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