The last thing to do before finishing up with Jupiter for now, is to determine its epoch positions.
Ptolemy does this by taking the intervals from the ancient observation we explored in the last post.
The interval between that observation and the beginning of the epoch, Ptolemy gives as
$506$ Egyptian years and approximately $316 \frac{3}{4}$ days.
If we add up the mean motions over that interval, I find an increase
in longitude of $258;13º$
in anomaly of $290;58º.$
My values here agree with Ptolemy.
We’ll then subtract these from the positions of Jupiter’s mean longitude and angle about the epicycle from the ancient observation. We determined these in the last post where we determined that the center of Jupiter’s epicycle was $74;19º$ before the apogee which is the same as $285;41º$ after it.
Thus, if we subtract $258;13º$ from that, we determine that, at the moment of epoch, Jupiter’s mean position was $27;28º$ after the apogee at least, with respect to where the apogee was during that time. Since we said that it was $7;13º$ into Virgo the means that Jupiter would have been located $4;41º$ into Libra.
We can also ask where apogee would have been at the epoch. Again, it was $7;13º$ into Virgo during the time of the ancient observation. And since the time between the observations was $\approx 507$ years, we can take the precession as $5;04º$. Thus, the apogee would have been $2;09º$ into Virgo.
Lastly, we can determine the epoch position of Jupiter about the epicycle. At the time of the ancient observation, we determined that Jupiter was $77;02º$ past apogee. Thus, if we subtract $290;58º$ we find that Jupiter was $146;04º$ after apogee about the epicycle.
And that’s it for Jupiter for now. We’ll do this all again for Saturn and then start working on how to deal with taking the position of all of the planets about their epicycles and turning that into the anomaly to apply to the mean position. Once that’s done, we’ll be done with Book XI!
