THE ELEMENTS. BOOK X 
407 
^eristics. 
rith x\ 
le with as]. 
) commen- 
'f) incom- 
]• 
mimensur- 
ncommen- 
y medial]. 
k} 
rational]. 
I + k 2 )) 
ind 
: another]. 
several 
;h only 
can be 
taken together. The twelve compound irrationals, with their 
names, are as follows : 
(A x ) [Binomial, p + \/k.p (Prop. 36) 
(A 2 ) [Apotome, p ** Vk.p (Prop. 73) 
(B x ) [ First bimedial 
(B 2 ) (First apotome of a medial 
kip + kip (Props. 37, 74) 
(A) 
(A) 
(A) 
(A) 
Second bimedial ) 7 , , A-p on , p . 
I JAp + —- (Props. 38, 75) 
Second apotome of a medial j ~ k* 
Major] p 
Minor Í ^ 2 
Vl+k 2 ' 
(Props. 39, 76) 
(jE x ) /Side of a rational plus 
a medial area I ./2(1 + k 2 ) 
(E 2 ) j That which ‘ produces ’ r 
| with a rational areaj — /2 (1 + k 2 ) 
V (Vl +k 2 + k) 
P 
a medial whole 
(F x ) /Side of the sum of two\ pA% 
V(Vl +k 2 -k) 
(Props. 40, 77) 
1 + 
k 
medial areas | V2 •s/1 + A 
s) 
(F 2 ) [That which ‘produces’- 
with a medial area 
a medial whole 
1 _ 
k 
è 
>/\ + k 2 ' 
(Props. 41, 78). 
As regards the above twelve compound irrationals, it is 
to be noted that 
A 1 , A,, are the positive roots of the equation 
a; 4 — 2 (1 + k) p 2 . x 2 + (1 — k) 2 p 4 = 0 ; 
B x , B 2 are the positive roots of the equation 
ic 4 — 2 Vk (1 + k) p 2 . x 2 + k (1 — &) 2 p 4 = 0 ; 
C'j, C 2 are the positive roots of the equation 
k + A 
~7k 
p 2 . x 2 + 
(k-xy 
P 4 = 0;