Page:The New International Encyclopædia 1st ed. v. 13.djvu/211

MATHEMATICS. Of the several mathematical papyri that have come lo light in recent years, the most elaborate is that transcribed by Ahmes about 1700, from one written probably some six or eight cen- turies earlier. Mathematics in Kgypt, however, made but slight progress beyond this point until the tireek ascendency in Alexandria. The Haby- lonians were the next to show signs of niathe- matiral power, particularly in the application of arithmetic and geometry to astronomy. To them is due the development of the sexagesimal system of fractions still connnonly used in angle and lime measurements. The extensive trade of the I'hienicians also developed a commercial arithmetic among them and their neighbors, but it did not lead to any general scicntitic progress.

The real beginning of mathematics as a steadily progressing science is to be found in Greece, and in particular in the establishing of the Ionian school of Thalcs about n.c. 000. Geometry as a science here makes its appearance. The next great step in the progress of mathematics was taken by Pythagoras in founding his famous school at Croton, in Southern Italy. Under his inlhience a considerable j)art of elementary geom- etry became developed, and a beginning was made in creating a theory of numbers. (See XuM- UEi;. I Considerable (jrogress had been made in geometry before the third epoch-making step was taken, the founding of the Athenian school aliout B.C. 420. Hippocrates of Chios began the move- ment that made Athens the mathematical centre for the next century and a half. It was Plato, however, who brought the school to the zenith of its fame. Although he was not, strictly S]ieak- ing, a mathematician, his ideas concerning the methods of establishing truths in philosophy and science gave a powerful impulse to the progress of mathematics. The third century B.C. saw the rise of the great Alexandrian school, where Euclid taught, and Archimedes, ApoUonius, and Eratosthenes studied, ^'ith that century closes the Hellenic ascendency in mathematics and philo.sophy, and thenceforth we find scientific progress sporadic and short-lived. By the second century of our era progress had jiractically ceased. Hero and Ptolemy were the greatest of the later Greek writers on applied geometry. The only new movement in mathematics made by the post-Christian Greeks was that of Diophan- tus, whose work on equations is the first of any pretensions ever compo.sed. The Romans did almost nothing in mathematics except in a pure- ly mercantile way, their only contribution being to the practical work of surveying. Among the later Romans the name of the philosoplier Boetliius stands out with some prominence for his text-book work in elementary mathematics, but he displayed no originality. The same must be said for such mediaeval writers as Alcuin, Gerbert ( see Sylve.ster ), and Bede.

Meanwhile mathematics had obtained a foothold in the East. The first definite trace of really satisfactory work among the Oriental peoples is that of Aryabhatta early in the sixth century . Aryaidiatta possessed considerable know- leilge of the theory of numbers, of algebra, and of the first principles of trigonometry. The next Hin<lu mathematician of great prominence was Braluuagupta. who lived in the seventh century, anil whose work on arithmetic and algebra and on the mensuration of solids is a distinct ad- vance on that of his predecessors. The list of prominent Hindu mathematicians closes with Bhaskara in the twelfth century, in whose work a fairly well developed algebraic symbolism, is found. It was among the Hindus, too, that our present numeral system was born, being by them transmitted, through the Arabs, to Euro]ie. (See XuMEi{.LS.) One of the most interesting peri- • ods in the development of mathematics is that of the Arab ascendency, and in particular that of the founding of the great school at Bagdad. In this school one of the first teachers was Al- kliow'arazmi, who gave the name to algebra in the ninth century. He was followed by several writers of prominence, but it is rather by their preservation of Greek and Hindu learning than by their own originality that they are note- w'orthy. Among the last of the Persian and Arab writers was the poet Omar Khayyam, whose work in algebra showed considerable power. The work of the Arabs in Sjiaiu was rather that of teaching than of contributing to scientific advance.

The first of the European writers to contribute in any large way to the advance of mathematics was Leonardo of Pisa, at the opening of the thir- teenth century. His Lihcr Ahlmci placed before Italian scholars the Hindu number system (al- ready slightly known), and the mathematical knowledge of "the world at that titne. The period of the Renaissance was one of great activity in mathematics. This activity was inaugurated in Austria by Regiomontanus and Peuerbach. and in Germany "by Widmann. In Italy, Paccioli was the first to publish, in 1404. any printed work of much importance on mathcnuitics, although several minor works had already appeared, nota- bly one on arithmetic printed at Treviso in 1478, and two printed at Baml)erg in 1482-83. During the sixteenth century the Italian alge- braists, notably Tartaglia, Ferro, Cardan, Fer- rari, and Bombelli, solved completely the cubic and biquadratic equations, and Vieta. in France, so improved the symbolism of algebra and so generalized the use" of letters as to put algebra upon substantially the present foundation. It ncM>ded only the symbolism sugge.sted by Descartes and a few of his contemporaries to bring ele- mentary algebra, about 1050, to the form fa- miliar "to students at the present day.

About the time that elementary algebra was becoming crystallized, a revival of interest in geometry took place. On the side of pure ge- ometry this was led by Kepler. Desargues. and Pascal, while to Descartes is due the invention of the method of analytic geometry. At the same period Fermat laid the foundatiim for the modern theory of numbers, and the new theory of logarithms (q.v.) became generally known. The greatest progress in the seventeenth century is, how-ever, represented by the invention of tlic fluxional calculus by Newton, and of the differ- ential calculus by "Leibnitz. These disciplines, essentially the same and so considered at present, revolutionized mathematics and its applications.

The period of the development of elementary mathematics closes with the seventeenth century. The eighteenth century was devoted largely to the investigations of the foundations of the new- analysis, to a consideration of its applications, to the study of infinite series (see Series), and to the understanding of the nature of complex numbers (q.v.). The thirteenth century saw the development of the so-called modern mathemat-