From the above, it has become clear to us that the sphere of the fixed stars, too, performs a rearward motion along the ecliptic, of approximately the amount indicated. Our next task is to determine the type of this motion, that is to say, whether it takes place about the poles of the equator or about the poles of the inclined circle of the ecliptic.
Now that Ptolemy has determined that precession does indeed happen at a rate that agrees with Hipparchus, he now asks whether that precession is happening in the same direction as the ecliptic or the celestial equator.
Since great circles drawn through the poles of either one of the above [ecliptic or celestial equator] cut off unequal arcs of the other, [the answer to] the above would become apparent merely from the motion in longitude, were it not for the fact that the motion in longitude over the time available [for comparison of observations] is so extremely small that the difference due to the above effect would be, as of yet, imperceptible.
Here, what Ptolemy is noting is that, if we divided the celestial sphere into equal chunks about the celestial equator, the length of the ecliptic cut off in each of those wedges would not be equal and vice versa. Thus, if we assumed that the precession was along the celestial equator and it was actually along the ecliptic, the amount of precession we would see in different wedges would be different as well. Unfortunately, because the precession is so small, the amount of difference is absolutely swamped by the inherent uncertainty in these measurements. How small is it?
Consider the following diagram:
If we consider the portion of the two great circles closest to one of the equinoxes (where the effect would be at its maximum) we can consider a star at the equinoxes. Then we’ll consider it to have moved by the amount of precession between Hipparchus and Ptolemy( $2;40º$). If the motion was along the ecliptic, then the star would move in ecliptic longitude by that amount. Then the corresponding change along the celestial equator1 would be $2;54º$, only $0;14º$ greater than the amount itself. Thus, it is indeed a hard measurement for the time.
The easiest way to detect [the answer] is through [comparison of] the positions [of the stars] in latitude in ancient times and now. For it is obvious that whichever of the two circles, [celestial] equator and ecliptic, it is from which they can be shown to maintain a constant distance in latitude, that is the circle about the poles of which the motion of their sphere will take place.
If we instead consider what the difference would be in latitude2 if the motion was along the ecliptic instead of celestial equator we could construct the following diagram:
If we considered a star that started at the equinox and the motion were along the ecliptic, then there would be a change in latitude of the object in comparison to the celestial equator of $1;06º$ which is readily noticeable.
Now, Hipparchus agrees with [the idea of] the motion taking place about the poles of the ecliptic. For in On the Displacement of the Solstical and Equinoctial Points, he deduces from the observations of Timocharis and himself that Spica (again) has maintained the same distance in latitude, not with respect to the equator, but with respect to the ecliptic, being $2º$ south of the ecliptic at both earlier and later periods. That is why in On the Length of the Year he assumes only the motion which takes place about the poles of the ecliptic, although he is still dubious, as he himself declares, both because the observations of the school of Timocharis are not trustworthy, having been made very crudely, and because the difference in time between the two [Timocharis and himself] is not sufficient to provide a secure result.
So Hipparchus evidently did precisely this type of experiment using Timocharis’ results as his baseline and found no change in latitude with respect to the ecliptic, but didn’t trust the original observations.
We, however, find the [latitudinal distances with respect to the ecliptic] preserved over the much longer interval [down to our times], and that for practically all fixed stars. For when we observe the latitudinal distance of any star with respect to the ecliptic, as measured along the great circles through the poles of the ecliptic, we find that it is practically the same as that computed from the records of Hipparchus, or if there is a discrepancy, it is of very small size, such as can be accounted for by small observational errors.
Ptolemy repeats the comparison using Hipparchus observations as his baseline and again there is no discrepancy when reckoned with respect to the ecliptic, in agreement with Hipparchus.
But, when we consider the distances [of the stars] from the [celestial] equator, as measured along great circles through the poles of the equator we find [1] that those observed by us do not agree with those recorded in the same way by Hipparchus, and [2] that the latter do not agree with those recorded even earlier by Timocharis and his associates’ rather, the constancy of their latitudes with respect to the ecliptic is confirmed even more by these very observations, since the distances from the equator of the stars located on the hemisphere from the winter solstice through the spring equinox to the summer solstice are found to be ever more northerly compared to those [of the same stars] in earlier periods, while for stars located on the opposite hemisphere they are ever more southerly. Furthermore, the differences [between earlier and later observations] are greater for stars near the equinoctial points, and less for stars near the solstices, and these differences are just about the same as the amount by which that section of the ecliptic to the rear [of the earliest longitude of any particular star] defined by the corresponding motion in longitude [during the period in question] produces a displacement to the north or south of the equator.
As one would expect, if the motion is not aligned with the celestial equator, then there should be a change in latitude over time and the further back one compares to, the larger it should be which is precisely what Ptolemy found comparing to Hipparchus’ and Timocharis’ records. Furthermore, because the ecliptic is inclined the most steeply to the celestial equator most steeply at the equinoxes, the effect should be strongest there, which Ptolemy again observes seeing that it is less pronounced at the solstices. Overall, a very strong case.
