Category: Almagest

Posted on

Almagest Book IV: Table of Lunar Equation of Anomaly (First Anomaly)

Although I don’t see any particular reason we couldn’t have done this several chapters earlier, Ptolemy has elected to hold off producing his table of Lunar Anomaly until almost the very end of Book IV1.

At the end of the previous chapter, he did toss in a few expository notes on this table noting that this will be useful in the future for “calculations concerning conjunctions and oppositions”2. To do so, Ptolemy doesn’t show his math, but notes that it was done “geometrically, in the same way as we already did for the sun.”

However, to use that method requires knowing the ratio of the radius of the deferent to that of the epicycle. Ptolemy states he takes a ratio of $60:5 \frac{1}{4}$. Using that, he calculates his table in the same manner as he did for the sun, which is in $6º$ intervals near apogee and $3º$ intervals near perigee. However, because this epicycle rotates opposite the direction of the deferent, we should subtract if the angle is in the first column, and add if it is in the second3

That being laid out, the table is below: Continue reading “Almagest Book IV: Table of Lunar Equation of Anomaly (First Anomaly)”

Posted on

Almagest Book IV: Determining the Lunar Epoch for Latitude

Now that we’ve squared away the most accurate motion in latitude Ptolemy can muster, it’s time to use it to reverse calculate the lunar latitude at the beginning of the epoch, but to do so, we’ll need to calculate the precise position in latitude at some point in time. To do so, Ptolemy again turns to eclipses, with all the same considerations as the last post, except we want ones at opposite nodes instead of near the same one. Continue reading “Almagest Book IV: Determining the Lunar Epoch for Latitude”

Posted on

Almagest Book IV: Correcting the Lunar Mean Motion in Latitude

Previously, we looked at how Ptolemy made corrections to the anomalistic motion for his lunar model. In this post, we’ll be doing something similar for the mean motion of lunar latitude.

Ptolemy explained that the value we originally noted was in error,

because we too adopted Hipparchus’ assumptions that [the diameter of] the moon goes approximately 650 times into its own orbit, and $2 \frac{1}{2}$ times into [the diameter of] the Earth’s shadow, when it is at mean distance in the syzygies.

In short, Hipparchus’ figures were a good starting point but now we can do better by

using more elegant methods which do not require any of the previous assumptions for the solution of the problem.

Continue reading “Almagest Book IV: Correcting the Lunar Mean Motion in Latitude”

Posted on

Almagest Book IV: The Epoch of the Mean Motions of the Moon in Longitude and Anomaly

As we saw for the sun, to be able to use the tables of mean motion to predict the position of the moon, we’ll need to know where the moon was at a specific point in time. Ptolemy chooses as that point in time as the beginning of the first year of the Nabonassar reign. To determine the position of the moon on this date, Ptolemy starts with the second eclipse of the Babylonian triple we discussed in the last chapter and then calculates backwards. Continue reading “Almagest Book IV: The Epoch of the Mean Motions of the Moon in Longitude and Anomaly”

Posted on

Almagest Book IV: Correction of the Mean Motions of the Moon in Longitude and Anomaly

Towards the beginning of Book IV, Ptolemy went through the methodology by which the various motions in the lunar mean motion table could be calculated. But if you were paying extra close attention, you may have noticed that the values that ended up in the mean motion table didn’t actually match what we derived. Specifically, for the daily increment in anomaly, we derived $13;3,53,56,29,38,38^{\frac{º}{day}}$. But in the table, we magically ended up with $13;3,53,56,17,51,59^{\frac{º}{day}}$. Identical until the 4th division.

So what gives? Why did Ptolemy derive one value and report another?

In Chapter 7, he gives the explanation: He found the value needed to be corrected and did so before he put it in the table. But before he could explain to us how, we needed to cover the eclipse triples we did in Chapter 6. So how do we apply them to check the mean motions? Continue reading “Almagest Book IV: Correction of the Mean Motions of the Moon in Longitude and Anomaly”

Posted on

Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon

We’re almost finished with chapter 6. All that’s left is to determine the position of the mean moon during one of the eclipses which will tell us the equation of anomaly at that point. To do so, we’ll add a few more points to the image we ended the last post with:

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon”

Posted on

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:

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Radius of the Epicycle”

Posted on

Almagest Book IV: Alexandrian Eclipse Triple – Solar/Lunar Positions & Epicyclic Anomaly

Modern commentary on Ptolemy often downplays the Almagest because it is certainly a work that relied heavily on the work that astronomers before him. While we no longer have a thorough record of those predecessors, it seems that few historians think much of the Almagest was truly novel4. But I would hasten to remind that, while Ptolemy stood on the shoulders of those who came before, he certainly climbed there on his own, not simply accepting their results, but doing his best to validate them.

And we’re about to get a big dosing of that, because all the work we’ve done in the past three posts, we’ll be redoing with a new set of eclipses observed by Ptolemy himself, allowing for an independent check on the important value of the radius of the epicycle.

Continue reading “Almagest Book IV: Alexandrian Eclipse Triple – Solar/Lunar Positions & Epicyclic Anomaly”

Posted on

Almagest Book IV: Babylonian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon

In the last post, we were able to determine the radius of the epicycle when the radius of the deferent is $60^p$. It took a lot of switching between demi-degrees contexts, but in the wake of all that math, we’re left with a mess of lines and arcs that we’ve already determined. So in this post, we’ll use that starting point to go just a little further and determine the position of the mean moon, specifically for the second eclipse. To do so, we’ll need to add a bit more to the configuration we ended with last time:

Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Equation of Anomaly & The Mean Moon”

Posted on

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 epicycle5

To begin, let’s sketch out the epicyclic lunar model6 with the three eclipses drawn on it.

Continue reading “Almagest Book IV: Babylonian Eclipse Triple Geometry – Radius of the Epicycle”