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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 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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