Now that we’ve worked out the system and the position of Mercury about the epicycle on two dates, we can use this to rewind and find the state of the system at the epoch time.
To do so, Ptolemy makes use of the more ancient of the two observations (i.e., the one we explored in the last post).
That observation, Ptolemy tell us, was a period of $483$ Egyptian years, $17$ days, and $18$ hours1 between the epoch time and that observation.
In the last post, we determined the position of Mercury about the epicycle to be $212;34º$.
We can now apply the mean motion over that interval of time to determine that it would have moved $190;39º$ in that time.
Thus, we need to subtract this which gives an epoch position of $21;55º$ after apogee.
The mean longitude, as stated before, matches that of the sun placing it $0;45º$ into Pisces.
Lastly, recall that the apogee of the eccentre for Mercury is taken to move over time. In the time of the previous observation, it was $6º$ into Libra. Ptolemy’s rate for this precession is $\frac{1}{100}º$ per year. So over a period of $483$ years, Ptolemy calls this $4 \frac{5}{6}º$ and subtracts it from the previous position to determine an epoch position of $1 \frac{1}{6}º$ into Libra.
And that’s the end of Book IX. However, it’s not the end of Mercury for us. Ptolemy still hasn’t dealt with the anomaly in latitude. So he’ll come back to that for Mercury (and other planets) in a later book.
