelogram equal to a given rectilinear figure and deficient 
(or exceeding) by a parallelogram similar to a given parallelogram. These 
constructions, like those of II. 5-6, are in reality solutions of the quadratic 
equations bx = ac ± xQ, subject to the restriction (implied in IX. 27) that 
the discriminant is not negative. 

The Elements of Euclid often is mistakenly thought of as restricted to 
geometry. We already have described two books (II and V) that are almost 
exclusively algebraic; three books (VII, VIII, and IX) are devoted to the 
theory of numbers. The word "number" to the Greeks always referred to 
what we call the natural numbers — the positive whole numbers or integers. 
Book VII opens with a list of twenty-two definitions distinguishing various 
types of number — odd and even, prime and composite, plane and solid (that 
is, those that are products of two or of three integers) — and finally defining 
a perfect number as "that which is equal to its own parts." The theorems in 
Books VII, VIII, and IX are likely to be familiar to the reader who has had 
an elementary course in the theory of numbers, but the language of the proofs 
will certainly be unfamiliar. Throughout these books each number is repre- 
sented by a line segment, so that Euclid will speak of a number as AB. (The 
discovery of the incommensurable had shown that not all line segments 
could be associated with whole numbers ; but the converse statement— that 
numbers can always be represented by line segments — obviously remains 
true.) Hence Euclid does not use the phrases "is a multiple of" or "is a factor 
of," for he replaces these by "is measured by" and "measures" respectively. 
That is, a number n is measured by another number m if there is a third 
number k such that n = km. 

Book VII opens with two propositions that constitute a celebrated rule 
in the theory of numbers, which today is known as "Euclid's algorithm" for 
finding the greatest common divisor (measure) of two numbers. It is a 
scheme suggestive of a repeated inverse application of the axiom of Eudoxus. 
Given two unequal numhers, one subtracts the smaller a from the larger b 


repeatedly until a remainder r t less than the smaller is obtained; then one 
repeatedly subtracts this remainder r l from a until a remainder r 2 <r l 
results; then one repeatedly subtracts r 2 from r t ; and so on. Ultimately the 
process will lead to a remainder r n which will measure r„_ 1 , hence all preced- 
ing remainders, as well as a and b ; this number r„ will be the greatest common 
divisor of a and b. Among succeeding propositions we find equivalents of 
familiar theorems in arithmetic. Thus Proposition 8 states that if an = bm 
and en = dm, then (a - c)n = (b — d)m ; Proposition 24 states that if a and 
b are prime to c, then ab is prime to c. The book closes with a rule (Proposition 
39) for finding the least common multiple of several numbers. 

Book VIII is one of the less rewarding of the thirteen books of the Elements. 
It opens with propositions on numbers in continued proportion (geometric 
progression) and then turns to some simple properties of squares and cubes, 
closing with Proposition 27 : "Similar solid numbers have to one another 
the ratio which a cube number has to a cube number." This statement means 
simply that if we have a "solid number" ma-mb- mc and a "similar solid 
number" na-nb- nc, then their ratio will be m 3 : n 3 — that is, as a cube is to 
a cube. 

Book IX, the last of the three books on theory of numbers, contains several 1 
theorems that are of special interest. Of these the most celebrated is Proposi- 
tion 20: "Prime numbers are more than any assigned multitude of prime 
numbers." That is, Euclid here gives the well-known elementary proof that 
the number of primes is infinite. The proof is indirect, for one shows that the 
assumption of a finite number of primes leads to a contradiction. Let P be 
the product of all the primes, assumed to be finite in number, and consider the 
number N = P + 1. Now, N cannot be prime, for this would contradict 
the assumption that P was the product of all primes. Hence N is composite 
and must be measured by some prime p. But p cannot be any of the prime 
factors in P, for then it would have to be a factor of 1. Hence p must be a 
prime different from all of those in the product P; therefore, the assumption 
that P was the product of all the primes must be false. 

Proposition 35 of this book contains a formula for the sum of numbers in 
geometric progression, expressed in elegant but unusual terms : 

If as many numbers as we please be in continued proportion, and there be 
subtracted from the second and the last numbers equal to the first, then as the 
excess of the second is to the first, so will the excess of the last be to all those 
before it. 

This statement is, of course, equivalent to the formula 

a„ + j — a x a 2 — a t 

a, + a-, + • ■ • + a„ a, 

which in turn is equivalent to 

The following and last proposition in Book IX is the well-known formula 
for perfect numbers : "If as many numbers as we please, beginning from 
unity, be set out continuously in double proportion until the sum of all 
becomes prime, and if the sum is multiplied by the last, the product will be 
perfect." That is, in modern notation, if S„ = 1 + 2 + 2 2 + • • • + 2"~ l = 
2" — 1 is prime, then 2" _1 (2" — 1) is a perfect number. The proof is easily 
established in terms of the definition of perfect number given in Book VII. 
The ancient Greeks knew the first four perfect numbers: 6, 28, 496, and 
8128. Euclid did not answer the converse question — whether or not his 
formula provides all perfect numbers. It is now known that all even per- 
fect numbers are of Euclid's type, but the question of the existence of odd 
perfect numbers remains an unsolved problem. 6 Of the two dozen per- 
fect numbers now known all are even, but to conclude by induction that all 
must be even would be hazardous. 

In Propositions 21 through 36 of Book IX there is a unity which suggests 
that these theorems were at one time a self-contained mathematical system, 
possibly the oldest in the history of mathematics and stemming presumably 
from the middle or early fifth century B.C. It has even been suggested that 
Propositions 1 through 36 of Book IX were taken over by Euclid, without 
essential change, from a Pythagorean textbook. 7 

1 1 Book X of the Elements was, before the advent of early modern algebra, 
the most admired — and the most feared. It is concerned with a systematic 
classification of incommensurable line segments of the forms a ± y/b, 

a ± sfb, Ja ± y/b, and J J a + jb, where a and b, when of the same 

dimension, are commensurable. Today we would be inclined to think of this 
as a book on irrational numbers of the types above, where a and b are rational 
numbers ; but Euclid regarded this book as a part of geometry, rather than 
of arithmetic. In fact, Propositions 2 and 3 of the book duplicate for 
geometrical magnitudes the first two propositions of Book VII, where the 
author had dealt with whole numbers. Here he proves that if to two unequal 
line segments one applies the process described above as Euclid's algorithm, 

6 For further details see L. E. Dickson, History of the Theory of Numbers (Washington, D.C., 
1919-1923, 3 vols.), I, 3-33. 

1 See Arpad Szabo, "The Transformation of Mathematics into Deductive Science and the 
Beginnings of Its Foundations on Definitions and Axioms," Scripta Mathematica, 27 (1964), 


and if the remainder never measures the one before it, the magnitudes are 
incommensurable. Proposition 3 shows that the algorithm, when applied to 
two commensurable magnitudes, will provide the greatest common measure 
of the segments. 

Book X contains 115 propositions — more than any other — most of which 
contain geometrical equivalents of what we now know arithmetically as 
surds. Among the theorems are counterparts of rationalizing denominators 
of fractions of the form a/(b ± N /c) and aj{-Jb ± N /c). Line segments given 
by square roots, or by square roots of sums of square roots, are about as 
easily constructed by straightedge and compasses as are rational combina- 
tions. One reason that the Greeks turned to a geometrical rather than an 
arithmetical algebra was that, in view of the lack of the real-number concept, 
the former appeared to be more general than the latter. The roots of 
ax — x 2 = b 2 , for example, can always be constructed (provided that 
a > 2b). Why, then, should Euclid have gone to great lengths to demonstrate, 
in Propositions 17 and 18 of Book X, the conditions under which the roots 
of this equation are commensurable with a! He showed that the roots are 
c ommensur able or incommensurable, with respect to a, according as 
sja 2 — 4b 2 and a are commensurable or incommensurable. It has been 
suggested 8 that such considerations indicate that the Greeks used their 
solutions of quadratic equations for numerical problems also, much as the 
Babylonians had in their system of equations x + y = a, xy = b 2 . In such 
cases it would be advantageous to know whether the roots will or will not 
be expressible as quotients of integers. A close study of Greek mathematics 
seems to give evidence that beneath the geometrical veneer there was more 
concern for logistic and numerical approximations than the surviving 
classical treatises portray. 

The material in Book XI, containing thirty-nine propositions on the 12 
geometry of three dimensions, will be largely familiar to one who has taken 
a course in the elements of solid geometry. Again the definitions are easily 
criticized, for Euclid defines a solid as "that which has length, breadth, and 
depth" and then tells us that "an extremity of a solid is a surface." The last 
four definitions are of four of the regular solids. The tetrahedron is not 
included, presumably because of an earlier definition of a pyramid as "a 
solid figure, contained by planes, which is constructed from one plane to any 
point." The eighteen propositions of Book XII are all related to the measure- 
ment of figures, using the method of exhaustion. The book opens with a 
careful proof of the theorem that areas of circles are to each other as squares 
on the diameters. Similar applications of the typical double reductio ad 

8 See Heath, Elements of Euclid, III, 43-45. 


absurdum method then are applied to the volumetric mensuration of 
pyramids, cones, cylinders, and spheres. Archimedes ascribed the rigorous 
proofs of these theorems to Eudoxus, from whom Euclid probably adapted 
much of this material. 

The last book is devoted entirely to properties of the five regular solids, a 
fact that has led some historians to say that the Elements was composed as 
a glorification of the cosmic or Platonic figures. Inasmuch as such a large 
proportion of the earlier material is far removed from anything relating to 
the regular polyhedra, such an assumption is quite gratuitous; but the 
closing theorems are a fitting climax to a remarkable treatise. Their object 
is to "comprehend" each of the regular solids in a sphere — that is, to find 
the ratio of an edge of the solid to the radius of the circumscribed sphere. 
Such computations are ascribed by Greek commentators to Theaetetus, to 
whom much of Book XIII is probably due. In preliminaries to these com- 
putations Euclid referred once more to the division of a line in mean and 
extreme ratio, showing that "the square on the greater segment added to 
half the whole is five times the square on the half"— as is easily verified by 
solving a/x = x/(a - x)— and citing other properties of the diagonals of a 
regular pentagon. Then in Proposition 10 Euclid proved the well-known 
theorem that a triangle whose sides are respectively sides of an equilateral 
pentagon, hexagon, and decagon inscribed in the same circle is a right 
triangle. Propositions 13 through 17 express the ratio of edge to diameter 
for each of the inscribed regular solids in turn : e/d is ^/f for the tetrahedron, 
^/l for the octahedron, ,/i for the cube or hexahedron, ^/{5 - y/5)/10 
for the icosahedron, and (^5 - 1)12^/1 for the dodecahedron. Finally, in 
Proposition 18, the last in the Elements, it is easily proved that there can be 
no regular polyhedron beyond these five. About 1900 years later the astron- 
omer Kepler was so struck by this fact that he built a cosmology on the 
five regular solids, believing that they must have been the creator's key to 
the structure of the heavens. 

13 In ancient times it was not uncommon to attribute to a celebrated author 
works that were not by him ; thus some versions of Euclid's Elements include 
a fourteenth and even a fifteenth book, both shown by later scholars to be 
apocryphal. The so-called Book XIV continues Euclid's comparison of the 
regular solids inscribed in a sphere, the chief results being that the ratio of 
the surfaces of the dodecahedron and icosahedron inscribed in the same 
sphere is the same as the ratio of their volumes, the ratio being that of the 
edge of the cube to the edge of the icosahedron — that is, V 10/[3(5 - </5)]. 
It is thought that this book may have been composed by Hypsicles on the 
basis of a treatise (now lost) by Apollonius comparing the dodecahedron 
and icosahedron. (Hypsicles, who probably lived in the second half of the 


second century B.C., is thought to be the author of an astronomical work, 
De ascensionibus, from which the division of the circle into 360 parts may 
have been adopted.) That the same circle circumscribes both the pentagon 
of the dodecahedron and the triangle of the icosahedron (inscribed in the 
same sphere) was said to have been proved by Aristaeus, roughly contem- 
poraneous with Euclid. 

The spurious Book XV, which is inferior, is thought to have been (at least 
in part) the work of Isidore of Miletus (fl. ca. a.d. 532), architect of the 
cathedral of Holy Wisdom (Hagia Sophia) at Constantinople. This book also 
deals with the regular solids, showing how to inscribe certain of them within 
others, counting the number of edges and solid angles in the solids, and 
finding the measures of the dihedral angles of faces meeting at an edge. It 
is of interest to note that despite such enumerations, the ancients all missed 
the so-called polyhedral formula enunciated by Euler in the eighteenth 

The Elements of Euclid not only was the earliest major Greek mathematical 1 4 
work to come down to us, but also the most influential textbook of all times. 
It was composed in about 300 B.C. and was copied and recopied repeatedly 
after that. Errors and variations inevitably crept in, and some later editors, 
notably Theon of Alexandria in the late fourth century, sought to improve 
on the original. Nevertheless, it has been possible to obtain a good impression 
of the content of the Euclidean version through a comparison of more than 
half a dozen Greek manuscript copies dating mostly from the tenth to the 
twelfth century. Later accretions, generally appearing as scholia, add supple- 
mentary information, often of an historical nature, and in most cases they 
are readily distinguished from the original. Copies of the Elements have come 
down to us also through Arabic translations, later turned into Latin in the 
twelfth century, and finally, in the sixteenth century, into the vernacular. 
The first printed version of the Elements appeared at Venice in 1482, one of 
the very earliest of mathematical books to be set in type ; it has been estimated 
that since then at least a thousand editions have been published. Perhaps no 
book other than the Bible can boast so many editions, and certainly no 
mathematical work has had an influence comparable with that of Euclid's 
Elements. How appropriate it was that Euclid's successors referred to him as 
"The Elementator!" 


Archibald, R. C, ed., Euclid's Book on Divisions of Figures (Cambridge : Cambridge 
University Press, 1915). 

Cohen, M. R., and I. E. Drabkin, A Source Book in Greek Science (New York : McGraw- 
Hill, 1948; reprinted Cambridge, Mass.: Harvard University Press, 1958). 


Frankland, W. B., The First Book of Euclid's Elements, with a Commentary Based 

Principally upon that of Proclus Diadochus (Cambridge: Cambridge University 

Press, 1905). 
Heath, T. L., History of Greek Mathematics (Oxford: Clarendon, 1921, 2 vols.). 
Heath, T. L., ed. The Thirteen Books of Euclid's Elements (Cambridge, 1908, 3 vols.; 

paperback ed., New York: Dover, 1956). 
Hultsch, F. O., "Eukleides," in Pauly-Wissowa, Real-Enzyclopadie der klassischen 

Altertumswissenschaft (Stuttgart, 1909), Vol. VI, columns 1003-1052. 
Loria, Gino, Storia delle matematiche (Turin: Sten, 1929-1933, 3 vols.). 
Sarton, George, Ancient Science and Modern Civilization (Lincoln, Nebr. : University of 

Nebraska Press, 1954). 
Szabo, Arpad, "Anfange des euklidischen Axiomensystems," Archive for History of 

Exact Sciences, 1 (1960), 37-106. 
Thomas, Ivor, ed., Selections Illustrating the History of Greek Mathematics (Cambridge, 

Mass. : Loeb Classical Library, 1939-1941, 2 vols.) 
Thomas-Stanford, Charles, Early Editions of Euclid's Elements (London : Bibliographical 

Society, 1926). 
Vogt, Heinrich, "Die Lebenzeit Euklids," Bibliotheca Mathematica (3), 13 (1913), 



1. Describe the sources Euclid probably used in writing the Elements; justify your presump- 

2. Which of the thirteen books of the Elements do you regard as the most important? Justify 
your answer. 

3. Which of the thirteen books of the Elements do you regard as most dispensable? Justify 

your answer. 

4. Given line segments a and b, construct, with straightedge and compasses alone, segments 
x and v satisfying the conditions x + y = a, xy = b 2 . 

5. Given line segments a and b, construct, with straightedge and compasses alone, a solution 
x of the equation x z = ax + b 2 . 

6. Use the Euclidean algorithm to find the greatest common divisor of 456 and 759. 

7. Use the Euclidean algorithm to find the greatest common divisor of 567 and 839 and 432. 

8. What is the greatest common measure of two line segments of lengths f and f respectively? 
Of two line segments of lengths a/b and c/d respectively, where a, b, c, d are relatively prime 

9. Given two unequal line segments a and b, prove that if a line segment c is obtained through 
the Euclidean algorithm, this is the greatest common measure of a and b. 

10. Provide all the details of the proof of the Pythagorean theorem by the "windmill" method. 

11. All even perfect numbers end in 6 or in 28 and, upon casting out nines, leave a remainder of 1 
(except in the case of the first perfect number). Verify these statements for the first four 
perfect numbers. 

12. Show how to construct a tangent to a circle from a point outside the circle. 

13. Justify Euclid's formula for the sum of terms in a geometric progression. 

14. The number 2 13 - 1 is a prime. Use this fact to find the fifth perfect number in order of 


15. Prove that there cannot be a regular convex solid other than those given by Euclid. 

16. Prove the law of cosines for an acute-angled triangle, indicating just how far Euclid could 
go in expressing this relationship. 

*17. In Elements IX. 14 it is proved that a number can be resolved into prime factors in only 

one way. Write out a proof of this proposition. 
*18. Prove Euclid's formula for perfect numbers. 
*19. Prove, by the method of exhaustion, that the volumes of spheres are to each other as cubes 

on their diameter (Elements XII. 18). 
*20. Prove that if a pentagon, a hexagon, and a decagon are inscribed in the same circle, a 

triangle made up of a side of the pentagon and a side of the hexagon and a side of the decagon 

is a right triangle (Elements XIII. 10). 
*21. Euclid's Division of Figures includes a construction of a line parallel to the bases of a 

trapezoid and dividing the trapezoid into two equal areas. Show how to carry out such a 

construction with straightedge and compasses alone. 


Archimedes of Syracuse 

There was more imagination in the head of Archimedes 
than in that of Homer. 


Throughout the Hellenistic Age the center of mathematical activity remained 
at Alexandria, but the leading mathematician of that age — and of all antiquity 
— was not a native of the city. Archimedes may have studied for a while at 
Alexandria under the students of Euclid, and he maintained communication 
with mathematicians there, but he lived and died at Syracuse. Details of his 
life are scarce, but we have some information about him from Plutarch's 
account of the life of Marcellus, the Roman general. During the Second 
Punic War the city of Syracuse was caught in the power struggle between 
Rome and Carthage ; having cast its lot with the latter, the city was besieged 
by the Romans during the years 214 to 212 B.C. We are told that throughout 
the siege Archimedes invented ingenious war machines to keep the enemy 
at bay — catapults to hurl stones; ropes, pulleys, and hooks to raise and 
smash the Roman ships ; devices to set fire to the ships. Ultimately, however, 
Syracuse fell through a "fifth column" ; in the sack of the city Archimedes was 
slain by a Roman soldier, despite orders from Marcellus that the life of the 
geometer be spared. Inasmuch as Archimedes at the time is reported to have 
been seventy-five years old, he was most likely born in 287 B.C. His father 
was an astronomer, and Archimedes also established a reputation in astron- 
omy. Marcellus is said to have reserved for himself, as booty, ingenious 
planetaria that Archimedes had constructed to portray the motions of the 
heavenly bodies. Accounts of the life of Archimedes are in agreement, how- 
ever, in depicting him as placing little value in his mechanical contrivances 
as compared with the products of his thought. Even when dealing with 
levers and other simple machines, he was far more concerned with general 
principles than with practical applications. 

Archimedes was not, of course, the first to use the lever, nor even the first 
to formulate the general law. Aristotelian works contain the statement that 



Mosaic representation of the death of Archimedes Once thought to have been from the 
floor of a room in Pompeii, it is now believed to be a sixteenth -century copy (or falsifica- 
tion). (Municipal Art Institute, Frankfurt am Main.) 

two weights on a lever balance when they are inversely proportional to their 
distances from the fulcrum; and the Peripatetics associated the law with 
their assumption that vertical rectilinear motion is the only natural ter- 
restrial motion. They pointed out that the extremities of unequal arms of a 
lever will, in their displacement about the fulcrum, trace out circles rather 
than straight lines; the extremity of the longer arm will move in the circle 
that is larger, hence the path will approach more nearly to the natural 
vertical rectilinear motion than will the extremity of the shorter arm. There- 
fore the law of the lever is a natural consequence of this kinematic principle. 
Archimedes, on the other hand, deduced the law from a more plausible static 
postulate- -that bilaterally symmetric bodies are in equilibrium. That is, let 
one assume that a weightless bar four units long and supporting three unit 
weights, one at either end and one in the middle (Fig. 8.1), is balanced by a 


FIG. 8.1 

fulcrum at the center. By the Archimedean axiom of symmetry the system is 
in equilibrium. But the principle of symmetry shows also that, considering 
only the right-hand half of the system, the balancing effect will remain the 
same if the two weights two units apart are brought together at the midpoint 
of the right-hand side. This means that a unit weight two units from the 
fulcrum will support on the other arm a weight of two units which is one unit 
from the fulcrum. Through a generalization of this procedure Archimedes 
established the law of the lever on static principles alone, without recourse 
to the Aristotelian kinematic argument. In the history of science during the 
Medieval period it will be found that a conjunction of static and kinematic 
views produced advances in both science and mathematics. 

The work of Archimedes on the law of the lever is part of his treatise, in 
two books, On the Equilibrium of Planes. This is not the oldest extant book 
on what may be called physical science, for Aristotle about a century earlier 
had published an influential work, in eight books, entitled Physics; but 
whereas the Aristotelian approach was speculative and nonmathematical, 
the Archimedean development was similar to the geometry of Euclid. From a 
set of simple postulates Archimedes deduced some very abstruse conclusions, 
establishing the close relationship between mathematics and mechanics 
that was to become so significant for both physics and mathematics. 1 The 
first book in the Equilibrium of Planes is concerned with rectilinear figures 
and closes with the centers of gravity of the triangle and the trapezoid. 
Book II concentrates attention on the center of gravity of a parabolic seg- 
ment and includes a proof of the fact that this center lies on the diameter of 
the segment and divides this diameter into segments in the ratio of 3 to 2. 
The procedure used is the familiar method of exhaustion, but a student 
familiar with the calculus and the principle of moments (or law of the lever) 
can easily verify the result. 

Archimedes can well be called the father of mathematical physics, not only 
for his On the Equilibrium of Planes, but also for another treatise, in two 
books, On Floating Bodies. Again, beginning from a simple postulate about 
the nature of fluid pressure, he obtains some very deep results. Among the 
earlier propositions are two that formulate the well-known Archimedean 
hydrostatic principle : 

1 See E. J. Dijksterhuis, Archimedes (1957), pp. 286 ff., where attention is called to differences 
of opinion concerning the rigor of Archimedes' proofs. 


Any solid lighter than a fluid will, if placed in a fluid, be so far immersed that 
the weight of the solid will be equal to the weight of the fluid displaced (I. 5). 
A solid heavier than a fluid will, if placed in it, descend to the bottom of the 
fluid, and the solid will, when weighed in the fluid, be lighter than its true 
weight by the weight of the fluid displaced (I. I). 2 

The mathematical derivation of this principle of buoyancy is undoubtedly 
the discovery that led the absentminded Archimedes to jump from his bath 
and run home naked, shouting "Eureka" ("I have found it"). It is also 
possible, although less likely, that the principle aided him in checking on 
the honesty of a goldsmith suspected of fraudulently substituting some silver 
for gold in a crown (or more likely a wreath) made for King Hiero of Syracuse, 
a friend (if not a relative) of Archimedes. Such fraud could easily have been 
detected by the simpler method of comparing the densities of gold, silver, and 
the crown by the simple device of measuring displacements of water when 
equal weights of each are in turn immersed in a vessel full of water. The later 
Roman architect, Vitruvius, attributed this method to Archimedes, whereas 
an anonymous Latin poetic account, De ponderibus et mensuris, written 
probably about a.d. 500, has Archimedes use the principle of buoyancy. 

The Archimedean treatise On Floating Bodies contains much more than 
the simple fluid properties so far described. Virtually the whole of Book II, 
for example, is concerned with the position of equilibrium of segments of 
paraboloids when placed in fluids, showing that the position of rest depends 
on the relative specific gravities of the solid paraboloid and the fluid in which 
it floats. Typical of these is Proposition 4 : 

Given a right segment of a paraboloid of revolution whose axis a is greater than 
fp (where p is the parameter), and whose specific gravity is less than that of a 
fluid but bears to it a ratio not less than (a - lp) 2 :a 2 , if the segment of the 
paraboloid be placed in the fluid with its axis at any inclination to the vertical, 
but so that its base does not touch the surface of the fluid, it will not remain in 
that position but will return to the position in which its axis is vertical. 

Still more complicated cases, with long proofs, follow. Archimedes could 
well have taught a theoretical course in naval architecture, although he 
probably would have preferred a graduate course in pure mathematics. 
No armchair scholar, he came to the rescue in mechanical emergencies. At 
one time, so it was reported, a ship had been built for King Hiero that was too 
heavy to be launched, but Archimedes, by a combination of levers and 
pulleys, accomplished the task. He is supposed to have boasted that if he were 
given a lever long enough, and a fulcrum on which to rest it, he could move 
the earth. It was probably at Alexandria that Archimedes became interested 

2 Translations in this chapter are taken from The Works of Archimedes, edited by T. L. Heath 


in the technical problem of raising water from the Nile River to irrigate the 
arable portions of the valley ; for this purpose he invented a device, now 
known as the Archimedean screw, made up of helical pipes or tubes fastened 
to an inclined axle with a handle by which it was rotated. 

A clear distinction was made in Greek antiquity not only between theory 
and application, but also between routine mechanical computation and the 
theoretical study of the properties of number. The former, for which Greek 
scholars are said to have shown scorn, was given the name logistic, while 
arithmetic, an honorable philosophical pursuit, was understood to be con- 
cerned solely with the latter. It has even been maintained that the classical 
attitude toward routine calculation mirrored the social structure of antiquity 
in which computations were relegated to slaves. Whatever truth there is in 
this view seems to have been exaggerated, for the Greeks took the trouble to 
replace their older Attic or Herodianic system of numeration by one distinctly 
more advantageous — the Ionian or alphabetic. Archimedes lived at about 
the time that the transition from Attic to Ionian numeration was made 
effective, 3 and this may account for the fact that he stooped to make a contri- 
bution to logistic. In a work entitled the Psammites ("Sand-Reckoner") 
Archimedes boasted that he could write down a number greater than the 
number of grains of sand required to fill the universe. In doing so he referred 
to one of the boldest astronomical speculations of antiquity — that in which 
Aristarchus of Samos, toward the middle of the third century B.C., proposed 
putting the earth in motion about the sun. Such an astronomical system 
would suggest that the relative positions of the fixed stars should change as 
the earth is displaced by many millions of miles while going around the sun. 
The absence of such parallactic displacement was the factor that led the 
greatest astronomers of antiquity (including, presumably, also Archimedes) 
to reject the heliocentric hypothesis ; but Aristarchus asserted that the lack 
of parallax can be attributed to the enormity of the distance of the fixed 
stars from the earth. Now, to make good his boast, Archimedes had perforce 
to provide against all possible dimensions for the universe, and so he showed 
that he could enumerate the grains of sand needed to fill even Aristarchus' 
immense world. Archimedes began with certain estimates that had been 
made in his day concerning the sizes of the earth, the moon, and the sun and 
the distances of the moon, the sun, and the stars. An estimate of the earth's 
circumference in his day, he reported, had been given as 300,000 stades 
(about 30,000 miles, for the stade generally used was roughly a tenth of a 
mile) ; Archimedes allowed for an underestimate and assumed a circumfer- 
ence of 3,000,000 stades. Moreover, Aristarchus had estimated the diameter 

3 However, O. Neugebauer, in Exact Sciences in Antiquity, 2nd ed. (Providence, R.l. : Brown 
University Press, 1957), p. 11, believes that the alphabetic system was in use several centuries 
before the time of Archimedes. 


of the sun as eighteen to twenty times that of the moon, which in turn is 
smaller than the earth. To play safe Archimedes took the diameter of the 
sun to be not more than thirty times that of the moon (or, a fortiori, of the 
earth). Next Archimedes assumed that the apparent size of the sun was 
greater than a thousandth part of a circle, for Aristarchus had estimated it 
to be about half a degree, a result confirmed by observation. Knowing an 
upper bound for the sun's actual size and a lower bound for its apparent 
size, an upper bound for its distance is easily established. Finally, Archimedes 
interpreted Aristarchus' universe to have a radius that is to the sun's distance 
as this distance is to the earth's radius. 4 From these assumptions Archimedes 
shows that the diameter of the ordinary universe as far as the sun is less than 
10 10 stades. Next he had to estimate the size of a grain of sand ; remaining on 
the safe side, he assumed that 10,000 grains of sand are not smaller than a 
poppy seed, that the diameter of a poppy seed is not less than one fortieth 
of a finger breadth, and that a stadium in turn is less than 10,000 finger 
breadths. Putting together all these inequalities, Archimedes concluded that 
the number of grains of sand required to fill the sphere of the then generally 
accepted universe is less than a number that we should write as 10 51 . For 
the universe of Aristarchus, which is to the ordinary universe as the latter 
is to the earth, Archimedes showed that not more than 10 63 grains of sand 
are required. Archimedes did not use this notation, but instead described 
the number as ten million units of the eighth order of numbers (where the 
numbers of second order begin with a myriad-myriads and the numbers of 
eighth order begin with the seventh power of a myriad-myriads). To show 
that he could express numbers ever so much larger even than this, Archimedes 
extended his terminology to call all numbers of order less than a myriad- 
myriads those of the first period, the second period consequently beginning 
with the number (10 8 ) 108 , one that would contain 800,000,000 ciphers. The 
periods of course continue through the 1 8 th period. That is, his system would 
go up to a myriad-myriad units of the myriad-myriadth order of the myriad- 
myriadth period — a number that would be written as one followed by some 
eighty thousand million millions of ciphers. It was in connection with this 
work on huge numbers that Archimedes mentioned, all too incidentally, a 
principle that later led to the invention of logarithms — the addition of 
"orders" of numbers (the equivalent of their exponents when the base is 
100,000,000) corresponds to finding the product of the numbers. 

In his approximate evaluation of the ratio of the circumference to diameter 
for a circle Archimedes again showed his skill in computation. Beginning 

4 The language in the Psammites is not clear at this point, but the interpretation adopted 
here seems to be appropriate. Erika and Rudolf von Erhardt, "Archimedes' Sand-Reckoner," 
Isis, 33 (1942), 578-602, question the authenticity of the Psammites, but this is defended by O. 
Neugebauer, "Archimedes and Aristarchus," Isis, 34 (1942), 4-6. 


with the inscribed regular hexagon, he computed the perimeters of polygons 
obtained by successively doubling the number of sides until one reached 
ninety-six sides. His iterative procedure for these polygons was related to 
what is sometimes called the Archimedean algorithm. One sets out the 
sequence P„, p„, P 2n , p 2n , P*„, P*„---, where P„ and p„ are the perimeters 
of the circumscribed and inscribed regular polygons of n sides. Beginning 
with the third term, one calculates any term from the two preceding terms 
by taking alternately their harmonic and geometric means. That is, P 2 „ = 
2p„P„/(p n + P„X Pin = y/fifT„ etc. If one prefers, one can use instead the 
sequence a n , A n , a 2n ,A 2n ,..., where a„ and A„ are the areas of the inscribed 
and circumscribed regular polygons of n sides. The third and successive 
terms are calculated by taking alternately the geometric and harmonic 
means, so that a 2n = Ja~^A„, A 2n = 2A„a 2 J(A n + a 2 „), etc. His method for 
computing square roots, in finding the perimeter of the circumscribed 
hexagon, and for the geometric means was similar to that used by the Babylon- 
ians. The result of the Archimedean computation on the circle was an 
approximation to the value of n expressed by the inequality 3^ < n < 3t§, 
a better estimate than those of the Egyptians and the Babylonians. (It should 
be borne in mind that neither Archimedes nor any other Greek mathemati- 
cian ever used our notation n for the ratio of circumference to diameter in a 
circle.) This result was given in Proposition 3 of the treatise On the Measure- 
ment of the Circle, one of the most popular of the Archimedean works during 
the Medieval period. This little work, probably incomplete as it has come 
down to us, includes only three propositions, of which one is the proof, by 
the method of exhaustion, that the area of the circle is the same as that of a 
right triangle having the circumference of the circle as one side and the 
radius of the circle as the other. It is unlikely that Archimedes was the dis- 
coverer of this theorem, for it is presupposed in the quadrature of the circle 
attributed to Dinostratus. 

Archimedes, like his predecessors, was attracted by the three famous 
problems of geometry, and the well-known Archimedean spiral provided 
solutions to two of these (but not, of course, with straightedge and com- 
passes alone). The spiral is defined as the plane locus of a point which, 
starting from the end point of a ray or half line, moves uniformly along this 
ray while the ray in turn rotates uniformly about its end point. In polar 
coordinates the equation of the spiral is r = aO. Given such a spiral, the 
trisection of an angle is easily accomplished. The angle is so placed that 
the vertex and initial side of the angle coincide with the initial point O of the 
spiral and the initial position OA of the rotating line. Segment OP, where P 
is the intersection of the terminal side of the angle with the spiral, is then 
trisected at points R and S (Fig. 8.2), and circles are drawn with O as center 



FIG. 8.2 

and OR and OS as radii. If these circles intersect the spiral in points U and V, 
lines OU and OV will trisect the angle AOP. 

Greek mathematics sometimes has been described as essentially static, 
with little regard for the notion of variability ; but Archimedes, in his study 
of the spiral, seems to have found the tangent to a curve through kinematic 
considerations akin to the differential calculus. Thinking of a point on the 
spiral r = ad as subjected to a double motion — a uniform radial motion 
away from the origin of coordinates and a circular motion about the origin — 
he seems to have found (through the parallelogram of velocities) the direction 
of motion (hence of the tangent to the curve) by noting the resultant of the 
two component motions. This seems to be the first instance in which a tangent 
was found to a curve other than a circle. 

Archimedes' study of the spiral, a curve that he ascribed to his friend 
Conon of Alexandria, was part of the Greek search for solutions of the three 
famous problems. The curve lends itself so readily to angle multisections 
that it may well have been devised by Conon for this purpose. As in the case 
of the quadratrix, however, it can serve also to square the circle, as Archi- 
medes showed. At point P let the tangent to the spiral OPR be drawn and 
let this tangent intersect in point Q the line through O that is perpendicular 
to OP. Then, Archimedes showed, the straight-line segment OQ (known as 
the polar subtangent for point P) is equal in length to the circular arc PS 
of the circle with center O and radius OP (Fig. 8.3) that is intercepted between 


the initial line (polar axis) and line OP (radius vector). This theorem, proved 
by Archimedes through a typical double reductio ad absurdum demonstration, 
can be verified by a student of the calculus who recalls that tan \p = r/r', where 
r = f(6) is the polar equation of a curve, r' is the derivative of r with respect 
to 0, and \\i is the angle between the radius vector at a point P and the tangent 
line to the curve at the point P. A large part of the work of Archimedes is 
such that it would now be included in a calculus course, which is particularly 
true of the work On Spirals. If point P on the spiral is chosen as the inter- 
section of the spiral with the 90° line in polar coordinates, the polar sub- 
tangent OQ will be precisely equal to quarter of the circumference of the 
circle of radius OP. Hence the entire circumference is easily constructed as 
four times the segment OQ, and by Archimedes' theorem a triangle equal in 
area to the area of the circle is found. A simple geometrical transformation 
will then produce a square in place of the triangle, and the quadrature of 
the circle is effected. 

Among the twenty-eight propositions in On Spirals are several concerning 
areas associated with the spiral. For example, it is shown in Proposition 24 
that the area swept out by the radius vector in its first complete rotation is 
one third of the area of the "first circle" — that is, the circle with center at 
the pole and radius equal to the length of the radius vector following the 
first complete rotation. Archimedes used the method of exhaustion, but again 
a student today can easily verify the result if he recalls that this area is 
\ ft? r 2 dO. Moreover, it can readily be shown by the calculus, as Archimedes 
did by the more difficult method of exhaustion, that on the next rotation the 
area of the additional ring R 2 (bounded by the first and second turns of the 
spiral and the portion of the polar axis between the two intercepts following 
the first and second rotations) is six times the region R l swept out in the 
first rotation. Areas of the additional rings added on successive rotations are 
given by the simple rule of succession R„ +1 = nRJ(n - 1), as Archimedes 

The work On Spirals was much admired but little read, for it was generally 
regarded as the most difficult of all Archimedean works. Of the treatises 
concerned chiefly with the method of exhaustion (that is, the integral calculus), 
the most popular was Quadrature of the Parabola. The conic sections had 
been known for almost a century when Archimedes wrote, yet no progress 
had been made in finding their areas. It took the greatest mathematician of 
antiquity to square a conic section — a segment of the parabola — which he 
accomplished in Proposition 17 of the work in which the quadrature was 
the goal. The proof by the standard method of exhaustion is long and in- 
volved, but Archimedes rigorously proved that the area K of a parabolic 
segment APBQC (Fig. 8.4) is four-thirds the area of a triangle T having the 


FIG. 8.4 

same base and equal height. In the succeeding (and last) seven propositions 
Archimedes gave a second but different proof of the same theorem. He first 
showed that the area of the largest inscribed triangle, ABC, on the base AC 
is four times the sum of the corresponding inscribed triangles on each of the 
lines AB and BC as base. By continuing the process suggested by this relation- 
ship, it becomes clear that the area K of the parabolic segment ABC is given 
by the sum of the infinite series T + T/4 + T/4 2 + • • • + T/4" + • • • , 
which, of course, is § T. Archimedes did not refer to the sum of the infinite 
series, for infinite processes were frowned on in his day ; instead he proved 
by a double reductio ad absurdum that K can be neither more nor less than § T. 
(Archimedes, like his predecessors, did not use the name "parabola," but 
the word "orthotome," or "section of a right cone.") 

In the preamble to the Quadrature of the Parabola we find the assumption 
or lemma that is usually known today as the axiom of Archimedes : "That 
the excess by which the greater of two unequal areas exceeds the less can, 
by being added to itself, be made to exceed any given finite area." This 
axiom in effect rules out the fixed infinitesimal or indivisible that had been 
much discussed in Plato's day. It is essentially the same as the axiom of 
exhaustion, and Archimedes freely admitted that 

The earlier geometers have also used this lemma, for it is by the use of this same lemma 
that they have shown that circles are to one another in the duplicate ratio of their 
diameters, and that spheres are to one another in the triplicate ratio of their diameters, 
and further that every pyramid is one third part of the prism which has the same base 
with the pyramid and equal height ; also, that every cone is one third part of the cylinder 
having the same base as the cone and equal height they proved by assuming a certain 
lemma similar to that aforesaid. 

The "earlier geometers" mentioned here presumably included Eudoxus 
and his successors. 

Archimedes apparently was unable to find the area of a general segment 8 
of an ellipse or hyperbola. Finding the area of a parabolic segment by modern 
integration involves nothing worse than polynomials, but the integrals 


arising in the quadrature of a segment of an ellipse or hyperbola (as well as 
the arcs of these curves or the parabola) require transcendental functions. 
Nevertheless, in his important treatise On Conoids and Spheroids Archimedes 
found the area of the entire ellipse : "The areas of ellipses are as the rectangles 
under their axes" (Proposition 6). This is, of course, the same as saying that 
the area of x 2 /a 2 + y 2 /b 2 = 1 is nab or that the area of an ellipse is the same 
as the area of a circle whose radius is the geometric mean of the semiaxes of 
the ellipse. Moreover, in the same treatise Archimedes showed how to find 
the volumes of segments cut from an ellipsoid or a paraboloid or a hyper- 
boloid (of two sheets) of revolution about the principal axis. The process 
that he used is so nearly the same as that in modern integration that we shall 
describe it for one case. Let ABC be a paraboloidal segment (or paraboloidal 
"conoid") and let its axis be CD (Fig. 8.5); about the solid circumscribe the 

circular cylinder ABFE, also having CD as axis. Divide the axis into n equal 
parts of length h, and through the points of division pass planes parallel to 
the base. On the circular sections that are cut from the paraboloid by these 
planes construct inscribed and circumscribed cylindrical fustra, as shown 
in the figure. It is then easy to establish, through the equation of the parabola 
and the sum of an arithmetic progression, the following proportions and 
inequalities : 

cylinder ABEF n 2 h n 2 h 

inscribed figure h + 2h + 3h + ■ ■ ■ + (n — l)h \n h 
cylinder ABEF n 2 h n 2 h 

circumscribed figure h + 2h + 3h + - ■ ■ + nh \n 2 h 

Archimedes had previously shown that the difference in volume between 
the circumscribed and inscribed figures was equal to the volume of the lowest 
slice of the circumscribed cylinder; by increasing the number n of sub- 
divisions on the axis, thereby making each slice thinner, the difference 


between the circumscribed and inscribed figures can be made less than any 
preassigned magnitude. Hence the inequalities lead to the necessary con- 
clusion that the volume of the cylinder is twice the volume of the conoidal 
segment. This work differs from the modern procedure in integral calculus 
chiefly in the lack of the concept of limit of a function — a concept that was 
so near at hand and yet was never formulated by the ancients, not even by 
Archimedes, the man who came closest to achieving it. 

Archimedes composed many marvelous treatises, of which his successors 
were inclined to admire most the one On Spirals. The author himself seems 
to have been partial to another, On the Sphere and Cylinder. Archimedes 
requested that on his tomb be carved a representation of a sphere inscribed 
in a right circular cylinder the height of which is equal to its diameter, for 
he had discovered, and proved, that the ratio of the volumes of cylinder and 
sphere is the same as the ratio of the areas — that is, three to two. This property, 
which Archimedes discovered subsequent to his Quadrature of the Parabola, 
remained unknown, he says, to geometers before him. It once had been 
thought 5 that the Egyptians knew how to find the area of a hemisphere ; but 
Archimedes appears now as the first one to have known, and proved, that 
the area of a sphere is just four times the area of a great circle of the sphere. 
Moreover, Archimedes showed that "the surface of any segment of a sphere 
is equal to a circle whose radius is equal to the straight line drawn from the 
vertex of the segment to the circumference of the circle which is the base of 
the segment." This, of course, is equivalent to the more familiar statement 
that the surface area of any segment of a sphere is equal to that of the curved 
surface of a cylinder whose radius is the same as that of the sphere and whose 
height is the same as that of the segment. That is, the surface area of the 
segment does not depend on the distance from the center of the sphere, but 
only on the altitude (or thickness) of the segment. The crucial theorem on the 
surface of the sphere appears in Proposition 33, following a long series of 
preliminary theorems, including one that is equivalent to an integration of 
the sine function : 

If a polygon be inscribed in a segment of a circle LAL so that all its sides 
excluding the base are equal and their number even, as LK ...A... K'L', 
A being the middle point of the segment; and if the lines BB', CC, . . . parallel 
to the base LL' and joining pairs of angular points be drawn, then (BB' + 
CC + ■■■ + LM):AM = A'B.BA, where M is the middle point of LL' and 
AA' is the diameter through M (Fig. 8.6). 

5 See, E. G. Archibald, Outline of the History of Mathematics, 6th ed., (The American Mathe- 
matical Monthly, Slaught Memorial Papers, No. 2, January, 1949), pp. 15-16. Cf. footnotes 
10 and 11 of Chapter 2. 







M \ 









FIG. 8.6 

This is the geometrical equivalent of the trigonometric equation 

9 .29 . n - 1 

sin - + sin !-••• + sin 1 

n n n 

j sin 

n6 1 — cos 9 



From this theorem it is easy to derive the modern expression $ sin x dx = 
1 - cos (f> by multiplying both sides of the equation above by 9/n and taking 
limits as n increases indefinitely. The left-hand side becomes 


lim Y sin X;Ax,- 

where x ; = i9/n for i= 1,2,... n, Ax ; = 9/n for i = 1, 2, . . . n - 1, and 
Ax„ = 9/2n. The right-hand side becomes 


cos 9) lim — cot — = 1 

n^oc 2« 2n 


The equivalent of the special case JS sin x dx = 1 - cos % = 2 had been 
given by Archimedes in the preceding proposition. 

The familiar formula for the volume of a sphere appears in On the Sphere 
and Cylinder I. 34: 

Any sphere is equal to four times the cone which has its base equal to the 
greatest circle in the sphere and its height equal to the radius of the sphere. 

The theorem is proved by the usual method of exhaustion, and the Archi- 
medean ratio for the volume and surface area of the sphere and circumscribed 
cylinder followed as an easy corollary. The sphere-in-a-cylinder diagram 
was indeed carved on the tomb of Archimedes, as we know from a report by 
Cicero. When he was quaestor in Sicily, the Roman orator found the neglected 
tomb with the engraving. He restored the tomb — almost the only contribution 


of a Roman to the history of mathematics — but all traces of it have since 

An interesting light on Greek geometrical algebra is cast by a problem 10 
in Book II of On the Sphere and Cylinder. In Proposition 2 Archimedes 
justified his formula for the volume of a segment of a given sphere ; in Proposi- 
tion 3 he showed that to cut a given sphere by a plane so that the surfaces 
of the segments are in a given ratio, one simply passes a plane perpendicular to 
a diameter through a point on the diameter which divides the diameter into 
two segments having the desired ratio. He then showed in Proposition 4 how 
to cut a given sphere so that the volumes of the two segments are in a given 
ratio — a far more difficult problem. In modern notation, Archimedes was 
led to the equation 

4a 2 (3a — x)(m + n) 
x 2 ma 

where m : n is the ratio of the segments. This is a cubic equation, and Archi- 
medes attacked its solution as had his predecessors in solving the Delian 
problem — through intersecting conies. Interestingly, the Greek approach to 
the cubic was quite different from that to the quadratic equation. By analogy 
with the "application of areas" in the latter case, we would anticipate an 
"application of volumes," but this was not adopted. Through substitutions 
Archimedes reduced his cubic equation to the form x 2 (c — x) = db 2 and 
promised to give separately a complete analysis of this cubic with respect 
to the number of positive roots. This analysis had apparently been lost for 
many centuries when Eutocius, an important commentator of the early sixth 
century, found a fragment that seems to contain the authentic Archimedean 
analysis. The solution was carried out by means of the intersection of the 
parabola ex 2 = b 2 y and the hyperbola (c — x)y = cd. Going further, he 
found a condition on the coefficients that determines the number of real 
roots satisfying the given requirements — a condition equivalent to finding 
the discriminant, 21b 2 d — 4c 3 , of the cubic equation b 2 d = x 2 {c — x). 
(This can easily be verified by the application of a little elementary calculus.) 
Inasmuch as all cubic equations can be transformed to the Archimedean type, 
we have here the essence of a complete analysis of the general cubic. Interest 
in the cubic equation disappeared shortly after Archimedes, to be revived 
for a while by Eutocius and then centuries later still by the Arabs. 

Most of the Archimedean treatises that we have described are a part of 11 
advanced mathematics, but the great Syracusan was not above proposing 
elementary problems. In his Book of Lemmas, for example, we find a study of 
the so-called arbelos or "shoemaker's knife." The shoemaker's knife is the 



region bounded by the three semicircles tangent in pairs in Fig. 8.7, the area 
in question being that which lies inside the largest semicircle and outside 
the two smallest. Archimedes showed in Proposition 4 that if CD is per- 
pendicular to AB, the area of the circle with CD as diameter is. equal to the 
area of the arbelos. In the next proposition it is shown that the two circles 
inscribed within the two regions into which CD divides the shoemaker's 
knife are equal. 

The Book of Lemmas contains also a theorem (Proposition 14) on what 
Archimedes called the salinon or "salt cellar." Draw semicircles with the 
segments AB, AD, DE, and EB as diameters (Fig. 8.8), with AD = EB. Then 

FIG. 8.7 

FIG. 8.8 

the total area bounded by the salinon (bounded entirely by semicircular 
arcs) is equal to the area of the circle having as its diameter the line of sym- 
metry of the figure, FOC. 

It is in the Book of Lemmas that we find also (as Proposition 8) the well- 
known Archimedean trisection of the angle. Let ABC be the angle to be 
trisected (Fig. 8.9). Then with B as center, draw a circle of any radius inter- 
secting AB in P and BC in Q, with BC extended in R. Then draw a line STP 

such that S lies on CQBR extended and T lies on the circle and such that 
ST = BQ = BP = BT. It is then readily shown, since triangles STB and 
TBP are isosceles, that angle BST is precisely a third of angle QBP, the 


angle that was to have been trisected. Archimedes and his contemporaries 
were of course aware that this is not a canonical trisection in the Platonic 
sense, for it involves what they called a neusis — that is, an insertion of a 
given length, in this case ST = BQ, between two figures, here the line QR 
extended, and the circle. 

The Book of Lemmas has not survived in the original Greek, but through 
Arabic translation that later was turned into Latin. (Hence it often is cited 
by its Latin title of Liber assumptorum.) In fact, the work as it has come down 
to us cannot be genuinely Archimedean, for his name is quoted several times 
within the text. However, even if the treatise is nothing more than a collection 
of miscellaneous theorems that were attributed by the Arabs to Archimedes, 
the work probably is substantially authentic. There is doubt also about the 
authenticity of the Cattle-problem, which is generally thought to be Archi- 
medean, and certainly dates back to within a few decades of his death. The 
Cattle-problem is a challenge to mathematicians to solve a set of indetermin- 
ate simultaneous equations in eight unknown quantities — the number of 
bulls and cows of each of four colors. There is some ambiguity in the formula- 
tion of the problem, but according to one interpretation it would take a 
volume of more than 600 pages to give the values for the eight unknowns 
contained in one of the possible solutions! The problem, which involves 
the solution of x 2 = 1 + 4729494v 2 , incidentally provides a first example of 
what later (see below) was to be known as a "Pell equation." 

It is certain that not all of the works of Archimedes have survived, for 1 2 
in a later commentary we learn (from Pappus) that Archimedes discovered 
all of the thirteen possible so-called semiregular solids. Whereas a regular 
solid or polyhedron has faces that are regular polygons of the same type, 
a semiregular solid is a convex polyhedron whose faces are regular polygons, 
but not all of the same type. For example, if from the eight corners of a cube 
a we cut off tetrahedra with edges a(2 - ,/2)/2, the resulting figure will be a 
semiregular or Archimedean solid with surfaces made up of eight equilateral 
triangles and six regular octagons. 

That quite a number of Archimedean works have been lost is clear from 
many references. Arabic scholars inform us that the familiar area formula 
for a tri angle in terms of its th ree sides, usually known as Heron's formula — 
K = yjs(s — a)(s — b)(s — c), where s is the semiperimeter — was known to 
Archimedes several centuries before Heron lived. Arabic scholars also 
attribute to Archimedes the "theorem on the broken chord" — if AB and BC 
make up any broken chord in a circle (with AB # BC) and if M is the mid- 
point of the arc ABC and F the foot of the perpendicular from M to the longer 
chord, F will be the midpoint of the broken chord ABC (Fig. 8.10). Archi- 
medes is reported by the Arabs to have given several proofs of the theorem, 



FIG. 8.10 

one of which is carried out by drawing in the dotted lines in the figure, 
making FC = FC, and proving that AMBC ^ AMBA. Hence BC = BA, 
and it therefore follows that C'F = AB + BF = FC. We do not know 
whether Archimedes saw any trigonometric significance in the theorem, but 
it has been suggested 6 that it served for him as a formula analogous to our 
sin(x - y) = sin x cos v - cos xjsin y. To show the equivalence we let 
MC = 2x and BM = 2y. Then AB = 2x - 2y. Now, the chords correspond- 
ing to these three arcs are respectively MC = 2 sin x, BM = 2 sin y, and 
AB = 2 sin (x - y). Moreover, the projections of MC and MB on BC are 
FC = 2 sin x cos y and FB = 2 sin y cos x. If, finally, we write the broken- 
chord theorem in the form AB = FC - FB, and if for these three chords we 
substitute their trigonometric equivalents, the formula for sin(x - y) results. 
Other trigonometric identities can, of course, be derived from the same 
broken-chord theorem, indicating that Archimedes may have found it a 
useful tool in his astronomical calculations. 

1 3 Unlike the Elements of Euclid, which have survived in many Greek and 
Arabic manuscripts, the treatises of Archimedes have reached us through a 
slender thread. Almost all copies are from a single Greek original which was 
in existence in the early sixteenth century and itself was copied from an 
original of about the ninth or tenth century. The Elements of Euclid has 
been familiar to mathematicians virtually without interruption since its 
composition, but Archimedean treatises have had a more checkered career. 
There have been times when few or even none of Archimedes' works were 
known. In the days of Eutocius, a first-rate scholar and skillful commentator 
of the sixth century, only three of the many Archimedean works were gener- 
ally known — On the Equilibrium of Planes, the incomplete Measurement of a 

6 See Johannes Tropfke, "Archimedes und die Trigonometrie," Archivfur die Geschichte der 
Mathematik, 10 (1927-1928), 432-463. 


Circle, and the admirable On the Sphere and Cylinder. Under the circum- 
stances it is a wonder that such a large proportion of what Archimedes wrote 
has survived to this day. Among the amazing aspects of the provenance of 
Archimedean works is the discovery within the twentieth century of one of 
the most important treatises — one which Archimedes simply called The 
Method and which had been lost since the early centuries of our era until 
its rediscovery in 1906. 

The Method of Archimedes is of particular significance because it discloses 
for us a facet of Archimedes' thought that is not found elsewhere. His other 
treatises are gems of logical precision, with little hint of the preliminary 
analysis that may have led to the definitive formulations. So thoroughly 
without motivation did his proofs appear to some writers of the seventeenth 
century that they suspected Archimedes of having concealed his method of 
approach in order that his work might be admired the more. How un- 
warranted such an ungenerous estimate of the great Syracusan was became 
clear in 1906 with the discovery of the manuscript containing The Method. 
Here Archimedes had published, for all the world to read, a description of 
the preliminary "mechanical" investigations that had led to many of his 
chief mathematical discoveries. He thought that his "method" in these cases 
lacked rigor, since it assumed an area, for example, to be a sum of line 

The Method, as we have it, contains most of the text of some fifteen proposi- 
tions sent in the form of a letter to Eratosthenes, mathematician and librarian 
at the university at Alexandria. The author opened by saying that it is easier 
to supply a proof of a theorem if we first have some knowledge of what is 
involved ; as an example he cites the proofs of Eudoxus on the cone and 
pyramid, which had been facilitated by the preliminary assertions, without 
proof, made by Democritus. Then Archimedes announced that he himself 
had a "mechanical" approach that paved the way for some of his proofs. 
The very first theorem that he discovered by this approach was the one on 
the area of a parabolic segment ; in Proposition 1 of The Method the author 
describes how he arrived at this theorem by balancing lines as one balances 
weights in mechanics. He thought of the areas of the parabolic segment 
ABC and the triangle AFC (where FC is tangent to the parabola at C) as 
the totality of a set of lines parallel to the diameter QB of the parabola, such 
as OP (Fig. 8.11) for the parabola and OM for the triangle. If, now, one were 
to place at H (where HK = KC) a line segment equal to OP, this would just 
balance the line OM where it now is, K being the fulcrum. (This can be 
shown through the law of the lever and the property of the parabola.) Hence 
the area of the parabola, if placed with its center of gravity at H, will just 
balance the triangle, whose center of gravity is along KC and a third of the 
way from K to C. From this one easily sees that the area of the parabolic 



FIG. 8.11 

segment is one-third the area of triangle AFC, or four-thirds the area of the 
inscribed triangle ABC. 

1 4 The favorite theorem of Archimedes, represented on his tomb, was also 
suggested to him by his mechanical method. It is described in Proposition 2 
of The Method: 

Any segment of a sphere has to the cone with the same base and height the 
ratio which the sum of the radius of the sphere and the height of the comple- 
mentary segment has to the height of the complementary segment. 

The theorem follows readily from a beautiful balancing property which 
Archimedes discovered (and which can be easily verified in terms of modern 
formulas). Let AQDCP be a cross section of a sphere with center O and 
diameter AC (Fig. 8.12) and let AUV be a plane section of a right circular 
cone with axis AC and UV as diameter of the base. Let IJUV be a right 
circular cylinder with axis AC and with UV = IJ as diameter and let AH = 
AC. If a plane is passed through any point S on the axis AC and perpendicular 
to AC, the plane will cut the cone, the sphere, and the cylinder in circles of 
radii r, = SR, r 2 = SP, and r 3 = SN respectively. If we call the areas of 
these circles A x , A 2 , and A 3 , then, Archimedes found, A x and -A 2 , when 
placed with their centers at H, will just balance A 3 where it now is, with A 
as the fulcrum. Hence if we call the volumes of the sphere, the cone, and the 
cylinder V,, V 2 , V 3 , it follows that V l + V 2 = \V 3 ; and since V 2 = \-V 3 , 
the sphere must be ^V 3 . Because the volume V 3 of the cylinder is known (from 
Democritus and Eudoxus), the volume of the sphere also is known — in 
modern notation, V = jnr 3 . By applying the same balancing technique to 
the spherical segment with base diameter BD, to the cone with base diameter 


















I M 
FIG. 8.12 

K U 

EF, and to the cylinder with base diameter KL, the volume of the spherical 
segment is found in the same manner as for the whole sphere. 

The method of equilibrium of circular sections about a vertex as fulcrum 
was applied by Archimedes to discover the volumes of the segments of 
three solids of revolution — the ellipsoid, the paraboloid, and the hypcrboloid, 
as well as the centers of gravity of the paraboloid (conoid), of any hemisphere, 
and of a semicircle. The Method closes with the determination of volumes of 
two solids that are favorites in modern calculus books — a wedge cut from a 
right circular cylinder by two planes (as in Fig. 8.13) and the volume common 


FIG. 8.13 


to two equal right circular cylinders intersecting at right angles. The work 
containing such marvelous results of more than 2000 years ago was recovered 
almost by accident in 1906. The indefatigable Danish scholar J. L. Heiberg 
had read that at Constantinople there was a palimpsest of mathematical 
content. (A palimpsest is a parchment the original writing on which has been 
only imperfectly washed off and replaced with a new and different text.) 
Close inspection showed him that the original manuscript had contained 
something by Archimedes, and through photographs he was able to read 
most of the Archimedean text. The manuscript consisted of 185 leaves, mostly 
of parchment but a few of paper, with the Archimedean text copied in a 
tenth-century hand. An attempt — fortunately, none too successful — had 
been made to expunge this text in order to use the parchment for a Eucho- 
logion (a collection of prayers and liturgies used in the Eastern Orthodox 
Church) written in about the thirteenth century. The mathematical text 
contained On the Sphere and Cylinder, most of the work On Spirals, part of the 
Measurement of a Circle and of On the Equilibrium of Planes, and On Floating 
Bodies, all of which have been preserved in other manuscripts ; most important 
of all, the palimpsest gives us the only surviving copy of The Method. In a 
sense the palimpsest is symbolic of the contribution of the Medieval Age. 
Intense preoccupation with religious concerns very nearly wiped out one of 
the most important works of the greatest mathematician of antiquity ; yet 
in the end it was Medieval scholarship that inadvertently preserved this, 
and much besides, which might otherwise have been lost. 


Bromwich, T. J., "The Methods Used by Archimedes for Approximating to Square 
Roots, The Mathematical Gazette, 14 (1928-1929), 253-257. 

Clagett, Marshall, Archimedes in the Middle Ages (Madison, Wis. : University of Wis- 
consin Press, 1964- , 2 vols.). 

Cohen, M. R., and I. E. Drabkin, A Source Book in Greek Science (New York : McGraw- 
Hill, 1948 ; reprinted Cambridge, Mass. : Harvard University Press, 1958). 

Davis, H. T., "Archimedes and Mathematics," School Science and Mathematics, 44 
(1944), 136-145, 213-221. 

Dijksterhuis, E. J., Archimedes (New York : Humanities Press, 1957). 

Erhardt, Erika von, and Rudolf von Erhardt, "Archimedes' Sand-Reckoner," Isis, 33 
(1942), 578-602. 

Heath, T. L., The Works of Archimedes (Cambridge, 1897; paperback reprint, including 
Archimedes' Method, New York : Dover, n.d.). 

Heiberg, J. L., Quaestiones archimedae (Copenhagen, 1879). 

Heiberg, J. L., "Le role d'Archimede dans le developpement des sciences exactes," 
Scientia, 20 (1916), 81-89. 

Heiberg, J. L., ed., Archimedes, Opera omnia (Leipzig, 1880-1881, 3 vols.). 


Heiberg, J. L., and H. G. Zeuthen, "Eine neue Schrift des Archimedes," Bibliotheca 

Mathematica (3). 7 (1 906-1907), 321-363. 
Hofmann, J. E., "Erklarungsversuche fur Archimeds Berechnung von y/3," Archiv 

fur die Geschichte der Mathematik, 12 (1929), 386-408. 
Hoppe, Edmund, "Die zweite Methode des Archimedes zur Berechnung von n" Archiv 

fur die Geschichte der Mathematik, 9 (1920-1922), 104-107. 
Midolo, P., Archimede e il suo tempo (Syracuse, 1912). 
Neugebauer, O., "Archimedes and Aristarchus," Isis, 34 (1942), 4-6. 
Smith, D. E., "A Newly Discovered Treatise of Archimedes," Monist, 19 (1909), 202-230. 
Thomas, Ivor, Selections Illustrating the History of Greek Mathematics (Cambridge, 

Mass. : Leob Classical Library, 1939-1941, 2 vols.). 
Tropfke, Johannes, "Archimedes und die Trigonometric," Archiv fur die Geschichte der 

Mathematik, 10 (1927-1928), 432^63. 
Weissenborn, Hermann, "Die irrationalen Quadratwurzeln bei Archimedes und 

Heron," Berliner Studien fur Klassische Philologie und Archaeologie, 1 (1884), 



1. Archimedes is sometimes regarded as the inventor of the integral calculus. To what extent 
do you agree or disagree with this view? 

2. Euclid depended heavily on the works of his predecessors. To what extent is this true also 
of Archimedes? 

3. Aristotle knew the law of the lever before Archimedes was born. Why, then, is the law 
sometimes attributed to Archimedes? Explain. 

4. Of the many treatises by Archimedes with which we are familiar, which do you regard as 
the most significant for the development of mathematics? Explain. 

5. Archimedes generally is legarded as the greatest mathematician of antiquity. Explain fully 
the justification for such a view, comparing his work with that of at least two potential 
earlier rivals. 

6. If Oj and A t are respectively the areas of regular polygons of / sides inscribed in and circum- 
scribed about a circle, prove the Archimedean recursion formulas a 2 „ = ^/a n A„ and 
A 2 „ = 2A n a 2 J{A„ + a 2 „). 

7. If p,- and P, are perimeters of regular polygons inscribed in and circums cribed about a circle, 
prove the Archimedean algorithm P 2n = 2P„p„/(P„ + p„) and p 2 „ = >/p„P2„. 

8. Beginning, as did Archimedes, with a regular hexagon inscribed in a circle, use an Archi- 
medean recursion algorithm to find either p 12 and P 12 or a l2 and A 12 . What value of it 
would be implied by the arithmetic mean of your answers? 

9. Find the area lying between the portions of the spiral r = a6 formed for < 8 < 2% and for 
2n <6 < Ait. 

10. Show clearly how to divide the surface area of a sphere by two parallel planes into three 
numerically equal areas. 

11. Prove the Archimedean theorem that the area of the "shoemaker's knife" is equal to the 
area of the circle with CD as diameter (Fig. 8.7). 

12. Prove the Archimedean trisection method described in the text. 

13. Either construct or draw diagrams of three Archimedean semiregular solids. 

*14. Find, for the Archimedean spiral r = aO, the length of the polar subtangent for 9 = In 
and show how this can be used to square the circle. 


*15. Prove the Archimedean theorem on the broken chord. 

*16. Using either the Archimedean balancing property or modern integration, give a proof of 

the formula for the volume of a segment of a sphere. 
*17. Prove the Archimedean theorem on the salinon. 
* 1 8. In the diagram of the Archimedean theorem on the broken chord (Fig. 8.10), use the equation 

BF + FC = BC to derive the familiar trigonometric identity for sin(x + y). 
*19. Can you, either exactly or approximately, divide the unit sphere by two parallel planes 

into three segments equal in volume? Explain. 
*20. Prove that the two circles inscribed in the two portions into which line CD divides the 

"shoemaker's knife" (Fig. 8.7) are equal. 


Apollonius of Perga 

It seems to me that all the evidence points to 
Apollonius as the founder of Greek mathematical 

Otto Neugebauer 

During the first century or so of the Hellenistic Age three mathematicians 
stood head and shoulders above all others of the time, as well as above most 
of their predecessors and successors. These men were Euclid, Archimedes, 
and Apollonius; it is their work that leads to the designation of the period 
from about 300 to 200 B.C. as the "Golden Age" of Greek mathematics. 
In a sense mathematics had lagged behind the arts and literature, for it was 
the Age of Pericles, in the middle of the fifth century B.C., that in the broader 
sense is known as the "Golden Age of Greece." Throughout the Hellenistic 
period the city of Alexandria remained the mathematical focus of the 
Western world, but Apollonius, like Archimedes, was not a native there. 
He was born at Perga in Pamphilia (southern Asia Minor); but he may have 
been educated at Alexandria, and he seems to have spent some time teaching 
there at the university. For a while he was at Pergamum, where there was a 
university and a library second only to that at Alexandria, through the 
patronage of Alexander's general, Lysimachus, and his successors. Inasmuch 
as the ancient world had many men named Apollonius (of these 129 with 
biographies are listed in Pauly-Wissowa, Real-Enzyclopadie der klassischen 
Altertumswissenschaft), our mathematician is distinguished from others by 
use of the full name, Apollonius of Perga. We do not know the precise dates 
of his life, but he is reported to have flourished during the reigns of Ptolemy 
Euergetes and Ptolemy Philopater; one report makes him a treasurer- 
general of Ptolemy Philadelphus, and it was said that he was twenty-five to 
forty years younger than Archimedes. The years 262 to 190 B.C. have been 
suggested for his life, about which little is known. He seems to have felt 
himself to be a rival of Archimedes ; he thus touched on several themes that 
we discussed in the preceding chapter. He developed a scheme of "tetrads" 
for expressing large numbers, using an equivalent of exponents of the single 
myriad, whereas Archimedes had used the double myriad as a base. The 



numerical scheme of Apollonius probably was the one of which part is 
described in the surviving last portion of Book II of the Mathematical 
Collection of Pappus. (All of Book I and the first part of Book II have been 
lost.) Here the number 5,462,360,064 x 10 6 is written as (i y ,eul;P ^ m n,"ev, 
where \i\ //, and \i* are the third, the second, and the first powers, respectively, 
of a myriad. 

Apollonius wrote a work (now lost) entitled Quick Delivery which seems 
to have taught speedy methods of calculation. In it the author is said to have 
calculated a closer approximation to n than that given by Archimedes — 
probably the value we know as 3.1416. We do not know how this value, 
which appeared later in Ptolemy and also in India, was arrived at. In fact, 
there are more unanswered questions about Apollonius and his work than 
about Euclid or Archimedes, for more of his works have disappeared. We 
have the titles of many lost works, such as one on Cutting-off of a Ratio, 
another on Cutting-off of an Area, one On Determinate Section, another on 
Tangencies (or Contacts), one on Vergings (or Inclinations), and one on 
Plane Loci. In some cases we know what the treatise was about, for Pappus 
later gave brief descriptions of a few. Six of the works of Apollonius were 
included, together with a couple of Euclid's more advanced treatises (now 
lost), in a collection known as the "Treasury of Analysis." Pappus described 
this as a special body of doctrine for those who, after going through the 
usual elements, wish to obtain power to solve problems involving curves. 
The "Treasury," made up largely of works by Apollonius, consequently 
must have included much of what we now call analytic geometry ; it was 
with good reason that Apollonius, rather than Euclid, was known in antiquity 
as "The Great Geometer." 

From the descriptions given by Pappus and others, it is possible to obtain 
a good idea of the contents of some of the lost Greek works, and when in the 
seventeenth century the game of reconstructing lost geometrical books was 
at its height, the treatises of Apollonius were among the favorites. 1 From 
restorations of the Plane Loci, for example, we infer that the following were 
two of the loci considered : (1) The locus of points the difference of the squares 
of whose distances from two fixed points is constant is a straight line perpen- 
dicular to the line joining the points ; (2) the locus of points the ratio of whose 
distances from two fixed points is constant (and not equal to one) is a circle. 
The latter locus is, in fact, now known as the "Circle of Apollonius," but this 
is a misnomer since it had been known to Aristotle who had used it to give 
a mathematical justification of the semicircular form of the rainbow. 2 

1 For an account of these "restorations" see the article on "Apollonius"' by T. L. Heath in the 
Encyclopaedia Britannica. 11th ed. (1910). 

2 See C. B. Boyer, The Rainbow (New York : Yoseloff, 1959), pp. 45-46. 


The Cut ting-off of a Ratio dealt with the various cases of a general prob- 
lem — given two straight lines and a point on each, draw through a third 
given point a straight line that cuts off on the given lines segments (measured 
from the fixed points on them respectively) that are in a given ratio. This 
problem is equivalent to solving a quadratic equation of the type ax — x 2 = 
be, that is, of applying to a line segment a rectangle equal to a rectangle and 
falling short by a square. In Cutting-off of an Area the problem is similar 
except that the intercepted segments are required to contain a given rectangle, 
rather than being in a given ratio. This problem leads to a quadratic of the 
form ax + x 2 = be, so that one has to apply to a segment a a rectangle equal 
to a rectangle and exceeding by a square. The Apollonian treatise On Deter- 
minate Section dealt with what might be called an analytic geometry of one 
dimension. It considered the following general problem, using the typical 
Greek algebraic analysis in geometric form : Given four points A, B, C, D 
on a straight line, determine a fifth point P on it such that the rectangle on 
AP and CP is in a given ratio to the rectangle on BP and DP. Here, too, the 
problem reduces easily to the solution of a quadratic ; and, as in other cases, 
Apollonius treated the question exhaustively, including the limits of possi- 
bility and the number of solutions. 

The treatise on Tangencies is of a different sort from the three above, for as 
Pappus describes it we see the problem familiarly known today as the "Prob- 
lem of Apollonius." Given three things, each of which may be a point, a line, 
or a circle, draw a circle that is tangent to each of the three given things (where 
tangency to a point is to be understood to mean that the circle passes through 
the point). This problem involves ten cases, from the two easiest (in which the 
three things are three points or three lines) to the most difficult of all (to 
draw a circle tangent to three circles). The two easiest had appeared in 
Euclid's Elements in connection with inscribed and circumscribed circles of a 
triangle ; another six cases were handled in Book I of Tangencies, and the case 
covering two lines and a circle, as well as the case of three circles, occupied all 
of Book II. We do not have the solutions of Apollonius, but they can be 
reconstructed on the basis of information from Pappus. Nevertheless, 
scholars of the sixteenth and seventeenth centuries generally were under the 
impression that Apollonius had not solved the last case ; hence they regarded 
this problem as a challenge to their abilities. Newton was among those who 
gave a solution, using straightedge and compasses alone. 3 

The trisection of the angle by Archimedes, in which a given length is 
inserted between a line and a circle along a straight line that is shifted so as 
to pass through a given point (point P in Fig. 8.9), is a typical example of a 
solution by means of a neusis (verging or inclination). Apollonius' treatise 

3 Arithmetica universalis. Problem XLVII. 



on Vergings considered the class of neusis problems that can be solved by 
"plane" methods — that is, by the use of compasses and straightedge only. 
(The Archimedean trisection, of course, is not such a problem, for in modern 
times it has been proved that the general angle cannot be trisected by "plane" 
methods.) According to Pappus, one of the problems dealt with in Vergings is 
the insertion within a given circle of a chord of given length verging to a 
given point. 

There were in antiquity allusions to still other works by Apollonius, 
including one on Comparison of the Dodecahedron and the Icosahedron. In 
this the author gave a proof of the theorem (known perhaps to Aristaeus) 
that the plane pentagonal faces of a dodecahedron are the same distance 
from the center of the circumscribing sphere as are the plane triangular 
faces of an icosahedron inscribed in the same sphere. The theorem in the 
spurious Book XIV of the Elements — that in this case the ratio of the areas of 
the icosahedron and the dodecahedron is equal to the ratio of their volumes — 
follows immediately from the Apollonian proposition; and it may be that 
the author of Elements XIV made use of the treatise of Apollonius. 

Apollonius was also a celebrated astronomer ; the favorite mathematical 
device in antiquity for the representation of the motions of the planets is 
apparently due to him. Whereas Eudoxus had used concentric spheres, 
Apollonius proposed instead two alternative systems, one made up of 
epicyclic motions and the other involving eccentric motions. In the first 
scheme a planet P was assumed to move uniformly about a small circle 
(epicycle), the center C of which in turn moved uniformly along the circum- 
ference of a larger circle (deferent) with center at the earth E (Fig. 9.1). In the 
eccentric scheme the planet P moves uniformly along the circumference of a 
large circle, the center C of which in turn moves uniformly in a small circle 

FIG. 9.1 


with center at E. If PC = C'E, the two geometric schemes will be equivalent, 
as Apollonius evidently knew. 4 While the theory of homocentric spheres 
had become, through the work of Aristotle, the favorite astronomical scheme 
of those satisfied by a gross representation of the approximate motions, the 
theory of cycles and epicycles, or of eccentrics, became, through the work of 
Ptolemy, the choice of mathematical astronomers who wanted refinement of 
detail and predictive precision. For some 1800 years the two schemes — the 
one of Eudoxus and the other of Apollonius — were friendly rivals vying for 
the favor of scholars. 

