GREEK SCIENCE IN ALEXANDRIA 111 



gested expressing the motions of the planets by combining uniform 

 circular motions, an idea afterwards elaborated by Hipparchus 

 and Ptolemy. How far his mathematical results were new, how 

 far he merely compiled and coordinated the work of others, notably 

 Euclid and Archimedes, cannot be precisely determined, but the 

 proportion of original work is certainly very large. 



On the arithmetical side he obtained a closer approximation 

 than Archimedes for the value of IT, invented an abridged method 

 of multiplication, and employed numbers of higher order in the 

 manner of Archimedes. This last experiment if followed out to 

 its logical conclusions might have had fundamental significance 

 for the future development of computation. In the words of 

 Gow: 



he, as well as Archimedes, lost the chance of giving to the world 

 once for all its numerical signs. That honor was reserved by the 

 irony of fate for a nameless Indian of an unknown time, and we know 

 not whom to thank for an invention which has been as important as 

 any to the general progress of intelligence. 



APOLLONIUS AND ARCHIMEDES. With Apollonius and Archi- 

 medes the ancient mathematics had accomplished whatever was 

 possible without the resources of analytic geometry and infinitesi- 

 mal calculus, which, though already foreshadowed, were not fully 

 realized until the seventeenth century. 



It is not only a decided preference for synthesis and a complete 

 denial of general methods which characterize the ancient mathematics 

 as against our newer science (modern mathematics) : besides this 

 external formal difference there is another real, more deeply seated, 

 contrast, which arises from the different attitudes which the two as- 

 sumed relative to the use of the concept of variability. For while the 

 ancients, on account of considerations which had been transmitted to 

 them from the philosophic school of the Eleatics, never employed the 

 concept of motion, the spatial expression for variability, in their 

 rigorous system, and made incidental use of it only in the treatment 

 of phoronomically generated curves, modern geometry dates from the 

 instant that Descartes left the purely algebraic treatment of equations 



