In Book III of the Almagest, Ptolemy developed a methodology by which the solar position could be predicted. This method had a few key components. This included determining the mean motion, the equation of anomaly, and calculating a start date/position to which those could be applied.
Unfortunately, trying to use Ptolemy’s solution would no longer work in the present day. While the mean motion and equation of anomaly are still fine, precession of the equinoxes and other small effects over the past ~1,900 years mean too much has changed. So while we can’t use the exact results of Ptolemy’s it should be possible to recreate his methodology with a more recent starting date again allowing for reasonably accurate predictions of the solar position.
So for my entry to this year’s Virtual Kingdom Arts & Science competition, I did.
UPDATE – 11/13/2021: As indicated below, I followed this paper by working on one to update the lunar model for present day. That method is calibrated based on eclipses which is dependent on the position of the sun and thus, the results of this paper. In using the results of this paper to determine the position of the sun on the modern dates, I realized calculated position was sufficiently far off that an eclipse should not have occurred. As such, I reviewed the paper and have found several errors. Most notably, I’d not recalculated the table of anomaly to account for the changing position of the center of the eccentre. In addition, I missed one of Ptolemy’s final steps in which he accounts for the anomaly at epoch. I’d also made more minor errors dealing with daylight savings time. Ultimately, I have decided to no longer share this version of the paper since the results are ultimately embarrassingly wrong. However, the final result of the corrections return impressively accurate results. A sample calculation for an eclipse in 2003 had the position of the sun off by less than $0.05º$. It is a good indication of why Ptolemy’s models were so influential.
Ultimately, instead of correcting this original paper, I have decided that all of these papers will get put into a single masterwork which I am compiling in LaTeX instead of a Google Doc. As such, when it is more polished, I will reshare this updated version, including only the introduction and the solar chapter, here.
I’m quite pleased with it, not necessarily for the paper itself, but because it proved an excellent capstone project and review of most of Book III.
The first part introduces Ptolemy’s two solar models, the eccentric and epicyclic, just as a way to introduce readers to the concepts. This basically covered all of III.3.
Next, I followed Ptolemy’s method to reverse calculate the position of the sun at the beginning of his epoch which was III.7.
With these two pieces, I could begin updating the method for modern times. However, one of the updates necessary would be to determine the modern position of the apogee. While this didn’t cover all of III.4, it covered a major portion of it.
Lastly, using the mean motion table and the table of anomaly, the future position from epoch could be determined as covered in III.8.
Given that these two tables were provided in the appendices, that also covered III.2 and III.6.
So what ended up not getting covered was III.1 which dealt with the definition of years and the calculation of the solar mean motion in such a time. I also skipped the derivation of the equation of anomaly from III.5.
Ultimately, this paper is a bit more like a Handy Tables guide on how to use the tables to make predictions without the full depth of knowledge. But given there’s many people interested in the astrological side of things1, this has practical applications for them. Or anyone who really just wants to calculate things.
My plan is to make this sort of paper a regular feature after finishing major topics in the Almagest. For Book II, I’d done something similar with another paper in which I demonstrated how to calculate rising signs, again making use of many of the major topics in the book. As I finish working out the models for the moon and planets, I intend to add them to this epoch so readers can easily calculate predictions in the modern era.