406 
EUCLID 
contained by them respectively have certain characteristics. 
The solutions are 
(a) x, y rational and commensurable in square only. 
Prop. 29 : p, pV{ 1 — k 2 ) [V(x 2 ~y 2 ) commensurable with x\ 
„ 30: p, p/ •/(! + k 2 ) [ V {x 2 — y 2 ) incommensurable with x\ 
(h) x, y medial and commensurable in square only. 
Prop. 31: p (1 — k 2 )*, p (1 — k 2 )* [xy rational, '/(^ — ^commen 
surable with x\; 
p/{\ + k 2 )*, p/{l +№)'* [xy rational, V(a? — y 2 ) incom 
mensurable with x\. 
,, 32: p\*, p\* V[l —k 2 ) [xy medial, '/(^ — ^commensur 
able with x\; 
p\*, p\*/V{^ +k 2 ) [xy medial, V(x 2 — y 2 ) incommen 
surable with x\. 
(c) x, y incommensurable in square. 
[{x 2 +y 2 ) rational, xy medial]. 
[;x 2 + y 2 medial, xy rational]. 
[x 2 + y 2 and xy both medial and 
incommensurable with one another]. 
With Proposition 36 begins Euclid’s exposition of the several 
compound irrationals, twelve in number Those which only 
differ in the sign separating the two component parts can be