Tracing the history of medieval astronomy
As the final post for this chapter, we’ll examine the equation of anomaly for angles measured from perigee in the epicyclic model. Again, we’ll start with a basic diagram for the epicyclic model and this time, we’ll drop a perpendicular onto $\overline{DA}$ from H to form $\overline{HK}$.
Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Epicyclic Hypothesis”
In the past couple Almagest posts, we’ve demonstrated that you can derive the equation of anomaly if you know the angle from apogee of either the mean or apparent motion using either the eccentric or epicyclic hypothesis1. Now, we’ll do the same using the angular distance from perigee again using 30º as our example. Continue reading “Almagest Book III: Equation of Anomaly from Perigee using Eccentric Hypothesis”
It’s been awhile since I’ve done a post on observing. This is in large part because I’ve been so busy with events as White Hawk herald for Their Majesties, that I’ve either been at an event, or so burnt out that I’ve been staying in. There was one small observing run between my last post and now, at Crown Tournament, but I only took 3 observations, so I didn’t feel it was worth an entire post2.
However, this month, I really wanted to get out for a long night of observing as I missed observing most of last year in November and December, so plugging that hole in my data was important to me. I ended up getting in about 3.5 hours of observing before it got too cold to want to be out anymore. Continue reading “Data: Stellar Quadrant Observations – 11/23/19”
Despite Ptolemy demonstrating the equivalence of the eccentric and epicyclic hypotheses many times now, he sets out to prove that we can arrive at the same equation of anomaly using the epicyclic hypothesis as we do the eccentric. As we did for the eccentric, we’ll again use the case when the mean motion is 30º from apogee. Continue reading “Almagest Book III: Equation of Anomaly from Apogee using Epicyclic Hypothesis”
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”
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”
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”
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”
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 apogee3 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”
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 times4. 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.