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 $184;01,25º$
This is readily seen from the Table of Oppositions1 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 2. 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.
- You may be wondering why we’re using this table instead of the Lunar Mean Motion Table. The reason is that when we found the distances eclipses could occur from the nodes, we found the distance along the lunar circle. The mean motion from the Lunar Mean Motion Table is for motion along the ecliptic. The lunar latitude is a function of the distance around the lunar circle and this, is the same as the mean motion on the moon’s own circle.
- Specifically that the moon was near apogee and thus the eclipse windows were as wide as absolutely possible