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

'
(I. 35. x = my, y
2
= nx.
1 1. 36. x = my, y
2
= ny.
DETERMINATE EQUATIONS 487
488 DIOPHANTUS OF ALEXANDRIA
IV. 1 5. (y + z) x = a, (z + x) y = b, (x + y)z — c.
[Dioph. takes the third number z as his unknown
thus
x + y = c/z.
Assume x = p/z, y = q/z. Then
„ x — my, x = 2 nx.
pq
„ x = my, x = 2 n{x + +P = a
y).
Z 2 >
„ x = my, x = 2 n(x — y).
-3+? = k
These equations are inconsistent unless p — q = a — b.
We have therefore to
[Solved by means of Lemma : see under (vi) Indeterminate
equations of the first degree.]
determine p q by dividing c into
}
two parts such that their difference = a — b (cf. I. 1).
A very interesting use of the false hypothesis
'
(Diophantus first takes two arbitrary numbers for p, q
(iv) Determinate systems reducible to equations of
such that p + q = c, and finds that the values taken have
second degree.
to be corrected).
The final equation being —L +p — (l} where p, q are
z
[Dioph. states the necessary condition, namely that
determined in the way described, z =
J a 2 — 2 pq/{& — p) r or
b must be a square, with the words eari Se tovto
irXaaiiaTLKov, which no doubt means
fpq/
'this is of the
(b — q), and the numbers a, b, c have to be such that
either of these expressions gives a square.]
nature of a formula (easily obtained)'. He puts
*-v = 2 £]
IV. 34. yz + (y + z) = a 2 — 1, zx + (z + x) = b 2 - 1,
[Necessary condition (with the same words) 4 b + a =
xy 2 -f (x + y) = c 2 — 1
a square, x + y is put = 2 £.]
[Dioph. states as the necessary condition for a rational
solution that each of the three constants to which the
[Necessary condition 2 b — a = 2 a square, x — y =
three expressions are to be equal must be some square
2 £.]
diminished by 1. The true condition is seen in our
notation by transforming the equations yz 4- (y + z) = (X,
[Dioph. puts x — y— 2^, whence x = \b + £, y = \b — £.
zx + (z + x) — /3, xy + (x + y) = y into
The numbers a, b are so chosen that (a — j6 3 )/3& is
(y+i)(z+i) = oc
a square.]
+ i,
(z+l)(x+l) = £+1,
[x + y = 21]
I. 37. x = my, y
2
— n(x + y).
I. 38. x = my, y
2
= n(x — y).
I. 38. Cor. x = my, x 2 = ny.
II. 6*. x — y = a, x 2 — y
2 = x — y + b.
IV. 36. yz — m(y + z), zx = n(z + x), xy = p(x + y).
I. 27. x + y = a, xy = b.
I. 30. x — y — a, xy = b.
I. 28. x + y = a, x 2 + y
2 = b.
(TV. 1. x* + y
z = a, x + y = b.
IV. 2. x 3 - y'* = a, x — y — b.
(x+\)(y+\) = y+1,