Tag: geometry

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Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee

In the last post, we looked at how much the total equation of anomaly would change during syzygy due to the eccentre we added to the lunar model in this book, when the moon was at its greatest base equation of anomaly. As Ptolemy told us, it wasn’t much. However, there was a second effect that can also change the equation of anomaly, which was based on where we measure the movement around the epicycle from. Namely, the mean apogee instead of the true apogee. This has its maximum effect when the moon is near apogee or perigee so in this post, we’ll again quantify how much.

Let’s start off by building our diagram:

Continue reading “Almagest Book V: The Difference at Syzygies – Lunar Apogee and Perigee”

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Almagest Book V: The Difference at Syzygies – Maximum Lunar Anomaly

Syzygy is one of those words that has popped up very little in the Almagest so every time it does, I’m always thrown off a bit1. Especially when Ptolemy is going to spend an entire chapter discussing a topic that has scarcely even come up. But here we have Ptolemy spending the entirety of chapter $10$, to demonstrate that these modifications we’ve made to the lunar model have a negligible effect because he fears readers might think it does since

the centre of the epicycle does not always … stand exactly at the apogee at those times, but can be removed from the apogee by an arc [of the eccentre] of considerable size, because location precisely at the apogee occurs at the mean syzygies, whereas the determination of true conjunction and opposition requires taking the anomalies of both luminaries into account.

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Almagest Book V: Constructing the Lunar Anomaly Table

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 moon2 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”

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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 epicycle3 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:

Continue reading “Almagest Book V: Determining True Position of the Moon Geometrically From Periodic Motions”

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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.

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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”

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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: 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: 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”