A history of Greek mathematics - Wilbourhall.org

.
474 DIOPHANTUS OF ALEXANDRIA
The species of the first
as follows.
class found in the Arithmetica are
1
Equation Ax 3 + Bx 2 + Gx + d 2 — y
2
.
As the absolute term is a square, we can assume
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 — z 3 x 2 + 3x + 1 = y' 2 (VI. 1 8),
and uses the first method.
2. Equation A # 4 -f Bx z + Ox 2 4- Dx .+ E — y
2
, where either A or
E is a square.
If A is a square ( = a 2 ), we may assume y = ax 2 H x + w,
determining ?i so that the term in x 2 in the resulting equation
may vanish. If E is a square (= e 2 ), we may assume
y = m# 2 + —x + e, determining m so that the term in x 2 in the
resulting equation may vanish.
obtain a simple equation in x.
We shall then, in either case,
3. Equation Ax^ + 0# 2 + E = y
2
all the coefficients are squares.
4. Equation Ax* + E = y
2
.
,
but in special cases only where
The case occurring in Diophantus is a? 4 + 97 — y
2
(V. 29).
Diophantus trie's one assumption, y = x 2 — 1 0, and finds that
this gives x 2 = 2%, which leads to no rational result. He
therefore goes back and alters his assumptions so that he
2
is able to replace the refractory equation by x + 337 = 4: y ,
and at the same time to find a suitable value for y, namely
y — x
2 — 25, which produces a rational result, x — --£-.
5. Equation of sixth degree in the special form
x 6 —Ax z + Bx + c 2 — y
2
.
Putting yz=zx z + c, we have — Ax 2 + B = 2cx 2 ,
and
B
B
x — 2 —. , which gives a rational solution if — A—— - is
A + 2c
5 A + 2c
.