Cajori History of mathematics 48 00376 4479 MA MATHEMATICS BY PLORIAN CAJORI FOHMEBLY PROFESSOR 03T APPLIED MATHEMATICS IN THE TULANE UNIVERSITY OF LOUISIANA; NOW PROCESSOR or PHYSICS IN COLORADO COLLEGE * I am sure that no subject loses more than mathematics "by any attempt to dissociate it from, its history." J. W. L. GLAISHMR gatfc MACMILLAN AND CO. ANB LONDON 1894 All riff Ms reserved COPYRIGHT, 189,1, BY MAOM1LLAN AND 00, * ITortooob prt ; J. S, Gushing & Co, -Berwick & Smith, Boston, Mass,, U.S.A. PREFACE. AN increased interest in the history of the exact sciences manifested in recent years by teachers everywhere, and the attention given to historical inquiry in the mathematical class-rooms and seminaries of our leading universities, cause me to believe that a brief general History of Mathematics will be found acceptable to teachers and students. The pages treating necessarily in a very condensed form of the progress made during the present century, are put forth with great diffidence, although I have spent much time in the effort to render them accurate and reasonably complete. Many valuable suggestions and criti cisms on the chapter on "B/ecent Times" have been made by ,I)r. E. W. Davis, of the University of Nebraska. The proof-shoots o f this chapter have also been submitted to Dr. J. E. Davies and Professor C. A. Van Velzer, both of the University of Wisconsin; to Dr. G-. B. Halsted, of the University of Texas ; Professor L. M. HosMns, of the Leland Stanford Jr. University ; and Professor Gr. D. Olds, of Amherst College, all of whom have afforded valuable assistance. 1 am specially indebted to Professor 3T. H. Loud, of Colorado College, who has read the proof-sheets throughout. To all the gentlemen above named, as well as to Dr. Carlo Veneziani v vi PKEFACE. of Salt Lake City, who read the first part of my work in. manuscript, I desire to express my hearty thanks. But in acknowledging their kindness, I trust that I shall not seem to lay upon them any share in the responsibility for errors which I may have introduced in subsequent revision of the FLORIAN CAJOBL COLORADO COLLEGE, December, 1893. TABLE OF CONTENTS. PAGE INTRODUCTION 1 , ANTIQUITY 5 THE BABYLONIANS 5 THE EGYPTIANS 9 THE GREEKS 16 Greek Geometry 16 The Ionic School 17 The School of Pythagoras 19 The Sophist School 23 The Platonic School 29 The First Alexandrian School 34 The Second Alexandrian School 54 Greek Arithmetic 63 TUB ROMANS 77 ^ MIDDLE AGES 84 THE HINDOOS 84 THE ARABS 100 EtJBOPE DURING THE MIDDLE AOES 117 Introduction of Roman Mathematics 117 Translation of Arabic Manuscripts 124 The First Awakening and its Sequel 128 MODERN EUROPE 138 THE RENAISSANCE : . . . . 189 VIETA TO DJCSOARTES ^ DBSGARTES TO NEWTON 183 NBWTGN TO EULBK 199 vii Viii TABLE OF CONTENTS. PAGE EULER, LAGRANGE, AND LAPLACE 246 The Origin of Modern Geometry 285 KECENT TIMES 291 SYNTHETIC GEOMETRY 293 ANALYTIC GEOMETRY 307 ALGEBRA 315 ANALYSIS 331 THEORY OP FUNCTIONS 347 THEORY OF NUMBERS 362 APPLIED MATHEMATICS 373 INDEX 405 BOOKS OF REFEKENCE. The following books, pamphlets, and articles have been used in the preparation of this history. Reference to any of them is made in the text by giving the respective number. Histories marked with a star are the only ones of which extensive use has been made. 1. GUNTHER, S. Ziele tmd Hesultate der neueren Mathematisch-his- torischen JForschung. Erlangen, 1876. 2. CAJTOEI, F. The Teaching and History of Mathematics in the U. S. Washington, 1890. 3. *CANToit, MORITZ. Vorlesungen uber Gfeschichte der MathematiJc. Leipzig. Bel I., 1880; Bd. II., 1892. 4. EPPING, J. Astronomisches aus Babylon. Unter Mitwirlcung von P. J. K. STUASSMAIER. Freiburg, 1889. 5. BituTHOHNKiDfflR, C. A. Die Qeometrie und die G-eometer vor Eukli- des. Leipzig, 1870. 6. * Gow, JAMES. A Short History of Greek Mathematics. Cambridge, 1884. 7. * HANKBL, HERMANN. Zur Gfeschichte der MathematiJc im Alterthum und Mittelalter. Leipzig, 1874. 8. *ALLMAN, G. J. G-reek G-eometr y from Thales to JEuclid. Dublin, 1889. 9. DB MORGAN, A. "Euclides" in Smith s Dictionary of Greek and Itoman Biography and Mythology. 10. HANKBL, HERMANN. Theorie der Complexen Zahlensysteme. Leip zig, 1807. 11. WmcwELL, WILLIAM. History of the Inductive Sciences. 12. XEUTIIISN, II. G. Die Lehre von den Kegelschnitten im Alterthum. KopQnlaagen, 1886. ix X BOOKS OF REFERENCE. 13. * CHASLES, M. G-eschichte der Geometric. Aus dem Franzosischen tibertragen durcli DR. L. A. SOHNCKE. Halle, 1839. 14. MARIE, MAXIMILIEN. Histoire des Sciences Matheniatiques et Phy siques. Tome I.-XII. Paris, 1883-1888. 15. COMTE, A. Philosophy of Mathematics, translated by W. M. GIL- LESPIE. 16. HANKEL, HERMANN. Die ISntwickelung der Mathematik in den letz- ten Jahrhunderten. Tubingen, 1884. 17. GUNTHER, SIEGMUND und WiNBELBAND, W. GesckicJite der antiJcen Naturwissenschaft und Philosophic. Nordlingen, 1888. 18. ARNETH, A. Geschichte der reinen Mathematik. Stuttgart, 1852. 19. CANTOR, MOIUTZ. Mathematische Beitrage zum Kulturleben der VoUcer. Halle, 1863. 20. MATTIIIESSEN, LTIDWIG. Grundzilge der Antiken und Modernen Algebra der Litteralen GUichungen. Leipzig, 1878. 21. OURTMANN und MULLER. Fort$chritte der Mathematik. 22. PEACOCK, GEORGE. Article " Aritlimetic, 1 in The Encyclopedia, of Pure Mathematics. London, 1847. 23. HERSCHEL, J. !F. W. Article * Mathematics," in Edinburgh Jfflncy- dopcedia. 24. SUTER, HEINRICH. Cfeschichte der Mathematischen Wissenschaften. Zurich, 1873-75. 25. QUETELET, A. Sciences Mathetna&iques et IViysiques ehe% les Beiges. Bruxelles, 1866. 26. PLAYFAIR, JOHN. Article u Progress of the Mathematical and Phys ical Sciences," in Encyclopedia Britannica, 7th editi6n, con tinued in the 8tlx edition by SIK JOHN LESLIE. 27. BE MORGAN, A. Arithmetical Books from the Invention of Printing to the Present Time. 28. NAPIER, MARK. Memoirs of John Napier of Merchiston. Edin burgh, 1834. 29. HALSTEB, G. B. "Note on the First English Euclid," American Journal of Mathematics, Vol. XL, 1879. 30. MADAME PERIER. The Life of Mr. Paschal. Translated into English by W. A., London, 1744. 31. MONTUCLA, J. F. Histoire des Mathematiques. Paps, 1802. 32. BtiHRiNG- E. Kritische Geschichte der allgemeimn Principien der Mechanik. Leipzig, 1887. 33. BREWSTER, D. The Memoirs of Nc.wton. Edinburgh, 1860. ^81. BALL, W. W. R. A Short Account of the History of Mathematics. London, 1888, 2nd edition, 189S, 35. DE MORGAN, A. "On the Early History of lEfiEitesixualB," in the Philosophical Magazine, November, 1852. BOOKS OF REFERENCE, xi 36. Bibliotheca Mathematica, herausgegeben von GUSTAP ENESTROM, Stockholm. 37. GUNTHER, SIEGMUND. Vermischte Untersuchtingen zur Geschichte der mathematischen Wissenschaften. Leipzig, 1876. 38. *GERHARDT, C. I. Geschichte der Mathematik in, Deutschland. Miinclien, 1877. 39. GERHARDT, C. I. SntdecJcung der Di/erenzialrechnung durch Leib niz. Halle, 1848. 40. GERIIARBT, K. I. " Leibniz in London," in jSitzirngsberichte der Koniglich Preussischen Academic der Wissenschaften zu Berlin, FeTbruar, 1891. 41. DB MoR(UK, A. Articles "Muxions" and u Commercimn Epistoli- cum," in tlie Penny Cyclopaedia, 42. *TODUUNTEK, I. A History of the Mathematical Theory of Probabil ity from the Time of Pascal to that of Laplace. Cambridge and London, 1865. 43. *Toi>iniNTBK, I. A History of the Theory of Elasticity and of the Strength of Materials. Edited and completed by KARL PEARSON. Cambridge, 1886. 44. TOBHUNTKR, I. " Note on tlie History of Certain Formulas in Spher ical Trigonometry," Philosophical Magazine, February, 1873. 46. Die JBasler Mathematiker, Daniel Bernoulli und Leonhard Euler. BaHol, 1884, 46. RKIFF, R. Gfeschichte der Unendlichen Heihen. Tubingen, 1889. 47. WALTKRSIIAUSKN , W. SAHTOUIXIS. Gauss , mm Q-ed&chniss. Leip zig, 1850. 48. BAUMCJART, OSWALO. Ueber das Quadratische J&eciprocitatsgesetz, Leipzig, 1885. 49. HATHAWAY, A. S. "Early History of the Potential," Bulletin of the N. Y. Mathematical Society, I. 3. 50. WOLF, RUDOLF. Cfeschichte der Astronomie. Mtinchen, 1887. 51. AUAOO, 1). F. J. " Eulogy on Laplace. 7 Translated by B. POWELL, Smitlisonian llPfiort, 1874, 52* BEAUMONT, M. I^LIK DB. "Memoir of Legendre." Translated by C. A. ALEXANDISR, Smithsonian Iteport, 1867. 58. AUAOO, I). F. X Joseph Fourier." Smithsonian Eeport, 1871. 54, WITHER, CnuiHTiAN. Lehrfatch der Darstellenden Gfeometrie. Leip zig, 1884. 55. *LoiA, GTKO. Die Ilmptsilchliehstm Theorien der Geometrie in ihrer fr dhtren und heutlgen fJntwicMnnff, ins deutsche tibertra- gen von Fitm SOHUTTB. Leipzig, 1888. Xll BOOKS OF REFERENCE. 56 . CAYLE Y, ARTHUR. Inaugural Address before the British Association, 1883. 57. SPOTTISWOODE, WILLIAM. Inaugural Address before the British Association, 1878. 58. GIBBS, J. WILLARD. " Multiple Algebra," Proceedings of the American Association for the Advancement of Science, 1886. 59. FINK, KARL. Geschichte der Elenientar-Mathematik. Tubingen, 1890. 60. WITTSTEIN, ARMIN. Zur Qeschichte des Malfatttf schen Problems. Nordlingen, 1878. 61. KLEIN, FELIX. Vergleichende Betrachtimgen uber neuere geome- trische Forschungen. Erlangen, 1872. 62. FORSYTH, A. R. Theory of Functions of a Complex Variable. Cambridge, 1893. 63. GRAHAM, R. H. Geometry of Position. London, 1891. 64. SCHMIDT, FRANZ. "Aus dem Leben zweier ungarischer Mathe- matiker Johann und Wolfgang Bolyai von Bolya." Grrunertfs Archiv, 48:2, 1868. 65. FAVARO, ANTON. Justus Bellavitis," Zeitschrift fur Mathematik und Physik, 26 : 5, 1881. 66. BRONICE, AD. Julius Plucker. Bonn, 1871. 67. BAUER, GUSTAV. Gfedachnissrede auf Otto Hesse. Miinchen, 1882. 68. ALFRED CLEBSCH. Versuch einer Darlegung und Wunligung seiner wissenschaftlichen Leistungen von einigen seiner Freunde. Leip zig, 1873. 69. HAAS, AUGUST. Versuch einer Darstellung der Geschichte dvs Krwnmungsmasses. Tubingen, 1881. 70. FINE, HENRY B. The Number- System of Algebra. Boston and New York, 1890. 71. SCHLEGEL, VICTOR. Hermann Gfrassmann, sein Leben und seine Werke. Leipzig, 1878. 72. ZAHN, W. v. " Einige Worte zum Andenkon an Hermann Ilankol," Mathematische Annalen, VII. 4, 1874. 73. MUIR, THOMAS. ^1 Treatise on Determinants*. 1882. 74. SALMON, GEORGE. "Arthur Cayley," Nature, 28:21, September, 1883. 75. CAYLEY, A. "James Joseph Sylvester," Nature, 39:10, January, 1889. 76. BURKHARDT, HEiNRicii. Die AnfUngo der Gruppontliooiie und Paolo Ikiffim," Zeitschrift der MathemaUk und Physik, Supple ment, 1892. BOOKS OF REFERENCE. xiii 77. SYLVESTER , J. J. Inaugural Presidential Address to the Mathe matical and Physical Section of the British Association at Exeter. 1869. 78. YALSON, C. A. La Vie et les travaux du Baron Cauchy. Tome I., II., Paris, 1868. 79. SACHSE, ARNOLD. Versuch einer Qeschichte der Darstellung will- kiirlicher Funktionen einer variablen durch trigonometrische Meihen. Gottingen, 1879. 80. BOIS-KEYMOND, PAUL DU. Zur G-eschichte der Trigonometrischen Heilien, Mine JBntgegnung. Tubingen. 81. POINCARE, HENRI. Notice sur les Travaux Scientifiques de Henri Poincare. Paris, 1886. 82. BJERKNES, C. A. Niels-HenriTc Abel, Tableau de sa vie et de son action scientifique. Paris, 1885. 83. TUCKER, R. "Carl Friedrich Gauss," Nature, April, 1877. 84. DIRICHLET, LEJEUNE. Gfedachnissrede auf Carl Gf-iistav Jacob Jacobi. 1852. 85. ENNEPER, ALFRED. JUlliptische JFunktionen. Theorie und Ge- schichte. Halle a/S., 1876. 86. HENRICI, O. "Theory of Functions," Nature, 43 : 14 and 15, 1891. 87. DARBOUX, GASTON. Notice stir les Travaux Scientijlques de M. Gas- ton Darboux. Paris, 1884. 88. KUMMER, E. E. Gfedachnissrede auf G-ustav Peter Lejeune-Dirichlet. Berlin, 1860. 89. SMITH, H. J. STEPHEN. "On the Present State and Prospects of Some Branches of Pure Mathematics," Proceedings of the London Mathematical Society, Vol. VIII, Nos. 104, 105, 1876. 90. GLAISUISH, J. W. L. " Henry John Stephen Smith, Monthly Notices of the Eoyal Astronomical Society, XLIV., 4, 1884. 91. Bessel als Bremer Ifandlungslehrling. Bremen, 1890. 92. FRANTZ, J*. Festrede aus Veranlassung von HesseVs hundertjahrigem Geburtstag. Konigsherg, 1884. 93. DZIOBEK, 0. Mathematical Theories of Planetary Motions. Translated into English by M. "W". Harrington and "W. J. Hussey. 94. HERMITB, Cn. "Discours prononc6 devant le president de la R6pii- Tblique," Bulletin des Sciences Mathematiques, XIV., Janvier, 1890. 95. SCHUSTER, ARTHUR. "The Influence of Mathematics on the Prog ress of Physics," Nature, 26: 17, 1882. 96. KERBEDJS, E. BE. "Sophie de KowalevsM," Bendiconti del Circolo Matematico di Palermo, V., 1891. ,97. VOIGT, W. Zum Gfeddchniss von G. Kirchhoff. Gottingen, 1888. xiv BOOKS O:F BEFBBBNCE, 08) Bdci-iER, MAXIME. u A Bit of Mathematical History," Bulletin of the 2V". T. Math. /SV>c., Vol. II., No. 5. 99. CAY:LEY, ARTHUR. Report on the Recent Progress of Theoretical Dynamics. 1857. 100. GLAZEBROOK, U. T. Report on Optical Theories. 1885. 101. ROSENBERGER, If. Geschichte tier Physik. Braunschweig, 1887-1890. A HISTORY OF MATHEMATICS. INTEODUCTION. THE contemplation of the various steps by which mankind has come into possession of the vast stock of mathematical knowledge can hardly fail to interest the mathematician. He takes pride in the fact that his science, more than any other, is an exact science, and that hardly anything ever done in jnatheBiati.es has proved to be useless. The chemist smiles at the childish, efforts of alchemists, but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of to-day. He is pleased to notice that though, in course of its develop ment, mathematics has had periods of slow growth, yet in the main it has been pre-eminently a progressive science. The history of mathematics may be instructive as well as agreeable 5 it may not only remind us of what we have, but inay also teach us how to increase our store. Says De Morgan, * The early history of the mind of men with regard to mathe matics leads us to point out our own errors; and in this * aspect it is well to pay attention to the history of mathe matics." It warns us against hasty conclusions ; it points out the importance of a good notation upon the progress of the science ; it discourages excessive specialisation on the part of 1 2 A HISTORY OF MATHEMATICS. investigators, by showing how apparently distinct brandies have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon prob lems which were, perhaps, solved long since; it discourages him from attacking an unsolved problem by the same method which has led other mathematicians to failure ; it teaches that fortifications can be taken in other ways than by direct attack, that when repulsed from a direct assault it is well to recon noitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken. 1 The importance of this strategic rule may be emphasised by citing a case in which it has been violated. (An untold amount of intellectual energy has been expended on the quadrature of the circle, yet no conquest has been made by direct assault. The circle-squarers have existed in crowds ever since the period of Archimedes. After innumerable fail ures to solve the problem at a time, even, when investigators possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, wMlo those who still persisted were completely ignorant of its Ms- tory and generally misunderstood the conditions of the prob lem.^ "Our problem," says De Morgan, "is to square the circle with the old allowance of means: Euclid s postulates and nothing more. We cannot remember an instance tyx a question to be solved by a definite method was tried by\$k6 best heads, and answered at last, by that method, after thou sands of complete failures." But progress was made on this problem by approaching it from a different direction and by newly discovered paths. Lambert proved in 1761 that ratio of the circumference of a circle to its diametot is iad0.ni- meiisurable. Some years ago, Linclomaim demonstrated that this ratio is also transcendental and that the quadrature <> the circle, by means of the ruler and compass only, is INTBODUCTION. 