EUDOXUS. MENAECHMUS 
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Then AL = AH sec 6 = AG sec 6 = AFsec 2 6. 
That is, if p = AL, p = — sec 2 6, 
and L is a point on the curve. 
Similarly any number of other points on the curve may be 
found. If the curve meets the circle ABC in M, the length 
AM is the same as that of AM in the figure of Archytas’s 
solution. 
And AM is the first of the two mean proportionals between 
A B and AC. The second (= AP in the figure of Archytas’s 
solution) is easily found from the relation AM 2 = AB . AP, 
and the problem is solved. 
It must be admitted that Tannery’s suggestion as to 
Eudoxus’s method is attractive ; but of course it is only a con 
jecture. To my mind the objection to it is that it is too close 
an adaptation of Archytas’s ideas. Eudoxus was, it is true, 
a pupil of Archytas, and there is a good deal of similarity 
of character between Archytas’s construction of the curve of 
double curvature and Eudoxus’s construction of the spherical 
lemniscate by means of revolving concentric spheres; but 
Eudoxus was, I think, too original a mathematician to con 
tent himself with a mere adaptation of Archytas’s method 
of solution. 
(5) Menaechmus. 
Two solutions by Menaechmus of the problem of finding 
two mean proportionals are described by Eutocius; both find 
a certain point as the intersection between two conics, in 
the one case two parabolas, in the other a parabola and 
a rectangular hyperbola. The solutions are referred to in 
Eratosthenes’s epigram : ‘ do not ’, says Eratosthenes, ‘ cut the 
cone in the triads of Menaechmus.’ From the solutions 
coupled with this remark it is inferred that Menaechmus 
was the discoverer of the conic sections. 
Menaechmus, brother of Dinostratus, who used the quadra- 
trix to square the circle, was a pupil of Eudoxus and flourished 
about the middle of the fourth century b. c. The most attrac 
tive form of the story about the geometer and the king who 
wanted a short cut to geometry is told of Menaechmus and