Despite his scholarly productivity, only two of the many treatises by 
Apollonius have in large part survived. All Greek versions of the Cutting-offof 
a Ratio were lost long ago, but not before an Arabic translation had been 
made. In 1706 Halley, Newton's friend, published a Latin translation of the 
work, and it has since appeared in vernacular tongues. Apart from this 
treatise, only one Apollonian work has substantially survived, which, 
however, was by all odds his chef-d'oeuvre — the Conies. Of this famous work 
only half— the first four of the original eight books — remains extant in Greek ; 
fortunately, an Arabic mathematician, Thabit ibn Qurra, had translated the 
next three books, and this version has survived. In 1710 Edmund Halley 
provided a Latin translation of the seven books, and editions in many 
languages have appeared since then. 

The conic sections had been known for about a century and a half when 
Apollonius composed his celebrated treatise on these curves. At least twice 
in the interval general surveys had been written — by Aristaeus and by 
Euclid — but just as Euclid's Elements had displaced earlier elementary 
textbooks, so on the more advanced level of the conic sections the Conies of 
Apollonius superseded all rivals in its field, including the Conies of Euclid, 
and no attempt to improve on it seems to have been made in antiquity. If 
survival is a measure of quality, the Elements of Euclid and the Conies of 
Apollonius were clearly the best works in their fields. 

Book I of the Conies opens with an account of the motivation for writing 
the work. While Apollonius was at Alexandria, he was visited by a geometer, 
named Naucrates, and it was at the latter's request that Apollonius wrote out 
a hasty draft of the Conies in eight books. Later at Pergamum the author took 
the time to polish the books one at a time, hence Books IV through VII open 
with greetings to Attalus, King of Pergamum. The first four books the author 
describes as forming an elementary introduction, and it has been assumed 
that much of this material had appeared in earlier treatises on conies. 
However, Apollonius expressly says that some of the theorems in Book III 

4 See O. Neugebauer, "Eccentric and Epicyclic Motion According to Apollonius," Scripta 
Mathematica, 24 (1959), 5-21. 


were his own, for Euclid had not completed the loci there considered. The 
last four books he describes as extensions of the subject beyond the essentials, 
and we shall see that in them the theory is advanced in more specialized 
directions. 5 

Before the time of Apollonius the ellipse, parabola, and hyperbola were 
derived as sections of three distinctly different types of right circular cones, 
according as the vertex angle was acute, right, or obtuse. Apollonius, ap- 
parently for the first time, systematically showed that it is not necessary to 
take sections perpendicular to an element of the cone and that from a single 
cone one can obtain all three varieties of conic section simply by varying 
the inclination of the cutting plane. This was an important step in linking 
the three types of curve. A second important generalization was made when 
Apollonius demonstrated that the cone need not be a right cone—that is, 
one whose axis is perpendicular to the circular base — but can equally well be 
an oblique or scalene circular cone. If Eutocius, in commenting on the Conies, 
was well informed, we can infer that Apollonius was the first geometer to 
show that the properties of the curves are not different according as they 
are cut from oblique cones or from right cones. Finally, Apollonius brought 
the ancient curves closer to the modern point of view by replacing the single- 
napped cone (somewhat like a modern ice-cream cone) by a double-napped 
cone (resembling two oppositely oriented indefinitely long ice-cream cones 
placed so that the vertices coincide and the axes are in a straight line). 
Apollonius gave, in fact, the same definition of a circular cone as that used 
today : 

If a straight line, indefinite in length and passing always through a fixed point 

be made to move around the circumference of a circle which is not in the same 

plane with the point so as to pass successively through every point of that 

circumference, the moving straight line will trace out the surface of a double 


This change made the hyperbola the double-branched curve familiar to us 

today. Geometers often referred to the "two hyperbolas," rather than to the 

"two branches" of a single hyperbola, but in either case the duality of the 

curve was recognized. 

Concepts are more important in the history of mathematics than is 
terminology, but there is more than ordinary significance in a change of 
name for the conic sections that was due to Apollonius. For about a century 
and a half the curves had had no more distinctive appellations than banal 
descriptions of the manner in which the curves had been discovered— sections 

5 See T. L. Heath, Apollonius ofPerga. Treatise on Conic Sections (1896), pp. xxvi-xxvii. Here, 
and throughout this chapter, we depend on Heath's valuable volume, from which passages in 
translation have been taken. 


of an acute-angled cone (oxytome), sections of a right-angled cone (ortho- 
tome), and sections of an obtuse-angled cone (amblytome). Archimedes had 
continued these names (although he is reported to have used also the word 
parabola as a synonym for section of a right-angled cone). It was Apollonius 
(possibly following up a suggestion of Archimedes) who introduced the names 
ellipse and hyperbola in connection with these curves. The words "ellipse," 
"parabola," and "hyperbola" were not newly coined for the occasion ; they 
were adapted from an earlier use, perhaps by the Pythagoreans, in the 
solution of quadratic equations through the application of areas. Ellipsis 
(meaning a deficiency) had been used when a rectangle of given area was 
applied to a given line segment and fell short by a square (or other specified 
figure), and the word hyperbola (a throwing beyond) had been adopted when 
the area exceeded the line segment. The word parabola (a placing beside or 
comparison) had indicated neither excess nor deficiency. Apollonius now 
applied these words in a new context as names for the conic sections. The 
familiar modern equation of the parabola with vertex at the origin is y 2 = Ix 
(where / is the "latus rectum" or parameter, now often represented by 2p, 
or occasionally by 4p). That is, the parabola has the property that no matter 
what point on the curve one chooses, the square on the ordinate is precisely 
equal to the rectangle on the abscissa x and the parameter /. The equations 
of the ellipse and hyperbola, similarly referred to a vertex as origin, are 
(x + afja 2 + y 2 /b 2 = 1, or y 2 = Ix + b 2 x 2 /a 2 (where / again is the latus 
rectum or parameter 2b 2 /a). That is, for the ellipse y 2 < Ix and for the hyper- 
bola y 2 > Ix, and it is the properties of the curves that are represented by these 
inequalities that prompted the names given by Apollonius more than two 
millennia ago and still firmly attached to them. 6 

In deriving all conic sections from a single double-napped oblique circular 
cone, and in giving them eminently appropriate names, Apollonius made an 
important contribution to geometry ; but he failed to go as far in generality 
as he might have. He could as well have begun with an elliptic cone — or with 
any quadric cone — and still have derived the same curves. That is, any plane 
section of Apollonius' "circular" cone could have served as the generating 
curve or "base" in his definition, and the designation "circular cone" is 
unnecessary. In fact, as Apollonius himself showed (Book I, Proposition 5), 
every oblique circular cone has not only an infinite number of circular 
sections parallel to the base, but also another infinite set of circular sections 
given by what he called subcontrary sections. Let BFC be the base of the 

The commentator Eutocius was responsible for an erroneous impression, still fairly wide- 
spread, that the words ellipse, parabola, and hyperbola were adopted by Apollonius to indicate 
that the cutting plane fell short of, or ran along with, or ran into the second nappe of the cone. 
This is not at all what Apollonius reported in the Conies. 


oblique circular cone and let ABC be a triangular section of the cone (Fig. 9.2). 
Let P be any point on a circular section DPE parallel to BFC and let HPK 
be a section by a plane such that triangles AHK and ABC are similar but 
oppositely oriented. Apollonius then called the section HPK a subcontrary 
section and showed that it is a circle. The proof is easily established in terms 
of the similarity of triangles HMD and EMK. from which it follows that 
HM >MK <= DM ■ ME = PM 2 , the characteristic property of a circle. (In the 
language of analytic geometry, if we let HM = x, HK = a, and PM = y, 
then y 2 = x(a - x) or x 2 + y 2 = ax, which is the equation of a circle.) 


FIG. 9 2 

Greek geometers divided curves into three categories. The first, known as 
"plane loci," consisted of all straight lines and circles; the second, known 
as "solid loci," was made up of all conic sections ; the third category, known as 
"linear loci," lumped together all other curves. The name applied to the 
second category undoubtedly was suggested by the fact that the conies were 
not defined as loci in a plane which satisfy a certain condition, as is done 
today ; they were described stereometrically as sections of a three-dimensional 
figure. Apollonius, like his predecessors, derived his curves from a cone in 
three-dimensional space, but he dispensed with the cone as promptly as 
possible. From the cone he derived a fundamental plane property or "symp- 
tome" for the section, and thereafter he proceeded with a purely planimetric 
study based on this property. This step, which we here illustrate for the 
ellipse (Book I, Proposition 13), probably was much the same as that used 
by his predecessors, including Menaechmus. Let ABC be a triangular section 


of an oblique circular cone (Fig. 9.3) and let P be any point on a section HPK 
cutting all elements of the cone. Extend HK to meet BC in G and through P 
pass a horizontal plane cutting the cone in the circle DPE and the plane 
HPK in the line PM. Draw DME, a diameter of the circle perpendicular to 
PM. Then from the similarity of triangles HDM and HBG we have 
DM/HM = BGjHG, and from the similarity of triangles MEK and KCG 
we have ME/MK = CG/KG. Now, from the property of the circle we have 
PM 2 = DM- ME: hence PM 2 = {HM • BG/HG) (MK ■ CG)/KG. If PM = 
y. HM = x, and HK = 2a, the property in the preceding sentence is equiva- 
lent to the equation y 2 = kx{2a — x), which we recognize as the equation of 
an ellipse with H as vertex and HK as major axis. In a similar manner 
Apollonius derived for the hyperbola the equivalent of the equation y 2 = 
kx(x + 2a). These forms are easily reconciled with the "name" forms above 
by taking k = b 2 /a 2 and i = 2b 2 /a. 

fig. 9.3 

After Apollonius had derived from a stereometric consideration of the 
cone the basic relationship between what we should now call the plane co- 
ordinates of a point on the curve — given by the three equations 
y 2 m=lx- b 2 x 2 /a 2 , y 2 = Ix, and y 1 = tx + b 2 x 2 /a 2 — he derived further 
properties from the plane equations without reference to the cone. The 
author of the Conies reported [hat in Book 1 he had worked out the funda- 
mental properties of the curves "more fully and generally than in the writings 
of other authors," The extent to which this statement holds true is suggested 
by the fact that here, in the very first book, the theory of conjugate diameters 
of a conic is developed. That is, Apollonius showed that the midpoints of a set 
of chords parallel to one diameter of an ellipse or hyperbola will constitute a 


second diameter, the two being called "conjugate diameters." In fact, whereas 
today we invariably refer a conic to a pair of mutually perpendicular lines 
as axes, Apollonius generally used a pair of conjugate diameters as equiva- 
lents of oblique coordinate axes. The system of conjugate diameters provided 
an exceptionally useful frame of reference for a conic, for Apollonius showed 
that if a line is drawn through an extremity of one diameter of an ellipse or 
hyperbola parallel to the conjugate diameter, the line "will touch the conic, 
and no other straight line can fall between it and the conic" — that is, the 
line will be tangent to the conic. Here we see clearly the Greek static concept 
of a tangent to a curve, in contrast to the Archimedean kinematic view. In 
fact, often in the Conies we find a diameter and a tangent at its extremity used 
as a coordinate frame of reference. 

Among the theorems in Book I are several (Propositions 41 through 49) 
that are tantamount to a transformation of coordinates from a system based 
on the tangent and diameter through a point P on the conic to a new system 
determined by a tangent and diameter at a second point Q on the same curve, 
together with the demonstration that a conic can be referred to any such 
system as axes. In particular, Apollonius was familiar with the properties of 
the hyperbola referred to its asymptotes as axes, given, for the equilateral 
hyperbola, by the equation xy = c 2 . He had no way of knowing, of course, 
that some day this relationship, equivalent to Boyle's law, would be funda- 
mental in the study of gases or that his study of the ellipse would be essential 
to modern astronomy. 

1 Book II continues the study of conjugate diameters and tangents. For 
example, if P is any point on any hyperbola, with center C, the tangent at P 
will cut the asymptotes in points L and L' (Fig. 9.4) that are equidistant from 
P (Propositions 8 and 10). Moreover (Propositions 11 and 16), any chord 
QQ' parallel to CP will meet the asymptotes in points K and K' such that 
QK = Q'K' and QK ■ QK' = CP 2 . (These properties were verified synthetic- 
ally, but the reader can convince himself of their validity by use of modern 
analytic methods.) Later propositions in Book II show how to draw tangents 

FIG. 9.4 


to a conic by making use of the theory of harmonic division. In the case of the 
ellipse (Proposition 49), for example, if Q is a point on the curve (Fig. 9.5), 
Apollonius dropped a perpendicular QN from Q to the axis AA' and found the 
harmonic conjugate T of N with respect to A and A'. (That is, he found the 
point Ton line A A' extended such that AT/A'T = AN/NA' ; in other words, 

FIG. 9.5 

he determined the point T that divides the segment AA' externally in the 
same ratio as N divides A A' internally.) The line through Tand Q then will 
be tangent to the ellipse. The case in which Q does not lie on the curve can be 
reduced to this through familiar properties of harmonic division. (It can be 
proved that there are no plane curves other than the conic sections such that, 
given the curve and a point, a tangent can be drawn, with straightedge and 
compasses, from the point to the curve ; but this was of course unknown to 

Apollonius apparently was especially proud of Book III, for in the General 1 1 
Preface to the Conies he wrote : 

The third book contains many remarkable theorems useful for the synthesis of solid 
loci and determinations of limits ; the most and prettiest of these theorems are new and, 
when I had discovered them, I observed that Euclid had not worked out the synthesis 
of the locus with respect to three and four lines, but only a chance portion of it and 
that not successfully : for it was not possible that the synthesis could have been completed 
without my additional discoveries. 

The three-and-four-line locus, to which reference is made, played an impor- 
tant role in mathematics from Euclid to Newton. Given three lines (or four 
lines) in a plane, find the locus of a point P that moves so that the square of the 
distance from P to one of these is proportional to the product of the distances 
to the other two (or, in the case of four lines, the product of the distances to 
two of them is proportional to the product of the distances to the other two), 
the distances being measured at given angles with respect to the lines. 
Through modern analytic methods, including the normal form of the 
straight line, it is easy to show that the locus is a conic section — real or 
imaginary, reducible or irreducible. If, for the three-line locus, equations of 
the given lines are A x x + B^y + C t = 0, A 2 x + B 2 y + C 2 = 0, and 


A 3 x + B 3 y + C 3 = 0, and if the angles at which the distances are to be 
measured are 9 t , 6 2 , and 6 3 , then the locus of P(x, y) is given by 

(A x x + B iy + C t ) 2 K{A 2 x + B 2 y + C 2 ) (A 3 x + B 3 y + C^ 
(V+ Bi^sin 2 ^! ~ JA 2 2 + B 2 2 sin6 2 ' ^/a 3 2 + B 3 2 sin 3 

This equation is, in general, of second degree in x and y ; hence the locus is a 
conic section. Our solution does not do justice to the treatment given by 
Apollonius in Book III, in which more than fifty carefully worded proposi- 
tions, all proved by synthetic methods, lead eventually to the required locus. 
Half a millennium later Pappus suggested a generalization of this theorem for 
n lines, where n > 4, and it was against this generalized problem that Des- 
cartes in 1637 tested his analytic geometry. Thus few problems have played 
as important a role in the history of mathematics as did the "locus to three 
and four lines." 

1 2 Book IV of the Conies is described by its author as showing "in how many 
ways the sections of cones meet one another," and he is especially proud of 
theorems, "none of which has been discussed by earlier writers," concerning 
the number of points in which a section of a cone meets the "opposite 
branches of a hyperbola." The idea of the hyperbola as a double-branched 
curve was new with Apollonius, and he thoroughly enjoyed the discovery 
and proof of theorems concerning it. For example, he showed (IV. 42) that 
if one branch of a hyperbola meets both branches of another hyperbola, 
the opposite branch of the first hyperbola will not meet either branch of the 
second hyperbola in two points ; or again (IV. 54), if a hyperbola is tangent to 
one of the branches of a second hyperbola with its concavity in the opposite 
direction, the opposite branch of the first will not meet the opposite branch 
of the second. It is in connection with the theorems in this book that Apol- 
lonius makes a statement implying that in his day, as in ours, there were 
narrow-minded opponents of pure mathematics who pejoratively inquired 
about the usefulness of such results. The author proudly asserted : "They are 
worthy of acceptance for the sake of the demonstrations themselves, in the 
same way as we accept many other things in mathematics for this and for no 
other reason." 7 

1 3 The preface to Book V, relating to maximum and minimum straight lines 
drawn to a conic, again argues that "the subject is one of those which seem 
worthy of study for their own sake." While one must admire the author for his 
lofty intellectual attitude, it may be pertinently pointed out that what in his 
day was beautiful theory, with no prospect of applicability to the science 

7 See Heath, Apollonius ofPerga. Treatise on Conic Sections, p. lxxiv. 


or engineering of his time, has since become fundamental in such fields as 
terrestrial dynamics and celestial mechanics. Apollonius' theorems on 
maxima and minima are in reality theorems on tangents and normals to 
conic sections. Without a knowledge of the properties of tangents to a 
parabola, an analysis of local trajectories would be impossible ; and a study 
of the paths of the planets is unthinkable without reference to the tangents 
to an ellipse. It is clear, in other words, that it was the pure mathematics of 
Apollonius that made possible, some 1800 years later, the Principia of 
Newton ; the latter, in turn, has given scientists of today the hope that some 
day a round-trip visit to the moon will be possible. Even in ancient Greece 
the Apollonian theorem that every oblique cone has two families of circular 
sections was applicable to cartography in the stereographic transformation, 
used by Ptolemy and possibly by Hipparchus, of a spherical region into a 
portion of a plane. It has often been true in the development of mathematics 
that topics that originally could be justified only as "worthy of study for their 
own sake" later became of inestimable value to the "practical man." 

Greek mathematicians had no satisfactory definition of tangent to a curve 
C at a point P, thinking of it as a line L such that no other line could be drawn 
through P between C and L. Perhaps it was dissatisfaction with this definition 
that led Apollonius to avoid defining a normal to a curve C from a point Q as 
a line through Q which cuts the curve C in a point P and is perpendicular 
to the tangent to C at P. Instead he made use of the fact that the normal from 
Q to C is a line such that the distance from Q to C is a relative maximum or 
minimum. In Conies V. 8, for example, Apollonius proved a theorem concern- 
ing the normal to a parabola which today generally is part of a course in the 
calculus. In modern terminology the theorem states that the subnormal 
of the parabola v 2 = 2px for any point P on the curve is constant and equal to 
p; in the language of Apollonius this property is expressed somewhat as 
follows : 

If A is the vertex of a parabola y 2 = px, and if G is a point on the axis such that 
AG > p, and, if AT is a point between A and G such that NG = p, and if NP is 
drawn perpendicular to the axis meeting the parabola in P (Fig. 9.6), then 
PG is the minimum straight line from G to the curve and hence is normal to the 
parabola at P). 

FIG. 9.6 


The proof by Apollonius is of the typical indirect kind — it is shown that if 
F is any other point on the parabola, FG increases as F moves further from 
P in either direction. A proof of the corresponding, but more involved, 
theorem concerning the normal to an ellipse or hyperbola from a point on 
the axis is then given ; and it is shown that if P is a point on a conic, only one 
normal can be drawn through P, whether the normal be regarded as a 
minimum or a maximum, and this normal is perpendicular to the tangent at 
P. Note that the perpendicularity that we take as a definition is here proved 
as a theorem, whereas the maximum-minimum property that we take as a 
theorem serves, for Apollonius, as a definition. Later propositions in Book V 
carry the topic of normals to a conic to such a point that the author gives 
criteria enabling one to tell how many normals can be drawn from a given 
point to a conic section. These criteria are tantamount to what we should 
describe as the equations of the evolutes to the conies. For the parabola 
y 2 = 2px Apollonius showed in essence that points whose coordinates satisfy 
the cubic equation 27 py 2 = 8(x - p) 3 are limiting positions of the point of 
intersection of normals to the parabola at points P and P' as F approaches 
P. That is, points on this cubic are the centers of curvature for points on 
the conic (that is, the centers of osculating circles for the parabola). In 
the case of the ellipse and the hyperbola, whose equations are respec- 
tively x 2 /a 2 ± y 2 /b 2 = 1, the corresponding equations of the evolute are 

(ax) % + (M % = (a 2 + b 2 f\ 

After giving the conditions for the evolute of a conic, Apollonius showed 
how to construct a normal to a conic section from a point Q. In the case of the 
parabola y 2 = 2px, and for Q outside the parabola and not on the axis, one 
drops a perpendicular QM to the axis AK, measures off MH = p, and erects 
HR perpendicular to HA (Fig. 9.7). Then through Q one draws the rectangular 

FIG. 9.7 

hyperbola with asymptotes HA and HR, intersecting the parabola in a point 
P. Line QP is the normal required, as one can prove by showing that 
NK = HM - p. If point Q lies inside the parabola, the construction is 


similar except that P lies between Q and R. Apollonius also gave construc- 
tions, likewise making use of an auxiliary hyperbola, for the normal from a 
point to a given ellipse or hyperbola. It should be noted that the construction 
of normals to the ellipse and hyperbola, unlike the construction of tangents, 
requires more than straightedge and compasses. As the ancients described 
the two problems, the drawing of a tangent to a conic is a "plane problem," 
for intersecting circles and straight lines suffice ; by contrast, the drawing of a 
normal from an arbitrary point in the plane to a given central conic is a 
"solid problem," for it cannot be accomplished by use of lines and circles 
alone, but can be done through the use of solid loci (in our case, a hyperbola). 
Pappus later severely criticized Apollonius for his construction of a normal to 
the parabola in that he treated it as a solid problem rather than a plane prob- 
lem. That is, the hyperbola that Apollonius used could have been replaced 
by a circle. Perhaps Apollonius felt that the line-and-circle fetish should give 
way, in his construction of normals, to a desire for uniformity of approach 
with respect to the three types of conic. 

When Apollonius sent King Attalus the sixth book of the Conies, he 1 4 
described it as embracing propositions about "segments of conies equal and 
unequal, similar and dissimilar, besides some other matters left out by those 
who have preceded me. In particular, you will find in this book how, in a 
given right cone, a section is to be cut equal to a given section." Two conies 
are said to be similar if the ordinates, when drawn to the axis at proportional 
distances from the vertex, are respectively proportional to the corresponding 
abscissas. Among the easier of the propositions in Book VI are those demon- 
strating that all parabolas are similar (VI. 11) and that a parabola cannot be 
similar to an ellipse or hyperbola nor an ellipse to a hyperbola (VI. 14, 15). 
Other propositions (VI. 26, 27) prove that if any cone is cut by two parallel 
planes making hyperbolic or elliptic sections, the sections will be similar but 
not equal. 

Book VII returns to the subject of conjugate diameters and "many new 
propositions concerning diameters of sections and the figures described 
upon them." Among these are some that are found in modern textbooks, such 
as the proof (VII. 12, 13, 29, 30) that 

In every ellipse the sum, and in every hyperbola the difference, of the squares 
on any two conjugate diameters is equal to the sum or difference respectively 
of the squares on the axes. 

There is also the proof of the familiar theorem that if tangents are drawn at 
the extremities of a pair of conjugate axes of an ellipse or hyperbola, the 
parallelogram formed by these four tangents will be equal to the rectangle 
on the axes. It has been conjectured that the lost Book VIII of the Conies 


continued with similar problems, for in the preface to Book VII the author 
wrote that the theorems of Book VII were used in Book VIII to solve deter- 
minate conic problems, so that the last book "is by way of an appendix." 

1 5 The Conies of Apollonius is a treatise of such extraordinary breadth and 
depth that we are startled to note the omission of some of the properties that 
to us appear so obviously fundamental. As the curves are now introduced in 
textbooks, the foci play a prominent role ; yet Apollonius had no name for 
these points, and he referred to them only indirectly. It is presumed that he, 
and perhaps also Aristaeus and Euclid, was indeed familiar with the focus- 
directrix property of the curves, but this is not even mentioned in the Conies. 
There is no numerical concept in the ancient treatment of conies correspond- 
ing to what we call the eccentricity, and although the focus of the parabola by 
implication appears in many an Apollonian theorem, it is not clear that the 
author was aware of the now familiar role of the directrix. He seems to have 
known how to determine a conic through five points, but this topic, which 
later loomed large in the Principia of Newton, is omitted in the Conies of 
Apollonius. It is quite possible, of course, that some or all of such tantalizing 
omissions resulted from the fact they had been treated elsewhere, in works 
no longer extant, by Apollonius or other authors. So much of ancient 
mathematics has been lost that an argument e silencio is precarious indeed. 
Moreover, the words of Leibniz should serve as a warning that one should 
not underestimate ancient accomplishments : "He who understands Archi- 
medes and Apollonius will admire less the achievements of the foremost men 
of later times." 

1 6 The methods of Apollonius in the Conies in many respects are so similar 
to the modern approach that his work sometimes is judged to be an analytic 
geometry anticipating that of Descartes by 1800 years. The application of 
reference lines in general, and of a diameter and a tangent at its extremity in 
particular, is of course not essentially different from the use of a coordinate 
frame, whether rectangular or, more generally, oblique. Distances measured 
along the diameter from the point of tangency are the abscissas, and segments 
parallel to the tangent and intercepted between the axis and the curve are the 
ordinates. The Apollonian relationships between these abscissas and the 
corresponding ordinates are nothing more nor less than rhetorical forms of 
the equations of the curves. However, Greek geometrical algebra did not 
provide for negative magnitudes; moreover, the coordinate system was in 
every case superimposed a posteriori upon a given curve in order to study its 
properties. There appear to be no cases in ancient geometry in which a 
coordinate frame of reference was laid down a priori for purposes of graphical 
representation of an equation or relationship, whether symbolically or 


rhetorically expressed. Of Greek geometry we may say that equations are 
determined by curves, but not that curves were defined by equations. Co- 
ordinates, variables, and equations were subsidiary notions derived from a 
specific geometrical situation ; and one gathers that in the Greek view it was 
not sufficient to define curves abstractly as loci satisfying given conditions 
on two coordinates. To guarantee that a locus was really a curve, the ancients 
felt it incumbent upon them to exhibit it stereometrically as a section of a solid 
or to describe a kinematic mode of construction. 

The Greek definition and study of curves compare quite unfavorably with 
the flexibility and extent of the modern treatment. Indeed, the ancients over- 
looked almost entirely the part that curves of various sorts played in the world 
about them. Aesthetically one of the most gifted people of all times, the only 
curves that they found in the heavens and on the earth were combinations 
of circles and straight lines. They did not even effectively exploit the two 
means of definition for curves that they recognized. The kinematic approach 
and the use of plane sections of surfaces are capable of far-reaching generaliza- 
tion, yet scarcely a dozen curves were familiar to the ancients. Even the 
cycloid, generated by a point on a circle that rolls along a straight line, seems 
to have escaped their notice. That Apollonius, the greatest geometer of 
antiquity, failed to develop analytic geometry, was probably the result of a 
poverty of curves rather than of thought. General methods are not necessary 
when problems concern always one of a limited number of particular cases. 
Moreover, the early modern inventors of analytic geometry had all Renais- 
sance algebra at their disposal, whereas Apollonius necessarily worked with 
the more rigorous but far more awkward tool of geometrical algebra. 


Apollonius of Perga, Les coniques, trans, by Paul Ver Eecke (Bruges : Desclee de 
Brouwer, 1924). 

Coolidge, J. L., History of the Conic Sections and Quadric Surfaces (Oxford • Clarendon 

Coolidge, J. L., History of Geometrical Methods (Oxford : Clarendon, 1940 ; paperback 

ed., New York : Dover, 1963). 
Coxeter, H. S. M, "The Problem of Apollonius," American Mathematical Monthly 

75 (1968), 5-15. 
Dingeldey, F., "Coniques," in Encyclopedic des sciences mathematiques, 3 (3), 1-256. 
Fladt, K., Geschichte und Theorie der Kegelschnitte und der Flachen zweiten Grades 

(Stuttgart, 1965). 
Heath, T. L., "Apollonius," in Encyclopaedia Britannica, 11th ed. (Cambridge 1910) 

II, 186-188. 
Heath, T. L., ed., Apollonius of Perga. Treatise on Conic Sections (Cambridge : Cambridge 

University Press, 1896; reprinted. New York: Barnes and Noble, 1961). 


Neugebauer, O., "Apollonius-Studien," Quellen und Studien zur Geschichte der Math- 
ematik, Part B, Studien, II (1932), 215-253. 

Neugebauer, O., "Eccentric and Epicyclic Motion According to Apollonius," Scripta 
Mathematics 24 (1959), 5-21. 

Taylor, Charles, An Introduction to the Ancient and Modern Geometry of Conies (Cam- 
bridge, 1881). 

Thomas, Ivor, Selections Illustrating the History of Greek Mathematics (Cambridge, 
Mass. : Loeb Classical Library, 1939-1941, 2 vols.). 

Van der Waerden, B. L., Science Awakening, trans, by Arnold Dresden (New York : 
Oxford, 1961 ; paperback ed., New York : Wiley, 1963). 

Zeuthen, H. G., Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886 and 


1. The names of Aristotle, Euclid, Archimedes, and Apollonius are associated respectively 
with those of four powerful rulers— Alexander, Ptolemy, Hiero, and Attalus. Tell where these 
men ruled and in what connection their names are associated with those of the scholars. 

2. Describe several respects in which the mathematics of Apollonius differs from that of 
Euclid and several respects in which their works are similar. 

3. In what respects does the work of Apollonius resemble that of Archimedes and in what ways 
do their works differ? 

4. Would you say that Apollonius used analytic geometry? Justify your answer, showing in 
what respects his methods resemble the modern subject and in what ways they differ. 

5. Write the number 12,345,678,987,654,321 as Apollonius would have written it. 

6. Prove the theorem of Apollonius that the locus of points the difference of the squares of 
whose distances from two fixed points is constant is a straight line perpendicular to the 
line joining the two fixed points. 

7. Prove the theorem concerning the "circle of Apollonius" ; that is, show that the locus of 
points whose distances from two fixed points are unequal, but are in a fixed ratio, is a circle. 

8. Given the points P t (3, 0), P 2 (0, 4), and P 3 (l, 2), find the equation of a line through P ? which 
intersects the x-axis in a point P 4 and the y-axis in a point P 5 such that (a) P^ is twice 
P 2 P 5 and (b) P^ x P 2 P 5 is 10. 

9. Solve the "Problem of Apollonius" for (a) the case of two points and a line and (b) the case 
of two lines and a point. 

10. Beginning from the standard equations of the ellipse, the parabola, and the hyperbola with 
a vertex at the origin, complete the proof of the "name property" of Apollonius. 

1 1. If one diameter of the ellipse x 2 /a 2 + y 2 /b 2 = 1 has slope m, find the slope of the conjugate 

12. Find the slope of the system of parallel chords of y 2 = 2px bisected by the "diameter" 

y = a. 

13. Given a diameter of a hyperbola, show precisely how, with straightedge and compasses, 
you would construct the conjugate diameter. 

14. Find equations of the tangents from the point (- 1, 2) to the parabola y 2 = 2px and show 
how to construct the tangents with compasses and straightedge. 

15. Find the coordinates of the feet of the four normals that can be drawn from the point (1,0) 
to the ellipse x 2 /25 + y 2 /16 = 1. How many normals can be drawn from (2,0) to this 


16. For what values of K can four normals be drawn from the point {K, 0) to the ellipse 
x 2 /a 2 + y 2 /h 2 = 1? 

17. Prove that the length of the subnormal to a parabola at a point P on the parabola is constant 
(hence independent of the position of the point P on the curve). 

18. Apollonius knew that a tangent to an ellipse or hyperbola at a point P on the curve makes 
equal angles with the focal radii through P. Prove this theorem. 

19. Prove the Apollonian theorem that the segment of a tangent to a hyperbola intercepted 
between the asymptotes is bisected by the point of tangency. 

*20. Find an equation of the locus of points P such that the product of the perpendicular distances 

of P from the coordinate axes is equal to the product of the perpendicular distances of P 

from the lines y = x and y = 1 - x. 
*21. Find an equation of the polar of the point (a, b) with respect to the parabola y 2 = 2px. 
*22. Prove, in the manner Apollonius used for the cone, that an oblique section of a circular 

cylinder is an ellipse. 
*23. Prove that if AA' is the major axis of an ellipse, if the tangent to the ellipse at any point P 

intersects this axis (extended) in T, and if N is the projection of P on AA', then (A A', TN') 

form a conjugate set of points. (See Fig. 9.5.) 
*24. How many normals can be drawn from the point (1, 2) to the parabola y 2 = 2x? Justify your 



Greek Trigonometry and 

When I trace at my pleasure the windings to and fro of 
the heavenly bodies, I no longer touch the earth with 
my feet : I stand in the presence of Zeus himself and take 
my fill of ambrosia, food of the gods. 


Trigonometry, like other branches of mathematics, was not the work of any 
one man — or nation. Theorems on ratios of the sides of similar triangles 
had been known to, and used by, the ancient Egyptians and Babylonians. 
In view of the pre-Hellenic lack of the concept of angle measure, such a 
study might better be called "trilaterometry," or the measure of three-sided 
polygons (trilaterals), than "trigonometry," the measure of parts of a triangle. 
With the Greeks we first find a systematic study of relationships between 
angles (or arcs) in a circle and the lengths of chords subtending these. Proper- 
ties of chords, as measures of central and inscribed angles in circles, were 
familiar to the Greeks of Hippocrates' day, and it is likely that Eudoxus had 
used ratios and angle measures in determining the size of the earth and the 
relative distances of the sun and the moon. In the works of Euclid there is 
no trigonometry in the strict sense of the word, but there are theorems 
equivalent to specific trigonometric laws or formulas. Propositions II. 12 and 
13 of the Elements, for example, are the laws of cosines for obtuse and acute 
angles respectively, stated in geometric rather than trigonometric language 
and proved by a method similar to that used by Euclid in connection with the 
Pythagorean theorem. Theorems on the lengths of chords are essentially 
applications of the modern law of sines. We have seen that Archimedes' 
theorem on the broken chord can readily be translated into trigonometric 
language analogous to formulas for sines of sums and differences of angles. 
More and more the astronomers of the Alexandrian Age — notably Eratos- 
thenes of Cyrene (ca. 276-ca. 194 B.C.) and Aristarchus of Samos (ca. 310- 
ca. 230 B.C.) — handled problems pointing to a need for more systematic 
relationsips between angles and chords. 



Aristarchus, according to Archimedes and Plutarch, proposed a helio- 
centric system, anticipating Copernicus by more than a millennium and a 
half; 1 but whatever he may have written on this scheme has been lost. 
Instead we have an Aristarchan treatise, perhaps composed earlier (ca. 
260 B.C.), On the Sizes and Distances of the Sun and Moon, which assumes a 
geocentric universe. 2 In this work Aristarchus made the observation that 
when the moon is just half-full, the angle between the lines of sight to the 
sun and the moon is less than a right angle by one-thirtieth of a quadrant. 
(The systematic introduction of the 360° circle came a little later.) In trigono- 
metric language of today this would mean that the ratio of the distance of 
the moon to that of the sun (the ratio ME to SE in Fig. 10.1) is sin 3°. Trigono- 
metric tables not having been developed yet, Aristarchus fell back upon a 

FIG. 10.1 

well-known geometrical theorem of the time which now would be expressed 
in the inequalities sin a/sin fi < tx/fi < tan a/tan /?, where 0° < ft < a < 90°. 
From these he derived the conclusion that ^ < sin 3° < -£%, hence he asserted 
that the sun is more than eighteen, but less than twenty, times as far from the 
earth as is the moon. This is far from the modern value — somewhat less than 
400 — but it is better than the values nine and twelve that Archimedes ascribed 
respectively to Eudoxus and to Phidias (Archimedes' father). Moreover, the 
method used by Aristarchus was unimpeachable, the result being vitiated 
only by the error of observation in measuring the angle MES as 87° (when 
in actuality it should have been about 89° 50'). 

Having determined the relative distances of the sun and moon, Aristarchus 
knew also that the sizes of the sun and moon were in the same ratio. This 
follows from the fact that the sun and moon have very nearly the same 
apparent size — that is, they subtend about the same angle at the eye of an 
observer on the earth. In the treatise in question, this angle is given as 2°, but 
Archimedes attributed to Aristarchus the much better value of \°. From 
this ratio Aristarchus was able to find an approximation for the sizes of the 
sun and moon as compared with the size of the earth. From lunar eclipse 

1 The most complete account of Aristarchus and his place in astronomy is found in T. L. 
Heath, Aristarchus ofSamos (1913). 

2 It is possible that Aristarchus had been anticipated, in determining these distances, by 
Eudoxus. See Paul Tannery, Memoires scientifiques, I, 371. 


observations he concluded that the breadth of the shadow cast by the earth 
at the distance of the moon was twice the width of the moon. Then if R s , 
R e , and R m are the radii of the sun, earth, and moon respectively and if D s 
and D m are the distances of the sun and moon from the earth, then from the 
similarity of triangles BCD and ABE (Fig. 10.2), one has the proportion 
{R e - 2R m )/(R s - RJ = DJD S . If in this equation one replaces D s and R s 
by the approximate values \9D m and \9R m , one obtains the equation 

FIG. 10.2 

(R e - 2RJ/(19R m - R e ) = tV or R m = ¥f R e- Here the actual computations 
of Aristarchus have been considerably simplified. His reasoning was in 
reality much more carefully carried out and led to the conclusion that 

108 R e 60 J 19 R s 43 

< — - < 77T an d -r- < — < ~r 

43 R m 19 3 R e 6 

All that was needed to arrive at an estimate of the actual sizes of the sun 
and moon was a measure of the radius of the earth. Aristotle had mentioned 
a figure equivalent to about 40,000 miles for the circumference of the earth 
(a figure possibly due to Eudoxus), and Archimedes reported that some of his 
contemporaries estimated the perimeter to be about 30,000 miles. 3 A much 
better calculation, and by far the most celebrated, was one due to Eratos- 
thenes, a younger contemporary of Archimedes and Aristarchus. Eratosthenes 
was a native of Cyrene who had spent much of his early life at Athens. He had 
achieved prominence in many fields — poetry, astronomy, history, mathe- 
matics, athletics — when, in middle life, he was called by Ptolemy III 
(Philopator) to Alexandria to tutor his son (later Ptolemy Philadelphus) and 
to serve as librarian of the university there. It was to Eratosthenes at Alex- 
andria that Archimedes had sent the treatise on Method. Today Eratosthenes 
is best remembered for his measurement of the earth — not the first or last such 
estimate made in antiquity, but by all odds the most successful. Eratosthenes 

3 A. Diller, "The Ancient Measurements of the Earth," Isis, 40 (1949), 6-9. 


observed that at noon on the day of the summer solstice the sun shone 
directly down a deep well at Syene. At the same time at Alexandria, taken to 
be on the same meridian and 5000 stades north of Syene, the sun was found 
to cast a shadow indicating that the sun's angular distance from the zenith 
was one fiftieth of a circle. From the equality of the corresponding angles 
S'AZ and S'OZ in Fig. 10.3 it is clear that the circumference of the earth 
must be fifty times the distance between Syene and Alexandria. This results 
in a perimeter of 250,000 stades, or, since a stade was about a tenth of a mile, 
of 25,000 miles. (Later accounts placed the figure at 252,000 stades, possibly 
in order to lead to the round figure of 700 stades per degree.) 

FIG. 10.3 

A contributor to many fields of learning, Eratosthenes is well known in 
mathematics for the "sieve of Eratosthenes," a systematic procedure for 
isolating the prime numbers. With all the natural numbers arranged in 
order, one simply strikes out every second number following the number 
two, every third number (in the original sequence) following the number 
three, every fifth number following the number five, and continues in this 
manner to strike out every nth number following the number n. The remaining 
numbers, from two on, will of course be primes. Eratosthenes wrote also 
works on means and on loci, but these have been lost. Even his treatise 
On the Measurement of the Earth is no longer extant, although some details 
from it have been preserved by others, including Heron and Ptolemy of 

For some two and a half centuries, from Hippocrates to Eratosthenes, 
Greek mathematicians had studied relationships between lines and circles 
and had applied these in a variety of astronomical problems, but no syste- 
matic trigonometry had resulted. Then, presumably during the second half 
of the second century B.C., the first trigonometric table apparently was com- 
piled by the astronomer Hipparchus of Nicaea (ca. 180-ca. 125 B.C.), who 
thus earned the right to be known as "the father of trigonometry." Aristarchus 
had known that in a given circle the ratio of arc to chord decreased as the 


angle decreases from 180° to 0°, tending toward a limit of 1. However, it 
appears that not until Hipparchus undertook the task had anyone tabulated 
corresponding values of arc and chord for a whole series of angles. 4 It has, 
however, been suggested that Apollonius may have anticipated Hipparchus 
in this respect, and that the contribution of the latter to trigonometry was 
simply the calculation of a better set of chords than had been drawn up by 
his predecessors. Hipparchus evidently drew up his tables for use in his 
astronomy, about the origin of which little is known. 5 Hipparchus was a 
transitional figure between Babylonian astronomy and the work of Ptolemy. 
Astronomy was flourishing in Mesopotamia when in about 270 B.C. Berossos, 
about the only Babylonian astronomer known by name, moved to the island 
of Cos, and it is not unlikely that the foundations of Near Eastern theory 
were transmitted to Greece by that time. The chief contributions attributed 
to Hipparchus in astronomy were his organization of the empirical data 
derived from the Babylonians, the drawing up of a star catalogue, improve- 
ment in important astronomical constants (such as the length of the month 
and year, the size of the moon, and the angle of obliquity of the ecliptic), and, 
finally, the discovery of the precession of the equinoxes. It generally has been 
assumed that he was largely responsible for the building of geometrical 
planetary systems, but this is uncertain because it is not clear to what extent 
Apollonius may have applied trigonometric methods to astronomy somewhat 

It is not known just when the systematic use of the 360° circle came into 
mathematics, but it seems to be due largely to Hipparchus in connection 
with his table of chords. It is possible that he took over from Hypsicles, who 
earlier had divided the day into 360 parts, a subdivision that may have been 
suggested by Babylonian astronomy. Just how Hipparchus made up his 
table is not known, for his works are not extant (except for a commentary 
on a popular astronomical poem by Aratus). It is likely that his methods were 
similar to those of Ptolemy, to be described below, for Theon of Alexandria, 
commenting on Ptolemy's table of chords, reported that Hipparchus earlier 
had written a treatise in twelve books on chords in a circle. 

Theon mentions also another treatise, in six books, by Menelaus of 
Alexandria (ca. 100) dealing with Chords in a Circle. Other mathematical 
and astronomical works by Menelaus are mentioned by later Greek and 
Arabic commentators, including an Elements of Geometry, but the only one 
that has survived — and only through the Arabic — is his Sphaerica. In 
Book I of this treatise Menelaus established a basis for spherical triangles 

4 See Paul Tannery, Recherches sur Vhistoire de Vastronomie ancienne (Paris, 1893), pp. 66 ff. 

5 How little is known is made clear in O. Neugebauer, The Exact Sciences in Antiquity, 
2nd ed. (Providence, R.I. ; Brown University Press, 1957), especially pp. 167-168. 



analogous to that of Euclid I for plane triangles. Included is a theorem without 
Euclidean analogue — that two spherical triangles are congruent if corres- 
ponding angles are equal (Menelaus did not distinguish between congruent 
and symmetric spherical triangles); and the theorem A + B + C > 180° is 
established. The second book of the Spherica describes the application of 
spherical geometry to astronomical phenomena and is of little mathematical 
interest. Book III, the last, contains the well-known "theorem of Menelaus" 
as part of what is essentially spherical trigonometry in the typical Greek 
form — a geometry or trigonometry of chords in a circle. In the circle in 
Fig. 10.4 we should write that chord AB is twice the sine of half the central 

FIG. 10.4 

angle AOB (multiplied by the radius of the circle). Menelaus and his Greek 
successors instead referred to AB simply as the chord corresponding to the arc 
AB. If BOB' is a diameter of the circle, then chord AB' is twice the cosine of half 
the angle AOB (multiplied by the radius of the circle). Hence the theorems 
of Thales and Pythagoras, which lead to the equation AB 2 + AB' 2 = 4r 2 , 
are equivalent to the modern trigonometric identity sin 2 6 + cos 2 0=1. 
Menelaus, as also probably Hipparchus before him, was familiar with other 
identities, two of which he used as lemmas in proving his theorem on trans- 
versals. The first of these lemmas may be stated in modern terminology as 
follows. If a chord AB in a circle with center O (Fig. 10.5) is cut in point C 

FIG. 10.5 


by a radius OD, then AC/CB = sin /ID/sin DB. The second lemma is similar: 
if the chord AB extended is cut in point C by a radius OD' extended, then 
AC'/BC = sin AD'/sin BD'. These lemmas were assumed by Menelaus with- 
out proof, presumably because they could be found in earlier works, possibly 
in Hipparchus' twelve books on chords. (The reader can prove the lemmas 
easily by drawing AO and BO, dropping perpendiculars from A and B to 
OD, and using similar triangles. 6 

It is probable that the "theorem of Menelaus" for the case of plane triangles 
had been known to Euclid, perhaps having appeared in the lost Porisms. The 
theorem in the plane states that if the sides AB, BC, CA of a triangle are cut 
by a transversal in points D, E, F respectively (Fig. 10.6), then AD ■ BE ■ CF = 
BD ■ CE ■ AF. In other words, any line cuts the sides of a triangle so that 

FIG. 10.6 

the product of three nonadjacent segments equals the product of the other 
three, as can readily be proved by elementary geometry or through the 
application of simple trigonometric relationships. This theorem was assumed 
by Menelaus to be well known to his contemporaries, but he went on to 
extend it to spherical triangles in a form equivalent to sin AD sin BE sin CF = 
sin BD sin CE sin AF. If sensed segments are used rather than absolute 
magnitudes, the two products are equal in magnitude but differ in sign. 

The theorem of Menelaus played a fundamental role in spherical trigono- 
metry and astronomy, but by far the most influential and significant trigono- 
metric work of all antiquity was the Mathematical Syntaxis, a work in 
thirteen books composed by Ptolemy of Alexandria about half a century 
after Menelaus. This celebrated "Mathematical Synthesis" was distinguished 
from another group of astronomical treatises by other authors (including 
Aristarchus) by referring to that of Ptolemy as the "greater" collection and 
to that of Aristarchus et al. as the "lesser" collection. From the frequent 
reference to the former as megiste, there arose later in Arabia the custom of 
calling Ptolemy's book Almagest ("the greatest"), and it is by this name 
that the work has since been known. 

6 See T. L. Heath, History of Greek mathematics (1921), II, 265-267. 


Of the life of its author we are as little informed as we are of that of the 
author of the Elements. We do not know when or where Euclid and Ptolemy 
were born. We know that Ptolemy made observations at Alexandria from 
127 to 151 and therefore assume that he was born at the end of the first 
century. Suidas, a writer who lived in the tenth century, reported that Ptolemy 
was still alive under Marcus Aurelius (emperor from 161 to 180). 

Ptolemy's Almagest is presumed to be heavily indebted for its methods 
to the Chords in a Circle of Hipparchus, but the extent of the indebtedness 
cannot be reliably assessed. It is clear that in astronomy Ptolemy made use 
of the catalogue of star positions bequeathed by Hipparchus, but whether or 
not Ptolemy's trigonometric tables were derived in large part from his 
distinguished predecessor cannot be determined. Fortunately, Ptolemy's 
Almagest has survived the ravages of time; hence we have not only his 
trigonometric tables but also an account of the methods used in their 
construction. Central to the calculation of Ptolemy's chords was a geo- 
metrical proposition still known as "Ptolemy's theorem" : If ABCD is a 
(convex) quadrilateral inscribed in a circle (Fig. 10.7), then ABCD + 
BCDA = ACBD; that is, the sum of the products of the opposite sides 

FIG. 10.7 

of a cyclic quadrilateral is equal to the product of the diagonals. The proof 
of this is easily carried through by drawing BE so that angle ABE is equal 
to angle DBC and noting the similarity of the triangles ABE and BCD. 
A special case of Ptolemy's theorem had appeared in Euclid's Data (Proposi- 
tion 93) : If ABC is a triangle inscribed in a circle, and if BD is a chord bisecting 
angle ABC, then {AB + BQ/BD = AC/ AD. 

Another, and more useful, special case of the general theorem of Ptolemy 
is that in which one side, say AD, is a diameter of the circle (Fig. 10.8). Then 
if AD = 2r, we have 2r ■ BC + AB ■ CD = AC ■ BD. If we let arc BD = 2a 
and arc CD = 20, then BC = 2r sin(a - P), AB = 2r sin(90° - a), BD = 2r 
sin a, CD = 2r sin p, and AC = 2r sin(90° - P). Ptolemy's theorem therefore 
leads to the result sin(a - P) = sin a cos P - cos a sin p. Similar reasoning 
leads to the formula sin(a + P) = sin a cos P + cos a sin p, and to the 




FIG. 10.8 

analogous pair cos(a ± j8) = cos a cos P + sin a sin p. These four sum-and- 
difference formulas consequently are often known today as Ptolemy's 

It was the formula for sine of the difference — or, more accurately, chord of 
the difference— that Ptolemy found especially useful in building up his 
tables. Another formula that served him effectively was the equivalent of 
our half-angle formula. Given the chord of an arc in a circle, Ptolemy found 
the chord of half the arc as follows. Let D be the midpoint of arc BC in a 
circle with diameter AC = 2r (Fig. 10.9), let AB = AE, and let DF bisect EC 

FIG. 10.9 

(perpendicularly). Then it is not difficult to show that FC = ^2r - AB). 
But from elementary geometry it is known that DC 2 = AC ■ FC, from which 
it follows that DC 2 = r(2r - /4B).IfweletarcBC ='2a,thenDC = 2r sin a/2 
and AB = 2r cos a ; hence we have the familiar modern formula sin a/2 = 
J {I - cos a)/2. In other words, if the chord of any arc is known, the chord 
of half the arc is also known. Now Ptolemy was equipped to build up a table 
of chords as accurate as might be desired, for he had the equivalent of our 
fundamental formulas. 

It should be recalled that from the days of Hipparchus until modern times 
there were no such things as trigonometric ratios. The Greeks, and after 
them the Hindus and the Arabs, used trigonometric lines. These at first 


took the form, as we have seen, of chords in a circle, and it became incumbent 
upon Ptolemy to associate numerical values (or approximations) with the 
chords. To do this two conventions were needed : (1) some scheme for sub- 
dividing the circumference of a circle and (2) some rule for subdividing the 
diameter. The division of a circumference into 360 degrees seems to have 
been in use in Greece since the days of Hipparchus, although it is not known 
just how the convention arose. It is not unlikely that the 360-degree measure 
was carried over from astronomy, where the zodiac had been divided into 
twelve "signs" or 36 "decans." A cycle of the seasons of roughly 360 days 
could readily be made to correspond to the system of zodiacal signs and 
decans by subdividing each sign into thirty parts and each decan into ten 
parts. Our common system of angle measure may stem from this corres- 
pondence. Moreover, since the Babylonian positional system for fractions 
was so obviously superior to the Egyptian unit fractions and the Greek com- 
mon fractions, it was natural for Ptolemy to subdivide his degrees into sixty 
"partes minutae primae," each of these latter into sixty "partes minutae 
secundae," and so on. It is from the Latin phrases that translators used in 
this connection that our words "minute" and "second" have been derived. 
It undoubtedly was the sexagesimal system that led Ptolemy to subdivide 
the diameter of his trigonometric circle into 120 parts ; each of these he further 
subdivided into sixty minutes and each minute of length into sixty seconds. 
Our trigonometric identities are easily converted into the language of 
Ptolemaic chords through the simple relationships 

chord 2x _, chord (180° - 2x) 

sin x = — — - — and cos x = 

120 120 

The formulas cos (x + y) = cos x cos y + sin x sin y become (chord is abbre- 
viated to cd) 

cd 2x cd 2y + cd 2x cd 2 v 

cd 2x ±2y = 

where a line over an arc (angle) indicates the supplementary arc. Note that 
not only angles and arcs, but also their chords were expressed sexagesimally. 
In fact, whenever scholars in antiquity wished an accurate system of approxi- 
mation, they turned to the sixty-scale for the fractional portion ; this led to 
the phrases "astronomers' fractions" and "physicists' fractions" to distin- 
guish sexagesimal from common fractions. 

Having decided upon his system of measurement, Ptolemy was ready to 8 
compute the chords of angles within the system. For example, since the radius 
of the circle of reference contained sixty parts, the chord of an arc of sixty 


degrees also contained sixty linear parts. The chord of 120° will be 60^/3 
or approximately 103 parts and 55 minutes and 33 seconds, or, in Ptolemy's 
Ionic or alphabetic notation, py p ve' Xy". Ptolemy could now have used his 
half-angle formula to find the chord of 30°, then the chord of 15°, and so on 
for still smaller angles. However, he preferred to delay the application of this 
formula, and computed instead the chords of 36° and of 72°. He used a 
theorem from Elements XIII. 9 which shows that a side of a regular pentagon, 
a side of a regular hexagon, and a side of a regular decagon, all being inscribed 
within the same circle, constitute the sides of a right triangle. Incidentally, 
this theorem from Euclid provides the justification for Ptolemy's elegant 
construction of a regular pentagon inscribed in a circle. Let be the center 
of a circle and AB a diameter (Fig. 10.10). Then if C is the midpoint of OB 

FIG. 10.10 

and OD is perpendicular to AB, and if CE is taken equal to CD, the sides 
of the right triangle EDO are the sides of the regular inscribed pentagon, 
hexagon, and decagon. Then if the radius OB contains 60 parts, from the 
properties of the pentagon and the golden section it follows that OE, the 
chord of 36°, is 30^ - 1) or about 37.083 or 37 p 4' 55" or AC" 8' ve". By 
the Pythagorean theorem the chord of 72° is 30^/10 - 2^/5 or approximately 
70.536 or 70 p 32' 3" or o p AjS' y". 

Knowing the chord of an arc of s degrees in a circle, one can easily find 
the chord of the arc 180° - s from the theorems of Thales and Pythagoras, 
for cd 2 s + cd 2 s = 120 2 . Hence Ptolemy knew the chords of the supple- 
ments of 36° and 72°. Moreover, from the chords of 72° and 60° he found 
chord 12° by means of his formula for the chord of the difference of two arcs. 
Then by successive applications of his half-angle formula he derived the 
chords of arcs of 6°, 3°, 1^°, and |°, the last two being F 34' 15" and P 47' 8" 
respectively. Through a linear interpolation between these values Ptolemy 
arrived at l p 2' 50" as the chord of 1°. By using the half-angle formula— or, 
since the angle is very small, simply dividing by two — he found the value of 
P 31'25" for the chord of 30'. This is equivalent to saying that sin 15' is 
0.00873, which is correct to almost half a dozen decimal places. 


Ptolemy's value of the chord of \° is, of course, the length of a side of a 
polygon of 720 sides inscribed in a circle of radius 60 units. Whereas 
Archimedes' polygon of 96 sides had led to 22/7 as an approximation to the 
value of 71, Ptolemy's is equivalent to 6(0 P 31' 25") or 3 ;8,30. This approxima- 
tion to Ti, used by Ptolemy in the Almagest, is the same as fU, which leads to a 
decimal equivalent of about 3.1416, a value that may have been given earlier 
by Apollonius. 

Armed with formulas for the chords of sums and differences and chords 
of half an arc, and having a good value of chord j°, Ptolemy went on to build 
up his table, correct to the nearest second, of chords of arcs from j° to 180° 
for every j°. This is virtually the same as a table of sines from 3° to 90°, 
proceeding by steps of |°. The table formed an integral part of Book I of 
the Almagest and remained an indispensable tool of astronomers for more 
than a thousand years. The remaining twelve books of this celebrated treatise 
contain, among other things, the beautifully developed theory of cycles and 
epicycles for the planets known as the Ptolemaic system. Like Archimedes, 
Hipparchus, and most other great thinkers of antiquity, Ptolemy postulated 
an essentially geocentric universe, for a moving earth appeared to be faced 
with difficulties — such as lack of apparent stellar parallax and seeming 
inconsistency with the phenomena of terrestrial dynamics. In comparison 
with these problems, the implausibility of an immense speed required for the 
daily rotation of the sphere of the "fixed" stars seemed to shrink into 
insignificance. Besides appealing to common sense, the Ptolemaic system 
had the advantage of easy representation. Planetaria generally are con- 
structed as though the universe were geocentric, for in this way the apparent 
motions are most easily reproduced. 

Plato had set for Eudoxus the astronomical problems of "saving the 
phenomena" — that is, producing a mathematical device, such as a combina- 
tion of uniform circular motions, which should serve as a model for the 
apparent motions of the planets. The Eudoxian system of homocentric 
spheres had been largely abandoned by mathematicians in favor of the 
system of cycles and epicycles of Apollonius and Hipparchus. Ptolemy in 
turn made an essential modification in the latter scheme. In the first place, 
he displaced the earth somewhat from the center of the deferent circle, so 
that he had eccentric orbits. Such changes had been made before him, but 
Ptolemy introduced a novelty so drastic in scientific implication that 
Copernicus later could not accept it, effective though the device, known as 
the equant, was in reproducing the planetary motions. Try as he would, 
Ptolemy had not been able to arrange a system of cycles, epicycles, and 
eccentrics in close agreement with the observed motions of the planets. His 
solution was to abandon the Greek insistence on uniformity of circular 


motions and to introduce instead a geometrical point, the equant £ collinear 
with the earth G and the center C of the deferent circle, such that the apparent 
angular motion of the center Q of the epicycle in which a planet P revolves 
is uniform as seen from E (Fig. 10.11). In this way Ptolemy achieved ac- 
curate representations of planetary motions, but of course the device was 
kinematic only and made no effort to answer the questions in dynamics 
raised by nonuniform circular movements. 

FIG. 10.11 

1 Ptolemy's fame today is associated largely with a single book, the A Imagest, 
but there are other Ptolemaic works as well. Among the more important 
was a Geography, in eight books, which was as much a bible to geographers 
of his day as the Almagest was to astronomers. The Geography of Ptolemy 
introduced the system of latitudes and longitudes as used today, described 
methods of cartographic projection, and catalogued some 8000 cities, rivers, 
and other important features of the earth. Unfortunately, there was at the 
time no satisfactory means of determining longitudes, hence substantial 
errors were inevitable. Even more significant was the fact that Ptolemy 
seems to have made a poor choice when it came to estimating the size of the 
earth. Instead of accepting the figure 252,000 stadia, given by Eratosthenes, 
he preferred the value 180,000 stadia proposed by Posidonius, a Stoic 
teacher of Pompey and Cicero. Hence Ptolemy thought that the known 
Eurasian world was a larger fraction of the circumference than it really is — 
more than 180° in longitude, instead of an actual figure of about 130°. This 
large error suggested to later navigators, including Columbus, that a voyage 
westward from Europe to India would not be nearly so far as it turned out 
to be. Had Columbus known how badly Ptolemy had underestimated the 
size of the earth, he might never have set sail. 

Ptolemy's geographical methods were better in theory than in practice, 
for in separate monographs, which have survived only through Latin 
translations from the Arabic, Ptolemy described two types of map projection. 
Orthographic projection is explained in the Analemma, the earliest account 


we have of this method, although it may have been used by Hipparchus. In 
this transformation from a sphere to a plane, points on the spherical surface 
are projected orthogonally upon three mutually perpendicular planes. In the 
Planisphaerium Ptolemy described the stereographic projection in which 
points on the sphere are projected by lines from a pole onto a plane — in 
Ptolemy's case from the south pole to the plane of the equator. He knew 
that under such a transformation a circle not through the pole of projection 
went into a circle in the plane, and that a circle through the pole was projected 
into a straight line. Ptolemy was aware also of the important fact that such a 
transformation is conformal — that is, angles are preserved. The importance 
of Ptolemy for geography can be gauged from the fact that the earliest maps 
in the Middle Ages that have come down to us in manuscripts, none before 
the thirteenth century, had as prototypes the maps made by Ptolemy more 
than a thousand years before. 7 

Ptolemy wrote also an Optics which has survived, imperfectly, through a 1 1 
Latin version of an Arabic translation. This deals with the physics and 
psychology of vision, with the geometry of mirrors, and with an early attempt 
at a law of refraction. From Ptolemy's table of angles of refraction from air to 
water (and also from air to glass and from water to glass) for angles of inci- 
dence from 10° to 80° at intervals of 10° we see that he assumed a law of the 
form r = ai + bi 2 , for the second differences in his values of r are constant. 
For angles of incidence of 10° and 80° he assumed angles of refraction of 8° 
and 50° respectively, and the second differences are all equal to j°. The 
second differences in the old Pythagorean formulas for polygonal numbers 
also were constant, and perhaps Ptolemy was influenced by these to seek a 
quadratic rather than a trigonometric law for refraction. Trigonometry for 
the first millennium and a half of its existence was almost exclusively an 
adjunct of astronomy and geography, and only in the seventeenth century 
were trigonometric applications in refraction and other parts of physics 

No account of Ptolemy's work would be complete without mention of his 
Tetrabiblos (or Quadripartitum), for it shows us a side of ancient scholarship 
that we are prone to overlook. Greek authors were not always the rational 
and clear-thinking men they are presumed to have been. The Almagest is 
indeed a model of good mathematics and accurate observational data put 
to work in building a sober scientific astronomy ; but the Tetrabiblos (or work 
in four books) represents a kind of sidereal religion to which much of the 
ancient world had succumbed. With the end of the Golden Age, Greek 
mathematics and philosophy became allies of Chaldean arithmetic and 
astrology, and the resulting pseudoreligion filled the gap left by repudiation 

7 See George Sarton, Ancient Science and Modern Civilization (1954), pp. 53-54. 


of the old mythology. Ptolemy seems to have shared the prejudices of his 
time; in the Tetrabiblos he argued that one should not, because of the 
possibility of error, discourage the astrologer any more than the physician. 
The further one reads in the work, the more dismayed one becomes, for the 
author showed no hesitation in accepting the superstitions of his day. 

The Tetrabiblos differs from the Almagest not only as astrology differs from 
astronomy ; the two works also make use of different types of mathematics. 
The latter is a sound and sophisticated work that makes good use of synthetic 
Greek geometry ; the former is typical of the pseudoscience of the day in 
the adoption of primitive Babylonian arithmetic devices. From the classical 
works of Euclid, Archimedes, and Apollonius one might obtain the im- 
pression that Greek mathematics was exclusively occupied with the highest 
levels of logical geometrical reasoning; but Ptolemy's Tetrabiblos suggests 
that the populace in general were more concerned with arithmetical computa- 
tion than with rational thought. At least from the days of Alexander the Great 
to the close of the classical world, there undoubtedly was much inter- 
communication between Greece and Mesopotamia, and it seems to be clear 
that the Babylonian arithmetic and algebraic geometry continued to exert 
considerable influence in the Hellenistic world. This aspect of mathematics, 
for example, appears so strongly in Heron of Alexandria (fl. ca. 100) that 
Heron once was thought to be Egyptian or Phoenician rather than Greek. 
Now it is thought that Heron portrays a type of mathematics that had long 
been present in Greece but does not find a representative among the greatest 
figures — except perhaps as betrayed by Ptolemy in the Tetrabiblos. Greek 
deductive geometry, on the other hand, seems not to have been welcomed in 
Mesopotamia until after the Arabic conquest. 

1 2 Heron of Alexandria is best known in the history of mathematics for the 
formula, bearing his name, for the area of a triangle : 

K = Js{s - a){s- b)(s - c) 

where a, b, c are the sides and s is half the sum of these sides, that is, the 
semiperimeter. The Arabs tell us that "Heron's formula" was known earlier 
to Archimedes, who undoubtedly had a proof of it, but the demonstration of 
it in Heron's Metrica is the earliest that we have. Although now the formula 
usually is derived trigonometrically, Heron's proof is conventionally 
geometric. The Metrica, like the Method of Archimedes, was long lost, until 
rediscovered at Constantinople in 1896 in a manuscript dating from about 
1100. The word "geometry" originally meant "earth measure," but classical 
geometry, such as that found in Euclid's Elements and Apollonius' Conies, 
was far removed from mundane surveying. Heron's work, on the other hand, 


shows us that not all mathematics in Greece was of the "classical" type. 
There evidently were two levels in the study of configurations — comparable 
to the distinction made in numerical context between arithmetic (or theory 
of numbers) and logistic (or techniques of computation) — one of which, 
eminently rational, might be known as geometry and the other, crassly 
practical, might better be described as geodesy. The Babylonians lacked the 
former, but were strong in the latter, and it was essentially the Babylonian 
type of mathematics that is found in Heron. It is true that in the Metrica an 
occasional demonstration is included, but the body of the work is concerned 
with numerical examples in mensuration of lengths, areas, and volumes. 
There are strong resemblances between his results and those found in ancient 
Mesopotamian problem texts. For example, Heron gave a tabulation 8 of the 
areas A„ of regular polygons of n sides in terms of the square of one side s„, 
beginning with A 3 = ^s 3 2 and continuing to A l2 = ^s 12 2 . As was the case 
in pre-Hellenic mathematics, Heron also made no distinction between results 
that are exact and those that are only approximations. For A s , for example, 
Heron gave two formulas— |s 5 2 and ^ s 5 2 — the first of which agrees with a 
value found in a Babylonian table, 9 but neither of which is precisely correct. 
For the hexagon Heron's ratio of A 6 to s 6 2 is ^, the Babylonian is 2; 37,30, 
whereas the true value lies between these and is of course irrational. In such 
calculations we should have expected Heron to use trigonometric tables such 
as Hipparchus had drawn up a couple of hundred years before, but apparently 
trigonometry was at the time largely the handmaid of the astronomer rather 
than of the practical man. 

The gap that separated classical geometry from Heronian mensuration is 
clearly illustrated by certain of the problems set and solved by Heron in 
another of his works, the Geometrica. One problem calls for the diameter, 
perimeter, and area of a circle, given the sum of these three magnitudes. The 
axiom of Eudoxus would rule out such a problem from theoretical con- 
sideration, for the three magnitudes are of unlike dimensions, but from an 
uncritical numerical point of view the problem makes sense. Moreover, Heron 
did not solve the problem in general terms but, taking a cue again from pre- 
Hellenic methods, chose the specific case in which the sum is 212 ; his solution 
is like the ancient recipes in which steps only, without reasons, are given. The 
diameter 14 is easily found by taking the Archimedean value for n and using 
the Babylonian method of completing the square to solve a quadratic equa- 
tion. Heron simply gives the laconic instructions, "Multiply 212 by 154, add 
841, take the square root and subtract 29, and divide by 11." This is scarcely 
the way to teach mathematics, but Heron's books were intended as manuals 
for the practitioner. 

8 See D. E. Smith, History of Mathematics (Boston: Ginn, 1923-1925, 2 vols.), II, 606. 

9 See Neugebauer : Exact Sciences in Antiquity, p. 47. 



Heron paid as little attention to the uniqueness of his answer as he did to 
the dimensionality of his magnitudes. In one problem he called for the sides 
of a right triangle if the sum of the area and perimeter is 280. This is, of 
course, an indeterminate problem, but Heron gave only one solution, making 
use of the Archimedean formula for area of a triangle. In modern notation, 
if s is the semiperimeter of the triangle and r the radius of the inscribed circle, 
then rs + 2s = s(r + 2) = 280. Following his own cookbook rule, "Always 
look for the factors," he chose r + 2 = 8 and s = 35. Then the area rs is 210. 
But the triangle is a right triangle, hence the hypotenuse c is equal to s - r 
or 35 - 6 or 29; the sum of the two sides a and b is equal tos + r or 41. 
The values of a and b are then easily found to be 20 and 21. Heron says 
nothing about other factorizations of 280, which of course would lead to 
other answers. 

Heron was interested in mensuration in all its forms — in optics and mech- 
anics, as well as in geodesy. The law of reflection for light had been known 
to Euclid and Aristotle (probably also to Plato) ; but it was Heron who showed 
by a simple geometrical argument, in a work on Catoptrics (or reflection), 
that the equality of the angles of incidence and reflection is a consequence 
of the Aristotelian principle that nature does nothing the hard way. That is, 
if light is to travel from a source S to a mirror MM' and then to the eye E Of 
an observer (Fig. 10.12), the shortest possible path SPE is that in which the 

FIG. 10.12 

angles SPM and EPM' are equal. That no other path SP'E can be as short as 
SPE is apparent on drawing SQS' perpendicular to MM\ with SQ = QS', 
and comparing the path SPE with the path SPE. Since paths SPE and SPE 
are equal in length to paths SPE and SPE respectively, and inasmuch as 
SPE is a straight line (because angle M'PE is equal to angle MPS), it follows 
that SPE is the shortest path. 


Heron is remembered in the history of science as the inventor of a 
primitive type of steam engine, described in his Pneumatics, of a forerunner 
of the thermometer, and of various toys and mechanical contrivances based 
on the properties of fluids and on the laws of the simple machines. He sug- 
gested in the Mechanics a law (clever but incorrect) of the simple machine 
whose principle had eluded even Archimedes — the inclined plane. His 
name is attached also to "Heron's algorithm" for finding square roots, but 
this method of iteration was in reality due to the Babylonians of 2000 years 
before his day. Although Heron evidently learned much of Mesopotamian 
mathematics, he seems not to have appreciated the importance of the posi- 
tional principle for fractions. Sexagesimal fractions had become the standard 
tool of scholars in astronomy and physics, but it is likely that they remained 
unfamiliar to the common man. Common fractions were used to some extent 
by the Greeks, at first with numerator placed below the denominator, later 
with the positions reversed (and without the bar separating the two), but 
Heron, writing for the practical man, seems to have preferred unit fractions. 
In dividing 25 by 13 he wrote the answer asl+^ + ^ + ^ + JL. The old 
Egyptian addiction to unit fractions continued in Europe for at least a 
thousand years after the time of Heron. 

The period from Hipparchus to Ptolemy, covering three centuries, was 14 
one in which applied mathematics was in the ascendant, and Heron's books 
resemble notes taken by a student at the equivalent of an institute of tech- 
nology at Alexandria. It sometimes is held 10 that mathematics develops 
most effectively when in close touch with the world's work ; but the period 
we have been considering would argue for the opposite thesis. The loss of 
nerve in religion and philosophy, which led the Greeks to pursue cults and 
mysticism, was paralleled in mathematics by a movement toward applica- 
tions which persisted for more than three centuries. From Hipparchus to 
Ptolemy there were advances in astronomy and geography, optics and 
mechanics, but no significant developments in mathematics. It is true that 
these centuries saw the development of trigonometry, but this subject, now 
an integral part of pure mathematics, was then at best a mensurational 
application of elementary geometry which met the needs of astronomy. 
Moreover, it is not even clear whether or not there was any significant 
advance in the trigonometry of Ptolemy in a.d. 150 over that of Hipparchus, 
in 1 50 B.C.— or even, perhaps, over that of Apollonius and Archimedes a 
century earlier still. It is evident that the rapid growth of mathematics from 
Eudoxus to Apollonius, when theoretical considerations were in the fore- 
front, had come to an end. Perhaps the trend toward applications was the 

10 Especially by Lancelot Hogben in his many works on mathematics and its history, such 
as Mathematics for the Million (New York : W. W. Norton, ca. 1937). 


result of the decline, rather than its cause, but in any case the two were con- 
comitant. Some 11 attribute the decline to the inadequacies and difficulties 
in Greek geometrical algebra, others 12 to the cold breath of Rome. In any 
case, the period during which trigonometry and mensuration came to the 
fore was characterized by lack of progress — if not actual decline ; yet it was 
precisely these aspects of Greek mathematics that most attracted the Hindu 
and Arabic scholars who served as a bridge to the modern world. Before we 
turn to these peoples, however, we must look at the Indian summer of Greek 
mathematics, sometimes known as the "Silver Age." 