3 sible. He thus showed by actual proof that which keen- minded mathematicians had long suspected ; namely, that the great army of circle-squarers have, for two thousand years, been assaulting a fortification which is as indestructible as the firmament of heaven. Another reason for the desirability of historical study is the value of historical knowledge to the teacher of mathe matics. The interest which pupils take in their studies may bo greatly increased if the solution of problems and the cold logic of geometrical demonstrations arc interspersed with historical remarks and anecdotes. A class in arithmetic will be pleased to hear about the Hindoos and their invention of the " Arabic notation " ; they will marvel at the thousands of years which elapsed before people had even thought of introducing into the numeral notation that Coluni bus-egg the zero j they will find it astounding that it should have taken so long to invent a notation which they themselves can now learn in a month. After the pupils have learned how to bisect a given angle, surprise them by telling of the many futile attempts which have been made to solve, by elementary geometry, the apparently very simple problem of the trisec- tion of an angle. When they know how to construct a square whose area is double the area of a given square, tell them about the duplication of the cube how the wrath of ^Apollo could be appeased only by the construction of a cubical altar double the given altar, and how mathematicians long wrestled with this problem. After the class have exhausted their ener gies on the theorem of the right triangle, tell them something about its discoverer how Pythagoras, jubilant over his great accomplishment, sacrificed a hecatomb to the Muses who in- him. When the value of mathematical training is in question, quote the inscription over the entrance into i academy of Plato, the philosopher : " Let no one who is 4 A HISTORY OF MATHEMATICS. unacquainted with geometry enter here." Students in analyt^ ical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Kewton, Leibniz, and Lagrange played in creating that science. In his historical talk it is possible for the teacher to make it plain to the student that mathematics is not a dead science, but a living one in which steady progress is made. 2 The history of mathematics is important also as a valuable contribution to the history of civilisation. Human progress is closely identified with scientific thought. Mathematical and physical researches are a reliable record of intellectual progress. The history of mathematics is one of the large windows through which the philosophic eye looks into past ages and traces the line of intellectual development. ANTIQUITY, THE BABYLONIANS, THE fertile valley of the Euphrates and Tigris was one of the primeval seats of human society. Authentic history of the peoples inhabiting this region begins only with the foun dation, in Chaldaoa and Babylonia, of a united kingdom out of tho previously disunited tribes. Much light has been thrown on their history by the discovery of the art of reading the cuneiform or wedge-shaped system of writing. In the study of Babylonian mathematics we begin with the notation of numbers, A vertical wedge If stood for 1, while the I characters" ^ and y>*. signified 10 and 100 respec tively. G-rotefend believes the character for 10 originally to been the picture of two hands, as held in prayer, the palniis being pressed together, the fingers close to each other, btiTOhe thumbs thrust out, In the Babylonian notation two ptincjiiples were employed the ^ditive) and multiplica tive. |i Numbers below 100 were expressed by symbols whose respt-Mctive values had to be added. ^ Thus, y stood for 2, |f )f | |or 3, XJJ1 for 4, <* for 23, ^ ^ < for 30 J Here the of higher order appear always to the left of those of I order. In writing the hundreds, on the other hand, a Ir symbol was placed to the left of the 100, and was, in fjase, to be multiplied by 100. Thus, s y ^^ signified the eaf 5 6 A HISTOKY OF MATHEMATICS. 10 times 100, or 1000. But this symbol for 1000 was itself taken for a new unit, which could take smaller coefficients to its left. Thus, ^ ^ f >*" denoted, not 20 times 100, but 10 times 1000. Of the largest numbers written in cuneiform symbols, which have hitherto been found, none go as high as a million. 3 If, as is believed by most specialists, the early Sumerians were the inventors of the cuneiform writing, then they were, in all probability, also familiar with the notation of numbers. Most surprising, in this connection, is the fact that Sumerian inscriptions disclose the use, not only of the above decimal system, biit also of a sexagesimal one. The latter was used chiefly in constructing tables for weights and measures. It is full of historical interest. Its consequential development, both for integers and fractions, reveals a high degree of mathematical insight. We possess two Babylonian tablets which exhibit its use. One of them, probably written between 2300 and 1600 B.C., contains a table of square numbers up to 601 The numbers 1, 4, 9, 16, ,25, 36, 49, are given as v the squares of the first seven integers respectively. We have next 1.4 = 8 s , 1.21 = 9 2 , 1.40 = 10 2 , 2.1 = 11*, etc. This"reinLfta unintelligible, taxless we assume the sexagesimal scale, wl xioh makes 1.4 = 60 + 4, 1.21 = 60 + 21, 2.1 = 2.60 + 1. The i tablet records the magnitude of the illuminated portion of moon s disc for every day from new to full moon, the wlxol being assumed to consist of 240 parts, The illuminated during the first five days are the series 5, 10/ 20, 40, (=80), which is a geometrical progression. From the series becomes an arithmetical progression, the from the fifth to the fifteenth day being respectively 1,20! 1.62, 2.8, 2.24, 2.40, 2.66, 3.12, 3.28, 3,44, 4. This only exhibits the use of the sexagesimal system, but cates the acquaintance of the Babylonians with THE BABYLONIANS. 7 Not to be overlooked is the fact that in the sexagesimal nota-. tion of integers the "principle of position" was employed. Thus, in 1.4 (=64), the 1 is made to stand for 60, the unit of the second order, by virtue of its position with respect to the 4. The introduction of this principle at so early a date is the more remarkable, because in the decimal notation it was not introduced till about the fifth or sixth century after Christ. The principle of position, in its general and syste matic application, requires a symbol for zero. We ask, Did the Babylonians possess one? Had they already taken the gigantic step of representing by a symbol the absence of units? Neither of the above tables answers this question, for they happen to contain no number in which there was occasion to use a zero. The sexagesimal system was used also in fractions. Thus, in the Babylonian inscriptions, | and | are designated by 30 and 20, the reader being expected, in his mind, to supply the word " sixtieths." The Greek geom eter Hypsicles and the Alexandrian astronomer Ptolemaeus borrowed the sexagesimal notation of fractions from the Babylonians and introduced it into Greece. From that time sexagesimal fractions held almost full sway in astronomical and mathematical calculations until the sixteenth century, when they finally yielded their place to the decimal fractions. It may be asked, What led to the invention of the sexagesi mal system ? Why was it that 60 parts were selected ? To this we have no positive answer. Ten was chosen, in the decimal system, because it represents the number of fingers. But nothing of the human body could have suggested 60. Cantor offers the following theory : At first the Babylonians reckoned the year at 360 days. This led to the division of the circle into 360 degrees, each degree representing the daily amount of the supposed yearly revolution of the sun around the earth. Now they were, very probably, familiar with the 8 A HISTOKY OF MATHEMATICS. fact that the radius can be applied to its cir%umference as a chord 6 times, and that each of these chords subtends an arc measuring exactly 60 degrees. Fixing their attention upon these degrees, the division into 60 parts may have suggested itself tp them. Thus, when greater precision necessitated a subdivision of the degree, it was partitioned into 60 minutes. In this way the sexagesimal notation may have originated. The division of the day into 24 hours, and of the hour into minutes and seconds on the scale of 60, is due to the Babylonians. It appears that the people in the Tigro-Exiphrates basin had made very creditable advance in arithmetic. Their knowledge of arithmetical and geometrical progressions has already been alluded to. lamblichus attributes to them also a knowledge of proportion, and even the invention of the so-called musical proportion. Though we possess- no conclusive proof, we have nevertheless reason to believe that in practical calculation they used the abacus. Among the races of middle Asia, even as far as China, the abacus is as old as fable. Now, Babylon, was once a great commercial centre, the -metropolis of many nations, and it is, therefore, not unreasonable to suppose that her merchants employed this most improved aid to calculation, In geometry the Babylonians accomplished almost nothing. Besides the division of the circumference into 6 parts by its radius, and into 360 degrees, they had some knowledge of geometrical figures, such as the triangle and quadrangle, which they used in their auguries. Like the Hebrews (1 Kin, 7 : 23), they took w = 3. Of geometrical demonstrations there is^ of course, no trace. "As a rule, in the Oriental mind the intui tive powers eclipse the severely rational and logical." The astronomy of the Babylonians has attracted much attention. They worshipped the heavenly bodies from tie earliest historic times, When Alexander the Great, after THE EGYPTIANS. 9 the battle of Arbela (331 B.C.), took possession of Babylon, Callisthenes found there on burned brick astronomical records reaching back as far as 2234 B.C. Porphyrius says that these were sent to Aristotle. Ptolemy, the Alexandrian astrono mer, possessed a Babylonian record of eclipses going back to 747 B.C. Eecently Epping and Strassmaier 4 threw considera ble light on Babylonian chronology and astronomy by explain ing two calendars of the years 123 B.C. and 111 B.C., taken from cuneiform tablets coining, presumably, from an old observatory. These scholars have succeeded in giving an account of the Babylonian calculation of the new and full moon, and have identified by calculations the Babylonian names of the planets, and of the twelve zodiacal signs and twenty-eight normal stars which correspond to some extent with the twenty-eight naksJiatras of the Hindoos. We append part of an Assyrian astronomical report, as translated by Oppert : "To the King, my lord, thy faithful servant, Mar-Istar." " . . . On the first day, as the new moon s day of the month Tham- muz declined, the moon was again visible over the planet Mercury, as I had already predicted to my master the King, I erred not." THE EGYPTIANS. Though there is great difference of opinion regarding the antiquity of Egyptian civilisation, yet all authorities agree in the statement that, however far back they go, they find no uncivilised state of society. " Menes, the first king, changes the course of the Wile, makes a great reservoir, and builds the temple of Phthah at Memphis." The Egyptians built the pyramids at a very early period. Surely a people engaging in 10 A HISTOBY OF MATHEMATICS. enterprises of such magnitude must have known something of mathematics at least of practical mathematics. All Greek writers are unanimous in ascribing, without envy, to Egypt the priority of invention in the mathematical sciences. Plato in Pho&drus says : " At the Egyptian city of Naucratis there was a famous old god whoso name was Theuth; the bird which is called the Ibis was sacred to him, and he was the inventor of many arts, such as arithmetic and calculation and geometry and astronomy and draughts and dice, but his great discovery was the use of letters/ 15 Aristotle says that mathematics had its birth in Egypt, because there the priestly class had the leisure needful for the study of it. Geometry, in particular, is said by Herodotus, Diodorus, Diogenes Laertius, lamblichus, and other ancient writers to have originated in Egypt. 5 In Herodotus wo find this (II. c. 109) : " They said also that this king [Sesostjris] divided the land among all Egyptians so as to give each 0110 a quadrangle of equal size and to draw from each Ms revenues, by imposing a tax to be levied yearly. But every one from whose part the river tore away anything, had to go to hh^i and notify what had happened; lie then sent the overseers, who had to measure out by how much th0 land lxad> become smaller,, in order that the owner might pay on what was left, in/ proportion to the entire tax imposed, Iti this wny/ifc appears to me, geometry originated, which passed thence to Hellas." "-y We abstain from introducing additional Greek opinion regarding Egyptian mathematics, or from indulging in wild conjectures. We rest our account on documentary evidottc^. A hieratic papyrus, included in the Rhine! collection of tha British Museum, was deciphered by Eisenlohr in 1877, and found to be a mathematical manual containing problems in arithmetic and geometry. It was written by Ataw THE GREEKS. 17 left behind no written records of their discoveries. A full jdstory of Greek geometry and astronomy during this period, written by Eudenus, a pupil of Aristotle, has been lost. It was well known to Proclus, who, in his commentaries on Euclid, gives a brief account of it. This abstract constitutes our most reliable information. We shall quote it frequently under the name of Eudemian Summary. The Ionic School To Thales of Miletus (640-546 B.C.), one of the "seven wise men," and the founder of the Ionic school, falls the honour of having introduced the study of geometry into Greece. During middle life he engaged in commercial pursuits, which took him to Egypt. He is said to have resided there, and to have studied the physical sciences and mathematics with the Egyp tian priests. Plutarch declares that Thales soon excelled his masters, and amazed King Amasis by measuring the heights of the pyramids from their shadows. According to Plutarch, this was dono by considering that the shadow cast by a verti cal staff of known length bears the same ratio to the shadow of the pyramid as the height of the staff bears to the height of the pyramid. This solution presupposes a knowledge of proportion ? and the Ahmes papyrus actually shows that the rudiments of proportion were known to the Egyptians. Ac cording to Diogenes Laertius, the pyramids were measured by Thales in. a different way ; viz. by finding the length of the shadow of the pyramid at the moment when the shadow of a staff was 0(jual to its own length. The JSud&mian Summary ascribes to Thales the invention of the theorems on the equality of vertical angles, the equality af the angles at the base of an isosceles triangle, the bisec tion of a circle by any diameter, and the congruence of two 18 A HISTORY OF MATHEMATICS. triangles having a side and the two adjacent angles equal re spectively. The last theorem he applied to the measurement of the distances of ships from the shore. Thus Thales was the first to apply theoretical geometry to practical uses. The theorem that all angles inscribed in a semicircle are right angles is attributed by some ancient writers to Thales, by others to Pythagoras. Thales was doubtless familiar with other theorems, not recorded by the ancients. It has been inferred that he knew the sum of the three angles of a tri angle to be equal to two right angles, and the sides of equi angular triangles to be proportional. 