A History
Mathematics
Carl R Boyer
WILEY INTERNATIONAL EDITION
MATHEMATICAL CENTERS IN THE THALASSIC AGE
1. Syracuse
Abdera
10
(Democritus)
2. Crotona
Alexandria
25
(Euclid, Heron, Ptolemy, Pappus
Menelaus and others)
3. Elea
Athens
8
(Plato, Theaetetus)
4. Rome
Byzantium
11
(Proclus)
5. Tarentum
Chalcedon
12
(Xenocrates)
6. Cyrene
Chalcis
23
(Iamblichus)
7. Elis
Chios
16
(Hippocrates)
8. Athens
Cnidus
20
(Eudoxus)
9. Stagira
Crotona
2
(Pythagoras)
10. Abdera
Cyrene
6
(Theodorus, Eratosthenes)
11. Byzantium
Cyzicus
14
(Callippus)
12. Chalcedon
Elea
3
(Parmenides, Zeno)
13. Nicaea
Elis
7
(Hippias)
14. Cyzicus
Gerasa
24
(Nicomachus)
15. Pergamum
Miletus
19
(Thales)
16. Chios
Nicaea
13
(Hipparchus)
17. Samos
Perga
22
(Apollonius)
18. Smyrna
Pergamum
15
(Apollonius)
19. Miletus
Rhodes
21
(Eudemus, Geminus)
20. Cnidus
Rome
4
(Boethius)
21. Rhodes
Samos
17
(Pythagoras, Conon, Aristarchus)
22. Perga
Smyrna
18
(Theon)
23. Chalcis
Stagira
9
(Aristotle)
24. Gerasa
Syene
26
(Eratosthenes)
25. Alexandria
Syracuse
1
(Archimedes)
26. Syene
Tarentum
5
(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
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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
and
Rebecca Catherine (Eisenhart) Boyer
Preface
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.
viii PREFACE
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
PREFACE ix
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
Contents
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.
xii CONTENTS
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.
CONTENTS xiii
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.
xiv CONTENTS
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.
CONTENTS xv
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
CHAPTER I
Primitive Origins
Did you bring me a man who cannot number his
fingers?
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.
A HISTORY OF MATHEMATICS
India Iran Hoopta* Syria Egypt Asia minor Greece Italy Spain
°]-S500
1500
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
PRIMITIVE ORIGINS
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
4 A HISTORY OF MATHEMATICS
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.
5 PRIMITIVE ORIGINS
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.
6 A HISTORY OF MATHEMATICS
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
7 PRIMITIVE ORIGINS
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.
BIBLIOGRAPHY
Conant, Levi, The Number Concept. Its Origin and Development (New York : Macmillan,
1923).
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.
8 A HISTORY OF MATHEMATICS
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),
44-^19.
EXERCISES
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.
CHAPTER II
Egypt
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.
Herodotus
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
10 A HISTORY OF MATHEMATICS
IA.II
I I innn
I I I- R
'"jtfjIlnA.
! ! ML7&*=>2£^7 <
mi.
-«&<=>-<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).
11 EGYPT
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
r^Wnnnni/,!
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.
12 A HISTORY OF MATHEMATICS
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.
13 EGYPT
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
14 A HISTORY OF MATHEMATICS
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
formula
2 11
n
n + I n(n + 1)
or from
2
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.
15 EGYPT
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.
16 A HISTORY OF MATHEMATICS
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
17 EGYPT
"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
18 A HISTORY OF MATHEMATICS
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
19 EGYPT
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
20 A HISTORY OF MATHEMATICS
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
56
4
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.
21
EGYPT
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.
22 A HISTORY OF MATHEMATICS
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.
23 EGYPT
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
Mesopotamia.
BIBLIOGRAPHY
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,
1950).
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),
359-382.
24 A HISTORY OF MATHEMATICS
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,
292-304.
EXERCISES
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
25 EGYPT
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 +
776-J
CHAPTER III
Mesopotamia
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
26
27 MESOPOTAMIA
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.).
28 A HISTORY OF MATHEMATICS
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
29 MESOPOTAMIA
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
30 A HISTORY OF MATHEMATICS
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
31 MESOPOTAMIA
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
2
30
3
20
4
15
5
12
6
10
8
7,30
9
6,40
6
2
5
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,
32 A HISTORY OF MATHEMATICS
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).
33 MESOPOTAMIA
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
mensuration.
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
34 A HISTORY OF MATHEMATICS
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-
tions.
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.
35 MESOPOTAMIA
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
Then
x + y
x + y
2
= 10:33,45
2
- xy = 3;3,45
and
Hence
and
x + y
2
- 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.
36 A HISTORY OF MATHEMATICS
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 .
37 MESOPOTAMIA
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.
