A history of Greek mathematics - Wilbourhall.org

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DETERMINATE EQUATIONS 487
(I. 35.
2
x = my, y = nx.
1 1. 36. x = my,
2
y = ny.
I. 37.
2
x = my, y — n(x + y).
I. 38.
2
x = my, y = n(x — y).
I. 38. Cor. x = my, x 2 = ny.
„ x — my, x 2 = nx.
„ x = my, x 2 = n{x + y).
„ x = my, x 2 = n(x — y).
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).
[Solved by means of Lemma : see under (vi) Indeterminate
equations of the first degree.]
(iv)
Determinate systems reducible to equations of
second degree.
I. 27. x + y = a, xy = b.
[Dioph. states the necessary condition, namely that
J a 2 — b must be a square, with the words eari Se tovto
irXaaiiaTLKov, which no doubt means 'this is of the
nature of a formula (easily obtained)'. He puts
*-v = 2 £]
I. 30. x — y — a, xy = b.
[Necessary condition (with the same words) 4 b + a 2 =
a square, x + y is put = 2 £.]
I. 28. x + y = a, x 2 + y
2 = b.
[Necessary condition 2 b — a 2 = a square, x — y = 2 £.]
(TV. 1. x* + y
z = a, x + y = b.
[Dioph. puts x — y— 2^, whence x = \b + £, y = \b — £.
The numbers a, b are so chosen that (a — j6 3 )/3& is
a square.]
IV. 2. x 3 - y'* = a, x — y — b.
[x + y = 21]