THE ELEMENTS. BOOK X 
405 
thus /Vp and XVp are commensurable in length, while ]Vp 
and VX . lAp are commensurable in square only : the rectangles 
formed by such pairs are in general medial, as XlVp 2 and 
VX . iVp 2 ; but if Vx — k'Vk in the second case, the rectangle 
(k'kp 2 ) is rational (Propositions 24,25). Proposition 26 proves 
that the difference between two medial areas cannot be 
rational ; as any two medial areas can be expressed in the 
form Vk.pt 2 , VX. p 2 , this is equivalent to proving, as we do in 
algebra, that {Vk— VX) cannot be equal to k'. Finally, 
Propositions 27, 28 find medial straight lines commensurable 
in square only (1) which contain u rational rectangle, viz. lAp, 
k*p, and (2) which contain a medial rectangle, viz. lAp, X ip/V. It 
should be observed that, as p may take either of the forms a 
or VA, a medial straight line may take the alternative forms 
V(a VB) or V(AB), and the pairs of medial straight lines just 
mentioned may take respectively the forms 
W '<»'*)• J(r& or У(АВ> • J( n Va) 
and (2) V(aVB), J{^B) or 
I shall henceforth omit reference to these obvious alternative 
forms. Next follow two lemmas the object of which is to find 
(1) two square numbers the sum of which is a square, Euclid’s 
solution being 
0 „ /mnp 2 ~-mnq\ 2 /mnp 2 + mnq\ 2 
mnp 2 . mnq 2 + ----- — \ — —j 5 
where mnp 2 , mnq 2 are either both odd or both even, and 
(2) two square numbers the sum of which is not square, 
Euclid’s solution being 
о о /mp 2 —mq 2 \ 2 
mp 2 .mq 2 , f —^— 1J • 
Propositions 29-35 are problems the object of which is to find 
(а) two rational straight lines commensurable in square only, 
(б) two medial straight lines commensurable in square only, 
(c) two straight lines incommensurable in square, such that 
the difference or sum of their squares and the rectangle