38
A HISTORY OF MATHEMATICS
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
39
MESOPOTAMIA
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,
40 A HISTORY OF MATHEMATICS
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.
41 MESOPOTAMIA
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.
42
A HISTORY OF MATHEMATICS
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.
43 MESOPOTAMIA
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.
44 A HISTORY OF MATHEMATICS
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.
45 MESOPOTAMIA
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.
46 A HISTORY OF MATHEMATICS
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
fun.
BIBLIOGRAPHY
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).
EXERCISES
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.
47 MESOPOTAMIA
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-
ians.
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
trapezoid.
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.
Aristotle
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.
48
49 IONIA AND THE PYTHAGOREANS
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
50 A HISTORY OF MATHEMATICS
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
quickened.
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.
51 IONIA AND THE PYTHAGOREANS
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
congruent.
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.
52 A HISTORY OF MATHEMATICS
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
master.
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.
53 IONIA AND THE PYTHAGOREANS
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.
54 A HISTORY OF MATHEMATICS
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.
55 IONIA AND THE PYTHAGOREANS
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
56
A HISTORY OF MATHEMATICS
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
G
H
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
57 IONIA AND THE PYTHAGOREANS
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.
58 A HISTORY OF MATHEMATICS
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.
59
IONIA AND THE PYTHAGOREANS
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
60 A HISTORY OF MATHEMATICS
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).
61 IONIA AND THE PYTHAGOREANS
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-
ematics.
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
(1)
-b
(2)
b-
c -
-a a
-b b
(3)
b -
c -
-a a
- b c
(4)
b-
c -
-a c
- b a
(5)
b-
c -
-a b
- b a
(6)
b — a c
c - b b
(7)
c — a c
b — a a
(8)
c — a c
c — b a
(9)
c — a b
b — a a
(10)
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,
62 A HISTORY OF MATHEMATICS
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
63 IONIA AND THE PYTHAGOREANS
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
as
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
64 A HISTORY OF MATHEMATICS
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 :
abtaefzhgikamn
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.
65 IONIA AND THE PYTHAGOREANS
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,
66 A HISTORY OF MATHEMATICS
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
67 IONIA AND THE PYTHAGOREANS
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.
BIBLIOGRAPHY
Allman, G. J., Greek Geometry from Thales to Euclid (Dublin : Dublin University Press,
1889).
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.).
EXERCISES
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
sides)?
5. Write the numbers 3456 and 4567 and their sum in the early Greek Attic notation and in the
Ionian or alphabetic system.
68 A HISTORY OF MATHEMATICS
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
pentagon.
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.
CHAPTER V
The Heroic Age
I would rather discover one cause than gain the king-
dom of Persia.
Democritus
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
69
70 A HISTORY OF MATHEMATICS
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).
71 THE HEROIC AGE
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
72 A HISTORY OF MATHEMATICS
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
73 THE HEROIC AGE
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.
74 A HISTORY OF MATHEMATICS
B
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
75 THE HEROIC AGE
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.
76
A HISTORY OF MATHEMATICS
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
77 THE HEROIC AGE
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
78 A HISTORY OF MATHEMATICS
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.
79 THE HEROIC AGE
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.
80 A HISTORY OF MATHEMATICS
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.
81
THE HEROIC AGE
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-
mensurable.
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
11
82 A HISTORY OF MATHEMATICS
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
absurdity.
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.
83
THE HEROIC AGE
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 A
Bx
B 2
B 3
B 4
Cx
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 :
Ax
A 2
A 3
A*
Bx
B 2
B 3
B A
Cx
C 2
c 3
C 4
84 A HISTORY OF MATHEMATICS
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;
1936V
' ' See van der Waerden, Science Awakening (1961), p. 89.
85
THE HEROIC AGE
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
13
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.
86
A HISTORY OF MATHEMATICS
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
P
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 =
ab
ac
ad
o2
ab
ab
6 2
FIG. 5.10
FIG. 5.11
87
THE HEROIC AGE
(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
a+b-
a—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
14
88 A HISTORY OF MATHEMATICS
"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
89 THE HEROIC AGE
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.
BIBLIOGRAPHY
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.
90 A HISTORY OF MATHEMATICS
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.
EXERCISES
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-
tion.
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.
CHAPTER VI
The Age of Plato and
Aristotle
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.
Eudoxus
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
91
92 A HISTORY OF MATHEMATICS
Plato and Aristotle in Raphael's "School of Athens."
93 THE AGE OF PLATO AND ARISTOTLE
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.
94 A HISTORY OF MATHEMATICS
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
Fire
tetrahedron
Water
icosahedron
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.
95 THE AGE OF PLATO AND ARISTOTLE
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
96 A HISTORY OF MATHEMATICS
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.).
