Now that these [mean motions] have been tabulated, our next task is to discuss the anomalies which occur in connection with the longitudinal positions of the five planets.
Having derived the mean and anomalistic motions, Ptolemy now turns to exploring the anomalies in more depth (as there’s going to be two of them), in order to derive the parameters necessary for configuring the model’s scale.
There are, as we said, two types of motion which are simplest and at the same time sufficient for our purpose, [namely] that produced by circles eccentric to [the center of] the ecliptic, and that produced by circles concentric with the ecliptic but carrying epicycles around.
Here, Ptolemy reiterates the two types of motion he intends to consider – the eccentric and epicyclic. He then indicates that there are two components to the anomaly we’ll need to consider:
There are likewise, two apparent anomalies for each planet: [$1$] that anomaly which varies according to its position in the ecliptic, and [$2$] that which varies according to its position relative to the sun.
Both of these anomalies Ptolemy introduced us to in IX.2.
Speaking in a modern astronomical context, there are two primary causes for such planetary anomalies. The first is that the planet’s orbit is not circular – they speed up closer to the sun and slow down further away. This effect always happens in the same portion of the ecliptic since apogee and perigee are relatively stable1. Thus, this is the cause of the first of the two anomalies.
The second is the motion of the earth causing an apparent change in position. Since earth’s motion is around the sun, the anomalistic effect caused by this would be directly tied to the position in relation to the sun as well, thus explaining the second of Ptolemy’s anomalies.
We might also ask about the impact of earth’s own eccentricity. However, this was accounted for in the solar model, so my suspicion is that it will be taken care of with the second anomaly described above.
Anyway, now that we know what causes these anomalies for our own edification, let’s get into what Ptolemy has to say about each, starting with the second:
For [$2$], we find, from a series of different [sun-planet] configurations observed round about the same part of the ecliptic, that in the case of the five planets, the time from greatest speed to mean is always greater than the time from mean speed to least.
As to how one would make this observation, Neugebauer explains:
In order to study the anomaly with respect to the sun one has to observe the sequence of synodic phenomena and compare it with the progress of the planet in longitude. The one finds that the time between mean progress and minimum (negative) progress is always shorter than from maximum velocity to the mean.
But to illustrate here, we can easily simulate some data.
Using an online ephemerides calculator, I generated a table of Jupiter’s position in RA/Dec for every day over several years. I then converted this to ecliptic longitude since that’s what Ptolemy works in. Next, since we’re talking about speed which is the derivative of position, I took the difference in position between days as a new column. Then, we can see where the min, max, and average all fall.
I used $2005$ for this test data. In that year, the minimum speed was $-0.129 \frac{º}{day}$2 on April $4$. The maximum speed was $0.218 \frac{º}{day}$ on October $25$. That’s a difference of $204$ days.
Now, the average speed is between the two extremes: $0.044 \frac{º}{day}$. Jupiter hit that speed on June $21$. That’s $78$ days since the minimum speed and $126$ days before the maximum speed, thus demonstrating that the time from mean speed to maximum is greater than from least to mean3.
As a quick aside, it’s always amazing to me that ancient astronomers were so aware of these things. Despite having a degree in the field, this is not a phenomena I was even aware of. That’s not to say that it isn’t something that’s easily explained by modern models4, but it’s yet another phenomena that’s gotten so well explained that we scarcely pay it any mind.
So what of it?
Ptolemy tells us that this will have implications on the configuration of the model because:
this feature cannot be a consequence of the eccentric hypothesis, in which exactly the opposite occurs, since the greatest speed takes place at the perigee in the eccentric hypothesis, while the arc from the perigee to the point of mean speed is less than the arc from the latter to the apogee in both [eccentric and epicyclic] hypotheses. But it can occur as a consequence of the epicyclic hypothesis, however, only when the greatest speed occurs, not at the perigee, as in the case of the moon, but at the apogee; that is to say, when the planet, starting from the apogee, moves, not as the moon does, in advance [with respect to the motion] of the universe, but instead towards the rear. Hence, we use the epicyclic hypothesis to represent this kind of anomaly.
In short, the eccentric hypothesis is unable to account for this sort of motion (specifically retrograde motion), so we’ll need to make use of epicycles5.
Anyway, that’s the second type of anomaly. Returning to the first (the “anomaly which varies according to its position in the ecliptic”),
we find from [observations of] the arcs of the ecliptic between [successive] phases or [sun-planet] configurations of the same kind that the opposite is true: the time from least speed to mean is always greater than the time from mean speed to greatest. This feature can indeed be a consequence of either of the two hypotheses (in the way we described in our discussion of the equivalence of the hypotheses at the beginning of our treatise on the sun [III.3]). But it is more appropriate6 to the eccentric hypothesis, and that is the hypothesis which we actually use to represent this kind of anomaly, so to speak, to the epicyclic hypothesis.
