A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

In the first
DETERMINATE EQUATIONS 465
and third of the last three cases the limits are not
accurate, but are integral limits which are a fortiori safe.
In the second f § should have been § J,
and it would have been
more correct to say that, if x is not greater than f
-£ and not
less than ff
, the given conditions are a fortiori satisfied.
For comparison with Diophantus's solutions of quadratic
equations we may refeV to a few of his solutions of
(3) Simultaneous equations involving quadratics.
In I. 27, 28, and 30 we have the following pairs of equations.
(a) ^rj = 2a] (/3) £ + v = 2a\ (y) £- V = 2a)
I use the Greek letters for the numbers required to be found
as distinct from the one unknown which Diophantus uses, and
which I shall call x.
In (a), he says, let £ — r] = 2x (£ > rj).
It follows, by addition and subtraction, that £ = a + x,
tj — a — x\
therefore £rj — (a + x) (a — x) = a 2 — x 2 = B,
and x is found from the pure quadratic equation.
In (/3) similarly he assumes £ — t] = 2x, and the resulting
equation is £ 2 + rj 2 — (a + x) 2 + (a — x)
2
= 2 (a 2 + x 2 )
= B.
In (y) he puts £ + ?; = 2x and solves as in the case of (a).
(4) Cubic equation.
Only one very particular case occurs.
leads to the equation
x 2 + 2x + 3 = x* + 3x — 3x 2 — 1.
In VI. 17 the problem
Diophantus says simply ' whence x is found to be 4 '. In fact
the equation reduces to
x z + x = 4x 2 + 4.
Diophantus no doubt detected, and divided out by, the common
factor x 2 + 1 , leaving x = 4.
1523.2 H ll
466 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 equations
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 -f Bx + G 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 G or both vanish, the equation can always
be solved rationally.
2
Form Bx = y
.
2
Form Bx + G = y
.
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 = y2
can be rationally solved according
to Diophantus
(a) when A is positive and a square, say a 2 ;
in this case we put a 2 x 2 + G = (ax ± m) 2 , whence
x= -\
C — m 2
(m and the sign being so chosen as to give x a positive value)