97 THE AGE OF PLATO AND ARISTOTLE
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.
98 A HISTORY OF MATHEMATICS
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.
99 THE AGE OF PLATO AND ARISTOTLE
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.
100 A HISTORY OF MATHEMATICS
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.
101 THE AGE OF PLATO AND ARISTOTLE
statement that lim M(l - rf = 0. Moreover, the Greeks made use of this
n-*co
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
102 A HISTORY OF MATHEMATICS
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.
103 THE AGE OF PLATO AND ARISTOTLE
(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
104 A HISTORY OF MATHEMATICS
A
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
equals,
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.
105
THE AGE OF PLATO AND ARISTOTLE
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
11
106
A HISTORY OF MATHEMATICS
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
e-*o
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.
107 THE AGE OF PLATO AND ARISTOTLE
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.
108 A HISTORY OF MATHEMATICS
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-
ematics.
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).
109 THE AGE OF PLATO AND ARISTOTLE
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.
BIBLIOGRAPHY
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)
93-107.
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
1955).
Zeuthen, H. G., Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886).
110 A HISTORY OF MATHEMATICS
EXERCISES
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
Elements.)
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
dodecahedron.
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.
CHAPTER VII
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
112 A HISTORY OF MATHEMATICS
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
constant.
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
113 EUCLID OF ALEXANDRIA
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).
114
A HISTORY OF MATHEMATICS
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
115 EUCLID OF ALEXANDRIA
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
116 A HISTORY OF MATHEMATICS
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.).
117 EUCLID OF ALEXANDRIA
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
118 A HISTORY OF MATHEMATICS
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
sound.
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
119
EUCLID OF ALEXANDRIA
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.
120
A HISTORY OF MATHEMATICS
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.]
121
EUCLID OF ALEXANDRIA
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.
122
A HISTORY OF MATHEMATICS
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
K
L
H
/
P
/
E G
FIG. 7.6
M
(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.
123
EUCLID OF ALEXANDRIA
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
D
K L
H
D
E
M
K
\
\
H G
E G
FIG. 7.7
B
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.
124 A HISTORY OF MATHEMATICS
D A
^ ^ ■
?
/
"^ ^
/
\ </^
/
^^.y^
V
/^ x
/
/ ^-^
/
/ ^
/
/ ^^
/
/ \
/
/ "
I M
Q o/
r-
4
-z
N
/R
\
\
1
V
X
*\
' \—
^
N
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.
125 EUCLID OF ALEXANDRIA
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.
126 A HISTORY OF MATHEMATICS
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
127 EUCLID OF ALEXANDRIA
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,
128 A HISTORY OF MATHEMATICS
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),
27-48A.
129 EUCLID OF ALEXANDRIA
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.
130 A HISTORY OF MATHEMATICS
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
131 EUCLID OF ALEXANDRIA
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
century.
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!"
BIBLIOGRAPHY
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).
132 A HISTORY OF MATHEMATICS
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),
193-202.
EXERCISES
1. Describe the sources Euclid probably used in writing the Elements; justify your presump-
tions.
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
integers?
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
magnitude.
133 EUCLID OF ALEXANDRIA
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.
CHAPTER VIII
Archimedes of Syracuse
There was more imagination in the head of Archimedes
than in that of Homer.
Voltaire
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
134
T35 ARCHIMEDES OF SYRACUSE
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
136 A HISTORY OF MATHEMATICS
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.
137 ARCHIMEDES OF SYRACUSE
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
(1897).
138 A HISTORY OF MATHEMATICS
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.
139 ARCHIMEDES OF SYRACUSE
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.
140 A HISTORY OF MATHEMATICS
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
141
ARCHIMEDES OF SYRACUSE
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
142 A HISTORY OF MATHEMATICS
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
showed.
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
143 ARCHIMEDES OF SYRACUSE
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
144 A HISTORY OF MATHEMATICS
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
145 ARCHIMEDES OF SYRACUSE
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.
146 A HISTORY OF MATHEMATICS
c
K
L
.?
/■
M \
1
J
3
\
-*==.
^=^
U
K'
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
cot
2n
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
n
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
(1
cos 9) lim — cot — = 1
n^oc 2« 2n
cos(
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
147 ARCHIMEDES OF SYRACUSE
of a Roman to the history of mathematics — but all traces of it have since
vanished.
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
148
A HISTORY OF MATHEMATICS
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
149 ARCHIMEDES OF SYRACUSE
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,
150 A HISTORY OF MATHEMATICS
Af
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.
151 ARCHIMEDES OF SYRACUSE
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
segments.
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
152 A HISTORY OF MATHEMATICS
H
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
153
ARCHIMEDES OF SYRACUSE
N
I!
