Almagest Book X: Mars’ Epoch Positions

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 7934 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;40º1, 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 434º.

So if we subtract 434º 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.



 

  1. Using Ptolemy’s values here.