A History of Mathematics

Front matter

A HISTORY OF MATHEMATICS

A HISTORY OF

MATHEMATICS

Errata

Pages xi to xiii, Nos. 47, 67, 84, 88, 97. For Gedächniss read Gedächtniss. No. 50. For 1887 read 1877. No. 65. For fur read für. No. 76. second line. For der read für. No. 96. For Bendiconti read Rendiconti.

Page 3, line 4 up. For accomplishment, sacrificed read accomplishment, is said to have sacrificed.

Page 31, line 10. For compass read compasses.

Page 36, line 2 up. For VII. read XII.

Page 39, lines 1 to 6 up. Chasles's or Simson's definition of a Porism is preferable to Proclus's, given in the text. See Gow, pp. 217-221.

Page 52, line 3. For About 155 B.C. read About 155 B.C., or, according to M. Cantor, about 100 B.C.

Page 62, line 1. For geometries read geometers.

Page 75, line 19. For read .

Page 106, line 1. For Mohammed ben Musa Hovarezmi read Mohammed ben Musa Al Hovarezmi, or read Muhammed ibn Musa Alchwarizmi.

Page 106, line 3. For 814 read 813.

Page 106, line 11. For Hovarezmi read Al Hovarezmi, or read Alchwarizmi.

Page 111, line 14. For Fahri des Al Karhi read Abu Bekr Mohammed ben Al Hasan Al Karhi.

Page 130, line 2. For Casso read Cassio.

Page 134, line 14. For small series read large series.

Page 142, line 18. For go read got.

Page 174, line 26. For Midorge read Mydorge.

Page 178, line 5; also, pp. 288, 289. For Branchion read Brianchon.

Page 220, line 12. For circle:—read circle, previously discovered by James Gregory:—

Page 226, line 5. For Thomas Craige read John Craig.

Page 233, line 6. For 1725 read 1726.

Page 241, line 19. For 1631—1708 read 1651—1708.

Page 285, line 3. For 1765—1845 read 1765—1842.

Page 308, line 3. For Untersuchungen read Entwicklungen.

Page 330, line 7. For Moor read Moore.

Page 335, line 12. For series was made by Gauss read series, excepting a test for convergence by Maclaurin (afterwards re-discovered by Cauchy), was made by Gauss.

Alphabetical Index. For Tchirnhausen read Tschirnhausen; under "Todhunter," for I., XIII. read XI. Insert the following new references: under "De Moivre" add 242, 245; under "Elements" (Euclid's) add 61; under "Laplace" add 257; under "Moivre, de" add 240, 242; under "Substitutions" add 329.

Preface

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 criticisms on the chapter on "Recent Times" have been made by Dr. E. W. Davis, of the University of Nebraska. The proof-sheets of 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. Hoskins, of the Leland Stanford Jr. University; and Professor G. D. Olds, of Amherst College,—all of whom have afforded valuable assistance. I am specially indebted to Professor F. 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 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 text.

FLORIAN CAJORI.

Colorado College, December, 1893.

Books of Reference

BOOKS OF REFERENCE.

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.

1Günther, S. Ziele und Resultate der neueren Mathematisch-historischen Forschung. Erlangen, 1876. 2Cajori, F. The Teaching and History of Mathematics in the U.S. Washington, 1890.

3Cantor, Moritz. Vorlesungen über Geschichte der Mathematik. Leipzig. Bd. I., 1880; Bd. II., 1892.

4Epping, J. Astronomisches aus Babylon. Unter Mitwirkung von P. J. R. Strassmaier. Freiburg, 1889. 5 Bretschneider, C. A. Die Geometrie und die Geometer vor Euklides. Leipzig, 1870.

6Gow, James. A Short History of Greek Mathematics. Cambridge, 1884.

7Hankel, Hermann. Zur Geschichte der Mathematik im Alterthum und Mittelalter. Leipzig, 1874.

8Allman, G. J. Greek Geometry from Thales to Euclid. Dublin, 1889.

9 De Morgan, A. "Euclides" in Smith's Dictionary of Greek and Roman Biography and Mythology. 10 Hankel, Hermann. Theorie der Complexen Zahlensysteme. Leipzig, 1867. 11 Whewell, William. History of the Inductive Sciences. 12 Zeuthen, H. G. Die Lehre von den Kegelschnitten im Alterthum. Kopenhagen, 1886.