8 The Egyptians must have made use of the above theorems on the straight line, in some of their constructions found in the Ahmes papyrus, but it was left for the Greek philosopher to give these truths, which others saw, but did not formulate into words, an explicit, abstract expression, and to put into scientific lan guage and subject to proof that which others merely felt to be true. Thales may be said to have created the geometry of lines, essentially abstract in its character, while the. Egyp tians studied only the geometry of surfaces and the rudiments of solid geometry, empirical in their character. 8 With Thales begins also the study of scientific astronomy. He acquired great celebrity by the prediction of a solar eclipse in 585 B.C. Whether he predicted the day of the occurrence, or simply the year, is not known. It is told of him that while contemplating the stars during an evening walk, he fell into a ditch. The good old woman attending him exclaimed, "How canst thou know what is doing in the heavens, when thou seest not what is at thy feet ? " The two most prominent pupils of Thales were Anaximander (b. 611 B.C.) and Anaximenes (b. 570 B.C.). They studied chiefly astronomy and physical philosophy. Of Anaxagoras, ;a pupil of Anaximenes, and the last philosopher of the Ionic THE GREEKS. 19 school, we know little, except that, while in prison, he passed his time attempting to square the circle. This is the first time, in the history of mathematics, that we find mention of the famous problem of the quadrature of the circle, that rock upon which so many reputations have been destroyed. It turns upon the determination of the exact value of IT. Approx imations to TT had been made by the Chinese, Babylonians, Hebrews, and Egyptians. But the invention of a method to find its exact value, is the knotty problem which has engaged the attention of many minds from the time of Anaxagoras down to our own. Anaxagoras did not of er any solution of it, und seems to have luckily escaped paralogisms. About the time of Anaxagoras, but isolated from the Ionic school, flourished (Enopides of Chios. Proclus ascribes to him the solution of the following problems : From a point without, to draw a perpendicular to a given line, and to draw an angle on. a line equal to a given augle. That a man could gain a reputation by solving problems so elementary as these, indi- eates that geometry was still in its infancy, and that the Greeks had not yet gotten far beyond the Egyptian con structions. The Ionic school lasted over one hundred years. The pt ogress of mathematics during that period was slow, as compared with its growth in a later epoch of Greek history. A new impetus to its progress was given by Pythagoras. TJie School of Pythagoras. Pyrthagoras (580 ?-500? B.C.) was one of those figures which impressed the imagination of succeeding /ffmes to such an eitenlt that their real histories have become difficult to be ,d&a| med through the mythical haze that envelops them. The jtello^dng account of Pythagoras excludes the most doubtful 20 A HISTORY OF MATHEMATICS. statements. He was a native of Samos, and was drawn by the fame of Pherecydes to the island of Syros. He then visited the ancient Thales, who incited him to stndy in Egypt. He sojourned in Egypt many years, and may have, visited Babylon. On his return to Samos, he found it under the tyranny of Polycrates. Failing in an attempt to found a school there, he quitted home again and, following the current of civilisation, removed to Magna Grsecia in South Italy. He settled at Croton, and founded the famous Pythagorean school. This was not merely an academy for the teaching of philosophy, mathematics, and natural science, but it was a brotherhood, the members of which were united for life. This brotherhood l|ad observances "approaching masonic peculiarity. Thejr wore forbidden to divulge the discoveries and doctrines of their school. Hence we are obliged to speak of the Pythagoreans as a body, and find it difficult to determine to whom each particular discovery is to be ascribed. The Pythagoreans themselves were in the habit of referring every discovep" back to the great founder of the sect. This school grew rapidly and gained considerable political ascendency. But the mystic and secret obseivaaptcfe, intro duced in imitation of Egyptian usages, and the a*|stooratic tendencies of the school, caused it to becoiae ai* object, of suspicion. The democratic party in Lower Itely revolted and destroyed the buildings of the Pythagorean school* ras fled to Tarentum and thence to Metapontum, murdered. Pythagoras has left behind no mathematical tventtees, and our sources of information are rather scanty. Certain it is that, in the Pythagorean school, mathematics was the study. Pythagoras raised mathematics to the taak of a so Arithmetic was courted by him as fervently ft&*geo$tetYj fact, arithmetic is the foundation of his philosophic icipal iencc. THE GrKEEKS. 21 The Eudemiart Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression. Like Egyptian geometry, the geometry of the Pythagoreans . is much concerned with areas. To Pythagoras is ascribed the important theorem that the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other , two sides/ He had probably learned from the Egyptians the truth of the theorem in the special case when the sides are 3, 4, 6, respectively. The story goes, that Pythagoras was so jubilant over this discovery that he sacrificed a hecatomb. Its authenticity is doubted, because the Pythagoreans believed in the transmigration of the soul and opposed, therefore, the shedding of blood. In the later traditions of the !N"eo-Pythago-" reans this objection is removed by replacing this bloody sacri fice by that of an ox made of flour " ! The proof of the law ,of three squares, given in Euclid s Elements, I. 47, is due to Euclid himself, and not to the Pythagoreans. What the Py thagorean method of proof was has been a favourite topic for conjecture. The theorem on the sum of the three angles of a triangle, presumably known to Thales, was proved bythe Pythagoreans after the manner of Euclid. They demonstrated also that the plane about a point is completely filled by six equilateral triangles, four squares, or three regular hexagons, so that it is possible to divide up a plane into figures of either kind. From the equilateral triangle and the square arise the solids, namely the tetraedron, octaedron, icosaedron, and the cube. These solids were, in all probability, known to the Egyptians, A HISTORY OF MATHEMATICS. excepting, perhaps, the icosaedron. In Pythagorean philos ophy, they represent respectively the four elements of the physical world; namely, fire, air, water, and earth. Later another regular solid was/ discovered, namely the dodecaedron, which, in absence of a/fifth element, was made to represent the universe itself. lamblichus states that Hippasus, a Py- thagorean, perished in the sea, "because he boasted that he first divulged " the sphere with the twelve pentagons." The star- f shaped pentagram was used as a symbol of recognition by the 1 Pythagoreans, and was called by them Health. Pythagoras called the sphere the most beautiful of all solids, and the circle the most beautifttl of all plane figures. The treatment of the subjects of proportion and of irrational quantities by him and his school will be taken up under the head of arithmetic. According to Eudemus, the Pythagoreans invented the prob-* lerns concerning the application of areas, including the cases ~f defect and excess, as in Euclid, VI. 28, 29. They were also familiar with the construction of a polygon iqual in area to a given polygon and similar to another given )olygon. This problem depends upon several important and somewhat advanced theorems, and testifies to the fact that t jhe Pythagoreans made no mean progress in geometry. Of the theorems generally ascribed to the Italian school, some cannot be attributed to Pythagoras himself, no* to his earliest successors. The progress from empirical to reasoned solutions must, of necessity, have been slow. It is worth noticing that on the circle no theorem of any importance *wa$ discovered by this school, , Though politics broke up the Pythagorean fraternity, yet the school continued to exist at least two centuries longer* Among the later Pythagoreans, Philolaus and Arckytas aw the most prominent. Philolaus wrote a book on the Pythago* THE GREEKS. 23 rean doctrines. By him were first given to tlie world tlie teachings of the Italian school, which had been kept secret for a whole century. The brilliant Archytas of Tarentum (428-347 B.C.), known as a great statesman and general, and universally admired for his virtues, was the only great geome ter among the Greeks when Plato opened his school. Archy- tas was the first to apply geometry to mechanics and to treat the latter subject methodically. He also found a very ingeni ous mechanical solution to the problem of the duplication of the cube. His solution involves clear notions on the genera tion of cones and cylinders. This problem reduces itself to finding two mean proportionals between two given lines. These mean proportionals were obtained by Archytas from the section of a half-cylinder. The doctrine of proportion was advanced through him. There is every reason to believe that the later Pythagoreans exercised a strong influence on the study and development of mathematics at Athens. The Sophists acquired geometry from Pythagorean sources. Plato bought the works of Philolaus, and had a warm friend in Archytas. The Sophist School After the defeat of the Persians under Xerxes at^the battle of Salamis, 480 B.C., a league was formed among the Greeks fco preserve the freedom of the now liberated Greek cities on bhe islands and coast of the JEgsean Sea. Of this league Athens soon became leader and dictator. She caused the separate treasury of the league to be merged into that of Athens, and then spent the money of her allies for her own tggrandisement. Athens was also a great commercial centre. Phus she became the richest and most beautiful city of an- iquity. All menial work was performed by slaves. The 24 * A HISTORY OF MATHEMATICS. citizen of Athens was well-to-do and enjoyed a large amount of leisure. The government being purely democratic, every citizen was a politician. To make his influence felt among his fellow-men he must, first of all, be educated. Thus there arose a demand for teachers. The supply came principally from Sicily, where Pythagorean doctrines had spread. These teachers were called Sophists, or "wise men." Unlike the Pythagoreans, they accepted pay for their teaching. Although rhetoric was the principal feature of their instruction, they also taught geometry, astronomy, and philosophy. Athens soon became the headquarters of Grecian men of letters, and of mathematicians in particular. The home of mathematics among the. Greeks was first in the Ionian Islands, then in Lower Italy, and during the time now under consideration, at Athens, \ The geometry of the circle, which had been entirely neglected by the Pythagoreans, was taken up by the Sophists. Nearly all their discoveries were made in connection with their innumerable attempts to solve the following three famous problems : (1) To trisect an arc or an angle. (2) To " double the cube," i.e. to find a cube whose volume is double that of a given cube. (3) To "square the circle," i.e. to find a square or some other rectilinear figure exactly equal in area to a given circle* These problems have probably been the subject of more discussion and research than any other problems m mathe matics. The bisection of an angle was one of the easiest problems in geometry. The trisection of an angle, on the other hand, presented unexpected difficulties. A right iwagle had been divided into three equal parts by the Pythagoreans, But the general problem, though easy in appearanee^ tran scended the power, of elementary geometry. Among the firfit THE GREEKS. 25 fco wrestle with it was Hippias of Blis, a contemporary of Socrates, and born about 460 B.C. Like all the later geome ters, he failed in effecting the trisection by means of a ruler and compass only. Prockts mentions a man, Hippias, presum ably Hippias of Elis, as the inventor of a transcendental curve which served to divide an angle not only into three, but into any number of equal parts. This same curve was used later by Deinostratus and others for the quadrature of the circle. On this account it is called the quadratrix. The Pythagoreans had shown that the diagonal of a square is the side of another square having double the area of the original one. This probably suggested the problem of the duplication of the cube, i.e. to find the edge of a cube having double the volume of a given cube. Eratosthenes ascribes to this problem a different origin. The Delians were once suf fering from a pestilence and were ordered by the oracle to double a certain cubical altar. Thoughtless workmen simply constructed a cube with edges twice as long, but this did not pacify the gods. The error being discovered, Plato was con sulted on the matter. He and his disciples searched eagerly for a solution to this "Delian Problem." Hippocrates of Chios (about 430 B.C.), a talented mathematician, but otherwise slow and stupid, was the first to show that the problem could be reduced to finding two mean proportionals between a given line and another twice as long. For, in the proportion a: a? = x : y = y : 2 a, since a? 2 = ay and y 2 = 2 ax and ce* = a 2 /, we have a; 4 = 2 cfx and a? 3 = 2 a 8 . But he failed to find the two mean proportionals. His attempt to square the pircl& was also a failure; for though lie made himself celebrated by squaring a kine, he committed an error in attempting to apply this result to the squaring of the circle. lujhis study of the quadrature and duplication-problems, contributed much to the geometry of the circle. 26 A HISTORY OF MATHEMATICS. The subject of similar figures was studied and partly developed by Hippocrates. This involved the theory of proportion. Proportion had, thus far, been used by the Greeks only in numbers. They never succeeded in uniting the notions of numbers and magnitudes. The term "number " was used by them in a restricted sense. What we call irrational numbers was not included under this notion. Not even rational fractions were called numbers. They used the word in the same sense as wo use "integers." Hence num bers were conceived as discontinuous, while magnitudes were continuous. The two notions appeared; therefore, entirely distinct. The chasm between them is exposed to full view in the statement of Euclid that "incommensurable magni tudes do not have the same ratio as numbers." In Euclid s Elements we find the theory of proportion of magnitudes developed and treated independent of that of numbers. The transfer of the theory of proportion from numbers to mag nitudes (and to lengths in particular) was a difficult and important step. Hippocrates added to his fame by writing a geometrical text-book, called the Elements. This publication shows that the Pythagorean habit of secrecy was being abandoned; secrecy was contrary to the spirit of Athenian life. The Sophist Antiphon, a contemporary of Hippocrates, intro duced the process of exhaustion for the purpose of solving the problem of the quadrature. Ho did himself credit by remarking that by inscribing in a circle a square, and oa its sides erecting isosceles triangles with their vertices itt the circumference, and on the sides of these triangles erecting new triangles, etc., one could obtain a succession of .regular polygons of 8, 16, 32, 64 sides, and so on, of "which eneh, approaches nearer to the circle than the pxeviot^. o&f until the circle is finally exhausted. Thais is obtained an iTD0 ^e THE GREEKS* 27 polygon whose sides coincide with the circumference. Since there can be found squares equal in area to any polygon, there also can be found a square equal to the last polygon inscribed, and therefore equal to the circle itself. Brys0n of Heraclea, a contemporary of Antiphon, advanced the prob lem of the quadrature considerably by circumscribing poly gons at the same time that he inscribed polygons.