360
THE DUPLICATION OF THE CUBE
inscription on the column; the construction was left to be
inferred from the single figure which corresponded to the
second above.
The epigram added by Eratosthenes was as follows;
‘ If. good friend, thou mindest to obtain from a small (cube),
a cube double of it, and duly to change any solid figure into
another, this is in thy power; thou canst find the measure of
a fold, a pit, or the broad basin of' a hollow well, by this
method, that is, if thou (thus) catch between two rulers' (two)
means with their extreme ends converging. 1 Do not thou seek
to do the difficult business of Archytas’s cylinders, or to cut the
cone in the triads of Menaechmus, or to compass such a curved
form oflines as is described by the god-fearing Eudoxus.
Nay thou couldst, on these tablets, easily find a myriad of
means, beginning from a small base. Happy art thou,
Ptolemy, in that, as a father the equal of his son in youthful
vigour, thou hast thyself given him all that is dear to Muses
and Kings, and may he in the future, 2 O Zeus, god of heaven,
also receive the sceptre at thy hands. Thus may it be, and
let any one who sees this offering say “This is the gift of
Eratosthenes of Cyrene
(77) Nicomedes.
The solution by Nicomedes was contained in his book on
conchoids, and, according to Eutocius, he was inordinately
proud of it, claiming for it much superiority over the method
of Eratosthenes, which he derided as being impracticable as
well as ungeometrical.
Nicomedes reduced the problem to a vevais which he solved
by means of the conchoid. Both Pappus and Eutocius explain
the method (the former twice over 3 ) with little variation.
Let A B, BG be the two straight lines between which two
means are to be found. Complete the parallelogram ABGL.
Bisect AB, BG in D and E.
Join LD, and produce it to meet GB produced in G.
Draw EF at right angles to BG and of such length that
GF = AD.
Join GF, and draw G1I parallel to it.
1 Lit. ‘ converging with their extreme ends ’ {reppaaiv aKpois <rw8po-
pa8as)-
2 Reading with v. Wilamowitz '6 8' es varepov.
3 Pappus, iii, pp. 58. 23-62. 13; iv, pp. 246. 20-250. 25.
