HISTORY OF ALGEBRA. 163 



If the analysis of the five first propositions of the 13th book of Euclid 

 were (as is believed) by Ttmoisr, they could not well be adduced in proof 

 of the ancient Greeks having a knowledge of Algebra ; because Theon is 

 supposed to have been nearly contemporary with Diophantus. He could 

 r,ot have been long before him, if it is true, that his daughter Hypatia 

 commented on a work of Diophantus. But, be this as it may, the ana- 

 lysis of the propositions in question is not at all Algebraical. It is the 

 common analysis of the ancient geometers, which is quite different from 

 Algebra; the former being geometrical and the latter arithmetical, 

 Wallis's reasoning, on the three propositions of Archimedes, to which 

 he refers, amounts to no more than this. The demonstrations, as they 

 now stand, are difficult ; they might have been done by Algebra with ease ; 

 therefore, it is probable they were done by Algebra. We know of no 

 Greek writer on Algebra, but Diophantus ; neither he, nor any known 

 author, of any age, or of any country, has spoken, directly or indirectly, 

 of any other Greek writer on Algebra, in any branch whatever ; the Greek 

 language has not even a term to designate the science. The instance of 

 Diophantus's treatise, with some indirect and disputable arguments, 

 drawn, by inference, from works on other subjects than Algebra, is not 

 sufficient. It is unlikely that the ravages of time and the depredations of 

 barbarians should have destoyed all the direct and indisputable proofs, 

 $uch causes might account for the deficiency of our information on cer- 

 tain particulars, but will not authorise forced constructions, to argue the 

 existence of a complete science, from its supposed demolition. The ge- 

 neral extent of the literature of the Greeks, especially in mathematics, is 

 well known ; and that they had Algebra, can be established only by clear 

 and positive evidence. For the different arguments which have been 

 used, and the authorities which have been quoted on this question, see 