- He erred, however, in assuming that the area of a circle was the arith metical mean between circumscribed and inscribed polygons. Unlike Bryson and the rest of Greek geometers, Antiphon seems to have believed it possible, by continually doubling the sides of an inscribed polygon, to obtain a polygon coin ciding with the circle. This question gave rise to lively disputes in Athens. If a polygon can coincide with the circle, then, says Simplicius, we must put aside the notion that magnitudes are divisible ad infinitum. Aristotle always supported the theory of tihe infinite divisibility, while Zeno, the Stoic, attempted to show its absurdity by proving that if magnitudes are infinitely divisible, motion is impossible. Zeno argues that Achilles could not overtake a tortoise; for while he hastened to the place where the tortoise had been when he started, the tortoise crept some distance ahead, and while Achilles reached that second spot, the tortoise again moved forward a little, and so on. Thus the tortoise was always in advance of Achilles. Such arguments greatly con founded Greek geometers. No wonder they were deterred by such paradoxes from introducing the idea of infinity into their geometry. It did not suit the rigour of their proofs. The process of Antiphon and Bryson gave rise to the cum brous but perfectly rigorous "method of exhaustion." In determining the ratio of the areas between two curvilinear plane i|jp,|% s&y/two circles, geometers first inscribed or Similar t>olverons, and then bv infyrAfl.ainar i A HISTOEY OF MATHEMATICS. the number of sides, nearly exhausted the spaces between the polygons and circumferences. IProm the theo rem that similar polygons inscribed in circles are to each othsr as the squares on their diameters, geometers may have divined the theorem attributed to Hippocrates of Chios that the circles, which differ but little from the last drawn poly gons, must be to each other as the squares on their diameters. But in order to exclude all vagueness and possibility of doubt, later Greek geometers applied reasoning like that in Euclid, XII. 2, as follows : Let and c, D and d be respectively the circles and diameters in question. Then if the proportion D 2 : d 2 = C : c is not true, suppose that D 2 : $ = : c . If d < c, then a polygon p can be inscribed in the circle c which conies nearer to it in area than does c f . If P be the corresponding polygon in C, then P : p = D 2 ; d 2 = G : c , and P : O = p : c . Since j> > c f , we have P>C, which is absurd. Next they proved by this same method of reductio ad absurdum the falsity of the supposition, that c f > c. Since c can be neither larger nor smaller than, c, it must be equal to it, QJE.D. Hankel refers this Method of Exhaustion back to Hippo crates of Chios, but the reasons for assigning it to this early writer, rather than to Eudoxus, seem insufficient. Though progress in geometry at this period is traceable only at Athens, yet Ionia, Sicily, Abdera in Thrace, and Gyrene produced mathematicians who made creditable contribution B to the science. We can mention here only Bemociitus of Abdera (about 460-370 B.C.), a pupil of Anaxagoras, a friend of Philolaus,- and an admirer of the Pythagoreans. He visited Egypt and perhaps even Persia. Ho was a successful geometer and wrote on incommensurable lines, on geometry, on numbers, and on perspective. Hone of these works are extant, He used to boast that in the construction of plane figures with proof no one had yet surpassed him, not even THE GREEKS. 29* the so-called harpedonaptae (" rope-stretchers ") of Egypt. By this assertion he pays a flattering compliment to the skill and ability of the Egyptians. TJie Platonic School. During the Peloponnesian War (431-404 B.C.) the progress of geometry was checked. After the war, Athens sank into the background as a minor political power, but advanced more and more to the front as the leader in philosophy, literature, and science. Plato was born at Athens in 429 B.C., the year of the great plague, and died in 348. He was a pupil and near friend of Socrates, but it was not from him that he acquired his taste for mathematics. After the death of Soc rates, Plato travelled extensively. In Cyrene he studied mathematics under Theodoras. He went to Egypt, then to Lower Italy and Sicily, where he came in contact with the Pythagoreans. Archytas of Tarentum and Timaeus of Locri became his intimate friends. On his return to Athens^ about 389 B.C., he founded his school in the groves of the Academia, and devoted the remainder of his life to teaching and writing. Plato s physical philosophy is partly based on that of the Pythagoreans. Like them, he sought in arithmetic and geometry the key to the universe. When questioned about the occupation of the Deity, Plato answered that " He geom- etrises continually." Accordingly, a knowledge of geometry is a necessary preparation for the study of philosophy. To show how great a value he put on mathematics and how necessary it is for higher speculation, Plato placed the inscrip tion over Ms porch, "Let no one who is unacquainted with geometry enter here," Xenocrates, a successor of Plato as teacher in the Academy, followed in his master s footsteps, by declining to admit a pupil who had no mathematical training, 30 A HISTOBY OF MATHEMATICS. with the remark, "Depart, for thou hast not the grip of philosophy. 1 Plato observed that geometry trained the mind for correct and vigorous thinking. Hence it was that the Eudemian Summary says, " He filled his writings with mathe matical discoveries, and exhibited on every occasion the re markable connection between mathematics and philosophy." With Plato as the head-master, we need not wonder that the Platonic school produced so large a number of mathemati cians. Plato did little real original work, but he made valuable improvements in the logic and methods employed in geometry. It is true that the Sophist geometers of the previous century were rigorous in their proofs, but as a rule they did not reflect on the inward nature of their methods. They used the axioms without giving them explicit expression, and the geometrical concepts, such as the point, line, surface, etc., without assigning to them formal definitions, The Py thagoreans called a point "unity in position/ 7 but this is a statement of a philosophical theory rather than a definition. Plato objected to calling a point a " geometrical fiction." He defined a point as the "beginning of a line" or as "an indivis ible line," and a line as " length without breadth." He called the point, line, surface, the boundaries of the line, surface, solid, respectively. Many of the definitions in Euclid are to be ascribed to the Platonic school. The same is probably true of Euclid s axioms. Aristotle refers to Plato the axiom that "equals subtracted from equals leave equals." 7 One of the greatest achievements of Plato and his school is the invention of analysis as a method of proof. To be sure, this method had been used unconsciously by Hippocrates and others ; but Plato, like a true philosopher^ turned the instinc tive logic into a conscious, legitimate method. The terms synthesis and analysis are used in mathematics in a more special sense than in logic. In ancient mathematics THE GREEKS. 31 they had a different meaning from what they now have. The oldest definition of mathematical analysis as opposed to syn thesis is that given in Euclid, XIII. 5, which in all probability was framed by Eudoxus : " Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth ; synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." The analytic method is not conclusive, unless all operations involved in it are known to be reversible. To remove all doubt, the Greeks, as a rule, added to the analytic process a synthetic one, consisting of a reversion of all operations occurring in the analysis. Thus the aim of analysis was to aid in the discovery of synthetic proofs or solutions. ; Plato is said to have solved the problem of the duplication of the cube. But the solution is open to the very same objec tion which he made to the solutions by Archytas, Eudoxus, and Menaeclmius. He called their solutions not geometrical, but mechanical, for they required the use of other instruments than the ruler and compass. He said that thereby " the good of geometry is set aside and destroyed, for we again reduce it to the world of sense, instead of elevating and imbuing it with the eternal and incorporeal images of thought, even as it is employed by God, for which reason He always is God." These objections indicate either that the solution is wrongly attrib uted to Plato or that he wished to show how easily non-geo metric solutions of that character can be found. It is now generally admitted that the duplication problem, as well as the trisection and quadrature problems, cannot be solved by means of the ruler and compass only. Plato gave a healthful stimulus to the study of stereometry, which until his time had been entirely neglected. The sphere and the regular solids had been studied to some extent, but the prism, pyramid, cylinder, and cone were hardly known to 32 A HISTOBY OF MATHEMATICS. exist. All these solids became the subjects of investigation by the Platonic school. One result of these inquiries was epoch-making. Menaechmus, an associate of Plato and pupil of Eudoxus, invented the conic sections, which, in course of only a century, raised geometry to the loftiest height which it was destined to reach during antiquity. Mensechmus cut three kinds of cones, the right-angled/ acute-angled/ and obtuse-angled/ by planes at right angles to a side of the cones, and thus obtained the three sections which we now call the parabola, ellipse, and hyperbola. Judging from the two very elegant solutions of the "Delian. Problem" by means of intersections of these curves, Mensechimis must have succeeded well in investigating their properties. Another great geometer was Dinostratus, the brother of Menaechmus and pupil of Plato. Celebrated is his mechanical solution of the quadrature of the circle, by means of the quad- ratri of Hippias. Perhaps the most brilliant mathematician of this period was Eudoxus. He was born at Cniclus about 408 B.O., studied under Archytas, and later, for two months, under Plato. He was imbued with a true spirit of scientific inquiry, and has beea called the father of scientific astronomical observation. From the fragmentary notices of his astronomical researches, found in later writers, Ideler and Schiaparolli succeeded in recon structing the system of Eudoxus with its celebrated representa tion of planetary motions by "concentric spheres*" Eudoxus had a school at Cyzicus, went with his pupils to Athens, visit ing Plato, and then returned to Cyzicxis, where ho died 355 B.C. The fame of the academy of Plato is to a large extent due to Eudoxtts s pupils of the school at Cyzicua, aiaong whom are Meneeclnnus, Dinostratus, Athensaus, and Helicon. Diogenes Laertius describes Eudoxus as astronomer, physician, legislator, as well as geometer. The Eudemimi Summary THE GREEKS. 33 says that Eudoxus " first increased the number of general theorems, added to the three proportions three more, aixd raised to a considerable quantity the learning, begun by Plato, on the subject of the section, to which he applied the analyt ical method." By this c section is meant, no doubt, the "golden section" (sectio aurea), which cuts a line in extreme and mean ratio. The first five propositions in Euclid XIII. relate to lines cut by this section, and are generally attributed to Eudoxus. Eudoxus added much to the knowledge of solid geometry. He proved, says Archimedes, that a pyramid is exactly one-third of a prism, and a cone one-third of a cylinder, having equal base and altitude. The proof that spheres are to each other as the cubes of their radii is probably due to him. He made frequent and skilful use of the method of exhaustion, of which he was in all probability the inventor. A scholiast on Euclid, thought to be Proclus, says further that Eudoxus practically invented the whole of Euclid s fifth book. Eudoxus also found two mean proportionals between two given lines, but the method of solution is not known. Plato has been called a maker of mathematicians. Besides the pupils already named, the Eudemian Summary men tions the following: Theaetetus of Athens, a man of great natural gifts, to whom, no\loubt, Euclid was greatly indebted in the composition of the 10th book ; 8 treating of incommensu- rables ; Leodamas of Thasos ; Feocleides and his pupil Leon, who added much to the work of their predecessors, for Leon wrote an Elements carefully designed, both in number and utility of its proofs; Theudius of Magnesia, who composed a very good book of Elements and generalised propositions, which had been confined to particular cases ; Hermotimus of Colophon, who discovered many propositions of the Elements and composed some on loci; and, finally, the names of Amyclas of Heraclea, Cyzicenus of Athens, and Philippus of Mende. 34 A HISTOBY OF MATHEMATICS, A skilful mathematician of whose life and works we have no details is Aristaelis, the elder, probably a senior contempo rary of Euclid. The fact that he wrote a work on conic sections tends to show that much progress had been made in their study during the time of Menaechmus. Aristous wrote also on regular solids and cultivated the analytic method. His works contained probably a summary of the researches of the Platonic school. 8 Aristotle (384-322 B.C.), the systematise! of deductive logic, though not a professed mathematician, promoted the science of geometry by improving some of the most difficult defini tions. His Physics contains passages with suggestive hints of the principle of virtual velocities. About his time there appeared a work called Mechanic, of which he is regarded by some as the author. Mechanics was totally neglected by the Platonic school. The First Alexandrian School, In the previous pages we have seen the birth of geometry in Egypt, its transference to the Ionian Islands, thence to Lower Italy and to Athens. Wo have witnessed its growth in Greece from feeble childhood to vigorous manhood, and now we shall see it return to the land of its birth and there derive new vigour. During her declining years, immediately following the Feloponnesian War, Athens produced the greatest scientists and philosophers of antiquity. It was the timo of Plato and Aristotle. In 338 B.C., at the battle of OUf&ronea, Athens was beaten, by Philip of Macedon, and her power was broken forever. Soon after, Alexander the Great, the son of Philip, started out to conquer the world. la eleven years he built up a great empire which broke to pieep ia a day* THE GREEKS. 