Almagest Book VII: On the Poles of Precession

From the above, it has become clear to us that the sphere of the fixed stars, too, performs a rearward motion along the ecliptic, of approximately the amount indicated. Our next task is to determine the type of this motion, that is to say, whether it takes place about the poles of the equator or about the poles of the inclined circle of the ecliptic.

Now that Ptolemy has determined that precession does indeed happen at a rate that agrees with Hipparchus, he now asks whether that precession is happening in the same direction as the ecliptic or the celestial equator.

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Almagest Book VI: Angles of Inclination at Eclipses

The final few chapters of Book VI are rather odd. Now that we’ve completed the discussion of eclipse prediction, Ptolemy wants to do an “examination of the inclination which are formed at eclipses.” However, he doesn’t appear to provide any motivation for doing so. Toomer and Neugebauer both indicate that the actual reason was likely weather prediction1, but the Almagest doesn’t contain any information on how this is to be used. Neugebauer indicates that,

[T]he technical term connected with this problem is “prosneusis”…developed from the original meaning of the verb νευειν (to nod, to incline the head, etc…). According to the terminology of hellenistic astrology, the planets or moon can, e.g., give their consent by “inclining” toward a certain position, i.e., by being found in a favorable configuration.

However, aside from these astrological purposes, these last few chapters are essentially left as a free-floating bit of material. Continue reading “Almagest Book VI: Angles of Inclination at Eclipses”

Almagest Book V: Parallaxes of the Sun and Moon

Now that Ptolemy has worked out his model for the sun and moon in earth radii, we can use this result to calculate parallax for any position in our model. To begin, we will calculate

the parallaxes with respect to the great circle drawn through the zenith and body.

This is essentially the reverse calculation of what we did in this post, so we can reuse the same diagram:

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Almagest Book V: On the Construction of an Astrolabe

Book IV was all about setting up a preliminary lunar model with a single anomaly which Ptolemy modeled using the epicyclic model. But throughout, Ptolemy kept referencing a second anomaly he discovered, without ever saying how. In his introduction to Book V, Ptolemy finally gives the answer:

We were led to awareness of and belief in this [second anomaly] by the observations of lunar positions recorded by Hipparchus, and also by our own observations, which were made by means of an instrument which we constructed for this purpose.

That instrument was, at the time, called an “astrolabe” which simply means “for taking the [position of] stars,”2 but today we would call it an armillary sphere. Ptolemy describes how one should be constructed which is what we’ll be exploring in this post. To help us, here’s the image of one labeled from Toomer’s translation3.

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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 chapter4, 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, AEG5. 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”