26 EPOCHS IN THE HISTORY OF THE CONTEMPLATION" 



the old Indian algebraists, namely, the establishment of a 

 conclusion by the successive advance from proposition tor 

 proposition. This fair inheritance, yet farther increased by 

 their own exertions, passed in the twelfth century from the 

 Arabs, to the European literature of the middle ages through 

 Johannes Hispalensis and Gerard of Cremona. ( 357 ) " In the 

 algebraical works of. the Indians we find the general solution . 

 of indeterminate equations of the first degree, and a far more 

 highly finished treatment of those of the second degree, than 

 in the writings of the Alexandrian school which have come 

 down to us.; there is. therefore no doubt, that if the works 

 of the Indian writers had been made known to Europeans: 

 two centuries earlier, instead of only in our own time,, they 

 must have aided the development of modern analysis." 



The Arabs in Persia and on the Euphrates, as well as in 

 Arabia, received in the 9th century,, the knowledge of the. 

 Indian numerical characters, through channels similar to those 

 whicli had led to their acquaintance with Indian algebra. Per- 

 sians were employed at that period as revenue collectors on 

 the Indus; and the use of Indian numbers became general 

 amongst the Arab revenue officers, and extended to Northern 

 Africa, opposite to the coast of, Sicily. Nevertheless, the 

 profound and important historical investigations to which a 

 distinguished mathematician, M. Chasles, was led, by his 

 correct interpretation of the so-called Pythagorean table in, 

 the geometry of Boethius, ( 358 ) render it more than probable 

 that the Christians in the West were acquainted even earlier, 

 than the Arabians with the Indian system of numeration; . 

 the ube of the nine figures, having their values determined-., 

 by position, being known by tiiem under the name of the 

 system of the Abacus, 