Aaboe, Asger, Episodes from the Early History of Mathematics (New York: Random 

House, 1964). 
Braunmuhl, Anton von, Vorlesungen iiber Geschichte der Trigonometrie (Leipzig, 

1900-1903, 2 vols.). 
Cohen, M. R., and I. E. Drabkin, Source Book in Greek Science (New York : McGraw- 
Hill, 1948 ; reprinted Cambridge, Mass. : Harvard University Press, 1958). 
Dantzig, Tobias, The Bequest of the Greeks (New York : Scribner, 1955). 
Heath, T. L., Aristarchus ofSamos (Oxford : Clarendon, 1913). 
Heath, T. L., A History of Greek Mathematics (Oxford : Clarendon, 1921, 2 vols.). 
Lammert, Friedrich, "Klaudios Ptolemaios," in Pauly-Wissowa, Real-Enzyclopadie 

der klassischen Altertumswissenschaft (Stuttgart, 1959), Vol. XXIII, Part 2, columns 

Manitius, Karl, Des Ptolemaus Handbuch der Astronomie (Leipzig, 1912-1913), 2 vols.). 
Peters, C. H. F., and E. B. Knobel, Ptolemy s Catalogue of Stars ; a Revision of the 

Almagest (Washington, D.C. : Carnegie Institution, 1915). 
Ptolemy, Claudius, L'optique, ed. by Albert Lejeune (Louvain, Belgium: Louvain 

University, 1956). 
Ptolemy, Claudius, Cosmographia, ed. by R. A. Skelton (Amsterdam: Meridian, 1963). 
Sarton, George, Ancient Science and Modern Civilization (Lincoln, Nebr. : University 

of Nebraska Press, 1954). 
Stahl, W. H, Ptolemy's Geography ; a Select Bibliography (New York : Bulletin of the 

New York Public Library, 1951-1952). 
Tannery, Paul, Memoires scientifiques (Toulouse, 1912, etc.), especially Vols. I and II. 
Thomas, Ivor, Selections Illustrating the History of Greek Mathematics (Cambridge, 

Mass. : Loeb Classical Library, 1939-1941, 2 vols.). 
Thomson, J. O, History of Ancient Geography (Cambridge, 1948). 
Van der Waerden, B. L., Science Awakening, trans, by Arnold Dresden (New York : 

Oxford, 1961 ; paperback ed., New York : Wiley, 1963). 

11 For example, B. L. van der Waerden in Science Awakening (1961), pp. 265-266. 

12 E. T. Bell in Development of Mathematics (New York : McGraw-Hill, 1940). 



1. How can one account for the fact that the period of the rise of Greek trigonometry was a 
time of decline in Greek geometry? 

2. Why did the ancients prefer a geocentric astronomical system to a heliocentric scheme? 
Explain clearly. 

3. How far would Columbus have had to sail from Gibraltar to India, assuming the latter to be 
accessible from the east by water, if Ptolemy's ideas on the size of the earth had been 

4. What happens to circles on a sphere if projected orthogonally on a plane? 

5. Using the information given in the text, find Ptolemy's law of refraction for rays going from 
air to water. 

6. Prove, either geometrically or trigonometrically, Heron's formula for the area of a triangle. 

7. Posidonius is said to have used observations of the stars to estimate the size of the earth. 
Show how this can be done. 

8. Which of Heron's formulas for the ratio of A 5 to s 5 2 is the better approximation? 

9. Heron gave the ratio of the area of a regular heptagon to the square of a side as -?|, and the 
Babylonians expressed this as 3; 41. Which is the better approximation? 

10. Find to the nearest tenth of a per cent the error in Heron's value *f for the ratio A 12 :s 12 2 . 

11. Complete the steps in Heron's solution of the problem of finding the diameter of a circle if 
the sum of the diameter and the perimeter and the area is 212. 

12. Prove Aristarchus' inequality ^ < sin 3° < Jg. 

13. Hipparchus knew from eclipse observations that the lunar parallax (that is, the angle sub- 
tended by the earth at a point on the moon) is about 2°. What lunar distance does this 

14. Write in Greek notation the chord of 45°. 

15. Find, without tables, sin 15° and from this write down in Greek alphabetic notation 
Ptolemy's value for chord 30°. 

16. Write in Greek notation the chord of 150°. 

17. If the Archimedean and Ptolemaic values of n are expressed as improper common frac- 
tions, and if a new fraction is formed by the difference of the two numerators over the 
difference of the two denominators, a better approximation, known to the Chinese, is found. 
How accurate is this new approximation? 

18. Prove the theorem of Aristarchus that if p < a < 90°, then sin a/sin P < a/p. 

19. Prove the two lemmas of Menelaus. 

20. Prove, either geometrically or trigonometrically, the theorem of Menelaus for plane triangles. 

21. Complete the proof of Ptolemy's theorem. 

22. Using the theorem of Ptolemy (with a diameter of the circle as one side of the quadrilateral), 
derive the formulas for sin(x + y) and cos(x ± y). 

23. Using Ptolemy's method for half angles, derive a formula for cos x/2. 

*24. Find exactly, in terms of radicals, the ratio of the area of a regular decagon to the square on a 
side. Is your value greater or less than the value ^ given by Heron? 


Revival and Decline of Greek 

Bees ... by virtue of a certain geometrical forethought 
. . . know that the hexagon is greater than the square 
and the triangle and will hold more honey for the same 
expenditure of material. 

Pappus of Alexandria 

Today we use the conventional phrase "Greek mathematics" as though it 
indicated a homogeneous and well-defined body of doctrine. Such a view can 
be very misleading, however, for it implies that the sophisticated geometry 
of the Archimedean-Apollonian type was the only sort that the Hellenes 
knew. We must remember that mathematics in the Greek world spanned a 
time interval from at least 600 b.c. to at least a.d. 600 and that it traveled from 
Ionia to the toe of Italy, to Athens, to Alexandria, and to other parts of the 
civilized world. The intervals in time and space alone produced changes in 
the depth and extent of mathematical activity, for Greek science did not have 
the sameness, century after century, that is found in pre-Hellenic thought. 
Moreover, even at any given time and place in the Greek world (as in our 
civilization today) there were sharp differences in the level of mathematical 
interest and accomplishment. We have seen how even in the work of a single 
individual, such as Ptolemy, there can be two types of scholarship — the 
Almagest for the "tough-minded" rationalists and the Tetrabiblos for the 
"tender-minded" mystics. It is probable that there always were at least two 
levels of mathematical understanding, but that the paucity of surviving works, 
especially on the lower level, tends to obscure this fact. The phrase used as the 
title for this chapter must itself be accepted with some hesitation, for although 
it is justified in the light of what we know about the Greek world, our know- 
ledge is far from complete. The period that we consider in this chapter, from 
Ptolemy to Proclus, covers almost four centuries (from the second to the 
sixth), but our account is based in large part on only two chief treatises, 
only portions of which are now extant, as well as on a number of works of 
lesser significance. 



Heron and Ptolemy were Greek scholars, but they lived in a world dom- 
inated politically by Rome. The death of Archimedes by the hand of a 
Roman soldier may have been inadvertent, but it was truly portentous. 
Throughout its long history, ancient Rome contributed little to science or 
philosophy and less to mathematics. Whether during the Republic or in the 
days of the Empire, Romans were little attracted to speculative or logical 
investigation. The practical arts of medicine and agriculture were cultivated 
with some eagerness, and descriptive geography met with favor. Impressive 
engineering projects and architectural monuments were related to the 
simpler aspects of science, but Roman builders were satisfied with elementary 
rule-of-thumb procedures that called for little in the way of understanding 
of the great corpus of Greek thought. The extent of Roman acquaintance with 
science may be judged from the De architectura of Vitruvius, written during 
the middle part of the Augustine Age and dedicated to the emperor. At one 
point the author describes what to him appeared to be the three greatest 
mathematical discoveries : the incommensurability of the side and diagonal 
of a cube ; the right triangle with sides 3, 4, and 5 ; and Archimedes' calculation 
on the composition of the king's crown. Marcus Vitruvius Pollio, the author, 
was especially interested in surveying instruments and in problems involving 
approximate mensurations. The perimeter of a wheel of diameter 4 feet is 
given by Vitruvius as 12| feet, implying a value of 3^ for n. This is not so good 
an approximation as that of Archimedes, with whose works Vitruvius was 
probably only slightly acquainted, but it is of a respectable degree of accuracy 
for Roman purposes. It is sometimes claimed that impressive works of 
engineering, such as the Egyptian pyramids and the Roman aqueducts, 
imply a high level of mathematical achievement, but historical evidence does 
not bear this out. Just as earlier Egyptian mathematics had been on a lower 
plane than that in Babylon of the same period, so Roman mathematics was 
on a much lower level than that in Greece during the same years. The Romans 
were almost completely lacking in mathematical drive, so that their best 
efforts, such as those of Vitruvius, were not comparable to the poorer results 
in Greece, as exemplified by the work of Heron. 1 

We have seen that Greek mathematics was not uniformly on a high level, 
for the glorious period of the third century B.C. had been followed by a decline, 
perhaps to some extent arrested in the days of Ptolemy, but not effectively 
reversed until the century of the "Silver Age," about a.d. 250 to 350. At the 
beginning of this period, also known as the Later Alexandrian Age, we find 
the leading Greek algebraist, Diophantus of Alexandria, and toward its close 
there appeared the last significant Greek geometer, Pappus of Alexandria. 

1 A devastating comparison of Roman science with that of Greece is presented by W. H. 
Stahl, Roman Science (1962). 


No other city has been the center of mathematical activity for so long a 
period as was Alexandria from the days of Euclid (ca. 300 B.C.) to the time of 
Hypatia (f 415). It was a very cosmopolitan center, and the mathematics that 
resulted from Alexandrian scholarship was not all of the same type. The results 
of Heron were markedly different from those of Euclid or Apollonius or 
Archimedes, and again there is an abrupt departure from the classical Greek 
tradition in the extant work of Diophantus. Little is known of Diophantus' 
life beyond a tradition that is reported in a collection of problems dating 
from the fifth or sixth century, known as the "Greek Anthology" (described 
below) : 

God granted him to be a boy for the sixth part of his life, and adding a twelfth part 
to this, He clothed his cheeks with down ; He lit him the light of wedlock after a seventh 
part, and five years after his marriage He granted him a son. Alas! late-born wretched 
child ; after attaining the measure of half his father's life, chill Fate took him. After 
consoling his grief by this science of numbers for four years he ended his life. 2 

If this conundrum is historically accurate, Diophantus lived to be eighty- 
four-years old. It should definitely not be taken as typical of the problems 
that interested Diophantus, for he paid little attention to equations of first 

Diophantus is often called the father of algebra, but we shall see that such a 
designation is not to be taken literally. His work is not at all the type of 
material forming the basis of modern elementary algebra ; nor is it yet similar 
to the geometric algebra found in Euclid. The chief Diophantine work known 
to us is the Arithmetica, a treatise originally in thirteen books, only the first 
six of which have survived. 3 It should be recalled that in ancient Greece the 
word arithmetic meant theory of numbers, rather than computation. Often 
Greek arithmetic had more in common with philosophy than with what we 
think of as mathematics ; hence the subject had played a large role in Neo- 
platonism during the Later Alexandrian Age. This had been particularly true 
of the Introductio arithmeticae of Nicomachus of Gerasa, a Neo-Pythagorean 
who lived not far from Jerusalem about the year 100. The author sometimes 
is held to be of Syrian background, but Greek philosophical tendencies 
certainly predominate in his work. The Introductio of Nicomachus, as we 
have it, contains only two books, and it is possible that this is only an abridged 
version of what originally was a more extensive treatise. At all events, the 
possible loss in this case is far less to be regretted than the loss of seven books 
of the Arithmetica of Diophantus, for there is a world of difference between 

2 Quoted from Cohen and Drabkin, Source Book in Greek Science (1958), p. 27. Uncertainty 
about the life of Diophantus is so great that we do not know definitely in which century he lived. 
Generally he is assumed to have flourished about 250, but dates a century or more earlier or 
later are sometimes suggested. 

3 For a full account see T. L. Heath, Diophantus of Alexandria (1910). 


Four antique mathematicians who contributed also to music: Boethius, Pythagoras, Plato, 
Nicomachus; from a Boethius manuscript, Cambridge, 


the two authors. Nicomachus had, so far as we can see, little mathematical 
competence and was concerned only with the most elementary properties 
of numbers. The level of the work may be judged from the fact that the author 
found it expedient to include a multiplication table up to i times t (that is, 
10 times 10). If this is genuine and not just a later interpolation, it is the oldest 
surviving Greek instance of such a table, although many older Babylonian 
multiplication tables are extant. 

The Introductio of Nicomachus opens with the anticipated Pythagorean 
classification of numbers into even and odd, then into evenly even (powers 
of two) and evenly odd (2" • p, where p is odd and p > 1 and n > 1) and oddly 
even (2 ■ p, where p is odd and p > 1). Prime, composite, and perfect numbers 
are defined, including a description of the sieve of Eratosthenes and a list 
of the first four perfect numbers (6 and 28 and 496 and 8128). The work 
includes also a classification of ratios and combinations of ratios (for ratios 
of integers are essential in the Pythagorean theory of musical intervals), an 
extensive treatment of figurate numbers (which had loomed so large in 
Pythagorean arithmetic) in both two and three dimensions, and a comprehen- 
sive account of the various means (again a favorite topic in Pythagorean 
philosophy). As some other writers, Nicomachus regarded the number three 
as the first number in the strict sense of the word, for one and two were really 
only the generators of the number system. For Nicomachus, numbers were 
endowed with such qualities as better or worse, younger or older ; and they 
could transmit characters, as parents to their progeny. Despite such arith- 
metical anthropomorphism as a background, the Introductio contains a mod- 
erately sophisticated theorem. Nicomachus noticed that if the odd integers 
are grouped in the pattern 1 ; 3 + 5 ; 7 + 9 + 11 ; 13 + 15 + 17 + 19 ; . . . , 
the successive sums are the cubes of the integers. This observation, coupled 
with the early Pythagorean recognition that the sum of the first n odd num- 
bers is n 2 , leads to the conclusion that the sum of the first n perfect cubes is 
equal to the square of the sum of the first n integers. 

The Introductio of Nicomachus 4 was neither a treatise on calculation nor 
one on algebra, but a handbook on those elements of mathematics that were 
essential to an understanding of Pythagorean and Platonic philosophy ; as 
such it served as a model for later imitators and commentators. Among these 
the best known were Theon of Smyrna (fl. ca. 125), who wrote his Expositio 
in Greek, and Boethius (T524), who wrote his Arithmetica, long afterward, 
in Latin. These men, like Nicomachus, were far more concerned about the 
application of arithmetic to music and Platonic philosophy than in advancing 

* For an English translation see Nicomachus of Gerasa, Introduction to Arithmetic, trans, by 
M. L. D'Ooge (1926). This very useful edition includes also an extensive introduction that places 
the work of Nicomachus in clear historical perspective. D'Ooge concluded from the evidence 
that Nicomachus was Greek rather than Syrian. 


the subject itself. The full title of the Expositio indicates, in fact, that it is an 
exposition of mathematical matters useful to an understanding of Plato. 5 
It explains, for example, that the tetractys consisting of the numbers 1, 2, 3, 
and 4 contains all the musical consonances inasmuch as it makes up the 
ratios 4:3, 3:2, 2:1, 3: 1, and 4: 1. The Arithmetica of Boethius is quite 
unoriginal, being almost a translation of the earlier work by Nicomachus. 6 

Quite different from the works of Nicomachus, Theon, and Boethius 
was the Arithmetica of Diophantus, a treatise characterized by a high degree 
of mathematical skill and ingenuity. In this respect the book can be compared 
with the great classics of the earlier Alexandrian Age ; yet it has practically 
nothing in common with these or, in fact, with any traditional Greek math- 
ematics. It represents essentially a new branch and makes use of a different 
approach. Being divorced from geometrical methods, it resembles Baby- 
lonian algebra to a large extent; but whereas Babylonian mathematicians 
had been concerned primarily with the approximate solution of determinate 
equations as far as the third degree, the Arithmetica of Diophantus (such as 
we have it) is almost entirely devoted to the exact solution of equations, both 
determinate and indeterminate. Because of the emphasis given in the Arith- 
metica to the solution of indeterminate problems, the subject dealing with 
this topic, sometimes known as indeterminate analysis, has since become 
known as Diophantine analysis. Since this type of work today is generally a 
part of courses in theory of numbers, rather than elementary algebra, it is 
not an appropriate basis for regarding Diophantus as the father of algebra. 
There is another respect, however, in which such a paternity is justified. 
Algebra now is based almost exclusively on symbolic forms of statement, 
rather than on the customary written language of ordinary communication 
in which earlier Greek mathematics, as well as Greek literature, had been 
expressed. It is generally held that three stages in the historical development 
of algebra can be recognized : (1) the rhetorical or early stage, in which every- 
thing is written out fully in words ; (2) a syncopated or intermediate stage, 
in which some abbreviations are adopted; and (3) a symbolic or final 
stage. Such an arbitrary division of the development of algebra into three 
stages is of course a facile oversimplification ; but it can serve effectively as a 
first approximation to what has happened, and within such a framework the 
Arithmetica of Diophantus is to be placed in the second category. 

Throughout the six surviving books of the Arithmetica there is a systematic 
use of abbreviations for powers of numbers and for relationships and opera- 
tions. An unknown number is represented by a symbol resembling the Greek 

5 There is an excerpt, in English translation, in Cohen and Drabkin, Source Book in Greek 
Science, pp. 294-298. 

" Marshall Clagett, Greek Science in Antiquity, pp. 185-186. 


letter .s (perhaps for the last letter of arithmos); the square of this appears as 
A 7 , the cube as K y , the fourth power, called square-square, as A y A, the fifth 
power or square-cube as AK r , and the sixth power or cube-cube as K y K. 
Diophantus was of course familiar with the rules of combination equivalent 
to our laws of exponents, and he had special names for the reciprocals of the 
first six powers of the unknowns, quantities equivalent to our negative 
powers. Numerical coefficients were written after the symbols for the powers 
with which they were associated ; addition of terms was understood in the 
appropriate juxtaposition of the symbols for the terms, and subtraction was 
represented by a single letter-abbreviation placed before the terms to be 
subtracted. With such a notation Diophantus was in a position to write 
polynomials in a single unknown almost as concisely as we do today. The 
expression 2x 4 + 3x 3 - 4x 2 + 5x - 6, for example, might appear in a form 
equivalent to SS2 C3 x5 M S4 u6, where the English letters S, C, x, M, and w 
have been used for "square," "cube," the "unknown," "minus," and "unit," 
and with our present numerals in place of the Greek alphabetic notation 
that was used in the days of Diophantus. Greek algebra now no longer was 
restricted to the first three powers or dimensions, and the identities 
(a 2 + b 2 )(c 2 + d 2 ) = (ac + bd) 2 + (ad - be) 2 = (ac - bd) 2 + (ad + be) 2 , 
which played important roles in Medieval algebra and modern trigonometry, 
appear in the work of Diophantus. The chief difference between the Dio- 
phantine syncopation and the modern algebraic notation is in the lack of 
special symbols for operations and relations, as well as of the exponential 
notation. These missing elements of notation were largely contributions of 
the period from the late fifteenth to the early seventeenth centuries in 

If we think primarily of matters of notation, Diophantus has a good claim 
to be known as the father of algebra, but in terms of motivation and concepts 
the claim is less appropriate. The Arithmetica is not a systematic exposition of 
the algebraic operations or of algebraic functions or of the solution of alge- 
braic equations. It is instead a collection of some 150 problems, all worked out 
in terms of specific numerical examples, although perhaps generality of 
method was intended. There is no postulational development, nor is an 
effort made to find all possible solutions. In the case of quadratic equations 
with two positive roots, only the larger is given, and negative roots are not 
recognized. No clear-cut distinction is made between determinate and 
indeterminate problems, and even for the latter, for which the number of 
solutions generally is unlimited, only a single answer is given. Diophantus 
solved problems involving several unknown numbers by skillfully expressing 
all unknown quantities, where possible, in terms of only one of them. Two 
problems from the Arithmetica will serve to illustrate the Diophantine 


approach. In finding two numbers such that their sum is 20 and the sum of 
their squares is 208, the numbers are not designated as x and y, but as 10 + x 
and 10 - x (in terms of our modern notation). Then (10 + x) 2 + (10 - x) 2 = 
208, hence x = 2; so the numbers sought are 8 and 12. Diophantus handled 
also the analogous problem in which the sum of the two numbers and the sum 
of the cubes of the numbers are given as 10 and 370 respectively. 

In these problems he is dealing with a determinate equation, but Diophan- 
tus used much the same approach in indeterminate analysis. In one problem 
it is required to find two numbers such that either when added to the square 
of the other will yield a perfect square. This is a typical instance of Diophan- 
tine analysis in which only rational numbers are acceptable as answers. In 
solving the problem Diophantus did not call the numbers x and y, but rather 
x and 2x + 1. Here the second, when added to the square of the first, will 
yield a perfect square no matter what value one chooses for x. Now, it is 
required also that (2x + l) 2 + x must be a perfect square. Here Diophantus 
does not point out the infinity of possible answers. He is satisfied to choose a 
particular case of a perfect square, in this instance the number (2x - 2) 2 , such 
that when equated to (2x + l) 2 + x an equation that is linear in x results. 
Here the result is x = ^, so that the other number, 2x + 1, is {f . One could, 
of course, have used (2x - 3) 2 or (2x - 4) 2 , or expressions of similar form, 
instead of (2x - 2) 2 , to arrive at other pairs of numbers having the desired 
property. Here we see an approach that comes close to a "method" in 
Diophantus' work : when two conditions are to be satisfied by two numbers, 
the two numbers are so chosen that one of the two conditions is satisfied ; 
and then one turns to the problem of satisfying the second condition. That 
is, instead of handling simultaneous equations on two unknowns, Diophantus 
operates with successive conditions so that only a single unknown number 
appears in the work. 

Among the indeterminate problems in the Arithmetica are some involving 
equations such as x 2 = 1 + 30y 2 and x 2 = 1 + 26y 2 , which are instances 
of the so-called "Pell equation" x 2 = 1 + py 2 ; again a single answer is 
thought to suffice. 7 In a sense it is not fair to criticize Diophantus for being 
satisfied with a single answer, for he was solving problems, not equations. 
In a sense the Arithmetica is not an algebra textbook, but a problem collection 
in the application of algebra. In this respect Diophantus is like the Babylonian 
algebraists ; and his work sometimes is regarded as "the finest flowering of 

7 See D. J. Struik, A Concise History of Mathematics, 3rd ed. (New York: Dover, 1967), p. 62. 
For a full account of the work of Diophantus see T. L. Heath, Diophantus of Alexandria. Cf. also 
J. A. Sanchez Perez: La arithmetica en Grecia (1947) and the article on Diophantus by F. O. 
Hultsch in Pauly-Wissowa, Real-Enzyclopadie der klassischen Altertumswissenschaft, Vol. V 
(Stuttgart: Metzler, 1905), columns 1051-1073. 


Babylonian algebra." 8 To some extent such a characterization is unfair to 
Diophantus, for his numbers are entirely abstract and do not refer to meas- 
ures of grain or dimensions of fields or monetary units, as was the case in 
Egyptian and Mesopotamian algebra. Moreover, he is interested only in 
exact rational solutions, whereas the Babylonians were computationally 
inclined and were willing to accept approximations to irrational solutions 
of equations. Hence cubic equations seldom enter in the work of Diophantus, 
whereas among the Babylonians attention had been given to the reduction of 
cubics to the standard form n 3 + n 2 = a in order to solve approximately 
through interpolation in a table of values of n 3 + n 2 . 

We do not know how many of the problems in the Arithmetica were original 
or whether Diophantus had borrowed from other similar collections. 
Possibly some of the problems or methods are traceable back to Babylonian 
sources, for puzzles and exercises have a way of reappearing generation after 
generation. To us today the Arithmetica of Diophantus looks strikingly 
original, but possibly this impression results from the loss of rival problem 
collections. Our view of Greek mathematics is derived from a relatively 
small number of surviving works, and conclusions derived from these 
necessarily are precarious. Indications that Diophantus may have been less 
isolated a figure than has been supposed are found in a collection of problems 
from about the early second century of our era (hence presumably antedating 
the Arithmetica) in which some Diophantine symbols appear. 9 Nevertheless, 
Diophantus has had a greater influence on modern number theory than any 
other nongeometric Greek algebraist. In particular, Fermat was led to his 
celebrated "great" or "last" theorem (see below) when he sought to generalize 
a problem that he had read in the Arithmetica of Diophantus (II. 8) : to divide 
a given square into two squares. 10 

The Arithmetica of Diophantus is a brilliant work worthy of the period of 
revival in which it was written, but it is, in motivation and content, far re- 
moved from the beautifully logical treatises of the great geometrical trium- 
virate of the earlier Alexandrian Age. Algebra seemed to be more appropriate 
for problem-solving than for deductive exposition, and the great work of 
Diophantus remained outside the mainstream of Greek mathematics. A 
minor work on polygonal numbers by Diophantus comes closer to the earlier 
Greek interests, but even this cannot be regarded as approaching the Greek 
logical ideal. Classical geometry had found no ardent supporter, with the 

8 See J. D. Swift, "Diophantus of Alexandria," American Mathematical Monthly, 43 (1956), 

9 See F. E. Robbins, "P. Mich. 620: A Series of Arithmetical Problems," Classical Philology, 
24 (1929), 321-329, and Kurt Vogel, "Die algebraischen Probleme des P. Mich. 620," Classical 
Philology, 25 (1930), 373-375. 

10 See Heath, Diophantus of Alexandria, pp. 144-145. 


possible exception of Menelaus, since the death of Apollonius some four 
hundred and more years before. But during the reign of Diocletian (284-305) 
there lived again at Alexandria a scholar who was moved by the spirit that 
had possessed Euclid, Archimedes, and Apollonius. Pappus of Alexandria 
in about 320 composed a work with the title Collection (Synagoge) which is 
important for several reasons. In the first place it provides a most valuable 
historical record of parts of Greek mathematics that otherwise would be 
unknown to us. For instance, it is in Book V of the Collection that we learn 
of Archimedes' discovery of the thirteen semiregular polyhedra or "Archi- 
median solids." Then, too, the Collection includes alternative proofs and 
supplementary lemmas for propositions in Euclid, Archimedes, Apollonius, 
and Ptolemy. Finally, the treatise includes new discoveries and generaliza- 
tions not found in any earlier work. The Collection, Pappus' most important 
treatise, contained eight books, but the first book and the first part of the 
second book are now lost. In this case the loss is less to be regretted than is that 
of the last books of Diophantus' Arithmetica, for it appears that the first 
two books of the Collection were chiefly concerned with the principles of 
Apollonius' system of tetrads in Greek numeration. Since we have, in the 
Sand-Reckoner, the corresponding system of octads from Archimedes, we 
can judge quite well what material has been lost from the exposition of 

Book III of the Collection shows that Pappus shared thoroughly the 
classical Greek appreciation of the niceties of logical precision in geometry. 
Here he distinguishes sharply between "plane," "solid," and "linear" 
problems — the first being constructible with circles and straight lines only, 
the second being solvable through the use of conic sections, and the last 
requiring curves other than lines, circles, and conies. Then Pappus describes 
some solutions of the three famous problems of antiquity, the duplication 
and trisection being problems in the second or solid category and the squaring 
of the circle being a linear problem. Pappus virtually here asserts the fact that 
the classical problems are impossible of solution under the Platonic condi- 
tions, for they do not belong among the plane problems ; but rigorous proofs 
were not given until the nineteenth century. 

In Book IV Pappus again is insistent that one should give for a problem a 
construction appropriate to it. That is, one should not use linear loci in the 
solution of a solid problem, nor solid or linear loci in the solution of a plane 
problem. Asserting that the trisection of an angle is a solid problem, he 
therefore suggests methods that make use of conic sections, whereas Archi- 
medes in one case had used a neusis or sliding-ruler type of construction 
and in another the spiral, which is a linear locus. One of the Pappus trisec- 
tions is as follows. Let the given angle AOB be placed in a circle with center 


FIG. 11.1 

O (Fig. 11.1) and let OC be the angle bisector. Draw the hyperbola having A 
as one focus, OC as the corresponding directrix, and with an eccentricity 
equal to 2. Then one branch of this hyperbola will cut the circumference of 
the circle in a point Tsuch that /_AOT is one-third L.AOB. 

A second trisection construction proposed by Pappus makes use of an 
equilateral hyperbola as follows. Let the side OB of the given angle AOB be a 
diagonal of a rectangle ABCO and through A draw the equilateral hyperbola 
having BC and OC (extended) as asymptotes (Fig. 11.2). With A as center 

FIG. 11.2 

and with radius twice OB draw a circle intersecting the hyperbola in P and 
from P drop the perpendicular FT to the line CB extended. Then it is readily 
proved, from the properties of the hyperbola, that the straight line through O 
and Tis parallel to AP and that /__AOT is one-third /_AOB. Pappus gives no 
source for his trisections, and we cannot help but wonder if this trisection was 
known to Archimedes. If we draw the semicircle passing through B, having 



QT as diameter and M as center, we have essentially the Archimedean neusis 
construction, for OB = QM = MT = MB. 

In Book III Pappus describes also the theory of means and gives an attrac- 
tive construction that includes the arithmetic, the geometric, and the har- 
monic mean within a single semicircle. Pappus shows that if in the semicircle 
ADC with center O (Fig. 11.3) one has DB ± AC and BF _L OD, then DO is 
the arithmetic mean, DB the geometric mean, and DF the harmonic mean of 
the magnitudes AB and BC. Here Pappus claims for himself only the proof, 
attributing the diagram to an unnamed geometer. Even when Pappus names 
his source, it sometimes is not otherwise known to us, indicating how in- 
adequate is our information on mathematicians of his day. 

The Collection of Pappus is replete with bits of interesting information and 
significant new results. In many cases the novelties take the form of generaliza- 
tions of earlier theorems, and a couple of these instances appear in Book IV. 
Here we find an elementary generalization of the Pythagorean theorem. If 
ABC is any triangle (Fig. 1 1 .4) and if ABDE and CBGF are any parallelograms 
constructed on two of the sides, then Pappus constructs on side AC a third 
parallelogram ACKL equal to the sum of the other two. This is easily 
accomplished by extending sides FG and ED to meet in H, then drawing HB 
and extending it to meet side AC in J, and finally drawing AL and CK parallel 



to HBJ. It is not known whether or not this generalization, usually bearing 
the name of Pappus was original with Pappus, and it has been suggested that 
possibly it was known earlier to Heron. 

Another instance of generalization in Book IV, also bearing Pappus' name, 
extends theorems of Archimedes on the shoemaker's knife. It asserts that 
if circles C t , C 2 , C 3 , C 4 , . . . , C„, . . . are inscribed successively as in Fig. 
11.5, all being tangent to the semicircles on AB and on AC, and successively 
to each other, the perpendicular distance from the center of the nth circle to 
the base line ABC is n times the diameter of the nth circle. 1 1 

FIG. 11.5 

1 Book V of the Collection was a favorite with later commentators, for it 
raised the question of the sagacity of bees. Inasmuch as Pappus showed that 
of two regular polygons having equal perimeters the one with the greater 
number of sides has the greater area, he concluded that bees demonstrated 
some degree of mathematical understanding in constructing their cells as 
hexagonal, rather than square or triangular, prisms. The book goes into 
other problems of isoperimetry, including a demonstration that the circle 
has a greater area, for a given perimeter, than does any regular polygon. Here 
Pappus seems to have been following closely a work On Isometric Figures 
written almost half a millennium earlier by Zenodorus (ca. 180 B.C.), some 
fragments of which were preserved by later commentators. Among the 
propositions in Zenodorus' treatise was one asserting that of all solid figures 
the surfaces of which are equal the sphere has the greatest volume, but only 
an incomplete justification was given. 12 

Books VI and VIII of the Collection are chiefly on applications of math- 
ematics to astronomy, optics, and mechanics (including an unsuccessful 

11 An indicated proof of the theorem will be found in R. A. Johnson, Modern Geometry 
(New York: Houghton Mifflin, 1929), p. 117. 

12 See Heath: History of Greek Mathematics (1921), II, 207 ff. A fascinating account of such 
matters is found in D'Arcy Wentworth Thompson : On Growth and Form, 2nd ed. (Cambridge 
University Press, 1942). 


attempt at finding the law of the inclined plane). Of far more significance in 
the history of mathematics is Book VII, in which, through his penchant for 
generalization, Pappus came close to the fundamental principle of analytic 
geometry. The only means recognized by the ancients for defining plane 
curves were (1) kinematic definitions in which a point moves subject to two 
• superimposed motions and (2) the section by a plane of a geometrical surface, 
such as a cone or sphere or cylinder. Among the latter curves were certain 
quartics known as spiric sections, described by Perseus (ca. 150 B.C.), obtained 
by cutting the anchor ring or torus by a plane. Occasionally a twisted curve 
caught the attention of the Greeks, including the cylindrical helix and an 
analogue of the Archimedean spiral described on a spherical surface, both 
of which were known to Pappus ; but Greek geometry was primarily restricted 
to the study of plane curves, in fact, to a very limited number of plane curves. 
It is significant to note, therefore, that in Book VII of the Collection Pappus 
proposed a generalized problem that implied infinitely many new types of 
curves. This problem, even in its simplest form, usually is known as the 
"Pappus problem," but the original statement, involving three or four lines, 
seems to go back to the days of Euclid. As first considered, the problem 
is referred to as "the locus to three or four lines," described above in connec- 
tion with the work of Apollonius. Euclid evidently had identified the locus 
for certain special cases only, but it appears that Apollonius, in a work now 
lost, had given a complete solution. Pappus nevertheless gave the impression 
that geometers had failed in attempts at a general solution and implied that 
it was he who had first shown the locus in all cases to be a conic section. 

More importantly, Pappus then went on to consider the analogous 
problem for more than four lines. For six lines in a plane he recognized that a 
curve is determined by the condition that the product of the distances from 
three of the lines shall be in a fixed ratio to the product of the distances to 
the other three lines. In this case a curve is defined by the fact that a solid is in 
a fixed ratio to another solid. Pappus hesitated to go on to cases involving 
more than six lines inasmuch as "there is not anything contained by more 
than three dimensions." But, he continued, "men a little before our time have 
allowed themselves to interpret such things, signifying nothing at all com- 
prehensible, speaking of the product of the content of such and such lines by 
the square of this or the content of those. These things might however be 
stated and shown generally by means of compounded proportions." The 
unnamed predecessors evidently were prepared to take a highly important 
step in the direction of an analytic geometry that should include curves of 
degree higher than three, just as Diophantus had used the expressions 
square-square and cube-cube for higher powers of numbers. Had Pappus 
pursued the suggestion further, he might have anticipated Descartes in a 
general classification and theory of curves far beyond the classical distinction 


between plane, solid, and linear loci. His recognition that, no matter what 
the number of lines in the Pappus problem, a specific curve is determined, 
is the most general observation on loci in all of ancient geometry, and the 
algebraic syncopations that Diophantus had developed would have been 
adequate to have disclosed some of the properties of the curves. But Pappus 
was at heart a geometer only, as Diophantus had been an algebraist only ; 
hence Pappus merely remarked with surprise that no one had made a syn- 
thesis of this problem for any case beyond that of four lines. Pappus himself 
made no deeper study of these loci, "of which one has no further knowledge 
and which are simply called curves." 13 What was needed for the next step 
in this connection was the appearance of a mathematician equally concerned 
for algebra and geometry ; it is significant to note that when such a figure 
appeared in the person of Descartes, it was this very problem of Pappus that 
served as the point of departure in the invention of analytic geometry. 

1 1 There are other important topics in Book VII of the Collection, apart from 
the Pappus problem. For one thing, there is a full description of what was 
called the method of analysis and of a collection of works known as the 
Treasury of Analysis. Pappus describes analysis as "a method of taking that 
which is sought as though it were admitted and passing from it through its 
consequences in order to something which is admitted as a result of syn- 
thesis." That is, he recognized analysis as a "reverse solution," the steps of 
which must be retraced in opposite order to constitute a valid demonstration. 
If analysis leads to something admitted to be impossible, the problem also 
will be impossible, for a false conclusion implies a false premise. Pappus 
explains that the method of analysis and synthesis is used by the authors 
whose works constitute the Treasury of Analysis : "This is a body of doctrine 
furnished for the use of those who, after going through the usual elements, 
wish to obtain power to solve problems set to them involving curves"; 
and Pappus lists among the works in the Treasury of Analysis the treatises 
on conies by Aristaeus, Euclid, and Apollonius. It is from Pappus' description 
that we learn that Apollonius' Conies contained 487 theorems. Since the 
seven books now extant comprise 382 propositions, we can conclude that 
the lost eighth book had 105 propositions. About half of the works listed by 
Pappus in the Treasury of Analysis are now lost, including Apollonius' 
Cutting-off of a Ratio, Eratosthenes' On Means, and Euclid's Porisms. It has 
been suggested that a porism was an antique equivalent of our equation of a 
curve or locus, indicating that Euclid and Pappus may not have been as far 
removed from what we call "analytic geometry" as generally is supposed. 

13 There is no English translation of the Collection of Pappus, but extensive accounts of it 
will be found in Heath, History of Greek Mathematics, and in I. Thomas, Selections Illustrating 
the History of Greek Mathematics. There is a convenient French translation of the Collection, 
made by Paul Ver Eecke, (Paris: Desclee de Brouwer, 1933, 2 vols.). 


Book VII of the Collection contains the first statement on record of the 1 2 
focus-directrix property of the three conic sections. It appears that Apollonius 
knew of the focal properties for central conies, but it is possible that the focus- 
directrix property for the parabola was not known before Pappus. Another 
theorem in Book VII that appears for the first time is one usually named for 
Paul Guldin, a seventeenth-century mathematician : If a closed plane curve 
is revolved about a line not passing through the curve, the volume of the 
solid generated is found by taking the product of the area bounded by the 
curve and the distance traversed during the revolution by the center of 
gravity of the area. Pappus was rightfully proud of this very general theorem, 
for it included "a large number of theorems of all sorts about curves, surfaces 
and solids, all of which are proved simultaneously by one demonstration." 
It is indeed the most general theorem involving the calculus to be found in 
antiquity. Pappus gave also the analogous theorem that the surface area 
generated by the revolution of a curve about a line not cutting the curve is 
equal to the product of the length of the curve and the distance traversed 
by the centroid of the curve during the revolution. 14 

The Collection of Pappus is the last truly significant ancient mathematical 
treatise, for the attempt of the author to revive geometry was not successful. 
Mathematical works continued to be written in Greek for about another 
thousand years, continuing an influence that had begun almost a millennium 
before, but authors following Pappus never again rose to his level. Their 
works are almost exclusively in the form of commentary on earlier treatises. 
Pappus himself is in part responsible for the ubiquitous commentaries that 
ensued, for he had composed commentaries on the Elements of Euclid and 
on the Almagest of Ptolemy, among others, only, fragments of which survive. 
Later commentaries, such as those of Theon of Alexandria (fl. 365), are more 
useful for historical information than for mathematical results. Theon was 
responsible also for an important edition of the Elements that has survived ; 
he is remembered also as the father of Hypatia, a learned young lady who 
wrote commentaries on Diophantus, Ptolemy, and Apollonius. An ardent 
devotee of pagan learning, Hypatia incurred the enmity of a fanatical 
Christian mob at whose hands she suffered a cruel death in 415. The dramatic 
impact of her death in Alexandria has caused that year to be taken by some 
to mark the end of ancient mathematics, but a more appropriate close is 
found another century later. 

Alexandria produced in Proclus (410-485) a young mathematical scholar 1 3 
who went to Athens, where be became the head of the Neoplatonic school. 

14 There is a possibility that the "Guldin theorem" represents an interpolation in the manu- 
script of the Collection. (See the Ver Eecke translation cited in footnote 12.) In any case, the 
theorem represents a striking advance by someone during or following the long period of 


Proclus was more the philosopher than the mathematician, but his remarks 
are often critical for the history of early Greek geometry. Of great significance 
is his Commentary on Book I of the Elements of Euclid, for, while writing 
this, Proclus undoubtedly had at hand a copy of the History of Geometry 
by Eudemus, now lost, as well as Pappus' Commentary on the Elements, 
largely lost. For our information on the history of geometry before Euclid 
we are heavily indebted to Proclus, who included in his Commentary a 
summary or substantial extract from Eudemus' History. This passage, which 
has come to be known as the Eudemian Summary, may be taken as Proclus' 
chief contribution to mathematics, although to him is ascribed the theorem 
that if a line segment of fixed length moves with its end points on two inter- 
secting lines, a point on the segment will describe a portion of an ellipse. 

1 4 During the years when Proclus was writing in Athens, the Roman Empire 
in the West was gradually collapsing. The end of the empire usually is 
placed at 476, for in this year the incumbent Roman emperor was displaced 
by Odoacer, a Goth. Some of the old Roman senatorial pride remained, but 
the senatorial party had lost political control. In this situation Boethius 
(ca. 480-524) found his position difficult, for he came of an old distinguished 
patrician family. He was not only a philosopher and mathematician, but also 
a statesman, and he probably viewed with distaste the rising Ostrogothic 
power. Although Boethius may have been the foremost mathematician 
produced by ancient Rome, the level of his work is a far cry from that charac- 
teristic of Greek writers. He was the author of textbooks for each of the four 
mathematical branches in the liberal arts, but these were jejune and exceed- 
ingly elementary abbreviations of earlier classics — an Arithmetic that was 
only an abridgement of the Introductio of Nicomachus ; a Geometry based 
on Euclid and including statements only, without proof, of some of the 
simpler portions of the first four books of the Elements ; an Astronomy derived 
from Ptolemy's Almagest ; and a Music that is indebted to the earlier works of 
Euclid, Nicomachus, and Ptolemy. In some cases these primers, used 
extensively in medieval monastic schools, may have suffered later interpola- 
tions, hence it is difficult to determine precisely what is genuinely due to 
Boethius himself. It is nevertheless clear that the author was concerned 
primarily with two aspects of mathematics : its relationship to philosophy and 
its applicability to simple problems of mensuration. Of mathematics as a 
logical structure there is little trace. 

Boethius seems to have been a statesman of high purpose and unques- 
tioned integrity. He and his sons in turn served as consuls, and Boethius was 
among the chief advisers of Theodoric; but for some reason, whether 
political or religious, the philosopher incurred the displeasure of the emperor. 
It has been suggested that Boethius was a Christian (as perhaps Pappus was 


also) and that he espoused Trinitarian views that alienated the Arian emperor. 
It is possible also that Boethius was too closely associated with political 
elements that looked to the Eastern Empire for help in restoring the old 
Roman order in the West. 15 In any case, Boethius was executed in 524 or 
525, following a long imprisonment. (Theodoric, incidentally, died only about 
a year later, in 526.) It was while in prison that he wrote his most celebrated 
work, De consolatione philosophiae. This essay, written in prose and verse 
while he faced death, discusses moral responsibility in the light of Aristotelian 
and Platonic philosophy. 

The death of Boethius may be taken to mark the end of ancient math- 1 5 
ematics in the Western Roman Empire, as the death of Hypatia had marked 
the close of Alexandria as a mathematical center; but work continued for a 
few years longer at Athens. There one found no great original mathematician, 
but the Peripatetic commentator Simplicius (fl. 520) was sufficiently con- 
cerned about Greek geometry to have preserved for us what may be the 
oldest fragment extant. Aristotle in the Physica had referred to the quadrature 
of the circle or of a segment, and Simplicius took this opportunity to quote 
"word for word" what Eudemus had written on the subject of the quadrature 
of lunes by Hippocrates. The account, several pages long, gives full details on 
the quadratures of lunes, quoted by Simplicius from Eudemus, who in turn 
is presumed to have given at least part of the proofs in Hippocrates' own 
words, especially where certain archaic forms of expression are used. This 
source is the closest we can come to direct contact with Greek mathematics 
before the days of Plato. 

Simplicius was primarily a philosopher, but in his day there circulated a 1 6 
work usually described as the Greek Anthology, the mathematical portions 
of which remind us strongly of the problems in the Ahmes Papyrus of more 
than two millennia earlier. The Anthology contained some six thousand 
epigrams ; of these more than forty are mathematical problems, collected 
presumably by Metrodorus, a grammarian of perhaps the fifth or sixth 
century. Most of them, including the epigram above on the age of Diophantus, 
lead to simple linear equations. For example, one is asked to find how many 
apples are in a collection if they are to be distributed among six persons so 
that the first person receives one third of the applies, the second receives one 
fourth, the third person receives one fifth, the fourth person receives one 
eighth, the fifth person receives ten apples, and there is one apple left for the 
last person. Another problem is typical of elementary algebra texts of our 

15 See Helen M. Barrett, Boethius. Some Aspects of His Times and Work (Cambridge Univer- 
sity Press, 1940). Brief extracts from works of Boethius are included in Cohen and Drabkin, 
Source Book in Greek Science, pp. 291-294, 298-299. 


day : If one pipe can fill a cistern in one day, a second in two days, a third in 
three days, and a fourth in four days, how long will it take all four running 
together to fill it? The problems presumably were not original with Metro- 
dorus, but were collected from various sources. Some probably go back 
before the days of Plato, reminding us that not all Greek mathematics was 
of the type that we think of as classical. 

1 7 Simplicius and Metrodorus were not the outstanding mathematicians of 
their day, for there were contemporary commentators with training adequate 
for an understanding of the works of Archimedes and Apollonius. Among 
these was Eutocius (born ca. 480), who commented on several Archimedean 
treatises and on the Apollonian Conies. It is to Eutocius that we owe the 
Archimedean solution of a cubic through intersecting conies, referred to in 
The Sphere and Cylinder but not otherwise extant except through the com- 
mentary of Eutocius. The commentary by Eutocius on the Conies of Apol- 
lonius was dedicated to Anthemius of Tralles (T534), an able mathematician 
and architect of St. Sophia of Constantinople, who described the string 
construction of the ellipse and wrote a work On Burning-mirrors in which the 
focal properties of the parabola are described. His colleague and successor 
in the building of St. Sophia, Isidore of Miletus (fl. 520), also was a math- 
ematician of some ability. It was Isidore who made known the commentaries 
of Eutocius and spurred a revival of interest in the works of Archimedes 
and Apollonius. To him perhaps we owe the familiar T-square and string 
construction of the parabola — and possibly also the apocryphal Book XV 
of Euclid's Elements. It may be in large measure due to the activities of the 
Constantinople group — Eutocius, Isidore, and Anthemius — that Greek 
versions of Archimedean works and of the first four books of Apollonius' 
Conies have survived to this day. 

Isidore of Miletus was one of the last directors of the Platonic Academy 
at Athens. The school had of course undergone many changes throughout 
its existence of more than 900 years, and during the days of Proclus it had 
become a center of Neoplatonic learning. When in 527 Justinian became 
emperor in the East, he evidently felt that the pagan learning of the Academy 
and other philosophical schools at Athens was a threat to orthodox Chris- 
tianity; hence in 529 the philosophical schools were closed and the 
scholars dispersed. Rome at the time was scarcely a very hospitable home for 
scholars, and Simplicius and some of the other philosophers looked to the 
East for a haven. This they found in Persia, where under King Chosroes they 
established what might be called the "Athenian Academy in Exile." 16 The 
date 529 may therefore be taken to mark the close of European mathematical 

16 See George Sarton, The History of Science (Cambridge, Mass. : Harvard University Press, 
1952-1959, 2 vols.), I, 400. 


development in antiquity. Henceforth the seeds of Greek science were to 
develop in Near and Far Eastern countries until, some 600 years later, the 
Latin world was in a more receptive mood. The date 529 has another signifi- 
cance that may be taken as symptomatic of a change in values — in this year 
the venerable monastery of Monte Cassino was established. Mathematics 
did not of course entirely disappear from Europe in 529, for undistinguished 
commentaries continued to be written in Greek in the Byzantine Empire and 
versions of the jejune Latin texts of Boethius continued in use in Western 
schools. The spirit of mathematics languished, however, while men argued 
less about the value of geometry and more about the way to salvation. For the 
next steps in mathematical development we must therefore turn our backs on 
Europe and look toward the East. 


Clagett, Marshall, Greek Science in Antiquity (New York: Abelard Schuman, 1955; 

paperback ed., Collier Books, 1963). 
Cohen, M. R., and I. E. Drabkin, Source Book in Greek Science (New York : McGraw- 
Hill, 1948; reprinted, Cambridge, Mass.: Harvard University Press, 1958). 
Chasles, Michel, Les trois livres de porismes d'Euclide, retablis . . . d'apres la notice . . . de 

Pappus (Paris: Mallet-Bachelier, 1860). 
Heath, T. L., Diophantus of Alexandria: A Study in the History of Greek Algebra, 2nd ed. 

(New York: Cambridge University Press, 1910; paperback ed., New York: Dover, 

Heath, T. L., History of Greek Mathematics (Oxford: Clarendon, 1921, 2 vols.). 
Nesselmann, G. H. F., Die Algebra der Griechen (Berlin, 1842). 
Nicomachus of Gerasa, Introduction to Arithmetic, trans, by M. L. D'Ooge, with studies 

in Greek arithmetic by F. E. Robbins and L. C. Karpinski (New York : Macmillan, 

Pappus of Alexandria, Collectionis quae supersunt, ed. by F. Hultsch (Berlin, 1876-1878, 

3 vols.). 
Pappus of Alexandria, La collection mathematique, trans, by Paul Ver Eecke (Paris, 

1933, 2 vols.). 
Proclus Diadochus, Les commentaires sur le premier livre des Elements d'Euclide, trans. 

by Paul Ver Eecke (Bruges: Desclee de Brouwer, 1948). 
Sanchez Perez, Jose Augusto, La aritmetica en Grecia (Madrid : Instituto Jorge Juan, 

Sanchez Perez, Jose Augusto, La aritmetica en Roma, en India y en Arabia (Madrid : 

Instituto Miguel Asin, 1949). 
Stahl, W. H, Roman Science (Madison, Wis. : University of Wisconsin Press, 1962). 
Swift, J. D., "Diophantus of Alexandria," American Mathematical Monthly, 43 (1956), 

Thomas, Ivor, ed., Selections Illustrating the History of Greek Mathematics (Cambridge, 

Mass. : Loeb Classical Library, 1939-1941, 2 vols.). 


Van der Waerden, B. L., Science Awakening, trans, by Arnold Dresden (New York : 

Oxford, 1961; paperback ed., New York: Wiley, 1963). 
Ziegler, Konrat, "Pappos," in Pauly-Wissowa, Real-Enzyclopadie der klassischen 

Wissenschaft (Stuttgart, 1949), Vol. XVIII, Part 3, columns 1084-1106. 


1. Do you think the conditions in Alexandria were more or less favorable for the development 
of mathematics in the days of Pappus than at the time of Ptolemy? Explain. 

2. How did the intellectual conditions at Alexandria compare with those at Rome in the days 
of Diophantus and Pappus? 

3. Would the development of mathematics have been essentially modified if Rome had not 
fallen in 476? Give reasons for your answer. 

4. If you were a mathematician living in the year 500, would you have chosen Alexandria, 
Rome, Athens, or Constantinople as your home? Give reasons for your answer. 

5. Show that the epigram concerning the age of Diophantus leads to the conclusion that he 
died at the age of eighty-four. 

6. Verify that the four numbers listed by Nicomachus as perfect are indeed perfect numbers. 

7. Solve the problem of Diophantus in which it is required to find two numbers such that their 
sum is 10 and the sum of their cubes is 370. 

8. Find two rational fractions, other than ^ and rs satisfying Diophantus' condition that 
either one when added to the square of the other will produce a perfect square. 

9. Prove that the lines OC, BD, and DF in Fig. 11.3 are indeed the arithmetic, the geometric, 
and the harmonic means, respectively, of AB and BC, as Pappus asserted. 

10. Prove Pappus' generalization of the Pythagorean theorem illustrated in Fig. 11.4. 

11. Draw carefully a diagram similar to Fig. 11.5 in which AB is 3 inches and BC is 2 inches 
and find approximately, by measurements, the diameter of the circle C 3 and the distance 
of its center from the line AC, thus verifying roughly the assertion of Pappus. 

12. Solve the problem of the distribution of apples described in the text. 

13. Solve the problem of the three pipes described in the text. 

14. Show analytically that the Pappus problem for six lines leads to a locus the equation of 
which is not higher than third degree. 

*15. Prove the first Pappus trisection given in the text. 

*16. Prove that OTis parallel to AP in Fig. 11.2. 

*17. Using the result in Exercise 16, complete the proof of the second Pappus trisection given 
in the text. 

* 18. Justify the Pappus theorem on solids of revolution. 

*19. Prove the theorem of Proclus on the generation of an ellipse for the case in which the inter- 
secting lines are mutually perpendicular. 


China and India 

A mixture of pearl shells and sour dates ... or of costly 
crystal and common pebbles. 

Al-Biruni's India 

The civilizations of China and India are of far greater antiquity than those 1 
of Greece and Rome, although not older than those in the Nile and Meso- 
potamian valleys. They go back to the Potamic Age, whereas the cultures 
of Greece and Rome were of the Thalassic Age. Civilizations along the 
Yangtze and Yellow rivers are comparable in age with those along the Nile 
or between the Tigris and Euphrates ; but chronological accounts in the case 
of China are less dependable than those for Egypt and Babylonia. Claims 
that the Chinese made astronomical observations of importance, or described 
the twelve signs of the zodiac, by the fifteenth millennium b.c. are certainly 
unfounded, but a tradition that places the first Chinese empire about 2750 B.C. 
is not unreasonable. More conservative views place the early civilizations 
of China nearer 1000 B.C. The dating of mathematical documents from China 
is far from easy, and estimates concerning the Chou Pei Suan Ching, generally 
considered to be the oldest of the mathematical classics, differ by almost a 
thousand years. The problem of its date is complicated by the fact that it 
may well have been the work of several men of differing periods. Some 
consider the Chou Pei to be a good record of Chinese mathematics of about 
1200 B.C. but others place the work in the first century before our era. A date 
of about 300 B.C. would appear reasonable, thus placing it in close competi- 
tion with another treatise, the Chiu-chang suan-shu, composed about 250 bc., 1 
that is, shortly before the Han dynasty (202 b.c). The words "Chou Pei" 
seem to refer to the use of the gnomon in studying the circular paths of the 
heavens, and the book of this title is concerned with astronomical calcula- 
tions, although it includes an introduction on the properties of the right 
triangle and some work on the use of fractions. The work is cast in the form 

Histories of mathematics generally devote little space to Chinese contributions. Exceptional 
in this respect are D. E. Smith, History of Mathematics (1923-1925), and J. E. Hofmann, Geschichte 
der Mathematik, 2nd ed. (Berlin, 1963), Vol. I. An unusually thorough and up-to-date account 
in the Near and Far East is given in A. P. Juschkewitsch, Geschichte der Mathematik im Mittel- 
alter (1964). 



of a dialogue between a prince and his minister concerning the calendar ; 
the minister tells his ruler that the art of numbers is derived from the circle 
and the square, the square pertaining to the earth and the circle belonging 
to the heavens. The Chou Pei indicates that in China, as Herodotus held in 
Egypt, geometry arose from mensuration; and, as in Babylonia, Chinese 
geometry was essentially only an exercise in arithmetic or algebra. There 
seem to be some indications in the Chou Pei of the Pythagorean theorem, a 
theorem treated algebraically by the Chinese. 

Almost as old as the Chou Pei, and perhaps the most influential of all 
Chinese mathematical books, 2 was the Chui-chang suan-shu, or Nine 
Chapters on the Mathematical Art. This book includes 246 problems on 
surveying, agriculture, partnerships, engineering, taxation, calculation, the 
solution of equations, and the properties of right triangles. Whereas the 
Greeks of this period were composing logically ordered and systematically 
expository treatises, the Chinese were repeating the old custom of the 
Babylonians and Egyptians of compiling sets of specific problems. The Nine 
Chapters resembles Egyptian mathematics also in its use of the method of 
"false position," but the invention of this scheme, like the origin of Chinese 
mathematics in general, seems to have been independent of Western influence. 

In Chinese works, as in Egyptian, one is struck by the juxtaposition of 
accurate and inaccurate, primitive and sophisticated results. Correct rules 
are used for the areas of triangles, rectangles, and trapezoids. The area of the 
circle was found by taking three fourths the square on the diameter or 
one-twelfth the square of the circumference — a correct result if the value 
three is adopted for n — but for the area of a segment of a circle the Nine 
Chapters uses the approximate results s(s + c)/2, where s is the sagitta (that 
is, the radius minus the apothem) and c the chord or base of the segment. 
There are problems that are solved by the rule of three; in others square and 
cube roots are found. Chapter eight of the Nine Chapters is significant for 
its solution of problems in simultaneous linear equations, using both positive 
and negative numbers. The last problem in the chapter involves four equations 
in five unknowns, and the topic of indeterminate equations was to remain a 
favorite among Oriental peoples. The ninth and last chapter includes prob- 
lems on right-angled triangles, some of which later reappeared in India and 
Europe. One of these asks for the depth of a pond 10 feet square if a reed 
growing in the center and extending 1 foot above the water just reaches the 
surface if drawn to the edge of the pond. Another of these well-known 
problems is that of the Broken bamboo: There is a bamboo 10 feet high, the 

2 See Joseph Needham, Science and Civilization in China (1959), Vol. Ill, pp. 24-25. For recent 
mathematical works see Tung-Li Yuan, Bibliography of Chinese Mathematics 1918-1960 
(Washington, DC, published by the author. 1963). 


upper end of which being broken reaches the ground 3 feet from the stem. 
Find the height of the break. 3 

The Chinese were especially fond of patterns ; hence it is not surprising 3 
that the first record (of ancient but unknown origin) of a magic square 
appeared there. The square 

was supposedly brought to man by a turtle from the River Lo in the days of 
the legendary Emperor Yii, reputed to be a hydraulic engineer. 4 The con- 
cern for such patterns led the author of the Nine Chapters to solve the 
system of simultaneous linear equations 

3x + 2y + z = 39 

2x + 3y + z = 34 

x + 2y + 3z = 26 

by performing column operations on the matrix 








to reduce it to 

36 1 





99 24 


The second form represented the equations 36z = 99, 5y + z = 24, and 
3x + 2y + z = 39, from which the values of z, y, and x are successively 
found with ease. 

Had Chinese mathematics enjoyed uninterrupted continuity of tradition, 
some of the striking anticipations of modern methods might have signifi- 
cantly modified the development of mathematics, but Chinese culture was 
seriously hampered by abrupt breaks. In 213 B.C., for example, the Chinese 

3 See Yoshio Mikami, The Development of Mathematics in China and Japan (1913), p.23. 

4 See D. J. Struik, "On Ancient Chinese Mathematics," The Mathematics Teacher, 56(1963), 


emperor ordered the burning of books. Some works must obviously have 
survived, either through the persistence of copies or through oral trans- 
mission; and learning did indeed persist, with mathematical emphasis on 
problems of commerce and the calendar. 

There seems to have been contact between India and China, as well as 
between China and the West, but scholars differ on the extent and direction 
of borrowing. The temptation to see Babylonian or Greek influence in China, 
for example, is faced with the problem that the Chinese did not make use 
of sexagesimal fractions. Chinese numeration remained essentially decimal, 
with notations rather strikingly different from those in other lands. In China, 
from early times, two schemes of notation were in use. In one the multiplica- 
tive principle predominated, in the other a form of positional notation was 
used. In the first of these there were distinct ciphers for the digits from one 
to ten and additional ciphers for the powers of ten, and in the written forms 
the digits in odd positions (from left to right or from bottom to top) were 
multiplied by their successor. Thus the number 678 would be written as a 
six followed by the symbol for one hundred, then a seven followed by the 
symbol for ten, and finally the symbol for eight. 

In the system of "rod numerals" the digits from one to nine appeared as 
I II III llli Hill T T TIT W and the first nine multiples of ten as 
— = = = H -L i i =. By the use of these eighteen symbols 
alternately in positions from right to left, numbers as large as desired could 
be represented. The number 56,789, for instance, would appear as 
Hill -LTTimr. As in Babylonia, a symbol for an empty position appeared only 
relatively late. In a work of 1247 the number 1,405,536 is written with a round 
zero symbol as I = = Hill =T. (Occasionally, as in the fourteenth-century 
form of the arithmetic triangle, the vertical and horizontal rods or strokes 
were interchanged.) 

The precise age of the original rod numerals cannot be determined, but 
they were certainly in use several hundred years before our era, that is, long 
before the positional notation had been adopted in India. The use of a 
centesimal, rather than a decimal, positional system in China was convenient 
for adaptation to computations with the counting board. Distinctive nota- 
tions for neighboring powers of ten enabled the Chinese to use, without 
confusion, a counting board with unmarked vertical columns. Before the 
eighth century the place in which a zero was required was simply left blank. 
Although in texts older than a.d. 300 the numbers and multiplication tables 
were written out in words, calculations actually were made with rod numerals 
on a counting board. 

The rod numerals of about 300 B.C. were not merely a notation for the 
written result of a computation. Actual bamboo, ivory, or iron rods were 


carried about in a bag by administrators and used as a calculating device. 
Counting rods were manipulated with such dexterity that an eleventh- 
century writer described them as "flying so quickly that the eye could not 
follow their movement." Cancellations probably were more rapidly carried 
out with rods on a counting board than in written calculations. So effective, 
in fact, was the use of the rods on a counting board that the abacus or rigid 
counting frame with movable markers on wires was not used so early as has 
been generally supposed. First clear descriptions of the modern forms, known 
in China as the suan phan and in Japan as the soroban, are of the sixteenth 
century; but anticipations would appear to have been in use perhaps a 
thousand years earlier. The word abacus probably is derived from the 
Semitic word abq or dust, indicating that in other lands, as well as in China, 
the device grew out of a dust or sand tray used as a counting board. It is 
possible, but by no means certain, that the use of the counting board in 
China antedates the European, but clear-cut and reliable dates are not 
available. In the National Museum in Athens there is a marble slab, dating 
probably from the fourth century B.C., which appears to be a counting board ; 
and when a century earlier Herodotus wrote, "The Egyptians move their 
hand from right to left in calculation, while the Greeks move it from left to 
right," he probably was referring to the use of some sort of counting board. 
Just when such devices gave way to the abacus proper is difficult to deter- 
mine ; fior can we tell whether or not the appearances of the abacus in 
China, Arabia, and Europe were independent inventions. The Arabic abacus 
had ten balls on each wire and no center bar, whereas the Chinese had five 
lower and two upper counters on each wire, separated by a bar. Each of the 
upper counters on a wire of the Chinese abacus is equivalent to five on the 
lower wire; a number is registered by sliding the appropriate counters 
against the separating bar. (See the accompanying illustration of an abacus.) 
No description of Chinese numeration would be complete without refer- 
ence to the use of fractions. The Chinese were familiar with operations on 
common fractions, in connection with which they found lowest common 
denominators. As in other contexts, they saw analogies with the differences 
in the sexes, referring to the numerator as the "son" and to the denominator 
as the "mother." Emphasis on yin and yang (opposites, especially in sex) 
made it easier to follow the rules for the manipulation of fractions. More 
important than these, however, was the tendency in China toward decimaliza- 
tion of fractions. As in Mesopotamia a sexagesimal metrology led to sexa- 
gesimal numeration, so also in China adherence to the decimal idea in 
weights and measures resulted in a decimal habit in the treatment of fractions 
that, it is said, can be traced back as far as the fourteenth century b.c. 5 

5 See Needham, op. cit., Ill, 89. 


Marble counting board, probably from the fourth century B.C. found on the island of 
Salamis and now in the National Museum in Athens. 




/J. yjU 

'■£. 'A 








An early printed picture of the abacus, from the Suan Fa Thung Tsung, 1 593. (Reproduced 
from Joseph Needham, Science and Civilization in China, III, 76.) 

Decimal devices in computation sometimes were adopted to lighten manipu- 
lations of fractions. In a first-century commentary on the Nine Chapters, 
for example, we find the use of the now familiar rules for square and cube 
roots, equivalent to yfa = N /l00a/10 and j/a = ^1000a/I0, which facilitate 
the decimalization of root extractions. 

The idea of negative numbers seems not to have occasioned much difficulty 
for the Chinese since they were accustomed to calculating with two sets of 
rods — a red set for positive coefficients or numbers and a black set for 
negatives. Nevertheless, they did not accept the notion that a negative 
number might be a solution of an equation. 

The earliest Chinese mathematics is so different from that of comparable 6 
periods in other parts of the world that the assumption of independent 
development would appear to be justified. At all events, it seems safe to say 


that if there was some intercommunication before 400, then more mathe- 
matics came out of China than went in. For later periods the question 
becomes more difficult. The use of the value three for n in early Chinese 
mathematics is scarcely an argument for dependence on Mesopotamia, 
especially since the search for more accurate values, from the first centuries 
of the Christian era, was more persistent in China than elsewhere. Values 
such as 3.1547, ^/To, 92/29, and 142/45 are found; and in the third century 
Liu Hui, an important commentator on the Nine Chapters, derived the 
figure 3.14 by use of a regular polygon of 96 sides and the approximation 
3.14159 by considering a polygon of 3072 sides. In Liu Hui's reworking of 
the Nine Chapters there are many problems in mensuration, including the 
correct determination of the volume of a frustum of a square pyramid. For 
a frustum of a circular cone a similar formula was applied, but with a value 
of three for n. Unusual is the rule that the volume of a tetrahedron with two 
opposite edges perpendicular to each other is one-sixth the product of these 
two edges and their common perpendicular. The method of false position 
is used in solving linear equations, but there are also more sophisticated 
results, such as the solution, through a matrix pattern, of a Diophantine 
problem involving four equations in five unknown quantities. The approxi- 
mate solution of equations of higher degree seems to have been carried out 
by a device similar to what we know as "Horner's method." Liu Hui also 
included, in his work on the Nine Chapters, numerous problems involving 
inaccessible towers and trees on hillsides. 6 

The Chinese fascination with the value of n reached its high point in the 
work of Tsu Ch'ung-chih (430-501). One of his values was the familiar 
Archimedean 22/7, described by Tsu Ch'ung-chih as "inexact"; his "ac- 
curate" value was 355/113. If one persists in seeking possible Western in- 
fluence, one can explain away this remarkably good approximation, not 
equaled anywhere until the fifteenth century, by subtracting the numerator 
and denominator, respectively, of the Archimedean value from the numerator 
and denominator of the Ptolemaic value 377/120. However, Tsu Ch'ung-chih 
went even further in his calculations, for he gave 3.1415927 as an "excess" 
value and 3.1415926 as a "deficit value." 7 The calculations by which he 
arrived at these bounds, apparently aided by his son Tsu Cheng-chih, were 
probably contained in one of his books, since lost. In any case, his results 
were remarkable for that age, and it is fitting that today a landmark on the 
moon bears his name. 

We should bear in mind that accuracy in the value of n is more a matter 

6 See the excellent article on Liu Hui, written by Ho Peng-Yoke, to appear in the forthcoming 
volumes of the Dictionary of Scientific Biography. 

1 See the article cited in footnote 6. There seems to be some confusion in the citation of this 
value by Mikami, op. cit., p. 50, by Smith, op. cit., II, 309, and Hofmann, op. cit., I, 76. 


of computational stamina than of theoretical insight. The Pythagorean 
theorem alone suffices to give as accurate an approximation as may be 
desired. Starting with the known perimeter of a regular polygon of n sides 
inscribed in a circle, the perimeter of the inscribed regular polygon of In 
sides can be calculated by two applications of the Pythagorean theorem. 
Let C be a circle with center O and radius r (Fig. 12.1) and let PQ = s be a 

fig. 12.1 

side of a regular inscribed polygon of n side s having a known perimeter. 
Then the apothem OM = u is given by u = Jr 2 - (s/2) 2 ; hence the sagitta 
MR = v = r - u is known. Then the si de RQ = w of the inscribed regular 
polygon of In sides is found from w = Jv 1 + (s/2) 2 ; hence the perimeter of 
this polygon is known. The calculation, as Liu Hui saw, can be shortened 
by noting that w 2 = 2rv. An iteration of the procedure will result in an ever 
closer approximation to the perimeter of the circle, in terms of which n is 

Chinese mathematical problems often appear to be more picturesque 
than practical, and yet Chinese civilization was responsible for a surprising 
number of technological innovations. The use of printing and gunpowder 
(eighth century) and of paper and the mariner's compass (eleventh century) 
was earlier in China than elsewhere, and earlier also than the high-water 
mark in Chinese mathematics that occurred in the thirteenth century, during 
the latter part of the Sung period. At that time there were mathematicians 
working in various parts of China; but relations between them seem to have 
been remote, and, as in the case of Greek mathematics, we evidently have 
relatively few of the treatises that once were available. The last and greatest 
of the Sung mathematicians was Chu Shih-chieh (fl. 1280-1303), yet we 
know little about him — not even when he was born or when he died. He was 
a resident of Yen-shan, near modern Peking, but he seems to have spent 
some twenty years as a wandering scholar who earned his living by teaching 
mathematics, even though he had the opportunity to write two treatises. The 


first of these, written in 1299, was the Suan-hsiieh cKi-meng ("Introduction 
to Mathematical Studies"), a relatively elementary work that strongly 
influenced Korea and Japan, although in China it was lost until it reappeared 
in the nineteenth century. 8 Of greater historical and mathematical interest 
is the Ssu-yuan yii-chien ("Precious Mirror of the Four Elements") of 1303. 
In the eighteenth century this too disappeared in China, only to be rediscov- 
ered in the next century. The four elements, called heaven, earth, man, and 
matter, are the representations of four unknown quantities in the same 
equation. The book marks the peak in the development of Chinese algebra, 
for it deals with simultaneous equations and with equations of degrees as 
high as fourteen. In it the author describes a transformation method that 
he calls fan fa, the elements of which seem to have arisen long before in 
China, but which generally bears the name of Horner, who lived half a 
millennium later. In solving the equation x 2 + 252x - 5292 = 0, for example, 
Chu Shih-chieh first obtained x = 19 as an approximation (a root lies 
between x = 19 and x = 20) and then used the fan-fa, in this case the trans- 
formation y = x - 19, to obtain the equation y 2 + 290y - 143 = (with 
a root between y = and y = 1). He then gave the root of the latter as 
(approximately) y = 143/(1 + 290); hence the corresponding value of x is 
19^f. For the equation x 3 - 574 = he used y = x - 8 to obtain 
^3 + 24y 2 + 192y - 62 = 0, and he gave the root as x = 8 + 62/(1 + 24 + 
192) or x = 8f. In some cases he found decimal approximations. 

That the so-called Horner method was a commonplace in China is indica- 
ted by the fact that at least three other mathematicians of the later Sung 
period made use of similar devices. One of these was Li Chih (or Li Yeh, 
1192-1279), a mathematician of Peking who was offered a government post 
by Khublai Khan in 1260, but politely found an excuse to decline it. His 
Ts'e-yuan hai-ching ("Sea-Mirror of the Circle Measurements") includes 170 
problems dealing with circles inscribed within, or escribed without, a right 
triangle and with determining the relationships between the sides and the 
radii, some of the problems leading to equations of fourth degree. Although 
he did not describe his method of solution of equations, including some of 
sixth degree, it appears that it was not very different from that used by Chu 
Shih-chieh and Horner. 9 Others who used the Horner method were Ch'in 
Chiu-shao (ca. 1202-ca. 1261) and Yang Hui (ft ca. 1261-1275). The former 
was an unprincipled governor and minister who acquired immense wealth 
within a hundred days of assuming office. His Shu-shu chiu-chang ("Mathe- 
matical Treatise in Nine Sections") marks the high point in Chinese indeter- 

8 See the extensive forthcoming article on Chu Shih-chieh by Ho Peng- Yoke to appear in the 
Dictionary of Scientific Biography. See also Needham, op. cit., Ill, 38-53. 
* See the article on Li Chih by Ho Peng- Yoke to appear in Dictionary of Scientific Biography. 


minate analysis, with the invention of routines for solving simultaneous 
congruences. In this work also he found the square root of 71,824 by steps 
paralleling those in the Horner method. With 200 as the first approximation 
to a root of x 2 - 71,824 = 0, he diminished the roots of this by 200 to obtain 
y 2 + 400y - 31,824 = 0. For the latter equation he found 60 as an approxi- 
mation, and diminished the roots by 60, arriving at a third equation, 
z 2 + 520z - 4224 = 0, of which 8 is a root. Hence the value of x is 268. In a 
similar way he solved cubic and quartic equations. The same "Horner" 
device was used by Yang Hui, about whose life almost nothing is known and 
whose work has survived only in part. Among his contributions that are 
extant are the earliest Chinese magic squares of order greater than three, 
including two each of orders four through eight and one each of orders nine 
and ten. 10 

Yang Hui's works included also results in the summation of series and the 
so-called Pascal triangle, things that were published and better known 
through the Precious Mirror of Chu Shih-chieh, with which the Golden 
Age of Chinese mathematics closed. A few of the many summations of series 
found in the Precious Mirror are the following : 

l 2 + 2 2 + 3 2 + • ■ • + n 2 = n(n + l)(2n + l)/3 ! 

1 + 8 + 30 + 80 + • • ■ + n 2 (n + \)(n + 2)/3! = n(n + \)(n + 2)(n + 3) 

x (An + l)/5! 

However, no proofs are given, nor does the topic seem to have been con- 
tinued again in China until about the nineteenth century. Chu Shih-chieh 
handled his summations through the method of finite differences, some 
elements of which seem to date in China from the seventh century; but 
shortly after his work the method disappeared for many centuries. 

