A Short Account of the History of Mathematics, by W. W. Rouse Ball pot

The Project Gutenberg EBook of A Short Account of the History of Mathematics, by W. W. Rouse Ball This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A Short Account of the History of Mathematics Author: W. W. Rouse Ball Release Date: May 28, 2010 [EBook #31246] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK MATHEMATICS *** A SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS BY W. W. ROUSE BALL FELLOW OF TRINITY COLLEGE, CAMBRIDGE DOVER PUBLICATIONS, INC. NEW YORK This new Dover edition, first published in 1960, is an unabridged and unaltered republication of the author’s last revision—the fourth edition which appeared in 1908. International Standard Book Number: 0-486-20630-0 Library of Congress Catalog Card Number: 60-3187 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N. Y. 10014 Produced by Greg Lindahl, Viv, Juliet Sutherland, Nigel Blower and the Online Distributed Proofreading Team at http://www.pgdp.net Transcriber’s Notes A small number of minor typographical errors and inconsistencies have been corrected. References to figures such as “on the next page” have been re- placed with text such as “below” which is more suited to an eBook. Such changes are documented in the L A T E X source: %[**TN: text of note] PREFACE. The subject-matter of this book is a historical summary of the development of mathematics, illustrated by the lives and discoveries of those to whom the progress of the science is mainly due. It may serve as an introduction to more elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the time, to study it systematically may yet desire to know. The first edition was substantially a transcript of some lectures which I delivered in the year 1888 with the object of giving a sketch of the history, previous to the nineteenth century, that should be intelli- gible to any one acquainted with the elements of mathematics. In the second edition, issued in 1893, I rearranged parts of it, and introduced a good deal of additional matter. The scheme of arrangement will be gathered from the table of con- tents at the end of this preface. Shortly it is as follows. The first chapter contains a brief statement of what is known concerning the mathemat- ics of the Egyptians and Phoenicians; this is introductory to the history of mathematics under Greek influence. The subsequent history is di- vided into three periods: first, that under Greek influence, chapters ii to vii; second, that of the middle ages and renaissance, chapters viii to xiii; and lastly that of modern times, chapters xiv to xix. In discussing the mathematics of these periods I have confined my- self to giving the leading events in the history, and frequently have passed in silence over men or works whose influence was comparatively unimportant. Doubtless an exaggerated view of the discoveries of those mathematicians who are mentioned may be caused by the non-allusion to minor writers who preceded and prepared the way for them, but in all historical sketches this is to some extent inevitable, and I have done my best to guard against it by interpolating remarks on the progress PREFACE v of the science at different times. Perhaps also I should here state that generally I have not referred to the results obtained by practical as- tronomers and physicists unless there was some mathematical interest in them. In quoting results I have commonly made use of modern no- tation; the reader must therefore recollect that, while the matter is the same as that of any writer to whom allusion is made, his proof is sometimes translated into a more convenient and familiar language. The greater part of my account is a compilation from existing histo- ries or memoirs, as indeed must be necessarily the case where the works discussed are so numerous and cover so much ground. When authori- ties disagree I have generally stated only that view which seems to me to be the most probable; but if the question be one of importance, I believe that I have always indicated that there is a difference of opinion about it. I think that it is undesirable to overload a popular account with a mass of detailed references or the authority for every particular fact mentioned. For the history previous to 1758, I need only refer, once for all, to the closely printed pages of M. Cantor’s monumental Vorlesungen ¨uber die Geschichte der Mathematik (hereafter alluded to as Cantor), which may be regarded as the standard treatise on the subject, but usually I have given references to the other leading authorities on which I have relied or with which I am acquainted. My account for the period subsequent to 1758 is generally based on the memoirs or monographs referred to in the footnotes, but the main facts to 1799 have been also enumerated in a supplementary volume issued by Prof. Cantor last year. I hope that my footnotes will supply the means of studying in detail the history of mathematics at any specified period should the reader desire to do so. My thanks are due to various friends and correspondents who have called my attention to points in the previous editions. I shall be grateful for notices of additions or corrections which may occur to any of my readers. W. W. ROUSE BALL. TRINITY COLLEGE, CAMBRIDGE. NOTE. The fourth edition was stereotyped in 1908, but no material changes have been made since the issue of the second edition in 1893, other duties having, for a few years, rendered it impossible for me to find time for any extensive revision. Such revision and incorporation of recent researches on the subject have now to be postponed till the cost of printing has fallen, though advantage has been taken of reprints to make trivial corrections and additions. W. W. R. B. TRINITY COLLEGE, CAMBRIDGE. vi vii TABLE OF CONTENTS. page Preface . