Now that we’ve covered the positions the sun needs to be in to avoid its anomaly influencing things, and the positions to avoid for the moon, so its anomaly doesn’t influence things, we’ll look into some positions which would make it the most obvious if the above were. Ptolemy states this saying,
we should select intervals [the ends of which are situated] so as to best indicate [whether the interval is or is not a period of anomaly] by displaying the discrepancy [between two intervals] when they do not contain an integer number of returns in anomaly.
So which are those?
According to Ptolemy,
Such is the case when the intervals begin from speeds which are not merely different, but greatly different either in size or in effect. By ‘in sidze’ I mean when in one interval [the moon] starts from its least speed and does not end at its greatest speed, while in the other it starts from its greatest speed and does not end at its least speed.
What we’re getting at here is that the first eclipse in one pair is at perigee, while the first eclipse from the second pair is at apogee (or vice versa). Let’s start with a picture:
Here, we have the first eclipse starting from apogee at $M_1$. Then the second eclipse in that same pair occurs at $M_2$ which is 90º clockwise from apogee.
Immediately we can tell from this top down view that this doesn’t represent a period of anomaly because $M_2$ isn’t at the same point on the epicycle as $M_1$. But we’re not trying for actual periods of anomaly. Instead, we’re trying to show the most optimal places for the first eclipse of each pair to start from that make the difference in their changes in ecliptic longitude largest.
Now let’s consider the second pair of eclipses. The first eclipse in this pair happens at $M_3$ and the second eclipse in that pair happens at $M_4$. Again, this clearly isn’t a period of eclipse since they aren’t at the same place on the ecliptic.
And looking at this, we can pretty well agree that we wouldn’t mistake it for one since $\angle{M_2 E M_1} \neq \angle{M_4 E M_3}$ and the requirement from our last post was that they be equal to even be considered as a period of the anomaly. But what we’re really looking at here is at what starting points would make the difference between $\angle{M_2 E M_1}$ and $\angle{M_4 E M_3}$ the greatest.
What I’ve illustrated here is nearly the worst case scenario: starting along the line of mean motion and ending at quadrature from it1. In that case, we would observe the difference between these two changes in ecliptic longitude as twice the equation of anomaly. However, if the first eclipse in one pair starts from apogee (or perigee) and the first eclipse in the second pair starts from another, then the difference in the ecliptic longitude between the two pairs will always be twice the equation of anomaly of the second eclipse. As Ptolemy puts it, these are “greatly different … in size”.
However, there’s also the possibility of the difference being greatest “in effect.” This happens when
[the moon] starts from mean speed in both positions, not, however from the same mean speed, but from the mean speed during the period of increasing speed at one interval, and from that during the period of decreasing speed at the other. Here too, if there is not a return in anomaly, there will be a great difference in the increment in longitude [over the two intervals].
Again, Ptolemy gives us a convoluted sentence, so let’s iron things out by illustrating:
As I’ve said a few times before, we’re not sure yet if the moon is going around the epicycle clockwise or counter-clockwise. If it’s clockwise, then the way I’ve drawn it has the first eclipse of the second pair, $M_3$ at the position of mean speed on the interval when it’s slowing (since greatest speed would have been at apogee), and and the first eclipse of the first pair, $M_1$ at mean speed on the interval of increasing speed2. If it’s counter clockwise, then flip these.
Regardless, this produces an even more pronounced effect than did the previous example. Again, I’ve drawn the most pronounced version of this effect here, where the second eclipse in each pair swaps with the first from the first pair. In this case we’ll have that the difference between $\angle{M_2 E M_1}$ and $\angle{M_4 E M_3}$ is four times the equation of anomaly.
This entire discussion of determining the period of anomaly is based on work done by astronomers prior to Ptolemy. Specifically, he cites Hipparchus stating,
Hipparchus too used his customary extreme care in the selection of the intervals adduced for his investigation of this question: he used [two intervals], in one of which the moon started from its greatest speed and did not end at its least speed, and in the other of which it started from its least speed and did not end at its greatest speed.
However, Ptolemy also notes that Hipparchus’ selection of eclipses was unable to eliminate the influence of the sun because
the sun fell short of an integer number of revolutions by about $\frac{1}{4}$ of a sign and this sign was different, and produced a different equation of anomaly, in each of the two intervals.
Ptolemy doesn’t give any information about what eclipses Hipparchus used or elaborate on any of his calculations, but this is explored in this article G. J. Toomer kindly shared with me. Regardless of the need for a correction, Ptolemy states he pointed this out
not to disparage the preceding method of determining the periodic returns, but to show that, while it can achieve its goal if applied with due care and the appropriate kind of calculations, if any of the conditions we set out above are omitted from consideration, even the least of them, it can fail utterly in its intended effect; and that, if one does use the proper criteria in making one’s selection of observation material, it is difficult to find corresponding [pairs of eclipse] observations which fulfil all the required conditions.
Overall, Ptolemy concludes that this selection did not impact his determination of the synodic month, but did result in
an error in the periods of anomaly and latitude, so considerable as to become quite apparent to us from the procedures we devised to check these values in simpler and more practical ways.
Ptolemy promises to return to all that in the future, but first, he’ll explore the
mean motions [of the moon] in longitude, anomaly, and latitude, in accordance with the above periods of their returns, and [also the mean motions] calculated on the basis of the corrections which we shall derive later.
- I drew it this way because that’s how Neugebauer had drawn it. However, I believe this is incorrect as the greatest equation of anomaly will occur when the moon is at mean speed which is quite close to this, but different as I have illustrated in the next example.
- If you’re confused about why the mean speed is at these points and not at a 90º angle from the line of mean motion, that was discussed in this post when we covered it for the epicyclic model for the sun.