/
/
ft
R
[C
^
A
k
S
<
°)
\
\
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
15
FIG. 8.13
154 A HISTORY OF MATHEMATICS
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.
BIBLIOGRAPHY
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.).
155 ARCHIMEDES OF SYRACUSE
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),
357^08.
EXERCISES
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.
156 A HISTORY OF MATHEMATICS
*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.
CHAPTER IX
Apollonius of Perga
It seems to me that all the evidence points to
Apollonius as the founder of Greek mathematical
astronomy.
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
157
158 A HISTORY OF MATHEMATICS
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.
159 APOLLONIUS OF PERGA
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.
160
A HISTORY OF MATHEMATICS
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
161 APOLLONIUS OF PERGA *
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.
162 A HISTORY OF MATHEMATICS
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
cone.
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.
163 APOLLONIUS OF PERGA
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.
164 A HISTORY OF MATHEMATICS
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.)
R<—
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
165 APOLLONIUS OF PERGA
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
166 A HISTORY OF MATHEMATICS
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
167 APOLLONIUS OF PERGA
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.)
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
168 A HISTORY OF MATHEMATICS
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.
169 APOLLONIUS OF PERGA
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
170 A HISTORY OF MATHEMATICS
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
171 APOLLONIUS OF PERGA
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
172 A HISTORY OF MATHEMATICS
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
173 APOLLONIUS OF PERGA
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.
BIBLIOGRAPHY
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
1945).
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).
174 A HISTORY OF MATHEMATICS
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
1902).
EXERCISES
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
diameter.
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
ellipse?
175 APOLLONIUS OF PERGA
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
answer.
CHAPTER X
Greek Trigonometry and
Mensuration
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.
Ptolemy
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.
176
177 GREEK TRIGONOMETRY AND MENSURATION
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.
178 A HISTORY OF MATHEMATICS
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.
179 GREEK TRIGONOMETRY AND MENSURATION
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
Alexandria.
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
180 A HISTORY OF MATHEMATICS
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
earlier.
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.
181
GREEK TRIGONOMETRY AND MENSURATION
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
182 A HISTORY OF MATHEMATICS
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.
183 GREEK TRIGONOMETRY AND MENSURATION
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
184 A HISTORY OF MATHEMATICS
B
7
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
formulas.
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
185 GREEK TRIGONOMETRY AND MENSURATION
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
120
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
186 A HISTORY OF MATHEMATICS
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.
187 GREEK TRIGONOMETRY AND MENSURATION
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
188 A HISTORY OF MATHEMATICS
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
189 GREEK TRIGONOMETRY AND MENSURATION
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
discovered.
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.
190 A HISTORY OF MATHEMATICS
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,
191 GREEK TRIGONOMETRY AND MENSURATION
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.
13
192 A HISTORY OF MATHEMATICS
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.
193 GREEK TRIGONOMETRY AND MENSURATION
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).
194 A HISTORY OF MATHEMATICS
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."
BIBLIOGRAPHY
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
1788-1858.
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).
195 GREEK TRIGONOMETRY AND MENSURATION
EXERCISES
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
correct?
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
imply?
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?
CHAPTER XI
Revival and Decline of Greek
Mathematics
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.
196
197 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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).
198 A HISTORY OF MATHEMATICS
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
degree.
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).
199 REVIVAL AND DECLINE OF GREEK MATHEMATICS
Four antique mathematicians who contributed also to music: Boethius, Pythagoras, Plato,
Nicomachus; from a Boethius manuscript, Cambridge,
200 A HISTORY OF MATHEMATICS
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.
201 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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.
202 A HISTORY OF MATHEMATICS
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
Europe.
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
203 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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.
204 A HISTORY OF MATHEMATICS
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),
163-170.
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.
205 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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
Pappus.
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
206 A HISTORY OF MATHEMATICS
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
207
REVIVAL AND DECLINE OF GREEK MATHEMATICS
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
^--»F
208 A HISTORY OF MATHEMATICS
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).
209 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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
210 A HISTORY OF MATHEMATICS
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.).
211 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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
decline.
212 A HISTORY OF MATHEMATICS
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
213 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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.
214 A HISTORY OF MATHEMATICS
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.
215 REVIVAL AND DECLINE OF GREEK MATHEMATICS
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.
BIBLIOGRAPHY
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,
1964).
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,
1926).
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,
1947).
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),
163-170.
Thomas, Ivor, ed., Selections Illustrating the History of Greek Mathematics (Cambridge,
Mass. : Loeb Classical Library, 1939-1941, 2 vols.).
216 A HISTORY OF MATHEMATICS
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.
EXERCISES
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.
CHAPTER XII
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).
217
218 A HISTORY OF MATHEMATICS
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).
