A CONCISE HISTORY OF A CONCISE HISTORY MATHEMATICS STRUIK OF MATHEMATICS DIRK J. STRUIK Professor Mathematics, BELL of Massachussetts Institute of Technology i Professor Struik has achieved the seemingly impossible task of compress- ing the history of mathematics into less than three hundred pages. By stressing the unfolding of a few main ideas and by minimizing references to other develop- ments, the author has been able to fol- low Egyptian, Babylonian, Chinese, Indian, Greek, Arabian, and Western mathematics from the earliest records to the beginning of the present century. He has based his account of nineteenth cen- tury advances on persons and schools rather than on subjects as the treatment by subjects has already been used in existing books. Important mathema- ticians whose work is analysed in detail are Euclid, Archimedes, Diophantos, Hammurabi, Bernoulli, Fermat, Euler, Newton, Leibniz, Laplace, Lagrange, Gauss, Jacobi, Riemann, Cremona, Betti, and others. Among the 47 illustra- tions arc portraits of many of these great figures. Each chapter is followed by a select bibliography. CHARLES WILSON A CONCISE HISTORY OF MATHEMATICS by DIRK. J. STRUIK Professor of Mathematics at the Massachusetts Institute of Technology LONDON G. BELL AND SONS LTD '954 Copyright by Dover Publications, Inc., U.S.A. FOR RUTH Printed in Great Britain by Butler & Tanner Ltd., Frame and London CONTENTS Introduction xi The Beginnings 1 The Ancient Orient 13 Greece 39 The Orient after the Decline of Greek Society 83 The Beginnings in Western Europe 98 The Seventeenth Century 125 The Eighteenth Century 163 The Nineteenth Century 201 Index 289 LIST OF ILLUSTRATIONS Geometrical Patterns Developed by American Indians 6 Neolithic Pottery Pattern 7 Egyptian Pottery Pattern 8 Pole House Dwellers Pattern 8 Patterns from Urns near Oedenburg 8 Predynastic Egyptian Pottery 10 Page from the Great Papyrus Rhind 20-21 One Side of a Cuneiform Text 27 Portion of a Page from the First Edition of Euclid's Elements, 1482 51 Two Pages from Archimedes Opera Omnia 62-63 Tomb of Omar Khayyam in Nishapur 90 Luca Pacioli with the Young Duke of Urbino 111 Francois Viete 117 John Napier 120 Johann Kepler 128 Galileo 130 Facsimile Page of Descartes' La Geometric 133 Ren6 Descartes 135 Pascal Explains to Descartes His Plans for Experiments in Jakob Steincr 247 Weight 137 Xicolai Ivanovitch Lobachevsky 252 Christiaan Huygens 143 Arthur Cayley 257 Reproduction of Leaflet Published by Pascal, 1(540 146 James Joseph Sylvester 261 Blaise Pascal 147 William Rowan Hamilton 264 Sir Isaac Newton 150 Felix Klein 270 Gottfried Wilhelm von Leibniz 155 Marius Sophus Lie 273 One Page from Leibniz's First Paper on the Calculus 159 T. J. Stieltjes 277 Leonard Euler 170 Henri Poincare 279 The Pages in Euler's Introductio where e* = cos x + i sin x is Introduced 172-173 Jean Le Rond D'Alembert 183 Joseph Louis Lagrange 189 Pierre Simon Laplace 193 Jean Etienne Montucla 197 Carl Friedrich Gauss 204 Adrien Marie Legendre 210 Gaspard Monge 214 Evariste Galois 224 Georg Friedrich Bemhard Riemann 233 Karl Weierstrass 238 Georg Cantor 242 — INTRODUCTION 1. Mathematics is a vast adventure in ideas; its his- tory reflects some of the nobiest thoughts of countless generations. It was possible to condense this history into a book of less than three hundred pages only by subjecting ourselves to strict discipline, sketching the unfolding of a few main ideas and minimizing reference to other developments. Bibliographical details had to be restricted to an outline; many relatively important authors—Roberval, Lambert, Schwarz—had to be by- passed. Perhaps the most crippling restriction was the insufficient reference to the general cultural and socio- logical atmosphere in which the mathematics of a period matured—or was stifled. Mathematics has been in- fluenced by agriculture, commerce and manufacture, by warfare, engineering and philosophy, by physics and by astronomy. The influence of hydrodynamics on func- tion theory, of Kantianism and of surveying on ge- ometry, of electromagnetism on differential equations, of Cartesianism on mechanics and of scholasticism on the calculus could only be indicated in a few sentences or perhaps a few words—yet an understanding of the course and content of mathematics can only be reached if all these determining factors are taken into considera- tion. Often a reference to the literature has had to re- place an historical analysis. Our story ends by 1900, for we do not feel competent to judge the work of our contemporaries. We hope that despite these restrictions we have been able to give a fairly honest description of the main ; xii INTRODUCTION INTRODUCTION XIII trends in the development of mathematics throughout spectively, to the commercial and the engineering period. the ages and of the social and cultural setting in which interests of the exposition of Nineteenth Century mathe- it took place. The selection of the material was, of 4. Base the course, not