A year ago today, I made my first post1 regarding the Almagest. And just a few days ago, I made the last post regarding Book II. It’s been a huge challenge so far, so I wanted to take a moment to review.
When telling people about my project, and the Almagest in particular, I often find myself pointing out how slow the progress has been. My initial estimate was that this book would take ~3 years to complete. However, at the current rate, it will take twice that, and the rate this year has largely been increased by a stint of unemployment that allowed me more time than I’ve had more recently to work. This has been abundantly evident simply from my post count, which was previously a post every ~2 days, but since starting my new job in November of 2018. dropped significantly.
Not only has the amount of time I’ve been able to dedicate to the project diminished, but the text has gotten significantly harder. The dearth of posts in April of this year was predominantly caused by a single calculation that I wasn’t able to replicate. Getting truly stuck has been happening more and more frequently.
Unfortunately, there is very little help available to turn to at such moments. At present, I’m only familiar with two sources which have been able to offer some help, A Survey of the Almagest, by Olaf Pedersen and A History of Ancient Mathematical Astronomy, by Neugebauer. However, I’ve found both sources have their limitations. The latter does not cover the Almagest entirely, plucking out only select topics. The former is much more comprehensive, but often does not follow the same arguments that Ptolemy does, resorting to modern methods to determine some values, and using a different set of variables from the Toomer translation of the Almagest. Using modern methods means that some of the calculations are not actually covered, and the change in notation often means having to learn the material a second time to try to understand Survey. As such, I’ve found both of these sources to be less helpful than one would hope.
The result is that getting stuck truly does feel like hitting a wall and is often very demoralizing. However, sometime this means that the satisfaction at getting over the hump is all the more satisfying. But not always. Sometimes I feel utterly stupid that I was hung up on something simple.
Still, a year in, I’m not really feeling much burnout. It’s rarer that I have time to work, but it’s still quite exciting. While I’ve had some training with spherical geometry during my degree, the approaches Ptolemy takes are entirely different. When I learned the material, we used trigonometry and Napier’s rules. Trigonometry was developed prior to Ptolemy, but he has not made any use of it. Napier’s rules weren’t developed until the 1600’s, so certainly outside the realm anything Ptolemy would have accessed. Instead, Ptolemy relies almost entirely on Menelaus’ theorem. The result is that I’ve had to engage with a fundamentally different way of thinking about spherical geometry. After almost a year of working with it, I feel like I’ve gotten pretty good about flipping between chords and their related arcs.
But after developing these skills, it looks like I’m about to set them aside. I’ve looked forward to book 3 somewhat, and fundamentally the diagrams are different. Instead of working on spheres, Book III looks to be top down diagrams of Ptolemy’s geocentric solar system. This certainly reduces the complexity of things, but it also means that some of what I’ve done thus far won’t be getting use for a good while. I expect I’ll be having to learn another way of thinking about these maths as well, since my normal approach would again be trigonometric.
I’m hoping that Book III ends up being a bit easier than Book II was. It is somewhat shorter at the least, and the first several pages are discussion as opposed to calculation, so that should get me a good jump into it in the next few days, but there’s still a long road to travel, and many years left to walk it.