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;