256 
THE DUPLICATION OF THE CUBE 
For, if we look at the figure of Menaechmus’s second solu 
tion, we shall see that the given straight lines and the two 
means between them are shown in cyclic order (clockwise) 
as straight lines radiating from 0 and separated by right 
angles. This is exactly the arrangement of the lines in 
‘ Plato’s ’ solution. Hence it seems probable that some one 
who had Menaechmus’s second solution before him wished 
to show how the same representation of the four straight 
lines could be got by a mechanical construction as an alterna 
tive to the use of conics. 
Drawing the two given straight lines with the means, that 
is to say, OA, OM, ON, OB, in cyclic clockwise order, as in 
Menaechmus’s second solution, we have 
AO:OM= OM:ON = ON: OB, 
and it is clear that, if AM, MN, NB are joined, the angles 
AMN, MNB are both right angles. The problem then is, 
given OA, OB at right angles to one another, to contrive the 
rest of the figure so that the angles at M, N are right. 
The instrument used is somewhat like that which a shoe 
maker uses to measure the length of the foot. FGH is a rigid 
right angle made, say, of wood. KL is a strut which, fastened, 
say, to a stick KF which slides along GF, can move while 
remaining always parallel to GH or at right angles to GF. 
Now place the rigid right angle FGH so that the leg GH 
passes through B, and turn it until the angle G lies on AO