ARCHYTAS. EUDOXUS
249
Compounding the ratios, we have
AG-.AB = {AM: ABf;
therefore the cube of side AM is to the cube of side AB as AC
is to AB.
In the particular case where AG = 2 AB, AM ? ’= 2 AB ? ’,
and the cube is doubled.
(y) Eudoxus.
Eutocius had evidently seen some document purporting to
give Eudoxus’s solution, but it is clear that it must have
been an erroneous version. The epigram of Eratosthenes
says that Eudoxus solved the problem by means of lines
of a ‘curved or bent form’ (KapiroXov eiSos tv ypap/xaLs).
According to Eutocius, while Eudoxus said in his preface
that he had discovered a solution by means of ‘ curved lines ’,
yet, when he came to the proof, he made no use of such
lines, and further he committed an obvious error in that he
treated a certain discrete proportion as if it were continuous. 1
It may be that, while Eudoxus made use of what was really
a curvilinear locus, he did not actually draw the whole curve
but only indicated a point or two upon it sufficient for his
purpose. This may explain the first part of Eutocius’s remark,
but in any case we cannot believe the second part ; Eudoxus
was too accomplished a mathematician to make any confusion
between a discrete and a continuous proportion. Presumably
the mistake which Eutocius found was made by some one
who wrongly transcribed the original ; but it cannot be too
much regretted, because it caused Eutocius to omit the solution
altogether from his account.
Tannery 2 made an ingenious suggestion to the effect that
Eudoxus’s construction was really adapted from that of
Archytas by what is practically projection on the plane
of the circle ABC in Archytas’s construction. It is not difficult
to represent the projection on that plane of the curve of
intersection between the cone and the tore, and, when this
curve is drawn in the plane ABC, its intersection with the
circle ABC itself gives the point M in Archytas’s figure.
1 Archimedes, ed. Heib., vol. iii, p. 56. 4-8.
2 Tannery, Mémoires scientifiques, vol. i, pp. 5B-61.
