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