A history of Greek mathematics - Wilbourhall.org

INDETERMINATE EQUATIONS 469
1 . Double equation of the first degree.
The equations are
a x + a = u 2 ,
.
ftx
+ b — w 2 .
Diophantus has one general method taking slightly different
forms according to the nature of the coefficients.
(a)
First method of solution.
This depends upon the identity
{i(p+q)V'-{i(p-q)V- = pq-
If the difference between the two expressions in x can be
separated into two factors p, q, the expressions themselves
are equated to {i(p +
2
q}} and { \ (p — 2 q) }
respectively. As
'
Diophantus himself says in II. 11, we equate either the square
of half the difference of the two factors to the lesser of the
expressions, or the square of half the sum to the greater',
We will consider the general case and investigate to what
particular classes of cases the method is applicable from
Diophantus's point of
view, remembering that the final quadratic
in x must always reduce to a single equation.
Subtracting, we have (oc — ft) x + (a — b) = u 2 — w 2 .
Separate (oc — ft) x + (a — b) into the factors
We write accordingly
p, {(a-ft)x + (a-b)} /p.
(oi—ft)x + (a-b)
n + w — —
-,
P
u + %v = p.
m, o i
{(oc~ft)x+(a—b) )
+ p[ ;
Thus u- = (xx + a = i \- — '
t
p s
therefore {(a- ft) X -\-a-b+p 2 2
=
} 4p
2
(ax + a).
This reduces to
2
(a-ft) 2 x 2 + 2x{(oc~ft)(u-b)-p 2 (oc + ft)}
+ [a - b)
2 - 2
p 2 (a + b)+ jA = 0.