268 
THE DUPLICATION OF THE CUBE 
Join RO, and produce it to meet the circle at S. Join 
DS, SC. 
Then, since RO = OS and RT = TQ, SQ is parallel to A B 
and meets OC in M. 
Now 
DM :MC= SM 2 : MG 2 = CM 2 : MQ 2 (since Z RGS is right). 
Multiply by the ratio CM : MQ ; 
therefore (DM: MG). {CM: MQ) = {CM 2 : MQ 2 ). {CM: MQ) 
or DM:MQ = CM 3 : MQ*. 
But DM:MQ = DO:OK, 
and CM: MQ = GO : OT. 
Therefore DO : OK = CO 3 : OT 3 = DO 3 : OT 3 . 
Therefore OT is the first of the two mean proportionals to 
DO, OK; the second is found by taking a third proportional 
to DO, OT. 
And a cube has been increased in any given ratio. 
(X) Approximation to a solution by plane methods only. 
There remains the procedure described by Pappus and 
criticized by him at length at the beginning of Book III of 
his Collection} It was suggested by some one ‘who was 
thought to be a great geometer but whose name .is not given. 
Pappus maintains that the author did not understand what 
he was about, ‘ for he claimed that he was in possession of 
a method of finding two mean proportionals between two 
straight lines by means of plane considerations only ’; he 
gave his construction to Pappus to examine and pronounce 
upon, while Hierius the philosopher and other friends of. his 
supported his request for Pappus’s opinion. The construction 
is as follows. 
Let the given straight lines be AB, AD placed at right 
angles to one another, AB being the greater. 
Draw BG parallel to AD and equal to AB. Join CD meeting 
BA produced in E. Produce BG to L, and draw EL' through 
E parallel to BL. Along CL cut off lengths GF, FG, GK, KL, 
1 Pappus, iii, pp. BO-48.