Almagest Book XII: Preliminary Lemma from Apollonius
In Books X and XI, we’ve begun developing a theory for the motion of the planets and have that pretty much complete for all of the planets in longitude; we haven’t begun addressing their latitude. But, before getting to that, Ptolemy wants to explore retrograde motion.
While the models already account for this, Ptolemy is wanting to be able to predict when it will happen1. That will be our focus for Book XII and we’ll get to latitudes in Book XIII.
Ptolemy tells us that he wants to “examine the greatest and least retrogradations associated with each of the five planets, and to show that the sizes of these, [as computed] from the above hypotheses, are in as close agreement as possible with those found from observations.” Continue reading “Almagest Book XII: Preliminary Lemma from Apollonius”
Understanding Ptolemy’s Almagest: Books I & II
I’m finally getting to some videos on the Almagest. So I suppose I need to get back to working on it, so I’m finished with it before the videos catch up…
Almagest Book XI: Calculating Planetary Longitude
With both the planetary mean motion tables, and the tables of anomaly, we’re now ready to calculate planetary longitude.
As with before, Ptolemy explains how this is to be done, but doesn’t provide an example. As such, I’ll follow along with example $14$ from Toomer (in Appendix A). Continue reading “Almagest Book XI: Calculating Planetary Longitude”
Almagest Book XI: Planetary Tables of Anomaly
Chapter $11$ of this book is simply the Planetary Equation Tables. I have transcribed them and make them available as a Google Sheet. Continue reading “Almagest Book XI: Planetary Tables of Anomaly”
Almagest Book XI: Constructing the Tables of Anomaly
In the last chapter, Ptolemy walked us through a light proof on how to find the apparent position of a planet. But, immediately in this chapter, Ptolemy tells us that going through that process every time is troublesome:
[A]lthough that method is the only one which provides a fully accurate solution to the problem, it is too cumbersome to be convenient for [astronomical] investigations.
What is Ptolemy’s solution?
As you probably expect, it’s to make tables that allow us to calculate quickly:
We have constructed, for each of the five planets, a table which is easy to use as we could devise, while at the same time being very close to full accuracy.
In this chapter, we’ll follow along as Ptolemy explains how they are laid out.
However, Ptolemy’s description is not well explained, so I will interweave this with a more comprehensive explanation based on Neugebauer’s History of Ancient Mathematical Astronomy. Continue reading “Almagest Book XI: Constructing the Tables of Anomaly”
Almagest Book XI: Finding the True Position
Our next goal will be to determine how,
given the arcs of the periodic [motions] on the eccentre which produces the uniform motion [i.e., the equant] and on the epicycle, one can readily obtain the apparent positions of the planets.
Continue reading “Almagest Book XI: Finding the True Position”
Almagest Book XI: Saturn’s Epoch Position
Our next step is to determine the epoch position for Saturn. Continue reading “Almagest Book XI: Saturn’s Epoch Position”
Almagest Book XI: Saturn’s Periodic Motion
Ptolemy now walks us through his supposed method for determining the periodic motions of Saturn. I say “supposed” because, again, Ptolemy’s final result doesn’t quite match what he tells us here. But we’ll follow along anyway.
As usual, he begins with an observation from the Babylonian astronomers: Continue reading “Almagest Book XI: Saturn’s Periodic Motion”
Almagest Book XI: Determining the Size of Saturn’s Epicycle
Next, we’ll turn our attention to determining the size of Saturn’s epicycle. As he did for Jupiter, Ptolemy introduces a new observation. Continue reading “Almagest Book XI: Determining the Size of Saturn’s Epicycle”