136 A SHORT HISTORY OF SCIENCE 



nevertheless been indebted, at least directly, to this brilliant genius 

 for few methods, because he was deficient in the speculative thought 

 which sees in the True more than the Correct. That is the general 

 impression which I have derived from a thorough and repeated study 

 of Diophantus ' arithmetic. Hankel. 



On the other hand Euler remarks : 



Diophantus himself, it is true, gives only the most special solu- 

 tions of all the questions which he treats, and he is generally content 

 with indicating numbers which furnish one single solution. But it 

 must not be supposed that his method was restricted to these very 

 special solutions. In his time the use of letters to denote undeter- 

 mined numbers was not yet established, and consequently the more 

 general solutions which we are now enabled to give by means of such 

 notation could not be expected from him. Nevertheless, the actual 

 methods which he uses for solving any of his problems are as general 

 as those which are in use to-day ; nay, we are obliged to admit that 

 there is hardly any method yet invented in this kind of analysis of 

 which there are not sufficiently distinct traces to be discovered in 

 Diophantus. 



With the very important process of reducing problems to equa- 

 tions he is relatively successful and often highly ingenious. For 

 example, " to find three numbers, so that the product of any two plus 

 the sum of the same two shall be given numbers, for example, 8, 15, 

 and 24." We should write : xy + x + y = 8; yz + y -f- z = 15 ; 

 zx + 2 + x = 24. 



Hence, by subtraction, x(z y) + z y =16, 



x + 1 = _li- , z (x - y) + x - y = 9, z + 1 = ^ , etc. 



- y x - y 



He, on the other hand, takes a 1 for one of the numbers and 



readily obtains 1 and 1 for the others, and a = 



a a 5 



He employs tentative assumptions with great effect. For 

 example, "To find a cube and its root such that if the same number 

 be added to each, the sums shall also be a cube and its root." If 