The Precious Mirror opens with a diagram of the arithmetic triangle, 
inappropriately known in the West as "Pascal's triangle." In Chu's arrange- 
ment we have the coefficients of binomial expansions through the eighth 
power, clearly given in rod numerals and a round zero symbol. Chu disclaims 
credit for the triangle, referring to it as a "diagram of the old method for 
finding eighth and lower powers." A similar arrangement of coefficients 
through the sixth power had appeared in the work of Yang Hui, but without 
the round zero symbol. There are references in Chinese works of about 1100 
to tabulation systems for binomial coefficients, and it is likely that the 
arithmetic triangle originated in China by about that date. It is interesting 
to note that the Chinese discovery of the binomial theorem for integral 

10 Excellent articles, including much more on the work of Ch'in Chiu-shao and Yang Hui, 
written by Ho Peng-Yoke, will appear in the forthcoming Dictionary of Scientific Biography. 



powers was associated in its origin with root extractions, rather than with 
powers. The equivalent of the theorem apparently was known to Omar 
Khayyam at about the time that it was being used in China, but the earliest 
extant Arabic work containing it is by Al-Kashi in the fifteenth century. By 
that time Chinese mathematics had failed to match achievements in Europe 

£ 4k * & 4f 







The "Pascal" Triangle as depicted in 1303 at the front of Chu Shih-Chieh's Ssu Yuan Yii 

Chien. It is entitled "The Old Method Chart of the Seven Multiplying Squares" and 

tabulates the binomial coefficients up to the eighth power. (Reproduced from Joseph 

Needham, Science and Civilization in China, III, 135.) 

and the Near East, and it is likely that by then more mathematics went into 
China than came out. Still to be answered is the thorny problem of determin- 
ing the relative influences of China and India on each other during the first 
millennium of our era. 


Archeological excavations at Mohenjo Daro give evidence of an old and 1 
highly cultured civilization in India during the era of the Egyptian pyramid 
builders, but we have no Indian mathematical documents from that age. 
Later the country was occupied by Aryan invaders who introduced the caste 
system and developed the Sanskrit literature. The great religious teacher, 
Buddha, was active in India at about the time that Pythagoras is said to 
have visited there, and it sometimes is suggested that Pythagoras learned 
his theorem from the Hindus. Recent studies make this highly unlikely in 
view of Babylonian familiarity with the theorem at least a thousand years 

The fall of the Western Roman Empire traditionally is placed in the year 
476 ; it was in this year that Aryabhata, author of one of the oldest Indian 
mathematical texts, was born. It is clear, however, that there had been 
mathematical activity in India long before this time— probably even before 
the mythical founding of Rome in 753 B.C. India, like Egypt, had its "rope- 
stretchers"; and the primitive geometrical lore acquired in connection with 
the laying out of temples and the measurement and construction of altars 
took the form of a body of knowledge known as the Sulvasutras or "rules 
of the cord." Sulva (or sulba) refers to cords used for measurements, and 
sutra means a book of rules or aphorisms relating to a ritual or science. The 
stretching of ropes is strikingly reminiscent of the origin of Egyptian geom- 
etry, and its association with temple functions reminds one of the possible 
ritual origin of mathematics. However, the difficulty of dating the rules is 
matched also by doubt concerning the influence they had on later Hindu 
mathematicians. Even more so than in the case of China, there is a striking 
lack of continuity of tradition in the mathematics of India; significant 
contributions are episodic events separated by intervals without achieve- 
ment. 11 

Three versions, all in verse, of the work referred to as the Sulvasutras are 1 1 
extant, the best-known being that bearing the name of Apastamba. In this 
primitive account, dating back perhaps as far as the time of Pythagoras, we 
find rules for the construction of right angles by means of triples of cords 
the lengths of which form Pythagorean triads, such as 3, 4, and 5, or 5, 12, 
and 13, or 8, 15, and 17, or 12, 35, and 37. However, all of these triads are 
easily derived from the old Babylonian rule ; hence Mesopotamian influence 
in the Sulvasutras is not unlikely. Apastamba knew that the square on the 
diagonal of a rectangle is equal to the sum of the squares on the two adjacent 

The reader should be forewarned that there are a number of books in which the contribu- 
tions from India are grossly overrated. One such instance is B. K. Sarkar, Hindu Achievements in 
Exact Science (New York, 1918). The two-volume History of Hindu Mathematics by B. Datta 
and A. N. Singh (1935-1938) is much more reliable, but even this must be qualified along the 
lines indicated by Solomon Gandz when he reviewed Volume I in Isis, 25 (1936), 478-488. 


sides, but this form of the Pythagorean theorem also may have been derived 
from Mesopotamia. Less easily explained is another rule given by Apastamba 
— one that strongly resembles some of the geometrical algebra in Book II of 
Euclid's Elements. To construct a square equal in area to the rectangle 
ABCD (Fig. 12.2), lay off the shorter sides on the longer, so that AF = AB = 
BE = CD, and draw HG bisecting segments CE and DF ; extend EF to K, 
GH to L, and AB to M so that FK = HL = FH = AM, and draw LKM. 





FIG. 12.2 

Now construct a rectangle with diagonal equal to LG and with shorter side 
HF. Then the longer side of this rectangle is the side of the square desired. 
So conjectural are the origin and period of the Sulvasutras that we cannot 
tell whether or not the rules are related to early Egyptian surveying or to the 
later Greek problem of altar doubling. They are variously dated within an 
interval of almost a thousand years stretching from the eighth century B.C. to 
the second century of our era. Chronology in ancient cultures of the Far 
East is scarcely reliable when orthodox Hindu tradition boasts of important 
astronomical work more than 2,000,000 years ago 12 and when calculations 
lead to billions of days from the beginning of the life of Brahman to about 
a.d. 400. 13 References to arithmetic and geometric series in Vedic literature 
that purport to go back to 2000 B.C. 14 may be more reliable, but there are 
no contemporary documents from India to confirm this. It has been claimed 
also that the first recognition of incommensurables is to be found in India 
during the Sulvasiitra period, 15 but such claims are not well substantiated. 

12 G. R. Kaye. "Indian Mathematics," I sis, 2 (1914), 326-356. 

13 AlbenmCs India, ed. by E. C. Sachan (London, 1960, 2 vols.), II, 32 f. 

14 A. N. Singh, "On the use of Series in Hindu Mathematics," Osiris, 1 (1936), 606-628. 

15 A. N. Singh, "A Review of Hindu Mathematics up to the Xllth Century," Archeion 18 
(1936), 43-62; Saradakanta Ganguli, "On the Indian Discovery of the Irrational at the Time of 
the Sulvasutras," Scripta Mathematica, 1 (1932), 135-141. 


The case for early Hindu awareness of incommensurable magnitudes is 
rendered most unlikely by the failure of Indian mathematicians to come to 
grips with fundamental concepts. 

The period of the Sulvasutras, which closed in about the second century, 1 2 
was followed by the age of the Siddhantas, or systems (of astronomy). The 
establishment of the dynasty of King Gupta (290) marked the beginning of 
a renaissance in Sanskrit culture, and the Siddhantas seem to have been an 
outcome of this revival. Five different versions of the Siddhantas are known 
by name, Paulisha Siddhanta, Surya Siddhanta, Vasisishta Siddhanta, 
Paitamaha Siddhanta, and Romanka Siddhanta. Of these, the Surya Siddhanta 
("System of the Sun"), written about 400, is the only one that seems to be 
completely extant. According to the text, written in epic stanzas, it is the 
work of Surya, the Sun God. 16 The main astronomical doctrines evidently 
are Greek, but with the retention of considerable old Hindu folklore. The 
Paulisha Siddhanta, which dates from about 380, was summarized by the 
Hindu mathematician Varahamihira (fl. 505) and was referred to frequently 
by the Arabic scholar Al-Biruni, who suggested a Greek origin or influence. 
Later writers report that the Siddhantas were in substantial agreement on 
substance, only the phraseology varying; hence we can assume that the 
others, like the Surya Siddhanta, were compendia of astronomy comprising 
cryptic rules in Sanskrit verse with little explanation and without proof. 

It is generally agreed that the Siddhantas stem from the late fourth or the 
early fifth century, but there is sharp disagreement about the origin of the 
knowledge that they contain. Hindu scholars insist on the originality and 
independence of the authors, whereas Western writers are inclined to see 
definite signs of Greek influence. It is not unlikely, for example, that the 
Paulisha Siddhanta was derived in considerable measure from the work of 
the astrologer Paul who lived at Alexandria shortly before the presumed 
date of composition of the Siddhantas. (Al-Biruni, in fact, explicitly attributes 
this Siddhanta to Paul of Alexandria.) This would account in a simple manner 
for the obvious similarities between portions of the Siddhantas and the 
trigonometry and astronomy of Ptolemy. The Paulisha Siddhanta, for 
example, uses the value 3 177/1250 for n, which is in essential agreement 
with the Ptolemaic sexagesimal value 3 ; 8,30. 

Even if the Hindus did acquire their knowledge of trigonometry from the 
cosmopolitan Hellenism at Alexandria, the material in their hands took on 
a significantly new form. Whereas the trigonometry of Ptolemy had been 
based on the functional relationship between the chords of a circle and the 

1 6 An English translation by Burgess and Whitney, together with extensive notes, was pub- 
lished in Journal of the American Oriental Society, 6 (1860), 141-498. See also George Sarton, 
An Introduction to the History of Science (1927), pp. 386-388. 


central angles they subtend, the writers of the Siddhantas converted this to a 
study of the correspondence between half of a chord of a circle and half of 
the angle subtended at the center by the whole chord. Thus was born, appar- 
ently in India, the predecessor of the modern trigonometric function known 
as the sine of an angle ; and the introduction of the sine function represents 
the chief contribution of the Siddhantas to the history of mathematics. 
Although it is generally assumed that the change from the whole chord to 
the half chord took place in India, it has been suggested by Paul Tannery, 
the leading historian of science at the turn of this century, that this transforma- 
tion of trigonometry may have occurred at Alexandria during the post- 
Ptolemaic period. Whether or not this suggestion has merit, there is no 
doubt that it was through the Hindus, and not the Greeks, that our use of 
the half chord has been derived ; and our word "sine", through misadventure 
in translation (see below), has descended from the Hindu name, jiva. 

1 3 During the sixth century, shortly after the composition of the Siddhantas, 
there lived two Hindu mathematicians who are known to have written books 
on the same type of material. The older, and more important, of the two was 
Aryabhata, whose best known work, written in 499 and entitled Aryabhatiya, 
is a slim volume, written in verse, covering astronomy and mathematics. 
The names of several Hindu mathematicians before this time are known, but 
nothing of their work has been preserved beyond a few fragments. In this 
respect, then, the position of the Aryabhatiya of Aryabhata in India is 
somewhat akin to that of the Elements of Euclid in Greece some eight 
centuries before. Both are summaries of earlier developments, compiled 
by a single author. There are, however, more striking differences than 
similarities between the two works. The Elements is a well-ordered synthesis 
of pure mathematics with a high degree of abstraction, a clear logical struc- 
ture, and an obvious pedagogical inclination; the Aryabhatiya is a brief 
descriptive work, in 123 metrical stanzas, intended to supplement rules of 
calculation used in astronomy and mensurational mathematics, with no 
feeling for logic or deductive methodology. About a third of the work is on 
ganitapada or mathematics. This section opens with the names of the powers 
often up to the tenth place and then proceeds to give instructions for square 
and cube roots of integers. Rules of mensuration follow, about half of which 
are erroneous. The area of a triangle is correctly given as half the product 
of the base and altitude, but the volume of a pyramid also is taken to be half 
the product of the base and altitude. 17 The area of a circle is found correctly 
as the product of the circumference and half the diameter, but the volume 
of a sphere is incorrectly stated to be the product of the area of a great circle 

17 The Aryabhatiya of Aryabhata, trans, by W. E. Clark (1930), p. 26. 


and the square root of this area. Again, in the calculation of areas of quadri- 
laterals, correct and incorrect rules appear side by side. The area of a trapezoid 
is expressed as half the sum of the parallel sides multiplied by the perpendicu- 
lar between them ; but then follows the incomprehensible assertion that the 
area of any plane figure is found by determining two sides and multiplying 
them. One statement in the Aryabhatiya to which Hindu scholars have 
pointed with pride is as follows: 18 

Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the 
circumference of a circle of which the diameter is 20,000. 

Here we see the equivalent of 3.1416 for n, but it should be recalled that this 
is essentially the value Ptolemy had used. The likelihood that Aryabhata 
here was influenced by Greek predecessors is strengthened by his adoption 
of the myriad, 10,000, as the number of units in the radius. 

A typical portion of the Aryabhatiya is that involving arithmetic pro- 
gressions, which contains arbitrary rules for finding the sum of the terms in a 
progression and for determining the number of terms in a progression when 
given the first term, the common difference, and the sum of the terms. The 
first rule had long been known by earlier writers. The second is a curiously 
complicated bit of exposition : 

Multiply the sum of the progression by eight times the common difference, add 
the square of the difference between twice the first term, and the common differ- 
ence, take the square root of this, subtract twice the first term, divide by the 
common difference, add one, divide by two. The result will be the number of 

Here, as elsewhere in the Aryabhatiya, no motivation or justification is given 
for the rule. It was probably arrived at through a solution of a quadratic 
equation, knowledge of which might have come from Mesopotamia or 
Greece. Following some complicated problems on compound interest (that 
is, geometrical progressions), the author turns, in flowery language, to the 
very elementary problem of finding the fourth term in a simple proportion : 

In the rule of three multiply the fruit by the desire and divide by the measure. 
The result will be the fruit of the desire. 

This, of course, is the familiar rule that if a/b - c/x, then x = bc/a, where 
a is the "measure," b the "fruit," c the "desire," and x the "fruit of the desire." 
The work of Aryabhata is indeed a potpourri of the simple and the complex, 
the correct and the incorrect. The Arabic scholar al-Biruni, half a millennium 
later, characterized Hindu mathematics as a mixture of common pebbles 
and costly crystals, a description quite appropriate to Aryabhatiya. 

18 Aryabhatiya, p. 28. Translations, here and below, are from the Clark edition cited in 
footnote 17. 


1 4 The second half of the Aryabhatiya is on the reckoning of time and on 
spherical trigonometry; here we note an element that was to leave a per- 
manent impress on the mathematics of later generations — the decimal 
place-value numeration. It is not known just how Aryabhata carried out his 
calculations, but his phrase "from place to place each is ten times the pre- 
ceding" is an indication that the application of the principle of position was 
in his mind. "Local value" had been an essential part of Babylonian numera- 
tion, and perhaps the Hindus were becoming aware of its applicability to 
the decimal notation for integers in use in India. The development of numeri- 
cal notations in India seems to have followed about the same pattern found 
in Greece. Inscriptions from the earliest period at Mohenjo Daro show at 
first simple vertical strokes, arranged into groups, but by the time of Asoka 
(third century B.C.) a system resembling the Herodianic was in use. In the 
newer scheme the repetitive principle was continued, but new symbols of 
higher order were adopted for four, ten, twenty, and one hundred. This 
so-called Karosthi script then gradually gave way to another notation, 
known as the Brahmi characters, which resembled the alphabetic cipheriza- 
tion in the Greek Ionian system ; one wonders if it was only a coincidence 
that the change in India took place shortly after the period when in Greece 
the Herodianic numerals were displaced by the Ionian. 

From the Brahmi ciphered numerals to our present-day notation for 
integers two short steps are needed. The first is a recognition that, through 
the use of the positional principle, the ciphers for the first nine units can 
serve also as the ciphers for the corresponding multiples of ten, or equally 
well as ciphers for the corresponding multiples of any power of ten. This 
recognition would make superfluous all of the Brahmi ciphers beyond the 
first nine. It is not known when the reduction to nine ciphers occurred, and 
it is likely that the transition to the more economical notation was made only 
gradually. It appears from extant evidence that the change took place in 
India, but the source of the inspiration for the change is uncertain. Possibly 
the so-called Hindu numerals were the result of internal development alone ; 
perhaps they developed first along the western interface between India and 
Persia, where remembrance of the Babylonian positional notation may have 
led to modification of the Brahmi system. It is possible that the newer system 
arose along the eastern interface with China where the pseudopositional rod 
numerals may have suggested the reduction to nine ciphers. There is also a 
theory that this reduction may first have been made at Alexandria within 
the Greek alphabetic system and that subsequently the idea spread to 
India. 19 During the later Alexandrian period the earlier Greek habit of 
writing common fractions with the numerator beneath the denominator was 

19 See Harriet P. Lattin. "The Origin of Our Present System of Notation According to the 
Theories of Nicholas Bubnov," Isis, 19 (1933), 181-194. 


reversed, and it is this form that was adopted by the Hindus, without the 
bar between the two. Unfortunately, the Hindus did not apply the new 
numeration for integers to the realm of decimal fractions ; hence the chief 
potential advantage of the change from Ionian notation was lost. 

The earliest specific reference to the Hindu numerals is found in 662 in 
the writings of Severus Sebokt, a Syrian bishop. After Justinian closed the 
Athenian philosophical schools some of the scholars moved to Syria, where 
they established centers of Greek learning. Sebokt evidently felt piqued by 
the disdain for non-Greek learning expressed by some associates ; hence he 
found it expedient to remind those who spoke Greek that "there are also 
others who know something." To illustrate his point he called attention to 
the Hindus and their "subtle discoveries in astronomy," especially "their 
valuable methods of calculation, and their computing that surpasses descrip- 
tion. I wish only to say that this computation is done by means of nine 
signs." 20 That the numerals had been in use for some time is indicated by 
the fact that the first Indian occurrence is on a plate of the year 595, where 
the date 346 is written in decimal place- value notation. 21 

It should be remarked that the reference to nine symbols, rather than ten, 1 5 
implies that the Hindus evidently had not yet taken the second step in the 
transition to the modern system of numeration — the introduction of a 
notation for a missing position, that is, a zero symbol. The history of mathe- 
matics holds many anomalies, and not the least of these is the fact that "the 
earliest undoubted occurrence of a zero in India is in an inscription of 876" 22 
— that is, more than two centuries after the first reference to the other nine 
numerals. It is not even established that the number zero (as distinct from a 
symbol for an empty position) arose in conjunction with the other nine 
Hindu numerals. It is quite possible that zero originated in the Greek world, 
perhaps at Alexandria, and that it was transmitted to India after the decimal 
positional system had been established there. 23 

The history of the zero placeholder in positional notation is further 
complicated by the fact that the concept appeared independently, well before 
the days of Columbus, in the western, as well as the eastern hemisphere. 
The Mayas of Yucatan, in their representation of time intervals between 
dates in their calendar, used a place-value numeration, generally with twenty 
as the primary base and with five as an auxiliary (corresponding to the 
Babylonian use of sixty and ten respectively). Units were represented by dots 
and fives by horizontal bars, so that the number seventeen, for example, 

20 Quoted from D. E. Smith. History of Mathematics, I, 167. 

2 ' See D. J. Struik. A Concise History of Mathematics, 3rd ed. (New York : Dover, 1967), p. 71. 

22 Smith, History of Mathematics, II, 69. 

23 See, for example, B. L. van der Waerden, Science Awakening (1961), pp. 56-58. 



would appear as = [that is. as 3(5) + 2]. A vertical positional arrangement 

was used, with the larger units of time above ; hence the notation ^ denoted 

352 [that is 17(20) + 12]. Because the system was primarily for counting 
days within a calendar having 360 days in a year, the third position usually 
did not represent multiples of (20) (20), as in a pure vigesimal system, but 
(18)(20). However, beyond this point the base twenty again prevailed. 
Within this positional notation the Mayas indicated missing positions 

From the Dresden Codex, of the Maya, displaying numbers. The second column on the 
left, from abovedown, displays the numbers 9, 9. 16, 0, 0, which standfor 9 x 144,000 + 
9 x 7200 + 16 x 360 + + = 1,366,560, In the third column are the numerals 
9, 9, 9, 16, 0, representing 1,364,360. The original appears in black and red colors, (Taken 
from M or ley. An Introduction to the Study of the Maya Hieroglyphs, p. 266.) 


through the use of a symbol, appearing in variant forms, somewhat resem- 
bling a half-open eye. In their scheme, then, the notation S, denoted 
17(2018-20) + 0(18-20) + 13(20) + 0. J 

With the introduction, in the Hindu notation, of the tenth numeral, a 
round goose egg for zero, the modern system of numeration for integers was 
completed. Although the Medieval Hindu forms of the ten numerals differ 
considerably from those in use today, the principles of the system were 
established. The new numeration, which we generally call the Hindu system, 
is merely a new combination of three basic principles, all of ancient origin: 
(1) a decimal base; (2) a positional notation; and (3) a ciphered form for 
each of the ten numerals. Not one of these three was due originally to the 
Hindus, but it presumably is due to them that the three were first linked to 
form the modern system of numeration. 

It may be well to say a word about the form of the Hindu symbol for zero— 
which is also ours. It once was assumed that the round form stemmed 
originally from the Greek letter omicron, initial letter in the word "ouden" 
or empty, but recent investigations seem to belie such an origin. Although the 
symbol for an empty position in some of the extant versions of Ptolemy's 
tables of chords does seem to resemble an omicron, the early zero symbols 
in Greek sexagesimal fractions are round forms variously embellished and 
differing markedly from a simple goose egg. Moreover, when in the fifteenth 
century in the Byzantine Empire a decimal positional system was fashioned 
out of the old alphabetic numerals by dropping the last eighteen letters and 
adding a zero symbol to the first nine letters, the zero sign took forms quite 
unlike an omicron. 24 Sometimes it resembled an inverted form of our small 
letter h, sometimes it appeared as a dot. 

The development of our system of notation for integers was one of the 1 6 
two most influential contributions of India to the history of mathematics. 
The other was the introduction of an equivalent of the sine function in 
trigonometry to replace the Greek tables of chords. The earliest tables of 
the sine relationship that have survived are those in the Siddhantas and the 
Aryabhatiya. Here the sines of angles up to 90° are given for twenty-four 
equal intervals of 3f° each. In order to express arc length and sine length in 
terms of the same unit, the radius was taken as 3438 and the circumference 
as 360 • 60 = 21,600. This implies a value of n agreeing to four significant 
figures with that of Ptolemy. In another connection Aryabhata used the value 
J\Q for n, which appeared so frequently in India that it sometimes is known 
as the Hindu value. 

24 See O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. (Providence, R.I.: Brown 
University Press, 1957), p. 14. 


For the sine of 3|° the Siddhantas and the Aryabhatiya took the number 
of units in the arc— that is, 60 x 3| or 225. In modern language, the sine of a 
small angle is very nearly equal to the radian measure of the angle (which is 
virtually what the Hindus were using). For further items in the sine table the 
Hindus used a recursion formula which may be expressed as follows. If 
the nth sine in the sequence from n = 1 to n = 24 is designated as s„, and if 
the sum of the first n sines is S„, then s„ +1 = s„ + Si - SJs^ From this 
rule one easily deduces that sin 7^° = 449, sin lli° = 671, sin 15° = 890, 
and so on up to sin 90° = 3438— the values listed in the table in the Sid- 
dhantas and the Aryabhatiya. Moreover, the table also includes values for 
what we call the versed sine of the angle— that is, 1 - cos 6 in modern 
trigonometry or 3438 (1 - cos 9) in Hindu trigonometry— from vers 
3|° = 7 to vers 90° = 3438. If we divide the items in the table by 3438, the 
results are found to be in close agreement with the corresponding values in 
modern trigonometric tables. 25 

1 7 Hindu trigonometry evidently was a useful and accurate tool in astronomy. 
How the Hindus arrived at results such as the recursion formula is uncertain, 
but it has been suggested 26 that an intuitive approach to difference equations 
and interpolation may have prompted such rules. Indian mathematics 
frequently is described as "intuitive," in contrast to the stern rationalism of 
Greek geometry. Although in Hindu trigonometry there is evidence of 
Greek influence, the Indians seem to have had no occasion to borrow Greek 
geometry, concerned as they were with simple mensurational rules. Of the 
classical geometrical problems, or the study of curves other than the circle, 
there is little evidence in India, and even the conic sections seem to have been 
overlooked by the Hindus, as by the Chinese. Hindu mathematicians were 
fascinated instead by work with numbers, whether it involved the ordinary 
arithmetic operations or the solution of determinate or indeterminate 
equations. Addition and multiplication were carried out in India much as 
they are by us today, except that they seem at first to have preferred to write 
numbers with the smaller units on the left, hence to work from left to right, 
using small blackboards with white removable paint or a board covered 
with sand or flour. Among the devices used for multiplications was one that 
is known under various names : lattice multiplication, gelosia multiplication, 
or cell or grating or quadrilateral multiplication. The scheme behind this is 
readily recognized in two examples. In the first example (Fig. 12.3) the 
number 456 is multiplied by 34. The multiplicand has been written above the 
lattice and the multiplier appears to the left, with the partial products 

25 The table from the Surya Siddhanta is reproduced in Smith, History of Mathematics, 11. 

26 E. S. Kennedy in the article "Trigonometry," to appear in the Yearbook on History of 
Mathematics of the National Council of Teachers of Mathematics. 



occupying the square cells. Digits in the diagonal rows are added, and the 
product 1 5,504 is read off at the bottom and the right. To indicate that other 
arrangements are possible, a second example is given in Fig. 12.4, in which 

4 5 6 


\ 6 


\ 4 


2 \ 

2 \ 

\ 2 

\ 5 

\ 8 


1 \ 



1 5 5 

FIG. 12.3 

5 3 7 

1 / 
/ ° 

/ 6 

1 / 
/ 4 

2 / 
/ ° 

1 / 
/ 2 

2 / 
/ 8 

FIG. 12.4 

the multiplicand 537 is placed at the top and the multiplier 24 is on the right, 
the product 12,888 appearing to the left and along the bottom. Still other 
modifications are easily devised. In fundamental principle gelosia multiplica- 
tion is of course the same as our own, the cell arrangement being merely a 
convenient device for relieving the mental concentration called for in 
"carrying over" from place to place the tens arising in the partial products. 
The only "carrying" required in lattice multiplication is in the final additions 
along the diagonals. 

It is not known when or where gelosia multiplication arose, but India 
seems to be the most likely source. It was used there at least by the twelfth 
century, and from India it seems to have been carried to China and Arabia. 
From the Arabs it passed over to Italy in the fourteenth and fifteenth centur- 
ies, where the name gelosia was attached to it because of the resemblance to 
gratings placed before windows in Venice and elsewhere. (The current word 
jalousie seems to stem from the Italian gelosia and is used for Venetian blinds 
in France, Germany, Holland, and Russia.) The Arabs (and through them the 
later Europeans) appear to have adopted most of their arithmetic devices 
from the Hindus, and so it is likely that the pattern of long division known 
as the "scratch method" or the "galley method" (from its resemblance to a 
boat) came ilso from India. To illustrate the method, let it be required to 
divide 44,977 by 382. In Fig. 12.5 we give the modern method, in Fig. 12.6 the 
galley method. 27 The latter parallels the former closely except that the 


27 For further description of the innumerable computational devices that have been used, 
see F. A. Yeldham, The Story of Reckoning in the Middle Ages (1926). 



Galley division, sixteenth century. From an unpublished manuscript of a Venetian monk. 

The title of the work is "Opus Arithmetic^ D. Honorati veneti monachj coenobij S. 

Lauretig." From Mr. Plimpton's library. 

dividend appears in the middle, for subtractions are performed by canceling 
digits and placing differences above rather than below the minuends. Hence 
the remainder, 283, appears above and to the right, rather than below. 








FIG. 12.5 


2 3 

3 9 8 

; 7 ? 3 

4 4 ? 7 7 

3 8 2 2 4 

3 8 7 


FIG. 12.6 



The process in Fig. 12.6 is easy to follow if we note that the digits in a given 
subtrahend, such as 2674, or in a given difference, such as 2957, are not 
necessarily all in the same row and that subtrahends are written below the 
middle and differences above the middle. Position in a column is significant, 
but not position in a row. The determination of roots of numbers probably 
followed a somewhat similar "galley" pattern, coupled in the later years 
with the binomial theorem in "Pascal triangle" form; but Hindu writers 
did not provide explanations for their calculations or proofs for their state- 
ments. It is possible that Babylonian and Chinese influences played a role 
in the problem of evolution or root extraction. It is often said that the 
"proof by nines," or the "casting out of nines," is a Hindu invention, but it 
appears that the Greeks knew earlier of this property, without using it 
extensively, and that the method came into common use only with the Arabs 
of the eleventh century. 

The last few paragraphs may leave the unwarranted impression that there 1 9 
was a uniformity in Hindu mathematics, for frequently we have localized 
developments as merely "of Indian origin," without specifying the period. 
The trouble is that there is a high degree of uncertainty in Hindu chronology. 
Material in the important Bakshali manuscript, containing an anonymous 
arithmetic, is supposed by some to date from the third or fourth century, 
by others from the sixth century, by others from the eighth or ninth century 
or later ; and there is a suggestion that it may not even be of Hindu origin. 28 
We have placed the work of Aryabhata around the year 500, but the date is 
doubtful since there were two mathematicians named Aryabhata and we 
cannot with certainty ascribe results to our Aryabhata, the elder. Hindu 
mathematics presents more historical problems than does Greek mathe- 
matics, for Indian authors referred to predecessors infrequently, and they 
exhibited surprising independence in mathematical approach. Thus it is 
that Brahmagupta (ft 628), who lived in Central India somewhat more than a 
century after Aryabhata, has little in common with his predecessor, who had 
lived in eastern India. Brahmagupta mentions two values of n — the "practical 
value" 3 and the "neat value" N /l0 — but not the more accurate value of 
Aryabhata ; in the trigonometry of his best-known work, the Brahmasphuta 
Siddhanta, he adopted a radius of 3270 instead of Aryabhata's 3438. In one 
respect he does resemble his predecessor — in the juxtaposition of good and 
bad results. He found the "gross" area of an isosceles triangle by multiplying 
half the base by one of the equal sides ; for the scalene triangle with base 
fourteen and sides thirteen and fifteen he found the "gross area" by multiply- 
ing half the base by the arithmetic mean of the other sides. In finding the 

28 See Florian Cajori, A History of Mathematics (1919), pp. 84-85; Smith. History of Mathe- 
matics, 1, 164; Hofmann, Geschichte iter Mathemalik, 1, 59. 


"exact" area he utilized the Archimedean-Heronian formula. For the radius 
of the circle circumscribed about a triangle he gave the equivalent of the cor- 
rect trigonometric result 2R = a/sin A = b/sin B = c/sin C, but this of course 
is only a reformulation of a result known to Ptolemy in the language of chords. 
Perhaps the most beautiful result in Brahmagupta's work is the generalization 
of "He ron's" formula in finding the area of a quadrilateral. This formula — 
K = y/(s - a)(s - b)(s - c)(s — d), where a, b, c, d are the sides and s is the 
semiperimeter — still bears his name; but the glory of his achievement is 
dimmed by failure to remark that the formula is correct only in the case of a 
cy clic quadrilateral. 29 (The correct formula fo r an arbitrary quadrilateral is 
N /(s - a)(s - b)(s — c)(s — d) — abed cos 2 a. where a is half the sum of two 
opposite angles.) As a rule for the "gross" area of a quadrilateral Brahma- 
gupta gave the pre-Hellenic formula, the product of the arithmetic means of 
the opposite sides. For the quadrilateral with sides a = 25, b = 25, c = 25, 
d = 39, for example, he found a "gross" area of 800. 

20 Brahmagupta's contributions to algebra are of a higher order than are 
his rules of mensuration, for here we find general solutions of quadratic 
equations, including two roots even in cases in which one of them is negative. 
The systematized arithmetic of negative numbers and zero is, in fact, first 
found in his work. The equivalents of rules on negative magnitudes were 
known through the Greek geometrical theorems on subtraction, such as 
(a - b)(c - d) = ac + bd - ad — be, but the Hindus converted these into 
numerical rules on positive and negative numbers. Moreover, although the 
Greeks had a concept of nothingness, they never interpreted this as a number, 
as did the Hindus. However, here again Brahmagupta spoiled matters some- 
what by asserting that 0^-0=0, and on the touchy matter of a -=- 0, for 
a # 0, he did not commit himself: 

Positive divided by positive, or negative by negative, is affirmative. Cipher divided 
by cipher is naught. Positive divided by negative is negative. Negative divided by 
affirmative is negative. Positive or negative divided by cipher is a fraction with that for 
denominator. 30 

It should be mentioned also that the Hindus, unlike the Greeks, regarded 
irrational roots of numbers as numbers. This was of enormous help in 
algebra, and Indian mathematicians have been much praised for taking this 
step ; but one must remember that the Hindu contribution in this case was 
the result of logical innocence rather than of mathematical insight. We have 

29 A proof of the formula can be found in R. A. Johnson, Modern Geometry (New York : 
Houghton Mifflin, 1929). pp. 81-82. 

30 See H. T. Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of 
Brahmagupta and Bhaskara (1817). 


seen the lack of nice distinction on the part of Hindu mathematicians between 
exact and inexact results, and it was only natural that they should not have 
taken seriously the difference between commensurable and incommensurable 
magnitudes. For them there was no impediment to the acceptance of irrational 
numbers, and later generations followed their lead uncritically until in the 
nineteenth century mathematicians established the real number system on a 
sound basis. 

Indian mathematics was, as we have said, a mixture of good and bad. But 
some of the good was superlatively good, and here Brahmagupta deserves 
high praise. Hindu algebra is especially noteworthy in its development of 
indeterminate analysis, to which Brahmagupta made several contributions. 
For one thing, in his work we find a rule for the formation of Pythagorean 
triads expressed in the form m, \(m 2 /n - n\ ^(m 2 /n + n) ; but this is only a 
modified form of the old Babylonian rule, with which he may have become 
familiar. Brahmagupta's area formula for a quadrilateral, mentioned above, 
was used by him in conjunction with the formulas 

J(ab + cd)(ac + bd)/(ad + be) and ^/{ac + bd)(ad + bc)/(ab + cd) 

for the diagonals, 31 to find quadrilaterals whose sides, diagonals, and areas 
are all rational. Among them was the quadrilateral with sides a = 52, b = 25, 
c = 39, d = 60, and diagonals 63 and 56. Brahmagupta gave the "gross" 
area as 1933|, despite the fact that his formula provides the exact area, 1764, 
in this case. 

Like many of his countrymen, Brahmagupta evidently loved mathematics 21 
for its own sake, for no practical-minded engineer would raise questions 
such as those Brahmagupta asked about quadrilaterals. One admires his 
mathematical attitude even more when one finds that apparently he was the 
first one to give a general solution of the linear Diophantine equation 
ax + by = c, where a, b. and c are integers. For this equation to have 
integral solutions, the greatest common divisor of a and b must divide c ; 
and Brahmagupta knew that if a and b are relatively prime, all solutions of 
the equation are given by x = p + mb, y = q - ma, where m is an arbitrary 
integer. He suggested also the Diophantine quadratic equation x 2 = 1 + py 2 , 
named mistakenly for John Pell (1611-1685), but first appearing in the 
Archimedean cattle problem. The Pell equation was solved for some cases 
by Brahmagupta's countryman, Bhaskara (1114-ca. 1185). 

It is greatly to the credit of Brahmagupta that he gave all integral solutions 
of the linear Diophantine equation, whereas Diophantus himself had been 

31 For indications of a proof of these formulas see Howard Eves. An Introduction to the History 
of Mathematics (1964). pp. 202-203. 


satisfied to give one particular solution of an indeterminate equation. Inas- 
much as Brahmagupta used some of the same examples as Diophantus, we 
see again the likelihood of Greek influence in India — or the possibility that 
they both made use of a common source, possibly from Babylonia. It is 
interesting to note also that the algebra of Brahmagupta, like that of Dio- 
phantus, was syncopated. Addition was indicated by juxtaposition, sub- 
traction by placing a dot over the subtrahend, and division by placing the 
divisor below the dividend, as in our fractional notation but without the 
bar. The operations of multiplication and evolution (the taking of roots), as 
well as unknown quantities, were represented by abbreviations of appropriate 

22 India produced a number of later Medieval mathematicians, but we shall 
describe the work of only one of these— Bhaskara (1 1 14-ca. 1185), the leading 
mathematician of the twelfth century. It was he who filled some of the gaps 
in Brahmagupta's work, as by giving a general solution of the Pell equation 
and by considering the problem of division by zero. Aristotle once had 
remarked that there is no ratio by which a number such as four exceeds the 
number zero; 32 but the arithmetic of zero had not been part of Greek 
mathematics, and Brahmagupta had been noncommittal on the division of a 
number other than zero by the number zero. It is therefore in Bhaskara's 
Vija-Ganita that we find the first statement that such a quotient is infinite. 

Statement: Dividend 3. Divisor 0. Quotient the fraction 3/0. This fraction 
of which the denominator is cipher, is termed an infinite quantity. In this 
quantity consisting of that which has cipher for a divisor, there is no altera- 
tion, though many be inserted or extracted ; as no change takes place in the 
infinite and immutable God. 

This statement sounds promising, but lack of clear understanding of the 
situation is suggested by Bhaskara's further assertion that a/0-0 = a. 

Bhaskara was the last significant Medieval mathematician from India, and 
his work represents the culmination of earlier Hindu contributions. In his 
best known treatise, the Lilavati, he compiled problems from Brahmagupta 
and others, adding new observations of his own. The very title of this book 
may be taken to indicate the uneven quality of Hindu thought, for the name 
in the title is that of Bhaskara's daughter who, according to legend, lost the 
opportunity to marry because of her father's confidence in his astrological 
predictions. Bhaskara had calculated that his daughter might propitiously 
marry only at one particular hour on a given day. On what was to have been 
her wedding day the eager girl was bending over the water clock, as the hour 

32 See C. B. Boyer, "An Early Reference to Division by Zero." American Mathematical 
Monthly, 50 (1943), 487-491. 


for the marriage approached, when a pearl from her headdress fell, quite 
unnoticed, and stopped the outflow of water. Before the mishap was noted, 
the propitious hour had passed. To console the unhappy girl, the father gave 
her name to the book we are describing. 

The Lilavati, like the Vija-Ganita, contains numerous problems dealing 23 
with favorite Hindu topics : linear and quadratic equations, both determinate 
and indeterminate, simple mensuration, arithmetic and geometric pro- 
gressions, surds, Pythagorean triads, and others. The "broken bamboo" 
problem, popular in China (and included also by Brahmagupta), appears in 
the following form : If a bamboo 32 cubits high is broken by the wind so that 
the tip meets the ground 16 cubits from the base, at what height above the 
ground was it broken? Also making use of the Pythagorean theorem is the 
following problem : A peacock is perched atop a pillar at the base of which 
is a snake's hole. Seeing the snake at a distance from the pillar which is three 
times the height of the pillar, the peacock pounced upon the snake in a 
straight line before it could reach its hole. If the peacock and the snake had 
gone equal distances, how many cubits from the hole did they meet? 

These two problems illustrate well the heterogeneous nature of the Lilavati, 
for despite their apparent similarity and the fact that only a single answer is 
required, one of the problems is determinate and the other is indeterminate. 
In treating of the circle and the sphere the Lilavati fails also to distinguish 
between exact and approximate statements. The area of the circle is correctly 
given as one-quarter the circumference multiplied by the diameter and the 
volume of the sphere as one-sixth the product of the surface area and the 
diameter, but for the ratio of circumference to diameter in a circle Bhaskara 
suggests either 3927 to 1 250 or the "gross" value 22/7. The former is equivalent 
to the ratio mentioned, but not used, by Aryabhata. There is no hint in 
Bhaskara or other Hindu writers that they were aware that all ratios that 
had been proposed were approximations only. However, Bhaskara severely 
condemns his predecessors for using the formulas of Brahmagupta for the 
area and diagonals of a general quadrilateral, because he saw that a quadri- 
lateral is not uniquely determined by its sides. Evidently he did not realize 
that the formulas are indeed exact for all cyclic quadrilaterals. 

Many of Bhaskara's problems in the Lilavati and the Vija-Ganita evidently 
were derived from earlier Hindu sources; hence it is no surprise to note that 
the author is at his best in dealing with indeterminate analysis. In connection 
with the Pell equation, x 2 = 1 + py 2 , proposed earlier by Brahmagupta, 
Bhaskara gave particular solutions for the five cases p = 8, 1 1, 32, 61, and 67. 
For x 2 = 1 + 61y 2 , for example, he gave the solution x = 1,776,319,049 
and y = 22,615,390. This is an impressive feat in calculation, and its verifica- 
tion alone will tax the efforts of the reader. 


Bhaskara's books are replete with other instances of Diophantine 
problems. 33 

24 Bhaskara died toward the end of the twelfth century, and for several 
hundred years there were few mathematicians in India of comparable stature. 
It is of interest to note, nevertheless, that Srinivasa Ramanujan (1887-1920), 
the twentieth-century Hindu genius, had the same uncanny manipulative 
ability in arithmetic and algebra that is found in Bhaskara. The British 
mathematician G. H. Hardy once visited Ramanujan in a hospital at Putney 
and mentioned to his ill friend that he had arrived in a taxi with the dull 
number 1729, whereupon Ramanujan without hesitation pointed out that 
this number is indeed interesting, for it is the least integer that can be repre- 
sented in two different ways as the sum of two cubes — l 3 + 12 3 = 1729 = 
9 3 + 10 3 . In Ramanujan's work we note also the disorganized character, 
the strength of intuitive reasoning, and the disregard for geometry that stood 
out so clearly in his predecessors. Although in Ramanujan these character- 
istics had perhaps developed largely because he was self-taught, we cannot 
help but see how strikingly different the development of mathematics in India 
has been from that in Greece. Even when the Hindus borrowed from their 
neighbors, they fashioned the material in their own peculiar manner. 
Although in attitude and interests they had more in common with the 
Chinese, they did not share the latter's fascination with accurate approxima- 
tions, such as led to Horner's method. And although they shared with the 
Mesopotamians a preponderately algebraic view, they tended to avoid 
sexagesimal numeration. In short, the eclectic Hindu mathematicians adopted 
and developed only such aspects as appealed to them. In one respect it was 
unfortunate that their first love should have been theory of numbers in 
general, and indeterminate analysis in particular, for it was not from these 
aspects that later developments in mathematics grew. Analytic geometry 
and calculus had Greek rather than Indian roots, and European algebra 
came from the Islamic countries rather than India. Nevertheless, in modern 
mathematics there are at least two reminders that mathematics owes its 
development to India, as well as to many other lands. The trigonometry 
of the sine function came presumably from India ; our own system of numera- 
tion for integers is appropriately called the Hindu-Arabic system to indicate 
its probable origin in India and its transmission through Arabia. 


Boyer, C. B., "Fundamental Steps in the Development of Numeration," his, 35 (1944), 

Cajori, Florian, A History of Mathematics, 2nd ed. (New York: Macmillan, 1919). 

33 An exceptionally full account of Bhaskara's work is found in J. F. Scott. A History of 
Mathematics (1958). See also Colebrooke, op. cit. 


Clark, W. E., ed., The Aryabhatiya of Aryabhata (Chicago: Open Court, 1930). 
Colebrooke, H. T., Algebra, with Arithmetic and Mensuration, from the Sanscrit of 

Brahmagupta and Bhaskara (London, 1817). 
Datta, B., and A. N. Singh, History of Hindu Mathematics (Lahore, 1935-1938, 2 vols. ; 

Bombay : Asia Publishing House, 1962). 
Eves, Howard, An Introduction to the History of Mathematics, 2nd ed. (New York: 

Holt, 1964). 
Goldschmidt, Victor, Die Entstehung unserer Ziffern (Heidelberg: C. Winter, 1932). 
Hill, G. F., The Development of Arabic Numerals in Europe (Oxford, 1915). 
Ho Peng-Yoke, articles on Liu Hui, Chu Shih-chieh, Ch'in Chiu-shao, Li Chih, and 

Yang Hui, in Dictionary of Scientific Biography (New York : Scribner's) in press. 
Juschkewitsch, A. P., Geschichte der Mathematik im Mittelalter (Leipzig: Teubner, 

Kaye, G. R, "Indian Mathematics," Isis, 2 (1914), 326-356. 
Lattin, Harriet P., "The Origin of Our Present System of Notation According to the 

Theories of Nicholas Bubnov," Isis, 19 (1933), 181-194. 
Loeffler, Eugen, Ziffern und Ziffernsysteme (Leipzig and Berlin : Teubner, 1912). 
Menninger, Karl, Zahlwort und Ziffer (2nd ed., Gottingen : Vandenhoeck and Ruprecht, 

1957-1958, 2 vols.). 
Mikami, Yoshio, The Development of Mathematics in China and Japan (1913 ; reprinted, 

New York : Chelsea, n.d.). 
Morley, S. G., An Introduction to the Study of Maya Hieroglyphics (Washington : 

Carnegie Institution, 1915). 
Needham, Joseph, Science and Civilization in China (Cambridge : Cambridge University 

Press, 1959), Vol. III. 
Rajagopal, C. T., and T. V. Vedamurthi Aiyar, "On the Hindu Proof of Gregory's 

Series," Scripta Mathematica, XVII (1951) 65-74. See also XV (1949) 201-209 and 

XVIII (1952) 25-30. 
Sarton, George, An Introduction to the History of Science (Baltimore : Carnegie Institu- 
tion of Washington, 1927-1948, 3 vols, in 5). 
Scott, J. F., A History of Mathematics (London: Taylor & Francis, 1958). 
Smith, D. E., History of Mathematics (Boston: Ginn, 1923-1925, 2 vols., paperback 

reprint, New York: Dover, 1958). 
Smith, D. E. and L. C. Karpinski: The Hindu-Arabic Numerals (Boston: Ginn, 1911). 
Struik, D. J., "On Ancient Chinese Mathematics," The Mathematics Teacher, 56 (1963), 

Winter, H. J. J., Eastern Science (London: John Murray, 1952). 
Yeldham, F. A., The Story of Reckoning in the Middle Ages (London : G. G. Harrap, 



1. Compare Hindu and Chinese mathematics with respect to favorite topics and level of 

2. Which had the greater influence on modern thought, Chinese or Hindu mathematics? 
Explain clearly. 


3. What evidences are there of Greek influence in Hindu mathematics? Are there evidences of 
Hindu influence in Greece? Explain. 

4. Is it likely that the ancient Chinese and Babylonian mathematicians borrowed from each 
other? Explain. 

5. How can one account for the Chinese and the Hindu indifference toward conic sections? 

6. Describe some respects in which Hindu algebra differed markedly from Greek algebra. 

7. Solve the system 

4x + y + z = 40 

2x + 3y + z = 30 

x + y + 2z = 20 

by the Chinese matrix method. 

8. Write the number 7,834,679 in Chinese rod numerals and in Mayan positional notation. 

9. Using the method of Ch'in Chiu-shao, find the square root of 29,584. 

10. Write in the notation of Chu Shih-chieh the coefficients in the expansion of the ninth power 
of a binomial. 

1 1 . Justify Aryabhata's rule for finding the number of terms in an arithmetic progression, given 
the first term, the common difference, and the sum of the terms. 

12. Find sin 15° by the Siddhanta recursion formula and compare this with the value found in 
modern tables. 

13. Use a gdosia pattern to find theproduct of 345 and 256. 

14. Divide 56,789 by 273, using the "galley" method. 

15. Check the multiplication in Exercise 1 3 by casting out nines in the multiplicand, the multi- 
plier, and the product. 

16. From Brahmagupta's formula for area deduce Heron's formula as a special case. 

17. Show that 21x + 14y = 3 has no solution in integers. 

18. From Brahmagupta's formulas for the diagonals of a (cyclic) quadrilateral deduce Ptolemy's 

19. Solve Bhaskara's broken-bamboo problem. 

20. Solve Bhaskara's peacock-and-the-snake problem. 

*21. Verify that Brahmagupta's quadrilateral with sides a = 52. h = 25. c = 39. d = 60 and 
diagonals e = 56 and/ = 63 is a cyclic quadrilateral. 

*22. Is it possible for Brahmagupta's quadrilateral with sides a = 25, b = 25, c = 25, d = 39 
to be cyclic? Explain. 

*23. Show that the formula of Liu Hui holds for the volume of the tetrahedron (0, 0, 0), (0, 0, a), 
(fc, 0, 0), (c, d, 0). Is the formula valid for all tetrahedra with a pair of opposite edges ortho- 
gonal? Explain. 


The Arabic Hegemony 

Ah, but my Computations, People say, Have squared 
the Year to human Compass, eh? If so, by striking from 
the Calendar Unborn To-morrow, and dead Yesterday. 
Omar Khayyam (Rubaiyat in the FitzGerald version) 

At the time that Brahmagupta was writing, the Sabean Empire of Arabia 
Felix had fallen and the peninsula was in a severe crisis. It was inhabited 
largely by desert nomads, known as Bedouins, who could neither read nor 
write ; among them was the prophet Mohammed, born at Mecca in about 
570. During his journeys Mohammed came in contact with Jews and Chris- 
tians, and the amalgam of religious feelings that were raised in his mind led 
him to regard himself as the apostle of God sent to lead his people. For some 
ten years he preached at Mecca, but in 622, faced by a plot on his life, he 
accepted an invitation to Medina. This "flight," known as the Hejira, marked 
the beginning of the Mohammedan era — one that was to exert a strong 
influence on the development of mathematics. Mohammed now became a 
military, as well as a religious leader. Ten years later he had established a 
Mohammedan state, with center at Mecca, within which Jews and Christians, 
being also monotheistic, were afforded protection and freedom of worship. 
In 632, while planning to move against the Byzantine Empire, Mohammed 
died at Medina. His sudden death in no way impeded the expansion of the 
Islamic state, for his followers overran neighboring territories with astonish- 
ing rapidity. Within a few years Damascus and Jerusalem and much of the 
Mesopotamian Valley fell to the conquerors ; by 641 Alexandria, which for 
many years had been the mathematical center of the world, was captured. 
There is a legend that the leader of the victorious troops, having asked what 
was to be done with the books in the library, was told to burn them ; for if 
they were in agreement with the Koran they were superfluous, if they were in 
disagreement they were worse than superfluous. However, stories that the 
baths were long heated by the fires of burning books undoubtedly are 
exaggerated. Following depredations by earlier military and religious 
fanatics, and long ages of sheer neglect, there probably were relatively few 
books in the library that once had been the greatest in the world. 



For more than a century the Arab conquerors fought among themselves 
and with their enemies, until by about 750 the warlike spirit subsided. By 
this time a schism had arisen between the western Arabs in Morocco and 
the eastern Arabs who, under the caliph al-Mansur, had established a new 
capital at Baghdad, a city that was shortly to become the new center for 
mathematics. However, the caliph at Baghdad could not command the 
allegiance even of all Moslems in the eastern half of his empire, although his 
name appeared on coins of the realm and was included in the prayers of his 
"subjects." The unity of the Arab world, in other words, was more economic 
and religious than it was political. Arabic was not necessarily the common 
language, although it was a kind of lingua franca for intellectuals. Hence it 
might be more appropriate to speak of the culture as Islamic, rather than 
Arabic, although we shall use the terms more or less interchangeably. 

During the first century of the Arabic conquests there had been political 
and intellectual confusion, and possibly this accounts for the difficulty in 
localizing the origin of the modern system of numeration. The Arabs were 
at first without intellectual interest, and they had little culture, beyond a 
language, to impose on the peoples they conquered. In this respect we see a 
repetition of the situation when Rome conquered Greece, of which it was 
said that, in a cultural sense, captive Greece took captive the captor Rome. 
By about 750 the Arabs were ready to have history repeat itself, for the con- 
querors became eager to absorb the learning of the civilizations they had 
overrun. By 766 we learn that an astronomical-mathematical work, known to 
the Arabs as the Sindhind was brought to Baghdad from India. It is generally 
thought that this was the Brahmasphuta Siddhanta, although it may have 
been the Surya Siddhanta. A few years later, perhaps about 775, this Siddhanta 
was translated into Arabic, and it was not long afterward (ca. 780) that 
Ptolemy's astrological Tetrabiblos was translated into Arabic from the 
Greek. Alchemy and astrology were among the first studies to appeal to 
the dawning intellectual interests of the conquerors. The "Arabic miracle" 
lies not so much in the rapidity with which the political empire rose as in the 
alacrity with which, their tastes once aroused, the Arabs absorbed the learn- 
ing of their neighbors. 

The first century of the Muslim empire had been devoid of scientific 
achievement. This period (from about 650 to 750) had been, in fact, perhaps 
the nadir in the development of mathematics, for the Arabs had not yet 
achieved intellectual drive, and concern for learning in other parts of the 
world had pretty much faded. Had it not been for the sudden cultural 
awakening in Islam during the second half of the eighth century, consid- 
erably more of ancient science and mathematics undoubtedly would have 
been lost. To Baghdad at that time were called scholars from Syria, Iran, and 


Mesopotamia, including Jews and Nestorian Christians ; under three great 
Abbasid patrons of learning — al-Mansur, Haroun al-Raschid, and al- 
Mamun — the city became a new Alexandria. During the reign of the second 
of these caliphs, familiar to us today through the Arabian Nights, part of 
Euclid was translated. It was during the caliphate of al-Mamun (809-833), 
however, that the Arabs fully indulged their passion for translation. The 
caliph is said to have had a dream in which Aristotle appeared, and as a 
consequence al-Mamun determined to have Arabic versions made of all the 
Greek works he could lay his hands on, including Ptolemy's Almagest and a 
complete version of Euclid's Elements. From the Byzantine Empire, with 
which the Arabs maintained an uneasy peace, Greek manuscripts were 
obtained through treaties. 

Al-Mamun established at Baghdad a "House of Wisdom" (Bait al-hikma) 
comparable to the ancient Museum at Alexandria. Among the faculty 
members was a mathematician and astronomer, Mohammed ibn-Musa 
al-Khowarizmi, whose name, like that of Euclid, later was to become a 
household word in Western Europe. 1 This scholar, who died sometime 
before 850, wrote more than half a dozen astronomical and mathematical 
works, of which the earliest were probably based on the Sindhind derived 
from India. Besides astronomical tables, and treatises on the astrolabe and 
the sundial, al-Khowarizmi wrote two books on arithmetic and algebra 
which played very important roles in the history of mathematics. One of 
these survives only in a unique copy of a Latin translation with the title 
De numero indorum ("Concerning the Hindu Art of Reckoning"), the original 
Arabic version having since been lost. In this work, based presumably on an 
Arabic translation of Brahmagupta, al-Khowarizmi gave so full an account 
of the Hindu numerals that he probably is responsible for the widespread 
but false impression that our system of numeration is Arabic in origin. 
Al-Khowarizmi made no claim to originality in connection with the system, 
the Hindu source of which he assumed as a matter of course ; but when 
subsequently Latin translations of his work appeared in Europe, careless 
readers began to attribute not only the book, but also the numeration, to 
the author. The new notation came to be known as that of al-Khowarizmi, 
or more carelessly, algorismi ; ultimately the scheme of numeration making 
use of the Hindu numerals came to be called simply algorism or algorithm, 
a word that, originally derived from the name al-Khowarizmi, now means, 
more generally, any peculiar rule of procedure or operation — such as the 
Euclidean method for finding the greatest common divisor. 

Through his arithmetic, al-Khowarizmi's name has become a common 3 
English word; through the title of his most important book, Al-jabr wal 

1 For two recent studies on the science of al-Khowarizmi, see Isis, 54 (1963), 97-119. 


muqabalah, he has supplied us with an even more popular household term. 
From this title has come the word algebra, for it is from this book that Europe 
later learned the branch of mathematics bearing this name. Diophantus 
sometimes is called "the father of algebra," but this title more appropriately 
belongs to al-Khowarizmi. It is true that in two respects the work of al- 
Khowarizmi represented a retrogression from that of Diophantus. First, it is 
on a far more elementary level than that found in the Diophantine problems 
and, second, the algebra of al-Khowarizmi is thoroughly rhetorical, with 
none of the syncopation found in the Greek Arithmetica or in Brahmagupta's 
work. Even numbers were written out in words rather than symbols ! It is 
quite unlikely that al-Khowarizmi knew of the work of Diophantus, but 
he must have been familiar with at least the astronomical and computational 
portions of Brahmagupta; yet neither al-Khowarizmi nor other Arabic 
scholars made use of syncopation or of negative numbers. Nevertheless, 
the Al-jabr comes closer to the elementary algebra of today than the works 
of either Diophantus or Brahmagupta, for the book is not concerned with 
difficult problems in indeterminate analysis but with a straightforward and 
elementary exposition of the solution of equations, especially of second 
degree. The Arabs in general loved a good clear argument from premise to 
conclusion, as well as systematic organization— respects in which neither 
Diophantus nor the Hindus excelled. The Hindus were strong'in association 
and analogy, in intuition and an aesthetic and imaginative flair, whereas 
the Arabs were more practical-minded and down-to-earth in their approach 
to mathematics. 

The Al-jabr has come down to us in two versions, Latin and Arabic, but 
in the Latin translation, Liber algebrae et almucabola, a considerable portion 
of the Arabic draft is missing. The Latin, for example, has no preface, perhaps 
because the author's preface in Arabic gave fulsome praise to Mohammed, 
the prophet, and to al-Mamun, "the Commander of the Faithful." Al- 
Khowarizmi wrote that the latter had encouraged him to 

compose a short work on Calculating by (the rules of) Completion and Reduction, 
confining it to what is easiest and most useful in arithmetic, such as men constantly 
require in cases of inheritance, legacies, partitions, law-suits, and trade, and in all their 
dealings with one another, or where the measuring of lands, the digging of canals, 
geometrical computation, and other objects of various sorts and kinds are concerned. 2 

It is not certain just what the terms "al-jabr" and "muqabalah" mean, but 
the usual interpretation is similar to that implied in the translation above. 
The word "al-jabr" presumably meant something like "restoration" or 
"completion" and seems to refer to the transposition of subtracted terms to 

2 See Robert of Chester's Latin Translation of the Algebra of al-Khowarizmi, ed. by L. C. 
Karpinski (1915), p. 46. Translations used by us are taken from this edition. 


the other side of an equation ; the word "muqabalah" is said to refer to 
"reduction" or "balancing"— that is, the cancellation of like terms on 
opposite sides of the equation. 3 Arabic influence in Spain long after the 
time of al-Khowarizmi is found in Don Quixote, where the word algebrista is 
used for a bone-setter, that is, a "restorer." 

The Latin translation of al-Khowarizmi's Algebra opens with a brief 
introductory statement of the positional principle for numbers and then 
proceeds to the solution, in six short chapters, of the six types of equations 
made up of the three kinds of quantities : roots, squares, and numbers 
(that is, x, x 2 , and numbers). As abu-Kamil Shoja ben Aslam. a slightly 
later textbook writer, expressed the situation. 

The first thing which is necessary for students in this science [algebra] is to understand 
the three species which are noted by Mohammed ibn Musa al-Khowarizmi in his book. 
These are roots, squares and numbers. 4 

Chapter I, in three short paragraphs, covers the case of squares equal to 
roots, expressed in modern notation as x 2 = 5x, x 2 /3 = 4x, and 5x 2 = lOx, 
giving the answers x = 5, x = 12. and x = 2 respectively. (The root x = 
was not recognized.) Chapter II covers the case of squares equal to numbers, 
and Chapter III solves the case of roots equal to numbers, again with three 
illustrations per chapter to cover the cases in which the coefficient of the 
variable term is equal to, more than, or less than one. Chapters IV, V, and VI 
are more interesting, for they cover in turn the three classical cases of three- 
term quadratic equations: (1) squares and roots equal to numbers, (2) 
squares and numbers equal to roots, and (3) roots and numbers equal to 
squares. The solutions are "cookbook" rules for "completing the square" 
applied to specific instances. Chapter IV, for example, includes the three 
illustrations x 2 + 10* = 39, 2x 2 + 10x = 48, and \x 2 + 5x = 28. In each 
case only the positive answer is given. In Chapter V only a single example — 
x 2 + 21 = lOx — is used, but both roots, 3 and 7, are given, corresponding 
to the rule x = 5 + V 2 ^ - 21. Al-Khowarizmi here calls attention to the 
fact that what we designate as the discriminant must be positive : 

You ought to understand also that when you take the half of the roots in this form of 
equation and then multiply the half by itself; if that which proceeds or results from the 
multiplication is less than the units above-mentioned as accompanying the square, 
you have an equation. 

3 It should be noted, however, that this interpretation has been questioned by Solomon 
Gandz, "The Origin of the Term 'Algebra - ," American Mathematical Monthly, 33 (1926), 
437-440. Gandz thinks that "jabr" was an Assyrian word for equation and that "al-muqabalah" 
is simply the Arabic translation of "al-jabr." 

4 L. C. Karpmski, "The Algebra of Abu Kamil," American Mathematical Monthly, 21 (1914) 


In Chapter VI the author again uses only a single example — 3x + 4 = x 2 — 
for whenever the coefficient of x 2 is not unity, the author reminds one to 
divide first by this coefficient (as in Chapter IV). Once more the steps in 
completing the square are meticulously indicated, without justificati on, the 
procedure being equivalent to the solution x = \\ + s/ilj) 2 + 4 - Again 
only one root is given, for the other is negative. 

The six cases of equations given above exhaust all possibilities for linear 
and quadratic equations having a positive root. So systematic and exhaustive 
was al-Khowarizmi's exposition that his readers must hav& had little 
difficulty in mastering the solutions. In this sense, then, al-Khowarizmi is 
entitled to be known as "the father of algebra." However, no branch of 
mathematics springs up fully grown, and we cannot help but ask where the 
inspiration for Arabic algebra came from. To this question no categorical 
answer can be given; but the arbitrariness of the rules and the strictly 
numerical form of the six chapters remind us of ancient Babylonian and 
medieval Indian mathematics. The exclusion of indeterminate analysis, a 
favorite Hindu topic, and the avoidance of any syncopation, such as is 
found in Brahmagupta, might suggest Mesopotamia as more likely a source 
than India. As we read beyond the sixth chapter, however, an entirely new 
light is thrown on the question. Al-Khowarizmi continued : 

We have said enough so far as numbers are concerned, about the six types of equations. 
Now, however, it is necessary that we should demonstrate geometrically the truth of the 
same problems which we have explained in numbers. 

The ring in this passage is obviously Greek rather than Babylonian or Indian. 
There are, therefore, three main schools of thought on the origin of Arabic 
algebra ; one emphasizes Hindu influences, another stresses the Mesopota- 
mian, or Syriac-Persian, tradition, and the third points to Greek inspiration. 3 
The truth is probably approached if we combine the three theories. The 
philosophers of Islam admired Aristotle to the point of aping him, but eclectic 
Mohammedan mathematicians seem to have chosen appropriate elements 
from various sources. 

The Algebra of al-Khowarizmi betrays unmistakable Hellenic elements, 
but the first geometrical demonstrations have little in common with classical 
Greek mathematics. For the equation x 2 + lOx = 39 al-Khowarizmi drew 
a square ab to represent x 2 , and on the four sides of this square he placed 
rectangles c, d, e, and /, each 2\ units wide. To complete the larger square one 

5 See Solomon Gandz. "The Sources of al-Khowarizmi's Algebra." Osiris, 1 (1936). 263-277 : 
also H. J. J. Winter, "Formative Influences in Islamic Science," Archives Internationales 
d'Histoire des Sciences, 6 (1953), 171-192. 



must add the four small corner squares (dotted in Fig. 13.1), each of which 
has an area of 6^ units. Hence to "complete the square" we add 4 times 65 
units or 25 units, thus obtaining a square of total area 39 + 25 = 64 units 

I 1 

! e ! 

1 1 

f d 

Li ■ L 

FIG. 13.1 

(as is clear from the right-hand side of the given equation). The side of the 
large square must therefore be 8 units, from which we subtract 2 times 2\ 
or 5 units to find that x = 3, thus proving that the answer found in Chapter IV 
is correct. 

The geometrical proofs for Chapters V and VI are somewhat more 
involved. For the equation x 2 + 21 = lCbc the author draws the square ab 
to represent x 2 and the rectangle bg to represent 21 units. Then the large 
rectangle, comprising the square and the rectangle bg, must have an area 
equal to lCbc, so that the side ag or hd must be 10 units. If, then, one bisects 
hd at e, draws et perpendicular to hd, extends te to c so that tc = tg, and 
completes the squares tclg and cmne (Fig. 1 3.2), the area tb is equal to area md. 
But square (/ is 25, and the gnomon tenmlg is 21 (since the gnomon is equal 
to the rectangle bg). Hence the square nc is 4, and its side ec is 2. Inasmuch as 
ec = be, and since he = 5, we see that x = hb = 5 - 2 or 3, which proves 
that the arithmetic solution given in Chapter V is correct. A modified diagram 

h I 



FIG. 13.2 


is given for the root x = 5 + 2 = 7, and an analogous type of figure is used 
to justify geometrically the result found algebraically in Chapter VI. 

A comparison of Fig. 13.2, taken from al-Khowarizmi's Algebra, with 
diagrams found in the Elements of Euclid in connection with Greek geomet- 
rical algebra (such as our Fig. 7.7) leads to the inevitable conclusion that 
Arabic algebra had much in common with Greek geometry; yet the first, 
or arithmetical part, of al-Khowarizmi's Algebra obviously is alien to Greek 
thought. What apparently happened in Baghdad was just what one would 
expect in a cosmopolitan intellectual center. Arabic scholars had great 
admiration for Greek astronomy, mathematics, medicine, and philosophy — 
subjects that they mastered as best they could. However, they could scarcely 
help but notice that, as the Nestorian Bishop Sebokt had observed when in 
662 he first called attention to the nine marvelous digits of the Hindus, "there 
are also others who know something." It is probable that al-Khowarizmi 
typified the Arabic electicism that will so frequently be observed in other 
cases. His system of numeration most likely came from India, his systematic 
aigebraic solution of equations may have been a development from Mesopo- 
tamia, and the logical geometric framework for his solutions palpably was 
derived from Greece. 

The Algebra of al-Khowarizmi contains more than the solution of equa- 
tions, material that occupies about the first half. There are, for example, 
rules for operations on binomial expressions, including products such as 
(10 + 2)(10 - I) and (10 + x)(10 - x). Although the Arabs rejected negative 
roots and absolute negative magnitudes, they were familiar with the rules 
governing what are now known as signed numbers. There are also alternative 
geometrical proofs of some of the author's six cases of equations. Finally, 
the Algebra includes a wide variety of problems illustrating the six chapters 
or cases. As an illustration of the fifth chapter, for example, al-Khowarizmi 
asks for the division of ten into two parts in such a way that "the sum of the 
products obtained by multiplying each part by itself is equal to fifty eight." 
The extant Arabic version, unlike the Latin, includes also an extended 
discussion of inheritance problems, such as the following: 

A man dies, leaving two sons behind him, and bequeathing one-third of his 
capital to a stranger. He leaves ten dirhems of property and a claim of ten 
dirhems upon one of the sons. 

The answer is not what one would expect, for the stranger gets only 5 dirhems. 
According to Arabic law, a son who owes to the estate of his father an amount 
greater than the son's portion of the estate retains the whole sum that he 
owes, one part being regarded as his share of the estate and the remainder as 
a gift from his father. To some extent it seems to have been the complicated 


nature of laws governing inheritance that encouraged the study of algebra 
in Arabia. 

A few of al-Khowarizmi's problems give rather clear evidence of Arabic 
dependence on the Babylonian-Heronian stream of mathematics. One of 
them presumably was taken directly from Heron, for the figure and dimen- 
sions are the same. Within an isosceles triangle having sides 10 yards and 
base 12 yards (Fig. 13.3) a square is to be inscribed, and the side of this square 

FIG. 13.3 

is called for. The author of the Algebra first finds through the Pythagorean 
theorem that the altitude of the triangle is 8 yards, so that the area of the 
triangle is 48 square yards. Calling the side of the square the "thing," he notes 
that the square of the "thing" will be found by taking from the area of the 
large triangle the areas of the three small triangles lying outside tha square 
but inside the large triangle. The sum of the areas of the two lower small 
triangles he knows to be the product of the "thing" by six less half the "thing" ; 
and the area of the upper small triangle is the product of eight less the "thing" 
by half the "thing." Hence he is led to the obvious conclusion that the 
"thing" is 4f yards — the side of the square. The chief difference between the 
form of this problem in Heron and that of al-Khowarizmi is that Heron 
had expressed the answer in terms of unit fractions as 4^} ^ . The similarities 
are so much more pronounced than the differences that we may take this 
case as confirmation of the general axiom that continuity in the history of 
mathematics is the rule rather than the exception. Where a discontinuity 
seems to arise, we should first consider the possibility that the apparent 
saltus may be explained by the loss of intervening documents. 

The Algebra of al-Khowarizmi usually is regarded as the first work on 
the subject, but a recent publication in Turkey raises some question about 
this. A manuscript of a work by abd-al-Hamid ibn-Turk, entitled "Logical 
Necessities in Mixed Equations," was part of a book on Al-jabr wa'l muqaba- 
lah which was evidently very much the same as that by al-Khowarizmi and 


was published at about the same time — possibly even earlier. The surviving 
chapters on "Logical Necessities" give precisely the same type of geometrical 
demonstration as al-Khowarizmi's Algebra and in one case the same illus- 
trative example x 2 + 21 = lOx. In one respect abd-al-Hamid's exposition is 
more thorough than that of al-Khowarizmi for he gives geometrical figures 
to prove that if the discriminant is negative, a quadratic equation has no 
solution. Similarities in the work of the two men and the systematic organiza- 
tion found in them seem to indicate that algebra in their day was not so 
recent a development as has usually been assumed. 6 When textbooks with a 
conventional and well-ordered exposition appear simultaneously, a subject is 
likely to be considerably beyond the formative stage. Successors of al- 
Khowarizmi were able to say, once a problem had been reduced to the form 
of an equation, "Operate according to the rules of algebra and almucabala." 
In any case, the survival of al-Khowarizmi's Algebra can be taken to indicate 
that it was one of the better textbooks typical of Arabic algebra of the time. 
It was to algebra what Euclid's Elements was to geometry — the best ele- 
mentary exposition available until modern times — but al-Khowarizmi's 
work had a serious deficiency that had to be removed before it could serve its 
purpose effectively in the modern world: a symbolic notation had to be 
developed to replace the rhetorical form. This step the Arabs never took, 
except for the replacement of number words by number signs. 

1 The ninth century was a glorious one in Arabic mathematics, for it pro- 
duced not only al-Khowarizmi in the first half of the century, but also Thabit 
ibn-Qurra (826-901) in the second half. If al-Khowarizmi resembled Euclid 
as an "elementator," then Thabit is the Arabic equivalent of Pappus, the 
commentator on higher mathematics. Thabit was the founder of a school of 
translators, especially from Greek and Syriac, and to him we owe an immense 
debt for translations into Arabic of works by Euclid, Archimedes, Apollonius, 
Ptolemy, and Eutocius. (Note the omission of Diophantus and Pappus, 
authors who evidently were not at first known in Arabia, although the 
Diophantine Arithmetica became familiar before the end of the tenth century.) 
Had it not been for his efforts, the number of Greek mathematical works 
extant today would be smaller. For example, we should have only the first 
four, rather than the first seven, books of Apollonius' Conies. Moreover, 
Thabit had so thoroughly mastered the content of the classics he translated 
that he suggested modifications and generalizations. To him is due a remark- 
able formula for amicable numbers : If p, q, and r are prime numbers, and if 
they are of the form p = 3 • 2" - 1, q = 3 • 2"" 1 - 1, and r = 9 • 2 2 "" 1 - 1, 
then 2"pq and 2"r are amicable numbers, for each is equal to the sum of the 

6 See Aydin Sayili, Logical Necessities in Mixed Equations by 'Abd al Hamid ibn Turk and the 
Algebra of His Time (1962). 



proper divisors of the other. Like Pappus, he also gave a generalization of the 
Pythagorean theorem that is applicable to all triangles, whether right or 
scalene. If from vertex A of any triangle ABC one draws lines intersecting 
BC in points B' and C such that angles ABB and AC'C are each equal to 
angle A (Fig. 13.4), then AB 2 + AC 2 = BQBB' + CC). Thabit gave no 


FIG. 13.4 

proof of the theorem, but this is easily supplied through theorems on similar 
triangles. In fact, the theorem provides a beautiful generalization of the 
pinwheel diagram used by Euclid in the proof of the Pythagorean theorem. 
If, for example, angle A is obtuse, then the square on side AB is equal to the 
rectangle BB'B'B'", and the square on AC is equal to the rectangle CC'C'C", 
where BB" = CC" = BC = B"C". That is, the sum of the squares on AB 
and AC is the square on BC less the rectangle B'CB'"C". If angle A is acute, 
then the positions of B' and C are reversed with respect to AP, where P is the 
projection of A on BC, and in this case the sum of the squares on AB and AC 
is equal to the square on BC increased by the rectangle B'C'B'"C". If A is a 
right angle, then B' and C coincide with P, and for this case Thabit's theorem 
becomes the Pythagorean theorem. (Thabit 7 did not draw the dotted lines 
that are shown in Fig. 13.4, but he did consider the several cases.) 

Alternative proofs of the Pythagorean theorem, works on parabolic and 
paraboloidal segments, a discussion of magic squares, angle trisections, and 
new astronomical theories are among Thabit's further contributions to 
scholarship. The Arabs sometimes are described as servile imitators of the 
Greeks in science and philosophy, but such accusations are exaggerated. 
Thabit, for instance, boldly added a ninth sphere to the eight previously 

7 See Aydin Sayili, "Thabit ibn Qurra's Generalization of the Pythagorean Theorem," 
I sis, 51 (1960), 35-37. See also Isis, 55 (1964), 68-70, and 57 (1 966), 56-66. 


assumed in simplified versions of Aristotelian-Ptolemaic astronomy ; and 
instead of the Hipparchan precession of the equinoxes in one direction or 
sense only, Thabit proposed a "trepidation of the equinoxes" in a reciprocat- 
ing type of motion. Such questioning of points in Greek astronomy may well 
have been a factor in paving the way for the revolution in astronomy initiated 
by Copernicus. 

