Now that we’ve taken the time to understand the model for the four planets other than Mercury, let’s start on the model for Mercury as that’s the focus of the remainder of this book.
Let the eccentre producing the anomaly be $ABG$ about centre $D$, and let the diameter through $D$ and centre $E$ of the ecliptic be $\overline{ADEG}$, [passing] through the apogee at $A$.
Again, let’s build this step by step.
The first step is quite similar to in the model for the other planets. We begin with an eccentric, centered on $D$ with the observer at $E$. This naturally creates an apogee, $A$, and a perigee $G$.
On $\overline{AG}$ take $\overline{DZ}$ towards the apogee, $A$, equal to $\overline{DE}$.
As with the previous model of the planets, we’ll now create the point, $Z$ about which our next circle will rotate. I’ve also drawn in $\overline{DB}$ which we’ll use momentarily.
But while some things will be similar to our last model, not everything will be.
Then everything else remains the same, namely the whole plane, [revolving] about centre $E$, shifts the apogee towards the rear by the same amount as for the other planets, the epicycle is revolved uniformly about centre $D$ towards the rear, as [here] by the line $\overline{DB}$, and furthermore the planet moves on the epicycle in the same way as the others.
No need for a new drawing just yet as Ptolemy is just explaining how things are working with what we have. There will still be a movement for the precession of the equinoxes which is about $E$ (although there’s no specific circle that drives this in this image).
The circle centered on $D$ will be the primary driver of the epicycle’s center (not yet drawn in as it will lie on another circle) and will, again, be moving rearwards, which is to say, counter-clockwise.
Lastly, it’s snuck in there given we haven’t yet drawn the epicycle, but Ptolemy tells us it will move in the same way as the others which was also counter-clockwise as described in the last post.
And here’s where things diverge.
But in this case, the centre of the other eccentre, which is, again, equal in size to the first eccentre, and on which the epicycle centre is always located, is carried around point $Z$ in the opposite sense to the motion of the epicycle, namely in advance [i.e. in the reverse order] of the signs, but uniformly and with the same speed as the epicycle, as [here] by the line $ZH \Theta$.
I’ll pause to draw things out again.
Here, we’ve added the eccentre that will actually carry Mercury. This eccentre rotates about, $Z$ but has $H$ as its center.
This reminds me much of the lunar model in which the center of the moon’s deferent was off center, which would bring the moon physically closer and further at times of the orbit to amplify the anomaly.
One important note here is that Ptolemy notes that this eccentre rotates in the opposite direction as the previous one, but with the same speed.
He states that it’s the same speed “as the epicycle”, which is an exceptionally poor phrasing as it makes it sound like he is saying that it has the same period as the epicycle itself. Rather, he means the speed of the eccentre that drives the center of the epicycle, centered on $D$. This escaped me on a first reading but it will become important as we explore some of the symmetries in a few posts.
A consequence of this is that $\angle ADB$ is always equal and opposite to $\angle AZ \Theta$. Again, this is not clear from what Ptolemy states here, but will become necessary when we discuss those symmetries.
Thus, in one year, each of the lines $\overline{DB}$ and $\overline{ZH \Theta}$ performs one return with respect to a [given] point of the ecliptic, but, with respect to each other, obviously, two returns.
In short, these two circles each rotate with the same speed, but because they do so in opposite directions, the points $B$ and $\Theta$ will each pass by each other twice per rotation.
This is another indication on what Ptolemy meant in the last quote. If he had meant that this second eccentre truly rotated with the same speed as the epicycle itself, then this would not necessarily be true since the speed at which the speed of the epicycle is lower than the mean speed as can be seen by the mean motion tables.
And the [centre of the eccentre centered on $H$] will always be at a constant distance from point $Z$, and that distance too will be both equal to both $\overline{ED}$ and $\overline{DZ}$ (as [here] $\overline{ZH}$).
This is just a reminder that $\overline{ZH} = \overline{ZD} = \overline{ED}$.
Thus, the small circle described by its motion in advance, with centre $Z$ and radius $\overline{ZH}$, always has on its boundary the point $D$ (the centre of the first, fixed eccentre) too;
A pretty straightforward consequence of the fact that $\overline{ZH} = \overline{ZD}$.
and the moving eccentre, at any given moment, can be described with centre $H$ and radius $\overline{H \Theta}$ equal to $\overline{DA}$ (as here $\overline{\Theta K}$), the epicycle always having its centre on it (as here at point $K$).
The first part reminds us that the eccentre centered on $H$ is equal in size to the one centered on $D$, before finally telling us that the epicycle (which we’ve not yet included will be centered on $K$. So let’s draw that in.
This essentially completes the model. But if you’re feeling lost, Ptolemy then tells us we’ll get a better understanding as we go through things:
We shall get an even clearer grasp of these hypotheses from the demonstrations we shall make [in determining] the parameters for each planet individually. In those demonstrations will also frequently become clear, [at least] in outline, the motives which somehow led us to adopt these hypotheses.
But before moving on, let’s see if we can review and simplify.
The epicycle itself is centered on $K$ and rotates around $D$ with a constant angular speed. However, it doesn’t ride around $D$ on the circle centered on $D$ (if it did, it would be centered on $B$). Rather, $K$ is on a second eccentre which has center $H$ but does not actually rotate around that center. Rather it rotates around point $Z$ with the same speed that $K$ rotates around $D$, but in the opposite direction1.
This will have the effect of bringing $K$ physically closer to the observer at $E$ at certain times of the cycle, thereby amplifying the effect of the anomaly in the same way we saw for the moon.
Ptolemy finishes off by promising that we’ll come to understand why he build these models the way he did. But before proceeding, he points out an oddity of this model that will be elaborated upon later:
[O]ne must make the preliminary point that the longitudinal periods do not bring the planet back to the same position both with respect to a point on the ecliptic and [simultaneously] with respect to the apogee or perigee of the eccentre; this is due to the shift in position which we assign to the later. Hence, the mean motions in longitude which we tabulated above represent, not the returns [of the planets] defined with respect to the apogees of the eccentres, but the returns defined with respect to the solstical and equinoctial points, agreeing with the length of the year as we have determined it.
Toomer explains that this is Ptolemy’s way of saying that the mean motions previously tabulated are for tropical2 as opposed to sidereal3. This means that, if the planet started at the apogee, then one return later, it would not return to that apogee, but rather, the same point in the ecliptic which would have moved from the apogee due to precession of the equinoxes.
I’ll pause here because this effectively ends the section in this chapter on explaining the models themselves. We’re not done as Ptolemy wants to explore the models further before we try to calibrate them, showing that there are various symmetries, but that’s going to get rather long, so we’ll take a break here.
- I think at this point, the concept of rotating spheres has rather gone out the window. As these spheres overlap, there’s no way they could pass through one another. This was a serious topic of discussion in the medieval period with astronomers debating whether or not the “spheres” were solid objects. I fail to understand how it could have been taken seriously given that they overlap like this but I haven’t had yet had the opportunity to dive into that discussion.
- I.e., making a return to the same point in the ecliptic.
- I.e., making a return to the same point with respect to the background stars. The difference here is the precession of the equinoxes.