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Arguments
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A somewhat garbled argument from Theon of Alexandria's __Commentary on
Ptolemy's Almages__t (pp. 398.6-399.8)

I claim that as the square on the diameter is to the circle so is the cube
to the cylinder with equal height.
(diagram 1) Let there
be set out circle AB with diameter GD, and let there be drawn the square
on diameter GD, namely EZ, (diagram
2) and let there be erected a cube on the square and a cylinder with
height equal to the cube. I say that as square EZ is to circle AB,
so is the cube to the cylinder.

(note that the letters are placed on top for
clarity)

(diagram
3) For let there be set out circle HQ on diameter KL equal to circle
AB, and the square on it. Clearly the square on it is equal to EZ.

(diagram 4) And so since
as square EZ is to square MN, so is circle AB to HQ as well as the cube
on EZ to the cube on MN and further as the cylinder on AB to the cylinder
on HQ, and alternando as square EZ is to circle AB so is square MN to circle
HQ.

And furthermore, as the cube on EZ to the cylinder on AB, so is the
cube MN to the cylinder from HQ,

(diagram 5) but all in
MN are equal to those in EZ.

There is clearly a gap here in the reasoning.
Perhaps this claim merely shows (what was given above) that:

cube MN = cube EZ.

with the conclusion derived in some other way,

or perhaps

all the circles in cylinder AB = all the circles
in cylinder HQ.

All the circles in cylinder AB : all the circles
in cylinder HQ = all the squares in cube EZ : all the squares in cube MN.

Hence, all the squares in cylinder EZ : all the
circles in cylinder AB = all the squares in cube MN : all the circles in
cylinder HQ.

Hence, square EZ : circle AB = all the squares
in cube EZ : all the circles in cylinder AB.

For a : b = ma : mb.

square EZ : circle AB = cube EZ : cylinder AB

Therefore as EZ square is to AB circle, so is the cube on EZ to the
cylinder on AB.