13Chasles, M. Geschichte der Geometrie. Aus dem Französischen tibertragen durch Dr. L. A. Sohncke. Halle, 1839.

14Marie, Maximilien. Histoire des Sciences Mathématiques et Physiques. Tome I.-XII. Paris, 1883-1888. 15Comte, A. Philosophy of Mathematics, translated by W. M. Gillespie. 16Hankel, Hermann. Die Entwickelung der Mathematik in den letzten Jahrhunderten. Tübingen, 1884. 17Günther, Siegmund und Windelband, W. Geschichte der antiken Naturwissenschaft und Philosophie. Nördlingen, 1888. 18Arneth, A. Geschichte der reinen Mathematik. Stuttgart, 1852. 19Cantor, Moritz. Mathematische Beiträge zum Kulturleben der Völker. Halle, 1863. 20Matthiessen, Ludwig. Grundzüge der Antiken und Modernen Algebra der Litteralen Gleichungen. Leipzig, 1878. 21Ohrtmann und Müller. Fortschritte der Mathematik. 22Peacock, George. Article "Arithmetic," in The Encyclopædia of Pure Mathematics. London, 1847. 23Herschel, J. F. W. Article "Mathematics," in Edinburgh Encyclopædia. 24Suter, Heinrich. Geschichte der Mathematischen Wissenschaften. Zürich, 1873-75. 25Quetelet, A. Sciences Mathématiques et Physiques chez les Belges. Bruxelles, 1866. 26Playfair, John. Article "Progress of the Mathematical and Physical Sciences," in Encyclopædia Britannica, 7th edition, continued in the 8th edition by Sir John Leslie. 27De Morgan, A. Arithmetical Books from the Invention of Printing to the Present Time. 28Napier, Mark. Memoirs of John Napier of Merchiston. Edinburgh, 1834. 29Halsted, G. B. "Note on the First English Euclid," American Journal of Mathematics, Vol. II., 1879. 30Madame Perier. The Life of Mr. Paschal. Translated into English by W. A., London, 1744. 31Montucla, J. F. Histoire des Mathématiques. Paris, 1802. 32Dühring E. Kritische Geschichte der allgemeinen Principien der Mechanik. Leipzig, 1887. 33Brewster, D. The Memoirs of Newton. Edinburgh, 1860. 34Ball, W. W. R. A Short Account of the History of Mathematics. London, 1888, 2nd edition, 1893. 35De Morgan, A. "On the Early History of Infinitesimals," in the Philosophical Magazine, November, 1852. 36Bibliotheca Mathematica, herausgegeben von Gustaf Eneström, Stockholm. 37Günther, Siegmund. Vermischte Untersuchungen zur Geschichte der mathematischen Wissenschaften. Leipzig, 1876.

38Gerhardt, C. I. Geschichte der Mathematik in Deutschland. München, 1877.

39Gerhardt, C. I. Entdeckung der Differenzialrechnung durch Leibniz. Halle, 1848. 40Gerhardt, K. I. "Leibniz in London," in Sitzungsberichte der Königlich Preussischen Academie der Wissenschaften zu Berlin, Februar, 1891. 41De Morgan, A. Articles "Fluxions" and "Commercium Epistolicum," in the Penny Cyclopædia.

42Todhunter, I. A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace. Cambridge and London, 1865.

43Todhunter, I. A History of the Theory of Elasticity and of the Strength of Materials. Edited and completed by Karl Pearson. Cambridge, 1886.

44Todhunter, I. "Note on the History of Certain Formulæ in Spherical Trigonometry," Philosophical Magazine, February, 1873. 45Die Basler Mathematiker, Daniel Bernoulli und Leonhard Euler. Basel, 1884. 46Reiff, R. Geschichte der Unendlichen Reihen. Tübingen, 1889. 47Waltershausen, W. Sartorius. Gauss, zum . Leipzig, 1856. 48Baumgart, Oswald. Ueber das Quadratische Reciprocitätsgesetz. Leipzig, 1885. 49Hathaway, A. S. "Early History of the Potential," Bulletin of the N. Y. Mathematical Society, I. 3. 50Wolf, Rudolf. Geschichte der Astronomie. München, . 51Arago, D. F. J. "Eulogy on Laplace." Translated by B. Powell, Smithsonian Report, 1874. 52Beaumont, M. Élie De. "Memoir of Legendre." Translated by C. A. Alexander, Smithsonian Report, 1867. 53Arago, D. F. J. "Joseph Fourier." Smithsonian Report, 1871. 54Wiener, Christian. Lehrbuch der Darstellenden Geometrie. Leipzig, 1884.

