Having thoroughly discussed what anomalies Ptolemy wants his model to account for as well as what hypotheses1 he intends to use for each, Ptolemy is ready to start laying out the basic models. Ultimately, there will be two models. One for the four planets other than Mercury, and a special one for Mercury.
In this post, we’ll explore the first of these models.
For the planets other than Mercury
imagine the eccentre $ABG$ about centre $D$, with $ADG$ as the diameter through $D$ and the centre of the ecliptic; on this let $E$ be taken as the centre of the ecliptic, i.e. the viewpoint of the observer, making $A$ the apogee and $G$ the perigee.
I’ll pause there to draw things out:
Next,
Let $DE$ be bisected at $Z$, and with centre $Z$ and radius $DA$, draw a circle $H \Theta K$, which must, clearly be equal to $ABG$.
Here’s that step:
Here, we’ve placed a circle of the same size as the previous one, centered on $Z$ which is half way between $D$ and $E$.
Then, Ptolemy invokes a point, $\Theta$, on this circle, but doesn’t tell us where to put it. I’ve followed Toomer’s lead here2.
Now for the final step:
Then, on centre $\Theta$, draw the epicycle $LM$ and join $L \Theta MD$.
And here’s where things start getting weird.
We’ve now added the epicycle. But it will rotate about $D$, which is the center of circle $ABG$, but the epicycle will instead be attached to circle $H \Theta K$.
We’ll get to what’s going on there in a moment. Before explaining, Ptolemy instead tells us about simplifying assumptions regarding the inclinations:
[A]lthough we assume that the plane of the eccentric circles is inclined to the plane of the ecliptic, and also that the plane of the epicycle is inclined to the plane of the eccentres, to account for the latitudinal motion of the planets, in accordance with what we shall demonstrate concerning that topic, nevertheless, for the motions in longitude, for the sake of convenience, let us imagine that all [those planes] lie in a single [plane], that of the ecliptic, since there will be no noticeable longitudinal difference, not at least when the inclinations are as small as those which will be brought to light for each of the planets.
In other words, Ptolemy assumes that the epicycle, $LM$ is inclined to the circle on which it rides, $H \Theta K$, and that this circle too is incline with respect to the ecliptic (not shown).
Next, we say that the whole plane [of the eccentre] moves uniformly about centre $E$ towards the rear [i.e. in the order] of the signs, shifting the position of apogee and perigee $1º$ in $100$ years, and that diameter $L \Theta M$ of the epicycle rotates uniformly about centre $D$, again towards the rear [i.e. in the order] of the signs, with a speed corresponding to the planet’s return in longitude, and that it carries with it point $L$ and $M$ of the epicycle, and centre $\Theta$ of the epicycle (which always moves on the eccentre $H \Theta K$) and also carries with it the planet.
Let’s pause there and take note of a few important points. First off, it’s the circle, $ABG$ that is centered on the earth and is going to help us with the aforementioned precession of the equinoxes. It does so by rotating $\overline{AK}$ with that rearward motion at the rate of precession, essentially rotating the circles that really matter at that rate. But otherwise, we can ignore this circle and just understand that the line, $\overline{DZE}$ is rotating about point $D$ at that rate.
Next, we have circle $H \Theta K$, that carries the epicycle. This circle is centered on $Z$ (as stated above in the second step, but Ptolemy now tells us it rotates about $D$. This seems horribly inconsistent3 but for an excellent explanation, check out this video:
What’s happening here is that Ptolemy has moved the point of rotation off of the center of the circle. After all, if we had a physical disk, we could drill a hole in it anywhere we’d like, put an axle through it, and spin it about that point.
So point $Z$ is the true center of the circle, $H \Theta K$, but we’re going to rotate it around an off center point, $D$. This is the point that would later become known as the “equant”. However, this is not a term that Ptolemy actually used4.
[t]he planet, for its part, moves with uniform motion on the epicycle $LM$ and performs its return always with respect to that diameter [of the epicycle] which points towards centre $D$, with a speed corresponding to the mean period of the synodic anomaly, and [a sense of rotation] such that its motion at the apogee $L$ takes place towards the rear.
In short, the epicycle behaves like an epicycle and rotates counter clockwise in this case.
And that’s it for the basic planetary model. However, as we noted, Mercury is a special case so we’ll be looking at how Mercury’s model differs here in the next post.
- Epicyclic vs eccentric.
- I’m also following Toomer’s lead that $B$ appears to be at the intersection of these two circles although I don’t see any support for it in Ptolemy’s layout. But since it appears to be, I’ve waited until now to add it.
- Especially because Ptolemy calls $D$ the “center” when it’s really not.
- Ptolemy is generally credited with developing the concept of the equant. However, there is evidence to suggest that other forms of the equant may have been used by astronomers between the time of Hipparchus and Ptolemy which would suggest Ptolemy did not develop the concept, but instead integrated it in a novel way.