exclusively based on objective factors, but matics on persons and schools rather than on sub- Klein's" history could be used as was influenced by the author's likes and dislikes, his jects [Here Felix guide. exposition by subjects can be knowledge and his ignorance. As to his ignorance, it a primary An Cajori and Bell, or with the was not always possible to consult all sources first-hand found in the books by too often second- or even third-hand sources had to be more technical details, in the "Encyklopajdie der Wissenschaften" (Leipzig, 1898- used. It is therefore good advice, not only with respect mathematischen in Pascal's "Repertorium der to this book, but with respect to all such histories, to 1935, 24 vols.) and check the statements as much as possible with the hoheren Mathematik" (Leipzig 1910-29, 5 vols.).] original sources. This is a good principle for more rea- sons than one. Our knowledge of authors such as Euclid, 2. We list here some of the most important books on Diophantos, Descartes, Laplace, Gauss, or Riemann the history of mathematics as a whole. Such a list is Sarton, should not be obtained exclusively from quotations or superfluous for those who can consult G. The History (Cambridge, 1936, histories describing their works. There is the same Study of the of Mathematics invigorating power in the original Euclid or Gauss as 103 pp.) which not only has an interesting introduction there is in the original Shakespeare, and there are places to our subject but also has complete bibliographical in Archimedes, in Format, or in Jacobi which are as information. beautiful as Horace or Emerson. English texts to be consulted are: R. C. Archibald, Outline of the History of Mathematics Among the principles which have led the author in (Amer. Math. Monthly 561, Jan. 1949, 6th ed.) many the presentation of his material are the following: This issue of 1 14 pp. contains an excellent summary and bibliographic references. 1. Stress the continuity and affinity of the Oriental civilizations, rather than the mechanical divisions F. Cajori, A History of Mathematics (London, 2nd between Egyptian, Babylonian, Chinese, Indian, ed., 1938). and Arabian cultures. This is a standard text of 514 pp. 2. Distinguish between established fact, hypothesis, !>• E. Smith, History Mathematics (London, 1923- and tradition, especially in Greek mathematics. of 25, 2 vols.). 3. Relate the two trends in Renaissance mathematics, This book is mainly restricted to elementary mathematics but the arithmetical-algebraic and the "fluctional," re- : riv INTRODUCTION INTRODUCTION XV has references concerning all leading mathematicians. It contains und MittelaUer (Copenhagen, 1896; French ed., many illustrations. Paris, 1902). E. T. Bell, Men. of Mathematics (Pelican Books, 1953). H. G. Zeuthen, Geschichte der Mathematik im XVI und Jahrhundert (Leipzig, 1903). E. T. Bell, The Development of Mathematics (New York- XVII London, 2nd ed., 1945). S. Giinther—H. Wieleitner, Geschichte der Mathematik These books two contain a wealth of material, both on the mathe- (Leipzig, 2 vols.). I (1908) written by Giinther; II maticians and on their works. The emphasis of the second book (1911-21, 2 parts) written by Wieleitner. Ed. by is on modern mathematics. Wieleitner (Berlin, 1939). Dealing mainly with elementary mathematics are: J. Tropfke, Geschichte der Elementar-Mathematik (Leip- V. Sanford, A Short History of Mathematics (London, zig, 2d ed., 1921-24, 7 vols.; vols. I-IV in 3d ed. 1930). 1930-34). W. W. Rouse Ball, A Short Account of the History of Die Kultur der Gegenwart III, 1 (Leipzig-Berlin, 1912); Mathematics (London, 6th. ed., 1915). contains: An older, very readable but antiquated text. H. G. Zeuthen, Die Mathematik im Altertum und im MittelaUer; A. Voss, Die Beziehungen der Mathematik zur allgemeinen Kultur; The standard work the history of is on mathematics H. E. Timerding, Die Verbreitung mathemalUchen Wissene und still: mathematUcher Auffassung. M. Cantor, Vorlesungen iiber Geschichte der Mathematik (Leipzig, 1900-08, 4 vols.). The oldest text book in the history of mathematics is in French: This enormous work, of which the fourth volume was written by a group of specialists under Cantor's direction, covers the history J. E. Montucla, Histoire des matMmatiques (Paris, new of mathematics until 1799. It is here and there antiquated and ed., 1799-1802, vols.). often incorrect in details, but it remains a good book for a first 4 This orientation. book, first published in 1758 (2 vols.), also deals with applied mathematics. It is still good reading. Corrections by G. Enestrdm a. o. in the volumes of "Bibliotheca mathematica." A good Italian book is: Other German books are G - Loria, Storia delle matematiche (Turin, 1929-33, H. G. Zeuthen, Geschichte der Mathematik im Altertum 3 vols.). : XVI INTRODUCTION INTRODUCTION xvu There also exist anthologies of mathematical works: (Milan, 1930, 2 vols.) ; German edition (previously published Leipzig, 1910-11, 2 vols.).