As described above, this anomaly is caused by the eccentricity of the planet itself. Neugebauer again explains the observations necessary:
In order to investigate the anomaly with respect to the ecliptic we have to consider positions in which the planet is always in the same situation with respect to the sun, e.g. at opposition or at a stationary point. Then the observations show that the spacing between such points is greater in one part of the ecliptic than in another and that the time from the smallest velocity to mean motion exceeds the time from mean motion to most rapid progress.
Returning to Ptolemy:
Now from prolonged application and comparison of observations of individual [planetary] positions with the results computed from the combination of both [the above] hypotheses, we find that it will not work simply to assume [as one has hitherto] that the plane in which we draw the eccentric circles is stationary, and that the straight line through both centres (the centre of the [planet’s] eccentre and the centre of the ecliptic), which defines apogee and perigee, remains at a constant distance from the solstical and equinoctial points;
Here, Ptolemy is stating that the plane of the planets should be allowed to rotate. In effect, the points at which the minimum and maximum ecliptic latitudes are will move. How Ptolemy will account for this without allowing the apogee and perigee to drift remains to be seen.
nor [to assume] that the eccentre on which the epicycle centre is carried is identical with the eccentre with respect to the centre of which the epicycle makes its uniform revolution towards the rear, cutting off equal angles in equal times at [that centre].
In this, Ptolemy is stating that the center of the planetary orbits need not be the same center as the sun’s center.
Rather, we find that the apogee of the eccentre performs a slow motion towards the rear with respect to the solstices, which is uniform about the centre of the ecliptic, and comes to about the same for each planet as the amount determined for the sphere of the fixed stars (i.e., $1º$ in $100$ years (at least, as far as can be estimated on the basis of available evidence).
This seems a lot like Ptolemy stating that the line of apses for each of the planets does drift, but what Ptolemy’s really getting at is that precession of the equinoxes can be observed by seeing where along the ecliptic apogee and perigee fall. Ptolemy is telling us that they drift at the same rate as the sphere of fixed stars7.
We find, too, that the epicycle centre is carried on an eccentre which, though equal in size to the eccentre which produces the anomaly, is not described about the same centre as the latter. For all planets except Mercury the centre [of the actual deferent] is the point bisecting the line joining the centre of the eccentre producing the anomaly to the centre of the ecliptic.
I’m not quite sure what Ptolemy is trying to drive at here, but what I think he’s saying is that his model will combine the epicyclic and eccentric models much like we did with the lunar model and that the center of the eccentre and the deferent will be different. I suspect this will become more clear as we start seeing the model itself.
For Mercury alone, [the centre of the deferent] is a point whose distance from the centre of the circle about which it rotates is equal to the distance of the latter point towards the apogee from the centre of the eccentre producing the anomaly, which in turn is the same distance towards the apogee from the point representing the observer; for also, in the case of this planet alone, we find that, just as for the moon, the eccentre is rotated by [the movement of] the above-mentioned centre in the opposite sense to the epicycle, [i.e.] in the advance direction, one rotation per year. [This must be so] because the planet itself appears twice in the perigee in the course of one revolution, just as the moon appears twice in the perigee in one [synodic] month.
In short, Mercury is a special case and its model will be slightly different than that for the other four planets because of it. But we’ll leave that until we get to said model.
- I should note that the line of apses does in fact precess for all planets. However, much like the precession of the equinoxes, this is a very slow motion. Saturn’s is the highest at a rate of $\approx 19.5$ arcseconds per year which means a full cycle should take approximately $6.6$ million years – Not particularly in the range of something Ptolemy would have noticed.
- Negative here indicating that the planet was in retrograde.
- I did things the other way around, but the sin curve is pretty symmetric (as seen in the graph below) so it holds. In this graph you can also see that the min is rather narrow whereas the max is rather wide. The result is that the mean gets pulled towards the minimums on both sides of each curve.
- I puzzled over why this is true for a good while, and think it has to do with the minimum speed being when earth is passing Jupiter at opposition. At that time, Jupiter’s speed is at its minimum since the earth’s motion makes Jupiter appear to move backwards (i.e., retrograde). However, because earth is so close, the angular difference its motion makes over a single day is considerably larger than the impact when earth’s motion causes an additive effect to Jupiter’s apparent motion – when Jupiter is opposite the sun from the earth. This can clearly be seen in this quick sketch in which I’ve greatly exaggerated the effect. But you can easily see that the change in motion when earth is closest to Jupiter (blue) is greater than when earth moves over the same arc when it is furthest from Jupiter (red).
- Toomer notes that this isn’t strictly true. He cites Neugebauer who points out that such motion could be explained if the line between apogee and perigee (the lines of apsides) is allowed to rotate as well. Indeed, this is something that Ptolemy will later refer to it in XII.1.
- Toomer points to the discussion in III.4 in which Ptolemy prefers this model as “simpler” for the argument behind this statement.
- Which is really the stars holding still but the point along the ecliptic of the equinoxes doing the drifting.
