Apollonius and Normals to Parabolas - amatyc

Duplication of the Cube
• Delian problem: To construct an alter double the size of an existing altar of Apollo.
• Hippocrates of Chios discovered this is equivalent to solving two mean proportionals
(perhaps because doubling a square is equivalent to solving one mean proportional).
• After this discovery, various solutions followed “two mean proportionals” tack.
According to Heath, “Two solutions by Menaechmus … 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.
a
x
= x
y
= y
b
=⇒ x 2 = ay, y 2 = bx, xy = ab
b = 2a =⇒ x 3 = 2a 3
• An epigram of Erastosthenes: “Do not seek to do the difficult business of the
cylinders of Archytas, or to cut the cones in the triads of Menaechmus [my emphasis], or
to draw such a curved form of lines as described by the god-fearing Eudoxus.”