11 We have mentioned several times that the Arabs were quick to absorb 
learning from the neighbors they conquered; it should be noted also that 
within the confines of the Arabic empire lived peoples of very varied ethnic 
backgrounds : Syrian, Greek, Egyptian, Persian, Turkish, and many others. 
Most of them shared a common faith, Islam, although Christians and Jews 
were tolerated ; very many shared a common language, Arabic, although 
Greek and Hebrew were sometimes used. Nevertheless, we should not expect 
a high degree of uniformity in learning. There was considerable factionalism 
at all times, and it sometimes erupted into conflict. Thabit himself lived in a 
pro-Greek community, which opposed him for his pro-Arabic sympathies. 
In Arabic mathematics such cultural differences occasionally became quite 
apparent, as in the works of the tenth- and eleventh-century scholars Abu'l- 
Wefa (940-998) and al-Karkhi (or al-Karagi, ca. 1029). In some of their works 
they used the Hindu numerals, which had reached Arabia through the 
astronomical Sindhind; at other times they adopted the Greek alphabetic 
pattern of numeration (with, of course, Arabic equivalents for the Greek 
letters). Ultimately the superior Hindu numerals won out, but even within 
the circle of those who used the Indian numeration, the forms of the numerals 
differed considerably. Variations had obviously been prevalent in India, 
but in Arabia variants were so striking that there are theories suggesting 
entirely different origins for forms used in the eastern and western halves of 
the Arabic world. Perhaps the numerals of the Saracens in the east came 
directly from India, while the numerals of the Moors in the west were derived 
from Greek or Roman forms. More likely the variants were the result of 
gradual changes taking place in space and time, for the Arabic numerals of 
today are strikingly different from the modern Devanagari (or "divine") 
numerals still in use in India. After all, it is the principles within the system of 
numeration that are important, and not the specific forms of the numerals. 
Our numerals often are known as Arabic, despite the fact that they bear little 
resemblance to those now in use in Egypt, Iraq, Syria, Arabia, Iran, and other 
lands within the Islamic culture — that is, the forms IITIQIVAV. We call 
our numerals Arabic because the principles in the two systems are the same 
and because our forms may have been derived from the Arabic. However, the 
principles behind the Arabic numerals presumably were derived from India ; 
hence it is better to call ours the Hindu or the Hindu-Arabic system. 



As in numeration there was competition between systems of Greek and 
Indian origin, so also in astronomical calculations there were at first in 
Arabia two types of trigonometry — the Greek geometry of chords, as found 
in the Almagest, and the Hindu tables of sines, as derived through the 
Sindhincl. Here, too, the conflict resulted in triumph for the Hindu aspect. 


- = = ¥• f 

Q> 1 h ? 


i ? 4 a V 

C •) V G\ e 

in (Gw 

m« v?\3K« 



Sanskrit-Devanagari (Indian) 



f «,;* r* <j i n I 3 

1 r r^fi h va 1 • 

West Arabic 


East Arabic 


b A 8 $ 

llth Cer 

ltury ( 


, z 3 a. q s "k 8 <f « 

1*5+5 fr7*9» 




16th Cen 

tury ( 


Genealogy of our digits. Following Karl Menninger, Zahlwort und Ziffer (Gottingen: 
Vanderhoeck & Ruprecht, 1957-1958, 2 vols), II, 233. 

and most Arabic trigonometry ultimately was built on the sine function. 
It was, in fact, again through the Arabs, rather than directly from the Hindus, 
that this trigonometry of the sine reached Europe. The astronomy of al- 
Battani (ca. 850-929), known in Europe as Albategnius, served as the primary 
vehicle of transmission, although Thabit ibn Qurra seems to have used sines 
somewhat earlier. In a book entitled On the Motion of the Stars Albategnius 
gave formulas, such as h = [a sin (90° - A)] I sin A (see Fig. 13.5), in which the 
sine and versed sine functions appear. By the time of Abu'1-Wefa, a century 
later, the tangent function was fairly well known, so that one could express 
the above relationship more simply as a = b tan A. Here one is in more 
immediate touch with modern trigonometry, for the Arabic tangent function. 


unlike the Hindu sine function, generally was given for a unit circle. Moreover, 
with Abu'1-Wefa trigonometry assumes a more systematic form in which such 
theorems as double and half-angle formulas are proved. Although the Hindu 
sine function had displaced the Greek chord, it was nevertheless the Almagest 
of Ptolemy that motivated the logical arrangement of trigonometric results. 
The law of sines had been known to Ptolemy in essence and is implied in the 
work of Brahmagupta, but it frequently is attributed to Abu'1-Wefa because 
of his clear-cut formulation of the law for spherical triangles. Abu'1-Wefa 
also made up a new sine table for angles differing by {", using the equivalent of 
eight decimal places. He contributed also a table of tangents and made use 
of all six of the common trigonometric functions, together with relations 
among them, but his use of the new functions seems not to have been followed 
widely in the medieval period. 

Sometimes attempts are made to attribute the functions tangent, cotangent, 
secant, and cosecant to specific times and even to specific individuals, but 
this cannot be done with any assurance. In India and Arabia there had been 
a general theory of shadow lengths, as related to a unit of length or gnomon, 
for varying solar altitudes. There was no one standard unit of length for the 
staff or gnomon used, although a handspan or a man's height was frequently 
adopted. The horizontal shadow, for a vertical gnomon of given length, was 
what we call the cotangent of the angle of elevation of the sun. The "reverse 
shadow" — that is, the shadow cast on a vertical wall by a stick or gnomon 
projecting horizontally from the wall — was what we know as the tangent of 
the solar elevation. The "hypotenuse of the shadow"— that is, the distance 
from the tip of the gnomon to the tip of the shadow— was the equivalent of 
our cosecant function ; and the "hypotenuse of the reverse shadow" played 
the role of our secant. This shadow tradition seems to have been well estab- 
lished in Asia by the time of Thabit ibn Qurra, 8 but values of the hypotenuse 
(secant or cosecant) were seldom tabulated. 

1 3 Abu'l-Wefa was a capable algebraist as well as a trigonometer. He com- 
mented on al-Khowarizmi's Algebra and translated from the Greek one 
of the last great classics — the Arithmetica of Diophantus. His successor 

8 See E. S. Kennedy, "Overview on Trigonometry," to appear in the Yearbook on History of 
Mathematics of the National Council of Teachers of Mathematics. 


al-Karkhi evidently used this translation to become an Arabic disciple of 
Diophantus— but without Diophantine analysis! That is, al-Karkhi was 
concerned with the algebra of al-Khowarizmi rather than the indeterminate 
analysis of the Hindus ; but like Diophantus (and unlike al-Khowarizmi) he 
did not limit himself to quadratic equations — despite the fact that he followed 
the Arabic custom of giving geometric proofs for quadratics. In particular, 
to al-Karkhi is attributed the first numerical solutions of equations of the 
form ax 2n + bx" = c (only equations with positive roots were considered), 
where the Diophantine restriction to rational numbers was abandoned. 
It was in just this direction, toward the algebraic solution (in terms of radicals) 
of equations of more than second degree, that the early developments in 
mathematics in the Renaissance were destined to take place. 

The time of al-Karkhi — the early eleventh century — was a brilliant era 
in the history of Arabic learning, and a number of his contemporaries deserve 
brief mention— brief not because they were less capable, but because they 
were not primarily mathematicians. Ibn-Sina (980-1037), better known to 
the West as Avicenna, was the foremost scholar and scientist in Islam, but 
in his encyclopedic interests mathematics played a smaller role than medicine 
and philosophy. He made a translation of Euclid and explained the casting- 
out of nines (which consequently is sometimes unwarrantedly attributed to 
him), but he is better remembered for his application of mathematics to 
astronomy and physics. As Avicenna reconciled Greek learning with Muslim 
thought, so his contemporary al-Biruni (973-1048) made the Arabs— hence 
us — familiar with Hindu mathematics and culture through his well-known 
book entitled India. An indefatigable traveler and a critical thinker, he gave 
a sympathetic but candid account, including full descriptions of the 
Siddhantas and the positional principle of numeration. It is he who tells us 
that Archimedes was familiar with Heron's formula and gives a proof of this 
and of Brahmagupta's formula, correctly insisting that the latter applies 
only to a cyclic quadrilateral. In inscribing a nonagon in a circle al-Biruni 
reduced the problem, through the trigonometric formula for cos 39, to 
solving the equation x 3 = 1 + 3x, and for this he gave the approximate 
solution in sexagesimal fractions as 1 ;52, 15, 17, 13— equivalent to more than 
six-place accuracy. 9 Al-Biruni also gave us, in a chapter on gnomon lengths, 
an account of the Hindu shadow reckoning. The boldness of his thought is 
illustrated by his discussion of whether or not the earth rotates on its axis, 
a question to which he did not give an answer. (Aryabhata seems earlier to 
have suggested a rotating earth at the center of space.) Al-Biruni contributed 
also to physics, especially through studies in specific gravity and the causes 

9 See Pierre Dedron and Jean hard, Mathematiques et mathematiciens (1959), p. 126. 



of artesian wells: but as a physicist and mathematician he was excelled by 
ibn-al-Haitham (ca. 965-1039), known to the West as Alhazen. The most 
important treatise written by Alhazen was the Treasury of Optics, a book 
which was inspired by work of Ptolemy on reflection and refraction and 
which in turn inspired scientists of medieval and early modern Europe. 
Among the questions that Alhazen considered were the structure of the eye, 
the apparent increase in the size of the moon when near the horizon, and an 
estimate, from the observation that twilight lasts until the sun is 19° below 
the horizon, of the height of the atmosphere. The problem of finding the 
point on a spherical mirror at which light from a source will be reflected to 
the eye of an observer is known to this day as "Alhazen's problem." It is a 
"solid problem" in the old Greek sense, solvable by conic sections, a subject 
with which Alhazen was quite familiar. He extended Archimedes' results on 
conoids by finding the volume generated by revolving about the tangent at 
the vertex the area bounded by a parabolic arc and the axis and an ordinate 
of the parabola. 

15 Arabic mathematics can with some propriety be divided into four parts: 
(l) an arithmetic derived presumably from India and based on the principle 
of position ; (2) an algebra which, although from Greek, Hindu, and Baby- 
lonian sources, nevertheless in Muslim hands assumed a characteristically 
new and systematic form; (3) a trigonometry the substance of which came 
chiefly from Greece but to which the Arabs applied the Hindu form and added 
new functions and formulas; (4) a geometry which came from Greece but 
to which the Arabs contributed generalizations here and there. In connection 
with (3) it should be noted that ibn-Yunus (T1008), Alhazen's contemporary 
and fellow countryman (they both lived in Egypt), introduced the formula 
2 cos .v cos v = cos (x + y) + cos (x - y). This is one of the four "product to 
sum" formulas that in sixteenth-century Europe served, before the invention 
of logarithms, to convert products to sums by the method known as "pros- 
thaphaeresis" (Greek for addition and subtraction). In connection with (4) 
there was a significant contribution about a century after Alhazen by a 
man who in the East is known as a scientist but whom the West recalls as 
one of the greatest Persian poets. Omar Khayyam (ca. 1050-1 123), the "tent- 
maker." wrote an Algebra 10 that went beyond that of al-Khowarizmi to 
include equations of third degree. Like his Arabic predecessors, Omar 
Khayyam provided for quadratic equations both arithmetic and geometric 
solutions; for general cubic equations, he believed (mistakenly, as the 
sixteenth century later showed), arithmetic solutions were impossible; hence 

10 See The Algebra of Omar Khayyam, ed. by D. S. Kasir (1931); also D. J. Struik, "Omar 
Khayyam, Mathematician," The Mathematics Teacher, 51 (1958), 280-285. 


he gave only geometric solutions. The scheme of using intersecting conies to 
solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazen, 
but Omar Khayyam took the praiseworthy step of generalizing the method 
to cover all third-degree equations (having positive roots). When in an earlier 
work he came across a cubic equation, he specifically remarked : "This cannot 
be solved by plane geometry [i.e.. using straightedge and compasses only] 
since it has a cube in it. For the solution we need conic sections."' ' 

For equations of higher degree than three Omar Khayyam evidently did 
not envision similar geometric methods, for space does not contain more than 
three dimensions, "what is called square-square by algebraists in continuous 
magnitude is a theoretical fact. It does not exist in reality in any way." 
The procedure that Omar Khayyam so tortuously — and so proudly — applied 
to cubic equations can be stated with far greater succinctness in modern 
notation and concepts as follows. Let the cubic be x 3 + ax 2 + b 2 x + c 3 = 0. 
Then if for x 2 in this equation we substitute 2py, we obtain (recalling that 
x 3 = x 2 ■ x) the result 2pxy + 2apy + b 2 x + c 3 = 0. Since the resulting 
equation represents an hyperbola, and the equality x 2 = 2py used in the 
substitution represents a parabola, it is clear that if the hyperbola and the 
parabola are sketched on the same set of coordinate axes, then the abscissas 
of the points of intersection of the two curves will be the roots of the cubic 
equation. Obviously many other pairs of conic sections can be used in a 
similar way to solve the cubic. 

Our exposition of Omar Khayyam's work does not do justice to his genius, 
for, lacking the concept of negative coefficients, he had to break the problem 
into many separate cases according as the parameters a, b, c are positive, 
negative, or zero. Moreover, he had to specifically identify his conic sections 
for each case, for the concept of a general parameter was not at hand in his 
day. Not all roots of a given cubic equation were given, for he did not accept 
the appropriateness of negative roots and did not note all intersections of the 
conic sections. It should be remarked also that in the earlier Greek geometric 
solutions of cubic equations the coefficients had been line segments, whereas 
in the work of Omar Khayyam they were specific numbers. One of the most 
fruitful contributions of Arabic eclecticism was the tendency to close the gap 
between numerical and geometrical algebra. The decisive step in this direction 
came much later with Descartes, but Omar Khayyam was moving in this 
direction when he wrote, "Whoever thinks algebra is a trick in obtaining 
unknowns has thought it in vain. No attention should be paid to the fact 
that algebra and geometry are different in appearance. Algebras are geometric 
facts which are proved." 12 In replacing Euclid's theory of proportions by a 

11 A. R. Amir-Moez, "A Paper of Omar Khayyam," Scripta Mathematica, 26(1963), 323-337, 
p. 328. 

12 Amir-Moez, op. cit., p. 329. 


numerical approach, he came close to a definition of the irrational and strug- 
gled with the concept of real number in general. 13 

16 In his Algebra Omar Khayyam wrote that he had set forth elsewhere a 
rule that he had discovered for finding fourth, fifth, sixth, and higher powers 
of a binomial, but such a work is not extant. It is presumed that he is referring 
to the Pascal triangle arrangement, one that seems to have appeared in 
China at about the same time. Such a coincidence is not easy to explain, 
but until further evidence is available, independence of discovery is to be 
assumed. Intercommunication between Arabia and China was not extensive 
at that time ; but there was a silk route connecting China with Persia, and 
information might have trickled along it. 

The Arabs were clearly more attracted to algebra and trigonometry than 
to geometry, but one aspect of geometry held a special fascination for them — 
the proof of Euclid's fifth postulate. Even among the Greeks the attempt to 
prove the postulate had become virtually a "fourth famous problem of 
geometry," and several Muslim mathematicians continued the effort. 
Alhazen had begun with a trirectangular quadrilateral (sometimes known as 
"Lambert's quadrangle" in recognition of efforts in the eighteenth century) 
and thought that he had proved that the fourth angle must also be a right 
angle. From this "theorem" on the quadrilateral the fifth postulate can easily 
be shown to follow. In his "proof" Alhazen had assumed that the locus of a 
point that moves so as to remain equidistant from a given line is necessarily 
a line parallel to the given line — an assumption shown in modern times to be 
equivalent to Euclid's postulate. Omar Khayyam criticized Alhazen's 
proof on the ground that Aristotle had condemned the use of motion in 
geometry. Omar Khayyam then began with a quadrilateral the two sides 
of which are equal and are both perpendicular to the base (usually known as a 
"Saccheri quadrilateral," again in recognition of eighteenth-century efforts), 
and he asked about the other (upper) angles of the quadrilateral, which 
necessarily are equal to each other. There are of course, three possibilities. 
The angles may be (1) acute, (2) right, or (3) obtuse. The first and third 
possibilities Omar Khayyam ruled out on the basis of a principle, which he 
attributed to Aristotle, that two converging lines must intersect — again an 
assumption equivalent to Euclid's parallel postulate. 

17 When Omar Khayyam died in 1123, Arabic science was in a state of 
decline. Excesses of political and religious factionalism — a condition that 
is well illustrated by the origin of our word "assassin" — would seem to be 

13 See D. J. Struik, "Omar Khayyam, Mathematician," The Mathematics Teacher, 51 (1958), 


among the causes of the decline. Islam never again was to reach the scholarly 
level of the glorious age of Avicenna and al-Karkhi, but Muslim contributions 
did not come to a sudden stop after Omar Khayyam. Both in the thirteenth 
century and again in the fifteenth century we find an Arabic mathematician of 
note. At Maragha, for example, Nasir Eddin al-Tusi (or at-Tusi, 1201-1274), 
astronomer to Hulagu Khan, grandson of the conqueror Genghis Khan and 
brother of Kublai Khan, continued efforts to prove the parallel postulate, 
starting from the usual three hypotheses on a Saccheri quadrilateral. His 
"proof" depends on the following hypothesis, again equivalent to Euclid's: 

If a line u is perpendicular to a line w at A, and if line v is oblique to w at B, then 
the perpendiculars drawn from u upon v are less than AB on the side on which v 
makes an acute angle with w and greater on the side on which v makes an obtuse 
angle with w. 14 

The views of Nasir Eddin, the last in the sequence of three Arabic precursors 
of non-Euclidean geometry, were translated and published by Wallis in the 
seventeenth century ; it appears that this work was the starting point for the 
developments by Saccheri in the first third of the eighteenth century. 

Nasir Eddin followed characteristic Arabic interests; hence he made 
contributions also to trigonometry and astronomy. Continuing the work 
of Abu'1-Wefa, he was responsible for the first systematic treatise on plane 
and spherical trigonometry, treating the material as an independent subject 
in its own right and not simply as the handmaid of astronomy, as had been 
the case in Greece and India. The six usual trigonometric functions are used, 
and rules for solving the various cases of plane and spherical triangles are 
given. Unfortunately, the work of Nasir Eddin had limited influence inasmuch 
as it did not become well known in Europe. In astronomy, however, Nasir 
Eddin made a contribution that may have come to the attention of Coperni- 
cus. The Arabs had adopted theories of both Aristotle and Ptolemy for the 
heavens; noticing elements of conflict between the cosmologies, they sought 
to reconcile them and to refine them. In this connection Nasir Eddin observed 
that a combination of two uniform circular motions in the usual epicyclic 
construction can produce a reciprocating rectilinear motion. That is, if a 
point moves with uniform circular motion clockwise around the epicycle 
while the center of the epicycle moves counterclockwise with half this speed 
along an equal deferent circle, the point will describe a straight-line segment. 
(In other words, if a circle rolls without slipping along the inside of a circle 
whose diameter is twice as great, the locus of a point on the circumference 
of the smaller circle will be a diameter of the larger circle.) This "theorem of 

u See Roberto Bonola, Non-Euclidean Geometry (New York : Dover reprint, 1955), p. 10. 
See also D. E. Smith, "Euclid, Omar Khayyam, and Saccheri," Scripta Mathematical 3 (1935), 


Nasir Eddin" became known to, or was rediscovered by, Copernicus and 
Cardan in the sixteenth century. 15 

18 Arabic mathematics continued to decline, following Nasir Eddin, but 
our account of the Muslim contribution would not be adequate without 
reference to the work of a figure in the early fifteenth century. Al-Kashi 
(tea. 1436) found a patron in the prince Ulugh Beg, grandson of the Mongol 
conqueror Tamerlane. At Samarkand, where he held his court, Ulugh Beg 
had built an observatory, and al-Kashi joined the group of scientists gathered 
there. In numerous works, written in Persian and Arabic, al-Kashi contrib- 
uted to mathematics and astronomy. Noteworthy is the accuracy of his 
computations, especially in connection with the solution of equations by 
Horner's method, derived perhaps from the Chinese. From China, too, al- 
Kashi may have taken the practice of using decimal fractions. Al-Kashi is an 
important figure in the history of decimal fractions, and he realized the 
significance of his contribution in this respect, regarding himself as the 
inventor of decimal fractions. 16 Although to some extent he had had precur- 
sors, he is perhaps the first user of sexagesimal fractions to suggest that 
decimals are just as convenient for problems requiring many-place accuracy. 
Nevertheless, in his systematic computations of roots he continued to make 
use of sexagesimals. In illustrating his method for finding the nth root of a 
number, he took the sixth root of the sexagesimal 

34,59,1,7,14,54,23,3,47,37 ;40 

This was a prodigious feat of computation, using the steps that we follow in 
Horner's method — locating the root, diminishing the roots, and stretching or 
multiplying the roots — and using a pattern similar to our synthetic division. 
Al-Kashi evidently delighted in long calculations, and he was justifiably 
proud of his approximation for n, which was more accurate than any of the 
values given by his predecessors. True to the penchant of the Arabs for 
alternative notations, he expressed his value of 2n in both sexagesimal and 
decimal forms. The former— 6;16,59,28,34,51,46,15,50— is more reminiscent 
of the past and the latter— 6.2831853071795865— in a sense presaged the 
future use of decimal fractions. No mathematician approached the accuracy 
in this tour de force of computation until the late sixteenth century. (The 

15 See C. B. Boyer, "Note on Epicycles and the Ellipse from Copernicus to Lahire," lsis, 38 

10 See Abdul-Kader Kakhel, Al-Kashi on Root Extraction (1960), p. 2. An unusually extensive 
account of some of the work of al-Kashi is found in P. Luckey, "Die Ausziehung der n-ten 
Wurzel und der binomische Lehrsatz in der islamischen Mathematik," Mathematische Annalen, 
120 (1948), 217-274. Very recently it has been pointed out that use of decimal fractions in 
Arabia is found in a work by abu-al-Hasan, Ahmad ibn-Ibrahim al-Uqlidist dating from 
952-953. See A. S. Saidan, "The Earliest Extant Arabic Arithmetic," his, 57 (1966), 475-490. 


following mnemonic device will aid in memorizing a good approximation 
to 7r: "How I want a drink, alcoholic of course, after the heavy lectures 
involving quantum mechanics." The number of letters in the words will 
provide the values for the successive digits in 3.14159265358979. and these 
will be found to be in full agreement with al-Kashi's value for 2n.) In al- 
Kashi the binomial theorem in "Pascal triangle" form again appears, just 
about a century after its publication in China and about a century before it 
was printed in European books. 

With the death of al-Kashi in about 1436 we can close the account of 
Arabic mathematics, for the cultural collapse of the Muslim world was more 
complete than the political disintegration of the empire. The number of 
significant Arabic contributors to mathematics before al-Kashi was con- 
siderably larger than our exposition would suggest, for we have concentrated 
only on major figures; 17 but following him the number is negligible. It was 
very fortunate indeed that when Arabic learning began to decline, scholarship 
in Europe was on the upgrade and was prepared to accept the intellectual 
legacy bequeathed by earlier ages. It is sometimes held that the Arabs had 
done little more than to put Greek science into "cold storage" until Europe 
was ready to accept it. But the account in this chapter has shown that at least 
in the case of mathematics the tradition handed over to the Latin world in 
the twelfth and thirteenth centuries was richer than that with which the 
unlettered Arabic conquerors had come into contact in the seventh century. 


Amir-Moez, A. R., "A Paper of Omar Khayyam," Scripta Mathematica, 26 (1963) 

Cajori, Florian, History of Mathematics, 2nd ed. (New York: Macmillan, 1919). 
Dedron, Pierre, and Jean Itard, Mathematiques et mathematiciens (Paris' Magnard 


Gandz. Solomon, "The Sources of al-Khowarizmi's Algebra." Osiris, 1 (1936), 263-277. 
Hill, G. F.. The Development of Arabic Numerals in Europe (Oxford : Clarendon, 1915). 
Kakhel, Abdul-Kader. Al-Kashi on Root Extraction (Lebanon, 1960). 
Kasir, D. S., ed., The Algebra of Omar Khayyam (New York : Columbia Teachers 

College, 1931). 
Karpinski, L. C, "The Algebra of Abu Kamil," American Mathematical Monthly 

21 (1914), 37-48. 
Karpinski, L. C, ed., Robert of Chester's Latin Translation of the Algebra of al-Khow- 

arizmi (New York: Macmillan, 1915). 
Kennedy, E. S., "Overview on Trigonometry," Yearbook on History of Mathematics, 

The National Council of Teachers of Mathematics (Washington, D.C), in press. 

17 See Heinrich Suter, Die Mathematiker unci Astronomer der Araher und ihre Werke (1900), 
for an account of more than 500 scholars. 


Levey, Martin, ed., The Algebra of Abu Kamil (Madison, Wis.: University of Wisconsin 

Press, 1966). 
Luckey, P., "Die Ausziehung der n-ten Wurzel und der binomische Lehrsatz in der 

islamischen Mathematik," Mathematische Annalen, 120 (1948), 217-274. 
Rosenfeld, B. A., and A. P. Youschkevitch, Omar Khayyam (in Russian, Moscow: 

Izdatelestvo "Nauka," 1965). 
Saidan. A. S., "The Earliest Extant Arabic Arithmetic," I sis, 57 (1966), 475-490. 
Sanchez Perez, Jose, La arithmetica en Roma, en India y en Arabia (Madrid : Instituto 

Miguel Asin, 1949). 
Sarton, George, Introduction to the History of Science (Baltimore: Carnegie Institution 

of Washington, 1927-1948, 3 vols, in 5). 
Sayili, Aydin, Logical Necessities in Mixed Equations by 'Abd al Hamid ibn Turk and 

the Algebra of His Time (Ankara, 1962). 
Sayili, Aydin, "Thabit ibn Qurra's Generalization of the Pythagorean Theorem," his, 

51 (1960), 35-37. 
Smith, D. E., History of Mathematics (Boston: Ginn, 1923-1925, 2 vols.; paperback 

reprint. New York : Dover, 1958). 
Smith, D. E., and L. C. Karpinski, The Hindu-Arabic Numerals (Boston, 1911). 
Struik, D. J., "Omar Khayyam, Mathematician," The Mathematics Teacher, 51 (1958), 

Suter, Heinrich, Die Mathematiker und Astronomer der Araber und ihre Werke (Leipzig, 

Vogel, Kurt, ed., Mohammed ibn Musa Alchwarizmis Algorismus (Aalen : O. Zeller, 1963). 
Winter, H. J. J., "Formative Influences in Islamic Science," Archives Internationales 

d'Histoire des Sciences, 6 (1953), 171-192. 


1. Compare, in its effect on learning, the Arabic conquest of neighboring lands with the earlier 
conquests of Alexander the Great and with the conquests of the Romans. 

2. Explain why al-Khowarizmi's Algebra contains no quadratic equation of the case squares 
and roots and numbers equal zero. 

3. Which of the numerals used in modern Arabia most closely resemble our own? Are there 
any advantages or disadvantages in the Arabic forms? 

4. Was it fortunate or unfortunate for the future of mathematics that Charles Martel turned 
back the Arabs at Tours in 732? Give reasons for your answer. 

5. How would you account for the fact that after 1500 the Arabs made virtually no further 
contribution to mathematics? 

6. Mention some parts of Greek mathematics that would be lost except for Arabic assistance. 

7. Compare Arabic and Hindu mathematics with respect to form, content, level, and influence. 

8. Compare the roles of logic and philosophy in Greek and Arabic mathematics. 

9. Using a geometrical diagram like that of al-Khowarizmi, solve x 2 + 12x = 64. 

10. Verify the answer given by al-Khowarizmi and Heron for the dimensions of a square 
inscribed in a triangle of sides 10, 10, and 12. 

11. Verify the theorem of Thabit ibn-Qurra on amicable numbers. 

12. Prove Thabit ibn-Qurra's generalization of the Pythagorean theorem. 

13. Solve al-Biruni's cubic x 3 = 1 + 3x for the positive root, correct to the nearest hundredth, 
and verify that to this extent your answer agrees with his. 


14. Prove the formula of ibn-Yunus 2 cos x cos y = cos (x + y) + cos (x — y). 

15. Use this formula to convert the product of 0.4567 and 0.5678 to a sum. 

16. Solve the equation x 3 = x 2 + 20 geometrically in the manner of Omar Khayyam. 

17. Solve the equation x 3 + x = 20 geometrically in the manner of Omar Khayyam. 

18. Using Alhazen's estimate for the length of twilight and taking the radius of the earth as 
4000 miles, find approximately the height of the atmosphere. (Twilight is caused by the 
reflection of the sun's rays in particles in the atmosphere.) 

19. Find the volume obtained by revolving about the j>-axis the area bounded by y 1 = 2px and 
the line x = a. Which of the Greeks and the Arabs were able to handle this problem? 

20. Show that the first three sexagesimals of al-Kashi's value of In are in agreement with the 
first five places in his decimal form. 

2 1 . Nasir Eddin showed that the sum of two odd squares cannot be a square. Prove this theorem, 
making use of properties of squares of odd and even numbers. 

*22. As a special case of Alhazen's problem, consider a spherical mirror with circular section 
given by the equation x 2 + y 2 = 1, let a source of light be at the point (0, 3), and let the 
eye be at the point (4, 0). Show that the point at which the light will be reflected by the mirror 
can be found through the intersection of the circle and a hyperbola. 


Europe in 

the Middle Ages 

Neglect of mathematics works injury to all knowledge, 
since he who is ignorant of it cannot know the other 
sciences or the things of this world. 

Roger Bacon 

Time and history are, of course, seamless wholes, like the continuum of 
mathematics, and any subdivision into periods is man's handiwork; but just 
as a coordinate framework is useful in geometry, so also the subdivision of 
events into periods or eras is convenient in history. For purposes of political 
history it has been customary to designate the fall of Rome in 476 as the 
beginning of the Middle Ages and the fall of Constantinople to the Turks in 
1453 as the end. Disregarding politics, it might be better to close the ancient 
period with the year 524, which is both the year of Boethius' death and the 
approximate time when the Roman abbot Dionysius Exiguus proposed the 
chronology based on the Christian era that has since come into common use. 
For the history of mathematics we indicated in Chapter II a preference for 
the year 529 as a marker for the beginning of the medieval period, and we 
shall somewhat arbitrarily designate the year 1436 as the close. 

The date 1436 is the probable year of death of al-Kashi, a very capable 
mathematician whom we already have described as somewhat Janus-faced 
— looking back on the old and in some respects anticipating the new. The 
year 1436 marks also the birth of another eminent mathematician, Johann 
Mtiller (1436-1476), better known under the name Regiomontanus, a 
Latinized form of his place of birth in Konigsberg. The year 1436, in other 
words, symbolizes the fact that during the Middle Ages those who excelled 
in mathematics wrote in Arabic and lived in Islamic Africa and Asia, whereas 
during the new age that was dawning the leading mathematicians wrote in 
Latin and lived in Christian Europe. 

An oversimplified view of the Middle Ages often results from a predomin- 
antly Europe-centered historical account; hence we remind readers that 
five great civilizations, writing in five different tongues, make up the bulk of 



the history of medieval mathematics. In the two preceding chapters we 
described contributions from China, India, and Arabia, three of the five 
leading medieval cultures. In this chapter we look at the mathematics of the 
other two : ( 1 ) the Eastern or Byzantine Empire, with center at Constantinople 
(or Byzantium), in which Greek was the official language ; and (2) the Western 
or Roman Empire, which had no one center and no single spoken language, 
but in which Latin was the lingua franca of scholars. 

When Justinian in 529 closed the pagan philosophical schools at Athens, 
the scholars were dispersed, and some of them made permanent homes in 
Syria, Persia, and elsewhere. Nonetheless, some of the scholars remained, 
and others returned some years later, with the result that there was no 
serious hiatus in Greek learning in the Byzantine world. We have men- 
tioned briefly the work of several Greek scholars of the sixth century: 
Eutocius, Simplicius, Isidore of Miletus, and Anthemius of Tralles. It was 
Justinian himself who put the building of Hagia Sophia in charge of the last 
two. To the list of Byzantine scholars should also be added the name of John 
Philoponus, who flourished at Alexandria in the early sixth century and was 
the leading physicist of his age anywhere in the world. Philoponus argued 
against the Aristotelian laws of motion and the impossibility of a vacuum, 
and he suggested the operation of a kind of inertia principle under which 
bodies in motion continued to move. Like Galileo later, he denied that the 
speed acquired by a freely falling body is proportional to its weight : 

If you let fall from the same height two weights of which one is many times as heavy 
as the other, you will see that the ratio of the times required for the motion does not 
depend on the ratio of the weights, but that the difference in time is a very small one. 1 

Philoponus was a Christian scientist (as were also perhaps Eutocius and 
Anthemius) who was making use of ancient pagan sources and whose ideas 
influenced later Islamic thinkers, thus indicating the continuity of the 
scientific tradition despite religious and political differences. 

Philoponus was not primarily a mathematician, but some of his work, 
such as his treatise on the astrolabe, can be thought of as applied mathematics. 
Most Byzantine contributions to mathematics were on an elementary level 
and consisted chiefly of commentaries on ancient classics. Byzantine mathe- 
matics, far more than Arabic, was a sort of holding action to preserve as 
much of antiquity as possible until the West was ready to carry on. Philo- 
ponus aided in this work through his commentary on the Introduction to 
Arithmetic of Nicomachus. Neoplatonic thought continued to exert a strong 
influence in the Eastern Empire, which accounts for the popularity of 
Nicomachus' treatise. Again in the eleventh century it was the subject of a 

1 Quoted from Marshall Clagett, The Science of Mechanics in the Middle Ages (1959), p. 546. 


commentary, this time by Michael Constantine Psellus (1018-1080?), a 
philosopher of Athens and Constantinople who counted among his pupils 
the Emperor Michael VII. Another of Psellus' works, a very elementary 
compendium on the quadrivium, enjoyed quite a vogue in the West during 
the sixteenth-century Renaissance period. Two centuries later we note another 
Greek summary of the mathematical quadrivium, this time by Georgios 
Pachymeres (1242-1316). Such compendia were significant only in showing 
that a thin thread of the old Greek tradition continued in the Eastern 
Empire to the very end of the medieval period. 

Pachymeres wrote also a commentary on the Arithmetic of Diophantus, as 
did his contemporary, Maximos Planudes (12557-1310). The latter, a Greek 
monk, was ambassador to Venice of the Emperor Andronicus II, indicating 
that there was some scholarly contact between the East and the West. 
Planudes wrote also a work on the Hindu system of numeration, which had 
finally reached the Greek world. In Byzantium, as might have been antici- 
pated, the alphabetic numerals were not wholly abandoned, for they have 
continued to our own day in Greece in legal, administrative, and ecclesiastical 
documents. Section LXXVIII of a document, for example is orj (that is, 
omicron eta) as in Alexandrian days. Moreover, even within the new Hindu 
system the Byzantine scholars of the fourteenth century retained the first 
nine letters of the old alphabetic scheme, adding to these a zero symbol, 
like an inverted h. The number 7890, for example, would be written as 
£^04, a form every bit as convenient as our own. Manuel Moschopoulos 
(fl. 1300), a disciple of Planudes, wrote on magic squares, and the account of 
Planudes on numeration was commented on by the arithmetician and 
geometer Nicholas Rhabdas (T1350). The latter composed also a work on 
finger reckoning ; but Byzantine mathematics, never very strong, by this time 
had become negligible. By the fourteenth century the Greek world had been 
clearly surpassed by the Latin world in the West, to which we now turn. 

I Chapter II included reference to the Latin treatises of Boethius at the 
end of the ancient period, with an indication of their very elementary level. 
Even from that level it was possible for mathematics to deteriorate still 
further, as we see in the trivial compendium on the liberal arts composed by 
Cassiodorus (ca. 480-ca. 575), a disciple of Boethius who spent his last years 
in a monastery that he had established. The primitive works of Cassiodorus 
served as textbooks in church schools in the early Middle Ages and some- 
times also as the source for still lower-level books, such as the Origines or 
Etymologies of Isidore of Seville (570-636), one book of the twenty being a 
brief summary of the arithmetic of Boethius. When we consider that his 
contemporaries regarded Isidore as the most learned man of his time, we 
can well appreciate the lament of his day that "the study of letters is dead in 


our midst." These were truly the "Dark Ages" of science, but we should not 
make the mistake of assuming that this was true of the Middle Ages as a 
whole. For the next two centuries the gloom continued to such an extent 
that it has been said that nothing scholarly could be heard in Europe but the 
scratching of the pen of the Venerable Bede (ca. 673-735) writing in England 
about the mathematics needed for the ecclesiastical calendar, or about the 
representation of numbers by means of the fingers. 

Alcuin of York (ca. 735-804) was born the year that Bede died; he was 
called by Charlemagne to revitalize education in France, and sufficient 
improvement was apparent to lead some historians to speak of a Carolingian 
Renaissance. Alcuin explained that the act of creation had taken six days 
because six was a perfect number; but beyond some arithmetic, geometry, 
and astronomy that Alcuin is reputed to have written for beginners, there 
was little mathematics in France or England for another two centuries. In 
Germany Hrabanus Maurus (784-856) continued the slight mathematical 
and astronomical efforts of Bede, especially in connection with the computa- 
tion of the date of Easter. But not for another century and a half was there 
any notable change in the mathematical climate in Western Europe, and 
then it came in the person of one who rose ultimately to become Pope 
Sylvester II. 

Gerbert (ca. 940-1003) was born in France and educated in Spain and 
Italy, and then served in Germany as tutor and later adviser to the Holy 
Roman Emperor, Otto III. Having served as archbishop, first at Reims and 
later at Ravenna, Gerbert in 999 was elevated to the papacy, taking the 
name Sylvester — possibly in recollection of an earlier pope who had been 
noted for scholarship, but more probably because Sylvester I, pope during 
the days of Constantine, symbolized the unity of papacy and empire. Gerbert 
was active in politics, both lay and ecclesiastical, but he had time also for 
educational matters. He wrote on both arithmetic and geometry, depending 
probably on the Boethian tradition, which had dominated the teaching in 
Western church schools and had not improved! More interesting than these 
expository works, however, is the fact that Gerbert was perhaps the first 
one in Europe to have taught the use of the Hindu-Arabic numerals. It is not 
clear how he came in contact with these. A possible explanation is that when 
he went to Spain in 967 he came in touch, perhaps at Barcelona, with Moorish 
learning, including Arabic numeration with the western or Gobar (dust) 
forms of the numerals, although there is little evidence of Arabic influence 
in extant documents. A Spanish copy of the Origines of Isidore, dating from 
992, contains the numerals, without the zero, and Gerbert probably never 
knew of this last part of the Hindu-Arabic system. In certain manuscripts of 
Boethius, however, similar numeral forms, or apices, appear as counters for use 


on a computing board or abacus ; and perhaps it was from these that Gerbert 
first learned of the new system. The Boethian apices, on the other hand, may 
themselves have been later interpolations. The situation with respect to the 
introduction of the numerals into Europe is about as confused as is that sur- 
rounding the invention of the system perhaps half a millennium earlier. More- 
over, it is not clear that there was any continued use of the new numerals in 
Europe during the two centuries following Gerbert. Not until the thirteenth 
century was the Hindu-Arabic system definitively introduced into Europe, 
and then the achievement was not the work of one man, but of several. 2 

Europe, before and during the time of Gerbert, was not yet ready for 
developments in mathematics. The Christian attitude, expressed by Tertul- 
lian, had at first been somewhat the same as that of early Islam, cited with 
respect to the library at Alexandria. Scientific research, Tertullian wrote, 
had become superfluous since the gospel of Jesus Christ had been received. 
The time of Gerbert was the high point of Muslim learning, but contemporary 
Latin scholars could scarcely have appreciated Arabic treatises if they had 
learned about them. By the early twelfth century the situation began to 
change in a direction reminiscent of the ninth century in Arabia. One cannot 
absorb the wisdom of one's neighbors if one cannot understand their 
language. The Moslems had broken down the language barrier to Greek 
culture in the ninth century, and the Latin Europeans overcame the language 
barrier to Arabic learning in the twelfth century. At the beginning of the 
twelfth century no European could expect to be a mathematician or an 
astronomer, in any real sense, without a good knowledge of Arabic; and 
Europe, during the earlier part of the twelfth century, could not boast of a 
mathematician who was not a Moor, a Jew, or a Greek. By the end of the 
century the leading and most original mathematician in the whole world 
came from Christian Italy. So obviously was the period one of transition from 
an older to a newer point of view that C. H. Haskins entitled his work The 
Renaissance of the Twelfth Century. 3 The revival of which he wrote began of 
necessity with a spate of translations. At first these were almost exclusively 
from Arabic into Latin, but by the thirteenth century there were many 
variants — Arabic to Spanish, Arabic to Hebrew, Greek to Latin, or com- 
binations such as Arabic to Hebrew to Latin. The Elements of Euclid was 
among the earliest of the mathematical classics to appear in Latin translation 
from the Arabic, the version being produced in 1 142 by Adelard of Bath (ca. 
1075-1160). It is not clear how the Englishman had come into contact with 
Muslim learning. There were at the time three chief bridges between Islam 

2 See G. F. Hill, The Development of Arabic Numerals in Europe (1915), and D. E. Smith and 
L. C. Karpinski, The Hindu-Arabic Numerals (1911). 

3 A paperback edition (New York: Meridian Books, 1957) is readily available. 


and the Christian world— Spain, Sicily, and the Eastern Empire— and of 
these the first was the most important. Adelard, however, seems not to have 
been one of the many who made use of the Spanish intellectual bridge. It is 
not easy to tell whether the religious crusades had a positive influence on the 
transmission of learning, but it is likely that they disrupted channels of 
communication more than they facilitated them. At all events, the channels 
through Spain and Sicily were the most important in the twelfth century, 
and these were largely undisturbed by the marauding armies of the crusaders 
from 1096 to 1272. The revival of learning in Latin Europe took place during 
the crusades, but probably in spite of the crusades. 

Adelard 's translation of the Elements did not become very influential for 
another century, but it was far from an isolated event. Adelard earlier (1 126) 
had translated al-Khowarizmi's astronomical tables from Arabic into Latin, 
and later (ca. 1155) Ptolemy's Almagest from Greek into Latin. Among the 
early translators, however, Adelard was an exception in that he was not one 
of the large group working in Spain. There, especially at Toledo, where the 
archbishop encouraged such work, a veritable school of translation was 
developing. The city, once a Visigothic capital and later in the hands of the 
Moors for several centuries before falling to the Christians, was an ideal 
spot for the transfer of learning. In Toledo libraries there was a wealth of 
Muslim manuscripts; and of the populace, including Christians, Moham- 
medans, and Jews, many spoke Arabic, facilitating the interlingual flow of 
information. The cosmopolitanism of the translators in Spain is evident 
from some of the names : Robert of Chester, Hermann the Dalmatian, Plato 
of Tivoli, Rudolph of Bruges, Gerard of Cremona, and John of Seville, the 
last a converted Jew. These are but a small portion of the men associated in 
the translation projects in Spain. 4 

Of the translators in Spain, perhaps the greatest was Gerard of Cremona 
(1114-1187). He had gone to Spain to learn Arabic in order to understand 
Ptolemy, but he devoted the rest of his life to translations from the Arabic. 
Among these was the translation into Latin of a revised version of Thabit ibn 
Qurra's Arabic of Euclid's Elements, a better piece of work than that of 
Adelard. In 1 175 Gerard translated the Almagest, and it was chiefly through 
this work that Ptolemy came to be known in the West. Translations of more 
than eighty-five works are ascribed to Gerard of Cremona, but only the 
translation of Ptolemy is dated. Among the works of Gerard was a Latin 
adaptation of the Algebra of al-Khowarizmi, but an earlier and more 
popular translation of the Algebra had been made in 1145 by Robert of 
Chester. This, the first translation of al-Khowarizmi's treatise (as Robert's 
translation of the Koran, a few years before, had marked another "first"), 
may be taken as marking the beginning of European algebra. 

4 For others see George Sarton, Introduction to the History of Science, II (U 1 13 ff, 338 ff. 


Robert of Chester returned to England in 1150, but the Spanish work of 
translation continued unabated through Gerard and others. The works of 
al-Khowarizmi evidently were among the more popular subjects of the time, 
and the names of Plato of Tivoli and John of Seville are attached to still 
other adaptations of the Algebra. Western Europe suddenly took far more 
favorably to Arabic mathematics than it ever had to Greek geometry. 
Perhaps part of the reason for this is that Arabic arithmetic and algebra 
were on a more elementary level than Greek geometry had been during the 
days of the Roman republic and empire. However, the Romans had never 
displayed much interest in Greek trigonometry, relatively useful and elemen- 
tary though it was ; yet Latin scholars of the twelfth century devoured Arabic 
trigonometry as it appeared in astronomical works. It was Robert of Chester's 
translation from the Arabic that resulted in our word "sine." The Hindus 
had given the name jiva to the half chord in trigonometry, and the Arabs 
had taken this over asjiba. In the Arabic language there is also a word jaib 
meaning "bay" or "inlet." When Robert of Chester came to translate the 
technical word jiba, he seems to have confused this with the word jaib 
(perhaps because vowels were omitted) ; hence he used the word sinus, the 
Latin word for "bay" or "inlet." Sometimes the more specific phrase sinus 
rectus, or "vertical sine," was used ; hence the phrase sinus versus, or our 
"versed sine," was applied to the "sagitta," or the "sine turned on its side." 
It was during the twelfth-century period of translation and the following 
century that the confusion arose concerning the name al-Khowarizmi and 
led to the word "algorithm," as explained in the preceding chapter. The 
Hindu numerals had been explained to Latin readers by Adelard of Bath and 
John of Seville at about the same time that an analogous scheme was intro- 
duced to the Jews by Abraham ibn Ezra (ca. 1090-1167), author of books on 
astrology, philosophy, and mathematics. As in the Byzantine culture the first 
nine Greek alphabetic numerals, supplemented by a special zero symbol, 
took the place of the Hindu numerals, so Ibn Ezra used the first nine Hebraic 
alphabetic numerals, and a circle for zero, in the decimal positional system 
for integers. Despite the numerous accounts of the Hindu-Arabic numerals, 
the transition from the Roman number scheme was surprisingly slow. 
Perhaps this was because computation with the abacus was quite common, 
and in this case the advantages of the new scheme are not nearly so apparent 
as in calculation with pen and paper only. For several centuries there was 
keen competition between the "abacists" and the "algorists," and the latter 
triumphed definitively only in the sixteenth century. 

6 It is sometimes claimed that in the later Middle Ages there were two 
classes of mathematicians— those in the church or university schools and 
those concerned with trade and commerce— and that rivalries are found 



between the two. There seems to be little basis for such a thesis ; certainly in 
the spread of the Hindu-Arabic numerals both groups shared in the dis- 
semination. Thirteenth-century authors from many walks of life helped to 
popularize "algorism," but we shall mention three in particular. One of 
them, Alexandre de Villedieu (ft ca. 1225), was a French Franciscan ; another, 

A woodcut from Gregor Reisch, Margarita Philosophica (Freiburg, 1503). Arithmetic is 
instructing the algorist and the abacist, here inaccurately represented by Boethius and 


John of Halifax (ca. 1200-1256), known also as Sacrobosco, was an English 
schoolman; and the third was Leonardo of Pisa (ca. 1180-1250), better 
known as Fibonacci, or "son of Bonaccio," an Italian merchant. The Carmen 
de algorismo of Alexandre is a poem in which the fundamental operations on 
integers are fully described, using the Hindu-Arabic numerals and treating 
zero as a number. The Algorismus vulgaris of Sacrobosco was a practical 
account of reckoning that rivaled in popularity his Sphaera, an elementary 


tract on astronomy used in the schools throughout the later Middle Ages. 
The book in which Fibonacci described the new algorism is a celebrated 
classic, completed in 1202, but it bears a misleading title — Liber abaci (or 
book of the abacus). It is not on the abacus; it is a very thorough treatise on 
algebraic methods and problems in which the use of the Hindu-Arabic 
numerals is strongly advocated. 

Leonardo's father was a Pisan engaged in business in northern Africa, 
and the son studied under a Muslim teacher and traveled in Egypt, Syria, and 
Greece. It therefore was natural that Fibonacci should have been steeped 
in Arabic algebraic methods, including, fortunately, the Hindu-Arabic 
numerals and, unfortunately, the rhetorical form of expression. The Liber 
abaci opens with an idea that sounds almost modern, but which was character- 
istic of both Islamic and Christian medieval thought — that arithmetic and 
geometry are connected and support each other. This view is, of course, 
reminiscent of al-Khowarizmi's Algebra, but it was equally accepted in the 
Latin Boethian tradition. The Liber abaci, nevertheless, is much more con- 
cerned with number than with geometry. It first describes "the nine Indian 
figures," together with the sign 0, "which is called zephirum in Arabic." 
Incidentally, it is from zephirum and its variants that our words "cipher" and 
"zero" are derived. Fibonacci's account of Hindu-Arabic numeration was 
important in the process of transmission ; but it was not, as we have seen, the 
first such exposition, nor did it achieve the popularity of the later but more 
elementary descriptions by Sacrobosco and Villedieu. The horizontal bar 
in fractions, for example, was used regularly by Fibonacci (and was known 
before in Arabia), but it was only in the sixteenth century that it came into 
general use. (The slanted solidus was suggested in 1845 by De Morgan.) 

The Liber abaci 5 is not a rewarding book for the modern reader, for after 
explanation of the usual algoristic or arithmetic processes, including the 
extraction of roots, it stresses problems in commercial transactions, using a 
complicated system of fractions in computing exchanges of currency. It is 
one of the ironies of history that the chief advantage of positional notation — 
its applicability to fractions — almost entirely escaped the users of the Hindu- 
Arabic numerals for the first thousand years of their existence. In this respect 
Fibonacci was as much to blame as anyone, for he used three types of 
fractions — common, sexagesimal, and unit — but not decimal fractions. In 
the Liber abaci, in fact, the two worst of these systems — unit fractions and 

5 There is no English translation of this important work, nor even a readily accessible Latin 
version. It is included in the Bullettino di Bibliografia e di Storia delle Scienze Mathematiche e 
Fisicheoi Baldassare Boncompagni (Rome, 1 868-1 887, 20 vols.). For notations used by Fibonacci 
and others, see Florian Cajori, A History of Mathematical Notations (Chicago, 1928-1929, 
2 vols.). 


common fractions — are extensively used. Moreover, problems of the follow- 
ing dull type abound : If 1 solidus imperial, which is 12 deniers imperial, is 
sold for 31 deniers Pisan, how many deniers Pisan should one obtain for 
11 deniers imperial? In a recipe type of exposition the answer is found 
laboriously to be ^ 28 (or, as we should write it, 28^). Fibonacci customarily 
placed the fractional part or parts of a mixed number before the integral part. 
Instead of writing llf , for example, he wrote ^ 11, with juxtaposition of 
unit fractions and integers implying addition. 

Fibonacci evidently was fond of unit fractions — or he thought his readers 
were — for the Liber abaci includes tables of conversion from common 
fractions to unit fractions. The fraction ^, for instance, is broken into 
Too so 5 4 i, and ^ appears as ^s i i 2 • An unusual quirk in his notation led 
him to express the sum of ^ J and ^ f as fh% 1, the notation H^> meaning 
in this case 

1 6 2 

+ — 

2-9-10 9 10 10 

Analogously in another of the many problems on monetary conversion in the 
Liber abaci we read that if H of a rotulus is worth \ £ f of a bizantium, then 
s-f-re of a bizantium is worth \ t 8 8 4 3 9 \\ of a rotulus. Pity the poor medieval 
businessman who had to operate with such a system ! 

Much of the Liber abaci makes dull reading, but some of the problems 8 
were so lively that they were used by later writers. Among these is a hardy 
perennial which may have been suggested by a similar problem in the Ahmes 
papyrus. As expressed by Fibonacci, it read : 

Seven old women went to Rome ; each woman had seven mules ; each mule 
carried seven sacks, each sack contained seven loaves; and with each loaf 
were seven knives ; each knife was put up in seven sheaths. 

Without doubt the problem in the Liber abaci that has most inspired 
future mathematicians was the following : 

How many pairs of rabbits will be produced in a year, beginning with a single 
pair, if in every month each pair bears a new pair which becomes productive 
from the second month on? 

This celebrated problem gives rise to the "Fibonacci sequence" 1, 1, 2, 3, 
5, 8, 13, 21, . . . , u„, . . . , where u„ = u„_ l + u„_ 2 , that is, where each term 
after the first two is the sum of the two terms immediately preceding it. This 
sequence has been found to have many beautiful and significant properties. 
For instance, it can be proved that any two successive terms are relatively 


prime and that lim u„_Ju n is the golden section ratio (^/s - l)/2. The 
sequence is applicable also to questions in phyllotaxy and organic growth. 

9 The Liber abaci was Fibonacci's best known book, appearing in another 
edition in 1228, but it evidently was not appreciated widely in the schools, 
and it did not appear in print until the nineteenth century. Leonardo of Pisa 
was without doubt the most original and most capable mathematician of 
the medieval Christian world, but much of his work was too advanced to 
be understood by his contemporaries. His treatises other than the Liber abaci 
also contain many good things. In the Flos, dating from 1225, there are 
indeterminate problems reminiscent of Diophantus and determinate 
problems reminiscent of Euclid, the Arabs, and the Chinese. 

Fibonacci evidently drew from many and varied sources. Especially 
interesting for its interplay of algorithm and logic is Fibonacci's treatment 
of the cubic equation x 3 + 2x 2 + lOx = 20. The author showed an attitude 
close to that of the modern period in first proving the impossibility of a root 
in the Euclidean sense, such as a ratio of integers, or a number of the form 
a + v /b ) where a and b are rational. As of that time, this meant that the 
equation could not be solved exactly by algebraic means. Fibonacci then 
went on to express the positive root approximately as a sexagesimal fraction 
to half a dozen places— 1 ; 22,7,42,33,4,40. This was a remarkable achieve- 
ment, but we do not know how he did it. Perhaps through the Arabs he had 
learned what we call "Horner's method," a device known before this time in 
China. This is the most accurate European approximation to an irrational 
root of an algebraic equation up to that time — or anywhere in Europe for 
another 300 years and more. It is characteristic of the time that Fibonacci 
should have used sexagesimal fractions in theoretical mathematical work but 
not in mercantile affairs. Perhaps this explains why the Hindu-Arabic 
numerals were not promptly used in astronomical tables, such as the Alfonsine 
Tables of the thirteenth century. Where the "Physicists' " (sexagesimal) frac- 
tions were in use, there was less urgency in displacing them than there was in 
connection with the common and unit fractions in commerce. 

10 In 1225 Leonardo of Pisa published not only the Flos, but also the Liber 
quadratorum, a brilliant work on indeterminate analysis. This, like Flos, 
contains a variety of problems, some of which stemmed from the mathematical 
contests held at the court of the emperor Frederick II, to which Fibonacci 
6 For some further mathematical properties see N. N. Vorob'ev, Fibonacci Numbers, trans, 
by H. Mors (New York: Blaisdell, 1961); S. M. Plotnick, "The Sum of Terms of the Fibonacci 
Series." Scripta Mathematics 9 (1943), 197. For the relevance of the sequence in biology, see 
D. W. Thompson. On Growth and Form, 2nd ed. (Cambridge University Press, 1952). See also 
issues of The Fibonacci Quarterly. Interesting applications, and further references, are given in 
H. S. M. Coxeter, "The Golden Section, Phyllotaxis, and Wythoff's Game," Scripta Mathematica, 
19(1953), 135-143. 


had been invited. One of the problems proposed strikingly resembles the 
type in which Diophantus had delighted — to find a rational number such 
that if five is added to, or subtracted from, the square of the number, the 
result will be the square of a rational number. Both the problem and a 
solution, 3-j^, are given in Liber quadratorum. The book makes frequent use 
of the identities 

(a 2 + b 2 )(c 2 + d 2 )=(ac + bdf + (be - ad) 2 

= {ad + be) 2 + (ac - bd) 2 

which had appeared in Diophantus and had been widely used by the Arabs. 
Fibonacci, in some of his problems and methods, seems to follow the Arabs 
closely. 7 

Fibonacci was primarily an algebraist, but he wrote also, in 1220, a book 
entitled Practica geometriae. This seems to be based on an Arabic version of 
Euclid's Division of Figures (now lost) and on Heron's works on mensuration. 
It contains among other things a proof that the medians of a triangle divide 
each other in the ratio 2 to 1, and a three-dimensional analogue of the 
Pythagorean theorem. Continuing a Babylonian and Arabic tendency, he 
used algebra to solve geometrical problems. 

It will be clear from the few illustrations we have given that Leonardo of 11 
Pisa was an unusually capable mathematician. It is true that he was without 
a worthy rival during the 900 years of medieval European culture, but he 
was not quite the isolated figure he is sometimes held to be. He had an able 
though less gifted younger contemporary in Jordanus Nemorarius (date 
uncertain). Some 8 identify this man with Jordanus Teutonicus or Jordanus 
of Saxony, leader of the Dominican Order, who died in 1237. In any case, our 
Jordanus Nemorarius, or Jordanus de Nemore, represents a more Aris- 
totelian aspect of science than others we have met in the thirteenth century, 
and he became the founder of what sometimes is known as the medieval 
school of mechanics. To him we owe the first correct formulation of the law 
of the inclined plane, a law that the ancients had sought in vain : the force 
along an oblique path is inversely proportional to the obliquity, where 
obliquity is measured by the ratio of a given segment of the oblique path to 
the amount of the vertical intercepted by that path 9 — that is, the "run" over 
the "rise." In the language of trigonometry this means that F: W = 1/csc 6, 

7 See L. C. Karpinski, "The Algebra of Abu Kamil," American Mathematical Monthly, 21 
(1914), 37-48. 

s See, for example, D. E. Smith, History oj Mathematics, I, 226, and George Sarton, Introduc- 
tion to the History of Science, II (2), 613 f. The identification is denied by Joseph Hoffmann, 
Geschichte der Mathematik, 2nd ed. (Berlin, 1963), I, 96. 

9 See Clagett, The Science of Mechanics in the Middle Ages, p. 74. 


which is equivalent of the modern formulation F = WsinO, where W is 
weight, F is force, and 8 is the angle of inclination. 

Jordanus was the author of books on arithmetic, geometry, and astronomy, 
as well as mechanics. His Arithmetica in particular was the basis of popular 
commentaries at the University of Paris as late as the sixteenth century: 
this was not a book on computation, but a quasi-philosophical work in the 
tradition of Nicomachus and Boethius. It contains such theoretical results 
as the theorem that any multiple of a perfect or abundant number is abundant 
and that a divisor of a perfect number is deficient. The Arithmetica is signifi- 
cant especially for the use of letters instead of numerals as numbers, thus 
making possible the statement of general algebraic theorems. In the arith- 
metical theorems in Euclid's Elements VII-IX, numbers had been represented 
by line segments to which letters had been attached, and the geometrical 
proofs in al-Khowarizmi's Algebra made use of lettered diagrams; but all 
coefficients in the equations used in the Algebra are specific numbers, 
whether represented by numerals or written out in words. The idea of 
generality is implied in al-Khowarizmi's exposition, but he had no scheme 
for expressing algebraically the general propositions that are so readily 
available in geometry. In the Arithmetica the use of letters suggests the 
concept of "parameter" ; but Jordanus' successors generally overlooked his 
scheme of letters. They seem to have been more interested in the Arabic 
aspects of algebra found in another Jordanian work, De numeris datis, a 
collection of algebraic rules for finding, from a given number, other numbers 
related to it according to certain conditions, or for showing that a number 
satisfying specific restrictions is determined. A typical instance is the follow- 
ing: If a given number is divided into two parts such that the product of one 
part by the other is given, then each of the two parts is necessarily determined. 
The rule is expressed awkwardly by Jordanus as follows: 

Let the given number be abc and let it be divided into two parts ab and c, and 
let d be the given product of the parts ab and c. Let the square of abc be e and let 
four times d be/ and let g be the result of taking /from e. Then g is the square 
of the difference between ab and c. Let h be the square root of g. Then h is the 
difference between ab and c. Since h is known, c and ab are determined. 10 

Note that Jordanus' use of letters is somewhat confusing, for, like Euclid, 
he sometimes uses two letters for a number and sometimes only a single 
letter. He- evidently followed Euclid in picturing the given number as a line 
segment ac and the two parts into which it is subdivided as ab and be; but 
he uses both end-point letters to designate the first part or number, and only 
the single letter c to represent the number of line segment be. It is greatly to 
his credit, however, that he first stated the rule, equivalent to the solution of a 

10 For an extensive account of many aspects of the work of Jordanus see Moritz Cantor, 
Vorlesungen uber Geschichte der Mathematik (1880-1908), II, 49-79. 


quadratic equation, completely in general form. Only later did he provide a 
specific example of it, expressed in Roman numerals : to divide the number X 
into two parts the product of which is to be XXI, Jordanus follows through 
the steps indicated above to find that the parts are III and VII. 

To Jordanus is attributed also an Algorismus (or Algorithmus) demonstra- 1 2 
tus, an exposition of arithmetic rules that was popular for three centuries. 
The Algorismus demonstratus again shows Boethian and Euclidean inspira- 
tion, as well as Arabic algebraic characteristics. Still greater preponderance 
of Euclidean influence is seen in the work of Johannes Campanus of Novara 
(fl. ca. 1260), chaplain to Pope Urban IV. To him the late medieval period 
owed the authoritative translation of Euclid from Arabic into Latin, the one 
that first appeared in printed form in 1482. In making the translation Campa- 
nus used various Arabic sources, as well as the earlier Latin version by 
Adelard. Both Jordanus and Campanus discussed the angle of contact, or 
horn angle, a topic that produced lively discussion in the later medieval period 
when mathematics took on a more philosophical and speculative aspect. 
Campanus noticed that if one compared the angle of contact — that is, the 
angle formed by an arc of a circle and the tangent at an end point— with the 
angle between two straight lines, there appears to be an inconsistency with 
Euclid's Elements X. 1, the fundamental proposition of the method of ex- 
haustion. The rectilineal angle is obviously greater than the horn angle. Then 
if from the larger angle we take away more than half, and if from the remainder 
we take away more than half, and if we continue in this way, each time taking 
away more than half, ultimately we should reach a rectilineal angle less than 
the horn angle ; but this obviously is not true. Campanus correctly concluded 
that the proposition applies to magnitudes of the same kind, and horn 
angles are different from rectilineal angles. 

Similarity in the interests of Jordanus and Campanus is seen in the fact 
that Campanus, at the end of Book IV of his translation of the Elements, 
describes an angle trisection which is exactly the same as that which had 
appeared in Jordanus' De triangulis. The only difference is that the lettering 
of the Campanus diagram is Latin, whereas that of Jordanus is Greco-Arabic. 
The trisection, unlike those in antiquity, is essentially as follows. Let the angle 
AOB that is to be trisected be placed with its vertex at the center of a circle 
of any radius OA = OB (Fig. 14.1). From O draw a radius OC 1 OB, and 
through A place a straight line AED in such a way that DE = OA. Finally, 
through draw line OF parallel to AED. Then /_FOB is one-third /_AOB, 
as required. 11 

11 Marshall Clagett, Archimedes in the Middle Ages (1964), I, 681. See also Moritz Cantor 
Vorlesungen iiber Geschichte der Mathematik, II, 75 f, 94. A more sophisticated trisection, using 
the hmacon, is attributed to Jordanus (see Clagett, The Science of Mechanics in the Middle Ages 
pp. 666-677). 


1 3 The thirteenth century presents such a striking advance over the earlier 
Middle Ages that it has occasionally been viewed, none too impartially, as 
"the greatest of centuries." 12 We have seen how, in the work of Leonardo 
of Pisa, Western Europe had come to rival other civilizations in the level of 
its mathematical achievement ; but this was only a small part of what was 
taking place in Latin culture as a whole. Many of the famous universities— 
Bologna, Paris, Oxford, and Cambridge— were established in the late 
twelfth and early thirteenth centuries, and this was the period in which 
great Gothic cathedrals— Chartres, Notre Dame, Westminster, Reims— were 
built. Aristotelian philosophy and science had been recovered and were 
taught in the universities and church schools. The thirteenth century is the 
period of great scholars and churchmen, such as Albertus Magnus, Robert 
Grosseteste, Thomas Aquinas, and Roger Bacon. Incidentally, two of these 
in particular, Grosseteste and Bacon, made strong pleas for the importance 
of mathematics in the curriculum, although neither was himself much of a 
mathematician. It was during the thirteenth century that many practical 
inventions became known in Europe — gunpowder and the compass, both 
perhaps from China, and spectacles from Italy, with mechanical clocks 
appearing only a little later. 

The twelfth century had seen the great tide of translation from Arabic into 
Latin, but there now were other crosscurrents of translations. Most of the 
works of Archimedes, for example, had been virtually unknown to the 
medieval West; but in 1269 William of Moerbeke (ca. 1215-1286) published 
a translation (the original manuscript of which was discovered in 1884 in 
the Vatican) from Greek into Latin of the chief Archimedean scientific and 
mathematical treatises. Moerbeke, who came from Flanders and was named 
Archbishop of Corinth, knew little mathematics; hence his excessively 
literal translation (helpful now in reconstructing the original Greek text) 

12 J. J. Walsh, The Thirteenth, Greatest of Centuries (New York, 1909). 


was of limited usefulness, but from this time on most of the works of Archi- 
medes were at least accessible. In fact, the Moerbeke translation included 
parts of Archimedes with which the Arabs evidently were not familiar, such 
as the treatise On Spirals, the Quadrature of the Parabola, and Conoids and 
Spheroids. Nevertheless, the Muslims had been able to make more progress 
in understanding the mathematics of Archimedes than did the Europeans 
during the medieval period. 

During the twelfth century the works of Archimedes had not completely 
escaped the attention of the indefatigable Gerard of Cremona, who had 
converted into Latin an Arabic version of the short work on Measurement 
of the Circle, which was used in Europe for several centuries. There had 
circulated also, before 1269, a portion of the Archimedean Sphere and 
Cylinder. These two examples could provide only a very inadequate idea of 
what Archimedes had done, and therefore the translation by Moerbeke was 
of the greatest importance, including as it did a number of major treatises. 
It is true that the version was only occasionally used during the next two 
centuries, but it at least remained extant. It was this translation that became 
known to Leonardo da Vinci and other Renaissance scholars, and it was 
Moerbeke's version that was first printed in the sixteenth century. 13 

The history of mathematics has not been a record of smooth and continu- 1 4 
ous development ; hence it should come as no surprise that the upward surge 
during the thirteenth century should have lost some of its momentum. There 
was no Latin equivalent of Pappus to stimulate a revival of classical higher 
geometry. The works of Pappus were not available in Latin or Arabic. Even 
Apollonius' Conies was little known, beyond some of the simplest properties 
of the parabola that arose in connection with the ubiquitous treatises on 
optics, a branch of science that fascinated the scholastic philosophers. The 
science of mechanics, too, appealed to the scholars of the thirteenth and 
fourteenth centuries, for now they had at hand both the statics of Archimedes 
and the kinematics of Aristotle. 

We noted earlier that the Aristotelian conclusions on motion had not gone 
unchallenged and modifications had been suggested, especially by Philopo- 
nus. During the fourteenth century the study of change in general, and of 
motion in particular, was a favorite topic in the universities, especially at 
Oxford and Paris. In Merton College at Oxford the scholastic philosophers 
had deduced a formulation for uniform rate of change which today generally 
is known as the Merton rule. Expressed in terms of distance and time, the 
rule says essentially that if a body moves with uniformly accelerated motion, 
then the distance covered will be that which another body would have 

1 3 For further details see Marshall Clagett, "The Impact of Archimedes on Medieval Science," 
his, 50 (1959), 419-429. See also Clagett's definitive work, Archimedes in the Middle Ages. 


covered had it been moving uniformly for the same length of time with a 
speed equal to that of the first body at the midpoint of the time interval. As 
we should formulate it, the average velocity is the arithmetic mean of the 
initial and terminal velocities. Meanwhile, at the University of Paris there 
was developed a more specific and clear-cut doctrine of impetus, in which 
we can recognize a concept akin to our inertia, than that proposed by 

1 5 The late medieval physicists comprised a large group of university teachers 
and churchmen, but we call attention to only two, for these were also promin- 
ent mathematicians. The first is Thomas Bradwardine (12907-1349), a 
philosopher, theologian, and mathematician who rose to the position of 
Archbishop of Canterbury; the second is Nicole Oresme (13237-1382), a 
Parisian scholar who became Bishop of Lisieux. To these two men was due 
a broadened view of proportionality. 14 The Elements of Euclid had included 
a logically sound theory of proportion, or the equality of ratios, and this had 
been applied by ancient and medieval scholars to scientific questions. For a 
given time, the distance covered in uniform motion is proportional to the 
speed; and for a given distance, the time is inversely proportional to the 
speed. Aristotle had thought, none too correctly, that the speed of an object 
subject to a moving force acting in a resisting medium is proportional to the 
force and inversely proportional to the resistance. In some respects this 
formulation seemed to later scholars to contradict common sense. When 
force F is equal to or less than resistance, a velocity V will be imparted 
accordingly to the law V = KF/R, where K is a nonzero constant of propor- 
tionality ; but when resistance balances or exceeds force, one should expect 
no velocity to be acquired. To avoid this absurdity Bradwardine made use 
of a generalized theory of proportions. In his Tractatus de proportionibus of 
1328, Bradwardine developed the Boethian theory of double or triple or, 
more generally, what we would call "n-tuple" proportion. His arguments are 
expressed in words, but in modern notation we would say that in these cases 
quantities vary as the second or third or nth power. In the same way the 
theory of proportions included subduple or subtriple or sub-n-tuple propor- 
tion, in which quantities vary as the second or third or nth root. Now Brad- 
wardine was ready to propose an alternative to the Aristotelian law of motion. 
To double a velocity that arises from some ratio or proportion F/R, he said, 
it was necessary to square the ratio F/R ; to triple the velocity, one must cube 
the "proportio" or ratio F/R ; to increase the velocity n-fold, one must take 
the nth power of the ratio F/R. This is tantamount to asserting that velocity 

14 See especially Nicole Oresme, De proportionibus proportionum and Ad pauca respicientes, 
ed. and trans, by Edward Grant (Madison, Wis.: University of Wisconsin Press, 1966). Cf. 
Edward Grant, "Part I of Nicole Oresme's Algorismus proportionum," Isis, 56 (1965), 327-341 . 


is given, in our notation, by the relationship V = K log F/R, for \og{F/R) n = 
n log F/R. That is, if V = log F /R , then V n = log(F /K )" = n log F /R = 
nV . Bradwardine himself evidently never sought experimental confirmation 
of his law, and it seems not to have been widely accepted. 

Bradwardine wrote also several other mathematical works, all pretty 
much in the spirit of the times. His Arithmetic and his Geometry show the 
influence of Boethius, Aristotle, Euclid, and Campanus. Bradwardine, 
known in his day as "Doctor profundus," was attracted also to topics such 
as the angle of contact and star polygons, both of which occur in Campanus 
and earlier works. Star polygons, which include regular polygons as special 
cases, go back to ancient times. A star polygon is formed by connecting with 
straight lines every mth point, starting from a given one, of the n points that 
divide the circumference of a circle into n equal parts, where n > 2 and m is 
prime to n. There is in the Geometry even a touch of Archimedes' Measurement 
of the Circle. The philosophical bent in all of Bradwardine's works is seen 
most clearly in the Geometrica speculative! and the Tractatus de continuo, in 
which he argued 15 that continuous magnitudes, although including an 
infinite number of indivisibles, are not made up of such mathematical atoms, 
but are composed instead of an infinite number of continua of the same kind. 
His views sometimes are said to resemble those of the modern intuitionists; 
at any rate, medieval speculations on the continuum, popular among 
Scholastic thinkers like St. Thomas Aquinas, later influenced the Cantorian 
infinite of the nineteenth century. 

