SPORUS AND PAPPUS 
267 
them in the same proportion as the given straight lines. 
Otherwise the construction is the same. 
A circle being drawn with centre 0 and radius DO, we join 
DK and produce it to meet the circle in I. 
Now conceive a ruler to pass through C and to be turned 
about G until it cuts DI, OB and the circumference of the 
circle in points Q, T, 11 such that QT = TR. Draw QM, BN 
perpendicular to DC. 
Then, since QT = TR, MO = ON, and MQ, NR are equi- 
distant from OB. Therefore in reality Q lies on the cissoid of 
Diodes, and, as in the first part of Diocles’s proof, we prove 
(since RN is equal to the ordinate through Q, the foot of 
which is M) that 
DM: RN = RN: MG = MG: MQ, 
and we have the two means between DM, MQ, so that we can 
easily construct the two means between DO, OK. 
But Sporus actually proves that the first of the two means 
between DO and OK is OT. This is obvious from the above 
relations, because 
RN: OT = CN: GO = DM: DO = MQ: OK. 
Sporus has an ah initio proof of the fact, but it is rather 
confused, and Pappus’s proof is better worth giving, especially 
as it includes the actual duplication of the cube. 
It is required to prove that DO: OK = DO 3 : OT 3 .