I am about to work with Ptolemy's Model for the motion of an upper Planet (Mars). I use a epicycle rolling on the deferent. As a first step, I am just interested in the shape of the trajectory, which is why I can ignore the equant, because the uniform motion arount the equant does not change the fact, that the epicycle rolls on the deferent, which is why $\phi_0^*$ can be expressed in terms of $\phi^*$, $r_A$ and $r_a$. I want to express the angle $\phi^*$ or $\phi_0^*$ in terms of $\gamma$. I assume $r_A$, $r_a$ and $c$ (distance from Earth A to the center of the deferent $B$ and distance from this center to the equant as well). How can I express $\phi^*$ geometrically?
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You misunderstood Ptolemy’s model: The epicycle doesn’t “roll on” the deferent; rather, the epicycle’s center is on deferent. Mind you, it’s only a minor detail in this situation.
In the diagram you drew up, it is impossible to know φ* in terms of γ alone, as φ₀ could take any value. Also, in Ptolemy’s model, φ₀ does not depend on φ*, so φ₀ ≠ δφ*. However, both vary uniformly, by quantities that we will here call ωₐ and ωₜ after Pedersen.¹ For example, in the case of Mars, ωₜ = 0°;31,26,36,53,51,33/day and ωₐ = 0°;27,41,40,19,20,58/day.
You are correct to presume that c, rₐ, and rA were known and specified in Ptolemy’s model—a good part of the Almagest is spent determining such values for rₐ and rA supposing that c = 60 units.
Finally, nowhere in the Almagest are φ* or φ₀ expressed in terms of γ, as both of them, as hinted two paragraphs above, are only time-dependent. Because none of the angular speeds given for ωₐ and ωₜ is a factor of 360°, this means that for a given γ, any value of φ* and φ₀ is possible, depending on how long has elapsed since the reference date (the “epoch”), which Ptolemy chooses, for historical reasons, to be February 26, −746 (aka 747 BCE), Noon Alexandria true solar time (Julian Day 1448637.91666…).
If you understand the language, I have translated the Almagest to French, with some diagrams animated or interactive. It’s freely available at https://ecliptiqc.ca.
¹Pedersen, O. A Survey of the Almagest. With Annotation and New Commentary by Alexander Jones. New York: Springer, 2011, 480 p., ISBN 978-0-387-84825-9.
