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Re: Astronomy - History



Joseph,

Have you seen my earlier posting yet? I derive the ellipse from his
description. I don't want to post it again (or a refutation of your
statement here) but I'll append a copy to this.

Leigh

Here's the relevant part:

3) One can produce an ellipse by using one epicycle, see below. Is this
well known?

It is obvious when one writes it down algebraically. It reduces to

x = R cos Wt + r cos (-Wt) = (R+r) cos Wt
y = R sin Wt + r sin (-Wt) = (R-r) sin Wt

This is the well known equation of a central ellipse.

Is there any relationship to the Ptolemaic Model?

No. In the Ptolemaic model both angular speeds are in the same sense.

Consider an epicycle (small circle) of radius r, whose center moves
on the the larger circle (of radius R) at the angular speed W.
Point P moves on the epicycle a the angular speed -W.
If one starts with the epicycle at +R on the X-axis, and P at R+r on
the X-axis, P will trace out an ellipse whose semi-major axis is R+r and
semi-minor axis, R-r.
An interesting result is that the line from the CENTER to P will trace
out equal areas in equal times.

Not if I've analyzed your construction correctly. I take W to be constant
as it is in Ptolemy's model. The rate of sweeping out of area is just

. 1 2 2 2
A = --- rho W where rho = x + y
2

rho, the distance from the center, is not a constant.

Leigh