# How did Copernicus establish the relative distance to the superior planets?

I understand that the relative distances to the planets had been calculated using various methods since ancient times, and, in particular, that the assumptions of the Copernican model of the Solar System allow one to easily establish the relative distances, $d$ (in AU), from the Sun to the inferior planets, Mercury and Venus, from their (easily measured) greatest elongations, $\varepsilon$ and simple trigonometry:

$$d=\sin\varepsilon$$

Hoever it is not clear to me what methods Copernicus or his contemporaries used to determine the relative distances to the superior planets. My texts say only that these were found by "slightly different geometrical methods" (p.44) or that "the analysis for a superior planet is more involved" (p.37).

What methods were those?

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Note that I understand that in principle Earth-orbit baseline parallax measurements could be made, etc., but I'm curious about the geometry actually used, esp. how the movement of planets between the two measurements that any method (that I can think of) was accounted for. I also see (again in incomplete descriptions) that the angular extent of each planet's retrograde motion can be used, but it is again not clear to me how that works. – raxacoricofallapatorius Sep 15 '12 at 1:02
I moved this comment to the answer – akhmeteli Sep 15 '12 at 1:07

It looks like the relevant procedure is described at the University of Nebraska's Astronomy Education site. It seems too long to copy it here.

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Excellent. Thanks! – raxacoricofallapatorius Sep 15 '12 at 1:13
Aha (and duh): I now see that this is (obviously) merely the inverse of the case for inferior planets: $d=(\sin\varepsilon)^{-1}$, where $\varepsilon = \pi/2-(\alpha-\beta)$ in the linked article. – raxacoricofallapatorius Sep 17 '12 at 18:00