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In his excellent treatment of the history of the science of astronomical distances and sizes, Albert van Helden says (p.29) that

The complicated [Ptolemaic] model of Mercury has the curious property of producing two perigees, each about 120° removed from the apogee.

But when I try to confirm this using the 88 day period Mercury's epicycle, I get approximately half the expected value.

In a given time (in days), $t$, Mercury will travel through an angle $\varepsilon = 2\pi t/88$ along its epicycle which will itself have traveled through an angle $\delta = 2\pi t/365$ along its deferent. In transitioning from apogee to perigee, it must be the case (since Mercury must go half way round the epicycle, and then an additional $\delta$ to "catch up" with the angle traveled by the center of the epicycle) that $$\varepsilon=\delta+\pi$$ solving which yields $$t=[2\cdot(1/88-1/365)]^{-1}\approx58$$ which corresponds to about 57°, roughly half the number expected.

What is missing from the above reasoning? Is there something about the definitions of epicyclic period I've missed, perhaps?

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I think this is better fit for history of science and math stack exchange (though it is on topic here). You might want to check out history of math and science stack exchange out for future questions. – Cicero Sep 30 '15 at 3:28
@Cicero: I doubt it. It's really about the details of the calculation. HSM hasn't proven to be very quantitative. – raxacoricofallapatorius Sep 30 '15 at 13:05

He isn't talking about the planet's motion on the epicycle (Second Anomaly), but about how Mercury's deferent moves, drawing the epicycle toward and away from the earth. To my knowledge, the equant is placed about .05 of the deferent's radius away in the direction of Scorpio, and the deferent center revolves clockwise around a point twice as far in that direction, so that at its closest it will co-incide with the equant, and at its farthest it will be three times the distance of the equant in the same direction. Since the clockwise revolution of the deferent center has a period of one year, Mercury's epicycle will always be a certain distance for any given degree of seperation from the apogee line; this ends up forming an elliptical shape with a greater apogee in Scorpio, a lesser apogee in Taurus, and two perigees in Cancer and Pisces. Mercury's true motion isn't exactly like this, but this is the effect of the elliptical action.

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