219 CHINA AND INDIA
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
1
2
3
3
2
3
3
1
2
1
to reduce it to
5
36 1
2
1
26
34
39
99 24
39
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),
424-432.
220 A HISTORY OF MATHEMATICS
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
221 CHINA AND INDIA
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.
222 A HISTORY OF MATHEMATICS
Marble counting board, probably from the fourth century B.C. found on the island of
Salamis and now in the National Museum in Athens.
223
CHINA AND INDIA
m$3i
/J. yjU
'■£. 'A
%
&
*4'fa
tmrnrntmi
4
it
y£
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
224 A HISTORY OF MATHEMATICS
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.
225 CHINA AND INDIA
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
defined.
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
226 A HISTORY OF MATHEMATICS
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.
227 CHINA AND INDIA
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.
228
A HISTORY OF MATHEMATICS
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
4*
fcfc'V*
<v^
«vs?
$<ftl«v*H^
m
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.
229 CHINA AND INDIA
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
earlier.
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.
230 A HISTORY OF MATHEMATICS
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.
D
H
F
M
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.
231 CHINA AND INDIA
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.
232 A HISTORY OF MATHEMATICS
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.
233 CHINA AND INDIA
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
terms.
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.
234 A HISTORY OF MATHEMATICS
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.
235 CHINA AND INDIA
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.
236
A HISTORY OF MATHEMATICS
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.)
237 CHINA AND INDIA
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.
238 A HISTORY OF MATHEMATICS
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.
239
CHINA AND INDIA
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
4
\ 6
\
\ 4
l\
2 \
2 \
\ 2
\ 5
\ 8
•I
1 \
l\
l\
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
18
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).
240
A HISTORY OF MATHEMATICS
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.
117
382)44977
382
~677
382
382
2957
2674
283
FIG. 12.5
2
2 3
3 9 8
; 7 ? 3
4 4 ? 7 7
3 8 2 2 4
3 8 7
2
FIG. 12.6
117
241 CHINA AND INDIA
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.
242 A HISTORY OF MATHEMATICS
"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).
243 CHINA AND INDIA
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.
244 A HISTORY OF MATHEMATICS
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
words.
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.
245 CHINA AND INDIA
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.
246 A HISTORY OF MATHEMATICS
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.
BIBLIOGRAPHY
Boyer, C. B., "Fundamental Steps in the Development of Numeration," his, 35 (1944),
153-168.
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.
247 CHINA AND INDIA
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,
1964).
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),
424^32.
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,
1926).
EXERCISES
1. Compare Hindu and Chinese mathematics with respect to favorite topics and level of
achievement.
2. Which had the greater influence on modern thought, Chinese or Hindu mathematics?
Explain clearly.
248 A HISTORY OF MATHEMATICS
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
theorem.
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.
CHAPTER XIII
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.
249
250 A HISTORY OF MATHEMATICS
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
251 THE ARABIC HEGEMONY
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.
252 A HISTORY OF MATHEMATICS
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.
253 THE ARABIC HEGEMONY
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)
37-48.
254 A HISTORY OF MATHEMATICS
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.
255
THE ARABIC HEGEMONY
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
a
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
e
n
FIG. 13.2
256 A HISTORY OF MATHEMATICS
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
257 THE ARABIC HEGEMONY
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
258 A HISTORY OF MATHEMATICS
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).
259
THE ARABIC HEGEMONY
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
r-^
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.
260 A HISTORY OF MATHEMATICS
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.
261
THE ARABIC HEGEMONY
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.
12
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Q> 1 h ?
Brahmi
1
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in (Gw
m« v?\3K«
'
'
Sanskrit-Devanagari (Indian)
>
■
f «,;* r* <j i n I 3
1 r r^fi h va 1 •
West Arabic
(Gobar)
East Arabic
IF^c^Q
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llth Cer
ltury (
Apices)
, z 3 a. q s "k 8 <f «
1*5+5 fr7*9»
15tr
Cen1
ury
16th Cen
tury (
Diirer)
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.
262 A HISTORY OF MATHEMATICS
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.
263 THE ARABIC HEGEMONY
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.
14
264 A HISTORY OF MATHEMATICS
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.
265 THE ARABIC HEGEMONY
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.
266 A HISTORY OF MATHEMATICS
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),
280-285.
267 THE ARABIC HEGEMONY
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),
5-10.
268 A HISTORY OF MATHEMATICS
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
(1947).
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.
269 THE ARABIC HEGEMONY
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.
BIBLIOGRAPHY
Amir-Moez, A. R., "A Paper of Omar Khayyam," Scripta Mathematica, 26 (1963)
323-337.
Cajori, Florian, History of Mathematics, 2nd ed. (New York: Macmillan, 1919).
