A history of Greek mathematics - Wilbourhall.org

DETERMINATE EQUATIONS 465
In the first
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 + 2 rj — (a + x) 2 + (a —
2 x) = 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
In VI. 17 the problem
x 2 + 2x + 3 = x* + 3x — 3x 2 — 1.
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.
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