So far in Book III, we’ve been looking at the mean sun1. However, the sun’s motion along the ecliptic is not even. In other words, its speed along the ecliptic is not constant. This poses a problem because, to Ptolemy:
the rearward displacements of the planets2 with respect to the heavens are, in every case, just like the motion of the universe in advance, by nature uniform and circular. That is to say, if we imagine the bodies or their circles being carried around by straight lines, in absolutely every case the straight line in question describes equal angles at the center of its revolution in equal times.
Here, Ptolemy states that he accepts uniform circular motion as true. So the question becomes how can something moving at a constant speed appear to have not constant speed?
Ptolemy’s answer:
The apparent irregularity (anomaly) in their motions is the result of the position and order of those circles in the sphere of each by means of which they carry out their movements.
Ptolemy explores two models which can both explain this inconsistent motion.
The first is the eccentric hypothesis. Consider the path that the object is travelling on (or being carried on) as ABGD which has its center as E. If we take the observer, and place them off center at Z, this means that the closest point to the observer (perigee) will be D, and A will be the furthest point (apogee).
Now we’ll choose two points, B and G, such that $arc \; AB = arc \; DG$, and draw in lines connecting them to both the center and the observer. We can immediately see that, while $arc \; AB$ and $arc \; DG$ will be traversed in equal time due to uniform circular motion, the angles they traverse in that time are unequal and $\angle \; AZB < \angle \; GZD$. Thus, their apparent motion must also be different.
The next model is the one that is most commonly associated with the geocentric theory, which is the use of epicycles.
In this model, circle ZHΘK is the epicycle on which the sun would ride around with uniform circular motion. The center of this circle, A, would travel around circle ABGD, also with uniform circular motion. That point, A, is the angular position of the mean sun. The effect of the epicycle is such that,
when the body is at points Z and Θ, it will appear to coincide with A, the center of the epicycle3, but when it is at other points it will not.
In this setup, the center of the epicycle will travel counter clockwise, toward B.
Thus when it is, e.g., at H, its motion will appear greater than the uniform motion [of the epicycle] by $arc \; AH$, and similarly when it is at K its motion will appear less than the uniform by $arc \; AK$.
Next, Ptolemy compares the apparently motion of the object between the two systems. In the first (the eccentric hypothesis), the apparent speed is rather obvious: It appears to move closest at perigee and slowest at apogee. But with the epicycle hypothesis, it’s not as straightforward. If we think in modern vector notation, it’s quite simple to see why: If the motion vectors of each circle align, it appears to be moving faster. If the epicycle vector opposed to the motion of the larger circle, known as the deferent4, then it appears to be moving slower. Ptolemy is slightly longer winded in his statement of this, saying
if the motion of the body on the epicycle is such that it too moves rearwards from the apogee5, that is from Z towards H, the greatest speed will occur at the apogee, since at that point both epicycle and body are moving in the same direction. But if the motion of the body from the apogee is in advance on the epicycle, that is from Z towards K, then the reverse will occur; the least speed will occur at the apogee, since at that point the body is moving in the opposite direction to the epicycle.
While either of these models is entirely sufficient for modeling the sun’s motion which only speeds up and slows down in a consistent way, Ptolemy notes that the motion (for other objects) can be further complicated by stacking these models, i.e. moving the Earth off the center of the deferent and putting the body on an epicycle. In this way, he hints that we will be able to explore the complex motions of the planets.
But in the case of the simpler sun, either model comes out the same so long as the ratios are the same. Specifically,
the ratio, in the eccentric hypothesis, of the distance between the center of vision and the center of the eccentre to the radius of the eccentre6, must be the same as the ratio, in the epicyclic hypothesis, of the radius of the epicycle to the radius of the deferent; and furthermore, that the time taken by the body, travelling towards the rear, to traverse the immovable eccentre, must be the same as the time taken by the epicycle, also travelling towards the rear, to traverse the circle with the observer as the center [the deferent], while the body moves with equal [angular] speed about the epicycle, but so that its motion at the apogee [of the epicycle] is in advance.
While Ptolemy doesn’t go through a full proof for this, he does illustrate this for a specific case which we’ll explore in the next post.
- Not this guy.
- The sun being included in here as well.
- Also the mean sun.
- “Deferent” isn’t actually a term Ptolemy used and the term was invented in medieval times. Apparently, Ptolemy didn’t have a single word term for this and instead called it things like “the concentric carrying the epicycle.” Because this term permeates so much of the literature on this topic, I will make use of it although the Toomer translation rarely does.
- Here “rearwards” is defined as counter-clockwise, which is the direction we’ll have the epicycle travelling on the deferent.
- Here, the “eccentre” is the circle. AGD in our above drawing of the eccentric model.