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Almagest Book VII: On the Rate of Precession from Other Greek Astronomer’s Observations

In the last post, we showed how we can determine the rate of precession if we know how much a star has changed its declination over a long period. In it, we used a baseline of $265$ years, corresponding to the time between Hipparchus and Ptolemy.

Next, Ptolemy wants to increase that baseline further and turns to the observations of three other Greek Astronomers: Timocharis, Agrippa1, and Menelaus. However, these astronomers did not give the position of the stars in equatorial coordinates. Rather, they described occultations of various stars by the moon. Thus, Ptolemy turns to the lunar model to determine the positions of these stars and instead of finding a change in declination, is able to directly compare the ecliptic longitude of them over time.

Unfortunately, this presents significant challenges. Toomer describes Ptolemy’s interpretation of the data as “strange” and has numerous footnotes calling out “errors resulting from Ptolemy’s miscomputations on the basis of his own theory.” Toomer points to several other sources that have studied this section in depth (although not always in the context of this section) including Schjellerup (1881)2, Fotheringham (1915) with a correction in 1923, and Britton (1967). With regards to Briton, Toomer notes that this analysis is the most complete and Britton concludes that the observations selected were likely chosen because they arrived at the value Ptolemy wanted3.

Since the interpretation of these observations is obviously complicated, for the purposes of this post, I’ll be somewhat less critical than normal as going through the complete lunar calculations and exploring all the commentary would be a project in and of itself. Rather, I’ll simply cite Toomer’s commentary4.

Ptolemy begins with an examination of occultations of the Pleiades.

Pleiades

Timocharis

[Firstly] Timocharis, who observed at Alexandria, records the following: In the $47^{th}$ year of the First Kallippic $76$ year period, on the eight of Anthesterion, which is Athyr $29$ in the Egyptian calendar, towards the end of the third hour [of the night], the southern half of the moon was seen to cover exactly either the rearmost third or [the rearmost] half of the Pleiades5. That moment is in the $465^{th}$ year from Nabonassar, Athyr [III] $29/30$ in the Egyptian calendar [$-282$ Jan $29/30$], $3$ seasonal hours before midnight, or $3 \frac{1}{3}$ equinoctial hours (since the sun was in about Aquarius $7º$). The interval reckoned in mean solar days comes to about the same number of equinoctial hours [$3 \frac{1}{3}$] before midnight. At that moment, according to the hypotheses we demonstrated previously, the position of the moon was as follows:

True longitude: $0;20º$ in Taurus (i.e., distance from the spring equinox of $30;20º$6)

[Latitude]: $+3;45º$ north of the ecliptic7

In Alexandri [i.e., adjusting to include parallax]:

Apparent longitude: $29;20º$8

Apparent latitude: $3;35º$ north of the ecliptic9

(for the culminating point was $\frac{2}{3}rds$ through Gemini).

Therefore, at that time, the rearmost end of the Pleiades was about $29 \frac{1}{2}º$ towards the rear from the spring equinox (for the moon’s centre was still in advanced of it), and was about $3 \frac{2}{3}º$ north of the ecliptic (for, again, it was a little north of the moon’s centre).

We can plug this into Stellarium to see what Timocharis was describing.

I’m always a bit skeptical that Stellarium is accurately showing things, but this image looks pretty spot on for Timocharis’ description at the time determined by Ptolemy, showing that the moon is indeed covering the rearward section of the Pleiades.

So Ptolemy uses his lunar model to calculate the ecliptic latitude and longitude, and then adjusts it for the parallax that should occur in Alexandria at that time since we need the apparent position as the true position wouldn’t actually cover the Pleiades at all. He finds the apparent ecliptic longitude to be $29;20º$ which Ptolemy then rounds this to $29;30º$10 for no apparent reason11.

Agrippa

Ptolemy then does the same calculation for an observation by Agrippa

who observed in Bithynia12 [and] records that in the twelfth year of Domitian, on the seventh of Metroos according to the calendar of that region, at the beginning of the third hour of night, the moon occulted the rearmost, southern part of the Pleiades with its southern horn. That moment is in the $840^{th}$ year from Nabonassar, Tybi [V] $2/3$ in the Egyptian calendar [$92$, Nov $29/30$], $4$ seasonal hours before midnight, or $5$ equinoctial hours (since the sun was in about Sagittarius $6º$). Therefore, reduced to the meridian of Alexandria, the observation occurred $5 \frac{1}{3}$ equinoctial hours before midnight, or $5 \frac{3}{4}$ hours with respect to mean solar days. At this moment, the positions of the centre of the moon were as follows:

True longitude: $3;07º$ into Taurus [i.e., $33;07º$ ecliptic longitude]13. into Taurus [i.e., distance from the spring equinox of $33;07º$]

[Latitude]: $4 \frac{5}{6}º$ north of the ecliptic14

In Bithynia:

Apparent longitude: $3;15º$ into Taurus [i.e., $33;15º$ ecliptic longitude]15

Apparent Latitude: $4º$ north of the ecliptic16

(for the culminating point was two-thirds through Pisces).