55Loria, Gino. Die Hauptsächlichsten Theorien der Geometrie in ihrer früheren und heutigen Entwickelung, ins deutsche übertragen von Fritz Schütte. Leipzig, 1888.

56Cayley, Arthur. Inaugural Address before the British Association, 1883. 57Spottiswoode, William. Inaugural Address before the British Association, 1878. 58Gibbs, J. Willard. "Multiple Algebra," Proceedings of the American Association for the Advancement of Science, 1886. 59Fink, Karl. Geschichte der Elementar-Mathematik. Tübingen, 1890. 60Wittstein, Armin. Zur Geschichte des Malfatti'schen Problems. Nördlingen, 1878. 61Klein, Felix. Vergleichende Betrachtungen über neuere geometrische Forschungen. Erlangen, 1872. 62Forsyth, A. R. Theory of Functions of a Complex Variable. Cambridge, 1893. 63Graham, R. H. Geometry of Position. London, 1891. 64Schmidt, Franz. "Aus dem Leben zweier ungarischer Mathematiker Johann und Wolfgang Bolyai von Bolya." Grunert's Archiv, 48:2, 1868. 65Favaro, Anton. "Justus Bellavitis," Zeitschrift Mathematik und Physik, 26:5, 1881. 66Dronke, Ad. Julius Plücker. Bonn, 1871. 67Bauer, Gustav. auf Otto Hesse. München, 1882. 68Alfred Clebsch. Versuch einer Darlegung und Würdigung seiner wissenschaftlichen Leistungen von einigen seiner Freunde. Leipzig, 1873. 69Haas, August. Versuch einer Darstellung der Geschichte des Krümmungsmasses. Tübingen, 1881. 70Fine, Henry B. The Number-System of Algebra. Boston and New York, 1890. 71Schlegel, Victor. Hermann Grassmann, sein Leben und seine Werke. Leipzig, 1878. 72Zahn, W. v. "Einige Worte zum Andenken an Hermann Hankel," Mathematische Annalen, VII. 4, 1874. 73Muir, Thomas. A Treatise on Determinants. 1882. 74Salmon, George. "Arthur Cayley," Nature, 28:21, September, 1883. 75Cayley, A. "James Joseph Sylvester," Nature, 39:10, January, 1889. 76Burkhardt, Heinrich. "Die Anfänge der Gruppentheorie und Paolo Ruffini," Zeitschrift Mathematik und Physik, Supplement, 1892. 77Sylvester, J. J. Inaugural Presidential Address to the Mathematical and Physical Section of the British Association at Exeter. 1869. 78Valson, C. A. La Vie et les travaux du Baron Cauchy. Tome I., II., Paris, 1868. 79Sachse, Arnold. Versuch einer Geschichte der Darstellung willkürlicher Funktionen einer variablen durch trigonometrische Reihen. Göttingen, 1879. 80Bois-Reymond, Paul du. Zur Geschichte der Trigonometrischen Reihen, Eine Entgegnung. Tübingen. 81Poincaré, Henri. Notice sur les Travaux Scientifiques de Henri Poincaré. Paris, 1886. 82Bjerknes, C. A. Niels-Henrik Abel, Tableau de sa vie et de son action scientifique. Paris, 1885. 83Tucker, R. "Carl Friedrich Gauss," Nature, April, 1877. 84Dirichlet, Lejeune. auf Carl Gustav Jacob Jacobi. 1852. 85Enneper, Alfred. Elliptische Funktionen. Theorie und Geschichte. Halle a/S., 1876. 86Henrici, O. "Theory of Functions," Nature, 43:14 and 15, 1891. 87Darboux, Gaston. Notice sur les Travaux Scientifiques de M. Gaston Darboux. Paris, 1884. 88Kummer, E. E. auf Gustav Peter Lejeune-Dirichlet. Berlin, 1860. 89Smith, 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. 90Glaisher, J. W. L. "Henry John Stephen Smith," Monthly Notices of the Royal Astronomical Society, XLIV., 4, 1884. 91Bessel als Bremer Handlungslehrling. Bremen, 1890. 92Frantz, J. Festrede aus Veranlassung von Bessel's hundertjährigem Geburtstag. Königsberg, 1884. 93Dziobek, O. Mathematical Theories of Planetary Motions. Translated into English by M. W. Harrington and W. J. Hussey. 94Hermite, Ch. "Discours prononcé devant le président de la République," Bulletin des Sciences Mathématiques, XIV., Janvier, 1890. 95Schuster, Arthur. "The Influence of Mathematics on the Progress of Physics," Nature, 25:17, 1882. 96Kerbedz, E. de. "Sophie de Kowalevski," del Circolo Matematico di Palermo, V., 1891. 97Voigt, W. Zum von G. Kirchhoff. Göttingen, 1888. 98Bôcher, Maxime. "A Bit of Mathematical History," Bulletin of the N. Y. Math. Soc., Vol. II., No. 5. 99Cayley, Arthur. Report on the Recent Progress of Theoretical Dynamics. 1857. 100Glazebrook, R. T. Report on Optical Theories. 1885. 101Rosenberger, F. Geschichte der Physik. Braunschweig, 1887-1890.

