In the last chapter, we introduced an instrument capable of determining the ecliptic latitude and longitude of an object so long as the position of the true sun or fixed star is known. Using this on the moon, Ptolemy found
that the distance of the moon from the sun was sometimes in agreement with that calculated from the above [Book IV] hypothesis, and sometimes in disagreement, the discrepancy being at some times small and at other times great.
How so? Ptolemy provides details.
[W]hen we paid more attention to the circumstances of the anomaly in question, and examined it more carefully over a continuous period, we discovered that at conjunction and opposition the discrepancy is either imperceptible or small, the difference being of a size explicable by lunar parallax.
This is why this second anomaly wasn’t discovered previously: Because Hipparchus only considered the position of the moon during eclipses where it is either opposite or in front of the sun and happens to be near zero. Any discrepancies with the first model are sufficiently small that they can be explained away because of the observer’s position on Earth. The lunar position when it’s not at conjunction or opposition is harder to determine since it does require more sophisticated instrumentation. Ptolemy states that this discrepancy was discovered through his own observations but also from observations of Hipparchus, indicating, but not outright stating, that perhaps Hipparchus was aware of something inconsistent about his theory as well.
So how does this anomaly work?
At both quadratures, while the discrepancy is very small or nothing when the moon is at apogee or perigee of the epicycle, it reaches a maximum when the moon is near its mean speed and [thus] the equation of first anomaly is also a maximum. Furthermore, at either quadrature, when the first anomaly is subtractive, the moon’s observed position is at an even smaller longitude than that calculated by subtraction the equation of the first anomaly, but when the first anomaly is additive its true position is even greater [than that calculated by adding the equation of the first anomaly], and the size of this discrepancy is closely related to the size of the equation of the first anomaly.
This is another word salad but there’s two key points in here. In general, if the effect of this second anomaly were zero at conjunction and opposition, we’d expect that they would likely be at maximums $90º$ away from those positions which are known as quadrature. However this isn’t precisely what Ptolemy found. Rather, he found that the deviations are
1) Still effectively zero if the moon is at quadrature near apogee or perigee (i.e., has an equation of anomaly near $0$).
2) Are at their maximums if the moon is near quadrature at the greatest distance from apogee/perigee, (i.e., the largest equation of anomaly).
In the second case, this new anomaly magnifies the effect of the previous anomaly. When the equation of anomaly predicts the moon should lag the mean moon, it lags even further. When the equation of anomaly predicts the moon should precede the mean moon, it precedes it even more.
So how to explain this?
Following what we’ve seen in the past two books, Ptolemy has two models at his disposal: Either the eccentric or epicyclic. This time he doesn’t play around at which one he’s using and immediately states that he will
suppose the moon’s epicycle to be carried on an eccentric circle, being farthest from the earth at conjunction and opposition, and nearest to the earth at both quadratures.
The phrasing here is important: “the moon’s epicycle to be carried on an eccentric circle”. This indicates that the eccentric is external to the epicycle making it much like the eccentric model we originally looked at for the sun. Except in this case, instead of the sun being carried about on this eccentre, it will be the center of the deferent1.
To begin developing this model, Ptolemy begins by reminding us of the layout of the first model:
Imagine the circle (in the inclined plane of the moon), concentric with the ecliptic and moving in advance… to represent the [motion in] latitude, about the poles of the ecliptic with a speed equal to the increment of the motion in latitude over the motion in longitude.
This is describing the motion of the nodes which, as we saw in this post, rotate “in advance” which means clockwise.
Imagine, again, the moon traversing the so-called epicycle (moving in advance on its apogee arc) with a speed corresponding to the return of the first anomaly.
Here he describes the motion on the epicycle, reminding us that this too is clockwise.
