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Data: Stellar Quadrant Observations – 11/23/19

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 post1.

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”

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Almagest Book III: Equation of Anomaly from Apogee using Epicyclic Hypothesis

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”

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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”

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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”

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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”

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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 apogee2 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”

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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 times3. 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.

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

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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 sun4. 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 planets5 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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Data: Stellar Quadrant Observations – 8/24/19

Life has been eating me pretty heavily lately so I haven’t had any time to work on the Almagest, and barely any time to observe. But last weekend, I took the quadrant to Baroness Wars over in the Midrealm. The location wasn’t all that great. The dark sky map said it was like being in the suburbs of St Louis, and the fact that they left some lights on illuminating a racetrack or something next to the county fairgrounds we were in also sucked, but I was still able to get a few observations in over the two nights I was there.

Continue reading “Data: Stellar Quadrant Observations – 8/24/19”