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”
Almagest Book IX: The Eccentricities of Mercury
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”
Almagest Book IX: Exploring Mercury’s Double Perigee
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”
Almagest Book IX: 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”
Almagest Book IX: The Line of Apsides of Mercury
Having demonstrated the conceptual model, Ptolemy now turns to determining
in what part of the ecliptic Mercury’s apogee lies by the following method. Continue reading “Almagest Book IX: The Line of Apsides of Mercury”
Almagest Book IX: Symmetries in Mercury’s Planetary Model
Now that we’ve demonstrated that the equation of anomaly is symmetric about the line of apsides for the general model, we’ll demonstrate the same for Mercury’s model. Or, as Ptolemy puts it,
we must prove that in this situation too the angles of the equation of ecliptic anomaly [are equal].
We’ll start by producing a diagram based on Mercury’s particular model. Continue reading “Almagest Book IX: Symmetries in Mercury’s Planetary Model”
Almagest Book IX: Symmetries in the General Planetary Model
Now that we’ve laid out the models for the planets except Mercury and the special case for Mercury, Ptolemy revisits them to explore some symmetries. We’ll start with the ones for the model for the planets other than Mercury first.
To start, let’s produce a new diagram based on that model. As usual, I’ll break up Ptolemy’s description into several steps to help make it a bit more digestible. Continue reading “Almagest Book IX: Symmetries in the General Planetary Model”
Almagest Book IX: Model for Mercury
Now that we’ve taken the time to understand the model for the four planets other than Mercury, let’s start on the model for Mercury as that’s the focus of the remainder of this book.
Let the eccentre producing the anomaly be $ABG$ about centre $D$, and let the diameter through $D$ and centre $E$ of the ecliptic be $\overline{ADEG}$, [passing] through the apogee at $A$. Continue reading “Almagest Book IX: Model for Mercury”
Almagest Book IX: Model for Planets Other than Mercury
Having thoroughly discussed what anomalies Ptolemy wants his model to account for as well as what hypotheses1 he intends to use for each, Ptolemy is ready to start laying out the basic models. Ultimately, there will be two models. One for the four planets other than Mercury, and a special one for Mercury.
In this post, we’ll explore the first of these models. Continue reading “Almagest Book IX: Model for Planets Other than Mercury”
Almagest Book IX: Preliminary Notions
Now that these [mean motions] have been tabulated, our next task is to discuss the anomalies which occur in connection with the longitudinal positions of the five planets.
Having derived the mean and anomalistic motions, Ptolemy now turns to exploring the anomalies in more depth (as there’s going to be two of them), in order to derive the parameters necessary for configuring the model’s scale. Continue reading “Almagest Book IX: Preliminary Notions”