Category: Almagest

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Almagest Book III: Hypotheses for Circular Motion – Similarities in Time Between Apogee and Mean Between Models

In the last post, we introduced two different models that could potentially explain the anomalous motion of the sun (or other objects). Specifically, the sun sometimes appears to move faster along the ecliptic than at other times1. The first model was the eccentric model in which the observer was placed off center. The second was the epicyclic in which the object would travel around the deferent on an epicycle.

Ptolemy stated that for the simple motion of the sun, either of these models would be sufficient. However, he wanted to demonstrate a key equivalence. Specifically that

for the eccentric hypothesis always, and for the epicyclic hypothesis when the motion at apogee is in advance, the time from least speed to mean is greater than the time from mean speed to greatest; for in both hypotheses the slower motion takes place at the apogee. But [for the epicyclic hypothesis] when the sense of revolution of the body is rearwards from the apogee on the epicycle, the reverse is true: the time from greatest speed to mean is greater than the time from mean to least, since in this case the greatest speed occurs at the apogee.

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Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses

So far in Book III, we’ve been looking at the mean sun2. However, the sun’s motion along the ecliptic is not even. In other words, its speed along the ecliptic is not constant. This poses a problem because, to Ptolemy:

the rearward displacements of the planets3 with respect to the heavens are, in every case, just like the motion of the universe in advance, by nature uniform and circular. That is to say, if we imagine the bodies or their circles being carried around by straight lines, in absolutely every case the straight line in question describes equal angles at the center of its revolution in equal times.

Here, Ptolemy states that he accepts uniform circular motion as true. So the question becomes how can something moving at a constant speed appear to have not constant speed? Continue reading “Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses”

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Almagest Book III: Table of the Mean Motion of the Sun

To help us determine where the sun is on a given date, Ptolemy sets out a table to allow us to more quickly look this up instead of needing to calculate how far around the ecliptic the sun has moved. This table is very similar to another set of tables Ptolemy would produce later: The Handy Tables. However, there are some notable differences between this table and the Handy Tables. Continue reading “Almagest Book III: Table of the Mean Motion of the Sun”

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Almagest Book III: On the Length of the Year

If I were to summarize the books of the Almagest so far, I’d say that Book I is a mathematical introduction to a key theorem4 and an introduction to the celestial sphere for the simplest case of phenomenon at sphaera recta. In Book II, much of that work is extended to sphaera obliqua, but in both cases, we’ve only dealt with more or less fixed points on the celestial sphere: The celestial equator, ecliptic, and points within the zodiacal constellations based on the immovable stars.

But the ultimate goal of the Almagest and my project isn’t to study the unchanging sky; it’s to understand the changing sky: The sun, moon, and planets. Ptolemy decides to start with the position of the sun is a prerequisite to understanding the phases of the moon, and planets are more complicated with their retrograde motions. And to kick off the investigation of the motion of the sun, Ptolemy first begins by carefully defining a “year” noting

when one examines the apparent returns [of the sun] to [the same] equinox of solstice, one finds that the length of the year exceeds 365 days by less than $\frac{1}{4}$-day, but when one examines its return to the fixed stars, it is greater [than 365 $\frac{1}{4}$-days].

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Almagest Book II: Table of Zenith Distances and Ecliptic Angles

Finally we’re at the end of Book II. In this final chapter6, Ptolemy presents a table in which a few of the calculations we’ve done in the past few chapters are repeated for all twelve of the zodiacal constellations, at different times before they reach the meridian, for seven different latitudes.

Computing this table must have been a massive undertaking. There’s close to 1,800  computed values in this table. I can’t even imagine the drudgery of having to compute these values so many times. It’s so large, I can’t even begin to reproduce it in this blog. Instead, I’ve made it into a Google Spreadsheet which can be found here.

First, let’s explore the structure.

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Almagest Book II: Angle Between Ecliptic And Altitude Circle – Calculations

In the last post, we started in on the angle between the ecliptic and an altitude circle, but only in an abstract manner, relating various things, but haven’t actually looked at how this angle would be found. Which is rather important because Ptolemy is about to put together a huge table of distances of the zenith from the ecliptic for all sorts of signs and latitudes. But to do so, we’ll need to do a bit more development of these ideas. So here’s a new diagram to get us going.

Here, we have the horizon, BED. The meridian is ABGD, and the ecliptic ZEH. We’ll put in the zenith (A) and nadir (G) and connect them with an altitude circle, AEG7. Although it’s not important at this precise moment, I’ve drawn it such that AEG has E at the point where the ecliptic is just rising. Continue reading “Almagest Book II: Angle Between Ecliptic And Altitude Circle – Calculations”

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Almagest Book II: Angle Between Ecliptic And Altitude Circle – Relationships

We’re almost to the end of book II. There’s really 2 chapters left, but the next one is almost entirely a table laying out the values we’ve been looking at here recently, so this is the last chapter in which we’ll be working out anything new.

In this chapter, we’ll tackle the angle between the ecliptic and a “circle through the poles of the horizon”. If you imagine standing outside, the zenith is directly overhead which is the pole for your local horizon. Directly opposite that, beneath you, is the nadir. If these two points are connected with a great circle, that’s the great circle we want to find the angle of with respect to the ecliptic. Because we measure upwards, from the horizon, along an arc of these great circles, to measure the altitude of a star, these are often called altitude circles.

But while we’re at it, Ptolemy promises that we’ll also determine “the size of the arc…cut off between the zenith and…the ecliptic.” In other words, because the ecliptic is tilted with respect to the horizon, the arcs between the two will be different.

To get us started, Ptolemy begins with the following diagram.

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Almagest Book II: Angle Between Ecliptic And Horizon – Calculations

We’ll continue on with our goal of finding the angle the ecliptic makes with the horizon. Fortunately, this task is simplified by the symmetries we worked out in the last post meaning we’ll only need to work out the values from Aries to Libra. Unfortunately, this value will change based on latitude as well as the position on the ecliptic, but we’ll still only do this for one location. And for that location, Ptolemy again uses Rhodes.

First we’ll start with angles at the equinoxes:

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