35 fell to the lot of Ptolemy Soter. Alexander had founded the seaport of Alexandria, which soon became "the noblest of all cities." Ptolemy made Alexandria the capital. The history of Egypt during the next three centuries is mainly the history of Alexandria. Literature, philosophy, and art were diligently cultivated. Ptolemy created the university of Alexandria. He founded the great Library and built labo ratories, museums, a zoological garden, and promenades. Alex andria soon became the great centre of learning. Demetrius Phalereus was invited from Athens to take charge of the Library, and it is probable, says Gow, that Euclid was invited with him to open the mathematical school. Euclid s greatest activity was during the time of the first Ptolemy, who- reigned from 306 to 283 B.C. Of the life of Euclid, little is known, except what is added by Proclus to the Eudemian Summary. Euclid, says Proclus, was younger than Plato and older than .Eratosthenes and Archimedes, the latter of whom mentions him. He was of the Platonic sect, and well read in its doctrines. He collected the Elements, put in order much that Eudoxus had prepared, completed many things of Theaetetus, and was the first who reduced to unob jectionable demonstration, the imperfect attempts of his prede cessors. When Ptolemy once asked him if geometry could not be mastered by an easier process than by studying the Elements, Euclid returned the answer, "There is no royal road to geometry." Pappus states that Euclid was distin guished by the fairness and kindness of his disposition, par ticularly toward those who could do anything to advance the mathematical sciences. Pappus is evidently making a contrast to Apollonius, of whom he more than insinuates the opposite character. 9 A pretty little story is related by Sto- baeus: 6 "A youth who had begun to read geometry with Euclid, when h had learnt the first proposition, inquired, 36 A HISTORY OF MATHEMATICS. < What do I get by learning these tilings ? So Euclid called his slave and said, Give him threepence, since he must make gain out of what he learns/ " These are about all the personal details preserved by Greek writers, Syrian and Arabian writers claim to know much more, but they are unre liable. At one time Euclid of Alexandria was universally confounded with Euclid of Megara, who lived a century earlier. The fame of Euclid has at all times rested mainly upon his book on geometry, called the Elements. This book was so far superior to the Elements written by Hippocrates, Loon, and Theudius, that the latter works soon perished in the straggle for existence. The Greeks gave Euclid the special title of ~ c the author of the .Elements" It is a remarkable fact in thei xistory of geometry, that the Elements of Euclid, written two thousand years ago, are still regarded by many as the best ntroduction to the mathematical sciences. In England they xre used at the present time extensively as a text-book in schools. Some editors of Euclid have, however, been inclined bo credit him with more than is his due. They would have us believe that a finished and unassailable system of geometry sprang at once from the brain of Euclid, " an armed Minerva from the head of Jupiter." They fail to mention the earlier eminent mathematicians from whom Euclid got his material. Comparatively few of the propositions and proofs in the Elements are his own discoveries. In fact, the proof of tlie " Theorem of Pythagoras " is the only one directly ascribed to him. Allman conjectures that the substance of Books I,, II, IV. comes from the Pythagoreans, that tlie substance of Book VI. is due to the Pythagoreans and Exidoatufy tlto latter con tributing the doctrine of proportion as applicable to ineom- mensurables and also the Method of Exhaustions (Book VII.), that Thesetetus contributed much toward Books X, and XIII, THE GREEKS. 37 that the principal part of the original work of Euclid himself is to be found in Book X. 8 Euclid was the greatest systema- tiser of his time. By careful selection from the material before him, and by logical arrangement of the propositions selected, he built up, from a few definitions and axioms, a proud and lofty structure. It would be erroneous to believe that he incorporated into his Elements all the elementary theorems known at. his time. Archimedes, Apollonius, and even he himself refer to theorems not included in his Ele ments, as being well-known truths. The text of the Elements now commonly used is Theon s edition. Theon of Alexandria, the father of Hypatia, brought out an edition, about 700 years after Euclid, with some altera tions in the text. As a consequence, later commentators, especially Robert Simson, who laboured under the idea that Euclid must be absolutely perfect, made Theon the scape goat for all the defects which they thought they could discover in the text as they knew it. But among the manuscripts sent by Napoleon I. from the Vatican to Paris was found a copy of the Elements believed to be anterior to Theon s recension. Many variations from Theon s version were noticed therein, but they were not at all important, and showed that Theon generally made only verbal changes. The defects in the Elements for which Theon was blamed must, therefore, be due to Euclid himself. The Elements has been considered as offering models of scrupulously rigoroxis demonstrations. It is certainly true that in point of rigour it compares favourably with its modern rivals ; but when examined in the light of strict mathematical logic, it has been pronounced by C. S. Peirce to be " riddled with fallacies." The results are correct only because the writer s experience keeps him on his guard. At the beginning of our editions of the Elements, under the head of definitions, are given the assumptions of such 38 A HISTORY OF MATHEMATICS. notions as the point, line, etc., and some verbal explanations. Then follow three postulates or demands, and twelve axioms. The term axiom 7 was used by Proclus, but not by Euclid. He speaks, instead, of common notions common either to all men or to all sciences. There has been much contro versy among ancient and modern critics on the postulates and axioms. An immense preponderance of manuscripts and the testimony of Proclus place the axioms ? about right angles and parallels (Axioms 11 and 12) among tho postulates. 9 10 This is indeed their proper place, for they arc really assump tions, and not common notions or axioms. Tho postulate about parallels plays an important role in the history of non* Euclidean geometry. The only postulate which Kxiolid missed was the one of superposition, according to which figures can, be moved about in space without any alteration in form or magnitude. The Moments contains thirteen books by Euclid, and two, of which it is stipposed that Hypsicles and Damasoms are the authors. The first four books are on plane geometry. The fifth book treats of the theory of proportion as applied to magnitudes in general. The sixth book develops the geometry of similar figures. The seventh, eighth, ninth booksy^re on the theory of numbers, or on arithmetic. In the ninth book is found the proof to the theorem that tha number of primes is infinite. The tenth book treats of the theory of incommensurables. The next three books are on stereometry. The eleventh contains its more elementary theorems ; the twelfth, the metrical relations of the pyramid, prism, cone, cylinder, and sphere. Tho thirteenth treats of the regular polygons, especially of the triangle and pentagon, and then uses them as faces of the five regular solids ; namely, the totraedron, octaedron, icosaedron, cube, and dodecaedron. The regular solids were studied so extensively by the.Platonists fhfrjfe they THE GEBEKS. 39 received the name of "Platonic figures." The statement of Proclns that the whole aim of Euclid in writing the Elements was to arrive at the construction of the regular solids, is obviously wrong. The fourteenth and fifteenth books, treat ing of solid geometry, are apocryphal. A remarkable feature of Euclid s, and of all Greek geometry before Archimedes is that it eschews mensuration. Thus the theorem that the area of a triangle equals half the product of its base and its altitude is foreign to Euclid. Another extant book of Euclid is the Data. It seems to have been written for those who, having completed the Ele ments, wish to acquire the power of solving new problems proposed to them. The Data is a course of practice in analy sis. It contains little or nothing that an intelligent student could not pick up from the Elements itself. Hence it contrib utes little to the stock of scientific knowledge. The following are the other extant works generally attributed to Euclid: Phenomena, a work on spherical geometry and astronomy; Optics, which develops the hypothesis that light proceeds from the eye, and not from the object seen; Catoptrica, con taining propositions on reflections from mirrors ; De Divisioni- ftus, a treatise on the division of plane figures into parts having to one another a given ratio ; Sectio Canonis, a work on musical intervals. His treatise on Porisms is lost ; but much learning has been expended by Eobert Sims on and M. Ohasles in restoring it from numerous notes found in the writings of Pappus. The term porism is vague in meaningl The aim of a porism is not to state some property or truth, like a theorem, nor to effect a construction, like a problem, but to find and bring to view a thing which necessarily exists with given numbers or a given construction, as, to find the centre of a given circle, or to find the G.C.D. of two given numbers. 6 His other lost works are Fallacies, containing 40 A HISTORY OF MATHEMATICS. exercises in detection of fallacies; Conic Sections, in four books, which are the foundation of a work on the same sub ject by Apollonius; and Loci on a Surface, the meaning of which title is not understood. Heiberg believes it to mean "loci which are surfaces." The immediate successors of Euclid in the mathematical school at Alexandria were probably Conon, Dositheus, and Zeuxippus, but little is known of them. \ Archimedes (287?~212 B.C.), the greatest mathematician of antiquity, was born in Syracuse. Plutarch calls him a rela tion of King Hieronj but more reliable is the statement of Oicero, who tells us he was of low birth. Diodorus says he visited Egypt, and, since he was a great friend of Conon and Eratosthenes, it is highly probable that he studied in. Alexan dria. This belief is strengthened by the fact that he had bhe most thorough acquaintance with all the work previously done in mathematics. He returned, however, to Syracuse, where he made himself useful to his admiring friend patron, King Hieron, by applying his extraordinary inventive genius to the construction of various war-engines, by wjbdch he inflicted much loss on the Romans during the siege of Marcellus. 1 The story that, by the use of mirrors reflecting bhe sun s rays, he set on fire the Roman ships, when they came within bow-shot of the walls, is probably a fiction. tJTIxe city was taken ait length "by the Romans, and Archimedes perished in the indiscriminate slaughter which followed. Ac cording to tradition, he was, at the time, studying the diagram bo some problem drawn in the sand. As a Roman soldier approached him, he called out, "Don t spoil my circles." The soldier, feeling insulted, rushed upon him and killed him. Jf No -blame attaches to [the Roman general Marcelltts, who admired his genius, and raised in his honour a tomb bearing the figure of a sphere inscribed in a cylinder. When THE GREEKS. 41 Cicero was in Syracuse, lie found the tomb buried under rubbish. Archimedes was admired by his fellow-citizens chiefly for""" his mechanical inventions ; he himself prized far more highly his discoveries in pure science. He declared that "every kind of art which was connected with daily needs was ignoble and vulgar-^" Some of his works have been lost. The following are the extant books, arranged approximately in chronological order : 1. Two books on Equiponderance of Planes or Centres of Plane Gravities, between which is inserted his treatise or. the Quadrature of the Parabola; 2. Two books on the Sphere and Cylinder; 3. The Measurement of the Circle ; 4. On Spirals; 5. Conoids and Spheroids; 6. The Sand-Counter; 7. Two books on Floating Bodies; 8. Fifteen Lemmas, j In the book on the Measurement of the Circle, Archimedes proves first that the area of a circle is equal to that of a right triangle having the length of the circumference for its b~se, and the radius for its altitude. In this he assumes that there exists a straight line equal in length to the circumference an assumption objected to by some ancient critics, on the ground that it is not evident that a straight line can equal a curved one. The finding of suct^ a line was the next prob lem. He fir^t finds an upper limit to the ratio of the circum ference to the diameter, or TT. To do this, he starts with an equilateral triangle of which the base is a tangent and the vertex is the centre of the circle. By successively bisecting the angle at the centre, by comparing ratios, and by taking the irrational square roots always a little too small, he finally arrived at the conclusion that ?r<3^. Next he finds a lower limit by inscribing in the circle regular polygons of 6, 12, 24, 48, 96 sides, finding for each successive polygon its perimeter, which is, of course, always less than the circumference. Thus he finally concludes that "the circumference of a circle ex- 42 A HISTORY OF MATHEMATICS. ceeds three times its diameter by a part which, is less than $ but more than f& of the diameter." This approximation is exact enough for most purposes. The Quadrature of the Parabola contains two solutions to the problem one mechanical, the other geometrical. The method of exhaustion is used in both. Archimedes studied also the ellipse and accomplished its quadrature, but to the hyperbola he seems to have paid less at tention. It is believed that he wrote a book on conic sections. J~-Of all his discoveries Archimedes prized most highly those in his Sphere and Cylinder. In it are proved the new theorems, that the surface of a sphere is equal to four times a great circle ; that the surface of a segment of a sphere is equal to a circle whose radius is the straight line drawn from the vertex of the segment to the circumference of its basal circle ; that the volume and the surface of a sphere are of, the volume and surface, respectively, of the cylinder circum scribed about the sphere. Archimedes desired that the figure to the last proposition be inscribed on his tomb. This was ordered done by Marcellus. } <CThe spiral now called the "spiral of Archimedes," and described in the book On Spirals, was discovered by Archi medes, and not, as some believe, by his friend Conon. 8 His treatise thereon is, perhaps, the most , wonderful of- all his works. Nowadays, subjects of this kind are made easy by . the use of the infinitesimal calculus. In its stead the aBteients, used the method of exhaustion. Nowhere is the fertility of his genius more grandly displayed than in his masterly use of this method. With Euclid and his predecessors the method of exhaustion was only the means of proving propositions which must have been seen anf : b&litved - , before they were proved. But in the hands of Arehtoete it lecame art instru ment of discovery, 9 THE GKEEKS. 