APPROXIMATION BY PLANE METHODS 269 
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each of which is equal to BG. Draw GO', FF', GG', KK', LL' 
parallel to BA. 
On LL', KK' take LM, KR equal to BA, and bisect LM 
in N. 
Take F, Q on LL' suc*h that L'L, L'N, L'P, L'Q are in con 
tinued proportion; join QR, RL, and through N draw NS 
parallel to QR meeting RL in S. 
Draw ST parallel to BL meeting GG' in T. 
To G'G, G'T take continued proportionals G'O, G'U, as before- 
Take W on FF' such that FW = BA, join U W, WG, and 
through T draw TI parallel io TJW meeting WG in I. 
Through I draw 1V parallel to BC meeting GG' in V. 
Take continued proportionals G'G, G'V, G'X, G'Y, and draw 
XZ, VZ' parallel to YD meeting EG in Z, Z'. Lastly draw 
ZX', Z'Y' parallel to BG. 
Then, says the author, it is required to prove that ZX', Z'Y' 
are two mean proportionals in continued proportion between 
AD, BC. 
Now, as Pappus noticed, the supposed conclusion is clearly 
not true unless DY is parallel to BG, which in general it is not. 
But what Pappus failed to observe is that, if the operation of 
taking the continued proportionals as described is repeated, 
not three times, but an infinite number of times, the length of 
the line G'Y tends continually towards equality with EA. 
Although, therefore, by continuing the construction we can 
never exactly determine the required means, the method gives 
an endless series of approximations tending towards the true 
lengths of the means.