410 
EUCLID 
Propositions 66-70 and Propositions 103-7 prove that 
straight lines commensurable in length with A 1} A 2 ... F 1 , F 2 
respectively are irrationals of the same type and order. 
Propositions 71, 72, 108-10 show that the irrationals 
arise severally as the sides of squares equal 
to the sum or difference of a rational and a medial area, or the 
sum or difference of two medial areas incommensurable with 
one another. Thus kp 2 + VX . p 2 is the sum or difference of a 
rational and a medial area, Vk. p 2 + Vx. p 2 is the sum or 
difference of two medial areas incommensurable with one 
another provided that Vk and Vx are incommensurable, and 
the propositions prove that 
V(kp 2 + VX . p 2 ) and V{Vk. p 2 ± Vx.p 2 ) 
take one or other of the forms A 1} A 2 ... F x , F 2 according to 
the different possible relations between k, X and the sign 
separating the two terms, but no other forms. 
Finally, it is proved at the end of Proposition 72, in Proposi 
tion 111 and the explanation following it that the thirteen 
irrational straight lines, the medial and the twelve other 
irrationals A 1} A 2 ... F x , F 2 , are all different from one another. 
E. g. (Proposition 111 ) a binomial straight line cannot also be 
an apotome; in other words, Vx+ Vy cannot be equal to 
Vx'—Vy', and x+Vy cannot be equal to x'—Vy'. We 
prove the latter proposition by squaring, and Euclid’s proce 
dure corresponds exactly to this. Propositions 112-14 prove 
that, if a rectangle equal to the square on a rational straight 
line be applied to a binomial, the other side containing it is an 
apotome of the same order, with terms commensurable with 
those of the|binornial and in the same ratio, and vice versa; 
also that a binomial and apotome of the same order and with 
terms commensurable respectively contain a rational rectangle. 
Here we have the equivalent of rationalizing the denominators 
of the fractions —¡— A ——^ or ^ by multiplying the 
VA± VB a±VB J 1 J 
numerator and denominator by VA + VB or a+ VB respec 
tively. Euclid in fact proves that 
a 2 /[p + Vk. p) = Xp—Vk .Xp {k < 1), 
and his method enables us to see that X = cr 2 /{p 2 — kp 2 ). 
Proposition 115 proves that from a medial straight line an