I !. 
252 
THE DUPLICATION OF THE CUBE 
Alexander: ‘ O king,’ said Menaechmus, ‘ for travelling over 
the country there are royal roads and roads for common 
citizens, but in geometry there is one road for all.’ 1 A similar 
story is indeed told of Euclid and Ptolemy; but there would 
he a temptation to transfer such a story at a later date to 
the more famous mathematician. Menaechmus was evidently 
a considerable mathematician; he is associated by Proclus with 
Amyclas of Heraclea, a friend of Plato, and with Dinostratus 
as having ‘ made the whole of geometry more perfect ’. 2 
Beyond, however, the fact that the discovery of the conic 
sections is attributed to him, we have very few notices relating 
to his work. He is mentioned along with Aristotle and 
Callippus as a supporter of the theory of concentric spheres 
invented by Eudoxus, but as postulating a larger number of 
spheres. 3 We gather from Proclus that he wrote on the 
technology of mathematics; he discussed for instance the 
difference between the broader meaning of the word element 
(in which any proposition leading to another may be said 
to be an element of it) and the stricter meaning of something 
simple and fundamental standing to consequences drawn from 
it in the relation of a principle, which is capable of being 
universally applied and enters into the proof of all manner 
of propositions. 4 Again, he did not agree in the distinction 
between theorems and problems, but would have it that they 
were all problems, though directed to two different objects 5 ; 
he also discussed the important question of the convertibility 
of theorems and the conditions necessary to it. 6 
If x, y are two mean proportionals between straight 
lines a, h, 
that is, if a:x = x:y = y :h, 
then clearly 
x 2 — ay, 
y 1 = hx, and xy = ah. 
It is easy for us to recognize here the Cartesian equations 
of two parabolas referred to a diameter and the tangent at its 
extremity, and of a hyperbola referred to its asymptotes. 
But Menaechmus appears to have had not only to recognize, 
1 Stobaeus, Eclogue, ii. 31, 115 (vol. ii, p. 228. 30, Wachsmuth). 
2 Proclus on Eucl. I, p. 67. 9. 
3 Theon of Smyrna, pp, 201. 22-202. 2. 
4 Proclus on Eucl. I, pp. 72, 23-73. 14. 5 Ih., p. 78. 8-13. 
6 lb., p. 254. 4-5.