Following Kepler

Having determined the position of Mercury about its epicycle in Ptolemy’s time, he now sets out to do the same for an ancient observation so that the can determine the rate Mercury moves about the epicycle.

Here’s the observation: Continue reading “Almagest Book IX: Position of Mercury About Epicycle 264 BCE Nov 14/15”

One of the biggest keys to understanding the model of Mercury¹, is that the eccentre which drives the main motion², is tied to the motion of the sun. Again, we can see this by comparing the values in the mean motion table for this rotation to those of the sun and seeing they’re identical.

However, we’ll also need to contend with the motion of the second eccentre which controls Mercury’s distance. We’ve talked a fair bit about this, but now it’s time to start seeing how it impacts the basic motion and anomaly. In short, it will offer a “correction” to these. So let’s get started. Continue reading “Almagest Book IX: Position of Mercury About Epicycle 139 CE, May 17/18”

Ptolemy’s next goal will be to determine the apparent size of the epicycle when at one of its two perigees. To do so, he sets up the following diagram: Continue reading “Almagest Book IX: Apparent Size of the Epicycle”

Now that Ptolemy has explained why the eccentric eccentre on which the center of the epicycle resides is necessary, Ptolemy informs us

we have still to demonstrate the position of the point on $\overline{AB}$ about which takes place the annual revolution of the epicycle in uniform motion towards the rear with respect to the signs, and the distance from $Z$ of the centre of that eccentre which performs its revolution in advance in the same period [as the previous].

In other words, it’s time to start figuring out all the parameters to calibrate Mercury’s model. In this post, we’ll explore the various eccentricities. Ptolemy has previously told us that they’re all equal, but now he will demonstrate this. Continue reading “Almagest Book IX: The Eccentricities of Mercury”

As a general rule, I try to stay away from using too much modern math as I work through the Almagest. The goal of this project is to try to understand how astronomers worked in a historical context – not simply examining their work through a modern lens.

However, Ptolemy’s discussion around Mercury has been greatly frustrating me. There’s several reasons for this. A large one is certainly that the language Ptolemy used is clunky which is challenging for a model that is so complex.

Therefore, I want to dig deeper into what’s happening with the double perigee and make sure I fully understand it. In particular, I previously showed a diagram from Pedersen which looked at the path the center of the epicycle would trace out for various eccentricities. This graphically showed the distance from earth over time, but I wanted something more quantitative, so in this post, we’ll derive an equation to determine the distance between the center of Mercury’s epicycle and earth as a function of  the angle from apogee.

This method comes straight from Pedersen but I’ll be doing it in the context of Toomer’s translation¹. Continue reading “Almagest Book IX: Exploring Mercury’s Double Perigee”

In our last post, I noted that Ptolemy had identified the line of apsides, but was very careful not to state which point was the apogee and which was the perigee. He’s now ready to start looking into that:

In accordance with the above, we investigated the size of the greatest elongations which occur when the mean longitude of the sun is exactly in the apogee, and again, when it is diametrically opposite that point.

Continue reading “Almagest Book IX: Mercury’s Double Perigee”