Nicole Oresme lived later than Bradwardine, and in the work of the former 1 6 
we see extensions of ideas of the latter. In De proportionibus proportionum, 
composed about 1 360, Oresme generalized Bradwardine's proportion theory 
to include any rational fractional power and to give rules for combining 
proportions that are the equivalents of our laws of exponents, now expressed 
in the notations x m -x" = x m+n and {x m f = x mn . For each rule specific 
instances are given; and the latter part of another work, the Algorismus 
proportionum, applies the rules in geometrical and physical problems. 
Oresme suggested also the use of special notations for fractional powers, for 
in his Algorismus proportionum there are expressions such as 





15 See Edward Stamm, "Tractatus de continuo von Thomas Bradwardina. Eine Handschrift 
aus dem XIV. Jahrhundert," I sis, 26 (1936), 13-32. 


to denote the "one and one-half proportion"— that is, the cube of the 
principal square root — and forms such as 

1 P- 1 


for $%. We now take for granted our symbolic notations for powers and 
roots, with little thought for the slowness with which these developed in the 
history of mathematics. Even more imaginative than Oresme's notations was 
his suggestion that irrational proportions are possible. Here he was striving 
toward what we should write as x- 71 , for example, which is perhaps the first 
hint in the history of mathematics of a higher transcendental function ; but 
lack of adequate terminology and notation prevented him from effectively 
developing his notion of irrational powers. 16 

1 7 The notion of irrational powers may have been Oresme's most brilliant 
idea, but it was not in this direction that he was most influential. For almost a 
century before his time Scholastic philosophers had been discussing the 
quantification of variable "forms," a concept of Aristotle roughly equivalent 
to qualities. Among these forms were such things as the velocity of a moving 
object and the variation in temperature from point to point in an object with 
nonuniform temperature. The discussions were interminably prolix, for the 
available tools of analysis were inappropriate. Despite this handicap the 
logicians at Merton College had reached, as we saw, an important theorem 
concerning the mean value of a "uniformly difform" form — that is, one in 
which the rate of change of the rate of change is constant. Oresme was well 
aware of this result, and to him occurred, some time before 1361, a brilliant 
thought— why not draw a picture or graph of the way in which things vary? 1 7 
Here we see, of course, an early suggestion of what we now describe as the 
graphical representation of functions. Everything measurable, Oresme wrote, 
is imaginable in the manner of continuous quantity ; hence he drew a velocity- 
time graph for a body moving with uniform acceleration. Along a hori- 
zontal line he marked points representing instants of time (or longitudes), 
and for each instant he drew perpendicular to the line of longitudes a line 

16 For an admirable account of this work see Edward Grant, "Nichole Oresme and his De 
proportionibus protortionum," Isis, 51 (1960), 293-314. Cf. Edward Grant, "Bradwardine and 
Galileo: Equality of Velocities in the Void," Archive for History of Exact Sciences, 2 (1965), 
344-364. See also references in footnote 14. 

17 We here imply, for simplicity of exposition, that Oresme was the first one to have this idea, 
but this is not necessarily the case. Marshall Clagett has found what looks like an earlier graph, 
drawn by Giovanni di Cosali, in which the line of longitude is placed in a vertical position. See 
Marshall Clagett, Science of Mechanics in the Middle Ages, pp. 332-333, 414. In any event, the 
exposition of Oresme surpasses that of Cosali in clarity and influence, and so our account does 
not do any real violence to history. 



segment (latitude) the length of which represented the velocity. The end 
points of these segments, he saw, lie along a straight line ; and if the uniformly 
accelerated motion starts from rest, the totality of velocity lines (which 
we call ordinates) will make up the area of a right triangle (see Fig. 14.2.) 

FIG. 14.2 

Inasmuch as this area represents the distance covered, Oresme has provided 
a geometrical verification of the Merton rule, for the velocity at the midpoint 
of the time interval is half the terminal velocity. Moreover, the diagram leads 
obviously to the law of motion generally ascribed to Galileo in the seven- 
teenth century. It is clear from the geometrical diagram that the area in the 
first half of the time is to that in the second half in the ratio 1 to 3. If we sub- 
divide the time into three equal parts, the distances covered (given by the 
areas) are in the ratio 1:3:5. For four equal subdivisions the distances are 
in the ratio 1 : 3 : 5 : 7. In general, as Galileo later observed, the distances are 
to each other as the odd numbers ; and since the sum of the first n consecutive 
odd numbers is the square of n , the total distance covered varies as the 
square of the time, the familiar Galilean law for falling bodies. 

The terms latitude and longitude that Oresme used are in a general sense 
equivalent to our ordinate and abscissa, and his graphical representation is 
akin to our analytic geometry. His use of coordinates was not, of course, 
new, for Apollonius, and others before him, had used coordinate systems, 
but his graphical representation of a variable quantity was novel. He seems 
to have grasped the essential principle that a function of one unknown can 
be represented as a curve, but he was unable to make any effective use of this 
observation except in the case of the linear function. Moreover, Oresme was 
chiefly interested in the area under the curve ; hence it is not very likely that 
he saw the other half of the fundamental principle of analytic geometry — 
that every plane curve can be represented, with respect to a coordinate 
system, as a function of one variable. Where we say that the velocity graph 
in uniformly accelerated motion is a straight line, Oresme wrote, "Any 
uniformly difform quality terminating in zero intensity is imagined as a 
right triangle." That is, Oresme was more concerned with the calculus 


aspects of the situation : (1) the way in which the function varies (that is, the 
differential equation of the curve), and (2) the way in which the area under 
the curve varies (that is, the integral of the function). He pointed out the 
constant-slope property for his graph of uniformly accelerated motion — an 
observation equivalent to the modern two-point equation of the line in 
analytic geometry and leading to the concept of the differential triangle. 
Moreover, in finding the distance function, the area, Oresme obviously is 
performing geometrically a simple integration that results in the Merton 
rule. He did not explain why the area under a velocity-time curve represents 
the distance covered, but it is probable that he thought of the area as made 
up of many vertical lines or indivisibles each of which represented a velocity 
that continued for a very short time. 

The graphical representation of functions, known then as the latitude of 
forms, remained a popular topic from the time of Oresme to that of Galileo. 
The Tractatus de latitudinibus formarum, written perhaps by a student of 
Oresme, if not by Oresme himself, appeared in numerous manuscript forms 
and was printed at least four times between 1482 and 1515 ; but this was only 
a precis of a larger work by Oresme entitled Tractatus defiguratione poten- 
tiarum et mensurarum. 18 Here Oresme went so far as to suggest a three- 
dimensional extension of his "latitude of forms" in which a function of two 
independent variables was pictured as a volume made up of all the ordinates 
erected according to a given rule at points in a portion of the reference plane. 
We even find a hint of a geometry of four dimensions when Oresme speaks 
of representing the intensity of a form for each point in a reference body or 
volume. What he really needed here was, of course, an algebraic geometry 
rather than a pictorial representation such as he had in mind ; but weakness 
in technique hampered Europe throughout the medieval period. 

1 8 Mathematicians of the Western world during the fourteenth century had 
imagination and precision of thought, but they were lacking in algebraic 
and geometrical facility ; hence their contributions lay not in extensions of 
classical work, but in new points of view. Among these was an occupation 
with infinite series, an essentially novel topic anticipated only by some 
ancient iterative algorithms and Archimedes' summation of an infinite 
geometrical progression. Where the Greeks had a horror inftniti, the late 
medieval Scholastic philosophers referred frequently to the infinite, both as a 
potentiality and as an actuality (or something "completed"). In England in 
the fourteenth century a logician by the name of Richard Suiseth (fl. ca. 

18 See especially two articles by Heinrich Wieleitner, "Der 'Tractatus de latitudinibus 
formarum' des Oresme," Bibliotheca Mathematica (3), 13 (1913), 113-145, and "Ueber den 
Funktionsbegriff und die graphische Darstellung bei Oresme," Bibliotheca Mathematica (3), 
14 (1914), 193-243. See also Marshall Clagett, Science of Mechanics in the Middle Ages. 


1350), but better known as Calculator, solved the following problem in the 

latitude of forms : 

If throughout the first half of a given time interval a variation continues at a 
certain intensity, throughout the next quarter of the interval at double this 
intensity, throughout the following eighth at triple the intensity and so ad 
infinitum; then the average intensity for the whole interval will be the intensity 
of the variation during the second subinterval (or double the initial intensity). 

This is equivalent to saying that the sum of the infinite series 

i + I + I + • • • + n/2" + • • • 

is 2. Calculator gave a long and tedious verbal proof, for he did not know 
about graphical representation, but Oresme used his graphical procedure 
to prove the theorem more easily. Oresme handled also other cases, such as 

H 2J 3J n-3 

4 + 16 +_ 64~ + '" + ^ r + '" 

in which the sum is f . Problems similar to these continued to occupy scholars 
during the next century and a half. 19 

Among other contributions of Oresme to infinite series was his proof, 
evidently the first in the history of mathematics, that the harmonic series is 
divergent. He grouped the successive terms in the series 

1111111 1 

placing the first term in the first group, the next two terms in the second group, 
the next four terms in the third group, and so on, the wth group containing 
2 m " l terms. Then it is obvious that we have infinitely many groups and that 
the sum of the terms within each group is at least \. Hence by adding together 
enough terms in order, we can exceed any given number. 20 

We have traced the history of mathematics in Europe through the Dark 1 9 
Ages of the early medieval centuries to the high point in the time of the 
Scholastics. From the nadir in the seventh century to the work of Fibonacci 
and Oresme in the thirteenth and fourteenth centuries the improvement had 
been striking; but the combined efforts of all medieval civilizations were in 
no sense comparable to the mathematical achievements in Ancient Greece. 
The progress of mathematics had not been steadily upward in any part of 
the world— Babylonia, Greece, China, India, Arabia, or the Roman world— 
and it should come as no surprise that in Western Europe a decline set in 

19 For more details see C. B. Boyer, History of the Calculus (1959), pp. 86-87, and H. Busard, 
"Uber unendliche Reihen im Mittelalter," L'Enseignement Mathematique, 8, Nos. 3^ (1962). 

20 See John Murdoch, "Oresme's Commentary on Euclid," Scripta Mathematica, 27 (1964) 


following the work of Bradwardine and Oresme. In 1349 Thomas Bradwar- 
dine had succumbed to the Black Death, the worst scourge ever to strike 
Europe. Estimates of the number of those who died of the plague within the 
short space of a year or two run between a third and a half of the population. 
This catastrophe inevitably caused severe dislocations and loss of morale. 
If we note that England and France, the nations that had seized the lead in 
mathematics in the fourteenth century, were further devastated in the 
fifteenth century by the Hundred Year's War and the Wars of the Roses, the 
decline in learning will be understandable. Italian, German, and Polish 
universities during the fifteenth century took over the lead in mathematics 
from the waning Scholasticism of Oxford and Paris, and it is primarily to 
representatives from these lands that we now turn. 


Boncompagni, Baldassare, ed., Bullettino di bibliografia e di storia delle scienze mathe- 

matische efisiche (Rome, 1 868-1 887, 20 vols ; reprint, New York : Johnson Reprint). 
Boyer, C. B., History of the Calculus (paperback ed., New York : Dover, 1959). 
Busard, H., "Uber unendliche Reihen im Mittelalter," L ' Enseignement Mathematique, 

8, Nos. 3^1 (1962). 
Cantor, Moritz, Vorlesungen uber Geschichte der Mathematik (Leipzig: Teubner 

1900-1908, 4 vols.). 
Clagett, Marshall, The Science of Mechanics in the Middle Ages (Madison, Wis. 

University of Wisconsin Press, 1959). 
Clagett, Marshall, Archimedes in the Middle Ages (Madison, Wis. : University of Wiscon- 
sin Press, 1964- , 2 vols.). 
Duhem, Pierre, Les origines de la statique (Paris, 1905-1906, 2 vols.). 
Ginsburg, Benjamin, "Duhem and Jordanus Nemorarius," Isis, 25 (1936), 340-362. 
Grant, Edward, "Bradwardine and Galileo: Equality of Velocities in the Void," 

Archive for History of Exact Sciences, 2 (1965), 344-364. 
Grant, Edward, "Nicole Oresme and his De proportionibus proportionum;' Isis, 51 

(1960), 293-314. 
Grant, Edward, "Part I of Nicole Oresme's Algorismus proportionum," Isis, 56 (1965), 

Grant, Edward, ed., Nicole Oresme: De proportionibus proportionum and Ad pauca 

respicientes (Madison, Wis.: University of Wisconsin Press, 1966). 
Hill, G. F., The Development of Arabic Numerals in Europe (Oxford : Clarendon, 1915). 
Murdoch, John, "Oresme's Commentary on Euclid," Scripta Mathematica, 27 (1964), 

Sarton, George, Introduction to the History of Science (Baltimore: Carnegie Institution 

of Washington, 1927-1948, 3 vols in 5). 
Smith, D. E., History of Mathematics (Boston: Ginn, 1923-1925, 2 vols.; paperback 

reprint. New York : Dover, 1 958). 


Smith, D. E., and L. C. Karpinski, The Hindu-Arabic Numerals (Boston : Ginn, 191 1). 

Sullivan, J. W. N., The History of Mathematics in Europe from the Fall of Greek Science 
to the Rise of the Conception of Mathematical Rigour (New York : Oxford Univer- 
sity Press, 1925). 

Wieleitner, Heinrich, "Der Tractatus de latitudinibus formarum' des Oresme," 
Bibliotheca Mathematica (3), 13 (1913), 113-145. 

Wieletner, Heinrich, "Ueber den Funktionsbegriff und die graphische Darstellung bei 
Oresme," Bibliotheca Mathematica (3), 14 (1914), 193-243. 

Wieleitner, Heinrich, "Zur Geschichte der unendlichen Reihen im christlichen Mittel- 
alter," Bibliotheca Mathematica (3), 14 (1914), 150-168. 

Youschkevitch, A. P., Geschichte der Mathematik im Mittelalter (Leipzig- Teubner 


1 . Compare the mathematical work of one representative, living in about the year 500, from 
each of the following civilizations : China, India, Rome, Greece. 

2. In what ways were the crusades likely to help or hinder the transmission of mathematics 
from Islam to the Christian world? 

3. Was Western Europe in 1 1 50 in closer touch with the Arabic or the Greek world? Which 
had relatively more to offer in mathematics? Give reasons for your answers. 

4. Which three of the following — Euclid, Archimedes, Apollonius, Diophantus, Boethius, 
al-Khowarizmi— would you think were the most influential mathematical authors in 
Europe in 1 250? Give reasons. 

5. Compare the sources of support for mathematicians in medieval Europe with those in 
medieval Arabia. 

6. Write the number 980,765 in the notation of Planudes. 

7. For a unit circle express the versed sine of an angle in terms of the sine of the same angle. 
Explain how the names sine and versed sine arose. 

8. Verify the answer given by Fibonacci in the problem (see text) of converting from a fractional 
part of a bizantium to a fractional part of a rotulus. 

9. Find the ratio of u 12 to u 13 in the Fibonacci sequence. To how many significant figures is 
this in agreement with the golden-section ratio? 

10. Prove that Fibonacci's cubic, x 1 + 2x 2 + Wx = 20, has no rational root. 

11. Prove that the equation in Exercise 10 has no root of the form a + ^/b, where a and b are 

12. Find to the nearest hundredth a root of the cubic in Exercise 10 and show that to this extent 
Fibonacci's answer and yours are in agreement. 

13. Verify Jordanus' rule (see text) for dividing a "given number abc." 

14. Prove the Jordanus-Campanus trisection construction. 

15. Using Bradwardine's law, and assuming that a force of 10 lb produces in a body a velocity 
of 20 ft/sec against a resistance of 2 lb, what velocity will be produced in the body against 
the same resistance by a force of 40 lb? 

16. Draw Bradwardine's star polygon for eleven points on a circle if we connect in order every 
seventh point. 

17. Prove for three equal subdivisions of the time interval that Oresme's ratio 1:3:5 for the 
distances covered is correct. . 


*18. Verify Calculator's summation of the series 

i -. 

.= ,2" 
*19. Verify Oresme's summation of the series 

.= i 4" 
*20. Prove, using Oresme's method, that the series 

1111 1 

is divergent. 


The Renaissance 

I will sette as I doe often in woorke use, a paire of 

paralleles, or Gemowe [twin] lines of one lengthe, thus : 

, bicause noe 2. thynges, can be moare equalle. 

Robert Recorde 

The fall of Constantinople in 1453 signaled the collapse of the Byzantine 
Empire, and in this respect it serves a convenient chronological placeholder 
in the history of political events. The significance of the date for the history 
of mathematics however, is, a moot point. It is frequently asserted that at that 
time refugees fled to Italy with treasured manuscripts of ancient Greek 
treatises, thereby putting the Western European world in touch with the 
works of antiquity. It is as likely, though, that the fall of the city had just 
the opposite effect : that now the West no longer could count on what had 
been a dependable source of manuscript material for ancient classics, both 
literary and mathematical. Whatever the ultimate decision may be on this 
matter, there can be no question that mathematical activity was again rising 
during the middle years of the fifteenth century. Europe was recovering from 
the physical and spiritual shock of the Black Death, and the then-recent 
invention of printing with movable type made it possible for learned works 
to become much more widely available than ever before. The earliest printed 
book from Western Europe is dated 1447, and by the end of the century 
over 30,000 editions of various works were available. Of these, few were 
mathematical ; but the few, coupled with existing manuscripts, provided a 
base for expansion. The recovery of unfamiliar Greek geometrical classics 
was at first less significant than the printing of medieval Latin translations 
of Arabic algebraic and arithmetic treatises, for few men of the fifteenth cen- 
tury either read Greek or were sufficiently proficient in mathematics to profit 
from the works of the better Greek geometers. A substantial portion of the 
treatises of Archimedes had, in fact, been accessible in Latin through the 
translation of William of Moerbeke, but to little avail, for there were few to 
appreciate classical mathematics. In this respect mathematics differed from 
literature, and even from the natural sciences. As Humanists of the fifteenth 


Title page of the first English version of Euclid's Elements (London. 1 570). The translation 

purports to be by Sir Henry Billingsley. later Lord Mayor of London, but part or all of rt 

may be by John Dee. writer of the preface. 


and sixteenth centuries fell ever more deeply in love with the newly redis- 
covered Greek treasures in science and the arts, their estimate of the im- 
mediately preceding Latin and Arabic achievements declined. Classical 
mathematics, except for the most elementary portions of Euclid, was an 
intensely esoteric discipline, accessible only to those with a high degree of 
preliminary training ; hence the disclosure of Greek treatises in this field 
did not at first seriously impinge on the continuing medieval mathematical 
tradition. Medieval Latin studies in elementary geometry and the theory of 
proportions, as well as Arabic contributions to arithmetic operations and 
algebraic methods, did not present difficulties comparable to those associated 
with the works of Archimedes and Apollonius. It was the more elementary 
branches that were to attract notice and to appear in printed works. 

Oresme had argued that everything measurable can be represented by a 
line (latitude); and a mathematics of mensuration, both from a theoretical 
and a practical standpoint, flourished during the early Renaissance period. 
A similar view was adopted by Nicholas of Cusa (1401-1464), a man who well 
represents the weaknesses of the age, for he was on the border line between 
medieval and modern times. (Cusa was a Latin place-name for a city on the 
Mosel.) Nicholas saw that a scholastic weakness in science had been a failure 
to measure; mens, he thought, was etymologically related to mensura, so 
that knowledge must be based on measurement. Cusa (or Cusanus, the Latin 
form) also was influenced by the Humanist concern for antiquity and 
espoused Neoplatonic views. Moreover, he had access to a translation of 
some of Archimedes work made in 1450 by Jacob of Cremona. But, alas, 
Nicholas of Cusa was better as an ecclesiastic than as a mathematician. ' 
In the Church he rose to the rank of cardinal, but in the field of mathematics 
he is known as a misguided circle-squarer. His philosophical doctrine of the 
"concordance of contraries" led him to believe that maxima and minima 
are related, hence that the circle (a polygon with the greatest possible number 
of sides) must be reconcilable with the triangle (the polygon with the smallest 
number of sides). He believed that through an ingenuous averaging of 
inscribed and circumscribed polygons he had arrived at a quadrature. That he 
was wrong was of less significance than that he was one of the first modern 
Europeans to attempt a problem that had fascinated the best minds of antiq- 
uity, and that his effort stimulated contemporaries to criticism of his work. 

Among those who pointed out the error in Cusa's reasoning was Regio- 
montanus (1436-1476), probably the most influential mathematician of the 

1 For an overappreciative account of his work see Max Simon, Cusanus als M athematiker 
(Strassburg, 191 1, in Festschrift Heinrich Weber, Leipzig and Berlin : Teubner, 1912, pp. 298-337). 
For a modern edition (in German) of the works of Nicholas of Cusa see his Mathematische 
Schriften, ed. by J. E. Hofmann, (Hamburg : F. Meiner, ca. 1952, 1950). 


Title page of Gregor Reisch, Margarita philosophica (1503). Around the three-headed 

figure in the center are grouped the seven liberal arts, with arithmetic seated in the middle 

and holding a counting board. 

fifteenth century, and one whose birth date might be taken to mark the 
beginning of the new age. Having studied at the universities of Leipzig and 
Vienna, where he developed a love for mathematics and astronomy, Regio- 
montanus accompanied Cardinal Bessarion to Rome, where he acquired a 
proficiency in Greek and became acquainted with the crosscurrents of 
scientific and philosophical thought. Bessarion, once Archbishop of Nicaea, 
had won a cardinal's hat from Pope Eugenius IV in Rome (1439) for efforts 
to unite the Greek and Latin churches. He thus became a link between the 
classical learning preserved at Constantinople and the young Renaissance 


movement in the West. It probably was his association with the cardinal 
that inspired in Regiomontanus the ambition to acquire, translate, and 
publish the scientific legacy of antiquity. After travel and study in Italy, 
Regiomontanus returned to Germany, where he set up a printing press and 
an observatory at Nuremberg in order to advance the interests of science 
and literature. He hoped to print translations of Archimedes, Apollonius, 
Heron, Ptolemy, and other scientists, but his tragic death at the early age of 
forty cut short his ambitious project. In 1475 he had been invited to Rome 
by Pope Sixtus IV to share in one of the perennial attempts to reform the 
calendar, but he died there (some said he was poisoned by enemies) shortly 
after he had arrived. The trade list of books he planned to print survives, 2 
and this indicates that the development of mathematics undoubtedly would 
have been accelerated had he survived. He was, in his wide and varied 
interests, a typical "Renaissance man," as his adopted name indicates. 
He was born "Johann Miiller of Konigsberg," but like others of his day he 
preferred to be known by the Latin form of his birthplace, the Germanic 
Konigsberg ("king's mountain") becoming Regiomontanus. 

Regiomontanus had become familiar, during his stay in Italy, with some 
of the leading figures of his day, and he entered into correspondence with 
others on current questions. His interests were broad, but he seems to have 
had little sympathy with the speculative thought of Nicholas of Cusa, which 
he criticized severely. In astronomy his chief contribution was the completion 
of a new Latin version, begun by his teacher at Vienna, Georg Peuerbach 
(1423-1469), of Ptolemy 's Almagest. Peuerbach 's Theoricae novae planetarum, 
a new textbook of astronomy, which was published in Regiomontanus' 
shop in 1472, was an improvement on the ubiquitous copies of the Sphere of 
Sacrobosco ; but Humanists felt the need for a better Latin edition of the 
Almagest than the medieval version that had been derived from the Arabic. 
(The Humanists insisted on elegance and purity in their classical languages ; 
hence they abhorred the barbarous medieval Latin, as well as the Arabic 
from which it often was derived.) Peuerbach had planned to make a trip to 
Italy with Regiomontanus to seek a good manuscript copy, but he died 
prematurely and the completion of the plan devolved upon his student. 
Regiomontanus' translation project resulted also in textbooks of his own. 
His Epitome of Ptolemy's Almagest is noteworthy for its emphasis on the 
mathematical portions that had often been omitted in commentaries dealing 
with elementary descriptive astronomy. Of greater significance for math- 
ematics, however, was his De triangulis omnimodis, a systematic account of 
the methods for solving triangles which marked the rebirth of trigonom- 

2 See George Sarton, "The Scientific Literature Transmitted Through the Incunabula " 
Osiris, 5 (1938), 41-247. 


New works on astronomy invariably had been accompanied by tables 
of trigonometric functions, and Peuerbach's works had included a new table 
of sines. In these cases, however, trigonometry was serving merely , as the 
handmatd of astronomy. In India, where the sine function eviden ly ^ had it 
birth, there had been little interest in this function apart from its role in the 
astronomical systems or Siddhantas. Even among the Arabs for whom 
trigonometry was second only to algebra in mathematical appeal, the subject 
had had no independent existence, except in the Treatise on the QuadrUateral 
tfNasir Eddin, a work that owed more to the Greeks than to the Hindus. 
The twelfth-century age of translation in Europe had included some Arab c 
trigonometry, but for several centuries Latin contributions were only pale 
imitations of the Arabic. The Practica geometriae of Fibonacci and the works 
of Bradwardine had contained some fundamentals of trigonometry gleaned 
from Muslim sources, but it was not until Regiomontanus began writing 
his De triangulis that Europe gained preeminence in this field It appears that 
Regiomontanus was acquainted with the work of Nasir Eddin, and this may 
have been the source of his desire to organize trigonometry as a discipline 
independent of astronomy. 

The first book of De triangulis, composed in about 1464, opens with 
fundamental notions, derived largely from Euclid, on magn itude » andrato£ 
then there are more than fifty propositions on the solution of triangles 
using the properties of right triangles. Book II begins with a clear statement 
and proof of the law of sines, and then includes problems on determining 
sides angles, and areas of plane triangles when given determinate conditions. 
Among fhe problems, for example, is the following : If the base of a triangle 
anTthe angle opposite are known, and if either the altitude to the base or 
the area is given men the sides can be found. Book III [contains theorems of 
the sort found in ancient Greek texts on "spherics" before the _ use of ^ g- 
onometry; Book IV is on spherical trigonometry, including the spherical 

^ The uTeof area "formulas," written out in words, was among the novelties 
in Regiomontanus' De triangulis, but in the avoidance of the tangent function 
the work falls short of Nasir Eddin's treatment. The tangent function never- 
theless was included in another trigonometric treatise by Regiomontanus- 
Tabulae directionum. Revisions of Ptolemy had suggested the need for new 
tables, and these were supplied by a number of fifteenth-century astronomers, 
of whom Regiomontanus was one. In order to avoid fractions it was custom- 
ary to adopt a large value for the radius of the circle, or the smus totus. For 
one of his sine tables Regiomontanus followed his ^mediate predecessors im 
using a radius of 600,000; for others he adopted 10,000,000 or 600 000,000^ 
For his tangent table in Tabulae directionum he chose 100,000. He does not 
call the function "tangent," but uses only the word "numerus for the 


entries, degree by degree, in a tabulation headed "Tabula fecunda" ("Pro- 
ductive Table"). The entry for 89° is 5,729,796, and for 90° simply infinite. 
The sudden death of Regiomontanus occurred before his two trigonometric 
works were published, and this considerably delayed their effect. The 
Tabulae directionum was published in 1490, but the more important treatise, 
De triangulis, appeared in print only in 1533 (and again in 1561). Nevertheless, 
the works were known in manuscript form to the circle of mathematicians at 
Nuremberg, where Regiomontanus was working, and it is very likely that 
they influenced work of the early sixteenth century. 3 For a hundred years 
after the fall of Constantinople, cities in central Europe, notably Vienna, 
Cracow, Prague, and Nuremberg, were leaders in astronomy and math- 
ematics. The last of these became a center for the printing of books (as well as 
for learning, art, and invention), and some of the greatest scientific classics 
were published there toward the middle of the sixteenth century. 

A general study of triangles led Regiomontanus to a consideration of 
problems of geometrical construction somewhat reminiscent of Euclid's 
Division of Figures. For example, one is asked to construct a triangle given 
one side, the altitude to this side, and the ratio of the other two sides. Here, 
however, we find a striking departure from ancient customs : whereas Euclid's 
problems invariably had been given in terms of general quantities, Regio- 
montanus gave his lines specific numerical values, even where he intended 
that his methods should be general. This enabled him to make use of the 
algorithmic methods developed by Arabic algebraists and transmitted to 
Europe in twelfth-century translations. In the construction problem above, 
one of the unknown sides can be expressed as a root of a quadratic equation 
with known numerical coefficients, and this root is constructible by devices 
familiar from Euclid's Elements, or Al-Khowarizmi's Algebra. (As Regio- 
montanus expressed it, he let one part be the "thing" and then solved by 
the rule of "thing" and "square" — that is, through quadratic equations.) 
Another problem in which Regiomontanus called for the construction of a 
cyclic quadrilateral, given the four sides, can be handled similarly. 

The algebra of Regiomontanus, like that of the Arabs, was rhetorical. The 
Arithmetica of Diophantus, in which some syncopation had been adopted, 
was known in Greek to Regiomontanus, who hoped ultimately to translate 
it ; but it was from al-Khowarizmi that Europe learned the routine algebraic 
procedures. The Arithmetica was, after all, concerned primarily with the more 
recondite aspects of number theory. Moreover, Regiomontanus did not get 

3 An extensive account of his work and influence is included in Sister Mary Claudia Zeller, 
The Development of Trigonometry from Regiomontanus to Pitiscus (1944). There is an English 
translation of De triangulis under the title Regiomontanus On Triangles, ed. by Barnabas Hughes 


around to publishing it, and few Latin scholars were aware of its contents for 
another century, until 1575 when it appeared in Latin. In fact, the influence 
of Regiomontanus in algebra was restricted not only by his adherence to the 
rhetorical form of expression and by his early death. His manuscripts, on 
his death, came into the hands of a Nuremberg patron who failed to make the 
work effectively accessible to posterity. Europe learned its algebra painfully 
and slowly from the thin Greek, Arabic, and Latin tradition that trickled 
down through the universities, the church scribes, the rising mercantile 
activities, and scholars from other fields. 

Regiomontanus stood at a critical juncture in the history of science, and he 
had the tastes and the abilities to make the most of this. His love of classical 
learning was shared by the Humanists, but unlike them he was strongly 
inclined toward the sciences. Moreover, he did not indulge in the Humanist 
contempt for Scholastic and Arabic learning, and he was a Renaissance man 
in his concern for the practical arts as well as for scholarship. What better 
combination could a modern scientist have had than a good library, an 
observatory, a printing press, and a love of knowledge? Regiomontanus was 
aware, through his contact with Averroists in the Italian universities, that 
the Arabic astronomers had been worried about inconsistencies between the 
schemes of Aristotle and Ptolemy; and he undoubtedly knew also that 
Oresme and Cusa had seriously raised the possibility of the earth's moving. 
It is reported that he planned to reform astronomy ; had he lived, he might 
have anticipated Copernicus. His premature death cut short all such schemes, 
and astronomy and mathematics had to look to others for the next steps, 
including in particular an isolated French figure outside of the mainstream 
of development. 

It was Germany and Italy that provided most of the early Renaissance 
mathematicians, but in France in 1484 a manuscript was composed which 
in level and significance was perhaps the most outstanding since the Liber 
abaci of Fibonacci, almost three centuries before and which, like the Liber 
abaci, was not printed until the nineteenth century. This work, entitled 
Triparty en la science des nombres, was by Nicolas Chuquet (T ca. 1500), 
about whom we know virtually nothing except that he was born at Paris, 
took his bachelor's degree in medicine, and practiced at Lyons. The Triparty 
does not closely resemble any earlier work in arithmetic or algebra, and the 
only writers the author mentions are Boethius and Campanus. There is 
evidence of Italian influence, which possibly resulted from acquaintance with 
Fibonacci's Liber abaci. 

The first of the "Three Parts" concerns the rational arithmetic operations 
on numbers, including an explanation of the Hindu-Arabic numerals. Of 


these Chuquet says that "the tenth figure does not have or signify a value, and 
hence it is called cipher or nothing or figure of no value." The work is essen- 
tially rhetorical, the four fundamental operations being indicated by the 
words and phrases plus, moins, multiplier par, and partyr par, the first two 
sometimes abbreviated in the medieval manner as p and in. In connection 
with the computation of averages, Chuquet gave a regie des nombres moyens 
according to which (a + c)/(b + d) lies between a/b and c/d if a, b, c, d are 
positive numbers. In the second part, concerning ro ots of numbe rs, there is 
some syncopation, so that the modern expression J\A - ^/llo appears in 
the not very dissimilar form R) 2 . 14 . in . R) 2 180. 

The last and by far the most important part of the Triparty concerns the 
"Regie des premiers"— that is, the rule of the unknown, or what we should call 
algebra. During the fifteenth and sixteenth centuries there were various names 
for the unknown thing, such as res (in Latin), or chose (in French) or cosa (in 
Italian) or coss (in German); Chuquet's word premier is unusual in this 
connection. The second power he called champs (whereas the Latin had been 
census), the third cubiez, and the fourth champs de champ. For multiples of 
these Chuquet invented an exponential notation of great significance. The 
denominacion or power of the unknown quantity was indicated by an ex- 
ponent associated with the coefficient of the term, so that our modern expres- 
sions 5x and 6x 2 and 10x 3 appeared in the Triparty as .5. 1 and .6. 2 and .10. 3 . 
Moreover, zero and negative exponents take their place along with the 
positive integral powers, so that our 9x° became .9.°, and 9x~ 2 was written 
as .9. 2m -, meaning .9. seconds moins. Such a notation laid bare the laws of 
exponents, with which Chuquet may have become familiar through the work 
of Oresme on proportions. Chuquet wrote, for example, that .72. l divided 
by .8. 3 is .9. 2 ■■»■— that is, 72x - 8x 3 = 9x" 2 . Related to these laws is his ob- 
servation of the relationships between the powers of the number two, and the 
indices of these powers set out in a table from to 20, in which sums of the 
indices correspond to products of the powers. Except for the magnitude of 
the gaps between entries, this constituted a miniature table of logarithms to 
the base two. Observations similar to those of Chuquet were to be repeated 
several times during the next century, and these undoubtedly played a role 
in the ultimate invention of logarithms. 

The second half of the last part of the Triparty is devoted to the solution of 
equations. Here are many of the problems that had appeared among his 
predecessors, but there is also at least one significant novelty. In writing 
A 1 egaulx a m.2.°— that is, 4x = -2— Chuquet was for the first time 
expressing an isolated negative number in an algebraic equation. Generally 
he rejected zero as a root of an equation, but on one occasion he remarked 
that the number sought was 0. In considering equations of the form 
ax m + bx m + n = cx m + 2n (where the coefficients and exponents are specific 


positive integers), he found that some implied imaginary solutions ; in these 
cases he simply added, "Tel nombre est ineperible." 4 

The Triparty of Chuquet, like the Collectio of Pappus, is a book in which 
the extent of the author's originality cannot be determined. Each undoubtedly 
was indebted to his immediate predecessors, but we are unable to identify 
any of them. Moreover, in the case of Chuquet we cannot determine his 
influence on later writers. The Triparty was not printed until 1880, and prob- 
ably was known to few mathematicians ; but one of those into whose hands 
it fell used so much of the material that he can be charged with plagiarism, 
even though he mentioned Chuquet's name. The Larismethique nouvellement 
composee, published at Lyons by Etienne de la Roche in 1 520, and again in 
1538, depended heavily, as we now know, on Chuquet ; hence it is safe to say 
that the Triparty was not without effect. 

The earliest Renaissance algebra, that of Chuquet, was the product of a 
Frenchman, but the best known algebra of that period was published ten 
years later in Italy. In fact, the Summa de arithmetica, geometrica, propor- 
tion et proportionalita of the friar Luca Pacioli (1445-1514) overshadowed 
the Triparty so thoroughly that older historical accounts of algebra leap 
directly from the Liber abaci of 1202 to the Summa of 1494 without mentioning 
the work of Chuquet or other intermediaries. The way for the Summa, how- 
ever, had been prepared by a generation of algebraists, for the Algebra of 
al-Khowarizmi was translated into Italian at least by 1464, the date of a 
manuscript copy in the Plimpton Collection in New York; the writer of 
this manuscript stated that he based his work on numerous predecessors in 
this field, naming some from the earlier fourteenth century. The Renaissance 
in science often is assumed to have been sparked by the recovery of ancient 
Greek works; but the Renaissance in mathematics was characterized 
especially by the rise of algebra, and in this respect it was but a continuation 
of the medieval tradition. Regiomontanus had been well versed in Greek ; 
but he had not shared the Humanists' apotheosis of Hellenism, and he had 
been ready to recognize the importance of medieval Arabic and Latin algebra. 
He obviously had been familiar with the works of al-Khowarizmi and 
Fibonacci and had planned to print the De numeris datis of Jordanus Nem- 
orarius. Had Regiomontanus achieved his plans for publication, the Summa 
of Pacioli (or Paciuolo) would certainly not today be regarded as the first 
printed work on algebra. 

4 Good accounts of this work are found in Ch. Lambo, S. J., "Une algebre francaise de 1484. 
Nicolas Chuquet," Revue des Questions Scientifiques, (3), 2 (1902), 442-472, and in Aristide 
Marre, "Notice sur Nicolas Chuquet et son Triparty en la science des nombres," Bullettino di 
Bibliografia e di Storia delle Scienze Matematiche e Fisiche, 13 (1880), 555-659, 693-814; 14 
(1881), 413-460. 


The Summa, the writing of which had been completed by 1487, was more 
influential than it was original. It is an impressive compilation (with sources of 
information not generally indicated) of material in four fields : arithmetic, 
algebra, very elementary Euclidean geometry, and double-entry bookkeep- 
ing. Pacioli (also known as Luca di Borgo) for a time had been tutor to the 
sons of a wealthy merchant at Venice, and he undoubtedly was familiar with 
the rising importance in Italy of commercial arithmetic. The earliest printed 
arithmetic, appearing anonymously at Treviso in 1478, had featured the 
fundamental operations, the rules of two and three, and business applications. 
Several more technical commercial arithmetics appeared shortly thereafter, 
and Pacioli borrowed freely from them. One of these, the Compendio de lo 
abaco of Francesco Pellos (fi. 1450-1500), which was published at Torino 
in the year Columbus discovered America, made use of a dot to denote the 
division of an integer by a power often, thus adumbrating our decimal point. 

The Summa, which like the Triparty was written in the vernacular, was a 
summing up of unpublished works that the author had composed earlier, 
as well as of general knowledge at the time. The portion on arithmetic is 
much concerned with devices for multiplication and for finding square roots ; 
the section on algebra includes the standard solution of linear and quadratic 
equations. Although it lacks the exponential notation of Chuquet, there is 
increased use of syncopation through abbreviations. The letters p and m were 
by this time widely used in Italy for addition and subtraction, and Pacioli 
used co, ce, and ae for cosa (the unknown), censo (the square of the unknown), 
and aequalis respectively. For the fourth power of the unknown he naturally 
used cece (for square-square). Echoing a sentiment of Omar Khayyam, he 
believed that cubic equations could not be solved algebraically. 

Pacioli's work in geometry in the Summa was not significant, although some 
of his geometrical problems remind one of the algebraic geometry of Regio- 
montanus, specific numerical cases being employed. For example, it is 
required to find the sides of a triangle if the radius of the inscribed circle is 
four and the segments into which one side is divided by the point of contact 
are six and eight. Although Pacioli's geometry did not attract much attention, 
so popular did the commercial aspect of the book become that the author 
generally is regarded as the father of double-entry bookkeeping. 

Pacioli, the first mathematician of whom we have an authentic portrait, 8 
in 1 509 tried his hand twice more at geometry, publishing an undistinguished 
edition of Euclid and a work with the impressive title De divina proportione. 
The latter concerns regular polygons and solids and the ratio later known as 
"the golden section." It is noteworthy for the excellence of the figures, 5 which 

5 For a further description of this work and of contemporary activity, see R. Emmett Taylor, 
No Royal Road. Luca Pacioli and His Times (1942). 


have been attributed to Leonardo da Vinci (1452-1519). Leonardo frequently 
is thought of as a mathematician, but his restless mind did not dwell on 
arithmetic or algebra or geometry long enough to make a significant contri- 
bution. In his notebooks we find quadratures of lunes, constructions of 
regular polygons, and thoughts on centers of gravity and on curves of double 
curvature ; but he is best known for his application of mathematics to science 
and the theory of perspective. Da Vinci is pictured as the typical all-round 
Renaissance man; and in fields other than mathematics there is much to 
support such a view. Leonardo was a genius of bold and original thought, 
a man of action as well as contemplation, at once an artist and an engineer ; 
but he appears not to have been in close touch with the chief mathematical 
trend of the time — the development of algebra. Few subjects depend as 
heavily on a continuous bookish tradition and long-continued concentration 
as does mathematics, and Leonardo was not one to maintain concentrated 
library research or even to pursue his own imaginative ideas to their con- 
clusions. Ultimately, hundreds of years later, Renaissance notions on math- 
ematical perspective were to blossom into a new branch of geometry, but 
these developments were not perceptibly influenced by the thoughts that 
the left-handed Leonardo entrusted to his notebooks in the form of mirror- 
written entries. 

The word Renaissance inevitably brings to mind Italian literary, artistic, 
and scientific treasures, for renewed interest in art and learning became appar- 
ent in Italy earlier than in the other parts of Europe. There, in a rough-and- 
tumble conflict of ideas, men learned to put greater trust in independent 
observations of nature and judgments of the mind. Moreover, Italy had been 
one of the two chief avenues along which Arabic learning, including algorism 
and algebra, had entered Europe. Nevertheless, other parts of Europe did not 
remain far behind, as the work of Regiomontanus and Chuquet shows. In 
Germany, for example, books on algebra became so numerous that for a time 
the Germanic word coss for the unknown triumphed in other parts of Europe, 
and the subject became known as the "cossic art." Moreover, the Germanic 
symbols for addition and subtraction ultimately displaced the Italian p and 
m. In 1489, before the publication of Pacioli's Summa, a German lecturer at 
Leipzig, Johann Widman (born ca. 1460), had published a commercial 
arithmetic, Rechenung auff alien Kauffmanschqffi, the oldest book in which 
our familiar + and — signs appear in print. At first used to indicate excess 
and deficiency in warehouse measures, they later became symbols of the famil- 
iar arithmetic operations. 6 Widman, incidentally, possessed a manuscript 

6 See J. W. L. Glaisher, "On the Early History of the Signs + and - and on the Early German 
Arithmeticians," Messenger of Mathematics, 51 (1921-1922), 1-148. 



copy of the Algebra of al-Khowarizmi, a work well known to other German 

Among the numerous Germanic algebras was Die Coss, written in 1524 by 
Germany's celebrated Rechenmeister, Adam Riese (1492-1559). The author 
was the most influential German writer in the move to replace the old com- 
putation (in terms of counters and Roman numerals) by the newer method 
(using the pen and Hindu-Arabic numerals) ; so effective were his numerous 
arithmetic books that the phrase "nach Adam Riese" still survives in 
Germany as a tribute to accuracy in arithmetic processes. Riese, in his Coss, 
mentions the Algebra of al-Khowarizmi and refers to a number of Germanic 
predecessors in the field. 

The first half of the sixteenth century saw a flurry of German algebras, 
among the most important of which were the Coss (1525) of Christoph 
Rudolff (ca. 1 500— ca. 1 545), the Rechnung (1 527) of Peter Apian (1495-1 552), 
and the Arithmetica Integra (1544) of Michael Stifel (ca. 1487-1567). The first 
is especially significant as one of the earliest printed works to make use of 
decimal fractions, as well as of the modern symbol for roots ; the second is 

Title page of an edition (1 529) of one of the Rechenbucher of Adam Riese, the celebrated 
Rechenmeister. It depicts a contest between an algorist and an abacist. 


worth recalling for the fact that here, in a commercial arithmetic, the so-called 
"Pascal triangle" was printed on the title page, almost a century before Pascal 
was born. The third work, Stifel's Arithmetica Integra, was the most important 
of all the sixteenth-century German algebras. It, too, includes the Pascal 
triangle, but it is more significant for its treatment of negative numbers, 
radicals, and powers. Through the use of negative coefficients in equations, 
Stifel was able to reduce the multiplicity of cases of quadratic equations to 
what appeared to be a single form ; but he had to explain, under a special rule, 
when to use + and when — . Moreover, even he failed to admit negative 
numbers as roots of an equation. Stifel, a onetime monk turned itinerant 
Lutheran preacher, and for a time Professor of Mathematics at Jena, was 
one of the many writers who popularized the "German" symbols + and — at 
the expense of the "Italian" p and m notation. He was thoroughly familiar 
with the properties of negative numbers, despite the fact that he called them 
"numeri absurdi." About irrational numbers he was somewhat hesitant, 
saying that they are "hidden under some sort of cloud of infinitude." Again 
calling attention to the relations between arithmetic and geometric progres- 
sions, as had Chuquet for powers of two from to 20, Stifel extended the table 
to include 2 _t = \ and 2" 2 = \ and 2" 3 = \ (without, however, using 
exponential notation). For powers of the unknown quantity in algebra Stifel 
in Arithmetica integra used abbreviations for the German words coss, zensus, 
cubus, and zenzizensus; but in a later treatise, De algorithmi numerorum 
cossicorum, he proposed using a single letter for the unknown and repeating 
the letter for higher powers of the unknown, a scheme later employed by 
Harriot. 7 

1 The Arithmetica integra was a thorough treatment of algebra as generally 
known up to 1544, but by the following year it was in a sense quite outmoded. 
Stifel gave many examples leading to quadratic equations, but none of his 
problems lead to mixed cubic equations, for the simple reason that he knew 
no more about the algebraic solution of the cubic than did Pacioli or Omar 
Khayyam. In 1545, however, the solution not only of the cubic but of the 
quartic as well became common knowledge through the publication of the 
Ars magna of Geronimo Cardano (1501-1576). Such a striking and unantici- 
pated development made so strong an impact on algebraists that the year 
1545 frequently is taken to mark the beginning of the modern period in math- 
ematics. It must be pointed out immediately, however, that Cardano (or 
Cardan) was not the original discoverer of the solution of either the cubic or 
the quartic. He himself candidly admitted this in his book. The hint for solving 

7 For accounts of early books in arithmetic and algebra see especially D. E. Smith, Rara 
arithmetica (1908). For lists of early arithmetics see J. E. Hofmann : Geschichte der Mathematik 
(1963), Vol. I, pp. 142-145. 


the cubic, he averred, he had obtained from Niccolo Tartaglia (ca. 1500- 
1557); the solution of the quartic was first discovered by Cardan's quondam 
amanuensis, Ludovico Ferrari (1522-1565). What Cardan failed to mention 
in Ars magna is the solemn oath he had sworn to Tartaglia that he would not 
disclose the secret, for the latter intended to make his reputation by publishing 
the solution of the cubic as the crowning part of his treatise on algebra. 

Lest one feel undue sympathy for Tartaglia, it may be noted that he had 
published an Archimedean translation (1 543), derived from Moerbeke, leaving 
the impression that it was his own, and in his Quesiti et inventioni diverse 
(Venice, 1546) he gave the law of the inclined plane, presumably derived from 
Jordanus Nemorarius, without proper credit. It is, in fact, possible that 
Tartaglia himself had received a hint concerning the solution of the cubic from 
an earlier source. Whatever may be the truth in a rather complicated and 
sordid controversy between proponents of Cardan and Tartaglia, it is clear 
that neither of the principals was first to make the discovery. The hero in the 
case evidently was one whose name is scarcely remembered today — Scipione 
del Ferro (ca. 1465-1526), professor of mathematics at Bologna, one of the 
oldest of the medieval universities and a school with a strong mathematical 
tradition. How or when del Ferro made his wonderful discovery is not known. 
He did not publish the solution, but before his death he had disclosed it to a 
student, Antonio Maria Fior (or Floridus in Latin), a mediocre mathemati- 

Word of the existence of an algebraic solution of the cubic seems to have 
gotten around, and Tartaglia tells us that knowledge of the possibility of 
solving the equation inspired him to devote himself to finding the method 
for himself. Whether independently or on the basis of a hint, Tartaglia did 
indeed learn, by 1541, how to solve cubic equations. When news of this 
spread, a mathematical contest between Fior and Tartaglia was arranged. 
Each contestant proposed thirty questions for the other to solve within a 
stated time interval. When the day for decision had arrived, Tartaglia had 
solved all questions posed by Fior, whereas the latter had not solved a single 
one set by his opponent. The explanation is relatively simple. Today we think 
of cubic equations as all essentially of one type and as amenable to a single 
unified method of solution. At that time, however, when negative coefficients 
were virtually unused, there were as many types of cubics as there are pos- 
sibilities in positive or negative signs for coefficients. Fior was able to solve 
only equations of the type in which cubes and roots equal a number — that is, 
those of the type x 3 + px = q, although at that time only specific numerical 
(positive) coefficients were used. Tartaglia meanwhile had learned how to 
solve also equations of the form where cubes and squares equal a «umber. 
It is likely that Tartaglia had learned how to reduce this case to Fior's by 
removing the squared term, for it became known by this time that if the 


leading coefficient is unity, then the coefficient of the squared term, when it 
appears on the other side of the equality sign, is the sum of the roots. 

News of Tartaglia's triumph reached Cardan, who promptly invited the 
winner to his home, with a hint that he would arrange to have him meet a 
prospective patron. Tartaglia had been without a substantial source of 
support, partly perhaps because of his speech impediment. As a child he 
had received a sabre cut in the fall of Brescia to the French in 1512, which 
impaired his speech. This earned him the nickname Tartaglia, or stammerer, 
a name that he thereafter used instead of the name Niccolo Fontana that 
had been given him at birth. Cardan, in contrast to Tartaglia, had achieved 
worldly success as a physician. So great was his fame that he was once called to 
Scotland to diagnose an ailment of the Archbishop of St. Andrews (evidently 
a case of asthma). By birth illegitimate, and by habit an astrologer, gambler, 
and heretic, Cardan nevertheless was a respected professor at Bologna and 
Milan, and ultimately he was granted a pension by the pope. One of his sons 
poisoned his own wife, the other son was a scoundrel, and Cardan's secretary 
Ferrari probably died of poison at the hands of his own sister. Despite such 
distractions, Cardan was a prolific writer on topics ranging from his own life 
and praise of gout to science and mathematics. 

In his chief scientific work, a ponderous volume with the title De subtilitate, 
Cardan is clearly a child of his age, discussing interminably the Aristotelian 
physics handed down through Scholastic philosophy, while at the same time 
he waxed enthusiastic about the new discoveries of the then-recent times. 
Much the same can be said of his mathematics, for this too was typical of 
the day. He knew little of Archimedes and less of Apollonius, but he was 
thoroughly familiar with algebra and trigonometry. He already had published 
a Practica arithmetice in 1 539, which included among other things the ration- 
alization of denominators containing cube roots. By the time he published the 
Ars magna, half a dozen years later, he probably was the ablest algebraist in 
Europe. Nevertheless, the Ars magna makes dull reading today. Case after 
case of the cubic equation is laboriously worked out in detail according as 
terms of the various degrees appear on the same or on opposite sides of the 
equality, for coefficients were necessarily positive. Despite the fact that he is 
dealing with equations on numbers, he followed al-Khowarizmi in thinking 
geometrically, so that we might refer to his method as "completing the cube." 
There are, of course, certain advantages in such an approach. For instance, 
since x 3 is a volume, 6x, in Cardan's equation below, must also be thought of 
as a volume. Hence the number 6 must have the dimensionality of an area, 
suggesting the type of substitution that Cardan used, as we shall shortly see. 

1 1 Cardan used little syncopation, being a true disciple of al-Khowarizmi, 
and, like the Arabs, he thought of his equations with specific numerical 



coefficients as representative of general categories. For example, when he 
wrote, "Let the cube and six times (he side be equal to 2CT (or x 3 + 6x = 20), 
he obviously was thinking of this equation as typical of all those having "a 
cube and thing equal to a number"— that is, of the form x* + px = q. The 
solution of this equation covers a couple of pages of rhetoric that wc should 
now put in symbols as follows : Substitute w - v for xand let wand v be related 

w *^te^^^^ 


■ ■ ■■:. 

/ .. 'mm/ 

Jerome Cardan. 


so that their product (thought of as an area) is one-third the x coefficient in 
the cubic equation — that is, uv = 2. Upon substitution in the equation, the 
result is u 3 - f 3 = 20; and, on eliminating v, we have u 6 = 20m 3 + 8, a 
quadratic in u 3 . Hence w 3 is known to be ^/lOS" + 10. From the relationship 
u 3 - v 3 = 20, we see th at v 3 = y/10 8 - 10 ; hence, from x = u — v, we have 
x = yyToTTTo - .yyToS - lO. Having carried through the method 
for this specific case, Cardan closes with a verbal formulation of the rule 
equivalent to our modern solution of x 3 + px = q as 

x = ^/(p/3) 3 + (q/2) 2 + q/2 - ^7(p/3) 3 + (q/2) 2 - q/2 

Cardan then went on to other cases, such as "cube equal to thing and num- 
ber." Here one makes the substitution x = u + v instead of x = u - v, the 
rest of the method remaining essentially the same. In this case, however, 
there is a difficulty. When the ru le is applied to x 3 = 15x + 4, for example, 
the result is x = ^2+^/^Ul + ^2 - v^-121- Cardan knew that there 
was no square root of a negative number, and yet he knew x = 4 to be a root. 
He was unable to understand how his rule could make sense in this situation. 
He had toyed with square roots of negative numbers in another connection 
when he asked that one divide 10 into two parts such that the prod uct o f the 
parts i s 40. The usual rules of algebra lead to the answers 5 + ,/-15 and 
5 - y/-l5(oT, in Cardan's notation, 5p:Rm:15 and 5m:Rm:15). Cardan 
referred to these square roots of negative numbers as "sophistic" and con- 
cluded that his result in this case was "as subtile as it is useless." Later writers 
were to show that such manipulations were indeed subtle but far from useless. 
It is to Cardan's credit that at least he paid some attention to this puzzling 
situation. 8 

12 Of the rule for solving quartic equations Cardan in the Ars magna wrote 
that it "is due to Luigi Ferrari, who invented it at my request." Again separate 
cases, twenty in all, are considered in turn, but for the modern reader one case 
will suffice. Let square-square and square and number be equal to side. 
(Cardan knew how to eliminate the cubic term by increasing or diminishing 
the roots by one-fourth the coefficient in the cubic term.) Then the steps in the 
solution of x 4 + 6x 2 + 36 = 60x are expressed by Cardan essentially as 
follows : 

1. First add enough squares and numbers to both sides to make the left- 
hand side a perfect square, in this case x* + \2x 2 + 36 or (x 2 + 6) 2 . 

8 There is no published English translation of the whole of the Ars magna, but a selection from 
it appears in D. E. Smith, A Source Book in Mathematics (1929). In a recent communication 
D. J. Struik informed me that there exists in manuscript an English translation of the Ars magna 
by J. R. Witner in Washington. This is to be published by the M. I .T. Press. 


2. Now add to both sides of the equation terms involving a new unknown 
y such that the left-hand side remains a perfect square, such as (x 2 + 6 + yf. 
The equation now becomes 

(x 2 + 6 + yf = 6x 2 + 60x + y 2 + 12y + 2yx 2 
= (2y + 6)x 2 + 60x + (y 2 + I2y) 

3. The next, and crucial, step is to choose y so that the trinomial on the 
right-hand side will be a perfect square. This is done, of course, by setting 
the discriminant equal to zero — an ancient and well-known rule equivalent 
in this case to 60 2 - 4(2y + 6)(y 2 + I2y) = 0. 

4. The result of step 3 is a cubic equation in y — y 3 + I5y 2 + 36y = 450 — 
today known as the "resolvent cubic" for the given quartic equation. This is 
now solved for y by the rules previously given for the solution of cubic 
equations, the result being 

y = ^287^ + y80449j + ^287^ - ^804491 - 5 

5. Substitute a value of y from step 4 into the equation for x in step 2 and 
take the square root of both sides. 

6. The result of step 5 is a quadratic equation, which must now be solved 
in order to find the value of x desired. 

The solution of cubic and quartic equations was perhaps the greatest 1 3 
contribution to algebra since the Babylonians, almost four millennia earlier, 
had learned how to complete the square for quadratic equations. No other 
discoveries had had quite the stimulus to algebraic development as did those 
disclosed in the Ars magna. The solutions of the cubic and quartic were in no 
sense the result of practical considerations, nor were they of any value to 
engineers or mathematical practitioners. Approximate solutions of some 
cubic equations had been known in antiquity, and al-Kashi a century before 
Cardan could have solved to any desired degree of accuracy any cubic 
equation resulting from a practical problem. The Tartaglia-Cardan formula 
is of great logical significance, but it is not nearly so useful to practical men 
as are methods of successive approximation. 

The most important outcome of the discoveries published in the Ars 
magna was the tremendous stimulus they gave to algebraic research in various 
directions. It was natural that study should be generalized to include poly- 
nomial equations of any order and that in particular a solution should be 
sought for the quintic. Here mathematicians of the next couple of centuries 
were faced with an unsolvable algebraic problem comparable to the classical 
geometrical problems of antiquity. Much good mathematics, but only a 
negative conclusion, was the outcome. Another immediate result of the solu- 
tion of the cubic was the first significant glance at a new kind of number. 


Irrational numbers had been accepted by the time of Cardan, even though 
they were not soundly based, for they are readily approximated by rational 
numbers. Negative numbers afforded more difficulty because they are not 
readily approximated by positive numbers, but the notion of sense (or 
direction on a line) made them plausible. Cardan used them even while calling 
them "numeri ficti." If an algebraist wished to deny the existence of irrational 
or negative numbers, he would simply say, as had the ancient Greeks, that 
the equations x 2 = 2 and x + 2 = are not solvable. In a similar way 
algebraists had been able to avoid imaginaries simply by saying that an equa- 
tion such as x 2 + 1 = is not solvable. There was no need for square roots 
of negative numbers. With the solution of the cubic equation, however, the 
situation became markedly different. Whenever the three roots of a cubic 
equation are real and different from zero, the Cardan-Tartaglia formula leads 
inevitably to square roots of negative numbers. The goal was known to be a 
real number, but it could not be reached without understanding something 
about imaginary numbers. The imaginary now had to be reckoned with even 
if one did agree to restrict oneself to real roots. 

At this stage another important Italian algebraist, Rafael Bombelli (ca. 
1526-1573), had what he called "a wild thought," for the whole matter 
"seemed to rest on sophistry." The two radicands of the cube roots resulting 
from the usual formula differ only in one sig n. We have see n th at the solution 

by formula of x 3 = 15x + 4 leads to x = J2 + J- 121 + J/2 - J- 121, 
whereas it is known by direct substitution that x = 4 is the only positive root 
of the equation. (Cardan had noted that when all terms on one side of 1 the 
equality sign are of higher degree than the terms on the other side, the equa- 
tion has one and only one positive root — an anticipation, in a small way, of 
part of Descartes' rule of signs.) Bombelli had the happy thought that the 
radicals themselves might be related in much the way that the radicands are 
related — that, as we should now say, they are conjugate imaginaries that 
lead to the real number 4. It is obvious that if the sum of the real parts is 4, then 
the real part of each is 2 ; and if a number of the form 2 -I- b^f—l is to be a 
cube root of 2 + 11^/^T, then it is easy to see that b must be 1. Hence 

x = 2 + lv^T + 2 - ly^T, or 4. 

Through his ingenious reasoning Bombelli had shown the important role 
that conjugate imaginary numbers were to play in the future ; but at that 
time the observation was of no help in the actual work of solving cubic 
equations, for Bombelli had had to know beforehand what one of the roots is. 
In this case the equation is already solved, and no formula is needed ; without 
such foreknowledge, Bombelli's approach fails. Any attempt to find alge- 
braically the cube roots of the imaginary numbers in the Cardan-Tartaglia 
rule leads to the very cubic in the solution of which the cube roots arose in the 


first place, so that one is back where he started from. Because this impasse 
arises whenever all three roots are real, this is known as the "irreducible 
case." Here an expression for the unknown is indeed provided by the formula, 
but the form in which this appears is useless for most purposes. 

Bombelli composed his Algebra 9 in about 1560, but it was not printed 
until 1572, about a year before he died, and then only in part. One of the 
significant things about this book is that it contains symbolisms reminiscent 
of those of Chuquet. Bombelli sometimes wrote 1 Z p.5Rm.4 (that is, 1 zenus 
plus 5 res minus 4) for x 2 + 5x - 4. But he used also another form of expres- 
sion — 1-ip • 5-i-m • 4 — in which the power of the unknown quantity is repre- 
sented simply as an Arabic numeral above a short circular arc, so that x, x 2 , x 3 
appear as J,,^ J, for example, influenced perhaps by de la Roche's Larisme- 
thique. Bombelli's Algebra of course uses the standard Italian symbols p and 
m for addition and subtraction, but he still had no symbol for equality. Our 
standard equality sign had been published before Bombelli wrote his book, 
but the symbol had appeared in a distant part of Europe — in England in 
1557 in the Whetstone ofWitte of Robert Recorde (1510-1558). 

Mathematics had not prospered in England during the period of almost 1 4 
two centuries since the death of Bradwardine, and what little work was 
done there in the early sixteenth century depended much on Italian writers 
such as Pacioli. Recorde was, in fact, just about the only mathematician of 
any stature in England throughout the century. He was born in Wales and 
studied and taught mathematics at both Oxford and Cambridge. In 1545 he 
received his medical degree at Cambridge, and thereafter he became physician 
to Edward VI and Queen Mary. One of the remarkable things about the 
period was the surprisingly large number of physicians who contributed 
outstandingly to mathematics, Chuquet, Cardan, and Recorde being three 
of the best known. It is likely that Recorde was the most influential of these 
three within his own country, for he virtually established the English math- 
ematical school. Like Chuquet and Pacioli before him, and Galileo after him, 
he wrote in the vernacular ; this may have limited his effect on the Continent, 
although the easy dialogue form that he adopted was used also, some time 
later, by Galileo. Recorde's first extant mathematical work was the Grounde 
of Artes (1541), a popular arithmetic containing computation by abacus and 
algorism, with commercial applications. The level and style of this book, 
dedicated to Edward VI and appearing in more than two dozen editions, may 
be judged from the following problem : 

Then what say you to this equation? If I sold unto you an horse having 4 
shoes, and in every shoe 6 nayles, with this condition, that you shall pay for 

9 There is no convenient edition. On Bombelli's life see articles by S. A. Jayawardine in Isis, 
54 (1963), 391-395; 56 (1965), 298-306. 


the first nayle one ob : for the second nayle two ob : for the third nayle foure 
ob : and so forth, doubling untill the end of all the nayles, now I ask you, how 
much would the price of the horse come unto? 10 

His Castle of Knowledge, an astronomy in which the Copernican system is 
cited with approval, and his Pathewaie to Knowledge, an abridgement of the 
Elements and the first geometry to appear in English, both appeared in 1551. 
The work of Recorde that is most often cited is The Whetstone of Witte, 
published in 1557, only a year before he died in prison. (Whether he was 
jailed for political or religious reasons or because of difficulties related to his 
position, from 1551 on, as Surveyor of the Mines and Monies of Ireland, is 
not known. 11 ) The title Whetstone evidently was a play on the word "coss," 
for cos is the Latin for whetstone, and the book is devoted to "the cossike 
practise" (that is, algebra). It did for England what Stifel had done for Ger- 
many — with one addition. The well-known equality sign first appeared in it, 
explained by Recorde in the quotation at the beginning of this chapter. 
However, it was to be a century or more before the sign triumphed over rival 

1 5 Recorde died in 1558, the year in which Queen Mary also died, and no 
comparable English mathematical author appeared during the long reign 
of Elizabeth I. It was France, rather than England, Germany, or Italy, that 
produced the outstanding mathematician of the Elizabethan Age ; but before 
we turn to his work in the next chapter, there are certain aspects of the earlier 
sixteenth century that should be clarified. The direction of greatest progress 
in mathematics during the sixteenth century was obviously in algebra, but 
developments in trigonometry were not far behind, although they were not 
nearly so spectacular. The construction of trigonometric tables is a dull task, 
but they are of great usefulness to astronomers and mathematicians ; here 
early sixteenth-century Poland and Germany were very helpful indeed. Most 
of us today think of Nicholas Copernicus (1473-1543) as an astronomer who 
revolutionized the world view by successfully putting the earth in motion 
about the sun (where Aristarchus had tried and failed); but an astronomer 
is almost inevitably a trigonometer as well, and we owe to Copernicus a 
mathematical obligation as well as an astronomical debt. 

During the lifetime of Regiomontanus, Poland had enjoyed a "Golden 
Age" of learning, and the University of Cracow, where Copernicus enrolled 
in 1491, enjoyed great prestige in mathematics and astronomy. After further 

10 See E. R. Ebert, "A Few Observations on Robert Recorde and his 'Grounde of Artes'," 
The Mathematics Teacher, 30 (1937), 110-121. See also Joy B. Easton, "A Tudor Euclid," 
Scripta Mathematica, 11 (1966), 339-355; F. R. Johnson and S. V. Larkey, "Robert Recorde's 
Mathematical Teaching and the Anti-Aristotelian Movement," Huntington Library Bulletin, 
7 (1935), 59-87. 

11 See F. M. Clarke, "New Light on Robert Recorde," his, 7 (1926), 50-70. 


The Jrte 

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A page from Robert Recorde's Whetstone of Witte (1557). Note that his symbols for 
equality are much longer than ours. 


studies in law, medicine, and astronomy at Bologna, Padua, and Ferrara, 
and after some teaching at Rome, Copernicus returned to Poland in 1510 to 
become Canon of Frauenburg. Despite multitudinous administrative obliga- 
tions, including currency reform and the curbing of the Teutonic Order, 
Copernicus completed the celebrated treatise, De revolutionibus orbium 
coelestium, published in 1543, the year he died. This contains substantial 
sections on trigonometry that had been separately published in the previous 
year under the title De lateribus et angulis triangulorum. The trigonometric 
material is similar to that in Regiomontanus' De triangulis, published at 
Nuremberg only a decade earlier; but Copernicus' trigonometric ideas seem 
to date from before 1533, at which time he probably did not know of the work 
of Regiomontanus. It is quite likely, nevertheless, that the final form of 
Copernicus' trigonometry was in part derived from Regiomontanus, for in 
1539 he received as a student the Prussian mathematician Georg Joachim 
Rheticus (or Rhaeticus, 1514-1576), a mathematician of Wittenberg who 
evidently had been in touch with Nuremberg mathematics. Rheticus worked 
with Copernicus for some three years, and it was he who, with his teacher's 
approval, published the first short account of Copernican astronomy in a 
work entitled Narratio prima (1540) and who made the first arrangements, 
completed by Andreas Osiander, for the printing of the celebrated De 
revolutionibus. It is likely, therefore, that the trigonometry in the classic 
work of Copernicus is closely related, through Rheticus, to that of Regiomon- 

We see the thorough trigonometric capabilities of Copernicus not only 
in the theorems included in De revolutionibus, but also in a proposition 
originally included by the author in an earlier manuscript version of the book, 
but not in the printed work. The deleted proposition is a generalization of the 
theorem of Nasir Eddin (which does appear in the book) on the rectilinear 
motion resulting from the compounding of two circular motions. The 
theorem of Copernicus is as follows : If a smaller circle rolls without slipping 
along the inside of a larger circle with diameter twice as great, then the locus 
of a point which is not on the circumference of the smaller circle, but which is 
fixed with respect to this smaller circle, is an ellipse. Cardan, incidentally, 
knew of the Nasir Eddin theorem, but not of the Copernican locus, a theorem 
rediscovered in the seventeenth century. 12 

16 Through the trigonometric theorems in De revolutionibus Copernicus 
spread the influence of Regiomontanus, but his student Rheticus went 
further. He combined the ideas of Regiomontanus and Copernicus, together 
with views of his own, in the most elaborate treatise composed up to that 

12 See C. B. Boyer, "Note on Epicycles and the Ellipse from Copernicus to Lahire," I sis, 
38 (1947), 54-56. 


time — the two-volume Opus palatinum de triangulis. Here trigonometry really 
came of age. The author discarded the traditional consideration of the func- 
tions with respect to the arc of a circle and focused instead on the lines in a 
right triangle. Moreover, all six trigonometric functions now came into full 
use, for Rheticus calculated elaborate tables of all of them. Decimal fractions 
still had not come into common use ; hence for the sine and cosine functions he 
used a hypotenuse (radius) of 10,000,000 and for the other four functions a 
base (or adjacent side or radius) of 10,000,000 parts, for intervals in the angle 
of 10". He began tables of tangents and secants with a base of 10 1 5 parts ; but 
he did not live to finish them, and the treatise was completed and edited, 
with additions, by his pupil Valentin Otho (ca. 1550-1605) in 1596. 13 

The work of Rheticus, who like Copernicus, Chuquet, Cardan, and Recorde 1 7 
had also studied medicine, was much admired by Pierre de la Ramee or 
Ramus (1515-1572), a man who contributed to mathematics in a pedagogical 
sense. At the College de Navarre he had in 1536 defended, for his master's 
degree, the audacious thesis that everything Aristotle had said was wrong— at 
a time when Peripateticism was the same as orthodoxy. In his intellectual 
criticism and pedagogical interests he may be compared with Recorde in 
England. Ramus was at odds with his age in many ways, and while his 
Humanist contemporaries had little use for mathematics, he had almost a 
blind faith in the subject. He proposed revisions in the university curricula 
so that logic and mathematics should receive more attention; his logic 
enjoyed considerable popularity in Protestant countries, in part because he 
died a martyr in the St. Bartholomew massacre. Not satisfied even with the 
Elements of Euclid, Ramus edited this with revisions. However, his compe- 
tence in geometry was very limited, and his suggested changes in mathematics 
were in the opposite direction from those in our day. Ramus had more con- 
fidence in practical elementary mathematics than in speculative higher 
algebra and geometry ; looking back on his age we see that the mathematics 
of that time seems already to have been excessively concerned with practical 
problems in arithmetic, while weakness in geometry was quite conspicuous. 

Pappus in about 320 had wished to initiate a geometrical revival, but he 1 8 
found no really capable successor in pure geometry in Greece. In China and 
India there never had been any real concern for geometry beyond problems 
in mensuration, but the Arabs, who appreciated demonstrative reasoning, 
used geometrical arguments in their algebra. In medieval Europe, as we have 
seen, there was a two-way tendency to relate algebra and geometry. In the 

See J. D. Bond, "The Development of Trigonometric Methods Down to the Close of the 
XVth Century," his, 4 (1921-1922), 295-323 ; also Sister Mary Claudia Zeller, The Development 
of Trigonometry from Regiomontanus to Pitiscus (1946). 


medieval tradition, Books IV and VI of Bombelli's Algebra were full of 
problems in geometry that are solved algebraically — somewhat in the 
manner of Regiomontanus, but making use of new symbolisms. For example, 
Bombelli asked for the side of a square inscribed in a triangle with sides 
ac = 13, cf = 14, fa = 15, so that one side lies on cf (Fig. 15.1), which he 
solved as follows: Let bg = 14-i-(that is, 14x). Then ag = 15-i-and ab = 13- 1 -. 
Now ah = 12^ and hi = 14-i-. Since ai = 12, we have 26 A = 12; then "cosa" 

or x is f$, so that hi, or the side of the square, must be 14 times f% or 6fj. Here 
a highly symbolic algebra has come to the aid of geometry ; but Bombelli 
worked in the other direction, too. In the Algebra, the algebraic solution of 
cubic equations is accompanied by geometric demonstrations in terms of the 
subdivision of the cube. Unfortunately for the future of geometry— and of 
mathematics in general — the last books of Bombelli's Algebra were not 
included in the publication of 1572, but remained in manuscript until 1929. 14 

1 9 Pure geometry in the sixteenth century was not entirely without representa- 
tives, for unspectacular contributions were made in Germany by Johannes 
Werner (1468-1522) and Albrecht Durer (1471-1528), and in Italy by 
Francesco Maurolico (1494-1575) and Pacioli. Once more we note the pre- 
eminence of these two countries in contributions to mathematics during the 
Renaissance. Werner had aided in preserving the trigonometry of Regio- 
montanus, but of more geometrical significance was his Latin work, in 
twenty-two books, on the Elements of Conies, printed at Nuremberg in 1522. 
This cannot be compared favorably with the Conies of Apollonius, almost 
entirely unknown in Werner's day, but it marks the renewal of interest in the 
curves for almost the first time since Pappus. Because the author was con- 
cerned primarily with the duplication of the cube, he concentrated on the 
parabola and the hyperbola, deriving the standard plane equations stereo- 
metrically from the cone, as had his predecessors in Greece ; but there seems 

14 See L'algebra. Opera di Rafael Bombelli da Bologna Books IV and V, comprising "La parte 
geometrica", ed. by Ettore Bortolotti (1929). 


to be an element of originality in his plane method for plotting points on a 
parabola with compasses and straightedge. One first draws a pencil of circles 
tangent to each other and intersecting the common normal in points c, d, 
e.f.g.... (Fig. 1 5.2). Then along the common normal one marks offa distance 

FIG. 15.2 

ab equal to a desired parameter. At b one erects the line bG perpendicular to 
ab and cutting the circles in points C, D, E,F,G,... respectively. Then at c 
one erects line segments cC and cC" perpendicular to ab and equal to bC ; 
at d one erects perpendicular segments dD' and dD" equal to bD ; at e one 
erects segments eE and eE" equal to bE, and so on. Then C, C", D', D", 
E, E", . . . will all lie on the parabola with vertex b, axis along ab, and having 
ab as the magnitude of the parameter — as is readily seen from the relation- 
ships (cC) 2 = ab ■ be, (dD') 2 = ab ■ bd, and so on. 15 

Werner's work is closely related to ancient studies of conies ; but meanwhile 
in Italy and Germany a relatively novel relationship between mathematics 
and art was developing. One important respect in which Renaissance art 
differed from art in the Middle Ages was in the use of perspective in the plane 
representation of objects in three-dimensional space. The Florentine architect 
Filippo Brunelleschi (1377-1446) is said to have given much attention to this 
problem, but the first formal account of some of the problems was given by 
Leon Battista Alberti (1404-1472) in a treatise of 1435 (printed in 1511) 
entitled Delia pictura. Alberti opens with a general discussion of the principles 
of foreshortening and then describes a method he had invented for repre- 
senting in a vertical "picture plane" a set of squares in a horizontal "ground 

15 See J. L. Coolidge, A History of the Conic Sections and Quadratic Surfaces (Oxford: 
Clarendon, 1945), pp. 26-28. 



plane." Let the eye be at a "station point" S that is h units above the ground 
plane and k units in front of the picture plane. The intersection of the 
ground plane and the picture plane is called the "groundline," the foot V of 
the perpendicular from S to the picture plane is called the "center of vision" 
(or the principal vanishing point), the line through V parallel to the ground- 
line is known as the "vanishing line" (or horizon line), and the points P 
and Q on this line which are k units from V are called the "distance points." 
If we take points A, B, C, D, E, F, G marking off equal distances along the 
groundline RT (Fig. 15.3), where D is the intersection of this line with the 
vertical plane through S and V, and if we draw lines connecting these points 
with V, then the projection of these last lines, with S as a center, upon the 


/ i i \ \ \ \ 

\ \\ \ \ V < / / I I I \ \ 

ground plane will be a set of parallel and equidistant lines. If P (or Q) is con- 
nected with the points B, C, D, E, F, G to form another set of lines intersecting 
AV in points H, I, J, K, L, M, and if through the latter points parallels are 
drawn to the groundline RT, then the set of trapezoids in the picture plane 
will correspond to a set of squares in the ground plane. 16 

A further step in the development of perspective was taken by the Italian 
painter of frescoes, Piero della Francesca (14107-1492), in De prospectiva 
pingendi (ca. 1478). Where Alberti had concentrated on representing on the 
picture plane figures in the ground plane, Piero handled the more complicated 
problem of depicting on the picture plane objects in three dimensions as seen 

16 Further details, as well as solid accounts of other work by Piero della Francesca, Leonardo 
da Vinci, and Albrecht Diirer, will be found in J. L. Coolidge, The Mathematics of Great Amateurs 
(Oxford : Clarendon, 1949), pp. 30-70. 



from a given station point. He wrote also a De corporibus regularibus where 
he noted the "divine proportion" in which diagonals of a regular pentagon 
cut each other and where he found the volume common to two equal circular 
cylinders whose axes cut each other at right angles (unaware of Archimedes' 
Method, which was unknown at the time). The connection between art and 

Albrecht Diirer's "Melancholia" (The British Museum). Note the four-celled magic square 
in the upper right-hand corner. 



mathematics was strong also in the work of Leonardo da Vinci. He wrote a 
work, now lost, on perspective; his Trattato della pittura opens with the 
admonition. "Let no one who is not a mathematician read my works." 17 
The same combination of mathematical and artistic interests is seen in 
Albrecht Diirer, a contemporary of Leonardo and a fellow townsman of 
Werner at Nuremberg. -In Diirer's work we see also the influence of Pacioli, 
especially in the celebrated engraving of 1514 entitled Melancholia. Here the 
magic square figures prominently. This often is regarded as the first use of a 

















magic square in the West, but Pacioli had left an unpublished manuscript, 
De viribus quantitatis, in which interest in such squares is indicated. Diirer's 
interests in mathematics, however, were far more geometrical than arithmetic, 
as the title of his most important book indicates; "Investigation of the 
measurement with circles and straight lines of plane and solid figures." This 
work, which appeared in several German and Latin editions from 1525 to 
1538, contains some striking novelties, of which the most important were his 
new curves. This is one direction in which the Renaissance could easily have 
improved on the work of the ancients, who had studied only a handful of types 
of curves. Diirer took a fixed point on a circle and then allowed the circle to 
roll along the circumference of another circle, generating an epicycloid ; but, 
not having the necessary algebraic tools, he did not study this analytically. 
The same was true of other plane curves that he obtained by projecting 
helical space curves onto a plane to form spirals. Too often those working 
in perspective were not familiar with the foundations of mathematics and 
failed to distinguish between exact and approximate results. In Diirer's 
work we find the Ptolemaic construction of the regular pentagon, which is 
exact, as well as another original construction that is only an approximation. 
For the heptagon and enneagon he also gave ingenious, but of course inexact, 
constructions. Diirer's construction of an approximately regular nonagon 
is as follows: Let O be the center of a circle ABC in which A, B, and C are 

17 Morris Kline, Mathematics in Western Culture (New York: Oxford University Press, 
1953). This book contains an eminently readable account of art as related to mathematics. 


