Tracing the history of medieval astronomy
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”
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”
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”
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”
Our next step is to determine the epoch position for Saturn. Continue reading “Almagest Book XI: Saturn’s Epoch Position”
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”
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”
Our next step in calibrating Saturn’s model, is to verify that the steps we’ve taken so far is able to reproduce the original observations. Continue reading “Almagest Book XI: Checking the Eccentricity and Apogee of Saturn”
With our corrections for the intervals from Saturn’s oppositions in hand, we’re ready to repeat the calculation. As with before, I’m doing this in a Google Sheet to speed things along. This means that I’ll be using modern trig and that the figures I give her may be subject to some rounding as we go since the Sheet is saving higher precision behind the curtain.
We’ll begin with the same diagram as before:
Continue reading “Almagest Book XI: Second Iteration for Saturn”
Now that we’ve done our initial determination of the eccentricity for Saturn, and the position of its line of apsides, we’ll work on making the first, and only, iterative correction to these. We’ll again start by finding corrections to the angles for each opposition beginning with a diagram that should look familiar. Continue reading “Almagest Book XI: Corrections for the Equant”