Sleanwhile mathematics had obtained a foot hold in the East. The first definite trace of real ly satisfactory work among the Oriental peoples is that of Aryabhatta early in the sixth century Aryabhatta possessed considerable know ledge of the theory of numbers. of algebra, and of the first principles of trigonometry. The next Hindu mathematician of great prominence was Brahmagupta. who lived in the seventh century. and 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 Ithaskam 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 Europe. (See NUMERALS.) 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 khow•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 preservatidn of Greek and Hindu learning than by their own originality that they are note worthy. 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 Spain 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 Liber .1 bbari placed before Italian scholars the Hindu number system (al ready slightly kuown), and the mathematical knowledge of the world at that time. The period of the Renaissance was one of great activity in mathematics. This activity was inaugurated in Austria by Regiomontanus and Penerbach, and in Germany by Widmann. In Italy, Paccioli was the first to publish, in 1494, any printed work of nmeh importance on mathematics, although several minor works had already appeared. nota
bly one on arithmetic printed at Treviso in 1478, and two printed at Bamberg in l4S2-83. During the sixteenth century the Italian alge braists, notably Tartaglia. Ferro. Cardan. Fer rari, and Bombelli, solved completely the cubic and biquadratie 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 needed only the symbolism suggested by Descartes and a few of his contemporaries to bring ele mentary* algebra, about 1650, 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, Descartes, and Pascal, while to Descartes is due the invention of the method of analytic geometry. At the same period Fermat laid the foundation 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. however, represented by the invention of the fluxional calculus by Newton, and of the differ ential calculus by Leibnitz. These essentially the same and so considered at present. revolutionized mathematics and its The period of the development of elementary mathematics closes with 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 ice, including subjects discmo-vd in the articles on sunsriTurioxs; QUATERN IONS :'+URFACES CURVE : COMPLEX Nym HER : DETERMINANTS; FI-NuTIoNS ; and the more general articles on ALGEBRA. GEOMETRY, TRIGONOMETRY, N LT31 RER, and CALCULI'S.