474 DIOPHANTUS OF ALEXANDRIA 
The species of the first class found in the Arithmetica are 
as follows. 
1. Equation Ax A + Bx 2 + Gx + d 2 = y 2 . 
As the absolute term is a square, we can assume 
G 
y=2d X + d ’ 
or we might assume y = m 2 x 2 + nx + d and determine m, n so 
that the coefficients of x, x 2 in the resulting equation both 
vanish. 
Diophantus has only one case, x 3 — 3îc 2 + 3x + 1 — y 1 (VI. 18), 
and uses the first method. 
2. Equation Ax 4 + Bx 3 + Gx 2 + Bx + E = y 2 , where either A or 
E is a square. 
B 
If A is a square ( — a 2 ), we may assume y = ax 2 + —-- x + n, 
cc 
determining n so that the term in x 2 in the resulting equa 
tion may vanish. If i? is a square (= e 2 ), we may assume 
y — mx 2 + —x + e, determining m so that the term in x 2 in the 
2 c 
resulting equation may vanish. We shall then, in either case, 
obtain a simple equation in x.. 
3. Equation Ax 4 + Gx 2 + E = y 2 , but in special cases only where 
all the coefficients are squares. 
4. Equation Ax 4 + E=y 2 . 
The case occurring in Diophantus is ce 4 + 97 — y 2 (Y. 29). 
Diophantus tries one assumption, y = x 2 — 10, and finds that 
this gives x 2 = 2 3 o, which leads to no rational result. He 
therefore goes back and alters his assumptions so that he 
is able to replace the refractory equation by a? 4 + 337 = y 2 , 
and at the same time to find a suitable value for y, namely 
y = x 2 — 25, which produces a rational result, x = -\ 2 -. 
5. Equation of sixth degree in the special form 
x G — A x 3 + Bx + c 2 — y 2 . 
Putting y = x 3 + c, we have —Ax 2 + B = 2cx 2 , and 
B B 
x 2 = -j , which gives a rational solution if -. is 
-A + 2 c yl + 2 G