Therefore at that time, the rearmost section of the Pleiades was, in longitude, $33 \frac{1}{4}º$ towards the rear from the spring equinox, and, [in latitude] $3 \frac{2}{3}º$ north of the ecliptic

Again, Ptolemy goes through his lunar calculations and this time, comes up with an apparent ecliptic longitude of $33;15º$, an increase of $3;45º$ over the interval of $375$ years between the two observations. That works out to exactly $1º$ per hundred years. This result is significantly in error given the gross miscalculations for Agrippa’s observations which Toomer describes as “disastrous” for this result17.

Ptolemy also points out that the ecliptic latitude is consistent between observations as we should expect since the direction of motion of parallax is directed along the ecliptic as we showed in the last post.

Spica

Timocharis #1

Ptolemy then tries to drive this point home even further by doing the same for three observations of occultations of the star Spica. The first was from Timocharis. Skipping the historic details, he finds that on March $9/10$, $293$  BCE, $4$ hours before midnight, the true longitude of the moon would have been $21;21º$ into Virgo ($171;21º$ ecliptic longitude) with a true ecliptic latitude of $-1 \frac{5}{6}º$. Adjusting for the parallax it would have an apparent ecliptic longitude of $172 \frac{1}{12}º$ and an apparent ecliptic latitude of $-2º$18.

Timocharis #2

The next observation is also from Timocharis and describes an occultation at $3 \frac{1}{8}$ equinoctial hours after midnight on Nov $8/9$, $282$ BCE. Then, the moon had a true ecliptic longitude of $171;30º$, and a true ecliptic latitude of $-2 \frac{1}{6}º$. Adjusting for parallax, the apparent ecliptic longitude he gives is $172 \frac{1}{2}º$ and the ecliptic latitude is $-2 \frac{1}{4}º$19.

Again, the ecliptic latitude stays pretty consistent as we should expect and in this interval of $12$ years as the longitude increases by $0;12º$ which Ptolemy rounds to $\frac{1}{6}º$. This is a very short interval so errors will be huge here. Indeed, this results in a rate of about $1;40º$ per century. Probably because Ptolemy knows that such a short interval is useless, instead doesn’t discuss this and gives a third observation of an occultation of Spica by “the geometer Menelaus” from Rome.

Menelaus

This occultation occurred $5$ hours after midnight on Jan $10/11$, $98$ CE. Ptolemy calculates the true ecliptic longitude to be $175 \frac{3}{4}º$, the true ecliptic latitude as $-1 \frac{1}{3}º$, the apparent ecliptic longitude as $176 \frac{1}{4}º$ and the apparent ecliptic latitude as $-2;00º$20

Ptolemy again notes that the ecliptic latitude remained consistent while the ecliptic longitude increased by $4;10º$ (for which Ptolemy somehow comes up with $3;55º$ for) over the period from the first of Timocharis’ observations. This is a period of $391$ years which again works out to $1;04º$ per century (or, basically exactly $1º$ per century if we use Ptolemy’s made up number).

If we do the same for the second of Timocharis’ observations I find a change in ecliptic longitude of $3;45º$ over the interval of $379º$ giving a rate of almost exactly $1º$ per century yet again.

β Sco

Ptolemy then goes through the exercise yet again with the “northernost of the stars in the forehead of Scorpius” (β Sco).

Timocharis #1

Ptolemy first cites an observation from Timocharis, $3 \frac{2}{5}$ equinoctial hour after midnight on Dec $20/21$, $294$ BCE for which he calculated an apparent ecliptic longitude of $212;00º$ and an apparent latitude of $1 \frac{1}{12}º$21.

Menelaus #1

He then cites Menelaus who observed an observation $6 \frac{1}{6}$ equinoctial hours after midnight on Jan $13/14$, $98$ CE. There, he computes apparent positions of $215;55º$ for ecliptic longitude and $1 \frac{1}{3}º$ ecliptic latitude22.

The change in ecliptic longitude is $3;18º$ over an interval of $391$ years for a rate of almost exactly $1º$ per century yet again23.

Conclusion

As we’ve seen, Ptolemy is working very hard to agree with Hipparchus’ rate of $1º$ per century. There is significant error in the numbers Ptolemy comes up with and I suspect there’s other sources that are more hidden as well. After all, the moon is $\approx \frac{1}{2}$ degree in diameter. Thus, if the occultation does not occur on the same part of the moon, this can result in a difference. And this is precisely the case for Spica. In the first observation (from Timocharis) he described the occultation as occurring “with the middle of that edge of the disk which is towards the equinoctial rising point [i.e., the east].” For the second observation from Timocharis, he describes it at “exactly touching the northern point [on the moon]”. In other words, not quite an actual occultation. And the difference between the edge of the moon on its east side and the tip of its northern side is about $\frac{1}{4}º$ in ecliptic longitude which I see no effort to take into account. No clear description is given for the third occultation of Spica. Rather, it is reported that Spica simply wasn’t visible and then some time later, it was visible “in advance of the moon’s center equidistant from the horns by an amount less than the moon’s diameter.” Thus, the timing is also suspect as well!

