A history of Greek mathematics - Wilbourhall.org

DETERMINATE EQUATIONS 485
I. 9. a — x = m(b — x).
I. 10. x + b = m{a — x).
I. 11. x + b = m(x — a).
1.39. (a + x)b+(b + x)a = 2(a + b)x, \
or (a + b) x + (b + x)a = 2 (a + a?) b, L (a > />)
or
(a + b)x + (a + x)b = 2 (b + x)a.)
Diophantus states this problem in this form, ' Given
two numbers (a, b), to find a third number (x) such that
the numbers
(a + x)b, (b + x)a, (a + b)x
are in arithmetical progression.'
The result is of course different according to the order
of magnitude of the three expressions. If a > b (5 and 3
are the numbers in Diophantus), then (a-\-x)b < (b + x)a;
there are consequently three alternatives, since (a + x)b
must be either the least or the middle, and (b + x) a either
the middle or the greatest of the three products. We may
have
(a + x) b < (a + b) x < (b + x)a,
or (a + b) x < {a + x) b < (b + x) a,
or (a + x) b < (b + x) a < (a + b) x,
and the corresponding equations are as set out above.
(ii)
Determinate systems of equations of the first degree.
I. 1. x + y = a, x — y = b.
(I. 2. x + y = a, x = my,
ll. 4. x — y = a, x = my.
I. 3. x + y =
11.
a, x = my + b.
/
I. 5. x + y = a, — x + - y — b, subject to necessary condition.
j
I. 6. x + y — a, — x y = b,
V lib lb