Our next step is to determine the epoch position for Saturn.
The procedure should be pretty familiar by now. We’ll determine the interval of time, look up the accumulated motions in the mean motion tables, and apply them to the positions from the observation in our last post.
Ptolemy gives the interval of time between the beginning of his epoch and the ancient observation from our last post as $518$ Egyptian years, $133 \frac{1}{4}$ days.
Taking this to the Planetary mean motion tables, I find an increase in longitude of 216;08,26º. Toomer notes that Ptolemy’s value has some uncertainty to it. Many of the Greek manuscripts have $216º$ or $216;00º$. However, some have $216;09º$ which would be consistent with how Ptolemy tends to round. However, Toomer takes Ptolemy to have meant $216;10º$ as this is the value implied by a calculation we’ll do shortly.
Turning to the anomaly, I find an increase of $149;15,26º$ which is consistent with Ptolemy’s value of $149;15º.$
These now get subtracted from the positions from the Babylonian astronomers observation. As a reminder, in the last post, we determined that Saturn was $2;53º$ into Virgo, and $183;17º$ about its epicycle from the apogee of the epicycle.
Thus, for longitude, if we subtract using the value of $216;10º$ that Toomer proposes, I find that Saturn would have been $26;43º$ into Capricorn which is the value Ptolemy gives and is found in all manuscripts, affirming that Toomer’s value for the increase in longitude over this interval is likely what Ptolemy intended, even if it is a bit rounded.
For the anomaly, I find that Saturn would have been $34;02º$ about its epicycle from apogee.
Lastly, we can also ask about the epoch position of apogee. As we established in the last post, at the time of the Babylonian observation, apogee would have been located $19 \frac{1}{3}º$ into Virgo. We then take a rate of $1º$ per century for the precession over the interval between epoch and this date to find it would have increased its position by $5;11º$. this would put the epoch position of apogee $14;09º$ into Virgo. However, Ptolemy comes up with $14;10º$ indicating he may have done some rounding, likely on the increase over this interval.
And that’s it for the series of steps that we effectively repeat for the superior planets. In the next post, we’ll explore how to calculate the true position from the periodic motions.