Overall, Ptolemy’s methods here are clever and not incorrect, but there is a lot to be desired in the meticulousness of it.

That almost takes us out of Book VII. In the next post, we’ll discuss how Ptolemy determines the positions of stars for the upcoming catalog to which I’ve frequently made reference, and then present the catalog itself!

 

  1. This is not the Roman general Agrippa who died before the observations Ptolemy cites were taken.
  2. This source is in French.
  3. Toomer states that Britton did not consider Ptolemy’s miscalculations and suggests that doing so would only strengthen Britton’s case. This is one of many such accusations about whether or not Ptolemy (at best) was biased in his presentation or (at worst) may have fabricated his data. This claim is most often leveled at the upcoming star catalog and those accusations are thoroughly disputed in Grasshoff’s History of Ptolemy’s Star Catalog. However, while I have not read that source yet, I did see if it had anything to say on the subject of this chapter of the Almagest and could not find anything of consequence. As such, the frequency with which Ptolemy makes errors in his calculations that just happen to result in achieving a consistent value makes me suspicious that Ptolemy was indeed working to achieve an end instead of presenting an honest discussion. That being said, the Almagest is intended as a teaching tool and as such, I find no fault with Ptolemy for choosing observations which best illustrate the theory presented. However, if he had to grossly fudge the math, they aren’t observations which best illustrate the theory in the first place and I can find fault with that, for if he was unable to find suitable observations in the first place, then it’s indicative of a flawed theory.
  4. I haven’t explored the sources cited above, but it’s interesting to note that none of the other texts that I often use (Neugebauer or Pedersen) comment on this section in any detail. Neugebauer apparently skips it entirely and Pedersen gives light summary without any exploration or criticism.
  5. Toomer notes that the meaning here is unclear. Was this uncertainty Ptolemy’s indicating there were multiple historical manuscripts available to him that differed? Or did this uncertainty arise after the Almagest was written?
  6. Toomer performs these calculations and comes up with a value of $30;11º$.
  7. Toomer finds the same value.
  8. Toomer: $29;00º$
  9. Toomer:$+3;38º$.
  10. You may be wondering why Ptolemy doesn’t immediately compare this value to his own position for the Pleiades from his upcoming star catalog. And… that’s a great question. My initial suspicion is that he might not have an equivalent position in his catalog for the “rearward portion” that is described by Timocharis. Yet he clearly does have a value for the “rearmost and narrowest end of the Pleiades” which sounds appropriate. Still, having two observations of occultations that are both the “rearmost portion” provides the most consistent context and is therefore presumably more reliable. However, I question how equivalent the rearmost portion can truly be between these two observations given that the Pleiades is an extended object.
  11. There is that bit right at the end where he states “for, again, it was a little north of the ecliptic” but it’s not obvious as to why that should choose him to round in such a way.
  12. Bithynia is in modern day Turkey on the southwest side of the Baltic sea.
  13. Here, Toomer has again performed the calculation and finds “gross errors”. For this value he finds the true longitude to be $32;13º$ – nearly a full degree less than what Ptolemy gives!
  14. Toomer: $+4;53º$.
  15. Toomer: $32;32º$ ecliptic longitude.
  16. Toomer: $+4;15º$.
  17. Indeed, using Toomer’s calculations would result in movement of $0;57º$ per century which is even lower than the value calculated in the last post!
  18. Toomer offers no significant corrections here.
  19. Toomer used this calculation as example problems $9$ and $10$ in Appendix A. I used the first of these, when we discussed calculation of the lunar position. There, I determined the true lunar longitude to be $171;42º$ (Toomer had $171;39º$) and the true ecliptic latitude as $-2;07º$ (in agreement with Toomer). I did not continue this example for the apparent positions as Ptolemy had an example already for the parallaxes. However, Toomer finds the apparent ecliptic longitude of $173;01º$ and the apparent ecliptic longitude to be $-2;20º$.
  20. Again, Toomer cites serious errors in the calculations. He derives values of $175;39º$ and $-2;10º$ for the apparent ecliptic longitude and latitude respectively. He has further discussion on potential sources for the error there for those interested.
  21. Toomer: $212;30º$ and $1;01º$ respectively.
  22. Toomer doesn’t offer corrections to the apparent positions. Rather, he gives his results of the calculations for the true position and then cites Neugebauer who performed the calculations for the parallax as an example in his text (p. $117-118$). Neugebauer gives final positions as $215;48º$ for apparent ecliptic longitude and $1;01º$ in ecliptic latitude.
  23. But only $0;51º$ per century if we use Toomer and Neugebauer’s values.