Almagest Book IV: The Lunar Anomaly and Eclipses

In the last post, we covered how the sun’s anomaly impacts things, but

we must pay no less attention to the moon’s [varying] speed. For if this is not taken into account, it will be possible for the moon, in many situations, to cover equal arcs in longitude in equal times which do not at all represent a return in lunar anomaly as well.

I’ll preface this section by saying this is, to date, by far the hardest section I’ve grappled with. I believe a large part of the difficulty came from the fact that Ptolemy is exceptionally unclear about what his goal is with this section. My initial belief was that it was to find the full period in which a the position of the sun and moon would “reset” as discussed in the last post. However, that’s something we’re going to have to work up to.

For now, we’re going to concentrate on just one of the various types of months. Namely, the “return in lunar anomaly” which is another way of saying the anomalistic month.

Before going any further, let’s define what a return in lunar anomaly really looks like. Although Ptolemy hasn’t yet said anything about what model he’ll use for the moon, I’ll go ahead and spoil it here: It’s the epicycle model. So let’s start getting used to things being drawn that way. First, let’s consider the moon at some position $M_1$.

Earth is at the center at $E$. The center of the epicycle is $C_1$, and $A_1$ is the apogee. At this time, the moon makes angle $A_1 E M_1$ with respect to the mean motion, $\overline{EA_1}$. This is the equation of anomaly.

Now we’ll let time progress. The moon goes around its epicycle¹ and the whole epicycle rotates counter-clockwise on the deferent². We’ll stop the clock again when the moon returns to the same position on the epicycle with respect to the apogee because this would be one full cycle of the anomaly or the anomalistic period which is what we’re after right now.

In the previous book where we dealt with the sun, that would mean that this configuration would repeat exactly, because the anomalistic period was the same as the sidereal period: One year. But for the moon, the anomalistic period and the sidereal period aren’t equal. So when it returns to the same anomalistic position, the position along the deferent will be different like so:

Here, I’ve horribly exaggerated things so we have some room to fit in our markings³. The moon is now at $M_2$, but it still forms the same angle, $\alpha$ with the line of mean motion, thus completing a return in anomaly.

But in addition, we can clearly see that the moon is in the same position with respect to the apogee such that $\angle{A_1 E M_1} = \angle{A_2 E M_2}$⁴

Now let’s look at how the ecliptic latitude of the mean motion⁵ changed over this period. It has changed by $\angle{A_2 E A_1}$. But if you add on $\angle{A_1 E M_1}$ and $\angle{A_2 E M_2}$ to $\overline{A_1 E}$ and $\overline{A_2 E}$ respectively, this immediately shows us that this angle, $\angle{A_2 E A_1} = \angle{M_2 E M_1}$.

What this means, is that if we can a period of time such that moon’s true change in position, $\angle{M_2 E M_1}$, is equal to the mean motion $\angle{A_2 E A_1}$, that is the period of the eclipse⁶.

There’s only one problem: We don’t know the mean motion of the moon yet.

So Ptolemy, following the path of the more ancient astronomers, employs a nifty little trick. Let’s consider another return in anomaly, this time somewhere else along the deferent and with a different anomaly:

Here’s that sketched out, although I almost didn’t. Because everything I said above still applies. Most notably $\angle{A_4 E A_3} = \angle{M_4 E M_3}$. But since the mean motion is consistent, it’s also true that $\angle{A_2 E A_1} = \angle{A_4 E A_3}$. Thus, we can do some substitution to arrive at the conclusion that $\angle{M_2 E M_1} = \angle{M_4 E M_3}$.

What’s important to see here is we’ve completely cut out the mean motion that we didn’t know. This essentially says that if we can find two periods that result in the same change in ecliptic longitude over the same time period, then we may well have found the period of anomaly.

However, we can’t just do this whenever because the instrumentation of the time was not precise enough to do so. Instead, as we noted in the first post on Book IV, we can only use eclipses because during lunar eclipses, the moon will be directly opposite in both ecliptic longitude and latitude. And since we worked out such a nice model of the sun in Book III, we know where the sun is very precisely.

This means that we aren’t simply going to have one return in anomaly. The moon will have gone about its course numerous times before the next eclipse. Thus, by using pairs of eclipses instead, we’ll be getting integer multiples of the period of anomaly.

However, there’s some cases in which we may follow the rules above but accidentally get a false positive. In other words, we may have to pairs of eclipses with the eclipses in each pair separated by the same amount of time and covering the same distance in ecliptic longitude, but are not actually returning to the same position along the epicycle with respect to the line of mean motion. So similar to the last post explored positions of the sun that are required to avoid its anomaly throwing things off, Ptolemy gives three cases for the moon which much be avoided.

As with the last post, Ptolemy didn’t bother explaining these at all. So I’ll walk through them more slowly. Ultimately, what these all boil down to is the fact that, assuming the epicycle model were correct, we wouldn’t be able to tell the difference between the moon being on the near side of the epicycle or the far side; there’s two positions in each trip around the epicycle in which they would have the same apparent latitude if the line of mean motion were the same.

So with that in mind, Ptolemy states we must be careful:

[1] if in both intervals, the moon starts from the same speed (either both increasing or both decreasing), but does not return to that speed

Here’s a picture to represent this:

Trying to figure out how to visualize things was one of the hardest parts about this post, so my apologies if I’m being overly verbose while I unpack what’s going on here.

First off, to help compare things I’ve rotated the second pair of eclipses so their lines of mean motion overlap. That may feel a bit strange, but remember, we’re not really interested in the true ecliptic longitudes here; just the differences between the first and second eclipse in each pair. So rotation about the center of the deferent makes no difference here but makes things cleaner in my opinion.