43 By the word conoid/ in his book on Conoids and Spheroids, is meant thQ solid produced by the revolution of a parabola or a hyperbola about its axis. Spheroids are produced by the revolution of an ellipse, and are long or flat, according as the ellipse revolves around the major or minor axis. The book leads up to the cubature of these solids. / We Rave now reviewed briefly all his extant works on geom etry. His arithmetical treatise and problems will be consid ered later. We shall now notice his works on mechanics. Archimedes is the author of the first sound knowledge on this subject. Archytas, Aristotle, and others attempted to form the known mechanical truths into a science, but failed. Aris totle knew the property of the lever, but could -not establish its true mathematical theory. The radical and fatal defect in the speculations of the Greeks, says Whewell, was "that though they had in their possession facts and ideas, the ideas were not distinct and appropriate to the facts. 93 For instance, Aristotle asserted that when a body at the end of a lever is moving, it may be considered as having two motions ; one in the direction of the tangent and one in the direction of the radius ; the former motion is, he says, according to nature, the latter contrary to nature. These inappropriate notions of natural 5 and unnatural motions, together with the habits of ^thought which dictated these speculations, made the per ception, of the true grounds of mechanical properties impos sible." It seems strange that even after Archimedes had entered upon the right path, this science should have remained absolutely stationary till the time of Galileo a period of nearly two thousand years. The proof of the property of the lever, given in his Equi- ponderance of Planes, holds its place in text-books to this day. His estimate of the efficiency of the lever is expressed in the 44 A HISTORY OF MATHEMATICS. saying attributed to him, "Give me a fulcrum on which to rest, and I will move the earth." / s "While the JSquiponderance treats of solids, or the equilib rium of solids, the book 011 Floating Bodies treats of hydro statics. His attention was first drawn to the subject of specific gravity when King Hieron asked him to test whether a crown, professed by the maker to be pure gold, was not alloyed with silver."] The story goes that our philosopher was in a bath when the true method of solution flashed on his mind. He immediately ran home, naked, shouting, " I have found it ! " llo solve the problem, he took a piece of gold and a piece of silver, each weighing the same as the crown. Ac cording to one author, he determined the volume of water displaced by the gold, silver, and crown respectively, and calculated from that the amount of gold and silver in the crown. According to another writer, he weighed separately the gold, silver, and crown, while immersed in water, thereby determining their loss of weight in water. Prom these data he easily found the solution. It is possible that Archimedes solved the problem by both methods. After examining the writings of Archimedes, one can well understand how, in ancient times, an < Archimedean problem ? came to mean a problem too deep for ordinary minds to solve, and how an i Archimedean proof came to be the synonym for unquestionable certainty. Archimedes wrote on a very wide range of subjects, and displayed great profundity in each. He is the Newton of antiquity/] Eratosthenes, eleven years younger than Archimedes, was a native of Cyrene. He was educated in, Alexandria under Callimachus the poet, whom he succeeded as custodian of the Alexandrian Library. His many-sided activity may be inferred from his works. He wrote on Cfood and Evil, Meas urement of the Earthy Comedy, Geography, Chronology, Constel- THE GREEKS. 45 lotions, and the Duplication of the Cube. He was also a philologian and a poet. He measured the obliquity of the ecliptic and invented a device for finding prime numbers. Of his geometrical writings we possess only a letter to Ptolemy Euergetes, giving a history of the duplication prob lem and also the description of a very ingenious mechanical contrivance of his own to solve it. In his old age he lost his eyesight, and on that account is said to have committed suicide by voluntary starvation. About forty years after Archimedes flourished Apollonius of Perga, whose genius nearly equalled that of his great prede cessor. He incontestably occupies the second place in dis tinction among ancient mathematicians. Apollonius was born in the reign of Ptolemy Euergetes and died under Ptolemy Philopator, who leigned 222-205 B.C. He studied at Alexandria under the successors of Euclid, and for some time/ also, at Perganmm, where he made the acquaintance of that Eudemus to whom he dedicated the first three books of his Conic Sections. The brilliancy of his great work brought him the title of the " Great Geometer." This is all that is known of his life. His Conic Sections were in eight books, of which the first four only have come down to us in the original Greek. The next three books were unknown in Europe till the middle of the seventeenth century, when an Arabic translation, made about 1250, was discovered. The eighth book has never been found. In 1710 Halley of Oxford published the Greek text of the first four books and a Latin translation of the remain ing three, together with his conjectural restoration of the eighth book, founded on the introductory lemmas, of Pappus. The first four books contain little more than the substance of what earlier geometers had done. Eutocius tells us that Heraclides, in his life of Archimedes, accused Apollonius of 46 A HISTOEY OF MATHEMATICS. having appropriated, in his Conic Sections, the unpublished discoveries of that great mathematician. It is difficult to believe that this charge rests upon good foundation. Eutocius quotes Geminus as replying that neither Archimedes nor Apollonius claimed to have invented the conic sections, but that Apollonius had introduced a real improvement. While the first three or four books were founded on the works of Menaechmus, Aristseus, Euclid, and Archimedes, the remaining ones consisted almost entirely of new matter. The first three books were sent to Eudemus at intervals, the other books (after Eudemus s death) to one Attalus. The preface of the second book is interesting as showing the mode in which Greek books were ( published ? at this time. It reads thus : " I have sent my son Apollonius to bring you (Eudemus) the second book of my Conies. Bead it carefully and communi cate it to such others as are worthy of it. If Philonides, the geometer, whom I introduced to you at Ephesus, comes into the neighbourhood of Pergamum, give it to him also." 12 The first book, says Apollonius in his preface to it, " con tains the mode of producing the three sections and the conju gate hyperbolas and their principal characteristics, more fully and generally worked out than in the writings of other authors." We remember that Mensechmus, and all his suc cessors down to Apollonius, considered only sections of right cones by a plane perpendicular to their sides, and that the three sections were obtained each from a different cone. Apollonius introduced an important generalisation. He pro duced all the sections from one and the same cone, whether right or scalene, and by sections which may or may not be perpendicular to its sides. The old names for the three curves were now no longer applicable. Instead of calling the three curves, sections of the^ acute-angled/ * right-angled/ and obtuse-angled cone, he called them ellipse, parabola, and THE GREEKS. 47 hyperbola, respectively. To be sure, we find the words < parab ola and ellipse ? in the works of Archimedes, but they are probably only interpolations. The word ellipse > was applied because y 2 <px,p being the parameter; the word parabola was introduced because y 2 =px, and the term < hyperbola because y*>px. The treatise of Apollonius rests on a unique property of conic sections, which is derived directly from the nature of the cone in which these sections are found. How this property forms the key to the system of the ancients is told in a mas terly way by M. Chasles. 13 "Conceive," says he, "an oblique cone on a circular base; the straight line drawn from its summit to the centre of the circle forming its base is called the axis of the cone. The plane jpassing through the axis, perpendicular to* its base; exit s" the cone along two lines and determines in the circle a diameter ; the triangle having this diameter for its base and the two lines, for its sides, is called the triangle through the axis. In the formation of his conic sections, Apollonius supposed the cutting plane to be perpen dicular to the plane of the triangle through the axis. The points in which this plane meets the two sides of this triangle are the vertices of the curve ; and the straight line which joins these two points is a diameter of it. Apollonius called this diameter latus transversum. At one of the two vertices of the curve erect a perpendicular (latus rectum) ;to the plane of the triangle through the axis, of a certain length, to be determined as we shall specify later, and from the extremity of this per pendicular draw a straight line to the other vertex of the curve ; now, through any point whatever of the diameter of the curve, draw at right angles an ordinate : the square of this ordinate, comprehended between the diameter and the curve, will be equal to the rectangle constructed on the portion of the ordinate comprised between the diameter and the straight 4:8 A HISTOBY OF MATHEMATICS. line, and the part of the diameter comprised between the first vertex and the foot of the ordinate. Such is the characteristic property which Apollonius recognises in his conic sections and which he uses for the purpose of inferring from it, by adroit transformations and deductions, nearly all the rest. It plays, as we shall see, in his hands, almost the same rdle as the equation of the second degree with two variables (abscissa and ordinate) in the system of analytic geometry of Descartes. "It will be observed from this that the diameter of the curve and the perpendicular erected at one of its extremities suffice to construct the curvj|r These are the two elements which the ancients used, with which to establish their theory of conies. The perpendicular in question was called by them latus erectum; the moderns changed this name first to that of latus rectum, and afterwards to that of parameter." The first book of the Conic Sections of Apollonius is almost wholly devoted to the generation of the three principal conic sections. The second book treats mainly* of asymptotes, axes, and diameters. The third book treats of the equality or proportionality of triangles, rectangles, or squares, of which the component parts are determined by portions of transversals, chords, asymptotes, or tangents, which are frequently subject to a great number of conditions. It also touches the subject of foci of the ellipse and hyperbola. In the fourth book, Apollonius discusses the harmonic divis ion of straight lines. He also examines a system of two conies, and shows that they cannot cut each other in more than four points. He investigates the various possible relative positions of two conies, as, for instance, when, they have one or two points of contact with each other. The fifth book reveals better than any other the giant THE GREEKS. 49 intellect of its author. Difficult questions of maxima and minima, of which, few examples are found in earlier works, are here treated most exhaustively. The subject investigated is, to find the longest and shortest lines that can he drawn from a given point to a conic. Here are also found the germs of the subject ofevolutes and centres of osculation. The sixth book is on the similarity of conies. The seventh book is on conjugate diameters. The eighth book, as restored by Halley, continues the sub ject of conjugate diameters. It is worthy of notice that Apollonius nowhere introduces the notion of directrix for a conic, and that, though he inciden tally discovered the focus of an ellipse and hyperbola, he did not discover the focus of a parabola. 6 Conspicuous in his geometry is also the absence of technical terms and symbols, which renders the proofs long and cumbrous. The discoveries of Archimedes and Apollonius, says M. Chasles, 13 marked the most brilliant epoch of ancient geometry. Two questions which have occupied geometers of all periods may be regarded as having originated with them. The first of these is the quadrature of curvilinear figures, which gave birth to the infinitesimal calculus. The second is the theory of conic sections, which was the prelude to the theory of geometrical curves of all degrees, and to that portion of geometry which considers only the forms and situations of figures, and uses only the intersection of lines and surfaces and the ratios of rectilineal distances. These two great divisions of geometry may be designated by the names of Geometry of Measurements and Geometry of Forms and Situa tions, or, Geometry of Archimedes and of Apollonius. Besides the Conic Sections, Pappus ascribes to Apollonius the following works: On Contacts, Plane Loci, Inclinations, Section of an Area, Determinate Section, and gives lemmas 50 A HISTOEY OF MATHEMATICS. from which attempts have been made to restore the lost originals. Two books on De Sectione Rationis have been found in the Arabic. The book on Contacts, as restored by Vieta, contains the so-called " Apollonian Problem " : Given three circles, to find a fourth which shall touch the three. Euclid, Archimedes, .and Apollonius brought geometry to as- high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer sym bolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent, during which they paused here and there to look around for details which had been passed by in the hasty ascent. 3 Among the earliest successors of Apollonius was Mcomedes. Nothing definite is known of him, except that he invented the conchoid (" mussel-like"). He devised a little machine by which the curve could be easily described. With aid of the conchoid he duplicated the cube. The curve can also be used for trisecting angles in a way much resembling that* in the eighth lemma of Archimedes. Proclus ascribes this mode of trisection to Nicomedes, but Pappus, on the other hand, claims it as his own. The conchoid was used by Newton in con structing curves of the third degree. About the time of Mcomedes, flourished also Diodes, the inventor of the cissoid ("ivy-like"). This curve he used for finding two mean proportionals between two given straight lines. About the life of Perseus we know as little as about that of Nicomedes and Diocles. He lived some time between 200 and 100 B.C. Prom Heron and Geminus we learn that he wjtote a THE GREEKS. 