Dedron, Pierre, and Jean Itard, Mathematiques et mathematiciens (Paris' Magnard
1959).
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.
270 A HISTORY OF MATHEMATICS
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),
280-285.
Suter, Heinrich, Die Mathematiker und Astronomer der Araber und ihre Werke (Leipzig,
1900).
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.
EXERCISES
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.
271 THE ARABIC HEGEMONY
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.
CHAPTER XIV
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
272
273 EUROPE IN THE MIDDLE AGES
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.
^74 A HISTORY OF MATHEMATICS
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
275 EUROPE IN THE MIDDLE AGES
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
276 A HISTORY OF MATHEMATICS
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.
277 EUROPE IN THE MIDDLE AGES
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.
278 A HISTORY OF MATHEMATICS
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
279
EUROPE IN THE MIDDLE AGES
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
Pythagoras.
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
280 A HISTORY OF MATHEMATICS
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.).
281 EUROPE IN THE MIDDLE AGES
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
282 A HISTORY OF MATHEMATICS
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.
283 EUROPE IN THE MIDDLE AGES
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.
284 A HISTORY OF MATHEMATICS
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.
285 EUROPE IN THE MIDDLE AGES
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).
286 A HISTORY OF MATHEMATICS
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).
287 EUROPE IN THE MIDDLE AGES
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.
288 A HISTORY OF MATHEMATICS
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
Philoponus.
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 .
289 EUROPE IN THE MIDDLE AGES
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
p
1
1
2
15 See Edward Stamm, "Tractatus de continuo von Thomas Bradwardina. Eine Handschrift
aus dem XIV. Jahrhundert," I sis, 26 (1936), 13-32.
290 A HISTORY OF MATHEMATICS
to denote the "one and one-half proportion"— that is, the cube of the
principal square root — and forms such as
1 P- 1
4-2-2
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.
291
EUROPE IN THE MIDDLE AGES
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
292 A HISTORY OF MATHEMATICS
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.
293 EUROPE 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)
67-91.
294 A HISTORY OF MATHEMATICS
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.
BIBLIOGRAPHY
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
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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),
327-341.
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).
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Sarton, George, Introduction to the History of Science (Baltimore: Carnegie Institution
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reprint. New York : Dover, 1 958).
295 EUROPE IN THE MIDDLE AGES
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
1964).
EXERCISES
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
rational.
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. .
296 A HISTORY OF MATHEMATICS
*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.
CHAPTER XV
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
297
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.
299 THE RENAISSANCE
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).
300 A HISTORY OF MATHEMATICS
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
301 THE 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-
etry.
2 See George Sarton, "The Scientific Literature Transmitted Through the Incunabula "
Osiris, 5 (1938), 41-247.
302 A HISTORY OF MATHEMATICS
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
303 THE RENAISSANCE
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
(1967).
304 A HISTORY OF MATHEMATICS
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
305 THE RENAISSANCE
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
306 A HISTORY OF MATHEMATICS
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.
307 THE RENAISSANCE
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).
308 A HISTORY OF MATHEMATICS
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.
309
THE RENAISSANCE
copy of the Algebra of al-Khowarizmi, a work well known to other German
mathematicians.
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.
310 A HISTORY OF MATHEMATICS
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.
311 THE RENAISSANCE
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-
cian.
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
312 A HISTORY OF MATHEMATICS
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
313
THE RENAISSANCE
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^^^^
w
■ ■ ■■:.
/ .. 'mm/
Jerome Cardan.
314 A HISTORY OF MATHEMATICS
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.
315 THE RENAISSANCE
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.
316 A HISTORY OF MATHEMATICS
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
317 THE RENAISSANCE
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.
318 A HISTORY OF MATHEMATICS
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
notations.
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.
319 THE RENAISSANCE
The Jrte
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tkuoo partes. ''&:)crcoftocnrft;i/j •»&?* one number u
equdle m*« ont other Ilia tt)C ftfOrtDc t <» Tfr&t n on* »•*;
Jrrr m cmps.ed ss tqtulle tnto ..other utmbers.
flliuaics unupug pou to rcnirbcr, tbat pou rcDttcc
pour nombcrs , to tfr.tr Uaftc Denomination* , anb
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trite Denomination G#£e, be ioineb to anp parte of a
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nombcr of tbe greatefte ftgnc alone, maieffanocas
cquallc to tbe rcttc.
ano tbts to all teat neaoetb to be taugbte , concern
npngtbtolttoojke.
