I 
468 
DIOPHANTUS OF ALEXANDRIA 
substituting in the original equation 1 + x for x and (q — kx) 
For y, where k is some integer, 
3. Form Ax 2 + Bx + G = y 2 : 
This can be reduced to the form in which the second term is 
wanting by replacing x by .s — -- ^ • 
Diophantus, however, treats this case separately and less 
fully. According to him, a rational solution of the equation 
Ax 2 + Bx + C = y 2 is only possible 
(a) when A is positive and a square, say a 2 ; 
(/3) when C is positive and a square, say c 2 ; 
(y) when ^B 2 — AC is positive and a square. 
In case (a) y is put equal to (ax — m), and in case (/3) y is put 
equal to (vix—c). 
Case (y) is not expressly enunciated, but occurs, as it 
were, accidentally (IV. 31). The equation to be solved is 
3 os + 18 — x 2 = y 2 . Diophantus first assumes 3 x + 18 — x 2 = 4 x 2 , 
which gives the quadratic 3a;+18 = 5a; 2 ; but this ‘is not 
rational ’. Therefore the assumption of 4 x 2 for y 2 will not do, 
‘and we must find a square [to replace 4] such that 18 times 
(this square +1) + (|) 2 may be a square’. The auxiliary 
equation is therefore 18(m 2 + 1) = y 2 , or 72m 2 + 81=a 
square, and Diophantus assumes 72 m 2 + 81 = (8 m + 9) 2 , whence 
m= 18. Then, assuming 2,x+ 18 — x 2 = (IS) 2 * 2 , he obtains the 
equation 325a; 2 — 3x— 18 = 0, whence x — that is, gV 
(2) Double equation. 
The Greek term is SLTrXoLcroTrjs, SnrXfj ia-orris or SinXy i'crcocns. 
Two different functions of the unknown have to be made 
simultaneously squares. The general case is to solve in 
rational numbers the equations 
mx 2 + ax + a = u 2 1 
nx 2 + /3x+ b = w 2 j 
The necessary preliminary condition is that each of the two 
expressions can be made a square. This is always possible 
when the first term (in x 2 ) is wanting. We take this simplest 
case first.