DECLINE OF ALEXANDRIAN SCIENCE 135 



ample, he gives but one root, even when both are positive. Nega- 

 tive numbers are for him unreal, and he also avoids the irrational. 

 He admits fractional results however, and is indeed the first Greek 

 for whom a fraction is a number rather than a mere ratio of two 

 numbers. For the solution of pure equations his rule is: "If a 

 problem leads to an equation containing the same powers of the 

 unknown on both sides but not with the same coefficients, you 

 must deduct like from like till only two equal terms remain. 

 But when on one side or both some terms are negative, you must 

 add the negative terms to both sides till all the terms are positive 

 and then deduct as before stated." 



His method for general quadratics is not given. He solves one 

 cubic equation, also particular cases of the indeterminate equation 

 Ax 2 + Bx + C = F 2 . The modern so-called Diophantine equa- 

 tions involving the solution in integers of one or more indeter- 

 minate equations, do not occur in his own extant work. 



In 130 indeterminate equations, which Diophantus treats, there 

 are more than 50 different classes. ... It is therefore difficult for a 

 modern, after studying 100 Diophantic equations, to solve the 101st ; 

 and if we have made the attempt, and after some vain endeavours read 

 Diophantus ' own solution, we shall be astonished to see how suddenly 

 he leaves the broad high-road, dashes into a side-path and with a quick 

 turn reaches the goal, often enough a goal with reaching which we 

 should not be content ; we expected to have to climb a toilsome path, 

 but to be rewarded at the end by an extensive view ; instead of which, 

 our guide leads by narrow, strange, but smooth ways to a small emi- 

 nence ; he has finished ! He lacks the calm and concentrated energy 

 for a deep plunge into a single important problem ; and in this way the 

 reader also hurries with inward unrest from problem to problem, as 

 in a game of riddles, without being able to enjoy the individual one. 

 Diophantus dazzles more than he delights. He is in a wonderful 

 measure shrewd, clever, quick-sighted, indefatigable, but does not 

 penetrate thoroughly or deeply into the root of the matter. As his 

 problems seem framed in obedience to no obvious scientific necessity, 

 but often only for the sake of the solution, the solution itself also 

 lacks completeness and deeper signification. He is a brilliant performer 

 in the art of indeterminate analysis invented by him, but the science has 



