Continuing in the theme of checking as few as possible syzygies for eclipses, Ptolemy now turns his attention towards
the problem of intervals at which, in general, it is possible for ecliptic syzygies to occur, so that, once we have determined a single example of of an ecliptic syzygy, we need not apply our examination to the [ecliptic] limits to every succeeding syzygy in turn, but only to those which are separated [from the first] by an interval of months at which it is possible for an eclipse to recur.
We’ll begin a series of several posts where we look at both lunar and solar eclipses over various intervals to determine whether or not it’s possible for them to occur. To begin, Ptolemy considers whether or not it’s possible for two lunar eclipses separated by six months to occur wherein we’re defining a “month” as a full cycle of phases; a lunation. In that amount of time,
the moons mean motion [in argument of] latitude… comes to $184;01,25º$
UPDATE 1/5/2026: While reviewing this chapter, I realized that, in the section covered in this post, Ptolemy has elected to use the argument of latitude. This implies that he is considering the motion along the moon’s inclined circle.
However, having reviewed how the eclipse limits were derived1, it looks like I originally believed that the limits were along the inclined circle, which would fit with what Ptolemy did in this section. However, I realized there was an error in the way I drew the diagram and, instead, Ptolemy was clearly determining the eclipse window in ecliptic longitude. As such, Ptolemy using the argument of latitude here makes no sense to me as it means he is comparing values along different circles. Thus, I believe Ptolemy made an error in his analysis here. Instead, he should have used the motion in ecliptic longitude for a period of six lunar synodic months ($174;38,18º$) which does not change the analysis of the remainder of this post as, even using the correct value, the argument still works. However, this will create confusion in later sections of this chapter, which I will cover in later posts.
None of the authors2 that have explored this section seem to comment on this and, indeed, Neugebauer was part of the source of my original confusion.
As to not deviate from Ptolemy, I continue with the incorrect value for the remainder of this post.
This is readily seen from the Table of Oppositions3 for a six month interval. So what of it? Let’s take a step back to the conclusions of our last series of posts regarding the eclipse limits.
For the moon, we determined it could be up to $15;12º$ to either side of the node and still have some form of eclipse occur giving a total window around each node of $30;24º$.
For the sun we found the window to be $11;22º$ to the south and $20;41º$ to the north.
Let’s sketch that out. First for the lunar eclipses:
Here, I’ve sketched the ranges we calculated for the lunar eclipse limits, showing a $30;24º$ window around each node. This means that there must be $149;36º$ between each window where lunar eclipses are not possible.
Now, let’s say an eclipse happens when the moon is at point $A$. Then we wait six months. Is the moon back in the opposite eclipse window?
As we stated above, its motion along its circle would be $184,01;25º$ which would put it just past $C$, but still before point $D$, indicating it’s possible that an eclipse should occur. However, this would not be the case if the first eclipse was at point $B$ as the second syzygy would be past $D$ and thus too far from the node. Thus, it is sometimes possible for another eclipse to occur, but not a guarantee.
Ptolemy states some rules for this a bit more academically, stating that eclipses are possible when
the arcs between ecliptic limits [at opposite nodes] … comprise less than the above amount ($184;01,25º$) if they are less than a semi-circle, and more than the above amount if they are greater than a semi-circle.
That’s a mouthful, but break down he’s saying. If $arc \; BC$ were greater than $184;01,25º$, then it would be impossible for the syzygy six months later to occur within the window because the next syzygy would occur before the window. Similarly, if $arc \; AD$ weren’t less than $184;01,25º$, then the syzygy six months later would occur after the window.
For the lunar eclipses, since the distance from the nodes that an eclipse can occur is the same to both the north and south, the same is true if we consider the first eclipse happening in the window from $C$ to $D$.
We can repeat this for solar eclipses:
Here, the same thing applies: $arc \; BC$ and $arc \; DA$ are both less than $184;01,25$ meaning we can be sure that the position will have had the opportunity to have reached the next eclipse window. In addition, $arc \; AD$ and $arc \; CB$ are both greater than $184;01,25$ which means there’s the opportunity not to always overshoot the window.
Thus, if either a solar or lunar eclipse occurs, it is possible than another of the same type happening six months later.
However, Ptolemy also wants to consider if it’s possible to happen $5$ or $7$ months. After all, $184;01,25º$ is pretty close to $180º$ and we’ve got well more wiggle room in the window than just that leftover $4;01,25º$.
To answer that will be a bit more involved than what we’ve done so far in this post as here, we were able to get away with a few simplifying assumptions 4. As such, I’ll be breaking up this discussion into a series of posts. In the next one, we’ll explore whether or not it’s possible to have two lunar eclipses happen five months apart.
- See my update at the bottom of that post.
- Toomer, Pedersen, or Neugebauer.
- You may be wondering why we’re using this table instead of the Lunar Mean Motion Table. The reason is that the Lunar Mean Motion table was calculated for months of 30 days each whereas the Table of Oppositions uses synodic months which are $29;31,50$ days long (i.e., $29$ days, $12$ hours, $44$ minutes).
- Specifically that the moon was near apogee and thus the eclipse windows were as wide as absolutely possible