Having laid out the general structure of the milky way, Ptolemy next describes the process of “the construction of the solid globe and the delineation of the constellations”. Continue reading “Almagest Book VIII: On the Construction of a Solid Globe”
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.
Continue reading “Almagest Book VII: On the Poles of Precession”
Class Materials
This post a collection of my class materials which will be posted below the fold. In general, this is a Powerpoint presentation and the script I follow is included in the file. Continue reading “Class Materials”
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”
Update – 2/5/21: New Classes Are Coming
It’s been a few weeks since I’ve posted anything but it’s not because I haven’t been busy on astronomy. My recent activity just hasn’t been in the Almagest or observing. Rather, I’ve been working on some classes.
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:
Continue reading “Almagest Book V: Parallaxes of the Sun and Moon”
Almagest Book V: Calculation of Lunar Distance
Now that we’ve demonstrated a parallax of $1;07º$ for the moon, we can use that do determine a distance to the moon. As a forewarning, some of the math may seem suspect here, so I’ll do my best to explain it.
To being, let’s start off with a new drawing:
Continue reading “Almagest Book V: Calculation of Lunar Distance”
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.
Continue reading “Almagest Book V: On the Construction of an Astrolabe”
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.
Continue reading “Almagest Book II: Table of Zenith Distances and Ecliptic Angles”
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”