Our goal in this chapter will be to determine a systematic way to the equation of anomaly for the sun at 6º intervals from apogee (and perigee) along the eccentre. In this post, we’ll explore the method for finding this value from apogee for a single value of 30º. Continue reading “Almagest Book III: Equation of Anomaly from Apogee using Eccentric Hypothesis”
Almagest Book III: On the Anomaly of the Sun – Difference Between Mean and Anomalistic Motion
In the last post, we determined the basic properties of the eccentric model to predict the motion of the sun. Now, we’ll use these properties in conjunction with the model itself to be able to predict “the greatest difference between mean and anomalistic motions” by referring back to our original model. Continue reading “Almagest Book III: On the Anomaly of the Sun – Difference Between Mean and Anomalistic Motion”
Almagest Book III: On the Anomaly of the Sun – Basic Parameters
Now that we’ve laid out how the two hypotheses work and explored how they sometimes function similarly, it’s time to use one of them for the sun. But which one? Or do we need both?
For the sun, Ptolemy states that the sun has
a single anomaly, of such a kind that the time taken from least speed to mean shall always be greater than the time from mean speed to greatest.
In other words, the sun fits both models, but only requires one. Ptolemy chooses the eccentric model due to its simplicity.
But now it’s time to take the hypothesis from a simple toy, in which we’ve just shown the basic properties, and to start attaching hard numbers to it to make it a predictive model. To that end, Ptolemy states,
Our first task is to find the ratio of the eccentricity of the sun’s circle, that is, the ratio of which the distance between the center of the eccentre and the center of the ecliptic (located at the observer) bears to the radius of the eccentre.
Continue reading “Almagest Book III: On the Anomaly of the Sun – Basic Parameters”
Almagest Book III: Hypotheses for Circular Motion – Similarities in Apparent Apogee and Perigee Distances
The last symmetry between the two models Ptolemy wants to point out is that,
where the apparent distance of the body from apogee [at one moment] equals its apparent distance from perigee [at another], the equation of anomaly will be the same at both positions.
Fortunately, the proofs for this are quite simple. Continue reading “Almagest Book III: Hypotheses for Circular Motion – Similarities in Apparent Apogee and Perigee Distances”
Almagest Book III: Hypotheses for Circular Motion – Similarities in the Equation of the Anomaly
So far in this chapter, we’ve been looking at the two different hypotheses to explain the non-constant angular motion of objects in the sky. Ptolemy claimed that these were equivalent under certain circumstances and, in the last post, we showed how they do indeed produce the same results in the specific case of the object travelling 90º in apparent motion from apogee1 and that it always takes longer for the object to go from slowest motion to mean, than it does mean to fastest.
But that’s not really a full demonstration that they’re functionally the same. So in this post, we’ll show that their apparent angular position from the mean (known as the equation of the anomaly) is always the same, so long as there’s a few things that are consistent between models. Continue reading “Almagest Book III: Hypotheses for Circular Motion – Similarities in the Equation of the Anomaly”
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 times2. 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.
Almagest Book III: Hypotheses for Circular Motion – The Two Hypotheses
So far in Book III, we’ve been looking at the mean sun3. 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 planets4 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”
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”
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 theorem5 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].
Continue reading “Almagest Book III: On the Length of the Year”
On Chronology and Calendars in the Almagest
In Book III, Ptolemy is going to start using actual observations to determine how things change over long periods of time. In order to do so, we will need to explore the calendar system that is used in the Almagest, namely the Egyptian calendar. Continue reading “On Chronology and Calendars in the Almagest”