Introduction

A HISTORY OF MATHEMATICS

INTRODUCTION.

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 mathematics 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 development, 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; it may not only remind us of what we have, but may also teach us how to increase our store. Says De Morgan, "The early history of the mind of men with regard to mathematics leads us to point out our own errors; and in this respect it is well to pay attention to the history of mathematics." 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 investigators, by showing how apparently distinct branches have been found to possess unexpected connecting links; it saves the student from wasting time and energy upon problems 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 reconnoitre and occupy the surrounding ground and to discover the secret paths by which the apparently unconquerable position can be taken.A History of Mathematics (1893)/Books of Reference#1|[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 failures to solve the problem at a time, even, when investigator possessed that most powerful tool, the differential calculus, persons versed in mathematics dropped the subject, while those who still persisted were completely ignorant of its history and generally misunderstood the conditions of the problem. "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 in which a question to be solved by a definite method was tried by the best heads, and answered at last, by that method, after thousands 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 the ratio of the circumference of a circle to its diameter is incommensurable. Some years ago, Lindemann demonstrated that this ratio is also transcendental and that the quadrature of the circle, by means of the ruler and compass only, is impos . 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 mathematics. The interest which pupils take in their studies may be greatly increased if the solution of problems and the cold logic of geometrical demonstrations are 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 Columbus-egg—the zero; 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 trisection 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 energies on the theorem of the right triangle, tell them the legend about its discoverer—how Pythagoras, jubilant over his great a hecatomb to the Muses who inspired him. When the value of mathematical training is called in question, quote the inscription over the entrance into the academy of Plato, the philosopher: "Let no one who is unacquainted with geometry enter here." Students in analytical geometry should know something of Descartes, and, after taking up the differential and integral calculus, they should become familiar with the parts that Newton, 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.A History of Mathematics (1893)/Books of Reference#2|[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

ANTIQUITY.

The Babylonians

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 foundation, in Chaldæa and Babylonia, of a united kingdom out of the 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 stood for 1, while the characters and signified 10 and 100 respectively. Grotefend believes the character for 10 originally to have been the picture of two hands, as held in prayer, the palms being pressed together, the fingers close to each other, but the thumbs thrust out. In the Babylonian notation two principles were employed—the additive and multiplicative. Numbers below 100 were expressed by symbols whose respective values had to be added. Thus, stood for 2, for 3, for 4, for 23, for 30. Here the symbols of higher order appear always to the left of those of lower order. In writing the hundreds, on the other hand, a smaller symbol was placed to the left of the 100, and was, in that case, to be multiplied by 100. Thus, signified