So what we have is our first eclipse when the moon is at $M_1$. Sometime later, there is a second eclipse when the Moon is at $M_2$. From this top down view, we can clearly see that $\alpha_1 = \alpha_2$, but from the Earth, we can’t see that. All we can see is $\angle{M_2 E M_1}$. So we wait⁷ for a second pair of eclipses separated by the same amount of time between them as between $M_1$ and $M_2$. We observe them happening at $M_3$ and $M_4$.

But again, we can’t see exactly where these points are on the epicycles. All we can see is $\angle{M_4 E M_3}$ which we observe to be the same as $\angle{M_2 E M_1}$. Thus, by our rules above, we could consider this a possibility for being an integer multiple of the period of anomaly.

However, from this top down view, we can clearly see that while $M_1$ and $M_2$ started off from the same position on the epicycle, which is to say $\alpha_1 = \alpha_3$, the second eclipse in each pair is not since $\alpha_2 \neq \alpha_4$.

We’ll talk more about how to avoid this later. Right now we’re just explaining how you might get fooled.

The second rule Ptolemy gives is actually a special case of the third, so I’ll skip it for now and come back to it. So Ptolemy’s third thing to watch out for is

[3] if the distance of [the position of] its speed at the beginning of one interval is the same distance from the [position of] greatest or least speed as [the position of] its speed at the end of the other interval, [but] on the other side

Here, Ptolemy starts giving a hint that he does indeed have his model for the moon in mind as the phrase “the other side” doesn’t make much sense for the eccentric model. It makes clear sense in the epicyclic model in which it’s on “the other side” of the line of mean motion⁸. So let’s draw this out:

Again we have a pair of eclipses which we could observe to take place with the same time difference between the eclipses in each pair, and again, $\angle{M_2 E M_1} = \angle{M_4 E M_3}$. But from this top down view, we can see that $\alpha_1 \neq \alpha_2$ and $\alpha_3 \neq \alpha_4$. This means that they did not actually return to the same position in either eclipse pair to be a full trip around the epicycle. Thus, it wouldn’t represent an integer number of returns in anomaly.

Lastly, we must be be wary

[2] if in one interval it starts from its greatest speed and ends at its least speed, while in the other interval it starts from its least speed and ends at its greatest speed

Here’s a diagram for that one.

At present, we don’t know when “greatest speed” occurs because we haven’t determined whether the moon will travel around the epicycle clockwise our counter-clockwise. As noted in one of the footnotes previously, we know it will travel counter-clockwise on the deferrent, so if the moon travels counter clockwise on the epicycle, greatest speed will occur at apogee because the motion vectors will be aligned. If clockwise, then greatest speed is at perigee.

Ultimately, as noted above, this is just a special case of the previous case when $\alpha_1= 180º$. From this top down view, we can immediately see that the moon is not returning to the same same position for the eclipses in the first pair, $M_1$ and $M_2$. Nor for the second. As such, this is another instance in which we can be fooled.

So now we’ve identified three situations in which we can potentially be fooled. Pairs of eclipses taking place over the same ecliptic longitude and with the same amount of time between each pair, but not truly returning to the same position on the epicycle. However, this was only apparent when we had the nice top-down views I’ve drawn here. In reality, we can’t observe this. Which leads us to ask the question: How could we tell if it’s a false positive?

This post is already getting quite long so I’ll save it for next time when we’ll explore the most opportune positions for the moon to be in.

Postscript: This section was exceptionally difficult to make it through. In large part because Ptolemy offered no diagrams. Neugebauer’s History of Ancient Mathematical Astronomy had them, but trying to decipher them actually led to me being more confused.

Specifically, Neugebauer includes the phrase “…two pairs of eclipses $E_1 E’_1$ and $E_2 E’_2$…”. This phrase was highly misleading since it implied $E_1$ and $E’_1$ were the first two eclipses (what we called $M_1$ and $M_2$ above). However, that was not the case. Working under this incorrect assumption is a large part of what caused me to fail to understand Neugebauer’s diagrams.

Ultimately, what helped was finding this paper which, while it covered the topic in a fundamentally different way⁹, it used a much easier to follow number system (1-2-3-4) which helped me reevaluate what Neugebauer was trying to say. The rest of the paper is interesting as well; it’s a translation and review of a 12th century Islamic text¹⁰, which actually criticizes Ptolemy’s understanding.

I’m hoping to spend some time between Ptolemy and Copernicus getting at least somewhat familiar with the Islamic golden age of astronomy, so this will certainly be a paper I return to!

The 4 month gap it took me to break down this section put a huge dent in my overall progress, taking the overall rate down sufficiently it added 1.5 years to the overall completion of this book as compared to my last post. Hopefully we’re over this hump!

We haven’t decided whether it goes clockwise or counter-clockwise at this point and it doesn’t actually matter for this topic.
We do know this is counter-clockwise because it, like the sun, is “rearwards” with respect to the stars.
The difference in the sidereal and anomalistic periods is only about 6 hours, so if drawn to scale, this would be quite close to overlapping.
These angles are the equation of anomaly for each eclipse.
Which is tracked by the line through the center of the epicycle
Possibly. We’ll be exploring exceptions to this later in this post.
Or more accurately, look through historical records.
Also known as the apsidal line.
Specifically, its methodology was entirely mathematical as opposed to the more qualitative method I’ve taken.
Interestingly, the paper was written in the Al-Andalus region of modern day Spain, which for much of the medieval period, was under Islamic rule.