488 
DIOPHANTUS OF ALEXANDRIA 
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IV. 15. (y + z)x = a, (z + x)y = b, {x + y)z = c. 
[Dioph. takes the third number 0 as his unknown; 
thus 
x + y = c/z. 
Assume x.= p / z, y = q/z. Then 
These equations are inconsistent unless p — q = a — b. 
We have therefore to determine p, q by dividing c i^to 
two parts such that their difference = a — b (c£. I. 1). 
A very interesting use of the ‘false hypothesis’ 
(Diophantus first takes two arbitrary numbers for p, q 
such that p + q = c, and finds that the values taken have 
to be corrected). 
The final equation being ~ +p = cl, where p, q are 
determined in the way described, z 1 — pq/{a — p) or 
pq/(b — q), and the numbers a, h, c have to be such that 
either of these expressions gives a square.] 
IV. 34. yz + {y + z) = a 2 — 1, zx +(z+ x) = b' z — 1, 
xy + {x + y) = c 2 —1. 
[Dioph. states as the necessary condition for a rational 
solution that each of the three constants to which the 
three expressions are to be equal must be some square 
diminished by 1. The true condition is seen in our 
notation by transforming the equations yz + (y + z) = a, 
zx + (z + x) = /3, xy + (x + y) = y into 
(?/+!) (2+1) = a+1, 
(z + 1) (ic + 1) — /3+1, 
(x+1) (y+ 1) = y + 1,