51 work omthe spire, a sort of anchor-ring surface described by Heron as being produced by the revolution of a circle around one of its chords as an axis. The sections of this surface yield peculiar curves called spiral sections, which, according to G-eminus, were thought out by Perseus. These curves appear to be the same as the Hippopede of Eudoxus. , Probably somewhat later than Perseus lived Zenodorus. He wrote an interesting treatise on a new subject; namely, iso- perimetncal figures. Fourteen propositions are preserved by Pappus and Theon. Here are a few of them : Of isoperimet- rical, regular polygons, the one having the largest number of angles has the greatest area; the circle has a greater area than any regular polygon of equal periphery ; of all isoperimetrical polygons of n sides, the regular is the greatest ; of all solids having surfaces equal in area, the sphere has TfieT^eatest** volume. Hypsicles (between 200 and 100 B.C.) was supposed to be the author of both the fourteenth and fifteenth books of Euclid, but recent critics are of opinion that the fifteenth book was written by an author who lived several centuries after Christ. The fourteenth book contains seven elegant theorems on regular solids. A treatise of Hypsicles on Risings is of interest because it is the first Greek work giving the division of the circumference into 360 degrees after the fash ion of the Babylonians. Hipparchus of Nicsea in Bithynia was the greatest astron omer of antiquity. He established inductively the famous theory of epicycles and eccentrics. As might be expected, he was interested in mathematics, not per se, but only as an aid to astronomical inquiry. No mathematical writings of his are extant, but Theon of Alexandria informs us that Hippar- chus originated the science of trigonometry, and that he calcu lated a " table of chords " in twelve books. Such calculations 52 A HISTORY OF MATHEMATICS. must have required a ready knowledge of arithm } Sal and algebraical operations. About 155 B.C. flourished Heron the Elder of Alexandria. He was the pupil of Ctesibius, who was celebrated for his ingenious mechanical inventions, such as the hydraulic organ, the water-clock, and catapult. It is believed by some that Heron was a son of Ctesibius. He exhibited talent of the same order as did his master by the invention of the eolipile and a curious mechanism known as "Heron s fountain." Great uncertainty exists concerning his writings. Most au thorities believe him to be the author of an important Treatise on the Dioptra, of which there exist three manuscript copies, quite dissimilar. But M. Marie u thinks that the Dioptra is the work of Heron the Younger, who lived in the seventh or eighth century after Christ, and that Geodesy, another book supposed to be by Heron, is only a corrupt and defective copy of the former work. Dioptra, contains the important formula for finding the area of a triangle expressed in terms of its sides ; its derivation is quite laborious and yet exceedingly ingenious. " It seems to me difficult to believe," says Chasles, "that so beautiful a theorem should be found in a work so ancient as that of Heron the Elder, without that some Greek geometer should have thought to cite it," Marie lays great stress on this .silence of the ancient writers, and argues from it that the true author must be Heron the Younger or some writer much more recent than Heron the Elder, But no reli able evidence has been found that there actually existed a second mathematician by the name of Herory "Dioptra," says Venturi, were instramejrfs which had great resemblance to our modern theodolites. ( The book Dioptra is a treatise on geodesy containing solutions, with aid of these ^instruments, of a large number of questions in geometry, such as to find the distance between two points, of which one only THE GEBEKS. 53 is accessible, or between two points which are visible but both inaccessible ; from a given point to draw a perpendicular to a line which cannot be approached; to find the difference of level between two points ; to measure the area of a field with out entering it. Heron was a practical surveyor. This may account for the fact that his writings bear so little resemblance to those of the Greek authors, who considered it degrading the science to apply geometry to surveying. The character of his geom etry is not Grecian, but decidedly Egyptian. This fact is the more surprising when we consider that Heron demonstrated his familiarity with Euclid by writing a commentary on the Elements. 21 Some of Heron s formulas point to an old Egyp tian origin. Thus, besides the above exact formula for the area of a triangle in terms of its sides, Heron gives the for mula a * "i" a * x -, which bears a striking likeness to the for- mula i 2 x -^_ 2 for finding the area of a quadrangle, jU found in the Edfu inscriptions. There are, moreover, points of resemblance between Heron s writings and the ancient Ahmes papyrus. Thus Ahmes used unit-fractions exclusively ; Heron uses them ^oftener than other fractions. Like Ahmes and the priests at Edfu, Heron divides complicated figures into simpler ones by drawing auxiliary lines; like them, he shows, throughout, a special fondness for the isosceles trapezoid. The writings of Heron satisfied a practical wan^ and for that reason were borrowed extensively by other peoples. We find traces of them in Rome, in the Occident during the Middle Ages, and even in India. Geminus of Khodes (about 70 B.C.) published an astronomi cal work still extant. He wrote also a book, now lost, on the Arrangement of Mathematics, which contained many valuable 54 . A HISTORY OF MATHEMATICS. ff notices of the early history of Greek mathematics. Froclus and Eutocius quote it frequently. Theodosius of Tripolis is the author of a book of little merit on the geometry of the sphere. Dionysodorus of Amisus in Pontus applied the inter section of a parabola and hyperbola to the solution of a prob lem which Archimedes, in his Sphere and Cylinder, had left incomplete. The problem is "to cut a sphere so that its seg ments shall be in a given ratio." We have now sketched the progress of geometry down to the time of Christ. Unfortunately, very little is known of the history of geometry between the time of Apollonius and the beginning of the Christian era. The names of quite a number of geometers have been mentioned, but very few of their works are now extant. It is certain, however, that there were no mathematicians of real genius from Apollonius to Ptolemy, excepting Hipparchus and perhaps Heron. The Second Alexandrian School. The close of the dynasty of the Lagides which ruled Egypt from the time of Ptolemy Soter, the builder of Alexandria, :or 300 years ; the absorption of Egypt into the Roman Em pire ; the closer commercial relations between peoples of the East and of the West ; the gradual decline of paganism and spread of Christianity, these events were of far-reaching influence on the progress of the sciences, which then had their home in Alexandria. Alexandria became a commercial and intellectual emporium. Traders of all nations met in her busy streets, and in her magnificent Library, museums, lecture- halls, scholars from the East mingled with those of the West; Greeks began to study older literatures and to com pare them with their own. In consequence of this interchange of ideas the Greek philosophy became fused with Oriental THE GKEEKS. 57 The foundation of this science was laid by the illustrious Hipparchus. The Almagest is in 13 books. Chapter 9 of the first book shows how to calculate tables of chords. The circle is divided into 360 degrees, each of which is halved. The diameter is divided into 120 divisions ; each of these into 60 parts, which are again subdivided into 60 smaller parts. In Latin, these parts were called partes minutes primce and paries mmutce secundcB. Hence our names, minutes and seconds. 73 The sexagesimal method of dividing the circle is of Babylonian origin, and was known to Geminus and Hipparchus. But Ptolemy s method of calculating chords seems "original with him. He first proved the proposition, now appended to Euclid VI. (D), that "the rectangle contained by the diag onals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides." He then shows how to find from the chords of two arcs the chords of their sum and difference, and from the chord of any arc that of its half. These theorems he applied to the calcu lation of his tables of chords. The proofs of these theorems are very pretty. Another chapter of the first book in the Almagest is devoted to trigonometry, and to spherical trigonometry in particular. Ptolemy proved the lemma of Menelaus/ and also the c regula sex quantitatum. Upon these propositions he built up his trigonometry. The fundamental theorem of plane trigonome try, that two sides of a triangle are to each other as the chords of double the arcs measuring the angles opposite the two sides, was not stated explicitly by him, but was contained implicitly in other theorems. More complete are the proposi tions in spherical trigonometry. The fact that trigonometry was cultivated not for its own sake, biit to aid astronomical inquiry, explains the rather 58 A HISTOKY OF MATHEMATICS. startling fact that spherical trigonometry came to exist in a developed state earlier than plane trigonometry. The remaining books of the Almagest are on astronomy. Ptolemy has written other works which have little or no bear ing on mathematics, except one on geometry. Extracts from this book, made by Proelus, indicate that Ptolemy did not regard the parallel-axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it. Two prominent mathematicians of this time were Nicoma- chus and Theon of Smyrna. Their favourite study was theory of numbers. The investigations in this science culminated later in the algebra of Diophantus. But no important geom eter appeared after Ptolemy for 150 years. The only occupant of this long gap was Sextus Julius Africanus, who wrote an unimportant work on geometry applied to the art of war, entitled Cestes. Pappus, probably bora about 340 A.D., in Alexandria, was the last great mathematician of the Alexandrian school. His genius was inferior to that of Archimedes, Apollonius, and Euclid, -who flourished over 500 years earlier. But living, as he did, at a period when interest in geometry was declin ing, he towered above his contemporaries "like the peak of Teneriffa above the Atlantic." He is the author of a Com mentary on the Almagest, a Commentary on JSucli& s JSlernents, a Commentary on the Analemma of Diodorm, a writer of whom nothing is known. All these works are lost. Proclus, probably quoting from the Commentary on EiicUd, says that Pappus objected to the statement that an, angle equal to a right angle is always itself a right angle. The only work of Pappus still extant is his Mathematical Collections. This was originally in eight books, but the firsi THE GBEBKS. 59 and portions of the second are now missing. The Mathemat ical Collections seems to have been written by Pappus to supply the geometers of his time with a succinct analysis of the most difficult mathematical works and to facilitate the study of them by explanatory lemmas. But these lemmas are selected very freely, and frequently have little or no connection with the subject on hand. However, he gives very accurate summaries of the works of which he treats. The Mathematical Collections is invaluable to us on account of the rich information it gives on various treatises by the foremost Greek mathemati cians, which are now lost. Mathematicians of the last century considered it possible to restore lost works from the resume by Pappus alone. We shall now cite the more important of those theorems in the Mathematical Collections which are supposed to be original with Pappus. First of all ranks the elegant theorem re-dis covered by Guldin, over 1000 years later, that pie volume generated by the revolution of a plane curve which lies wholly on one side of the axis, equals the area of the curve multiplied by the circumference described by its centre of gravity. Pappus proved also that the centre of gravity of a triangle is that of another triangle whose vertices lie upon the sides of the first and divide its three sides in the same ratio. In, the fourth book are new and brilliant proposition^ on the quac|ra- trix which indicate *&& intimate -acqnafitai!K^- Wifeii curvs^i surfaces.^ He generates the quadratrix as follows : Let a spiral line be drawn upon a right circular cylinder ; then the perpendiculars to the axis of the cylinder drawn from each point of* the spiral line form the surface of a screw. A plane passed through one of these perpendiculars, making any con venient angle with the base of the cylinder, cuts the screw- surface in a curve, the orthogonal projection of which upon the base is the quadratrix. A. second mode of generation is 60 A HISTORY OF MATHEMATICS. no less admirable : If we make the spiral of Archimedes the base of a right cylinder, and imagine a cone of revolution having for its axis the side of the cylinder passing through the initial point of the spiral, then this cone cuts the cylinder in a curve of double curvature. The perpendiculars to the axis drawn through every point in this curve form the surface of a screw which Pappus here calls the plectoidal surface. A plane passed through one of the perpendiculars at any con venient angle cuts that surface in a curve whose orthogonal projection upon the plane of the spiral is the required quadra- trix. Pappus considers curves of double curvature still further. He produces a spherical spiral by a point moving uniformly along the circumference of a great circle of a sphere, while the great circle itself revolves uniformly around its diameter. He then finds the area of that portion of the surface of the sphere determined by the spherical spiral, "a complanation which claims the more lively admiration, if we consider that, although the entire surface of the sphere was known since Archimedes time, to measure portions thereof, such as spher ical triangles, was then and for a long time afterwards an unsolved problem." 8 A question which was brought into prominence jby Descartes and Hewton is the "problem of Pappus." ijGriven several straight lines in a plane, to find the locus of a point such that when perpendiculars (or ? more generally, straight lines at given angles) are drawn from it to the given lines, the product of certain ones of them shall be in a given ratio to the product of the remaining ones. It is worth noticing that it was Pappus who first found the focus of the parabola, suggested the iise of the directrix,! and pro pounded the theory of the involution of points. He solved the problem to draw through three points lying in the same straight line, three straig% lines wiiich shaft form a triangle inscribed in a given circle.* Prom the Mathematical Collections THE GREEKS. 61 many more equally difficult theorems might be quoted which are original with Pappus as far as we know. It ought to be remarked; however, that he is known in three instances to have copied theorems without giving due credit, and that he may have done the same thing in other cases in which we have no data by which to ascertain the real discoverer. About the time of Pappus lived Theon of Alexandria. He brought out an edition of Euclid s Elements with notes, which he probably used as a text-book in his classes. His commen tary on the Almagest is valuable for the many historical notices, and especially for the specimens of Greek arithmetic which it contains. Theon s daughter Hypatia, a woman celebrated for her beauty and modesty, was the last Alexandrian teacher of reputation, and is said to have been an abler philosopher and mathematician than her father. Her notes on the works of Diophantus and Apollonius have been lost. Her tragic death in 415 A.D. is vividly described in Kingsley s Hypatia. From now on, mathematics ceased to be cultivated in Alexandria. The leading subject of men s thoughts was Christian theology. Paganism disappeared, and with it pagan learning. The Neo-Platonic school at Athens struggled on a century longer. Proclus, Isidorus, and others kept up the " golden chain of Platonic succession." Proclus, the successor of Syrianus, at the Athenian school, wrote a commentary on Euclid s Elements. We possess only that on the first book, which is valuable for the information it contains on the history of geometry. Damascius of Damascus, the pupil of Tsidorus, is now believed to be the author of the fifteenth book of Euclid. Another pupil of Isidorus was Eutocius of Ascalon, the commentator of Apollonius and Archimedes. Simplicius wrote a commentary on Aristotle s De Oodo. In the year 529, Justinian, disapproving heathen learning, finally closed by imperial edict the schools at Athens. 