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DIOPHANTUS OF ALEXANDRIA 
(B) Indeterminate equations. 
Diophantus says nothing of indeterminate equations of the 
first degree. The reason is perhaps that it is a principle with 
him to admit rational fractional as well as integral solutions, 
whereas the whole point of indeterminate equations of the 
first degree is to obtain a solution in integral numbers. 
Without this limitation (foreign to Diophantus) such equa 
tions have no significance. 
(a) Indeterminate equations of the second degree. 
The form in which these equations occur is invariably this; 
one or two (but never more) functions of x of the form 
Ax 2 4- Bx 4- C or simpler forms are to be made rational square 
numbers by finding a suitable value for x. That is, we have 
to solve, in the most general case, one or two equations of the 
form Ax 2 + Bx + C = y 2 . 
(1) Single equation. 
The solutions take different forms according to the particular 
values of the coefficients. Special cases arise when one or 
more of them vanish or they satisfy certain conditions. 
1. When A or C or both vanish, the equation can always 
be solved rationally. 
Form Bx = y 2 . 
Form Bx + C = y 2 . 
Diophantus puts for y 2 any determinate square m 2 , and x is 
immediately found. 
Form Ax 2 + Bx = y 2 . 
Diophantus puts for y any multiple of x, as — x. 
2. The equation Ax 2 + C = y 2 can be rationally solved accord 
ing to Diophantus: 
(a) when A is positive and a square, say a 2 ; 
in this case we put a 2 x 2 + C = {ax + m) 2 , whence 
C—m 2 
x = 4 
~ 2 ma 
(m and the sign being so chosen as to give x a positive value);