Having completed the preliminary models for the inferior planets, Mercury and Venus, Ptolemy now turns his attention towards the superior ones: Mars, Jupiter, and Saturn. Continue reading “Almagest Book X: Preliminary Comments Regarding the Models for Superior Planets”
Almagest Book VI: Predicting Solar Eclipses
Now that we understand how to predict lunar eclipses, we’ll turn our attention towards solar eclipses. However, Ptolemy warns us that these will be
more complicated to predict because of lunar parallax1.
Toomer again provides an example that we can follow along with2. This will be Example $12$ from Appendix A. Surprisingly, nowhere in the Almagest does Ptolemy describe the details of a solar eclipse. As such, Toomer has selected his own example. In this case, we are to determine the details of the solar eclipse of June $16$, $364$ CE (Nabonassar $1112$ in the month of Thoth), which was observed by Theon of Alexandria3. Upon observing the eclipse, Theon then followed Ptolemy’s methods in the Almagest and Handy Tables to compare the predictions against observations and his calculation are what Toomer follows as an example using Ptolemy’s methods4. Continue reading “Almagest Book VI: Predicting Solar Eclipses”
Almagest Book VI: Solar Eclipses Separated by One Month
We have finally reached the final in this run of eclipse timing feasibility checks. In it Ptolemy wants to demonstrate that it is impossible to have two eclipses separated by one month
even if one assumes a combination of conditions which could not in fact all hold true at the same time, but which may be lumped together in a vain attempt to provide a possibility of the event in question happening.
In short, we’re going to assume an overly ambitious “best case” scenario which can’t actually happen because some of these best case conditions contradict one another. Continue reading “Almagest Book VI: Solar Eclipses Separated by One Month”
Almagest Book VI: Solar Eclipses Separated by Seven Months
Having established that two solar eclipses separated by the five months from the same location are just barely possible, Ptolemy then works on whether it will be possible for the same to occur over a period of seven months concluding that it is possible, provided it happen in the “shortest $7$-month interval”1. Continue reading “Almagest Book VI: Solar Eclipses Separated by Seven Months”
Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax
Now that we’ve determined how far away from the nodes a lunar eclipse can occur, we’ll work on doing the same for a solar eclipse1. But before diving in, I want to say that this has been one of the most, if not the most challenging section of the the Almagest so far. One of the primary reasons is that Ptolemy shows no work and gives almost no explanation on how he did this. When such things happen, I often turn to Neugebauer’s History of Ancient Mathematical Astronomy which I did in this case. There, Neugebauer refers to Pappus of Alexandria, a fourth century mathematician who did commentary on the Almagest and walks through a process that arrives at the same values as Ptolemy.
However, there was a very large amount to unpack in just a few pages there and, unlike most cases where I can simply work along with it and see where things are going, this time I had to really understand the whole process before the first steps made any sense. This led me to agonize over what was going on with those first steps, amounting to several days of effort and rewriting this post from scratch several times. The result is twofold. First because I feel this section can only be approached by understanding the methodology before diving into the math, there’s going to be far more exposition than normal and, as a result, this is likely to be one of my longer posts. Second, the struggles I had with trying to understand the method and rewriting this post so many times has left me with a lot of fragments of thoughts in my brain and in the blog editor. I’ve done my best to clean it up, and maybe it’s just those thoughts swirling around in my brain, but this post just doesn’t feel as coherent as I like. Apologies in advance if you struggle to follow. Know I did as well.
Anyway, moving on to the topic at hand.
Normally, I like to start with a quote from Ptolemy to give us some direction, but I think Ptolemy did such a poor job of laying this section out, I’m going to avoid doing so for the majority of the post. Instead, let’s try to understand the process by recalling what we did with the moon and discussing how things will change. Continue reading “Almagest Book VI: Eclipse Limits for Solar Eclipses – Latitudinal Parallax”
Almagest Book VI: How to Determine the Mean and True Syzygies
Now that we’ve created our table of conjunctions and oppositions, how do we go about using it? As usual, Ptolemy walks through the process in a vacuum, so to help, I’ll follow along with the example Neugebauer does in History of Ancient Mathematical Astronomy on pages $123-124$, although somewhat slimmed down. In particular, I’ll walk through finding the true opposition from the year $718$ in the epoch for the first opposition in the year. Continue reading “Almagest Book VI: How to Determine the Mean and True Syzygies”
Almagest Book VI: Table of Mean Syzygies
Now that we’ve walked through how it’s calculated, here is Ptolemy’s Table of Mean Syzygies. As with previous tables, I’ve made this into a Google Sheet. I’ve also broken it into tabs to make the formatting easier since the first section has two tables and gives actual positions whereas the ones for years and months gives increments.
Almagest Book VI: Construction of the Table of Mean Syzygies
As promised in the last chapter, Ptolemy’s first task in eclipse prediction is going to be laying out a table of mean syzygies around which eclipses might be possible, so we can check those to see if an eclipse might occur instead of performing useless calculations where the sun and moon are nowhere near a syzygy. In this post, we’ll go over the construction of that table! Continue reading “Almagest Book VI: Construction of the Table of Mean Syzygies”