And here we are. Final chapter of Book VI. Here, we’ll finish out Ptolemy’s discussion of the inclinations of eclipses. In the previous post, we discussed a few things we’ll need to determine that1, but we really haven’t discussed is what the “inclination” means. So I’ll begin by doing that as Ptolemy doesn’t really do a clear job of doing so.
First, let’s consider a view of the observer’s horizon, looking down on it from above for a lunar eclipse happening with the moon’s shadow near the observer’s zenith:
Here, we have the four cardinal directions laid out, east at $E$, south at $S$, west at $W$ and north at $N$. Then, $arc \; AB$ is the ecliptic with the center of earth’s shadow at $Z$, which coincides with the observer’s zenith for purposes of this illustration2.
Then we have the moon at the various positions discussed in this post all of which are south of the ecliptic with the dashed line as its path. The first ($1$) at the first moment the shadow touches the moon, then $2$ when totality begins, then $4$ when totality ends, and $5$ when the eclipse ends. You’ve probably noticed I skipped point $3$ (mid-eclipse) and we’ll return to why in a bit.
Now, let’s draw in the great circle that goes through the center of the moon and the center of the earth’s shadow at the first of these positions:
Here, we can see this great circle intersects the horizon at two points which I’ve called $P_{1E}$ and $P_{1W}$ as this is for the first position and there’s one in the east and one in the west.
These points are the ones that Ptolemy is actually concerned with. Essentially, where on the horizon is the great circle between the sun/moon or earth’s shadow/moon pointing on the horizon? This is something that would presumably be calculated as an azimuth position.
In theory we can find these points using what we discussed in this post. Specifically, there we first created the circular horizon diagram which gives the angular distance along the horizon between a point of the ecliptic and the celestial equator. Since the celestial equator always intersects the horizon due east or west (i.e., at $E$ or $W$), and the point on the ecliptic would then be $A$ or $B$, this table is essentially giving us $arc \; EA$ or $arc \; BW$.
In that post we also calculated the angle between the great circle between the sun/moon or earth’s shadow/moon and the ecliptic. In other words, the arcs between the various $P$s and $E$ or $W$. Putting all that together with the azimuths of east ($90º$) and west ($270º$), we can effectively determine the azimuths of the various $P$s.
However, there’s a huge caveat here: This only works in the case when the center of the earth’s shadow or the sun is directly at the zenith. However, Ptolemy glosses over this entirely, treating the angle as if it could work in any case. Neugebauer notes that this glaring error was known in antiquity and points to Pappus’ commentary and also notes that Ptolemy attempted to address this in his later work, the Handy Tables. But as far the Almagest goes, Neugebauer excoriates Ptolemy stating
that the results lose all astronomical meaning and degenerate to some sort of empty game which associated non-existent “angles of inclination” with eclipses, simply in order to produce some numerical result when one gets tired of computing correctly, although there exists no theoretical difficulty in obtaining the exact result in all cases.
In other words, Ptolemy dropped the ball big on this one.
Regardless, this final chapter is essentially telling us, for solar and lunar eclipses, which points on the horizon are the “significant” ones3. Since each phase of the eclipse creates two, he will pick one and this will change depending on whether the it’s a solar or lunar eclipse and whether the moon is on, north or south of the horizon.
“As a preliminary” Ptolemy tells us we should calculate the specifics of the given solar or lunar eclipse. Once that’s done
we’ll determine… from the times [of the eclipse], those points on the ecliptic which are rising and setting at those moment
Ptolemy doesn’t go into how to determine which points of the ecliptic are on the horizon…
and, from [the horizon diagram], the situation [of the ecliptic] with respect to the horizon.
In other words, once we know which points of the ecliptic are on the horizon, we can use the circular diagram to determine the arc between the ecliptic and the east/west points. As described above, this, along with the information from the table we calculated will tell us where along the horizon these points will be.
From there, he goes on to explain which points are the important ones. However, his description is a bit of a garbled run on sentence in which he tries to fit his instructions for both a solar and lunar eclipse into one sentence. I’ll quote it in its entirety and then try to make sense of it afterwards.
[W]hen the center of the moon (the apparent center at solar eclipses and the true center at lunar eclipses) is exactly on the ecliptic, we get the inclination for a solar eclipse at the beginning of the eclipse, and the inclination for a lunar eclipse at the end of the partial phase and also at the end of emersion, from the situation on the horizon of the point of the ecliptic setting at the moment in question; we get the inclination for a solar eclipse at the end of the eclipse and the inclination for a lunar eclipse at the beginning of the eclipse and the beginning of emersion [i.e., end of totality], from the [horizon situation] of the rising-point of the ecliptic.
