408 
EUCLID 
ft, ft are the positive roots of the equation 
x i -2 p 2 .£c 2 + = 0: 
r 1 + k* ^ 
ft, ft are the positive roots of the equation 
V(1 +№) 
p*. a 2 + 
(1 + k 2 f 
ft, ft are the positive roots of the equation 
Propositions 42-7 prove that each of the above straight lines, 
made up of the sum of two terms, is divisible into its terms 
in only one way. In particular, Proposition 42 proves the 
equivalent of the well-known theorem in algebra that, 
if a+Vb = x+Vy, then a = x, b — y\ 
and if Va + Vb = Vx + Vy, 
then a = x, h — y (or a— y, b = x). 
Propositions 79-84 prove corresponding facts in regard to 
the corresponding irrationals with the negative sign between 
the terms: in particular Proposition 79 shows that, 
if a— Vb = x~ Vy, then a = x, b = y ; 
and if Va— Vb = Vx— Vy, then a — x, b = y. 
The next sections of the Book deal with binomials and 
apotomes classified according to the relation of their terms to 
another given rational straight line. There are six kinds, 
which are first defined and then constructed, as follows: 
(oq) ( First binomial 
(a 2 ) | First apotome 
| kp±kpV{ 1—X 2 ); (Props. 48, 85) 
(ft) [Second binomial 
(/3 2 ) [Second apotome 
kp 
V {l — A 2 ) 
±kp ; (Props. 49, 86) 
(7i) j Third binomial 
(y 2 ) ( Third apotome 
i m Vk. p + m Vk. p V(1 — A 2 ) ; 
(Props. 50, 87)