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . vii Chapter I. Egyptian and Phoenician Mathematics. The history of mathematics begins with that of the Ionian Greeks . . 1 Greek indebtedness to Egyptians and Phoenicians . . . . . 1 Knowledge of the science of numbers possessed by the Phoenicians . . 2 Knowledge of the science of numbers possessed by the Egyptians . . 2 Knowledge of the science of geometry possessed by the Egyptians . . 4 Note on ignorance of mathematics shewn by the Chinese . . . . 7 First Period. Mathematics under Greek Influence. This period begins with the teaching of Thales, circ. 600 b.c., and ends with the capture of Alexandria by the Mohammedans in or about 641 a.d. The characteristic feature of this period is the development of geometry. Chapter II. The Ionian and Pythagorean Schools. Circ. 600 b.c.–400 b.c. Authorities . . . . . . . . . . . . 10 The Ionian School . . . . . . . . . . . 11 Thales, 640–550 b.c. . . . . . . . . . . 11 His geometrical discoveries . . . . . . . . 11 His astronomical teaching . . . . . . . . . 13 Anaximander. Anaximenes. Mamercus. Mandryatus . . . . 14 The Pythagorean School . . . . . . . . . . 15 Pythagoras, 569–500 b.c. . . . . . . . . . 15 The Pythagorean teaching . . . . . . . . . 15 The Pythagorean geometry . . . . . . . . 17 TABLE OF CONTENTS viii The Pythagorean theory of numbers . . . . . . . 19 Epicharmus. Hippasus. Philolaus. Archippus. Lysis . . . . . 22 Archytas, circ. 400 b.c. . . . . . . . . . . 22 His solution of the duplication of a cube . . . . . . 23 Theodorus. Timaeus. Bryso . . . . . . . . . 24 Other Greek Mathematical Schools in the Fifth Century b.c. . . . 24 Oenopides of Chios . . . . . . . . . . . 24 Zeno of Elea. Democritus of Abdera . . . . . . . . 25 Chapter III. The Schools of Athens and Cyzicus. Circ. 420–300 b.c. Authorities . . . . . . . . . . . . 27 Mathematical teachers at Athens prior to 420 b.c. . . . . . 27 Anaxagoras. The Sophists. Hippias (The quadratrix). . . . 27 Antipho . . . . . . . . . . . . 29 Three problems in which these schools were specially interested . . 30 Hippocrates of Chios, circ. 420 b.c. . . . . . . . 31 Letters used to describe geometrical diagrams . . . . . 31 Introduction in geometry of the method of reduction . . . 32 The quadrature of certain lunes . . . . . . . . 32 The problem of the duplication of the cube . . . . . 34 Plato, 429–348 b.c. . . . . . . . . . . . 34 Introduction in geometry of the method of analysis . . . . 35 Theorem on the duplication of the cube . . . . . . 36 Eudoxus, 408–355 b.c. . . . . . . . . . . 36 Theorems on the golden section . . . . . . . . 36 Introduction of the method of exhaustions . . . . . 37 Pupils of Plato and Eudoxus . . . . . . . . . 38 Menaechmus, circ. 340 b.c. . . . . . . . . . 38 Discussion of the conic sections . . . . . . . . 38 His two solutions of the duplication of the cube . . . . 38 Aristaeus. Theaetetus . . . . . . . . . . 39 Aristotle, 384–322 b.c. . . . . . . . . . . 39 Questions on mechanics. Letters used to indicate magnitudes . . . 40 Chapter IV. The First Alexandrian School. Circ. 300–30 b.c. Authorities . . . . . . . . . . . . 41 Foundation of Alexandria . . . . . . . . . . 41 The Third Century before Christ . . . . . . . . 43 Euclid, circ. 330–275 b.c. . . . . . . . . . 43 Euclid’s Elements . . . . . . . . . . 44 The Elements as a text-book of geometry . . . . . . 44 The Elements as a text-book of the theory of numbers . . . 47 TABLE OF CONTENTS ix Euclid’s other works . . . . . . . . . . 49 Aristarchus, circ. 310–250 b.c. . . . . . . . . . 51 Method of determining the distance of the sun . . . . . 51 Conon. Dositheus. Zeuxippus. Nicoteles . . . . . . . 52 Archimedes, 287–212 b.c. . . . . . . . . . 53 His works on plane geometry . . . . . . . . 55 His works on geometry of three dimensions . . . . . 58 His two papers on arithmetic, and the “cattle problem” . . . 59 His works on the statics of solids and fluids . . . . . 60 His astronomy . . . . . . . . . . . 63 The principles of geometry assumed by Archimedes . . . . 63 Apollonius, circ. 260–200 b.c. . . . . . . . . 63 His conic sections . . . . . . . . . . 64 His other works . . . . . . . . . . . 66 His solution of the duplication of the cube . . . . . 67 Contrast between his geometry and that of Archimedes . . . 68 Eratosthenes, 275–194 b.c. . . . . . . . . . 69 The Sieve of Eratosthenes . . . . . . . . . 69 The Second Century before Christ . . . . . . . . 70 Hypsicles (Euclid, book xiv). Nicomedes. Diocles . . . . . 70 Perseus. Zenodorus . . . . . . . . . . . 71 Hipparchus, circ. 130 b.c. . . . . . . . . . 71 Foundation of scientific astronomy . . . . . . . 72 Foundation of trigonometry . . . . . . . . 73 Hero of Alexandria, circ. 125 b.c. . . . . . . . . 73 Foundation of scientific engineering and of land-surveying . . . 73 Area of a triangle determined in terms of its sides . . . . 74 Features of Hero’s works . . . . . . . . . 