270 
THE DUPLICATION OF THE CUBE 
Let LL' = BE = a, AB — h, L'N — a (for there is no 
necessity to take N at the middle point of LM). 
Then L'Q = a 3 /“ 2 ’ 
therefore LQ = (a 3 —a 3 )/a 2 . 
TG SL NL (a-a) a 2 . 
.RK~ RL~ QL a 3 -a 3 ’ 
(a — a)a 2 b 
TG = 
GT — a — 
a" — oc J 
(a—a)a 2 b 
And 
therefore 
and accordingly 
^ a° — or 
Now let a n be the length corresponding to Cr'T after u 
operations; then it is clear that 
_ (a — oi n )a~b 
Ct (X™ i i — o o 
1 ar—(x n 6 
a n must approach some finite limit when n == co. Taking £ 
as this limit, we have 
(a — £)a 2 b 
a-i = 
and, £ = cc not being a root of this equation, we get at once 
£ :! = o?—a 2 b = a 2 (a — b). 
Therefore, ultimately C'V is one of the mean proportionals 
between EA and EB, whence Y'Z' will be one of the mean 
proportionals between AD, BG, that is, between AD and AB. 
The above was pointed out for the first time by R. Pendle- 
bury, 1 and I have followed his way of stating the matter. 
1 Messenger of Mathematics, ser. 2, vol. ii (1873), pp. 166—8.