Tag: algebra

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Almagest Book V: Lunar Distance Adjustments for Epicycle

In the last post, we explored how to calculate parallax if the distance to an object is known and its distance from the zenith. This was done for the sun and the moon at four different distances. However, because the moon varies so widely in distance in Ptolemy’s model, we need a way to estimate between those positions and we’ll begin by looking at the effect the epicycle has on distance for various points throughout its cycle. To help us, we’ll start with a new diagram:

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Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle

Continuing on with Ptolemy’s check on the radius of the epicycle, we’ll produce a new diagram based on the positions of the Alexandrian eclipses. However, instead of doing it piece-by-piece as I did when we explored the Babylonian eclipses, I’ll drop everything into a single diagram since we already have some experience and the configuration for this triple is a bit more for forgiving on the spacing:

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Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle

In the last post, we introduced three eclipses from Babylonian times which we used to build a couple intervals: The first eclipse to the second, and the second to the third. Using those, we used lunar and solar mean motion tables to figure out the solar position, as well as its change. Since the moon must be opposite the sun in ecliptic longitude for an eclipse to occur, we used the change in solar position to determine the true change in lunar position in these intervals. From that, we could compare that to the mean motion to determine how much of it must be caused by the lunar anomaly. But while we’ve determined this component, we haven’t done anything with them yet.

So in this post, we’ll start using these to answer several questions that will build out the details of the model. Specifically, we want answers to questions like what is the radius of the epicycle? Where, in relation to the ecliptic was the mean moon during these eclipses and what was the equation of anomaly? That’s a lot of information to extract so I’m going to try to break it up a bit and in this post, we’ll only tackle the radius of the epicycle1

To begin, let’s sketch out the epicyclic lunar model2 with the three eclipses drawn on it.

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

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 $\overline{HK}$.

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Almagest Book III: Equation of Anomaly from Perigee using Eccentric 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 hypothesis3. 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”

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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 – 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.

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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, AEG4. 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”