75 The First Century before Christ . . . . . . . . 76 Theodosius . . . . . . . . . . . . 76 Dionysodorus . . . . . . . . . . . . 76 End of the First Alexandrian School . . . . . . . . 76 Egypt constituted a Roman province . . . . . . . 76 Chapter V. The Second Alexandrian School. 30 b.c.–641 a.d. Authorities . . . . . . . . . . . . 78 The First Century after Christ . . . . . . . . . 79 Serenus. Menelaus . . . . . . . . . . . 79 Nicomachus . . . . . . . . . . . . 79 Introduction of the arithmetic current in medieval Europe . . 79 The Second Century after Christ . . . . . . . . 80 Theon of Smyrna. Thymaridas . . . . . . . . . 80 Ptolemy, died in 168 . . . . . . . . . . 80 The Almagest . . . . . . . . . . . 80 Ptolemy’s astronomy . . . . . . . . . . 80 [...]... find the height of a pyramid In a dialogue given by Plutarch, the speaker, addressing Thales, says, “Placing your stick at the end of the shadow of the pyramid, you made by the sun’s rays two triangles, and so proved that the [height of the] pyramid was to the [length of the] stick as the shadow of the pyramid to the shadow of the stick.” It would seem that the theorem was unknown to the Egyptians, and... from the (Aryan) Hindoos Arya-Bhata, circ 530 His algebra and trigonometry (in his Aryabhathiya) Brahmagupta, circ 640 His algebra and geometry (in his Siddhanta) Bhaskara, circ 1140 The Lilavati or arithmetic; decimal numeration used The Bija Ganita or algebra Development of Mathematics in Arabia ¯ Alkarismi or Al-Khwarizm¯ circ 830 i, His Al-gebr we’ l mukabala His... swept away the landmarks in the valley of the river, and by altering its course increased or decreased the taxable value of the adjoining lands) rendered a tolerably accurate system of surveying indispensable, and thus led to a systematic study of the subject by the priests We have no reason to think that any special attention was paid to geometry by the Phoenicians, or other neighbours of the Egyptians... place, finding that one of the most important government departments was known as the Board of Mathematics, they supposed that its function was to promote and superintend mathematical studies in the empire Its duties were really confined to the annual preparation of an almanack, the dates and predictions in which regulated many a airs both in public and domestic life All extant specimens of these almanacks... historical references We have also a fragment of the General View of Mathematics written by Geminus about 50 b.c., in which the methods of proof used by the early Greek geometricians are compared with those current at a later date In addition to these general statements we have biographies of a few of the leading mathematicians, and some scattered notes in various writers in which allusions are made to the. .. period Modern Mathematics This period begins with the invention of analytical geometry and the infinitesimal calculus The mathematics is far more complex than that produced in either of the preceding periods: but it may be generally described as characterized by the development of analysis, and its application to the phenomena of nature Chapter XIV The History of Modern Mathematics Treatment of the subject... marking the lines of shadow cast by the style at sunrise and sunset on the same day, and taking the plane bisecting the angle so formed); and thence, by observing the time of year when the noon-altitude of the sun was greatest and least, he got the solstices; thence, by taking half the sum of the noon-altitudes of the sun at the two solstices, he found the inclination of the equator to the horizon (which... its translation it was commonly thought that these statements exaggerated the acquirements of the Egyptians, and its discovery must increase the weight to be attached to the testimony of these authorities We know nothing of the applied mathematics (if there were any) of the Egyptians or Phoenicians The astronomical attainments of the Egyptians and Chaldaeans were no doubt considerable, though they were... almanacks are defective and, in many respects, inaccurate The only geometrical theorem with which we can be certain that the ancient Chinese were acquainted is that in certain cases (namely, √ when the ratio of the sides is 3 : 4 : 5, or 1 : 1 : 2) the area of the square described on the hypotenuse of a right-angled triangle is equal to the sum of the areas of the squares described on the sides It is barely... B¨ckh says that they o regularly supplied the weights and measures used in Babylon Now the Chaldaeans had certainly paid some attention to arithmetic and geometry, as is shown by their astronomical calculations; and, whatever was the extent of their attainments in arithmetic, it is almost certain that the Phoenicians were equally proficient, while it is likely that the knowledge of the latter, such as it . The Project Gutenberg EBook of A Short Account of the History of Mathematics, by W. W. Rouse Ball This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever elaborate works on the subject, but primarily it is intended to give a short and popular account of those leading facts in the history of mathematics which many who are unwilling, or have not the. deal of additional matter. The scheme of arrangement will be gathered from the table of con- tents at the end of this preface. Shortly it is as follows. The first chapter contains a brief statement