Here, Ptolemy is using the “inclination” to mean the position on the horizon. In short, he’s telling us what stages of the eclipse we’ll need to worry about for each type of eclipse staring with the case when the moon is directly on the ecliptic.
Let’s start with the solar eclipse and draw that out:
When the moon is on the ecliptic, the great circle between the moon and the sun with necessarily be coincident with the ecliptic itself which means the points on the horizon will be the rising/setting points of the ecliptic. As such, we won’t need to concern ourselves with Table VI.12 – The Table of Angles of Immersion of Eclipses, and will only need to determine point $A$ and $B$ using the horizon diagram. The value that comes from this is what Ptolemy is referring to with the “horizon situation”.
Here, Ptolemy says that for the beginning of the eclipse (i.e., point $1$) we take the point on the horizon from the point of the ecliptic setting (i.e., $B$) and for the end of the eclipse, we use the point rising (i.e., $A$). Again, for a solar eclipse, point $2$, $3$, and $4$ are all basically the same which is why we can skip $2$ and $4$, but why skip $3$? Well, Ptolemy doesn’t say, but he never gives a rule for what to do with $3$ in any of these cases. And in this one, it doesn’t even make sense since the two points that define the great circle in question are coincident.
Now for a lunar eclipse on the ecliptic. In this case, he tells us that for points $2$ and $5$ we should use the setting point, and for points $1$ and $4$ we should use the rising points. Sketching that out, we get the following:
Moving on:
When the moon’s center is not exactly on the ecliptic, we take from the table the angles corresponding to the relevant magnitude [of the eclipse] in digits, and apply those angles to the intersection of the horizon and ecliptic.
Now Ptolemy asks what happens if the moon isn’t on the ecliptic. In that case, we’ll need to refer to Table VI.12 to get the extra angle.
If the moon’s center is north of the ecliptic, we set off the angle to the north of the setting-point for eclipse-beginning in solar eclipses and for the end of the partial phase in lunar eclipses; we set it off to the north of the rising-point for the end of emersion in solar eclipses and [the beginning of] emersion in lunar eclipses; furthermore we set it off to the south of the rising-point for eclipse-beginning in lunar eclipses and to the south of the setting-point for eclipse-end in lunar eclipses.
Again, Ptolemy has rolled a bunch of instructions into one run-on sentence so I’ll pull them apart:
If the moon’s center is north of the ecliptic:
For solar eclipses, we again only need consider points $1$ and $5$ and in both cases, we use the points on the horizon north of the ecliptic rise/set point. Sketching this out:
For lunar eclipses the diagram gets a bit busy, so let me reorganize what Ptolemy wrote:
North of the setting point for…the end of the partial phase ($2$)
North of the rising point for…[beginning of] emersion ($4$)
South of the rising point for… eclipse beginning ($1$)
South of the setting point for…eclipse end ($5$)
And here’s a sketch:
Next up, when the moon is south of the ecliptic:
If the moon’s center is south of the ecliptic, we set the angle off to the south of the setting-point for eclipse-beginning in solar eclipses and for end of the partial phase in lunar eclipses; to the south of the rising-point for the eclipse end in solar eclipses and for the beginning of emersion in lunar eclipses; to the north of the rising-point for eclipse-beginning in lunar eclipses; and to the north of the setting-point for eclipse-end in lunar eclipses.
Again, separating out the solar and lunar.
For solar eclipses, we’ll use the points south of the ecliptic in both cases:
For the lunar eclipses I’ll again abbreviate the rules:
South of the setting point for…end of the partial phase ($2$)
South of the rising point for…beginning of emersion ($4$)
North of the rising point for…eclipse-beginning ($1$)
North of the setting point for…eclipse-end ($5$)
And again, a handy diagram:
The result of this procedure will give us the point on the horizon towards which (speaking roughly, as we said), are inclined those point of the luminaries comprising the significant [moments of the phases], namely the beginning and end of the eclipse and of total phase.
And that’s the end of Book VI. We are now entirely done with the lunar and solar models and their interactions.
Next, we’ll be moving on to the stars which Ptolemy breaks into two books: One for the stars in the northern hemisphere, and one for the southern.
- Specifically the angle between the great circle between the centers of the sun/moon or earth’s shadow/moon and the ecliptic as well as the distance of the points of the ecliptic along the horizon from east and west.
- Since this eclipse is happening so high in the sky, this can only ever happen between the tropics.
- No reasoning is given to why he picks the ones in the east vs the ones in the west, but then again, this is astrology.