FIG. 15.4 

vertices of the inscribed equilateral triangle (Fig. 15.4). Through A, O, and 
C draw a circular arc, draw similar arcs through B, 0, and C and through 
B, 0, and A. Let AO be trisected at points D and E, and through E draw a 
circle with center at and cutting arcs AFO and AGO in points F and G 
respectively. Then the straight line segment FG will be very nearly equal to 
the side of the regular nonagon inscribed in this smaller circle, the angle FOG 
differing from 40° by less than 1°. 18 The relation of art and geometry might 
have been very productive indeed, had it gained the attention of profession- 
ally minded mathematicians, but in this respect it failed for more than a cen- 
tury after Diirer's time. 

Diirer's contemporaries in pure mathematics failed to appreciate the 
future of geometric transformations, but projections of various sorts are 
essential to cartographers. Geographical explorations had widened horizons 
and created a need for better maps, but Scholasticism and Humanism were 
of little help here since new discoveries had outmoded medieval and ancient 
maps. One of the most important of the innovators was a German math- 
ematician and astronomer Peter Apian (or Bienewitz, 1495-1552). In 1520 
he published perhaps the earliest map of the Old World and the New World 
in which the name "America" was used ; in 1527 he issued a business arith- 
metic in which, on the title page, the arithmetic or "Pascal" triangle appeared 
in print for the first time. The maps of Apian were well done, but they followed 
Ptolemy closely wherever possible. For that novelty which is thought to be so 
characteristic of the Renaissance it is better to look instead to a Flemish 
geographer, Gerard Mercator (or Gerhard Kremer, 1512-1594), who was 
for a time associated with the court of Charles V at Brussels. Mercator may 
be said to have broken with Ptolemy in geography as Copernicus had revolted 
against Ptolemaic astronomy. 

18 See Moritz Cantor, Voriesungen iiber Geschichte der Mathematik (Leipzig : Teubner, 1900- 
1908, 4 vols.), II, 425. 




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Ingolstadt, 1527, more than a century before Pascal investigated the properties of the 


For the first half of his life Mercator depended heavily on Ptolemy, but by 
1554 he had emancipated himself sufficiently to cut down the Ptolemaic 
estimate of the width of the Mediterranean from 62° to 53°. (Actually it is 
close to 40°.) More importantly, in 1569 he published the first map, Nova et 
aucta orbis terrae descriptio, drawn up on a new principle. Maps in common 
use in Mercator's day were usually based upon a rectangular grid made up 
of two sets of equidistant parallel lines, one set for latitudes, the other for 


longitudes. The length of a degree of longitude, however, varies with the 
parallel of latitude along which it is measured, an inequality disregarded in 
common practice and resulting in distortion of shape and in errors of direc- 
tion on the part of navigators who based a course upon the straight line drawn 
between two points on the map. The Ptolemaic stereographic projection 
preserved shapes, but it did not use the common grid of lines. In order to bring 
theory and practice into some accord, Mercator introduced the projection 
that bears his name and, with later improvement, has been basic in carto- 
graphy ever since. The first step in the Mercator projection is to think of a 
spherical earth inscribed within an indefinitely long right circular cylinder 
touching the earth along the equator (or some other great circle), and to 
project, from the center of the earth, points on the surface of the earth onto 
the cylinder. If the cylinder then is cut along an element and flattened out, the 
meridians and parallels on the earth will have been transformed into a 
rectangular network of lines. Distances between successive meridian lines 
will be equal, but not distances between successive lines of latitude. In fact, 
the latter distances increase so rapidly, as one moves away from the equator, 
that distortions of shape and direction occur; but Mercator found that 
through an empirically determined modification of these distances preserva- 
tion of direction and shape (although not of size) was possible. 19 In 1599 
Edward Wright (1558-1615), a fellow at Cambridge, tutor to Henry, Prince 
of Wales, and a good sailor, developed the theoretical basis of the Mercator 
projection by computing the functional relationship D = a In tan(<^/2 + 45°) 
between map distance D from the equator and latitude <j). 

Mathematics during the Renaissance had been widely applied — to book- 
keeping, mechanics, surveying, art, cartography, optics — and there were 
numerous books devoted to the practical arts. Nevertheless, interest in the 
classical works of antiquity continued strong, as we see in the case of Mauro- 
lico, a priest of Greek parentage who was born, lived, and died in Sicily. 
Maurolico was a scholarly geometer who did much to revive interest in the 
more advanced of the antique works. 20 Geometry in the first half of the 
sixteenth century had been far too heavily dependent on the elementary 
properties found in Euclid. Werner had been an exception to this rule, but 
few others were really familiar with the geometry of Archimedes, Apollonius, 
and Pappus. The reason for this was simple — Latin translations of these did 
not become generally available until the middle of the century. In this process 
of translation Maurolico was joined by an ardent Italian scholar, Federigo 

19 A compendious historical account of this and other projections is provided in the article 
"Map" by E. G. Ravenstein et al. in Encyclopaedia Britannica, 1 1th ed., 17, 629-663. 

20 Some idea of the extent of his writings and the difficulty of dating them can be gained from 
Edward Rosen, "The Editions of Maurolico's Mathematical Works," Scripta Mathematica 
24(1959), 59-76. 


Commandino, who died in the same year— 1575. We have mentioned Tar- 
taglia's borrowed translation of Archimedes printed in 1543; this was 
followed by a Greek edition of 1544 and a Latin translation by Commandino 
at Venice in 1558. 

Four books of the Conies of Apollonius had survived in Greek, and these 
had been translated into Latin and printed at Venice in 1537. Maurolico's 
translation, completed in 1548, was not published for more than a century, 
appearing in 1654, but another translation by Commandino was printed at 
Bologna in 1566. The Mathematical Collection of Pappus had been virtually 
unknown to the Arabs and the medieval Europeans, but this, too, was 
translated by the indefatigable Commandino, although it was not printed 
until 1588. Maurolico was acquainted with the vast treasures of ancient 
geometry that were becoming available, for he read Greek as well as Latin. 
In fact, from some indications in Pappus of Apollonius' work on maxima and 
minima— that is, on normals to the conic sections— Maurolico tried his 
hand at a reconstruction of the then-lost Book V of the Conies. In this respect 
he represented a vogue that was to be one of the chief stimuli to geometry 
before Descartes — the reconstruction of lost works in general and of the last 
four books of the Conies in particular. During the interval from Maurolico's 
death in 1575 to the publication of La geometrie by Descartes in 1637, geom- 
etry was marking time until developments in algebra had reached a level 
making algebraic geometry possible. The Renaissance could well have 
developed pure geometry in the direction suggested by art and perspective, 
but the possibility went unheeded until almost precisely the same time that 
algebraic geometry was created. Between Maurolico and Descartes, mean- 
while, mathematics developed in several nongeometrical directions, and it is 
to these that we now turn. 


Bond, J. D., "The Development of Trigonometric Methods Down to the Close of the 

XVth Century," Isis, 4 (1921-1922), 295-323. 
Bortolotti, Ettore, Studi e ricerche sulla storia della matematica in Italia nei secoli 

XVI e XVII (Bologna, 1928). 
Bortolotti, Ettore, ed., L 'algebra. Opera di Rafael Bombelli da Bologna (Bologna, 1929). 
Cardan, Jerome, The Book of My Life, trans, by Jean Stoner (paperback ed., New York : 

Dover, 1963). 
Clarke, F. M., "New Light on Robert Recorde," Isis, 7 (1926), 50-70. 
Easton, Joy B, "A Tudor Euclid," Scripta Mathematica, 27 (1966), 339-355. 
Ebert, E. R., "A Few Observations on Robert Recorde and his 'Grounde of Aries'," 

The Mathematics Teacher, 30 (1937), 110-121. 
Glaisher, J. W. L., "On the Early History of the Signs + and - and on the Early German 

Arithmeticians," Messenger of Mathematics, 51 (1921-1922), 1-148. 


Hoftnann, J. E., Geschichte der Mathematik, 2nd ed. (Berlin : Walter de Gruyter, 1963), 

Vol. I. 
Hughes, Barnabas, ed., Regiomontanus on Triangles (Madison, Wis.: University of 

Wisconsin Press, 1967). 
Lambo, Ch., S. J., "Une algebre francaise de 1484. Nicolas Chuquet," Revue des Ques- 
tions Scientifiques (3), 2 (1902), 442^72. 
Marre, Aristide, "Notice sur Nicolas Chuquet et son Triparty en la science des nombres," 

Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche, 13 (1880), 

555-659, 693-814; 14 (1881), 413^160. 
Ore, Oystein, Cardano, the Gambling Scholar (Princeton, N. J. : Princeton University 

Press, 1953). 
Sarton, George, "The Scientific Literature Transmitted Through the Incunabula," 

Osiris, 5 (1938), 41-247. 
Simon, Max, Cusanus als Mathematiker (Strassburg, 1911, in Festschrift Heinrich Weber, 

Leipzig and Berlin : Teubner, 1912, pp. 298-337). 
Smith, D. E., Rara arithmetica (Boston : Ginn, 1908). 
Smith, D. E., ed., A Source Book in Mathematics (New York : McGraw-Hill, 1929 ; 

paperback ed., New York: Dover, 1959, 2 vols.). 
Sullivan, J. W. N, The History of Mathematics in Europe, from the Fall of Greek Science 

to the Rise of the Conception of Mathematical Rigour (New York : Oxford Univer- 
sity Press, 1925). 
Taylor, R. Emmett, No Royal Road. Luca Pacioli and His Times (Chapel Hill, N.C. : 

University of North Carolina Press, 1942). 
Waters, W. G., Jerome Cardan, a Biographical Study (London, 1 898). 
Zeller, Sister Mary Claudia, The Development of Trigonometry from Regiomontanus 

to Pitiscus (Ann Arbor, Mich.: University of Michigan, Ph. D. thesis, 1944). 


1. Which of the following factors were important in the development of Renaissance math- 
ematics : (a) the fall of Constantinople, (b) the Protestant Reformation, (c) the rise of Human- 
ism, (d) the invention of printing, (e) the rising mercantile class? Explain. 

2. How do you account for the fact that algebra and trigonometry developed more rapidly 
than geometry during the Renaissance? 

3. Why was the solution of the cubic so important for the development of imaginary numbers? 

4. How would you account for the fact that many of the leading mathematicians in the six- 
teenth century were physicians? 

5. Which countries took the lead, during the Renaissance, in the development of (a) algebra, 
(b) trigonometry, (c) geometry? Mention specific contributions in each case. 

6. How does Regiomontanus' value for tan 89° compare with that in modern tables? How 
might he have found his value? 

7. Construct, with compasses and straightedge, a triangle in which one side is 5, the altitude to 
this side is 3, and the ratio of the other two sides is y/l:l. {Suggestion: apply the algebraic 
approach of Regiomontanus and Bombelli.) 

8. Solve Pacioli's problem in which it is required to find the sides of a triangle if the radius of 
the inscribed circle is 4, and if the segments into which one side is divided by the point of 
contact are 6 and 8. 


9. Derive a solution of Bombelli's equation x 3 = 1 5x + 4 as a sum or difference of cube roots 

of imaginary numbers. 
10. Reduce the solution of Ferrari's quartic x 4 + 6x 2 + 36 = 60x to the solution of a cubic 

Verify Bombelli's statement that 4 + J^l is a cube root of 52 + x/-2209. 

12. Recorde's Grounde of Artes contains the following "simplified" scheme for multiplying two 
one-digit numbers (both more than 6) : First subtract each number from 10. The product of 
these differences is the units digit in the product of the original numbers, and if either differ- 
ence is subtracted from the other original number this will be the tens digit in the product of 
the original numbers. Prove this rule. 

13. Form the cubic equation with roots 1 + ,/! and - 3 and then apply the method of Cardan 
and Tartaglia to solve this cubic. 

14. Solve Cardan's problem in dividing 10 into two parts the product of which shall be 40. 
Verify your answer. 

*15. Justify Werner's construction of the parabola, indicating where the directrix lies. 

*16. How do you know that Durer's construction of a regular nonagon is not exact? Find to the 

nearest minute the arc FG. 
*17. Show how to construct the cyclic quadrilateral with successive sides a = 25, b = 33, c = 60, 

d = 16. 
*18. Prove the Copernican theorem on the epicyclic generation of the ellipse. 
*19. Justify Alberti's method for representing in the picture plane a set of squares in the ground 




to Modern Mathematics 

In mathematics I can report no deficiency, except it 
be that men do not sufficiently understand the excellent 
use of the Pure Mathematics. 

Francis Bacon 

When in 1575 Maurolico and Commandino died, Western Europe had 
recovered most of the major mathematical works of antiquity now extant. 
Arabic algebra had been thoroughly mastered and improved upon, both 
through the solution of the cubic and quartic and through a partial use of 
symbolism ; and trigonometry had become an independent discipline. The 
time was almost ripe for rapid strides beyond ancient, medieval, and Ren- 
aissance contributions— but not quite. There is in the history of mathematics 
a high degree of continuity from one age to the next ; the transition from the 
Renaissance to the modern world was also made through a large number of 
intermediate figures, a few of the more important of whom we shall now 
consider. Two of these men, Galileo Galilei (1564-1642) and Bonaventura 
Cavalieri (1598-1647), came from Italy ; several more, such as Henry Briggs 
(1561-1639), Thomas Harriot (1560-1621), and William Oughtred (1574- 
1660), were English; two of them, Simon Stevin (1548-1620) and Albert 
Girard (1590-1633), were Flemish; others came from varied lands— John 
Napier ( 1 550-1 6 1 7) from Scotland, Jobst Burgi ( 1 552-1 632) from Switzerland, 
and Johann Kepler (1571-1630) from Germany. Most of Western Europe 
now was involved in the advance of mathematics, but the central and most 
magnificent figure in the transition was a Frenchman, Francois Viete (1540- 
1603) — or, in Latin, Franciscus Vieta. 

Viete was not a mathematician by vocation. As a young man he studied 
and practiced law, becoming a member of the Bretagne parlement ; later 
he became a member of the king's council, serving first under Henry III 
and later under Henry IV. It was during his service with the latter, Henry of 
Navarre, that he became so successful in deciphering cryptic enemy messages 
that the Spanish accused him of being in league with the devil. Only Viete's 
leisure time was devoted to mathematics, yet he made contributions to 



arithmetic, algebra, trigonometry, and geometry. There was a period of 
almost half a dozen years, before the accession of Henry IV, during which 
Viete was out of favor, and these years he spent largely on mathematical 
studies. In arithmetic he should be remembered for his plea for the use of 
decimal, rather than sexagesimal, fractions. In one of his earliest works, the 
Canon-mathematicus of 1579, he wrote: 

Sexagesimals and sixties are to be used sparingly or never in mathematics, and 
thousandths and thousands, hundredths and hundreds, tenths and tens, and similar 
progressions, ascending and descending, are to be used frequently or exclusively. 1 

In the tables and computations he adhered to his word and used decimal 
fractions The sides of the squares inscribed in and circumscribed about a 
circle of diameter 200,000 he wrote as 141,421,^^ and 200,000,^^, and 
their mean as 177,245,^^^. A few pages further on he wrote the semi- 
circumference as 314,159, ? 6 5 rig, 5 oo , and still later this figure appeared as 
314,159,265,36, with the integral portion in boldface type. Occasionally he 
used a vertical stroke to separate the integral and fractional portions, as 
when writing the apothem of the 96-sided regular polygon, in a circle of 
diameter 200,000, as about 99,946| 458,75. 

The use of a decimal point separatrix generally is attributed either to G. A. 
Magini (1555-1617), a map-making friend of Kepler and rival of Galileo for a 
chair at Bologna, in his De planis triangulis of 1592, or to Christoph Clavius 
(1537-1612), a Jesuit friend of Kepler, in a table of sines of 1593. But the 
decimal point did not become popular until Napier used it more than twenty 
years later. 

Without doubt it was in algebra that Viete made his most estimable 
contributions, for it was here that he came closest to modern views. Mathe- 
matics is a form of reasoning, and not a bag of tricks, such as Diophantus had 
possessed ; yet algebra, during the Arabic and early modern periods, had 
not gone far in freeing itself from the treatment of special cases. There could 
be little advance in algebraic theory so long as the chief preoccupation was 
with finding "the thing" in an equation with specific numerical coefficients. 
Symbols and abbreviations for an unknown, and for powers of the unknown, 
as well as for operations and for the relationship of equality, had been 
developed. Stifel had gone so far as to write A AAA for the fourth power of an 
unknown quantity ; yet he had no scheme for writing an equation that might 
represent any one of a whole class of equations — of all quadratics, say, or of 
all cubics. A geometer, by means of a diagram, could let ABC represent all 
triangles, but an algebraist had no counterpart for writing down all equations 

1 For further details see C. B. Boyer, "Viete's Use of Decimal Fractions," The Mathematics 
Teacher, 55 (1962), 123-127. 


of second degree. Letters had indeed been used to represent magnitudes, 
known or unknown, since the days of Euclid, and Jordanus had done this 
freely ; but there had been no way of distinguishing magnitudes assumed to 
be known from those unknown quantities that are to be found. Here Viete 
introduced a convention as simple as it was fruitful. He used a vowel to 
represent the quantity in algebra that was assumed to be unknown or un- 
determined and a consonant to represent a magnitude or number assumed 
to be known or given. Here we find for the first time in algebra a clear-cut 
distinction between the important concept of a parameter and the idea of 
an unknown quantity. 

Had Viete adopted other symbolisms extant in his day, he might have 
written all quadratic equations in the single form BA 2 + CA + D = 0, 
where A is the unknown and B, C, and D are parameters ; but unfortunately 
he was modern only in some ways and ancient and medieval in others. His 
algebra is fundamentally syncopated rather than symbolic, for although he 
wisely adopted the Germanic symbols for addition and subtraction and, 
still more wisely, used differing symbols for parameters and unknowns, the 
remainder of his algebra consisted of words and abbreviations. The third 
power of the unknown quantity was not A 3 , or even AAA, but A cubus, and 
the second power was A quadratus. Multiplication was signified by the Latin 
word in, division was indicated by the fraction line, and for equality Viete 
used an abbreviation for the Latin aequalis. It is not given for one man to 
make the whole of a given change; it must come in steps. 

One of the steps beyond the work of Viete was taken by Harriot when he 
revived the idea Stifel had had of writing the cube of the unknown as AAA. 
This notation was used systematically by Harriot in his posthumous book 
entitled Artis analyticae praxis and printed in 1631. Its title had been sug- 
gested by the earlier work of Viete, who had disliked the Arabic name algebra. 
In looking for a substitute Viete noted that in problems involving the "cosa" 
or unknown quantity, one generally proceeds in a manner that Pappus and 
the ancients had described as analysis. That is, instead of reasoning from what 
is known to what was to be demonstrated, algebraists invariably reasoned 
from the assumption that the unknown was given and deduced a necessary 
conclusion from which the unknown can be determined. In modern symbols, 
if we wish to solve x 2 — 3x + 2 = 0, for example, we proceed on the premise 
that there is a value of x satisfying this equation ; from this assumption we 
draw the necessary conclusion that (x — 2)(x — 1) = 0, so that either 
x — 2 = Oorx — 1 = (or both) is satisfied, hence that x necessarily is 2 or 1. 
However, this does not mean that one or both of these numbers will satisfy 
the equation unless we can reverse the steps in the reasoning process. That is, 
the analysis must be followed by the synthetic demonstration. 


In view of the type of reasoning so frequently used in algebra, Viete called 
the subject "the analytic art." Moreover, he had a clear awareness of the 
broad scope of the subject, realizing that the unknown quantity need not be 
either a number or a geometrical line. Algebra reasons about "types" or 
species, hence Viete contrasted logistica speciosa with logistica numerosa. His 
algebra was presented in the Isagoge (or Introduction), printed in 1591, but 
his several other algebraic works did not appear until many years after his 
death. In all of these he maintained a principle of homogeneity in equations, 
so that in an equation such as x 3 + lax = b the a is designated as planum 
and the b as solidum. This suggests a certain inflexibility, which Descartes 
removed a generation later ; but homogeneity also has certain advantages, as 
Viete undoubtedly saw. 

4 The algebra of Viete is noteworthy for the generality of its expression, but 
there are also other novel aspects. For one thing, Viete suggested a new 
approach to the solution of the cubic. Having reduced it to the standard form 
equivalent to x 3 + 3ax = b, he introduced a new unknown quantity y that 
was related to x through the equation y 2 + xy = a. This changes the cubic 
in x into a quadratic equation in y 3 , for which the solution is readily obtained. 
Moreover, Viete was aware of some of the relations between roots and 
coefficients of an equation, although here he was hampered by his failure to 
allow the coefficients and roots to be negative. He realized, for example, that 
if x 3 + b = 3ax has two positive roots, x y and x 2 , then 3a = x t 2 + x t x 2 + 
x 2 2 and b = x 1 x 2 2 + x 2 x 2 . This is, of course, a special case of our theorem 
that the coefficient of the term in x, in a cubic with leading coefficient unity, 
is the sum of the products of the roots taken two at a time, and the constant 
term is the negative of the product of the roots. Viete, in other words, was 
close to the subject of symmetric functions of the roots in the theory of 
equations. It remained for Girard in 1629, in Invention nouvelle en I'algebre, 
to state clearly the relations between roots and coefficients, for he allowed 
for negative and imaginary roots, whereas Viete had recognized only the 
positive roots. In a general way Girard realized that negative roots are 
directed in a sense opposite to that for positive numbers, thus anticipating 
the idea of the number line. "The negative in geometry indicates a retro- 
gression," he said, "where the positive is an advance." To him also seems to 
be largely due the realization that an equation can have as many roots as is 
indicated by the degree of the equation. Girard retained imaginary roots of 
equations because they show the general principles in the formation of an 
equation from its roots. 

5 Discoveries much like those of Girard had been made even earlier by 
Thomas Harriot, but these did not appear in print until ten years after Harriot 


had died of cancer in 1621. Harriot had been hampered in publication by 
conflicting political currents during the closing years of the reign of Queen 
Elizabeth I. He had been sent by Sir Walter Raleigh as a surveyor on the 
latter's expeditions to the New World in 1585, becoming thus the first 
substantial mathematician to set foot in North America. (Brother Juan Diaz, 
a young chaplain with some mathematical training, had earlier joined 
Cortes on an expedition to Yucatan in 1518.) On his return he published A 
Briefe and True Report of the New-found Land of Virginia (1586). When his 
patron lost favor with the queen and was executed, Harriot was granted a 
pension of £300 a year by Henry, Earl of Northumberland ; but in 1606 the 
earl was committed to the Tower by Elizabeth's successor, James I. Harriot 
continued to meet with Henry in the Tower, and distractions and poor health 
contributed to his failure to publish results. 

Harriot knew of relationships between roots and coefficients and between 
roots and factors, but like Viete he was hampered by failure to take note of 
negative and imaginary roots. In notations, however, he advanced the use 
of symbolism, being responsible for the signs > and < for "greater than" 
and "less than." 2 It was partly also his use of Recorde's equality sign that 
led to the ultimate adoption of this sign. Harriot showed much more modera- 
tion in the use of new notations than did his younger contemporary, William 
Oughtred. The latter published his Clavis mathematicae in the same year, 
1631, in which Harriot's Praxis was printed. In the Clavis the notation for 
powers was a step back toward Viete, for where Harriot had written 
AAAAAAA, for example, Oughted used Aqqc (that is, A squared squared 
cubed). Of all Oughtred's new notations, only one is now widely used — the 
cross x for multiplication. 3 

The homogeneous form of his equations shows that Viete's thought was 
always close to geometry, but his geometry was not on the elementary level 
of so many of his predecessors ; it was on the higher level of Apollonius and 
Pappus. Interpreting the fundamental algebraic operations geometrically, 
Viete realized that straightedge and compasses suffice up through square 
roots. However, if one permits the interpolation of two geometric means 
between two magnitudes, one can construct cube roots, or, a fortiori, solve 
geometrically any cubic equation. In this case one can, Viete showed, 

2 See J. A. Lohne, "Thomas Harriot als Mathematiker," Centaurus, 11 (1965), 19-45. For the 
life of Harriot see the article by Agnes M. Clerke in Dictionary of National Biography, XXIV 
(1890), 437-439. See also R. C. H. Tanner, "On the Role of Equality and Inequality in the 
History of Mathematics," British Journal of the History of Science, 1 (1962), 159-169; Robert 
Kargon, "Thomas Harriot, the Northumberland Circle and Early Atomism in England," 
Journal of the History of Ideas, 27 (1966), 128-136. 

3 For a life of Oughtred and further references, see Florian Cajori, William Oughtred, a Great 
Seventeenth-Century Teacher of Mathematics (1916), and the article on Oughtred by J. B. 
Mullinger in Dictionary of National Biography, XLII (1895), 356-358. On matters of symbolism 
one should be sure to consult Florian Cajori, A History of Mathematical Notations (1929). 


construct the regular heptagon, for this construction leads to a cubic of the 
form x 3 = ax + a. In fact, every cubic or quartic equation is solvable by 
angle trisections and the insertion of two geometric means between two 
magnitudes. Here we see clearly a very significant trend — the association of 
the new higher algebra with the ancient higher geometry. Analytic geometry 
could not, then, be far away, and Viete might have discovered this branch 
had he not avoided the geometrical study of indeterminate equations. The 
mathematical interests of Viete were unusually broad, hence he had read 
Diophantus' Arithmetical but when a geometrical problem led Viete to a 
final equation in two unknown quantities, he dismissed it with the casual 
observation that the problem is indeterminate. One wishes that, with his 
general point of view, he had inquired into the geometrical properties of the 

In many respects the work of Viete is greatly undervalued, but in one case 
it is possible that he has been given undue credit for a method known long 
before in China. In one of his later works, the De numerosa potestatum . . . 
resolutione (1600), he gave a method for the approximate solution of equations 
which is virtually that known today as Horner's method. To solve x 2 + Ix = 
60,750, for example, Viete found as a first lower approximation for x the 
value Xj = 200. Then upon substituting x = 200 + x 2 in the original equa- 
tion (or, as we should say, reducing the roots by 200), he found x 2 2 + 407x 2 = 
19,350. This equation now leads to a second approximation x 2 = 40. Now 
substituting x 2 = 40 + x 3 , the equation x 3 2 + 487x 3 = 1470 results, and 
the positive root of this is x 3 = 3. Hence x 2 = 43 and x = 243. This illustra- 
tive equation taken from Viete (but written in modern notation) could of 
course have been solved by completing the square ; but the author solved 
in the same manner other cases in which no simple alternative was at hand, 
finding, for example, a solution of x 6 + 6000x = 191,246,976. One of the 
beauties of the method is that it is applicable to any polynomial equation 
with real coefficients and a real root. 

The trigonometry of Viete, like his algebra, was characterized by a height- 
ened emphasis on generality and breadth of view. As Viete was the effective 
founder of a literal algebra, so he may with some justification be called the 
father of a generalized analytic approach to trigonometry that sometimes is 
known as goniometry. Here too, of course, Viete started from the work of his 
predecessors, notably, Regiomontanus and Rheticus. Like the former, he 
thought of trigonometry as an independent branch of mathematics ; like the 
latter, he generally worked without direct reference to half chords in a circle. 
Viete in the Canon mathematicus (1579) prepared extensive tables of all six 
functions for angles to the nearest minute. We have seen that he had urged 



the use of decimal, rather than sexagesimal, fractions; but to avoid all 
fractions as much as possible, Viete chose a "sinus totus" or hypotenuse of 
100,000 parts for the sine and cosine table and a "basis" or "perpendiculum" 
of 100,000 parts for the tangent, cotangent, secant, and cosecant tables. 
(Except for the sine function, he did not, however, use these names.) 

In solving oblique triangles, Viete in the Canon mathematicus broke them 
down into right triangles, but in another work a few years later, Variorum de 
rebus mathematicis (1593), there is a statement equivalent to our law of 
tangents : 

(a + b) 






Though Viete may have been the first to use this formula, it was first pub- 
lished by a more obscure figure, Thomas Finck, in 1583 in Geometria rotundi. 
Trigonometric identities of various sorts were appearing about this time 
in all parts of Europe, resulting in reduced emphasis on computation in the 
solution of triangles and more on analytic functional relationships. Among 
these were a group of formulas known as the prosthaphaeretic rules — that is, 
formulas that would convert a product of functions into a sum or difference 
(hence the name prosthaphaeresis, a Greek word meaning addition and 
subtraction). From the following type of diagram, for example, Viete derived 

y t _. .:_ .. ^ B ( Fig 161) 
x - y 

, r , . „ . x + y x 

the formula sin x + sin y = 2 sin — - — cos - 

2 2 

and sin y = CD. Then sin x + sin y = AB + CD 

. Let sin x 

AE = AC cos : 

On making the substitutions {x + y)/2 = A and 

x + y x - y 
2 sin cos . 

2 2 

(x ~ y)/2 = B, we have the more useful form sin (A + B) + sin (A — B) = 
2 sin A cos B. In a similar manner one derives sin (A + B) — sin (A — B) = 
2 cos A sin B by placing the angles x and y on the same side of the radius OD. 


FIG. 16.1 


The formulas 2 cos A cos B = cos (A + B) + cos (A - B) and 2 sin A sin B 
= cos {A — B) — cos (A + B) are somewhat similarly derived. 

The rules above sometimes bear the name "formulas of Werner," for they 
seem to have been used by Werner to simplify astronomical calculations. At 
least one of these, that converting a product of cosines to a sum of cosines, 
had been known to the Arabs in the time of ibn-Yunus, but it was only in the 
sixteenth century, and more particularly near the end of the century, that 
the method of prosthaphaeresis came to be widely used. If, for example, one 
wished to multiply 98,436 by 79,253, one could let cos .4 = 49,218 (that is, 
98,436/2) and cos B = 79,253. (In modern notation we would place a decimal 
point, temporarily, before each of the numbers and adjust the decimal point 
in the answer.) Then from the table of trigonometric functions one reads off 
angles A and B, and from the table one looks up cos (A + B) and cos (A — B), 
the sum of these being the product desired. Note that the product is found 
without any multiplication having been performed. In our example of 
prosthaphaeretic multiplication there is not a great saving of time and energy ; 
but when we recall that at that time trigonometric tables of a dozen or 
fifteen significant figures were not uncommon, the laborsaving possibilities 
of prosthaphaeresis become more pronounced. The device was adopted at 
major astronomical observatories, including that of Tycho Brahe (1546- 
1601) in Denmark, from where word of it was carried to Napier in Scotland. 
Quotients are handled in the same manner by using a table of secants and 

Perhaps nowhere is Viete's generalization of trigonometry into gonio- 
metry more pronounced than in connection with his multiple-angle formulas. 
The double-angle formulas for the sine and cosine had of course been known 
to Ptolemy, and the triple-angle formulas are then easily derived from 
Ptolemy's formulas for the sine and cosine of the sum of two angles. By 
continuing to use the Ptolemy formulas recursively, a formula for sin nx or 
cos nx can be derived, but only with great effort. Viete used an ingenious 
manipulation of right triangles and the well-known identity 

(a 2 + b 2 )(c 2 + d 2 ) = (ad + be) 2 + (bd - ac) 2 = (ad - be) 2 + (bd + ac) 2 

to arrive at formulas for multiple angles equivalent to what we should now 
write as 

n(n — 1) _, . , 

cos nx — cos" x cos" x sin x 

1 -2 

n(n - l)(n - 2)(» - 3) „ 

H cos 




„-i • n ( n ~ !)(« _ 2) „_, . , 

sin nx = n cos" x sin x cos x sin x + • • • 


where the signs alternate and the coefficients are in magnitude the alternate 
numbers in the appropriate line of the arithmetic triangle. Here we see a 
striking link between trigonometry and the theory of numbers. 

Viete noted also an important link between his formulas and the solution 8 
of the cubic equation. Trigonometry could serve as a handmaid to algebra 
where the latter had run up against a stone wall — in the irreducible case of 
the cubic. This evidently occurred to Viete when he noticed that the angle 
trisection problem led to a cubic equation. If in the equation x 3 + 3px + q = 
one substitutes mx = y (to obtain a degree of freedom in the later selection 
of a value form), the result is y 3 + 3m 2 py + m 3 q = 0. Comparing this with the 
formula cos 3 9 - | cos 9 - i cos 30 = 0, one notes that if y = cos 9, and if 
3m 2 p = — |, then — jcos 3d = m 3 q. Since p is given, m is now known (and 
will be real whenever the three roots are real). Hence 39 is readily determined, 
since q is known ; hence cos 9 is known. Therefore y, and from it x, will be 
known. Moreover, by considering all possible angles satisfying the conditions, 
all three real roots will be found. This trigonometric solution of the irreducible 
case of the cubic, suggested by Viete, was carried out in detail later by Girard 
in 1629 in Invention nouvelle en Valgebre. 

Viete in 1593 found an unusual opportunity to use his multiple-angle 
formulas. A Belgian mathematician, Adriaen van Roomen (1561-1615) or 
Romanus, had issued a public challenge to anyone to solve an equation of 
forty-fifth degree : 

x 45 - 45x 43 + 945x 41 3795x 3 + 45x = K 

The ambassador from the Low Countries to the court of Henry IV boasted 
that France had no mathematician capable of solving the problem proposed 
by his countryman. Viete, called upon to defend the honor of his countrymen, 
noted that the proposed equation was one that arises in expressing K = 
sin 450 in terms of x=2sin 6, and he promptly found the positive roots. The 
achievement so impressed van Roomen that he paid Viete a special visit. 

In applying trigonometry to arithmetic and algebraic problems, Viete was 
broadening the scope of the subject. 4 Moreover, his multiple-angle formulas 

4 There is no generally accessible edition of the works of Viete, nor even a good general 
account in English of his life and work. His Opera mathematical, ed. by Fr. van Schooten (Leiden, 
1646) is rare, as are most of Viete's published books. Useful is Frederic Ritter, Franfois Viete 
(1895). A recent communication from Professor D. J. Struik informs me that there exists an 
English translation (typeset) of Viete's Isagoge by J. Winfree Smith, St. John's College, Annapolis, 
Md., 1955. 


should have disclosed the periodicity of the goniometric functions, but it 
probably was his hesitancy with respect to negative numbers that prevented 
him — or his contemporaries — from going as far as this. There was consider- 
able enthusiasm for trigonometry in the late sixteenth and early seventeenth 
centuries, but this took the form primarily of synthesis and textbooks. It 
was during this period that the name "trigonometry" came to be attached 
to the subject. It was used as the title of an exposition by Bartholomaeus 
Pitiscus (1561-1613), which was first published in 1595 as a supplement to a 
book on spherics and again independently in 1600, 1606, and 1612. 5 Coinci- 
dentally the development of logarithms, ever since a close ally of trigonom- 
etry, was also taking place during these years. 

John Napier (or Neper), like Viete, was not a professional mathematician. 
He was a Scottish laird, the Baron of Murchiston, who managed his large 
estates and wrote on varied topics. In a commentary on the Book of Revela- 
tions, for example, he argued that the pope at Rome was the anti-Christ. He 
was interested in certain aspects of mathematics only, chiefly those relating 
to computation and trigonometry. "Napier's rods" or "bones" were sticks 
on which items of the multiplication tables were carved in a form ready to be 
applied to lattice multiplication ; "Napier's analogies" and "Napier's rule 
of circular parts" were devices to aid the memory in connection with spherical 

Napier tells us that he had been working on his invention of logarithms for 
twenty years before he published his results, a statement that would place 
the origin of his ideas about 1 594. He evidently had been thinking of the 
sequences, which had been published now and then, of successive powers of 
a given number — as in Stifel's Arithmetica Integra fifty years before and as 
in the works of Archimedes. In such sequences it was obvious that sums and 
differences of indices of the powers corresponded to products and quotients 
of the powers themselves ; but a sequence of integral powers of a base, such 
as two, could not be used for computational purposes because the large 
gaps between successive terms made interpolation too inaccurate. While 
Napier was pondering the matter, Dr. John Craig, physician to James VI of 
Scotland, called on him and told him of the use in Denmark of prosthaphaer- 
esis. Craig presumably had been in the party when James VI of Scotland in 
1590 had sailed with a delegation for Denmark to meet his bride-to-be, Anne 
of Denmark. The party had been forced by storms to land on the shore not 
far away from the observatory of Tycho Brahe, where, while awaiting more 
favorable weather, they were entertained by the astronomer. Reference 
apparently was made to the marvelous device of prosthaphaeresis, freely 

5 For a full account of this work see Sister Mary Claudia Zeller, The Development of Trigono- 
metry from Regiomontanus to Pitiscus (1944). 


used in the computations at the observatory ; and word of this encouraged 
Napier to redouble his efforts and ultimately to publish in 1614 the Mirifici 
logarithmorum canonis descriptio ("A Description of the Marvelous Rule of 

The key to Napier's work can be explained very simply. To keep the terms 1 
in a geometrical progression of integral powers of a given number close 
together, it is necessary to take as the given number something quite close to 
one. Napier therefore chose to use 1 — 10~ 7 (or .9999999) as his given 
number. Now the terms in the progression of increasing powers are indeed 
close together—too close, in fact. To achieve a balance and to avoid decimals 
Napier multiplied each power by 10 7 . That is, if N = 10 7 (1 - 1/10 7 ) L , then 
L is Napier's "logarithm" of the number N. Thus his logarithm of It)' 7 is 0, 
his logarithm of 10 7 (1 - 1/10 7 ) = 9999999 is 1, and so on. If his numbers 
and his logarithms were to be divided by 10 7 , one would have virtually a 
system of logarithms to the base 1/e, for (1 - 1/10 7 ) 107 is close to 
hm (1 - l/rif = 1/e. It must be remembered, however, that Napier had 

no concept of a base for a system of logarithms, for his definition was different 
from ours. The principles of his work were explained in geometrical terms 
as follows. Let a line segment AB and a half line or ray CDE ... be given 
(Fig. 16.2). Let a point P start from A and move along AB with variable 
speed decreasing in proportion to its distance from B ; during the same time 
let a point Q start from C and move along CDE . . . with uniform speed equal 
to the rate with which point P began its motion. Napier called this variable 
distance CQ the logarithm of the distance PB. 



D Q 

FIG. 16.2 

Napier's geometrical definition is of course in agreement with the numerical 
description given above. To show this, let PB = x and CQ = y. If AB is 
taken as 10 7 , and if the initial speed of P is also taken as 10 7 , then in modern 
calculus notations we have dx/dt = -x and dy/dt = 10 7 , x = 10 7 , v = 0. 
Then dy/dx = - 10 7 /x, or y = - 10 7 In ex, where c is found from the initial 
boundary conditions to be 10 -7 . Hence y = - 10 7 In (x/10 7 ) or y/10 7 = 
logi /e (x/10 7 ). That is, if the distances PB and CQ were divided by 10 7 , the 
definition of Napier would lead precisely to a system of logarithms to the 


base 1/e, as mentioned earlier. Needless to say, Napier built up his tables 
numerically rather than geometrically, as the word "logarithm," which he 
coined, implies. At first he called his power indices "artificial numbers," but 
later he made up the compound of the two Greek words Logos (or ratio) and 
arithmos (or number). 

Napier did not think of a base for his system, but his tables nevertheless 
were compiled through repeated multiplications, equivalent to powers of 
.9999999. Obviously the power (or number) decreases as the index (or 
logarithm) increases. This is to be expected, because he was essentially using 
a base 1/e which is less than one. A more striking difference between his 
logarithms and ours lies in the fact that his logarithm of a product (or quo- 
tient) generally was not equal to the sum (or difference) of the logarithms. 
If Li = Log JVi and L 2 = Log N 2 , then A^ = 10 7 (1 - lO" 7 )^ and N 2 = 
10 7 (1 - 10' 1 ) L \ whence N.NJW = 10 7 (1 - KT 7 ) Ll+t2 , so that the sum 
of Napier's logarithms will be the logarithm not of N 1 N 2 but of N t N 2 /l0 7 . 
Similar modifications hold, of course, for logarithms of quotients, powers, 
and roots. If L = LogAf, for instance, then nL = Log A^/IO 71 " -11 . These 
differences are not too significant, for they merely involve shifting a decimal 
point. That Napier was thoroughly familiar with rules for products and 
powers is seen in his remark that all numbers (he called them "sines") in the 
ratio of 2 to 1 have differences of 6,931,469.22 in logarithms, and all those 
in the proportion of 10 to 1 have differences of 23,025,842.34 in logarithms. 
In these differences we see, if we shift the decimal point, the natural logarithms 
of the numbers two and ten. Hence it is not unreasonable to use the name 
"Napierian" for natural logarithms, even though these logarithms are not 
strictly the ones that Napier had in mind. 

The concept of the logarithmic function is implied in Napier's definition 
and in all of his work with logarithms, but this relationship was not upper- 
most in his mind. He had laboriously built up his system for one purpose — 
the simplification of computations, especially of products and quotients. 
Moreover, that he had trigonometric computations in view is made clear by 
the fact that what we for simplification of exposition referred to as Napier's 
logarithm of a number, he actually called the logarithm of a sine. In Fig. 16.2, 
the line CQ was called the logarithm of the sine PB. This makes no real 
difference either in theory or in practice. 

11 The publication in 1614 of the system of logarithms was greeted with 

prompt recognition, and among the most enthusiastic admirers was Henry 
Briggs, the first Savilian professor of geometry at Oxford. In 1615 he visited 
Napier at his home in Scotland, and there they discussed possible modifica- 
tions in the method of logarithms. Briggs proposed that powers often should 
be used, and Napier said he had thought of this and was in agreement. 



Napier at one time had proposed a table using log 1 = and log 10 = 10 
(to avoid fractions). The two men finally concluded that the logarithm of one 
should be zero and that the logarithm often should be one. Napier, however, 
no longer had the energy to put their ideas into practice. He died in 1617, the 
year in which his Rhabdologia, with its description of his rods, appeared. The 
second of his classic treatises on logarithms, the Mirifici logarithmorum 
canonis construction in which he gave a full account of the methods he used 
in building up his tables, appeared posthumously in 1619. 6 To Briggs, there- 
fore, fell the task of making up the first table of common, or Briggsian, 
logarithms. Instead of taking powers of a number close to one, as had Napier, 
Briggs began with log 10 = 1 and then found other logarithms by taking 
successive roots. By finding v /l0 = 3.162277, for example, Briggs had 
log 3.162277 = .5000000, and from 10* = ^3 1.62277 = 5.623413, he had 
log 5.623413 = .7500000. Continuing in this manner, he computed other 
common logarithms. In the year of Napier's death, 1617, Briggs published 
his Logarithmorum chilias prima — that is, the logarithms of numbers from 1 
to 1000, each carried out to fourteen places. In 1624, in Arithmetica logarith- 
mica, Briggs extended the table to include common logarithms of numbers 
from 1 to 20,000 and from 90,000 to 100,000, again to fourteen places. Work 
with logarithms now could be carried out just as it is today, for all the usual 
laws of logarithms applied in connection with Briggs' tables. Incidentally, 
it is from Briggs' book of 1624 that our words "mantissa" and "character- 
istic" are derived. While Briggs was working out tables of common log- 
arithms, a contemporary, John Speidell, drew up natural logarithms of 
trigonometric functions, publishing these in his New Logarithmes of 1619. A 
few natural logarithms had, in fact, appeared earlier in 1616 in an English 
translation by Edward Wright (1559-1615) of Napier's first work on log- 
arithms. Seldom has a new discovery "caught on" so rapidly as did the inven- 
tion of logarithms, and the result was the prompt appearance of tables of 
logarithms which were more than adequate for that time. 

It has been implied, up to this point, that the invention of logarithms was 1 2 
the work of one man alone, but such an impression must not be permitted 
to remain. Napier was indeed the first one to publish a work on logarithms, 

6 There are many good accounts of Napier's work. Among the best is the article on "Log- 
arithms" by J. W. L. Glaisher in the Encyclopaedia Britannica, 1 1th ed., Vol. 16, pp. 868-877. Also 
excellent is the article by Florian Cajori, "History of the Exponential and Logarithmic Con- 
cepts," American Mathematical Monthly, 20 (1913), 5-14, 35-^7, 75-84, 107-117, 148-151, 
173-182, 205-210, as well as the article by Glaisher, "On Early Tables of Logarithms and Early 
History of Logarithms," Quarterly Journal of Pure and Applied Mathematics, 48 (1920), 151-192. 
See also E. W. Hobson, John Napier and the Invention of Logarithms (1914), and the Napier 
Tercentary Memorial Volume, ed. by C. G. Knott (1915). The latter, however, gives Napier more 
credit than is his due. 


but very similar ideas were developed independently in Switzerland by 
Jobst Biirgi at about the same time. In fact, it is possible that the idea of 
logarithms had occurred to Biirgi 7 as early as 1588, which would be half a 
dozen years before Napier began work in the same direction. However, 
Biirgi printed his results only in 1620, half a dozen years after Napier had 
published his Descriptio. Biirgi 's work appeared at Prague in a book entitled 
Arithmetische und geometrische Progress-Tabulen, and this indicates that the 
influences leading to his work were similar to those operating in the case of 
Napier. Both men proceeded from the properties of arithmetic and geometric 
sequences, spurred, probably, by the method of prosthaphaeresis. The 
differences between the work of the two men lie chiefly in their terminology 
and in the numerical values they used ; the fundamental principles were the 
same. Instead of proceeding from a number a little less than one (as had 
Napier, who used 1 — 10 7 ), Biirgi chose a number a little greater than one — 
the number 1 + 10~ 4 ; and instead of multiplying powers of this number by 
10 7 , Biirgi multiplied by 10 8 . There was one other minor difference : Biirgi 
multiplied all of his power indices by ten in his tabulation. That is, if N = 
10 8 (1 + 10~ 4 ) L , Biirgi called 10L the "red" number corresponding to the 
"black" number N. If in this scheme we were to divide all the black numbers 
by 10 8 and all red numbers by 10 5 , we should have virtually a system of 
natural logarithms. For instance, Biirgi gave for the black number 
1,000,000,000 the red number 230,270.022, which, on shifting decimal points, 
is equivalent to saying that In 10 = 2.30270022. This is not a bad approxima- 
tion to the modern value, especially when we recall that (1 + 10" 4 ) 10 " is 
not quite the same as lim (1 + 1/n)", although the values agree to four 
significant figures. "^ co 

In publishing his tables, Biirgi placed his red numbers on the side of the 
page and his black numbers in the body of the table, hence he had what we 
should describe as an antilogarithmic table ; but this is a minor matter. The 
essence of the principle of logarithms is there, and Biirgi must be regarded 
as an independent discoverer who lost credit for the invention because of 
Napier's priority in publication. In one respect his logarithms come closer 
to ours than do Napier's, for as Biirgi's black numbers increase, so do the 
red numbers ; but the two systems share the disadvantage that the logarithm 
of a product or quotient is not the sum or difference of the logarithms. 

1 3 The invention of logarithms ultimately had a tremendous impact on the 
structure of mathematics, but at that time it could not be compared in 
theoretical significance with the work, say, of Viete. Logarithms were hailed 
gladly by Kepler not as a contribution to thought, but because they vastly 

7 See J. E. Hofmann: Geschichte der Mathematik (1963), p. 167. 


increased the computational power of the astronomer. Viete was not exactly 
a "voice crying in the wilderness" ; it is nevertheless true that most of his 
contemporaries were primarily concerned with the practical aspects of 
mathematics. Biirgi was a clockmaker, Galileo was a physicist and astrono- 
mer, and Stevin was an engineer. It was inevitable that these men should have 
preferred parts of mathematics that gave promise of applicability to their 
fields. Biirgi and Stevin, for example, aided in the development of decimal 
fractions, and Biirgi and Galileo were rivals in the manufacture and sale of a 
practical computing device known as the proportional compass. The so- 
called Renaissance in science, illustrated by the work of such men as Leonardo 
da Vinci and Copernicus, had been a ferment that to a large extent grew 
out of contact between old ideas and new and between the views of artisans 
and those of scholars. 

In mathematics of the sixteenth century there were diverse and conflicting 
tendencies ; but we can perceive there, as well as in science, the results of a 
confrontation of established ideas by new concepts and of theoretical views 
by the exigencies of practical problems. We have seen that the work of Viete 
grew out of two factors in particular: (1) the recovery of ancient Greek 
classics and (2) the relatively new developments in medieval and early 
modern algebra. Throughout the sixteenth century both professional and 
amateur theoretical mathematicians showed concern for the practical tech- 
niques of computation, which contrasted strongly with the dichotomy 
emphasized two millennia earlier by Plato. Viete, the outstanding mathe- 
matician in France, in 1579 had urged the replacement of sexagesimal frac- 
tions by decimal fractions. In 1585 an even stronger plea for the use of the 
ten-scale for fractions, as well as for integers, was made by the leading 
mathematician in the Low Countries, Simon Stevin of Bruges. 

Stevin, a supporter of the Protestant faction under William of Orange in 
the struggle against Catholic Spain, was tolerant, if not indifferent, in religion. 
Under Prince Maurice of Nassau he served as quartermaster and as com- 
missioner of public works, and for a time he tutored the prince in mathe- 
matics. Although Stevin was a great admirer of the theoretical treatises of 
Archimedes, there runs through the works of the Flemish engineer a strain 
of practicality that is more characteristic of the Renaissance period than of 
classical antiquity. Thus Stevin was largely responsible for the introduction 
into the Low Countries of double-entry bookkeeping fashioned after that of 
Pacioli in Italy almost a century earlier. 8 Of far more widespread influence 
in economic practice, in engineering, and in mathematical notations was 
Stevin's little book with the Flemish title De thiende ("The Tenth"), published 
at Leyden in 1585. A French version entitled La disme appeared in the same 

8 For an account of Stevin's life and work see The Principal Works of Simon Stevin (edited by 
E. J. Dijksterhuis, D. J. Struik, and others), Amsterdam, 1955-1958. 


year, and the popularity of the book was such that its place in the development 
of mathematics has been often misunderstood. 

It is clear that Stevin was in no sense the inventor of decimal fractions, nor 
was he the first systematic user of them. More than incidental use of decimal 
fractions is found in ancient China, in medieval Arabia, and in Renaissance 
Europe ; by the time of Viete's forthright advocacy of decimal fractions in 
1579 they were generally accepted by mathematicians on the frontiers of 
research. Among the common people, however, and even among mathe- 
matical practitioners, decimal fractions became widely known only when 
Stevin undertook to explain the system in full and elementary detail. He 
wished to teach everyone "how to perform with an ease, unheard of, all 
computations necessary between men by integers without fractions." That 
is, oddly enough Stevin was concentrating on his tenths, hundredths, 
thousandths, and so on, as integral numerators, much as we do in the com- 
mon measure of time in minutes and seconds. How many of us think of 
3 minutes and 4 seconds, say, as a fraction? We are far more likely to think of 
3 minutes as an integer than as 3/60 of an hour ; and this was precisely 
Stevin's view. For this reason he did not write his decimal expressions with 
denominators, as Viete had ; instead, in a circle above or after each digit he 
wrote the power of ten assumed as a divisor. Thus the value of n, approxi- 
mately, appeared as 

® ® © © © 

3® 1® 4© 1® 6© or 3 14 16 

Instead of the words "tenth," "hundredth," and so on, he used "prime," 
"second," and so on, somewhat as we still designate the places in sexagesimal 
fractions. 9 

Stevin obviously had the right idea about decimal fractions, but his 
Bombelli-inspired notation for places was more appropriate for algebra 
than for arithmetic. Fortunately, the modern notation was not long delayed. 
In the 1616 English translation of Napier's Descriptio decimal fractions 
appear as today, with a decimal point separating the integral and fractional 
portions. In 1617 in the Rhabdologia, in which he described computation 
using his rods, Napier referred to Stevin's decimal arithmetic and proposed 
a point or a comma as the decimal separatrix. In the Napierian Constructio 
of 1619 the decimal point became standard in England, but many European 
countries continue to this day to use the decimal comma. Stevin urged also 

9 See D. J. Struik, "Simon Stevin and the Decimal Fractions," The Mathematics Teacher, 
52(1959), 474-478; also George Sarton, "Simon Stevin of Bruges (1548-1620)," Isis, 21 (1934), 
241-303, and "The First Explanation of Decimal Fractions and Measures (1585)," Isis, 23 
(1935), 153-244. See also D. E. Smith, "The Invention of the Decimal Fraction," Teachers 
College Bulletin, 5 (1910), 11-21. 


Thiende. I) 


der Thiende vande 



^ffefendegbegevenThiendetdente ver- 
gaderen: hare Somme te vmden. 

T'Ghegheven. Hct up drie oirdcns van 
Thicndetalen, welckereer|te 27 @ 8 (1.: 4(3) 
7 .'$ , dc twcede , 37.(0) 6 (f, 7 S 5 ($ , de de&le, 
S7$@7(i}S(iji(i<, Pbegheeiv de. Wy 
moccen hacr Somme vinden . Wircxinc 
Men (al dc ghegheven ghc- © (f) (2 Q) 

talen in oirden ftellen ab 27847 

hierncven, die vergaderen- , _ g _ - 

denaerdeghemeene manic 87*782 

re der vergaderinghe van ■ 

hcelegetalenaldus: 9 4' J ° 4 

Comt in Somme (door het 1 . probleme onfer 
Franfcher Arith.) 941304 dat fijn (t'welck de 
teeckenen boven de ghetalen ftaende, anwijfen) 
9 4 1 © 3 (?. o (2, 4 Q. Ick fc^ghc de (elve te wefen 
de ware begheerde Somme. B e wy s. Dc ghegc- 
ven 2 7 @ 8 (1) 4 (2 ;. 7 Qf , doen (doordc y. he-pa- 
ling) 27^, TVS • TO , maeck * t,famcn i7i 8 ^rr. 
Ende door de/elvcreden Allien de 37(3^; (r.'7&) 
5Q, weerdich fijn 37^-; En <k de 875(0)7(1) 


A page from Stevin's work (1 634 edition) showing Stevin's notations for decimal fractions. 

a decimal system of weights and measures, but this part of his work has not 
yet triumphed in England and America. 

In the history of science, as well as in mathematics, Stevin is an important 
figure. He and a friend dropped two spheres of lead, one ten times the weight 
of the other, from a height of 30 feet onto a board and found the sounds of 
their striking the board to be almost simultaneous. But Stevin's published 
report (in Flemish in 1 586) of the experiment has received far less notice than 
the similar and later experiment attributed, on very doubtful evidence, 


to Galileo. On the other hand, Stevin usually receives credit for the discovery 
of the law of the inclined plane, justified by his familiar "wreath of spheres" 
diagram, whereas this law had been given earlier by Jordanus Nemorarius. 10 

1 4 Stevin was a practical-minded mathematician who saw little point in the 
more speculative aspects of the subject. Of imaginary numbers he wrote : 
"There are enough legitimate things to work on without need to get busy on 
uncertain matter." Nevertheless, he was not narrow-minded, and his reading 
of Diophantus impressed him with the importance of appropriate notations 
. as an aid to thought. Although he followed the custom of Viete and other 
contemporaries in writing out some words, such as that for equality, he 
preferred a purely symbolic notation for powers. Carrying over to algebra 
his positional notation for decimal fractions, he wrote (5) instead of Q (or 
square), (3) for C (or cube), for QQ (or square-square), and so on. This 
notation may well have been suggested by Bombelli's Algebra. It also 
paralleled a notation of Biirgi who indicated powers of an unknown by 
placing Roman numerals over the coefficients. Thus x 4 + 3x 2 - 7x, for 
example, would be written by Biirgi as 

iv ii i 


and by Stevin as 

© © (D 

1 + 3-7 

Stevin went further than Bombelli or Biirgi in proposing that such notations 
be extended to fractional powers. (It is interesting to note that although 
Oresme had used both fractional-power indices and coordinate methods in 
geometry, these seem to have had only a very indirect influence, if any, on 
the progress of mathematics in the Low Countries and in France in the early 
seventeenth century.) Even though Stevin had no occasion to use the frac- 
tional index notation, he clearly stated that \ in a circle would mean square 
root and that f in a circle would indicate the square root of the cube. A little 
later Girard, editor of Stevin' s works, adopted the circled-numerical notation 
for powers, and he, too, indicated that this could be used for roots instead 
of such symbols as J and ^J . Symbolic algebra was developing apace, 
and it reached its maturity, only eight years after Girard 's Invention nouvelle, 
in Descartes' La geometrie. 

10 See Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, Wis.: 
University of Wisconsin Press, 1959). 


Simon Stevin was a typical mathematician of his day in that he enjoyed 1 5 
the elementary applications of the subject ; in this respect he was like Galileo. 
Galileo had originally intended to take a degree in medicine, but a taste for 
Euclid and Archimedes led him instead to become a professor of mathe- 
matics, first at Pisa and later at Padua. This does not mean, however, that 
he taught on the level of the authors he admired. Little mathematics was 
included in university curricula of the time, and a large proportion of what 
was taught in Galileo's courses would now be classified as physics or 
astronomy or engineering applications. Moreover, Galileo was not a 
"mathematician's mathematician," as was Viete; he came close to being . 
what we should call a mathematical practitioner. This we see in his interest 
in computational techniques that led him in 1597 to construct and market a 
device that he called his "geometric and military compasses." 

In a pamphlet of 1606 with the title Le operazioni del compasso geometrico 
et militare, he described in detail the way in which the instrument could be 
used to perform a variety of computations quickly without pen or paper or 
an abacus. The theory behind this was extremely elementary, and the degree 
of accuracy was very limited, but the financial success of Galileo's device 
shows that military engineers and other practitioners found a need for such 
an aid in calculation. Burgi had constructed a similar device, but Galileo 
had a better entrepreneurial sense, one that gave him an advantage. The 
Galilean compasses consisted of two arms pivoted as in the ordinary com- 
passes of today, but each of the arms was engraved with graduated scales of 
varying types. Fig. 16.3 shows only the simple equispaced markings up to 
250, and only the simplest of the many possible computations, the first one 
explained by Galileo, is described here. If, for instance, one wishes to 
divide a given line segment into five equal parts, one opens a pair of ordinary 
compasses (or divider) to the length of the line segment. Then one opens the 
geometric compasses so that the distance between the points of the divider 
just spans the distance between two markings, one on each arm of the 

i l i i i i | i i i i 1 1 i i i | i i i i | 

50 100 150 200 250 

FIG. 16.3 


geometric compasses, which are simple integral multiples of five — say, the 
number 200 on each scale. Then, if one holds the opening of the geometric 
compasses fixed and places the ends of the divider on the mark for 40 on 
each scale, the distance between the divider points will be the desired fifth 
of the length of the original line segment. The instructions Galileo provided 
with his compasses included many other operations, from changing the 
scale of a drawing to computing amounts of money under compound 
interest. 1 x 

16 Elementary though it was, Galileo's 1606 pamphlet on the geometric 
compasses, published when he was over forty years old, was his only strictly 
mathematical treatise. Nevertheless, it was far from his only contribution to 
the field. More significant are the many appeals in his astronomical and 
physical works to mathematical reasoning, and here he was frequently close 
to developments leading to the calculus. Much the same can be said also of 
Stevin and Kepler. Physics and astronomy had reached the point where 
there was increasing need for arguments concerning the infinitely large and 
small — the subject now known as analysis. Viete had been one of the first 
to use the word "analysis" as a synonym for algebra, but he was one of the 
earliest analysts also in the more modern sense of one who studies infinite 

Before the time of Viete there had been many good and bad approximations 
for the ratio of circumference to diameter in a circle, 12 such as that of V. 
Otho and A. Anthonisz who, evidently independently, rediscovered (about 
1573) the approximation n « 355/113 by subtracting numerators and 
denominators of the Ptolemaic and Archimedean values, 377/120 and 22/7 
respectively. Viete worked out n correctly to ten significant figures, apparently 
unaware of al-Kashi's still better approximation. The most impressive 
achievement of this type was by Ludolph van Ceulen (1540-1610). First he 
published in 1596 a twenty-place value obtained by starting with a polygon 
of fifteen sides and doubling the number of sides thirty-seven times. Using a 
still larger number of sides, he ultimately achieved a thirty-five place approxi- 
mation, which his widow had engraved on his tombstone. This feat of 
computation so impressed his successors that n frequently has been known 
as the "Ludolphine constant." Such tours de force, however, have no 
theoretical significance. An exact expression was far more to be desired; 
and it is in this respect that Viete gave the first theoretically precise numerical 

1 ' A brief excerpt, in English translation, from he operazioni del compasso geometrico et 
militare is included in D. E. Smith, Source Book in Mathematics (1929), pp. 186-191. The Italian 
original appears in Galileo's Opere (Florence, 1890-1909, 20 vols.), II, 335-424. 

12 An extensive list of very many of these values is given by H. C. Schepler, "The Chronology 
of Pi," Mathematics Magazine,23 (1949-1950), 162-170, 216-228, 279-283. 


expression for n~ an infinite product that can be written as 

In a sense Viete's approach is not novel. His product is easily derived by 
inscribing a square within a circle, then applying the recursive trigono- 
metric formula a 2n = a n sec n/n, where a n is the area of the inscribed regular 
polygon of n sides, and finally allowing n to increase indefinitely. Moreover, 
the same infinite product is readily derived by calculating radius vectors of 
points on the quadratrix of Hippias, r sin 9 = 29, for successive bisections 
of the angle, beginning with 9 = n/2 and noting that rjr„_ , = cos n/2" 
and that ^lim r„ = 2/n. Nevertheless, it was Viete who first expressed n 

analytically, a significant result because arithmetic, algebraic, and trigono- 
metric notations were more and more invading the realm of the infinitely 
large and the infinitely small, a field once almost exclusively dominated by 

Viete's last years were embittered by a controversy largely of his own 
making. « Christopher Clavius (1537-1612), a well-known contemporary 
mathematician, had advised Pope Gregory XIII on the reform of the calendar, 
and Viete attacked the accuracy of this. The bitterness of Viete's statement 
may have resulted from resentment that his opponents failed to evaluate 
correctly the significance of the new "logistica speciosa." Viete had a few 
ardent disciples, one of whom, Alexander Anderson (1582-ca. 1620) of 
Scotland, published some of his work in 1615, but it was not until the 1630s 
that the "Analytic Art" began to receive the attention it deserved. This delay is 
in sharp contrast to the rapidity with which logarithms became widely 

Viete was primarily an analyst, but he contributed also to pure geometry. 1 7 
Here his work centered chiefly on problems raised in the works of Apollonius. 
Regiomontanus had doubted that the celebrated Apollonian problem 
(proposed in the lost book On Tangencies) of constructing a circle tangent 
to three circles could be solved with compass and straightedge ; van Rooman 
therefore solved it by means of two intersecting hyperbolas. Viete knew 
through a reference in Pappus' Collection that an elementary construction 
was indeed possible, and in his Varia responsa of 1600 he published his 
solution. In a reconstruction of what he thought Apollonius' book may 
have contained, Viete proceeded through the simpler cases, in which one or 
more of the three circles are replaced by points or lines, until he had reached 
the tenth and most difficult case— that of three circles. This construction was 
one of Viete's most beautiful contributions to mathematics. Such problems 
in geometry later had a significant attraction for Descartes, but Viete's 


immediate successors were far less attracted to the theoretical results of 
Apollonius than to the applicability of Archimedes' work. 

1 8 Stevin, Kepler, and Galileo all had need for Archimedean methods, being 
practical men, but they wished to avoid the logical niceties of the method of 
exhaustion. It was largely the resulting modifications of the ancient in- 
finitesimal methods that ultimately led to the calculus, and Stevin was one 
of the first to suggest changes. In his Statics of 1586, almost exactly a century 
before Newton and Leibniz published their calculus, the engineer of Bruges 
demonstrated as follows that the center of gravity of a triangle lies on its 
median. In the triangle ABC inscribe a number of parallograms of equal 
height whose sides are pairwise parallel to one side and to the median drawn 
to this side (Fig. 16.4). The center of gravity of the inscribed figures will lie 

FIG. 16.4 

on the median, by the Archimedean principle that bilaterally symmetrical 
figures are in equilibrium. However, we may inscribe in the triangle an infinite 
number of such parallelograms, and the greater the number of parallelograms, 
the smaller will be the difference between the inscribed figure and the triangle. 
Inasmuch as the difference can be made as small as one pleases, the center of 
gravity of the triangle also lies on the median. In some of the propositions on 
fluid pressure Stevin supplemented this geometrical approach by a "demon- 
stration by numbers" in which a sequence of numbers tended to a limiting 
value ; but the "Dutch Archimedes" had more confidence in a geometrical 
proof than an arithmetic one. 1J 

1 9 Whereas Stevin was interested in physical applications of infinitely many 
infinitely small elements, Kepler had need for astronomical applications, 
especially in connection with his elliptical orbits of 1609. As early as 1604 
Kepler had become involved with conic sections through work in optics 
and the properties of parabolic mirrors. Whereas Apollonius had been 
inclined to think of the conies as three distinct types of curves — ellipses, 

13 For further details see C. B. Boyer, The Concepts of the Calculus (1939), pp. 99-104. 


Johann Kepler. 

parabolas, and hyperbolas Kepler preferred to think of five species of 
conies, all belonging to a single family or genus. With a strong imagination 
and a Pythagorean feeling for mathematical harmony, Kepler developed for 
conies in 1604 (in his Ad Vitellionem paralipomena, that is, "Introduction to 


Vitello's Optics") what we call the principle of continuity. From the conic 
section made up simply of two intersecting lines, in which the two foci 
coincide at the point of intersection, we pass gradually through infinitely 
many hyperbolas as one focus moves farther and farther from the other. 
When the one focus is infinitely far away, we no longer have the double- 
branched hyperbola, but the parabola. As the moving focus passes beyond 
infinity and approaches again from the other side, we pass through infinitely 
many ellipses until, when the foci again coincide, we reach the circle. 

The idea that a parabola has two foci, one at infinity, is due to Kepler, as 
is also the word "focus" (Latin for "hearthside") ; we find this bold and 
fruitful speculation on "points at infinity" extended a generation later in 
the geometry of Desargues. Meanwhile, Kepler found a useful approach to 
the problem of the infinitely small in astronomy. In his Astronomia nova of 
1609 he announced his first two laws of astronomy. (1) the planets move 
about the sun in elliptical orbits with the sun at one focus, and (2) the radius 
vector joining a planet to the sun sweeps out equal areas in equal times. In 
handling problems of areas such as these, Kepler thought of the area as made 
up of infinitely small triangles with one vertex at the sun and the other two 
vertices at points infinitely close together along the orbit. In this way he was 
able to use a crude type of integral calculus resembling that of Oresme. 
The area of a circle, for example, is found in this way by noting that the 
altitudes of the infinitely thin triangles (Fig. 16.5) are equal to the radius. 
If we call the infinitely small bases, lying along the circumference, b lt b 2 , 
..., b„,.. ., then the area of the circle — that is, the sum of the areas of the 
triangles — will be \bj + jb 2 r + ■ • ■ + \b n r + ■■■ or ^(b^ + b 2 + ■ ■ ■ + 
b„ + ■■ •). Inasmuch as the sum of the b's is the circumference C, the area A 
will be given by A = \rC, the well-known ancient theorem which Archimedes 
had proved more carefully. 

By analogous reasoning Kepler knew the area of the ellipse — a result of 
Archimedes not then extant. The ellipse can be obtained from a circle of 
radius a through a transformation under which the ordinate of the circle at 

FIG. 16.5 


each point is shortened according to a given ratio, say b:a. Then, following 
Oresme, one can think of the area of the ellipse and the area of the circle as 
made up of all the ordinates for points on the curves (Fig. 16.6) ; but inasmuch 
as the ratio of the components of the areas are in the ratio b : a, the areas 
themselves must have the same ratio. However, the area of the circle is known 
to be na 2 ; hence the area of the ellipse x 2 /a 2 + y 2 /b 2 = 1 must be nab. This 
result is correct ; but the best that Kepler could do for the circumference of 
the ellipse was to give the approximate formula n(a + b). Lengths of curves 
in general, and of the ellipse in particular, were to elude mathematicians for 
another half a century. 

FIG. 16.6 

Kepler had worked with Tycho Brahe, first in Denmark and later at Prague, 
where, following Brahe's death, Kepler became mathematician to the Em- 
peror Rudolph II. One of his duties was the casting of horoscopes ; mathe- 
maticians, whether for emperors or at universities, found various applica- 
tions for their talents, as Kepler discovered while he was at Linz, in Austria. 
The year 1612 had been a very good one for wine, and Kepler began to 
meditate at this time on the crude methods then in use for estimating the 
volumes of wine casks. He compared these with the methods of Archimedes 
on the volumes of conoids and spheroids, and then he proceeded to find the 
volumes of various solids of revolution not previously considered by Archi- 
medes. For example, he revolved a segment of a circle about its chord, calling 
the result a citron if the segment was less than a semicircle and an apple if 
the segment exceeded a semicircle. His volumetric method consisted in 
regarding the solids as composed of infinitely many infinitesimal elements, 
and he proceeded much as we have indicated above for areas. He dispensed 
with the Archimedean double reductio ad absurdum, and in this he was 
followed by most mathematicians from that time to the present. 14 

4 See D. J. Struik, "Kepler as a Mathematician," in Johann Kepler, 1571-1630. A Tercentenary 
Commemoration of His Life and Works, ed. by F. E. Brasch (1931). 


