Re: Astronomy - History

[Carlos.arribas@alc.servicom.es]

Leigh:

I think that you are wrong in an aspect. The rate of the line from the
center (0,0) to point P  it sweeps out area, is:

      A' = 1/2 |r x v|

if
     x = (R+r) cos wt
     y = (R-r) sin wt

and
     x' = (R+r) (-w) sin wt
     y' = (R-r) w cos wt

then if you make the vectorial product r x v

    A' = 1/2 |r x v| = 1/2 (R^2-r^2) w

that it is a *constant* if we take angular speed w constant ( and it is not
A' = 1/2 rho w, like you wrote)

The ellipse with semimajor axis R+r and  semi-minor axis R-r have a focal
distance c = 2 (R r)^(1/2). 

If we take the center of line in the focus of ellipse (-c, 0) (the analogous
ocurrs if we take the other focus), the line from focus to point (x,y)
sweeps out area with a speed:

A' = (R^2-r^2) w + 2 ((R r)^(1/2))(R - r) w Cos wt

that it is not constant.

Yours sincerely

Carlos
--
> 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
-==============================================
Carlos Arribas Ugarte
Carlota Pasaron 23-3r. ALACANT (PAIS VALENCIA)
03005 SPAIN
Tfno: (96) 5924441
E-mail: Carlos.Arribas@alc.servicom.es