Diagonal Argument

From Kepler to Ptolemy 11

March 23, 2025 5:31 pm

Here is the basic model for Venus:

The deferent now represents the Sun’s orbit and the epicycle Venus’s orbit around the Sun. As with the outer planets, Ptolemy uses an eccentric circle with equant for the deferent, but a concentric circle with uniform rotation for the epicycle¹. Given the relative eccentricities, this makes sense in hindsight. Of course, from Ptolemy’s perspective, he was just doing his best to be consistent.

The final result looks like this.

Venus

The vector $\scriptstyle\overrightarrow{ED}$ is always parallel to $\scriptstyle\overrightarrow{O\overline{S}}$ . (S is the mean Sun.) Note that both vectors rotate uniformly.

[1] Linton (p.78) observes, “It is interesting to note that in his model for Venus, Ptolemy in effect introduced an equant for the motion of the Sun, something he had not found to be necessary in the solar theory itself.”

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