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, 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.A History of Mathematics (1893)/Books of Reference#3|[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, but 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 . The numbers 1, 4, 9, 16, 25, 36, 49, are given as the squares of the first seven integers respectively. We have next , , , , etc. This remains unintelligible, unless we assume the sexagesimal scale, which makes , , . The second tablet records the magnitude of the illuminated portion of the moon's disc for every day from new to full moon, the whole disc being assumed to consist of 240 parts. The illuminated parts during the first five days are the series 5, 10, 20, 40, 1.20 (), which is a geometrical progression. From here on the series becomes an arithmetical progression, the numbers from the fifth to the fifteenth day being respectively 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4. This table not only exhibits the use of the sexagesimal system, but also indicates the acquaintance of the Babylonians with progressions. Not to be overlooked is the fact that in the sexagesimal notation of integers the "principle of position" was employed. Thus, in 1.4 (), 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 systematic 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 geometer Hypsicles and the Alexandrian astronomer Ptolemæus 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 sexagesimal 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 fact that the radius can be applied to its circumference 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 to 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-Euphrates basin had made very creditable advance in arithmetic. Their knowledge of arithmetical and geometrical progressions has already been alluded to. Iamblichus 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 . Of geometrical demonstrations there is, of course, no trace. "As a rule, in the Oriental mind the intuitive powers eclipse the severely rational and logical."

The astronomy of the Babylonians has attracted much attention. They worshipped the heavenly bodies from the earliest historic times. When Alexander the Great, after 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 astronomer, possessed a Babylonian record of eclipses going back to 747 B.C. Recently Epping and StrassmaierA History of Mathematics (1893)/Books of Reference#4|[4] threw considerable light on Babylonian chronology and astronomy by explaining two calendars of the years 123 B.C. and 111 B.C., taken from cuneiform tablets coming, 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 nakshatras 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

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 Nile, 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 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 Phœdrus says: "At the Egyptian city of Naucratis there was a famous old god whose 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."

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, Iamblichus, and other ancient writers to have originated in Egypt.A History of Mathematics (1893)/Books of Reference#5|[5] In Herodotus we find this (II. c. 109): "They said also that this king [Sesostris] divided the land among all Egyptians so as to give each one a quadrangle of equal size and to draw from each his revenues, by imposing a tax to be levied yearly. But every one from whose part the river tore away anything, had to go to him and notify what had happened; he then sent the overseers, who had to measure out by how much the land had become smaller, in order that the owner might pay on what was left, in proportion to the entire tax imposed. In this way, it appears to me, geometry originated, which passed thence to Hellas."

We abstain from introducing additional Greek opinion regarding Egyptian mathematics, or from indulging in wild conjectures. We rest our account on documentary evidence. A hieratic papyrus, included in the Rhind collection of the 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 Ahmes some time before 1700 B.C., and was founded on an older work believed by Birch to date back as far as 3400 B.C.! This curious papyrus—the most ancient mathematical handbook known to us—puts us at once in contact with the mathematical thought in Egypt of three or five thousand years ago. It is entitled "Directions for obtaining the Knowledge of all Dark Things." We see from it that the Egyptians cared but little for theoretical results. Theorems are not found in it at all. It contains "hardly any general rules of procedure, but chiefly mere statements of results intended possibly to be explained by a teacher to his pupils."A History of Mathematics (1893)/Books of Reference#6|[6] In geometry the forte of the Egyptians lay in making constructions and determining areas. The area of an isosceles triangle, of which the sides measure 10 ruths and the base 4 ruths, was erroneously given as 20 square ruths, or half the product of the base by one side. The area of an isosceles trapezoid is found, similarly, by multiplying half the sum of the parallel sides by one of the non-parallel sides. The area of a circle is found by deducting from the diameter of its length and squaring the remainder. Here is taken , a very fair approximation.A History of Mathematics (1893)/Books of Reference#6|[6] The papyrus explains also such problems as these,—To mark out in the field a right triangle whose sides are 10 and 4 units; or a trapezoid whose parallel sides are 6 and 4, and the non-parallel sides each 20 units.

Some problems in this papyrus seem to imply a rudimentary knowledge of proportion.

The base-lines of the pyramids run north and south, and east and west, but probably only the lines running north and south were determined by astronomical observations. This, coupled with the fact that the word harpedonaptæ, applied to Egyptian geometers, means "rope-stretchers," would point to the conclusion that the Egyptian, like the Indian and Chinese geometers, constructed a right triangle upon a given line, by stretching around three pegs a rope consisting of three parts in the ratios 3:4:5, and thus forming a right triangle.A History of Mathematics (1893)/Books of Reference#3|[3] If this explanation is correct, then the Egyptians were familiar, 2000 years B.C., with the well-known property of the right triangle, for the special case at least when the sides are in the ratio 3:4:5.