Now, in this inclined plane, we suppose two motions take place, in opposite directions, both uniform with respect to the center of the ecliptic: one of these carries the centre of the epicycle towards the rear through the signs with the speed of motion in latitude
This is the basic rearwards motion of the mean moon, counter-clockwise which completes the sets of motions covered in our first model. To that he adds a second motion which
carries the centre and apogee of the eccentre, which we assume located in the same [inclined] plane, (the centre of the epicycle will at all times be located on this eccentre), in advance through the signs, by an amount corresponding to the difference between the motion in latitude and the double elongation (the elongation being the amount by which the moon’s mean motion in longitude exceeds the sun’s mean motion).
And here we finally get to the new portion. The new eccentric circle has a forwards motion, moving clockwise which we haven’t seen before. Another way to state this is that the center of the eccentre, instead of being fixed like in the solar model, will orbit the observer in its own small circle.
Next Ptolemy gives an example of how this would function:
In one day, the centre of the epicycle traverses about $13;14º$ in motion of latitude towards the rear through the signs…
This is the increment in latitude from the lunar mean motions table.
but appears to have traversed $13;11º$ in longitude on the ecliptic since the whole inclined circle [of the moon] traverses the difference of $0;3º$ in the opposite direction [i.e.] in advance.
This is again stating what I covered at the end of this post in which I covered how the mean moon actually moves $13;14º$ per day along the deferent, but because the deferent itself is rotating $~0;3º$ per day in the other direction, we only perceive $13;11º$ of motion.
The apogee of the eccentre, in turn, travels $11;9º$ in the opposite direction (again in advance).
This is the new part2. What he is saying here is that the apogee of the eccentre appears to move $11;9º$ per day clockwise. So where did Ptolemy come up with this number? He tries to explain:
this is the amount by which the double elongation, $24;23º$, exceeds the motion in latitude, $13;14º$.
Well, the math certainly checks out, but where did he come up with the $24;23º$?
To understand, we must recall the observation that this new anomaly essentially disappears at conjunction and opposition with the sun. This means that this new anomaly must have its full cycle as precisely half that of the synodic month3. Ptolemy’s period for the synodic month was $29;31,50,8,20$ days. So half of that is $14;45,55,4,10$ days for the center of this new eccentre to complete a full $360º$ around the observer. Thus, dividing we get $24;23º$ per day, again in a clockwise direction.
Alternatively, we can arrive at this by considering the daily change in the elongation, again from the mean motion table, which was $12;11,26º$ per day. Since this new eccentre must be moving twice as fast, we can again arrive at $24;23º$ per day.
Going back to the big picture, the deferent itself is rotating clockwise around the observer, so we subtract that out to get the apparent motion of $11;9º$ per day.
That’s a bit much to understand verbally, so Ptolemy creates a diagram.
In this image, we’ll temporarily ignore the epicycle and only concern ourselves with the mean moon. It travels around the smaller of the two solid circles here which is the deferent, now centered on point $Z$ instead of the observer since this introduces the eccentric. The larger circle will represent the ecliptic, still centered on the observer at $E$.
Let’s start by considering the mean moon to be at the same ecliptic longitude as the vernal equinox. In addition, this same point is the apogee for the eccentre. From there, let’s apply each of the motions we’ve discussed.
First I’ll mention is the rotation of the northernmost point. However, this motion isn’t just of the single point. It also rotates the nodes as we saw in this post and is really rotating the entire lunar assembly clockwise $0;3º$ per day. So this motion, if taken alone, would move the mean moon and the apogee of the eccentre to point $A$4.
However, the mean moon isn’t holding still. The deferent is rotating counter-clockwise about its own center, $Z$, and gives an apparent motion from the point of view of the observer of $13;14º$ per day which means it would end up that distance from point $A$ or $13;11º$ from the vernal equinox putting the apparent position of the moon at $B$ for ecliptic longitude5.
Lastly the new motion is the rotation of the center of the deferent around the observer, clockwise, which would move the apogee of the eccentre, which the moon has moved away from, from point $A$ to $D$.
