In our last post, we showed how it is possible to determine the equation of anomaly by knowing the motion around the epicycle and the double elongation. This, combined with the position of the mean moon1 gives the true position of the moon. As usual, Ptolemy is going to give us a new table to make this relatively easy to look up. But before doing so, Ptolemy wants to explain what this table is going to look like. Continue reading “Almagest Book V: Constructing the Lunar Anomaly Table”
Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions
Now that we’ve revised our lunar model to include the position of the “mean apogee” from which we’ll measure motion around the epicycle, we need to discuss how we can use this to determine the true position of the moon.
As a general statement, we know how to do this: Take the position of the mean moon, determined by adding the motion since the beginning of the epoch, and add or subtract the equation of anomaly. The problem is that our revisions in this book mean the table for the lunar equation of anomaly we built in Book IV is no longer correct.
Instead, to determine the equation of anomaly, we’ll start with the motion around the epicycle2 and need to factor in the double elongation of the moon from the sun.
To see how to do so, let’s get started on a new diagram:
Almagest Book V: Second Determination of Direction of Epicycle
In the last post, we followed along as Ptolemy determined that the position of “apogee” used for calculating the motion around the epicycle is not the continuation of the line from the center of the ecliptic or center of the eccentre through the center of the epicycle. Rather, motion should be measured from the “mean apogee” which is defined from a third point opposite the center of the ecliptic from the center of the eccentre.
Ptolemy doesn’t give a rigorous proof for this and instead relies on proof by example. So in that last post, we went through one example, but in this post, we’ll do a second one
in order to show that we get the same result at the opposite sides of the eccentre and epicycle.
Continue reading “Almagest Book V: Second Determination of Direction of Epicycle”
Almagest Book V: The Direction of the Moon’s Epicycle
When we built the first lunar model, it was done using observations only at opposition, which is to say, during eclipses which only happen during the full phase. In the last few chapters, we looked at quadrature, which is to say, during first and third quarter moon and derived a second anomaly. But what happens if we consider the moon when it’s somewhere between those phases?
Ptolemy gives the answer:
[W]e find that the moon has a peculiar characteristic associated with the direction in which the epicycle points.
So what does that mean? Continue reading “Almagest Book V: The Direction of the Moon’s Epicycle”
Almagest Book V: Second Anomaly Eccentricity
So far, we’ve stated that the effect of the second anomaly is to magnify the first anomaly. In the last chapter, we worked out how much larger. Since this second model works by bringing the moon physically closer and further by offsetting the center of the lunar orbit with an eccentric and having that eccentre orbit the Earth, we can determine how far that center must be. In other words, the eccentricity of this second anomaly. Continue reading “Almagest Book V: Second Anomaly Eccentricity”
Almagest Book V: Size of the Second Anomaly
So far, what we know about Ptolemy’s second anomaly is that it doesn’t have an effect at conjunction or opposition. Its at its maximum at quadrature, which is to say, a $\frac{1}{4}$ and $\frac{3}{4}$ of the way through each synodic month3. Its effect is to re-enforce whatever anomaly was present from the first anomaly. Ptolemy laid out a conceptual model in Chapter 2, but to determine the parameters of the model, we’ll need to first explore how much this second anomaly impacts things.
Continue reading “Almagest Book V: Size of the Second Anomaly”
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”
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”
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”
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”