Now that we’ve ironed out Mars’ position for the time of the Babylonian observation we discussed in the last post, and we’ve revised the periodic motions, it’s time to calculate the epoch position.
[T]he interval from the first year of Nabonassar, Thoth $1$ in the Egyptian calendar, noon, until the above observation is $475$ Egyptian years and approximately $79 \frac{3}{4}$ days, and that interval comprises increments of $180;40º$ in longitude, and $142;29º$ in anomaly.
Ptolemy quickly gives us the increments here, but let’s check them.
| Interval | Longitude | Anomaly |
| $468$ years | $39;52,48º$ | $6;20,18º$ |
| $7$ years | $258;58,21º$ | $99;19,32º$ |
| $60$ days | $31;26,37º$ | $27;41,40º$ |
| $19$ days | $9;57,56º$ | $8;46,12º$ |
| $18$ hours | $0;23,35º$ | $0;20,46º$ |
| Total | $180;38,46º$ | $142;28,29º$ |
As you can see, my numbers are slightly different than Ptolemy’s, but easily within reason for rounding.
Thus, we need to subtract these totals from each of the values.
For Mars’ position in longitude, we determined that, at the time of the Babylonian observation in question, Mars should have been $4;12º$ into Libra which is $182;12º$ ecliptic longitude. Thus, if we subtract $180;40º$1, we find that Mars was $3;32º$ into Aries at the beginning of the epoch.
Similarly, Mars was $109;42º$ about its epicycle from perigee at the time of the Babylonian observation. Thus, if we subtract off $142;29º$, I find it would have been $327;13º$ after its apogee on its epicycle.
Lastly, the latitude of apogee changes over time in this model. During the Babylonian observation, we determined it was $21;25º$ into Cancer. So how much to adjust it?
Recall that Mars’ apogee isn’t really changing. What is changing is our coordinate system due to precession of the equinoxes. Thus, the apogee for all the planets changes at this same rate which Ptolemy takes as $1º$ per century.
Thus,
for the shift of the apogee in $475$ years, one gets by computation $4 \frac{3}{4}º.$
So if we subtract $4 \frac{3}{4}º$ from $21;25º$ into Cancer, we find Mars’ apogee would have been $16;40º$ into Cancer at the time of epoch.
And that concludes Book X.
You may be a bit confused at this point because, while we now know how to calculate the position of Mars about its eccentre and its epicycle, we haven’t discussed what the effect is. In short, what amount of anomaly does the position about the epicycle impart? And how does the equant and center of mean distance work with it?
We’ll come to that in Book XI. But first, we’ll repeat what we’ve just done for Mars for Jupiter and Saturn so we can tackle those questions for all three planets at the same time.