So to review, this gives us quite a few arcs we can describe:
- $arc \; VA = 0;3º$6
- $arc \; AD = 11;9º$
- $arc \; VD = 11;12º$
- $arc \; BV = 13;11º$
- $arc \; BA = 13;14º$
- $arc \; BD = 24;23º$
Next, let’s consider briefly where these points will be at some important points. We’ve already considered them overlapping, so let’s figure out what happens a quarter of the way through the synodic month which is $7;22,57,32,5$ days. Since the mean moon, represented by $B$ is moving $13;11º$ per day away from the vernal equinox that’s a total of $97;20º$.
Meanwhile, the apogee of the eccentre, represented by point $D$, is moving $11;12º$ per day clockwise, so in that same amount of time, it would have moved $82;41º$ from the vernal equinox. Let’s sketch that out:
These two points are diametrically opposite one another from the point of view of the observer7. This will happen a quarter and three-quarters of the way through each synodic month.
This has the interesting effect that, since the mean moon, being on the deferent which is also the eccentre, is directly opposite the apogee of the eccentre. That means that the mean moon at point $H$ will then be at the perigee of the eccentre.
We can quickly determine the location of each half way through the synodic month simply by multiplying each by two. At this point in time, $B$ would have moved $194;40º$ counter clockwise from the vernal equinox while $D$ would have moved $165;22º$ clockwise from the vernal equinox. Again, within the rounding error, this means the two would again overlap.
But how does all this impact anything we did in the last book when we developed the first model? As Ptolemy puts it:
the eccentre itself will not produce any correction to the mean motion. For the uniform motion of the line $\overline{EB}$ is counted, not along $arc \; DH$ of the eccentre, but along $arc \; DB$ of the ecliptic, since it rotates, not about the center of the eccentre, $Z$, but about $E$.
In other words, we didn’t build the first model with a spinning deferent/eccentre in mind. We built it with the mind of a static one centered on the observer. In this case, that’s the same thing a the ecliptic. So as long as we keep things in that context, we don’t actually need to make any adjustments to the model even though this would imply that the moon is moving much more quickly around the deferent.
Rather where this will matter is when we consider the moon’s position on the epicycle since the apogee and perigee of the eccentre will amplify the effect at quadratures. To understand why, let’s take that last picture and reintroduce the epicycles. One on the ecliptic at $B$, which would represent the original model, and a second on the deferent/eccentre at $H$. I’ll color code things to make this easier to see:
Here, we can quite easily see that $\angle{YEH} > \angle{XEB}$ even though they’re epicycles of the same size centered at the same angular position. The fact that the eccentre being off center and pulling this closer to the observer at $E$ simply increases the angle. Since perigee is when the center of the epicycle would be pulled closest, this would mean that the effect is greatest at these times which happens $\frac{1}{4}$ and $\frac{3}{4}$ of the way through each synodic month, or roughly at first and third quarter moon.
However, if I drew this same picture when $B$ and $H$ overlapped which, as we saw happens twice through each synodic period, then the two epicycles would similarly overlap and thus there would be no effect since the angle produced would be the same.
- Back when discussing the different models, we called the circle that was off-center like this an “eccentre”. Since the deferent is now an eccentre, I will be using these words rather interchangeably as they describe the same thing.
- I’m following the order Ptolemy went in here which I don’t necessarily think is the best. He starts by talking through things and then draws us a picture. I think having the picture first would have helped. So if this next bit doesn’t make sense, feel free to skip until you see the diagram.
- If you need a refresher on the types of lunar months see this post.
- I’ve exaggerated the distance between the vernal equinox and $A$ here because if I truly drew $0;3º$ it would have been undistinguishable at this scale.
- Its true position would be $H$, but that’s along the same line of sight for the observer so it’s not a huge distinction here.
- I’m using V here for the vernal equinox because the LaTeX plugin I’m using doesn’t include the package for astronomical symbols.
- There’s a bit of rounding error here which is why they don’t quite add up to $180º$.