]»otobcit,fo;eauc alteratio cfeftutwu.j toill p;o*
pounoc a fetoe eraplesibicanfe tbe ertraoton of tbeir
rootcs,.maie tbe tnojc aptlp bee tojougbte. and to a*
ttotoc tbe teoioufe repetition of tbefe looo;bes : ise*
qualle to : 3 toill fette as 3 toe often in tooo&c bfe,a
patrc of paralleles,o; ©cmoUic lines of one lengtbc,
tbus: -— *=— =,bicaufe noe.2rlbpnges,ran be moare
e qualle. flno nolo marsc tbefe nombers*
1.
3*
4. 1 9.*£ — h-l 9 2.f —— 1 o j^ — f — lc«f 1 9 J£
f. 1 8.2^ — }~ 2 4.^ .-=— 8.5-t — ! —
6 * 345, I2S£~= 4oS£--f-48of — 9.^
1. 3fn tbe firftc tbert appearetb. 2 ♦ nombcr* , tbat is
14.^.
A page from Robert Recorde's Whetstone of Witte (1557). Note that his symbols for
equality are much longer than ours.
320 A HISTORY OF MATHEMATICS
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-
tanus.
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.
321 THE RENAISSANCE
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).
322 A HISTORY OF MATHEMATICS
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).
323 THE RENAISSANCE
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.
20
324 A HISTORY OF MATHEMATICS
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
~7F~
/ 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.
325
THE RENAISSANCE
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.
326
A HISTORY OF MATHEMATICS
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
16
3
2
13
5
10
11
8
9
6
7
12
4
15
14
1
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.
327 THE RENAISSANCE
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.
21
328
A HISTORY OF MATHEMATICS
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Pascal Triangle as first printed, 1527. Title page of the arithmetic of Petrus Apianus,
Ingolstadt, 1527, more than a century before Pascal investigated the properties of the
triangle.
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
329 THE RENAISSANCE
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.
330 A HISTORY OF MATHEMATICS
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.
BIBLIOGRAPHY
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.
331 THE RENAISSANCE
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).
EXERCISES
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.
332 A HISTORY OF MATHEMATICS
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
equation.
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
plane.
CHAPTER XVI
Prelude
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
333
334 A HISTORY OF MATHEMATICS
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.
335 PRELUDE TO MODERN MATHEMATICS
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.
336 A HISTORY OF MATHEMATICS
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
337 PRELUDE TO MODERN MATHEMATICS
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).
338 A HISTORY OF MATHEMATICS
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
indeterminacy.
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
339
PRELUDE TO MODERN MATHEMATICS
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)
tan-
B
(a-b)
tan-
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.
x-y
FIG. 16.1
340 A HISTORY OF MATHEMATICS
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
cosecants.
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
1-2-3-4
341 PRELUDE TO MODERN MATHEMATICS
and
„-i • n ( n ~ !)(« _ 2) „_, . ,
sin nx = n cos" x sin x cos x sin x + • • •
1-2-3
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.
342 A HISTORY OF MATHEMATICS
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
trigonometry.
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).
343 PRELUDE TO MODERN MATHEMATICS
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
Logarithms").
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.
B
_L
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
344 A HISTORY OF MATHEMATICS
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.
345 PRELUDE TO MODERN MATHEMATICS
10
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.
346 A HISTORY OF MATHEMATICS
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.
347 PRELUDE TO MODERN MATHEMATICS
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.
348 A HISTORY OF MATHEMATICS
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.
349 PRELUDE TO MODERN MATHEMATICS
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der Thiende vande
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8^
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,
350 A HISTORY OF MATHEMATICS
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
1+3-7
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).
351 PRELUDE TO MODERN MATHEMATICS
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
352 A HISTORY OF MATHEMATICS
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
processes.
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.
353 PRELUDE TO MODERN MATHEMATICS
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
geometry.
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
known.
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
354 A HISTORY OF MATHEMATICS
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.
355 PRELUDE TO MODERN MATHEMATICS
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
356 A HISTORY OF MATHEMATICS
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
357 PRELUDE TO MODERN MATHEMATICS
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).
358 A HISTORY OF MATHEMATICS
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
359 PRELUDE TO MODERN MATHEMATICS
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
360 A HISTORY OF MATHEMATICS
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).
361 PRELUDE TO MODERN MATHEMATICS
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).
362 A HISTORY OF MATHEMATICS
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
procedures.
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
calculus
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.
363 PRELUDE TO MODERN MATHEMATICS
A,
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,
r
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
of
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.
364 A HISTORY OF MATHEMATICS
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.
BIBLIOGRAPHY
Boyer, C. B., The Concepts of the Calculus (New York, 1939 ; paperback ed., New York :
Dover, 1959).
Brasch, F. E., ed., Johann Kepler, 1571-1630. A Tercentenary Commemoration of His
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).
365 PRELUDE TO MODERN MATHEMATICS
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).
Cajori, Florian, A History of Mathematical Notations (Chicago: Open Court, 1929, 2
vols.).
