A history of Greek mathematics - Wilbourhall.org

492 DIOPHANTUS OF ALEXANDRIA
II. 1 6. x = my, a 2 + x = u 2 , a
2 + y = v 2 .
II. 19. x 2 -y 2 = m(y 2 -z 2 ).
II. 20. x 2 + y = u 2 , y 2 + x = v 2 .
[Assume y = 2fmx + m 2 ,
and one condition is satisfied.]
II. 21. x 2 — y = u 2 ,
y 2 — x — v
2
.
[Assume x = £ + m, y = 2 m £ + m 2 , *and one condition
is satisfied.]
ill. 22. o? 2 + (a? + y) = u 2 2
, 2/
+ (a; + 2/)
= v 2 .
J [Put a; + 2/
= 2mx + m 2 .'\
Jl. 23. o? 2 — (a; + 2/)
= u 2 ,
y 2 — (x + y) = v 2 .
2
II 24. (o3 + 2/) + ^ = ^ (x+y) 2 +y = v 2 .
[Assume x — (m 2 —
2
1)£ , y — {n
2 2
— 1)£ , 02 + 3/
= £.]
2
II. 25. (a3 + 2/) — x = u 2 2
(x
, + y)
— y = v 2 .
II. 26. xy + x = u 2 , xy + y = v 2 , u
+ v = a.
[Put y = m 2 x — 1.]
II. 27. xy — x = u 2 , xy — y = v 2 , u
+ v = a.
II. 28. a; 2 2
2/
+ a; = 2 u 2 , x 2 y 2 2
+ y = ^ 2 .
JL 29. x 2 y 2 — x = 2 it 2 , x 2 y 2 2
— y = v 2 .
II. 30. xy + (x + y) = u 2 , xy — (x + y) — v 2 .
[Since m 2 + w, 2 + 2 mw is a square, assume
xy = (m 2 + n 2 )£ 2 and x + y = 2mwf;
put a; = £>£, y = qg, where _pg = m 2 + ^2 ;
then
(£> + #)£ = 2mn£ 2 .]
II. 31. xy + (x + y) = u 2 , xy — (x + y) = v 2 , x + y = w 2 .
[Suppose w 2 — 2.2m. m, which is a square, and use
formula
(2 m) 2 + m 2 ±2.2m.m = a square.]
2
'II. 32. 2/
+ 2 = u 2 , s 2 + os = v 2 , x 2 + y = w 2 .
[V = I * = (2a| + a 2 ), a; = 2&(2a£ + a 2 ) + 6 2 .]
II. 33. y 2 — z — U 2 , z 2 — x = v 2 , x — 2 y — w 2 .