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While looking up the old Copernican model for orbits, I encountered the following image (courtesy Wikipedia):

enter image description here

This seems... weird. Not only would it be an odd thing to come up with or express in mathematics, it also doesn't make much sense intuitively. Even if they weren't using rigorous mathematics to develop the model... why would they ever consider this a reasonable orbit?

I understand that not much about orbiting bodies was known at the time, but it still leads me to wonder why they would consider this viable. In particular:

  • In such a model, wouldn't Mars change size in the sky?
  • Wouldn't Mars tend to delay at a particular point in the sky for a measurable length of time?

Why would these things not be immediately obviously wrong to anyone who could observe these bodies? Of note, I'm assuming they had some method of observing the location of Mars, since the model includes its existence.

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  • $\begingroup$ Because this is the (approximate) path of Mars as seen from the Earth. See e.g. Thomas Kuhn, The Copernican Revolution : Planetary Astronomy in the Development of Western Thought (1957), page 65. $\endgroup$
    – Mauro ALLEGRANZA
    Jul 29, 2015 at 11:33
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    $\begingroup$ I'm confused as to why this earned a downvote. Would it be possible for me to impose on someone to explain briefly? $\endgroup$
    – user2285
    Jul 29, 2015 at 15:28
  • $\begingroup$ No, it is impossible to "impose" anything on anonymous dowvoters. Despite multiple requests by newbies. And the proposal to require downvoters to comment has been rejected. $\endgroup$
    – Gerald Edgar
    Jul 31, 2015 at 16:29
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    $\begingroup$ @Gerald I realize. I'm a 30k Stack Exchange user and a mod. I was asking because sometimes people do clarify, and that would be helpful feedback in the event there's something wrong with this post. I'm not even asking the downvoter, I'm asking anyone who might have thoughts on why. $\endgroup$
    – user2285
    Jul 31, 2015 at 16:43
  • $\begingroup$ The Ptolemaic epicycle model presupposes the constraint that planets move along tiny circular paths at a constant rate which are themselves moving along a greater circle at a constant rate, and so on and so forth. Now, when we observe Mars in retrograde, we see it slow down until it changes direction, retraces it steps for a bit, and reverse direction again. This problem is that Greek theory doesn't allow for variable rates along a given circle. Any apparent change in the lateral motion of the planet must be compensated for by a change in the motion towards the observer. $\endgroup$
    – David H
    Jul 31, 2015 at 17:35

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The right hand side picture shows qualitatively what you obtain from Ptolemy's model. The actual curve is a cycloid, in the first approximation. It uses rigorous mathematics. Moreover it describes quite well the actual motion of planets, including Mars. Just look at the actual motion of Mars in the sky (or in the Internet:-)

To answer your specific questions: yes Mars actually changes its visible size when you look at it. (And this is explained in both models) Yes, Mars tends to delay and stop at certain moments (called stationary points) and then moves in the opposite direction (retrogradation). All these phenomena are very well familiar to anyone who looks at the sky, and Ptolemy's model was designed exactly for the purpose to explain them.

EDIT. Moreover, if you look at the left hand side picture, and imagine how the same motion of Mars looks FROM THE EARTH, you obtain the right hand side picture. If it is difficult to imagine, write a computer program and make a plot. Or make a plot by hand. In fact both descriptions (the one of Copernicus and the one of Ptolemy) are equivalent, if you ignore smaller effects which are not visible in these pictures anyway. As it should be, because both pictures describe the reality correctly.

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