METHOD OF LIMITS 
477 
some function of x a value intermediate between the values 
of two other functions of x. 
Ex. 1. In IV. 25 a value of x is required such that 8/(x 2 + x) 
shall lie between x and x+l. 
One part of the condition gives 8 >x 6 + x A . Diophantus 
accordingly assumes 8 = {x + §) 3 = x’ + x 2 + ^x + -gV> which is 
> + x 2 . Thus x + ^ = 2 or x — f satisfies one part of 
the condition. Incidentally it satisfies the other, namely 
8/{x 2 + x) < x+l. This is a piece of luck, and Diophantus 
is satisfied with it, saying nothing more. 
Ex. 2. We have seen how Diophantus concludes that, if 
(x 2 — 60) > x > |(ic 2 —60), 
then x is not less than 11 and not greater than 12 (V. 30). 
The problem further requires that x 2 — 60 shall be a square. 
Assuming x 2 — 60 = (x—m) 2 , we find x — (m 2 + 60)/2m. 
Since x > 11 and <12, says Diophantus, it follows that 
24m > on 2 + 60 > 22 m; 
from which he concludes that m lies between 19 and 21. 
Putting m = 20, he finds x = 11^. 
III. Method of approximation to Limits. 
Here we have a very distinctive method called by Diophantus 
tTotpLcroTijs or 7rapuroTTjTos dycoyrj. I he object is to solve such 
problems as that of finding two or three square numbers the 
sum of which is a given number, while each of them either 
approximates to one and the same number, or is subject to 
limits which may be the same or different. 
Two examples will best show the method. 
Ex. 1. Divide 13 into two squares each of which > 6 (V. 9). 
Take half of 13, i.e. 6£, and find what small fraction 1 /x 2 
added to it will give a square; 
6^ + -i;? or 26 + must be a square. 
2 x 2 yl 
thus