Caspar, Max, Kepler, trans, by C. Doris Hellman (New York : Abelard-Schuman, 1959).
Dedron, Pierre, and Jean hard, Mathematiques et mathematiciens (Paris- Magnard
1959).
Dijksterhuis, E. J., and D. J. Struik, eds., The Principal Works of Simon Stevin (Amster-
dam : Swets and Zeitlinger, 1955-1958).
Galilei, Galileo, Dialogue Concerning Two New Sciences, ed. by Henry Crew and Alfonso
de Salvio (paperback ed., New York : Dover, no date).
Galilei, Galileo, Discourses on the Two Chief Systems, ed. by Stillman Drake (Berkeley,
Calif. : University of California Press, 1953).
Galilei, Galileo, Discourses on the Two Chief Systems, ed. by Giorgio de Santillana
(Chicago : University of Chicago Press, 1953).
Glaisher, J. W. L., "Logarithms," Encyclopaedia Britannica, 11th ed., (1910-1911)
Vol. XVI, pp. 868-877.
Glaisher, J. W. L., "On Early Tables of Logarithms and Early History of Logarithms,"
Quarterly Journal of Pure and Applied Mathematics, 48 (1920), 151-192.
Hobson, E. W., John Napier and the Invention of Logarithms (Cambridge, 1914).
Hofmann, J. E, Geschichte der Mathematik, 2nd ed. (Berlin : Walter de Gruyter, 1963).
Kepler, Johann, Gesammelte Werke, ed. by Walther von Dyck and Max Caspar (Mun-
chen : C. H. Beck, 1937- ).
Knott, C. G., Napier Tercentenary Memorial Volume (London: Longmans Green
1915).
Ritter, Frederic, Francois Viete (Paris, 1895).
Sarton, George, "Simon Stevin of Bruges (1548-1620)," Isis, 21 (1934), 241-303.
Sarton, George, "The First Explanation of Decimal Fractions and Measures (1585),"
Isis, 23 (1935), 153-244.
Smith, D. E., A Source Book in Mathematics (New York : McGraw-Hill, 1929 ; paperback
ed., New York : Dover, 1959, 2 vols.).
Tropfke, Johannes, Geschichte der Elementar-Mathematik (Berlin: De Gruyter 1923)
Vol. V.
Turnbull, H. W., The Great Mathematicians (New York : New York University Press
1961).
Zeller, Sister Mary Claudia, The Development of Trigonometry from Regiomontanus to
Pitiscus (Ann Arbor, Mich. : University of Michigan, Ph.D. thesis, 1944).
EXERCISES
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.
366 A HISTORY OF MATHEMATICS
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.
CHAPTER XVII
The Time
of Fermat and Descartes
Fermat, the true inventor of the differential calculus.
Laplace
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,
367
368
A HISTORY OF MATHEMATICS
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,
369 THE TIME OF FERMAT AND DESCARTES
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
science.
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
370 A HISTORY OF MATHEMATICS
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,
371 THE TIME OF FERMAT AND DESCARTES
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
372
A HISTORY OF MATHEMATICS
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
known.
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.
373 THE TIME OF FERMAT AND DESCARTES
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
374 A HISTORY OF MATHEMATICS
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
375 THE TIME OF FERMAT AND DESCARTES
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 :
376
A HISTORY OF MATHEMATICS
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.
377 THE TIME OF FERMAT AND DESCARTES
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.
378 A HISTORY OF MATHEMATICS
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
379 THE TIME OF FERMAT AND DESCARTES
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
380 A HISTORY OF MATHEMATICS
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
381 THE TIME OF FERMAT AND DESCARTES
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).
382 A HISTORY OF MATHEMATICS
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)
hm
E^O E
and setting this equal to zero. Hence it is appropriate to follow Laplace in
4 See Fermat, Oeuvres, I, 186-187.
383 THE TIME OF FERMAT AND DESCARTES
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
found.
Fermat's procedure amounts to saying that
lim
E-0
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
384
A HISTORY OF MATHEMATICS
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.
385 THE TIME OF FERMAT AND DESCARTES
a"E 2n (aE 2 - aE 3 ), .... The sum to infinity of these terms is
a n+1 (l - E)
or
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
q
p + q
t-«V8
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
386 A HISTORY OF MATHEMATICS
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.
387 THE TIME OF FERMAT AND DESCARTES
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
false.
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
388 A HISTORY OF MATHEMATICS
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.
389 THE TIME OF FERMAT AND DESCARTES
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).
390 A HISTORY OF MATHEMATICS
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).
391 THE TIME OF FERMAT AND DESCARTES
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-
coveries.
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
392 A HISTORY OF MATHEMATICS
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
393 THE TIME OF FERMAT AND DESCARTES
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 for so