But Ptolemy doesn’t ask us to take his word for it. He then goes through a series of “easily recognizable stars” to illustrate the point. For each of the stars he explores how their latitudes changed both with respect to the ecliptic and the celestial equator. The raw data is displayed in two tables giving the latitudes with respect to the celestial equator showing that they do indeed change. The first table is for stars from the summer solstice to the winter solstice:
Ptolemy then gives a second table for stars from the winter solstice to the summer solstice:
In both cases Ptolemy neglects to tell us where in the celestial sphere these stars are and Toomer only gives partial help, referencing their place in the upcoming star catalogue so we can look up their designations there. So to make things easier, let’s summarize both tables into something a bit easier to read. Here, I’ve included the designations and ecliptic longitude as Ptolemy gives them in the upcoming stellar catalog.
| Star | Ecliptic Longitude | Δ Timocharis | Δ Hipparchus |
| α Aql | $273 \frac{5}{6}º$ | $\frac{1}{6}º$ | $\frac{1}{6}º$ |
| Pleiades | $32 \frac{1}{6}º$ | $1 \frac{3}{4}º$ | $1 \frac{1}{12}º$ |
| α Tau | $42 \frac{2}{3}º$ | $2 \frac{1}{4}º$ | $1 \frac{1}{4}º$ |
| α Aur | $55º$ | $1 \frac{1}{6}º$ | $\frac{23}{30}º$ |
| γ Ori | $84º$ | $1 \frac{3}{10}º$ | $\frac{7}{10}º$ |
| α Ori | $62º$ | $1 \frac{5}{12}º$ | $\frac{11}{12}º$ |
| α CMa | $77 \frac{2}{3}º$ | $\frac{7}{12}º$ | $\frac{1}{4}º$ |
| α Gem | $83 \frac{1}{3}º$ | $\frac{2}{5}º$ | $\frac{7}{30}º$ |
| β Gem | $86 \frac{2}{3}º$ | $\frac{1}{6}º$ | $\frac{1}{6}º$ |
| α Leo | $122 \frac{1}{2}º$ | $- 1 \frac{1}{2}º$ | $- \frac{15}{18}º$ |
| α Vir | $176 \frac{2}{3}º$ | $- 1 \frac{9}{10}º$ | $- 1 \frac{1}{10}º$ |
| η UMa | $149 \frac{5}{6}º$ | $- 1 \frac{5}{6}º$ | $- 1 \frac{1}{12}º$ |
| ζ UMa | $138º$ | $- 2 \frac{1}{4}º$ | $- 1 \frac{1}{2}º$ |
| ε UMa | $132 \frac{1}{6}º$ | $- 2 \frac{1}[{4}º$ | $- 1 \frac{7}{20}º$ |
| α Boo | $177º$ | $-1 \frac{2}{3}º$ | $-1 \frac{2}{3}º$ |
| α Lib | $180 \frac{2}{3}º$ | $- 2 \frac{1}{6}º$ | $- 1 \frac{17}{30}º$ |
| β Lib | $202 \frac{1}{6}º$ | $- 2 \frac{1}{5}º$ | $- 1 \frac{2}{5}º$ |
| α Sco | $222 \frac{2}{3}º$ | $- 1 \frac{11}{12}º$ | $ – 1 \frac{1}{4}º$ |
In the case of all the [stars from the first table], which are located (to speak of their longitudinal position) on that that one of the above-defined hemispheres which contains the spring equinox, the vertical distances from the equator which are later in time are all more northerly than the earlier, and for those stars very near the solstical points [the difference] is very small, while for those near the equinoxes it is quite considerable: this accords with a rearward motion about the poles of the ecliptic, for if one takes successive sections of this semi-circle [of the ecliptic] going towards the rear, each is more northerly than the one in advance of it, and the difference [between successive equal sections] is again greater near the equinoxes and less near the solstices.
[And in the case of the stars from the second table], the reverse [of the above] is true, as one would logically expect: the later vertical distances from the equator are more southerly than the earlier, in proportion [to the time intervals and locations].
If we plot out the changes as a function of the ecliptic longitude we can see what Ptolemy is referring to quite easily:
In this graph, I’ve plotted the change in declination as a function of ecliptic longitude. We can clearly see that the change is greatest towards the equinoxes at $0º$ and $180º$ and at their minimums near the solstices at $90º$ and $270º$ where they cross the x-axis. In addition, the variance is larger since the time of Timocharis as compared to that of Hipparchus.
[O]ne can can conclude from these data that the rearward motion in longitude of the sphere of the fixed stars is, as we said previously, $1º$ in about $100$ years, or $2 \frac{2}{3}º$ in the $265$ years between Hipparchus’ and our observations. It is particularly [easy to do this] from the differences in declination found for those stars near the equinoctial points.
Ptolemy then states that the rate of precession can also be derived from the above data:
Furthermore one can conclude from these data that the rearward motion in longitude of the sphere of fixed stars is, as we have said previously, $1º$ in about $100$ years, or $2 \frac{2}{3}º$ in the $265$ years between Hipparchus and our observations.
We’ll explore how in the next post.
- Using a modern value of the angle between the two of $23;30º$.
- Toomer correctly notes here that the term “latitude” is being used somewhat loosely, just meaning “at a right angle to whatever great circle is under consideration.” In the example I’m providing here, it would be the latitude for the celestial equator which is more properly known as declination.