THE HISTORY OF MATHEMATICS 393 



being that of Diophantus, whose main contribution was a study of in- 

 determinate equations. Algebra seems to have been developing very 

 slowly and naturally, first by abbreviation of the words of a statement, 

 next by a typical word (heap) and later by a symbol to represent the 

 unknown, and finally by the adoption of symbols for common opera- 

 tions, like those of addition, equality, and so on. Diophantus took a 

 great step in this direction, and he may be regarded as the father of 

 Algebra in the main stream of the development of mathematics, in the 

 sense that he placed it on a basis which rendered it capable of develop- 

 ment. However, his work received no recognition at the time and was 

 not revived for more than a thousand years. 



We must now say something about some of the tributaries which 

 paralleled the main Grecian stream and connected with it through the 

 Moorish invasion of Europe during the 7th and 8th centuries. During 

 the time of their dominance the Romans, if they contributed, little, at 

 least allowed development to continue in the centers of civilization. 

 The Arabs, however, following the example set by the theologians, who 

 by this time were beginning to come into power, had little use for 

 scientific works and investigation, and between them they destroyed 

 completely the greatest library and museum of antiquity, that of Alex- 

 andria, so that most of the written learning of that and of previous 

 ages was lost for all time. They did however form a valuable connec- 

 tion with the Hindu mathematics of their time and appeared not only 

 to have brought it to Europe, but to have assimilated it themselves to 

 some considerable extent. 



The origin of Hindu mathematics seems very uncertain. Apparently 

 some centuries before our era, they had developed arithmetic further 

 than the Grecians of the same period, but we do not know to what ex- 

 tent the Persian conquest might have been responsible for introducing 

 early Greek learning into India. It is however true, that from what- 

 ever source they started, they developed arithmetic and geometry from 

 the mensuration point of view further than the Greeks who had con- 

 sidered geometry from an abstract standpoint. In illustration of this, 

 it may be mentioned that they had found good approximations to 

 V2 and to -n. But their greatest contribution is our present number 

 system, which came to Europe through the Arabs and hence got its 

 name. From the seventh century it does not appear that they formed 

 an independent school. Hindu mathematics seems to have been largely 

 improved by the needs of astronomy. Their greatest exponents were 

 Brahmagupta, who lived in the seventh century A. D., and Bhaskara, 

 some five centuries later. Their work, again, is chiefly arithmetical. 

 While the so-called arabic numbers were probably used by the former, 

 the latter was the first who is known to have given a systematic treat- 

 ment of the decimal system. Although the Arabs dispersed the western 