62 A HISTORY OF MATHEMATICS. As a rule, the geometries of the last 500 years showed a lack of creative power. They were commentators rather than discoverers. The principal characteristics of ancient geometry are : (1) A wonderful clearness and defmiteness of its concepts and an almost perfect logical rigour of its conclusions. (2) A complete want of general principles and methods. Ancient geometry is decidedly special Thus the Greeks possessed no general method of drawing tangents. "The determination of the tangents to the three conic sections did not furnish any rational assistance for drawing the tangent to any other new curve, such as the conchoid, the cissoid, etc." 35 In the demonstration of a theorem, there wore, for the ancient geometers, as many different cases requiring separate proof as there were different positions for the lines. The greatest geometers considered it necessary to treat all possible cases Independently of each other, and to prove each with equal fulness. To devise methods by which the various eases could all be disposed of by one stroke, was beyond the power of the ancients. "If we compare a mathematical problem with a huge rock, into the interior of which we desire to penetrate, then the work of the Greek mathematicians appears to us like that of a vigorous stonecutter who, with chisel and hammer, begins with indefatigable perseverance, from without, to crumble the rock slowly into fragments 5 the modern mathe matician appears like an excellent minor, wlio first bores through the rock some few passages, from which he then bursts it into pieces with one powerful blast, and brings to light the treasures within." I6 THE GREEKS. 63 GREEK ARITHMETIC. G-reek mathematicians were in the habit of discriminating between the science of numbers and the art of calculation. The former they called arithmetical, the latter logistica. The drawing of this distinction between the two was very natural and proper. The difference between them is as marked as that between theory and practice. Among the Sophists -the art of calculation was a favourite study. Plato, on the other hand, gave considerable attention to philosophical arithmetic, but pronounced calculation a vulgar and childish art. In sketching the history of Greek calculation, we shall first give a brief account of the Greek mode of counting and of writing numbers. Like the Egyptians and Eastern nations, the earliest Greeks counted on their fingers or with pebbles. In case of large numbers, the pebbles- were probably ar ranged in parallel vertical lines. Pebbles on the first line represented units, those on the second tens, those on the third hundreds, and so on. Later, frames came into use/ in which strings or wires took the place of lines. According to tra dition, Pythagoras, who travelled in Egypt and, perhaps, in India, first introduced this valuable instrument into Greece. The abacus, ais it is called, existed among different peoples and at different tim$s, in various stages of perfection. An abacus is still employe! by the Chinese under the name of Sivan-pan. We possess no specific information as to how the Greek abacus looked or how it was used. Boethius says that the Pytha goreans used with the abacus certain nine signs called apices, which resembled in form the nine " Arabic numerals." But the correctness of this assertion is subject to grave doubts. The oldest Grecian numerical symbols were the so-called Herodianic signs (after Herodianus, a Byzantine grammarian of about 200 A.D., who describes them). These signs occur fre- 64 A HISTORY OF MATHEMATICS. quently in Athenian inscriptions and are, on that account, now generally called Attic. For some unknown reason these sym bols were afterwards replaced by the alphabetic numerals, in which the letters of the Greek alphabet were used, together with three strange and antique letters & 9 , and 5), and the symbol M. This change was decidedly for the worse, for the old Attic numerals were less burdensome on the memory, inas much as they contained fewer symbols and were better adapted to show forth analogies in numerical operations. The follow ing table shows the Greek alphabetic numerals and their respective values : 1 2 8 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 p<TTV<xV rcw ^/ a J /y etc. 100 200 300 400 500 600 700 800 900 1000 2000 3000 ft v M M M etc. 10,000 20,000 30,000 It will be noticed that at 1000, the alphabet is begun over again, but, to prevent confusion, a stroke is now placed before the letter and generally somewhat bolow it, A horizontal line drawn over a number served to distinguish it more readily from words. The coefficient for M was sometimes placed before or behind instead of over the M. Thus 43,678 was written SM^yx 07 ?- ^ * s * * )e observed that the Greeks had no zero. Fractions were denoted by first writing the numerator marked with an accent, then the denominator marked with two accents and written twice. Thus, ly tO^nO" |^|. In case of fractions having unity for the numerator, the a was omitted and the denominator was written only once. Thus /x8" = -$%* THE GBEBKS. 65 Greek writers seldom refer to calculation with alphabetic numerals. Addition, subtraction, and even multiplication were probably performed on the abacus. Expert mathematicians may have used the symbols. Thus Eutocius, a commentator of the sixth century after Christ, gives a great many multipli cations of which the following is a specimen : 6 The operation is ex plained sufficiently by the modern numerals append ed. In case of mixed numbers, the process was still more clumsy. Divis 265 265 8 a MM M cr/c e 40000, 12000, 1000 12000, 3600, 300 1000, 300, 25 70225 ions are found in Theon of Alexandria s commen tary on the Almagest. As might be expected, the process is long and tedious. We have seen in geometry that the more advanced mathe maticians frequently had occasion to extract the square root. Thus Archimedes in his Mensuration of the Circle gives a large number of square roots. He states, for instance, that V3 < l^y- and VS > f -f-f, but he gives no clue to the method by which he obtained these approximations. It is not im probable that the earlier Greek mathematicians found the square root by trial only. Eutocius say^ that the method of extracting it wsts given by Heron, Pappus, Theon, and other commentators on the Almagest. Theon s is the only ancient method known to us. It is the same as the one used nowa days, except that sexagesimal fractions are employed in place of our decimals. What the mode of procedure actually was when sexagesimal fractions were not used, lias been the sub ject of conjecture on the part of numerous modern writers. 17 Of interest, in connection with arithmetical symbolism, is the Sand-Counter (Arenarius), an essay addressed by Archi- 66 A HISTORY OF MATHEMATICS. medes to Gelon, king of Syracuse. In it Archimedes shows that people are in error who think the sand cannot be counted, or that if it can be counted, the number cannot be expressed by arithmetical symbols. He shows that the number of grains in a heap of sand not only as large as the whole earth, but as large as the entire universe, can be arithmetically expressed. Assuming that 10,000 grains of sand suffice to make a little solid of the magnitude of a poppy-seed, and that the diameter of a poppy-seed be not smaller than ^ part of a finger s breadth; assuming further, that the diameter of the universe (supposed to extend to the sun) be less than 10,000 diameters of the earth, and that the latter be less than 1,000,000 stadia, Archimedes finds a number which would exceed the number of grains of sancl in the sphere of the universe. He goes on even further. Supposing the universe to reach out to the fixed stars, he finds that the sphere, having the distance from the earth s centre to the fixed stars for its radius, would contain a number of grains of sancl less than 1000 myriads of tho eighth octad. In our notation, this number would be 10 (I3 or 1 with 63 ciphers after it. It can hardly be cioubtod that one object which Archimedes had in view in making this calcula tion was the improvement of the Greek symbolism. It is not known whether he invented some short notation by which to represent the above number or not. We judge from fragments in the second book of "Pappus that Apollonius proposed an improvement in the Greek method o writing numbers, but its nature wo do not know. Thus we see that the Greeks never possessed tho boon of a clear, com prehensive symbolism. The honour of giving suoli to the world, once for all, was reserved by tho irony of fate for a namdcBB Indian of an unknown time, and we. know not whom to thank for an invention of such importance to the general progress of intelligence, 6 THE GREEKS. 75 suggestions of algebraic notation, and of the solution of equations, then his Arithmetica is the earliest treatise on algebra now extant. In this work is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods. He is, as far as we know, the first to state that " a negative number multiplied by a negative num ber gives a positive number." This is applied to the multi plication of differences, such as (x l)(x 2). It must be remarked, however, that Diophantus had no notion whatever of negative numbers standing by themselves. All he knew were differences, such as (2 x 10), in which 2 x could not be smaller than 10 without leading to an absurdity. He appears to be the first who could perform such operations as (x 1) x(x 2) without reference to geometry. Such identities as (a + 6) 2 = a 2 + 2 ab + 6 2 , which with Euclid appear in the ele vated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation. His sign for subtraction was ^/, for equality i. For unknown quantities he had only one symbol, ?. He had no sign for addition except juxtaposition. Diophantus used but few sym bols, and sometimes ignored even these by describing an oper ation in words when the symbol would have answered just as well. In the solution of simultaneous equations Diophantus adroitly managed with only one symbol for the unknown quantities and arrived at answers, most commonly, by the method of tentative assumption, which consists in assigning to some of the unknown quantities preliminary values, that satisfy only one or two of the conditions. These values lead to expressions palpably wrong, but which generally suggest some stratagem by which r^lues can be secured satisfying all the conditions of the >roblem. 76 A HISTORY OF MATHEMATICS. Diophantus also solved determinate equations of the second degree. We are ignorant of Ms method, for he nowhere goes through with the whole process of solution, but merely states the result. Thus, " 84 x 2 + 7 x = 7, whence x is found = ." Notice he gives only one root. His failure to observe that a quadratic equatioti has two roots, even when both roots are positive, rather surprises us. It must be remembered, how ever, that this same inability to perceive more than one out of the several solutions to which a problem may point is common to all Greek mathematicians. Another point to be observed is that he never accepts as an answer a quantity which is negative or irrational. Diophantus devotes only the first book of his Arithmetica to the solution of determinate equations. The remaining- books extant treat mainly of indeterminate quadratic equations of the form J.& 2 +JS& 4-0=?/ 2 , or of two simultaneous equations of the same form. He considers several but not all the possible cases which may arise in these equations. The opinion of Nesselmann on the method of Diophantus, as stated by Gow, is as follows : " (1) Indeterminate equations of the second degree are treated completely only when the quadratic or the absolute term is wanting: his solution of the equations Ax*-\- (7= f and Ax 2 +Bx+ (7= ;?/ 2 is in many respects cramped. (2) Eor the double equation of the second degree he has a definite rule only when the quadratic term is wanting in both expressions : even then his solution is not general. More com plicated expressions occur only under specially favourable circumstances." Thus, he solves B% + C = ?/ 2 , B$s + d 2 = y*. The extraordinary ability of Diophantus lies rather in another direction, namely, in his wonderful ingenuity to re duce all sorts of equations to particular forms which ho knoW Jiow to solve. Very great is the variety of problems considered! The 130 problems found in the great work of Diophantus COB/- THE ROMANS* 77 tain over 50 different classes of problems, which, are strung together without any attempt at classification. But still more multifarious than the problems are the solutions. General methods are unknown to Diophantus. Each problem has its own distinct method, which is often useless for the most closely related problems. "It is, therefore, difficult for a modern, after studying 100 Diophantine solutions, to solve the 101st." 7 That which robs his work of much of its scientific value is the fact that he always feels satisfied with one solution, though his equation may admit of an indefinite number of values. Another great defect is the absence of general methods. Mod ern mathematicians, such as Euler, La Grange, Gauss, had to begin the study of indeterminate analysis anew and received no direct aid from Diophantus in the formulation of methods. In spite of these defects we cannot fail to admire the work for the wonderful ingenuity exhibited therein in the solution of particular equations. It is still an open question and one of great difficulty whether Diophantus derived portions of his algebra from Hindoo sources or not. THE BOMANS. Nowhere is the contrast .between the Greek and Eoman mind shown forth more distinctly than in their attitude toward the mathematical science. The sway of the Greek was a flowering time for mathematics, but that of the Eoman a period of sterility. In philosophy, poetry, and art the Eoman was an imitator. But in mathematics he did not even rise to the desire for imitation. The mathematical fruits of Greek genius lay before him untasted. In him a science which had 78 A HISTOJEtY OF MATHEMATICS. no direct bearing on practical life could awake no interest. As a consequence, not only the higher geometry of Archimedes and Apollonius, but even the Elements of Euclid, were en tirely neglected. What little mathematics the Romans pos sessed did not come from the Greeks, but from more ancient sources. Exactly where and how it originated is a matter of doubt. It seems most probable that the " Roman notation," as well as the practical geometry of the Romans, came from the old Etruscans, who, at the earliest period to which our knowledge of them extends, inhabited the district between the Arno and Tiber. Livy tells us that the Etruscans were in the habit of repre senting the number of years elapsed, by driving yearly a nail into the sanctuary of Minerva, and that the Romans continued this practice. A less primitive mode of designating numbers, presumably of Etruscan origin, was a notation resembling the present " Roman notation." This system is noteworthy from the fact that a principle is involved in it which is not met with in any other ; namely, the principle of subtraction. If a letter be placed before another of greater value, its value is not to be added to, but subtracted from, that of the greater. In the designation of large numbers a horizontal bar placed over a letter was made to increase its value one thousand fold. In fractions the Romans used the duodecimal system. Of arithmetical calculations, the Romans cm ploy od three different kinds : Reckoning on the fingers, upon the abacus, and by tables prepared for the purpose, 8 Finger-symbolism was known as early as the time of King Nuina, for he