266 
THE DUPLICATION OF THE CUBE 
Suppose now that the cissoid has been drawn as shown by 
the dotted line in the figure, and that we are required to find 
two mean proportionals between two straight lines a, h. 
Take the point K on OB such that DO : OK = a:b. 
Join DK, and produce it to meet the cissoid in Q. 
Through Q draw the ordinate LM perpendicular to DC. 
Then, by the property of the cissoid, LM, MG are the two 
mean proportionals between DM, MQ. And 
DM: MQ = DO: OK = a:b. 
In order, then, to obtain the two mean proportionals between 
a and b, we have only to take straight lines which bear respec 
tively the same ratio to DM, IjM, MG, MQ as a bears to DM. 
The extremes are then a, h, and the two mean proportionals 
are found. 
(k) Sporus and Pappus. 
The solutions of Sporus and Pappus are really the same as 
that of Diodes, the only difference being that, instead of using 
the cissoid, they use a ruler which they turn about a certain 
point until certain intercepts which it cuts off between two 
pairs of lines are equal. 
In order to show the identity of the solutions, I shall draw 
Sporus’s figure with- the same lettering as above for corre 
sponding points, and I shall add dotted lines to show the 
additional auxiliary lines used by Pappus. 1 (Compared with 
my figure, Sporus’s is the other way up, and so is Pappus’s 
where it occurs in his own Synagoge, though not in Eutocius.) 
Sporus was known to Pappus, as we have gathered from 
Pappus’s reference to his criticisms on the quadratrix, and 
it is not unlikely that Sporus was either Pappus’s master or 
a fellow-student of his. But when Pappus gives (though in 
better form, if we may judge by Eutocius’s reproduction of 
Sporus) the same solution as that of Sporus, and calls it 
a solution Kad’ gyas, he clearly means ‘according to my 
method ’, not ‘ our method ’, and it appears therefore that he 
claimed the credit of it for himself. 
Sporus makes DO, OK (at right angles to one another) the 
actual given straight lines; Pappus, like Diodes, only takes 
1 Pappus, hi, pp, 64-8 ; viii, pp. 1070-2.