In the last post, Ptolemy concluded that the motion of the fixed stars over time, known as precession, happens about the poles of the ecliptic. He determined this by stating that the longitude of stars with relation to the ecliptic remained consistent over a long interval of time but varied with respect to the celestial equator. That post concluded with Ptolemy’s promise that we would be able to determine rate of that precession using the same data he presented previously.
It is particularly [easy to demonstrate] from the differences in declination found for those stars near the equinoctial points.
The reason that the equinoctial points will present the best opportunity is that the slope of the curve showing how the longitudinal position of the stars changed that we created in the last post, is steepest at the equinoctial points. Thus the effect will be the most apparent.
[T]he middle of the Pleiades, which was found to be $15 \frac{1}{6}º$ north of the equator in Hipparchus’ time, and $16 \frac{1}{4}º$ in our time, has [thus] moved $1 \frac{1}{12}º$ northward in the interval between us: this is nearly the same as the difference in declination from the equator between [both ends of] the $2 \frac{2}{3}º$ of the ecliptic near the end of Aries which represents the rearward motion in longitude over that interval.
Ptolemy’s explanation here isn’t great and it doesn’t help he’s pretty fuzzy with the math.
First, let’s consider what’s happening here. As we established in the last post, the sphere of the fixed stars seems to have a “rearwards” motion along the ecliptic. Or, conversely, this would mean that the coordinate system we’re considering, the ecliptic coordinate system, has a forward motion. Because of that, the coordinates of the star will change. In the ecliptic coordinate system, they will only change in ecliptic longitude, but in the equatorial coordinate system, they will change both in right ascension and declination as the coordinate system moves. As we briefly noted in the last post, Ptolemy discusses the change in “latitude” of the star, but what he’s really referring to is the declination. We’ll use the observed change in declination1 to determine the change in ecliptic longitude since they’re related.
To do so, Ptolemy refers to the Table of Inclinations and looked up what arc of the ecliptic corresponds to such a change in latitude2. However, you need to know where in the table to look as you can create any number of arcs corresponding to that change in latitude. He notes that he is looking in the table “near the end of Aries” which is reasonable if we consider the position of the Pleiades.
Here’s a screencap from Stellarium set for Ptomey’s time. Here, we can see that the Pleiades are not particularly far into Taurus. Maybe by about $5º$. Ptolemy, evidently just considers them at the beginning of Taurus (which is the end of Aries) or $30º$ ecliptic longitude. Thus, from there, we can ask how far we’d need to go along the ecliptic to produce a change of $1 \frac{1}{12}º$ ($1;05º$).
A change from $30º$ to $32º$ produces an arc of $0;46,31º$ and a change from $30º$ to $33º$ produces an arc of $1;07,29º$. Thus, the answer is somewhere between $2-3º$. Doing a bit of interpolation, I get that it would require an arc of the ecliptic of $32;52,53º$, or an increase of $2;52,53º$. This is notably higher than Ptolemy’s value of $2 \frac{2}{3}º$ ($2;40º$). Toomer notes this as well indicating, if we used Ptolemy’s value it would have only corresponded to an arc of the ecliptic of $0;57,30º$. So, as we note, Ptolemy’s math is a bit loose here.
As a quick thought, we can ask what would have happened if Ptolemy had used a more accurate value for the ecliptic longitude of the Pleiades, say, around $35º$? There, we can easily see that the arc of the ecliptic needed to produce such a change in latitude is over $3º$ of ecliptic longitude which is even further off from Ptolemy’s value3. Thus, it’s hard to explain where Ptolemy came up with his value4.
He then repeats this came calculation for
Capella, which was found to be $40 \frac{2}{5}º$ north of the equator in Hipparchus’ time, and $41 \frac{1}{6}º$ in our time, has [thus] moved northward $\frac{4}{5}º$ [$0;48º$].
Again, Ptolemy is doing a bit of rounding here. A more accurate computing of the change in latitude would be $0;46º$ but Ptolemy is on a kick with fractions so we’ll stick with his number.
Ptolemy places Capella “near the middle of Taurus” or $45º$ ecliptic longitude5. Going through the same estimation above I find that an increase in the latitude of $0;48º$ corresponds to an increase (from $45º$) of $2;44,36º$ which is reasonably close to the $2 \frac{2}{3}º$ that Ptolemy is trying to sell us6.
This calculation then gets repeated for
the star on the advance shoulder of Orion [γ Ori], which was found to be $1 \frac{4}{5}º$ north of the equator in Hipparchus’ time, and $2 \frac{1}{2}º$ in our time, [and thus] has moved northward by about $\frac{2}{3}º$ [$0;40º$].