20 Kepler collected his volumetric thoughts in a book that appeared in 1615 
under the title Stereometria doliorum ("Volume-measurement of Barrels"). 
For a score of years it seemed to have excited no great interest, but in 1635 
the Keplerian ideas were systematically expanded in a celebrated book 
entitled Geometria indivisibilibus, written by Cavalieri, a disciple of Galileo. 
While Kepler had been studying wine barrels, Galileo had been scanning the 
heavens with his telescope and rolling balls down inclined planes. The 
results of Galileo's efforts were two famous treatises, one astronomical and 
the other physical. They were both written in Italian, but we shall refer to 
them in English as The Two Chief Systems (1632) and The Two New Sciences 
(1638). The first was a dialogue concerning the relative merits of the Ptolemaic 
and Copernican views of the universe, carried on by three men : Salviati (a 
scientifically informed scholar), Sagredo (an intelligent layman), and Simpli- 
cio (an obtuse Aristotelian). In the dialogue Galileo left little doubt about 
where his preferences lay, and the consequences were his trial and imprison- 
ment. During the years of his detention he nevertheless prepared The Two 
New Sciences, a dialogue concerning dynamics and the strength of materials, 
carried out by the same three characters. Although neither of the two great 
Galilean treatises was in a strict sense mathematical, there are in both of 
them many points at which appeal is made to mathematics, frequently to 
the properties of the infinitely large and the infinitely small. 

The infinitely small was of more immediate relevance to Galileo than the 
infinitely large, for he found it essential in his dynamics. Galileo gave the 
impression that dynamics was a totally new science created by him, and all 
too many writers since have agreed with this claim. It is virtually certain, 
however, that he was thoroughly familiar with the work of Oresme on the 
latitude of forms, and several times in the Two New Sciences Galileo had 
occasion to use a diagram of velocities similar to the triangle graph of 
Oresme. Nevertheless, Galileo organized the ideas of Oresme and gave them 
a mathematical precision that had been lacking. Among the new contribu- 
tions to dynamics was Galileo's analysis of projectile motion into a uniform 
horizontal component and a uniformly accelerated vertical component. As 
a result he was able to show that the path of a projectile, disregarding air 
resistance, is a parabola. It is a striking fact that the conic sections had been 
studied for almost 2000 years before two of them almost simultaneously 
found applicability in science — the ellipse in astronomy and the parabola 
in physics. Galileo mistakenly thought he had found a further application of 
the parabola in the curve of suspension of a flexible rope or wire or chain 
(catena); but mathematicians later in the century proved that this curve, 
the catenary, not only is not a parabola, it is not even algebraic. 

Galileo resembled Diirer in that they both were quick to notice new curves, 
but neither was mathematically equipped to analyze them. Galileo had 


Galileo Galilei. 

noted the curve now known as the cycloid, traced out by a point on the rim 
of a wheel as it rolls along a horizontal path, and he tried to find the area 
under one arch of it. The best he could do was to trace the curve on paper, 
cut out an areh, and weigh it, concluding that the area was a little less than 
three times the area of the generating circle. (French and Italian mathe- 
maticians later showed that the area of the arch is precisely three times the 
area of the circle.) Galileo abandoned study of the curve, suggesting only that 
the cycloid would make an attractive arch for a bridge; many years Sater 
his disciple Torrieelli took up the study of the curve with great success. 

A more important contribution to mathematics was made by Galileo in 21 
the Two Chief Systems of 1632 at a point on the "third day" when Salviati 


adumbrated the idea of an infinitesimal of higher order. Simplicio had 
argued that an object on a rotating earth should be thrown off tangentially 
by the motion; but Salviati argued that the distance QR through which an 
object has to fall to remain on the earth, while the latter rotates through a 
small angle (Fig. 16.7), is infinitely small compared with the tangential 
distance PQ through which the object travels horizontally. Hence even a 
very small downward tendency, as compared with the forward impetus, will 
be sufficient to hold the object on the earth. 15 Galileo's argument here is 
equivalent to saying that PS = vers 8 is an infinitesimal of higher order with 
respect to lines PQ or RS or arc PR. 

FIG. 16.7 

A similar bit of reasoning arises also in Galileo's Two New Sciences of 
1638, a very influential treatise on dynamics and the strength of materials. 
Here the author used the infinitely small sometimes to the point of whimsy, 
as when Salviati assures Simplicio that it is as easy to resolve a line segment 
into an infinite number of parts as it is to divide the line into finite parts. First 
he gets Simplicio to admit that one need not separate the parts, but merely 
to mark the points of division. If, for example, a line segment is bent into the 
form of a square or a regular octagon, one has resolved it into four or eight 
equal parts. Salviati then concluded that by bending the line segment into 
the shape of a circle, he has "reduced to actuality that infinite number of 
parts into which you claimed, while it was straight, were contained in it only 
potentially," for the circle is a polygon of an infinite number of sides. On 
another occasion, however, Galileo has Salviati assert that infinities and 
indivisibles "transcend our finite understanding, the former on account of 
their magnitude, the latter because of their smallness ; Imagine what they 
are when combined." 

From the infinite in geometry Salviati led Simplicio to the infinite in 
arithmetic, pointing out that a one-to-one correspondence can be set up 

15 There are two excellent English editions of The Two Chief Systems, one edited by Stillman 
Drake (1953), the other by Giorgio de Santillana (1953). 


between all the integers and the perfect squares, despite the fact that the 
further one proceeds in the sequence of integers, the scarcer the perfect 
squares become. Through the simple expedient of counting the perfect 
squares, a one-to-one correspondence is established in which each integer 
inevitably is matched against a perfect square, and vice versa. Even though 
there are many whole numbers that are not perfect squares (and the propor- 
tion of these increases as we consider larger and larger numbers), "we must 
say that there are as many squares as there are numbers." Galileo here was 
face-to-face with the fundamental property of an infinite set — that a part of 
the set can be equal to the whole set— but Galileo did not draw this con- 
clusion. While Salviati correctly concluded that the number of perfect 
squares is not less than the number of integers, he could not bring himself 
to make the statement that they are equal. Instead, he simply concluded that 
"the attributes 'equal,' 'greater,' and 'less' are not applicable to infinite, but 
only to finite quantities." He even asserted (incorrectly, we now know) that 
one cannot say that one infinite number is greater than another infinite 
number, or even that an infinite number is greater than a finite number. 
Galileo, like Moses, came within sight of the promised land, but he could not 
enter it. 16 

Galileo had intended to write a treatise on the infinite in mathematics, 22 
but it has not been found. Meanwhile his disciple Cavalieri was spurred 
by Kepler's Stereometria, as well as by ancient and medieval views and by 
Galileo's encouragement, to organize his thoughts on infinitesimals in the 
form of a book. Cavalieri was a member of a religious order (a Jesuate, not a 
Jesuit as is frequently but incorrectly stated) who lived at Milan and Rome 
before becoming professor of mathematics at Bologna in 1629. Character- 
istically for that time, he wrote on many aspects of pure and applied mathe- 
matics — geometry, trigonometry, astronomy, and optics — and he was the 
first Italian writer to appreciate the value of logarithms. In his Directorium 
universale uranometricum of 1632 he published tables of sines, tangents, 
secants, and versed sines, together with their logarithms, to eight places; 
but the world remembers him instead for one of the most influential books of 
the early modern period, the Geometria indivisibilibus continuorum, pub- 
lised in 1635. 

The argument on which the book is based is essentially that implied by 
Oresme, Kepler, and Galileo—that an area can be thought of as made up 
of lines or "indivisibles" and that a solid volume can be regarded similarly 
as composed of areas that are indivisible or quasi-atomic volumes. Although 
Cavalieri at the time could scarcely have realized it, he was following in very 

16 Galileo's Dialogue Concerning Two New Sciences (in English translation) is readily available 
in a paperback edition from Dover Publications, New York (no date). 


respectable footsteps indeed, for this is precisely the type of reasoning that 
Archimedes had used in the Method, then lost. But Cavalieri, unlike Archi- 
medes, felt no compunction about the logical deficiencies behind such 

The general principle that in an equation involving infinitesimals those of 
higher order are to be discarded because they have no effect on the final 
result is frequently erroneously attributed to Cavalieri's Geometria in- 
divisibilibus. The author undoubtedly was familiar with such an idea, for it 
is implied in some of the work of Galileo, and it appeared more specifically 
in results of contemporary French mathematicians ; but Cavalieri assumed 
almost the opposite of this principle. There was in Cavalieri's method no 
process of continued approximation, nor any omission of terms, for he used a 
strict one-to-one pairing of the elements in two configurations. No elements 
are discarded, no matter what the dimension. The general approach and the 
specious plausibility of the method of indivisibles is well illustrated by the 
proposition still known in many solid geometry books as "the theorem of 
Cavalieri" : 

If two solids have equal altitudes, and if sections made by planes parallel to the 
bases and at equal distances from them are always in a given ratio, then the 
volumes of the solids also are in this ratio. 17 

Cavalieri evidently had developed his method by 1626, for in that year 
he wrote to Galileo that he was going to publish a book on the subject. 
Galileo himself had once planned to write a book on the infinite, and perhaps 
Cavalieri delayed publishing his own work in deference to Galileo. However, 
Galileo's book undoubtedly would have been more philosophical and spec- 
ulative, with emphasis on the nature of the infinitely large and small, a 
theme that Cavalieri avoided. Instead, Cavalieri concentrated on an extremely 
useful geometrical theorem equivalent to the modern statement in the 

The statement and the proof of the theorem are very different from those 
with which a modern reader is familiar, for Cavalieri compared powers of 
the lines in a parallelogram parallel to the base with the corresponding 
powers of lines in either of the two triangles into which a diagonal divides 
the parallelogram. Let the parallelogram AF DC be divided into two triangles 
by the diagonal CF (Fig. 16.8) and let HE be an indivisible of triangle CDF 
which is parallel to the base CD. Then upon taking BC = FE and drawing 

17 D. E. Smith, Source Book in Mathematics, pp. 605-609. 



FIG. 16.8 

BM parallel to CD, it is easy to show that the indivisible EM in triangle 
ACF will be equal to HE. Hence one can pair all of the indivisibles of tri- 
angle CDF with equal indivisibles in triangle ACF, and therefore the two 
triangles are equal. Inasmuch as the parallelogram is the sum of the indivis- 
ibles in the two triangles, it is clear that the sum of the first powers of the lines 
in one of the constituent triangles is half the sum of the first powers of the 
lines in the parallelogram ; in other words, 


xdx = — 
o 2 

Through a similar but considerably more involved argument Cavalieri 
showed that the sum of the squares of the lines in the triangle is one-third the 
sum of the squares of the lines in the parallelogram. 18 For the cubes of the 
lines he found the ratio to be 1/4. Later he carried the proof to higher powers, 
finally asserting, in Exercitationes geometricae sex (that is, "Six Geometrical 
Exercises") of 1647, the important generalization that for the nth powers the 
ratio will be l/(n + 1). This was known at the same time to French mathe- 
maticians, but Cavalieri was first to publish this theorem — one that was to 
open the way to many algorithms in the calculus. Geometrica indivisibilibus, 
which so greatly facilitated the problem of quadratures, appeared again in a 
second edition in 1653, but by that time mathematicians had achieved 
remarkable results in new directions that outmoded Cavalieri's laborious 
geometric approach. 

The most significant theorem by far in Cavalieri's work was his equivalent 23 

x"dx = 

'o n + 1 

but another contribution was also to lead to important results. The spiral 

18 For further details see C. B. Boyer, "Cavalieri, Limits and Discarded Infinitesimals," 
Script a Mathematica, 8 (1941), 79-91. 


r = aQ and the parabola x 2 = ay had been known since antiquity without 
anyone's having previously noted a relationship between them, until Cavalieri 
thought of comparing straight-line indivisibles with curvilinear indivisibles. 
If, for example, one were to twist the parabola x 2 = ay (Fig. 16.9) around like 
a watch spring so that vertex O remains fixed while point P becomes point P', 

FIG. 16.9 

then the ordinates of the parabola can be thought of as transformed into 
radius vectors through the relationships x = r and y = r0 between what 
we now call rectangular and polar coordinates. The points on the Apollonian 
parabola x 2 = ay then will lie on the Archimedian spiral r = a8. Cavalieri 
noted further that if PP' is taken equal to the circumference of the circle of 
radius OP', the area within the first turn of the spiral is exactly equal to the 
area between the parabolic arc OP and the radius vector OP. Here we see 
work that amounts to analytic geometry and the calculus ; yet Cavalieri was 
writing before either of these subjects had been formally invented. As in 
other parts of the history of mathematics, we see that great milestones do 
not appear suddenly, but are merely the more clear-cut formulations along 
the thorny path of uneven development. 


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Life and Works (Baltimore : Williams and Wilkins, 1931). 
Braunmuhl, Anton von, Vorlesungen iiber Geschichte der Trigonometrie (Leipzig: 

Teubner, 1900, 2 vols.). 
Cajori, Florian, A History of the Logarithmic Slide Rule and Allied Instruments (New 

York: McGraw-Hill, 1909). 


Cajori, Florian, "History of the Exponential and Logarithmic Concepts," American 
Mathematical Monthly, 20 (1913), 5-14, 35-47, 75-84, 107-117. 

Cajori, Florian, William Oughtred, a Great Seventeenth-Century Teacher of Mathematics 
(Chicago : Open Court, 1916). 

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Caspar, Max, Kepler, trans, by C. Doris Hellman (New York : Abelard-Schuman, 1959). 

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Calif. : University of California Press, 1953). 
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(Chicago : University of Chicago Press, 1953). 
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Vol. XVI, pp. 868-877. 
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Quarterly Journal of Pure and Applied Mathematics, 48 (1920), 151-192. 
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1. Compare the contributions to mathematics of Stevin with those of Biirgi. 

2. Why is Viete sometimes called the first really modern mathematician? Explain clearly. 

3. What were the first two curves, other than straight-line and circle, or combinations of these, 

to find application in science? Explain how they came to be applied. 


4. What advantages do decimal fractions have over sexagesimal fractions? What reasons can 
you give for the late appearance of the former in Europe? 

5. What is a parameter? Can you find instances of parameters before Viete? Explain. 

6. Compare Viete's use of analytic method with that of Euclid. 

7. What are the relative advantages and disadvantages of the algebraic notations of Viete 
and Harriot? 

8. Prove Viete's observation that if x t and x 2 are positive roots of x + b = 3ax, then 
3a = x, 2 + x,x 2 + x 2 2 and b = x,x 2 2 + x^ 2 . 

9. Using Viete's method, solve x 3 = 232x 2 + 465x + 702 for the positive root (which lies 
between 200 and 300). 

10. Prove Viete's form of the law of tangents. 

11. Using Viete's method, prove that 

x + y . x — y 
sin x — sin y = 2 cos — - — sin — - — 

12. Using Viete's method, prove that 

x + y x - y 
cos x + cos y = 2 cos — - — cos — - — 

13. Multiply 8743 by 5692 prosthaphaeretically. 

14. Divide 8743 by 5692 prosthaphaeretically. 

15. Write sin lOx and cos lOx in terms of powers of sin x and cos x. 

16. Using Napier's system of logarithms, what is the relationship between Log x, Log y, and 
Log x/y? Justify your answer. 

17. Find approximately the number of which Napier's Log is 3. 

18. Find approximately the number of which Biirgi's Log is 4. 

19. What is the difference between Napier's logarithms of two numbers in the ratio of 3 to 1? 

20. Using Briggs' method, find antilog 0.2500 to four decimal places. 

*21. Using Biirgi's logarithms, what is the relationship between log x, log y, log z, and log xy/z? 
22. Use Kepler's type of reasoning to prove that the volume of a sphere is one-third the surface 

area times the radius. 
*23. Verify Galileo's observation that 

vers 6 
lim —— = 

*24. Verify Cavalieri's comparison of the areas of the spiral and the parabola. 

*25. Prove Viete's infinite product for n by starting with an inscribed polygon of four sides and 

doubling the number of sides successively. 
*26. Use the trigonometric method of Viete and Girard to solve the equation x 

7 = for one root correct to the nearest thousandth. 


The Time 

of Fermat and Descartes 

Fermat, the true inventor of the differential calculus. 


The year 1647 in which Cavalieri died marked the death also of another 
disciple of Galileo, the young Evangelista Torricelli (1608-1647); but in 
many ways Torricelli represented the new generation of mathematicians who 
were building rapidly on the infinitesimal foundation that Cavalieri had 
sketched all too vaguely. Had Torricelli not died so prematurely, Italy might 
have continued to share the lead in new developments; as it turned out, 
France was the undisputed mathematical center during the second third of 
the seventeenth century. The leading figures were Rene Descartes (1596- 
1650) and Pierre de Fermat (1601-1665), but three other contemporary 
Frenchmen also made important contributions, in addition to Torricelli — 
Gilles Persone de Roberval (1602-1675), Girard Desargues (1591-1661), and 
Blaise Pascal (1623-1662). This chapter, covering one of the most critical 
periods in the history of mathematics, focuses attention on these six men, 
not only as individuals, but also collectively, for not since the days of Plato 
had there been such mathematical intercommunication as during the 
seventeenth century. 

No professional mathematical organizations yet existed, but in Italy, 
France, and England there were loosely organized scientific groups: the 
Accademia dei Lincei (to which Galileo belonged) and the Accademia del 
Cimento in Italy, the Cabinet DuPuy in France, and the Invisible College in 
England. There was in addition an individual who, during the period we are 
now considering, served through correspondence as a clearing house for 
mathematical information. This was the Minimite friar, Marin Mersenne 
(1588-1648), a close friend of Descartes and Fermat, as of many another 
mathematician of the time. Had Mersenne lived a century earlier the delay 
in information concerning the solution of the cubic might not have occurred, 
for when Mersenne knew something, the whole of the "Republic of Letters" 
was shortly informed about it. From the seventeenth century on, therefore, 




Rene Descartes. 

mathematics developed more in terms of inner logic than through economic, 
social, or technological forces, as is apparent particularly in the work of 
Descartes, the best-known mathematician of the period. 

Descartes was born of a good family and received a thorough education 
at the Jesuit college at La Fleche, where the textbooks of Clavius were 
fundamental. Later he took a degree at Poitier, where he had studied law, 


without much enthusiasm. For a number of years he traveled about in 
conjunction with varied military campaigns, first in Holland with Maurice, 
Prince of Nassau, then with Duke Maximillian I of Bavaria, and later still 
with the French army at the siege of LaRochelle. Descartes was not really a 
professional soldier, and his brief periods of service in connection with 
campaigns were separated by intervals of independent travel and study 
during which he met some of the leading scholars in various parts of Europe — 
Faulhaber in Germany and Desargues in France, for example. At Paris he 
met Mersenne and a circle of scientists who freely discussed criticisms of 
Peripatetic thought ; from such stimulation Descartes went on to become the 
"father of modern philosophy", to present a changed scientific world view, 
and to establish a new branch of mathematics. In his most celebrated treatise, 
the Discours de la methode pour bien conduire sa raison et chercher la verite 
dans les sciences ("Discourse on the Method of Reasoning Well and Seeking 
Truth in the Sciences") of 1637, he announced his program for philosophical 
research. In this he hoped, through systematic doubt, to reach clear and 
distinct ideas from which it would then be possible to deduce innumerably 
many valid conclusions. This approach to science led him to assume that 
everything was explainable in terms of matter (or extension) and motion. 
The entire universe, he postulated, was made up of matter in ceaseless motion 
in vortices, and all phenomena were to be explained mechanically in terms 
of forces exerted by contiguous matter. Cartesian science enjoyed great 
popularity for almost a century, but it then necessarily gave way to the 
mathematical reasoning of Newton. Ironically, it was in large part the 
mathematics of Descartes that later made possible the defeat of Cartesian 

The philosophy and science of Descartes were almost revolutionary in 
their break with the past; his mathematics, by contrast, was linked with 
earlier traditions. To some extent this may have resulted from the commonly 
accepted humanistic heritage — a belief that there had been a Golden Age 
in the past, a "reign of Saturn," the great ideas of which remained to be 
rediscovered. Probably in larger measure it was the natural result of the fact 
that the growth of mathematics is more cumulatively progressive than is the 
development of other branches of learning. Mathematics grows by accretions, 
with very little need to slough off irrelevancies, whereas science grows 
largely through substitutions when better replacements are found. It should 
come as no surprise, therefore, to see that Descartes' chief contribution to 
mathematics, the foundation of analytic geometry, was motivated by an 
attempt to return to the past. 

Descartes had become seriously interested in mathematics by the time he 
spent the cold winter of 1619 with the Bavarian army, where he lay abed 


until ten in the morning, thinking out problems. It was during this early 
period in his life that he discovered the polyhedral formula usually named 
for Euler — v + f = e + 2, where v, f, and e are the number of vertices, faces, 
and edges, respectively, of a simple polyhedron. Nine years later Descartes 
wrote to a friend in Holland that he had made such strides in arithmetic and 
geometry that he had no more to wish for. Just what the strides were is not 
known, for Descartes had published nothing ; but the direction of his thoughts 
is indicated in a letter of 1628 to his Dutch friend where he gave a rule for the 
construction of the roots of any cubic or quartic equation by means of a 
parabola. This is, of course, essentially the type of thing that Menaechmus 
had done for the duplication of the cube some 2000 years earlier and that 
Omar Khayyam had carried out for cubics in general around the year 1100. 
Whether or not Descartes by 1628 was in full possession of his analytic 
geometry is not clear, but the effective date for the invention of Cartesian 
geometry cannot be much later than that. At this time Descartes left France 
for Holland, where he spent the next twenty years. Three or four years after 
settling down there, his attention was called by another Dutch friend, a 
classicist, to the three-and-four-line problem of Pappus. Under the mistaken 
impression that the ancients had been unable to solve this problem, Descartes 
applied his new methods to it and succeeded without difficulty. This made 
him aware of the power and generality of his point of view, and he conse- 
quently wrote the well-known work, La geometrie, which made analytic 
geometry known to his contemporaries. 

La geometrie was not presented to the world as a separate treatise, but 
as one of three appendices to the Discours de la methode in which he thought 
to give illustrations of his general philosophical method. The other two 
appendices were La dioptrique, containing the first publication of the law of 
refraction (discovered earlier by Snell), and Les meteores, including, among 
other things, the first generally satisfactory quantitative explanation of the 
rainbow. Descartes' successors had difficulty seeing just how the three 
appendices were related to his general method, and in subsequent editions of 
the Discours they frequently were omitted. The original edition of the 
Discours was published without the name of the author, but the authorship 
of the work was generally known. 

Cartesian geometry now is synonymous with analytic geometry, but the 
fundamental purpose of Descartes was far removed from that of modern 
textbooks. The theme is set by the opening sentence : 

Any problem in geometry can easily be reduced to such terms that a knowledge of 
the lengths of certain lines is sufficient for its construction. 

As this statement indicates, the goal is generally a geometric construction, 


and not necessarily the reduction of geometry to algebra. The work of 
Descartes far too often is described simply as the application of algebra to 
geometry, whereas actually it could be characterized equally well as the 
translation of the algebraic operations into the language of geometry. The 
very first section of La geometrie is entitled "How the calculations of arith- 
metic are related to the operations of geometry" ; the second section describes 
"How multiplication, division, and the extraction of square roots are 
performed geometrically." Here Descartes was doing what had to some 
extent been done from al-Khowarizmi to Oughtred — furnishing a geometric 
background for the algebraic operations. The five arithmetic operations are 
shown to correspond to simple constructions with straightedge and com- 
passes, thus justifying the introduction of arithmetical terms in geometry. 

Descartes was more thorough in his symbolic algebra, and in the geometric 
interpretation of algebra, than any of his predecessors. Formal algebra had 
been advancing steadily since the Renaissance, and it found its culmination 
in Descartes' La geometrie, the earliest mathematical text that a present-day 
student of algebra can follow without encountering difficulties in notation. 
About the only archaic symbol in the book is the use of x> instead of = for 
equality. The Cartesian use of letters near the beginning of the alphabet for 
parameters and those near the end as unknown quantities, the adaptation of 
exponential notation to these, and the use of the Germanic symbols + and — , 
all combined to make Descartes' algebraic notation look like ours, for of 
course we took ours from him. There was, nevertheless, an important differ- 
ence in view, for where we think of the parameters and unknowns as numbers, 
Descartes thought of them as line segments. In one essential respect he 
broke from Greek tradition, for instead of considering x 2 and x 3 , for example, 
as an area and a volume, he interpreted them also as lines. This permitted 
him to abandon the principle of homogeneity, at least explicitly, and yet 
retain geometrical meaning. Descartes could write an expression such as 
a 2 b 2 — b, for, as he expressed it, one "must consider the quantity a 2 b 2 
divided once by unity (that is, the unit line segment), and the quantity b 
multiplied twice by unity." It is clear that Descartes substituted homo- 
geneity in thought for homogeneity in form, a step that made his geometric 
algebra more flexible — so flexible indeed that today we read xx as "x- 
squared" without ever seeing a square in our mind's eye. 

Book I includes detailed instructions on the solution of quadratic equa- 
tions, not in the algebraic sense of the ancient Babylonians, but geometrically, 
somewhat in the manner of the ancient Greeks. To solve the equation 
z 2 = az + b 2 , for example, Descartes proceeded as follows. Draw a line 
segment LM of length b (Fig. 17.1) and at L erect a segment NL equal to 
a/2 and perpendicular to LM. With center N construct a circle of radius a/2 



FIG. 17.1 

and draw the line through M and N intersecting the circle at O and P. Then 
z = OM is the line desired. (Descartes ignored the root PM of the equation 
because it is "false," that is negative.) Similar constructions are given for 
z 2 = az — b 2 and for z 2 + az = b 2 , the only other quadratic equations with 
positive roots. 

Having shown how algebraic operations, including the solution of 
quadratics, are interpreted geometrically, Descartes turned to the applica- 
tion of algebra to determinate geometrical problems, formulating far more 
clearly than the Renaissance cossists the general approach : 

If, then, we wish to solve any problem, we first suppose the solution already effected, 
and give names to all the lines that seem needful for its construction — to those that are 
unknown as well as to those that are known. Then, making no distinction between 
known and unknown lines, we must unravel the difficulty in any way that shows most 
naturally the relations between these lines, until we find it possible to express a single 
quantity in two ways. This will constitute an equation (in a single unknown), since the 
terms of the one of these two expressions are together equal to the terms of the other. ' 

Throughout Books I and III of La geometrie Descartes is concerned primarily 
with this type of geometrical problem, in which the final algebraic equation 
can contain only one unknown quantity. Descartes was well aware that it 
was the degree of this resulting algebraic equation that determined the 
geometrical means by which the required geometric construction can be 
carried out. 

If it can be solved by ordinary geometry, that is, by the use of straight lines and circles 
traced on a plane surface, when the last equation shall have been entirely solved there 
will remain at most only the square of an unknown quantity, equal to the product of its 
root by some known quantity, increased or diminished by some other quantity also 

1 Translations of passages from La geometrie here and elsewhere are from The Geometry of 
Rene Descartes, trans, by D. E. Smith and Marcia L. Latham (New York : Dover reprint, 1954). 
See pp. 7-9 for the above passage. 


Here we see a clear-cut statement that what the Greeks had called "plane 
problems" lead to nothing worse than a quadratic equation. Since Viete 
already had shown that the duplication of the cube and the trisection of the 
angle lead to cubic equations, Descartes stated, with inadequate proof, that 
these cannot be solved with straightedge and compasses. Of the three ancient 
problems, therefore, only the squaring of the circle remained open to question. 
The title La geometrie should not mislead one into thinking that the 
treatise is primarily geometrical. Already in the Discourse, to which the 
Geometry had been appended, Descartes had discussed the relative merits 
of algebra and geometry, without being partial to either. He charged the 
latter with relying too heavily on diagrams that unnecessarily fatigue the 
imagination, and he stigmatized the former as a confused and obscure art 
that embarrasses the mind. The aim of his method, therefore, was twofold : 
(1) through algebraic procedure to free geometry from the use of diagrams 
and (2) to give meaning to the operations of algebra through geometric 
interpretation. Descartes was convinced that all mathematical sciences 
proceed from the same basic principles, and he decided to use the best of 
each branch. His procedure in La geometrie, then, was to begin with a 
geometrical problem, to convert it to the language of an algebraic equation, 
and then, having simplified the equation as far as possible, to solve this 
equation geometrically, in a manner similar to that which he had used for 
the quadratics. Following Pappus, Descartes insisted that one should use in 
the geometric solution of an equation only the simplest means appropriate 
to the degree of the equation. For quadratic equations, lines and circles 
suffice; for cubics and quartics, conic sections are adequate. Now Descartes 
was ready to move beyond the point at which the Greeks had stopped. 

Descartes was much impressed by the power of his method in handling 
the three-and-four-line locus, and so he moved on to generalizations of this 
problem — a problem that runs like a thread of Ariadne through the three 
books of La geometrie. He knew that Pappus had been unable to tell anything 
about the loci when the number of lines was increased to six or eight or more ; 
so Descartes proceeded to study such cases. He was aware that for five 
or six lines, the locus is a cubic, for seven or eight, it is a quartic, and so on. 
But Descartes showed no real interest in the shape of these loci, for he was 
obsessed with the question of the means needed to construct geometrically 
the ordinates corresponding to given abscissas. For five lines, for example, 
he remarked triumphantly that if they are not all parallel, then the locus is 
elementary in the sense that, given a value for x, the line representing y is 
constructible by ruler and compasses alone. If four of the lines are parallel 
and equal distances a apart and the fifth is perpendicular to the others (Fig. 
17.2), and if the constant of proportionality in the Pappus problem is taken 


FIG. 17.2 

as this same constant a, then the locus is given by (a + x)(a - x)(2a — x) = 
axy, a cubic that Newton later called the Cartesian parabola or trident — 
x 3 - lax 2 — a 2 x + 2a 3 = axy. This curve comes up repeatedly in La 
geometrie, yet Descartes at no point gave a complete sketch of it. His interest 
in the curve was threefold : (1) deriving its equation as a Pappus locus, (2) 
showing its generation through the motion of curves of lower degree, and 
(3) using it in turn to construct the roots of equations of higher degree. 

Descartes considered the trident constructible by plane means alone 
inasmuch as, for each point x on the axis of abscissas, the ordinate y can 
be drawn with ruler and compasses alone. This is not in general possible for 
five or more lines taken at random in the Pappus problem. In the case of 
not more than eight lines, the locus is a polynomial in x and y such that, for a 
given point on the x-axis, the construction of the corresponding ordinate y 
requires the geometric solution of a cubic or quartic equation which, as 
we have seen, usually calls for the use of conic sections. For not more than 
twelve lines in the Pappus problem, the locus is a polynomial in x and y of 
not more than sixth degree, and the construction in general requires curves 
beyond the conic sections. Here Descartes made an important advance 
beyond the Greeks in problems of geometric constructibility. The ancients 
had never really legitimized constructions that made use of curves other than 
straight lines or circles, although they sometimes reluctantly recognized, as 
Pappus did, the classes that they called solid problems and linear problems. 
The second category in particular was a catchall class of problems with no 
real standing. 

Descartes now took the step of specifying an orthodox classification of 
determinate geometrical problems. Those that lead to quadratic equations, 
and can therefore be constructed by lines and circles, he placed in class one ; 
those leading to cubic and quartic equations, the roots of which can be 
constructed by means of conic sections, he placed in class two ; those leading 
to equations of degree five or six can be constructed by introducing a cubic 
curve, such as the trident or the higher parabola y = x 3 , and these he placed 


in class three. Descartes continued in this manner, grouping geometric 
problems and algebraic equations into classes, assuming that the construction 
of the roots of an equation of degree In or In — 1 was a problem of class n. 
The Cartesian classification by pairs of degrees seemed to be confirmed by 
algebraic considerations. It was known that the solution of the quartic was 
reducible to that of the resolvent cubic, and Descartes extrapolated prema- 
turely to assume that the solution of an equation of degree In can be reduced 
to that of a resolvent equation of degree 2n — 1. Many years later it was 
shown that Descartes' tempting generalization does not hold. A number of 
his contemporaries were only too eager to point out a more serious error 
made by Descartes, for it is clear from the theory of algebraic elimination 
that curves of degree n suffice to solve equations not up to degree In only, 
but up to n 2 . His classification, therefore, lost validity, but his work did have 
the salutary effect of encouraging the relaxation of the rules on constructi- 
bility so that higher plane curves might be used. 

It will be noted that the Cartesian classification of geometric problems 
included some, but not all, of those that Pappus had lumped together as 
"linear." In introducing the new curves that he needed for geometric con- 
structions beyond the fourth degree, Descartes added to the usual axioms 
of geometry one more axiom : 

Two or more lines (or curves) can be moved, one upon the other, determining by 
their intersection other curves. 

This in itself is not unlike what the Greeks had actually done in their kine- 
matic generation of curves such as the quadratrix, the cissoid, the conchoid, 
and the spiral ; but whereas the ancients had lumped these together, Descartes 
now carefully distinguished between those, such as the cissoid and the con- 
choid, that we should call algebraic, and others, such as the quadratrix and 
the spiral, that are now known as transcendental. To the first type Descartes 
gave full-fledged geometrical status, along with the line, the circle, and the 
conies, calling all of these the "geometrical curves" ; the second type he ruled 
out of geometry entirely, stigmatizing them as "mechanical curves." The 
basis upon which Descartes made this decision was "exactness of reasoning." 
Mechanical curves, he said, "must be conceived of as described by two 
separate movements whose relation does not admit of exact determination" 
— such as the ratio of circumference to diameter of a circle in the case of the 
motions describing the quadratrix and the spiral. In other words, Descartes 
thought of algebraic curves as exactly described and of transcendental curves 
as inexactly described, for the latter generally are defined in terms of arc 
lengths. On this matter he wrote, in La geometrie : 



Geometry should not include lines (or curves) that are like strings, in that they are 
sometimes straight and sometimes curved, since the ratios between straight and curved 
lines are not known, and I believe cannot be discovered by human minds, and therefore 
no conclusion based upon such ratios can be accepted as rigorous and exact. 

Descartes here is simply reiterating the dogma, suggested by Aristotle and 
affirmed by Averroes, that no algebraic curve can be exactly rectified. 
Interestingly enough, in 1638, the year after the publication of La geometrie, 
Descartes ran across a "mechanical" curve that turned out to be rectifiable. 
Through Mersenne, Galileo's representative in France, the question, raised 
in the Two New Sciences, of the path of fall of an object on a rotating earth 
(assuming the earth permeable) was widely discussed, and this led Descartes 
to the equiangular or logarithmic spiral r = ae M as the possible path. 2 Had 
Descartes not been so firm in his rejection of such nongeometrical curves, he 
might have anticipated Torricelli in discovering, in 1645, the first modern 
rectification of a curve. Torricelli showed, by infinitesimal methods that he 
had learned from Archimedes, Galileo, and Cavalieri, that the total length 
of the logarithmic spiral from 9 = as it winds backward about the pole O 
is exactly equal to the length of the polar tangent PT (Fig. 17.3) at the point 
for which 9 = 0. This striking result did not, of course, disprove the Cartesian 
doctrine of the nonrectifiability of algebraic curves. In fact, Descartes could 
have asserted not only that the curve was not exactly determined, being 
mechanical, but also that the arc of the curve has an asymptotic point at the 
pole, which it never reaches. 

Virtually the whole of La geometrie is devoted to a thoroughgoing applica- 
tion of algebra to geometry and of geometry to algebra ; but there is little in 
the treatise that resembles what usually is thought of today as analytic 
geometry. There is nothing systematic about rectangular coordinates, for 
oblique ordinates usually are taken for granted ; hence there are no formulas 

2 See Oeuvres de Descartes, ed. by Charles Adam and Paul Tannery (1897-1913), II, 222-245. 


for distance, slope, point of division, angle between two lines, or other similar 
introductory material. Moreover, in the whole of the work there is not a single 
new curve plotted directly from its equation, and the author took so little 
interest in curve sketching that he never fully understood the meaning of 
negative coordinates. He knew in a general sort of way that negative ordinates 
are directed in a sense opposite to that taken as positive, but he never made 
use of negative abscissas. Moreover, the fundamental principle of analytic 
geometry — the discovery that indeterminate equations in two unknowns 
correspond to loci — does not appear until the second book, and then only 
somewhat incidentally. 

The solution of any one of these problems of loci is nothing more than the finding of a 

point for whose complete determination one condition is wanting In every such 

case an equation can be obtained containing two unknown quantities. 

In one case only did Descartes examine a locus in detail, and this was in 
connection with the three-and-four-line locus problem of Pappus for which 
Descartes derived the equation y 2 = ay — bxy + ex — dx 2 . This is the 
general equation of a conic passing through the origin ; even though the 
literal coefficients are understood to be positive, this is by far the most com- 
prehensive approach ever made to the analysis of the family of conic sections. 
Descartes indicated conditions on the coefficients for which the conic is a 
straight line, a parabola, an ellipse, or a hyperbola, the analysis being in a 
sense equivalent to a recognition of the characteristic of the equation of the 
conic. The author knew that by a proper choice of the origin and axes 
the simplest form of the equation is obtained, but he did not give any of the 
canonical forms. The omission of much of the elementary detail made the 
work exceedingly difficult for his contemporaries to follow. In concluding 
remarks Descartes sought to justify inadequacy of exposition by the in- 
congruous assertion that he had left much unsaid in order not to rob the 
reader of the joy of discovery. A genius himself, he could not appreciate the 
difficulty that others were to have in understanding his new and profound 
thoughts. It is small wonder that the number of editions of La geometrie, 
apart from those with considerable amplification, was small in the seventeenth 
century and has been still smaller since then. 

Inadequate though the exposition is, it is Book II of La geometrie that 
comes closest to modern views of analytic geometry. There is even a statement 
of a fundamental principle of solid analytic geometry : 

If two conditions for the determination of a point are lacking, the locus of 
the point is a surface. 

However, Descartes did not give any illustrations of such equations or 
expand the brief hint of analytic geometry of three dimensions. 


Descartes was so fully aware of the significance of his work that he regarded 
it as bearing to ancient geometry somewhat the same relationship as the 
rhetoric of Cicero bears to the a, b, c's of children. His mistake, from our point 
of view, was in emphasizing determinate equations rather than indeterminate. 
He realized that all the properties of a curve, such as the magnitude of its 
area, or the direction of its tangent, are fully determined when an equation 
in two unknowns is given, but he did not take full advantage of this recogni- 
tion. He wrote: 

I shall have given here a sufficient introduction to the study of curves when I shall 
have given a general method of drawing a straight line making right angles with a curve 
at an arbitrarily chosen point upon it. And I dare say that this is not only the most 
useful and most general problem in geometry that I know, but even that I have ever 
desired to know. 

Descartes was quite right that the problem of finding the normal (or the 
tangent) to a curve was of great importance, but the method that he published 
in La geometrie was less expeditious than that which Fermat had developed 
at about the same time. Descartes suggested that to find the normal to an 
algebraic curve at a fixed point P on the curve, one should take a second 
variable point Q on the curve, then find the equation of the circle with center 
on the coordinate axis (for he used only an axis of abscissas) and passing 
through P and Q. Now, by setting equal to zero the discriminant of the 
equation that determines the intersections of the circle with the curve, one 
finds the center of the circle where Q coincides with P. The center being known, 
the tangent and normal to the curve at P are then easily found. 

Book II of La geometrie contains also much material on the "ovals of 
Descartes," which are very useful in optics and are obtained by generalizing 
the "gardner's method" for constructing an ellipse by means of strings. If 
D, and D 2 are the distances of a variable point P from two fixed points F l and 
F 2 respectively, and if m and n are positive integers and K is any positive 
constant, then the locus of P such that mD t + nD 2 = K is now known as an 
oval of Descartes ; but the author did not use the equations of the curves. 
Descartes realized that his methods can be extended to "all those curves 
which can be conceived of as generated by the regular movement of the 
points of a body in three-dimensional space," but he did not carry out any 
details. The sentence with which Book II concludes, "And so I think I have 
omitted nothing essential to an understanding of curved lines," is presump- 
tuous indeed. 

The third and last book of La geometrie resumes the topic of Book I — 
the construction of the roots of determinate equations. Here the author 
warned that in such constructions "We should always choose with care the 
simplest curve that can be used in the solution of a problem." This means, of 


course, that one must be fully aware of the nature of the roots of the equation 
under consideration, and in particular one must know whether or not the 
equation is reducible. For this reason, Book III is virtually a course in the 
elementary theory of equations. It tells how to discover rational roots, if 
any, how to depress the degree of the equation when a root is known, how 
to increase and decrease the roots of an equation by any amount, or to 
multiply or divide them by a number, how to eliminate the second term, how 
to determine the number of possible "true" and "false" roots (that is, 
positive and negative roots) through the well-known "Descartes' rule of 
signs," and how to find the algebraic solution of cubic and quartic equations. 
In closing, the author reminds the reader that he has given the simplest 
constructions possible for problems in the various classes mentioned earlier. 
In particular, the trisection of the angle and the duplication of the cube are in 
class two, requiring more than circles and lines for their construction. 

Our account of Descartes' analytic geometry should make clear how far 1 
removed the author's thought was from the practical considerations that 
are now so often associated with the use of coordinates. He did not lay down 
a coordinate frame in order to locate points as a surveyor or geographer 
might do, nor were his coordinates thought of as number pairs. In this 
respect the phrase "Cartesian product," so often used today, is an anachron- 
ism. La geometrie was in its day just as much a triumph of impractical theory 
as was the Conies of Apollonius in antiquity, despite the inordinately useful 
role that both were ultimately destined to play. Moreover, the use of oblique 
coordinates was much the same in both cases, thus confirming that the origin 
of modern analytic geometry lies in antiquity rather than in the medieval 
latitude of forms. The coordinates of Oresme, which influenced Galileo, are 
closer, both in motive and in appearance, to the modern point of view than 
are those of Apollonius and Descartes. Even if Descartes was familiar with 
Oresme 's graphical representation of functions, and this is not evident, there 
is nothing in Cartesian thought to indicate that he would have seen any 
similarity between the purpose of the latitude of forms and his own classifica- 
tion of geometric constructions. The theory of functions ultimately profited 
greatly from the work of Descartes, but the notion of a form or function 
played no apparent role in leading to Cartesian geometry. 

In terms of mathematical ability Descartes probably was the most able 
thinker of his day, but he was at heart not really a mathematician. His 
geometry was only an episode in a life devoted to science and philosophy, 
and although occasionally in later years he contributed to mathematics 
through correspondence, he left no other great work in this field. In 1649 he 
accepted an invitation from Queen Christina of Sweden to instruct her in 
philosophy and to establish an academy of sciences at Stockholm. Descartes 


had never enjoyed robust health, and the rigors of a Scandinavian winter 
were too much for him; he died early in 1650. 

11 If Descartes had a rival in mathematical ability, it was Fermat, but the 
latter was in no sense a professional mathematician. Fermat studied law at 
Toulouse, where he then served in the local parlement, first as a lawyer and 
later as councillor. This meant that he was a busy man ; yet he seems to have 
had time to enjoy as an avocation a taste for classical literature, including 
science and mathematics. The result was that by 1629 he began to make 
discoveries of capital importance in mathematics. In this year he joined in 
one of the favorite sports of the time — the "restoration" of lost works of 
antiquity on the basis of information found in extant classical treatises. 
Fermat undertook to reconstruct the Plane Loci of Apollonius, depending on 
allusions contained in the Mathematical Collection of Pappus. A by-product 
of this effort was the discovery, at least by 1636, of the fundamental principle 
of analytic geometry : 

Whenever in a final equation two unknown quantities are found, we have a 
locus, the extremity of one of these describing a line, straight or curved. 

This profound statement, written a year before the appearance of Descartes' 
Geometry, seems to have grown out of Fermat's application of the analysis 
of Viete to the study of loci in Apollonius. In this case, as also in that of 
Descartes, the use of coordinates did not arise from practical considerations, 
nor from the medieval graphical representation of functions. It came about 
through the application of Renaissance algebra to problems from ancient 
geometry. However, Fermat's point of view was not entirely in conformity 
with that of Descartes, for Fermat emphasized the sketching of solutions of 
indeterminate equations, instead of the geometrical construction of the roots 
of determinate algebraic equations. Moreover, where Descartes had built his 
Geometry around the difficult Pappus problem, Fermat limited his exposition, 
in the short treatise entitled Ad locus pianos et solidos isagoge ("Introduction 
to Plane and Solid Loci"), to the simplest loci only. Where Descartes had 
begun with the three-and-four-line locus, using one of the lines as an axis of 
abscissas, Fermat began with the linear equation and chose an arbitrary 
coordinate system upon which to sketch it. 

Using the notation of Viete, Fermat sketched first the simplest case of a 
linear equation — given in Latin as "D in A aequetur B in E" (that is, Dx = By 
in modern symbolism). The graph, is of course, a straight line through the 
origin of coordinates — or rather a half line with the origin as end point, for 
Fermat, like Descartes, did not use negative abscissas. The more general 
linear equation ax + by = c 2 (for Fermat retained Viete's homogeneity) he 
sketched as a line segment in the first quadrant terminated by the coordinate 


axes. Next, to show the power of his method for handling loci, Fermat 
announced the following problem that he had discovered by the new 
approach : 

Given any number of fixed lines, in a plane, the locus of a point such that the 
sum of any multiples of the segments drawn at given angles from the point 
to the given lines is constant, is a straight line. 

That is, of course, a simple corollary of the fact that the segments are linear 
functions of the coordinates, and of Fermat 's proposition that every equation 
of first degree represents a straight line. 3 

Fermat next showed that xy = k 2 is a hyperbola and that an equation of 
the form xy + a 2 = bx + cy can be reduced to one of the form xy = k 2 (by 
a translation of axes). The equation x 2 = y 2 he considered as a single straight 
line (or ray), for he operated only in the first quadrant, and he reduced other 
homogeneous equations of second degree to this form. Then he showed that 
a 2 + x 2 = by is a parabola, that x 2 + y 2 + lax + 2by = c 2 is a circle, 
that a 2 — x 2 = ky 2 is an ellipse, and that a 2 + x 2 = ky 2 is a hyperbola (for 
which he gave both branches). To more general quadratic equations, in 
which the several second-degree terms appear, Fermat applied a rotation of 
axes to reduce them to the earlier forms. As the "crowning point" of his 
treatise, Fermat considered the following proposition : 

Given any number of fixed lines, the locus of a point such that the sum of the 
squares of the segments drawn at given angles from the point to the lines is 
constant, is a solid locus. 

This proposition is obvious in terms of Fermat's exhaustive analysis of the 
various cases of quadratic equations in two unknowns. As an appendix to 
the Introduction to Loci, Fermat added "The Solution of Solid Problems by 
Means of Loci," pointing out that determinate cubic and quartic equations 
can be solved by conies, the theme that had loomed so large in the geometry 
of Descartes. 

Fermat's Introduction to Loci was not published during the author's 12 
lifetime ; hence analytic geometry in the minds of many was regarded as the 
unique invention of Descartes. It is now clear that Fermat had discovered 
essentially the same method well before the appearance of La geometrie and 
that his work circulated in manuscript form until its publication in 1679 in 
Varia opera mathematica. It is a pity that Fermat published almost nothing 
during his lifetime, for his exposition was much more systematic and didactic 
than that of Descartes. Moreover, his analytic geometry was somewhat 

3 For this and other aspects of Fermat's work see his Oeuvres, ed. by Paul Tannery and 
Charles Henry (1891-1922). 


closer to ours in that ordinates usually are taken at right angles to the line of 
abscissas. Like Descartes, Fermat was aware of an analytic geometry of 
more than two dimensions, for in another connection he wrote : 

There are certain problems which involve only one unknown, and which can be 
called determinate, to distinguish them from the problems of loci. There are certain 
others which involve two unknowns and which can never be reduced to a single one ; 
these are the problems of loci. In the first problems we seek a unique point, in the latter 
a curve. But if the proposed problem involves three unknowns, one has to find, to 
satisfy the equation, not only a point or a curve, but an entire surface. In this way surface 
loci arise, etc. 4 

Here in the final "etc." there is a hint of geometry of more than three dimen- 
sions, but if Fermat really had this in mind, he did not carry it further. Even 
the geometry of three dimensions had to wait until the eighteenth century 
for its effective development. 

13 It is possible that Fermat was in possession of his analytic geometry as 
early as 1629, for about this time he made two significant discoveries that 
are closely related to his work on loci. The more important of these was 
described a few years later in a treatise, again unpublished in his lifetime, 
entitled Method of Finding Maxima and Minima. Fermat had been consider- 
ing loci given (in modern notation) by equations of the form y = x" ; hence 
today they are often known as "parabolas of Fermat" if n is positive or 
"hyperbolas of Fermat" if n is negative. Here we have an analytic geometry 
of higher plane curves ; but Fermat went further. For polynomial curves of 
the form y = f(x) he noted a very ingenious method of finding points at 
which the function takes on a maximum or a minimum value. He compared 
the value of f(x) at a point with the value f(x + £) at a neighboring point. 
Ordinarily these values will be distinctly different, but at the top or bottom 
of a smooth curve the change will be almost imperceptible. Hence to find 
maximum and minimum points Fermat equated fix) and f(x + E), realizing 
that the values, although not exactly the same, are almost equal. The smaller 
the interval E between the two points, the nearer the pseudoequality comes 
to being a true equation ; so Fermat, after dividing through by E, set E = 0. 
The results gave him the abscissas of the maximum and minimum point of the 
polynomial. Here in essence is the process now called differentiation, for 
the method of Fermat is equivalent to finding 

.. f(x + E) - f{x) 

E^O E 

and setting this equal to zero. Hence it is appropriate to follow Laplace in 
4 See Fermat, Oeuvres, I, 186-187. 


acclaiming Fermat as the discoverer of the differential calculus, as well as a 
codiscoverer of analytic geometry. Obviously Fermat was not in possession 
of the limit concept, but otherwise his method of maxima and minima parallels 
that used in the calculus today, except that now the symbol h or Ax is custom- 
arily used in place of Fermat's E. Fermat's process of changing the variable 
slightly and considering neighboring values has ever since been the essence 
of infinitesimal analysis. 

During the very years in which Fermat was developing his analytic 
georoetry, he discovered also how to apply his neighborhood process to find 
the tangent to an algebraic curve of the form y = f(x). If P is a point on the 
curve y = f(x) at which the tangent is desired, and if the coordinates of P are 
(a, b), then a neighboring point on the curve with coordinates x = a + E, 
y = f(a + E) will lie so close to the tangent that one can think of it as 
approximately on the tangent as well as on the curve. If, therefore, the sub- 
tangent at the point P is TQ = c (Fig. 17.4), the triangles TPQ and TP'Q' 
can be taken as being virtually similar. Hence one has the proportion 

b _ f(a + £) 

c c + E 

Upon cross-multiplying, canceling like terms, recalling that b = f(a), then 
dividing through by E, and finally setting E = 0, the subtangent c is readily 

Fermat's procedure amounts to saying that 



f(a + E) - f(a) 

is the slope of the curve at x = a ; but Fermat did not explain his procedure 
satisfactorily, saying simply that it was similar to his method of maxima and 
minima. Descartes in particular, when the method was reported to him in 
1638 by Mersenne, attacked it as not generally valid. He proposed as a 
challenge the curve ever since known as the "folium of Descartes"— 
x 3 + y 3 = 3axy. That mathematicians of the time were quite unfamiliar 

FIG. 17.4 



with negative coordinates is apparent in that the curve was drawn as but a 
single folium or "leaf" in the first quadrant — or sometimes as a four-leaf 
clover, with one leaf in each quadrant ! Ultimately Descartes grudgingly 
conceded the validity of Fermat's tangent method, but Fermat never was 
accorded the esteem to which he was entitled. 

1 4 Mersenne, through correspondence and in his own printed works, made 
some of Fermat's results known in France and Italy, but it would have been 
ever so much better if Fermat had published his marvelous discoveries. 
Fermat not only had a method for finding the tangent to curves of the form 
y = x m , he also, some time after 1629, hit upon a theorem on the area under 
these curves — the theorem that Cavalieri published in 1635 and 1647. In 
finding the area Fermat at first seems to have used formulas for the sums of 
powers of the integers, or inequalities of the form 

r + 2 m + 3 m + 

+ n m > 

m + 1 

> l m + 2 m + 3 m +■■■ +(n - 1)" 

to establish the result for all positive integral values of m. This in itself was an 
advance over the work of Cavalieri, who limited himself to the cases from 
m = 1 to m = 9 ; but later Fermat developed a better method for handling the 
problem, 5 which was applicable to fractional as well as integral values of m. 
Let the curve be y = x", and let the area under the curve from x = to 
'x = a be desired. Then Fermat subdivided the interval from x = to x = a 
into infinitely many subintervals by taking the points with abscissas a, aE, 
aE 2 , aE 3 , ..., where £ is a quantity less than one. At these points he erected 
ordinates to the curve and then approximated to the area under the curve 
by means of rectangles (as indicated in Fig. 17.5). The areas of the successive 

FIG. 17.5 

approximating circumscribed rectangles, beginning with the largest, are 
given by the terms in geometric progression a"(a - aE), a"E"(aE — aE 2 ), 

5 See Fermat, Oeuvres, I, 255-288; III, 216-240. 


a"E 2n (aE 2 - aE 3 ), .... The sum to infinity of these terms is 
a n+1 (l - E) 


l-£" +1 1 +E + E 2 + ••■ + E n 

As E tends toward one — that is, as the rectangles become narrower — the 
sum of the areas of the rectangles approaches the area under the curve. 
Upon letting E = 1 in the formula above for the sum of the rectangles, we 
obtain (a n+ l )/{n + 1), the desired area under the curve y = x" from x = to 
x = a. To show that this holds also for rational fractional values, p/q, let 
n = p/q. The sum of the geometric progression then is 

,(„+,)/,( 1 ~ l* \ = fl(P+4 ,J l + E + E +... + £•- 

[1 -£"+«/ \l +E + E 2 + ... + E p+q - 1 

and when E = 1, this becomes 


p + q 


If, in modern notation, we wish to obtain J* x" dx, it is only necessary to 
observe that this is J* x" dx - fox" dx. 

For negative values of n (except n = - 1) Fermat used a similar procedure, 
except that E is taken as greater than one and tends toward one from above, 
the area found being that beneath the curve from x = a to infinity. To find 
J* x ~ " dx, then, it was only necessary to note that this is J*^ x ~ " dx - ]™ x ~ " dx. 

For n = - 1 the procedure fails ; but Fermat's older contemporary, 1 5 
Gregory of St. Vincent (1584-1667) disposed of this case in his Opus geo- 
metrician quadraturae circuli et sectionum coni ("Geometrical Work on the 
Squaring of the Circle and of Conic Sections"). Much of this work had been 
completed before the time that Fermat was working on tangents and areas, 
perhaps as early as 1622-1625, although it was not published until 1647. 
Gregory of St. Vincent, born at Ghent, was a Jesuit teacher at Rome and 
Prague and later became a tutor at the court of Philip IV of Spain. Through his 
travels he became separated from his papers, with the result that the appear- 
ance of the Opus geometricum was long delayed. In this treatise Gregory 
had shown that if along the x-axis one marks off from x = a points the 
intervals between which are increasing in continued geometric proportion, 
and if at these points ordinates are erected to the hyperbola xy = 1, then 
the areas under the curve intercepted between successive ordinates are equal. 
That is, as the abscissa increases geometrically, the area under the curve 
increases arithmetically. Hence the equivalent off* x~ l dx = In b - In a was 


known to Gregory and his contemporaries. Unfortunately, a faulty applica- 
tion of the method of indivisibles had led Gregory of St. Vincent to believe 
that he had squared the circle, an error that damaged his reputation. 

Fermat had been concerned with many aspects of infinitesimal analysis- 
tangents, quadratures, volumes, lengths of curves, centers of gravity. He 
could scarcely have failed to notice that in finding tangents to y = kx" one 
multiplies the coefficient by the exponent and lowers the exponent by one, 
whereas in finding areas one raises the exponent and divides by the new 
exponent. Could the inverse nature of these two problems have escaped him? 
Although this seems unlikely, it nevertheless appears that he nowhere 
called attention to the relationship now known as the fundamental theorem 
of the calculus. Perhaps he recognized the inverse nature of the problems 
but saw no great significance in this. Integration of x", about the only function 
that he really considered, was, after all, almost as easy for him as differentia- 
tion—and chronologically, at least for positive integral values of n, the 
former may have preceded the latter in Fermat's work. Thus also in the work 
of Gregory of St. Vincent the integral calculus came before the differential 
calculus for the logarithmic function. 

The inverse relationship between area and tangent problems should have 
been apparent from a comparison of Gregory of St. Vincent's area under 
the hyperbola and Descartes' analysis of inverse tangent problems proposed 
through Mersenne in 1638. The problems had been set by Florimond 
Debeaune (1601-1652), a jurist at Blois who was also an accomplished 
mathematician, for whom even Descartes expressed admiration. One of 
the problems called for the determination of a curve whose tangent had the 
property now expressed by the differential equation a dy/dx = x - y. 
Descartes recognized the solution as nonalgebraic, but he evidently just 
missed seeing that logarithms were involved. 6 

1 6 Fermat's contributions to analytic geometry and to infinitesimal analysis 
were but two aspects of his work— and probably not his favorite topics. In 
1621 the Arithmetica of Diophantus had come to life again through the 
Greek and Latin edition by Claude Gaspard de Bachet (1591-1639), a 
member of an informal group of scientists in Paris. Diophantus' Arithmetica 
had not been unknown, for Regiomontanus had thought of printing it ; several 
translations had appeared in the sixteenth century, with little result for the 
theory of numbers. Perhaps the work of Diophantus was too impractical for 
the practitioners and too algorithmic for the speculatively inclined ; but it 
appealed strongly to Fermat, who became the founder of the modern theory 
of numbers. Many aspects of the subject caught his fancy, including perfect 

6 See C. J. Scriba, "Zur Losung des 2. Debeauneschen Problems durch Descartes," Archive 
for History of Exact Sciences, 1 (1961), 406-419. 


and amicable numbers, figurate numbers, magic squares, Pythagorean triads, 
divisibility, and, above all, prime numbers. Some of his theorems he proved 
by a method that he called his "infinite descent" — a sort of inverted mathe- 
matical induction, a process that Fermat was among the first to use. As an 
illustration of his process of infinite descent, let us apply it to an old and 
familiar problem — the proof that ^3 is not rational. Let us assume that 
'3 = ajb±, where a Y and b x are positive integers with a 1 > b x . Since 

3 + 1 

upon replacing the first y/3 by its equal a 1 /b 1 , we have 

= 3ft i - a 1 

In view of the inequality f < ajb^ < 2, it is clear that 3ft x - a t and a x - b 1 
are positive integers, a 2 and b 2 , each less than a l and ftj respectively, and 
such that y/3 = a 2 /b 2 ■ This reasoning can be repeated indefinitely, leading 
to an infinite descent in which a„ and ft„ are ever smaller integers such that 
yjl = ajb„. This implies the false conclusion that there is no smallest 
positive integer. Hence the premise that v /3 is a quotient of integers must be 

Using his method of infinite descent, Fermat was able to prove Girard's 
assertion that every prime number of the form An + 1 can be written in one 
and only one way as the sum of two squares. He showed that if An + 1 is 
not the sum of two squares, there always is a smaller integer of this form that 
is not the sum of two squares. Using this recursive relationship backward 
leads to the false conclusion that the smallest integer of this type, 5, is not the 
sum of two squares (whereas 5 = l 2 + 2 2 ). Hence the general theorem is 
proved to be true. Since it is easy to show that no integer of the form An — 1 
can be the sum of two squares and since all primes except 2 are of the form 
An + 1 or An — 1, by Fermat 's theorem one can easily classify prime numbers 
into those that are and those that are not the sum of two squares. The prime 
23, for example, cannot be so divided, whereas the prime 29 can be written 
as 2 2 + 5 2 . Fermat knew that a prime of either form can be expressed as the 
difference of two squares in one and only one way. 

Fermat used his method of infinite descent to prove that there is no cube 1 7 
that is divisible into two cubes — that is, that there are no positive integers 
x, y, and z such that x 3 + y 3 = z 3 . Going further, Fermat stated the general 
proposition that for n an integer greater than two, there are no positive 
integral values x, y, and z such that x" + y" = z". He wrote in the margin of 


his copy of Bachet's Diophantus that he had a truly marvelous proof of this 
celebrated theorem, which since has become known as Fermat's "last" or 
"great" theorem. Fermat, most unfortunately, did not give his proof, but 
described it only as one "which this margin is too narrow to contain." If 
Fermat did indeed have such a proof, it has remained lost to this day. Despite 
all efforts to find a proof, once stimulated by a pre-World War I prize offer 
of 100,000 marks for a solution, the problem remains unsolved. However, 
the search for solutions has led to even more good mathematics than that 
which in antiquity resulted from efforts to solve the three classical and un- 
solvable geometrical problems. Like Horace Walpole's three princes of 
Serendip, mathematicians seem to have had the gift of finding along the way 
agreeable things not sought for. 

Whether or not Fermat was correct in stating his "great" theorem is not 
yet known, but decisions have been reached on two of his other conjectures 
in the theory of numbers. Perhaps two millennia before his day there had 
been a "Chinese hypothesis" which held that n is prime if and only if 2" - 2 
is divisible by n, where n is an integer greater than one. Half of this conjecture 
now is known to be false, for 2 341 - 2 is divisible by 341, and 341 = 11-31 
is composite; but the other half is indeed valid, and Fermat's "lesser" 
theorem is a generalization of this. A consideration of many cases of numbers 
of the form a"' 1 - 1, including 2 36 - 1, suggested that whenever p is prime 
and a is prime to p, then a p ~ 1 — 1 is divisible by p. On the basis of an induction 
from only five cases (n = 0, 1, 2, 3, and 4), Fermat formulated a second con- 
jecture — that integers of the form 2 2 " + 1, now known as "Fermat numbers," 
always are prime. Euler a century later showed this conjecture to be false, 
for 2 25 + 1 is composite. In fact, it is known now that 2 2 " -1- 1 is not prime 
for n between five and sixteen inclusive, and we begin to wonder if there is 
even one more prime Fermat number beyond those that Fermat knew. 7 

Fermat's lesser theorem fared better than his conjecture on prime Fermat 
numbers. A proof of the theorem was left in manuscript by Leibniz, and 
another elegant and elementary demonstration was published by Euler in 
1736. The proof by Euler makes ingenious use of mathematical induction, 
a device with which Fermat, as well as Pascal, was quite familiar. In fact, 
mathematical induction, or reasoning by recurrence, sometimes is referred 
to as "Fermatian induction," to distinguish it from scientific or "Baconian" 
induction. (Today the former sometimes is known also as "complete induc- 
tion," the latter as "incomplete induction.") 

1 8 Fermat was truly "the prince of amateurs" in mathematics. No professional 
mathematician of his day made greater discoveries or contributed more to 

7 W. Sierpinski, "L'induction incomplete dans la theorie des nombres," Scripta M athematica 
28 (1967), 5-13. 


the subject ; yet Fermat was so modest that he published virtually nothing. 
He was content to write of his thoughts to Mersenne (whose name, inci- 
dentally, is preserved in connection with the "Mersenne numbers," that is, 
primes of the form 2" - 1) and thus lost priority credit for much of his work. 
In this respect he shared the fate of one of his most capable friends and con- 
temporaries — the unamiable professor Roberval, a member of the "Mer- 
senne group" and the only truly professional mathematician among the 
Frenchmen whom we discuss in this chapter. Appointment to the chair of 
Ramus at the College Royal, which Roberval held for some forty years, was 
determined every three years on the basis of a competitive examination, the 
questions for which were set by the incumbent. In 1634 Roberval won the 
contest, probably because he had developed a method of indivisibles similar 
to that of Cavalieri ; by not disclosing his method to others, he successfully 
retained his position in the chair until his death in 1675. This meant, how- 
ever, that he lost credit for most of his discoveries and that he became em- 
broiled in numerous quarrels with respect to priority. The bitterest of these 
controversies concerned the cycloid, to which the phrase "the Helen of 
geometers" came to be applied because of the frequency with which it 
provoked quarrels during the seventeenth century. Mersenne in 1615 had 
called the attention of mathematicians to the cycloid, perhaps having heard 
of the curve through Galileo; in 1628, when Roberval arrived in Paris, 
Mersenne proposed to the young man that he study the curve. By 1634, 
Roberval was able to show that the area under one arch of the curve is 
exactly three times the area of the generating circle. By 1638 he had found 
how to draw the tangent to the curve at any point (a problem solved at about 
the same time also by Fermat and Descartes) and had found the volumes 
generated when the area under an arch is revolved about the base line. Later 
still he found the volumes generated by revolving the area about the axis of 
symmetry or about the tangent at the vertex. 8 

Roberval did not publish his discoveries concerning the cycloid (which 1 9 
he named the "trochoid," from the Greek word for wheel), for he may have 
wished to set similar questions for prospective candidates for his chair. 
Meanwhile Tofricelli became interested in the cycloid, possibly on the 
suggestion of Mersenne, perhaps through Galileo, whom Torricelli, like 
Mersenne, greatly admired. In 1643 Torricelli sent Mersenne the quadrature 
of the cycloid, and in 1644 he published a work with the title De par abole to 
which he appended both the quadrature of the cycloid and the construction 
of the tangent. Torricelli made no mention of the fact that Roberval had 

8 An excellent account of all of this work and of the place of Roberval in the mathematics of 
the time is found in Evelyn Walker, A Study of the Traite des Indivisibles of Gilles Persone de 
Roberval (1932). 


arrived at these results before him, and so in 1646 Roberval wrote a letter 
accusing Torricelli of plagiarism from him and from Fermat (on maxima 
and minima). It is clear now that priority of discovery belongs to Roberval, 
but priority in publication goes to Torricelli, who probably rediscovered the 
area and tangent independently. Roberval had used the method of indivisibles 
for the area problem ; Torricelli gave two quadratures, one making use of 
Cavalieri's method of indivisibles and the other of the ancient method of 
exhaustion. For finding the tangent to the curve both men employed a 
composition of motions reminiscent of Archimedes' tangent to his spiral. 
Roberval thought of a point P on the cycloid as subject to two equal notions, 
one a motion of translation, the other a rotary motion. As the generating 
circle rolls along the base line AB (Fig. 17.6), P is carried horizontally, at the 
same time rotating about O, the center of the circle. Through P one therefore 
draws a horizontal line PS, for the motion of translation, and a line PR 
tangent to the generating circle, for the rotary component. Inasmuch as the 
motion of translation is equal to that of rotation, the bisector PT of the 
angle SPR is the required tangent to the cycloid. 

The idea of the composition of movements was not original with Roberval, 
for Archimedes, Galileo, Descartes, and others had used it. Torricelli might 
have derived the idea from any one of these men ; hence his application of the 
principle to the cycloid need not have been plagiarism from Roberval. Both 
Torricelli and Roberval applied the kinematic method to other curves as 
well. A point on the parabola, for example, moves away from the focus at 
the same rate at which it moves away from the directrix, hence the tangent 
will be the bisector of the angle between lines in these two directions. A 
similar argument holds for the ellipse, in which the motion away from one 
focus is equal to the motion toward the other focus. Torricelli made use also 
of Fermat's method of tangents for the higher parabolas, knowledge of 
which is known to have reached Italy. 9 

20 The works of Roberval and Torricelli include many excellent results, 
only a few of which can be mentioned here. Among the contributions of 
Roberval was the first sketch, in 1635, of half an arch of a sine curve. This 
was important as an indication that trigonometry gradually was moving 
away from the computational emphasis, which had dominated thought in 
that branch, toward a functional approach. By means of his method of 
indivisibles, Roberval was able to show the equivalent of J* sin xdx = 
cos a — cos b, again indicating that area problems tended at that time to be 
easier to handle than tangent questions. Roberval and Torricelli, working 

9 There is no good account in English of the work of Torricelli, but certain aspects of it, 
especially tangents, are well treated in Evelyn Walker, A Study of the Traite des Indivisibles of 
Gilles Persone de Roberval. For other aspects see Torricelli's Opere (1919-1944). 


FIG. 17.6 

independently but along remarkably similar lines, extended Cavalieri's 
comparison of the parabola and the spiral by considering arc length as well 
as area. In the 1640s they showed that the length of the first rotation of the 
spiral r = ad is equal to the length of the parabola x 2 = lay from x = to 
x = Ina. Interest in the spiral at the time may have arisen from correspond- 
ence between Galileo and Mersenne concerning the path of a freely falling 
object on a moving earth, but the discussion soon greatly broadened. 
Fermat, ever one to seek generalizations, introduced the higher spirals 
r" = a6 and compared the arcs of these with the lengths of his higher para- 
bolas x"' 1 = lay. Torricelli studied spirals of various kinds, discovering 
the rectification of the logarithmic spiral, as we have seen. There was a 
remarkable unity in the mathematical interests of the period from about 
1630 to 1650, attributable in part to the intercommunication through 
Mersenne. Problems involving infinitesimals were by far the most popular 
at the time, and Torricelli in particular delighted in these. In the De dimen- 
sione parabolae, for instance, Torricelli gave twenty-one different proofs of 
the quadrature of the parabola, using approaches about evenly divided 
between the use of indivisibles and the method of exhaustion. One in the 
first category is almost identical with the mechanical quadrature given by 
Archimedes in his Method, presumably not then extant ; as might be antici- 
pated, one in the second category is virtually that given in Archimedes' 
treatise On the Quadrature of the Parabola, extant and well-known at the 
time. Had Torricelli arithmetized his procedures in this connection, he 
would have been very close to the modern limit concept, but he remained 
under the heavily geometrical influence of Cavalieri. Nevertheless, Torricelli 
far outdid his master in the flexible use of indivisibles to achieve new dis- 

One novel result of 1641 that greatly pleased Torricelli was his proof that 
if an infinite area, such as that bounded by the hyperbola xy = a 2 , an 
ordinate x = b, and the axis of abscissas, is revolved about the x-axis, the 


volume of the solid generated may be finite. Torricelli believed that he was 
first to discover that a figure with infinite dimensions can have a finite 
magnitude; but in this respect he may have been anticipated by Fermat's 
work on the areas under the higher hyperbolas, or possibly by Roberval, 
and certainly by Oresme in the fourteenth century. 

Among the problems that Torricelli handled just before his premature 
death in 1647 was one in which he sketched the curve whose equation we 
should write as x = log y — perhaps the first graph of a logarithmic function, 
thirty years after the death of the discoverer of logarithms as a computational 
device. Torricelli found the area bounded by the curve, its asymptote, and 
an ordinate, as well as the volume of the solid obtained upon revolving the 
area about the x-axis. 

Torricelli was one of the most promising mathematicians of the seventeenth 
century — often referred to as the century of genius. Mersenne had made the 
work of Fermat, Descartes, and Roberval known in Italy, both through 
correspondence with Galileo dating from 1635 and during a pilgrimage to 
Rome in 1644; Torricelli promptly mastered the new methods, although he 
always favored the geometric approach over the algebraic. Torricelli's 
brief association with the blind and aged Galileo in 1614-1642 had aroused 
in the young student an interest in physical science also, and today he is 
probably better recalled as the inventor of the barometer than as a mathe- 
matician. He studied the parabolic paths of projectiles fired from a point 
with fixed initial speeds but with varying angles of elevation, finding that the 
envelope of the parabolas is another parabola. In going from an equation for 
distance in terms of time to that for speed as a function of time, and inversely, 
Torricelli saw the inverse character of quadrature and tangent problems. 
Had he enjoyed the normal span of years, it is possible that he would have 
become the inventor of the calculus ; but a cruel fate cut short his life in 
Florence only a few days after his thirty-ninth birthday. 

21 The great developments in mathematics during the days of Descartes 
and Fermat were in analytic geometry and infinitesimal analysis. It is 
likely that it was the very success in these branches that made men of the 
time relatively oblivious to other aspects of mathematics. We already have 
seen that Fermat found no one to share his fascination with the theory of 
numbers; pure geometry likewise suffered a wholly undeserved neglect in 
the same period. The Conies of Apollonius once had been among Fermat's 
favorite works, but analytic methods redirected his views. Meanwhile, the 
Conies had attracted the attention of a practical man with a very impractical 
imagination — Girard Desargues, an architect and military engineer of 
Lyons. For some years Desargues had been at Paris, where he was part of 
the group of mathematicians that we have been considering; but his very 


unorthodox views on the role of perspective in architecture and geometry 
met with little favor, and he returned to Lyons to work out his new type of 
mathematics largely by himself. The result was one of the most unsuccessful 
great books ever produced. Even the ponderous title was repulsive — Brouillon 
projet d'une atteinte aux evenemens des rencontres d'un cone avec un plan 
(Paris, 1639). This may be translated as "Rough Draft of an Attempt to Deal 
with the Outcome of a Meeting of a Cone with a Plane," the barbarity of 
which stands in sharp contrast to the brevity and simplicity of Apollonius' 
title, Conies. The thought on which Desargues' work is based nevertheless is 
simplicity itself— a thought derived from perspective in Renaissance art and 
from Kepler's principle of continuity. Everyone knows that a circle, when 
viewed obliquely, looks like an ellipse, or that the outline of the shadow of a 
lampshade will be a circle or a hyperbola according as it is projected upon 
the ceiling or a wall. Shapes and sizes change according to the plane of 
incidence that cuts the cone of visual rays or of light rays ; but certain proper- 
ties remain the same throughout such changes, and it is these properties that 
Desargues studied. For one thing, a conic section remains a conic section no 
matter how many times it undergoes a projection. The conies form a single 
close-knit family, as Kepler had suggested fo