A history of Greek mathematics - Wilbourhall.org

i
490 DIOPHANTUS OF ALEXANDRIA
(v) Systems of equations apparently indeterminate but
really reduced, by arbitrary assumptions, to determinate
equations of the first degree.
I. 14. xy = m (x + y). [Value of y arbitrarily assumed.]
II. 3*. xy = m(x + y) }
and xy = m(x — y).
*
II. 1*. (cf. I. 31). x 2 + y
2
= m(x + y).
\
I II. 2*. (cf. I. 34). x 2 — y
2 = m(x — y).
> [x assumed = 2 ?/.]
II. 4*. (cf. I. 32). x 2 -\-y 2 =m(x-y).
II. 5*. (cf. I. 33). x 2 -y 2 =m(x + y). t
II. 7*. x 2 — y
2
= m(x — y) +a. [Diopb. assumes x — y= 2.]
/ T ^
1 1 11
11
m p n m _p 7i
I. 22. x a?+ -z = y y -\ x = z— - z+ ~y.
[Value of y assumed.]
i
1.23.
a; a? H—
w =. y y H a5 = # H— y
m q n m p n
= ty WH— 0. [Value of */ assumed.]
L ° J
q p
1 1 1
1.24. sc + — (y + z) = y+-(z + x) = z+-(x + y).
[Value of y + z assumed.]
1.25. x + — (y + z + ty) = y + -(z + w + x)
= z + - (10 + aj + y) = w + - (x + y +z).
p\ at
q \
u
[Value of y +
+ w assumed.]
11.17*. (cf. I. 22). x-(--x+a\ + (~z + c\
[Ratio of 35 to 2/ assumed.]