Tag: moon

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Almagest Book V: Model for the Second Anomaly

In the last chapter, we introduced an instrument capable of determining the ecliptic latitude and longitude of an object so long as the position of the true sun or fixed star is known. Using this on the moon, Ptolemy found

that the distance of the moon from the sun was sometimes in agreement with that calculated from the above [Book IV] hypothesis, and sometimes in disagreement, the discrepancy being at some times small and at other times great.

How so? Ptolemy provides details. Continue reading “Almagest Book V: Model for the Second Anomaly”

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Almagest Book IV: Hipparchus’ Two Values of Lunar Anomaly Second Triad

In the last post, we followed Ptolemy as he reviewed three eclipses Hipparchus used to determine the parameters for his model showing they were different from Ptolemy’s. In this post, we’ll repeat the procedure for the second set of three eclipses, again showing that Hipparchus’ calculations did not match those of Ptolemy.

Continue reading “Almagest Book IV: Hipparchus’ Two Values of Lunar Anomaly Second Triad”

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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)”

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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”

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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”

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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”

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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”

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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”

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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:

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