Again, Ptolemy is using a fraction. The change should more accurately be described as $0;42º$. This star, Ptolemy places “two-thirds through Taurus” or $\approx 50º$ ecliptic longitude. As with Capella, this is incorrect as it should be about $55º$. However, this is much closer to being correct than Capella.
Walking through the calculation I get that an increase of $0;42º$ in latitude corresponds to an increase of longitude by $2;58,35º$. Ptolemy, by virtue of his prodigious rounding again comes up with $2 \frac{2}{3}º$.
These three stars we have just examined are in the hemisphere from the summer to winter solstice. Ptolemy then examines three stars from the other hemisphere: Spica, the star in the tip of the tail of Ursa Major (η UMa), and Arcturus.
For Spica, it changed by $1;06º$. According to Ptolemy’s star catalog, Spica should have a ecliptic longitude of $\approx 177º$7. Here, the Table of Inclinations gets a bit funny to use as it only covers $90º$. But we have to remember the symmetry: From $0 – 90º$ it goes up, peaks at the summer solstice, and then repeats in reverse from $90 – 180º$ before doing the same, but with negative values.
Doing the math, I find that a change of $1;06º$ in latitude corresponds to a change along the ecliptic of $2;43,21º$ which is indeed reasonably close to Ptolemy’s $2 \frac{2}{3}º$.
For η UMa, it changed by $1 \frac{1}{12}º$ ($1;05º$). Ptolemy places it at the “beginning of Libra” which is $180º$ ecliptic longitude. This is in very good agreement with the position he gives in the star catalog of $29 \frac{5}{6}º$ into Leo. So using his figures I come up with a change of $1;05º$ corresponding to a change of $2;24,15º$. Again, Ptolemy simply gives a value of $2 \frac{2}{3}º$.
Finally, for Arcturus, it changed by $1 \frac{1}{6}º$ ($1;10º$) in latitude. According to Ptolemy’s star catalog, this should have a ecliptic longitude of $177º$, but he again gives it as “near the beginning of Libra” ($180º$). Using the more accurate value I find that a change of $1;10º$ in latitude corresponds to a change of $2;53,15º$ which is a bit higher then Ptolemy’s value of $2 \frac{2}{3}º$.
Although Ptolemy doesn’t state it here, this is sufficient to determine the rate of precession. If we take his value of $2 \frac{2}{3}º$ which occurred over a period of $265$ years that’s a rate of roughly $1º$ per hundred years.
Ultimately, the value of $2 \frac{2}{3}º$ that Ptolemy is pushing isn’t great as we saw by performing more precise calculations above. However, if I instead use the values I calculated and average them, I come up with an average motion in longitude of $2;52,38º$ in $265$ years which corresponds to a rate of $1;05;52º$ per century. Inversely, that’s $1º$ every $\approx 92$ years8. This is a nice improvement, but not enough that Ptolemy would likely have felt compelled to refine the result.
From here, Ptolemy is going to repeat this set of calculations, but over a longer interval of time, using observations from Timarchus, Agrippa, and Menelaus. However, these observations are not so readily usable as Hipparchus’ and Ptolemy is going to have to spend significantly more effort to interpret them. As such, we’ll save that for the next post!
- Which I’ll refer to going forward as latitude to follow Ptolemy.
- There, the column is the “Arc of the meridian” for we performed the calculation based on when this arc was on the observer’s meridian.
- This makes perfect sense since the further one gets from the equinoctial point, the more parallel the lines become resulting in a longer arc of the ecliptic needed to produce the same change in the latitude.
- Toomer suggests that Ptolemy may have used a very sloppy linear approximation from $0º – 30º$ ecliptic longitude. In that case, the change over $30º$ is $\approx 11;40º$ and $2 \frac{2}{3}$ of that would produce a change in latitude of $\approx 1;02º$ which is reasonably close. Still, this is uncharacteristically sloppy.
- This is incorrect. In Ptolemy’s time, Capella would have had an ecliptic longitude of $\approx 56º$. Ptolemy’s own star catalog (which we haven’t gotten to yet) gives a value of $55º$.
- If we instead use Ptolemy’s position for Capella from his star catalog of $55º$, I come up with a change in ecliptic longitude necessary of $3;23,59º$.
- Ptolemy describes it s position “near the end of Virgo” implying $180º$ ecliptic longitude, which is reasonably close. However, for my calculations, I’ll use the more accurate value.
- The correct value is about $1º$ every $72$ years.