In our last post we had established that it was impossible for Ptolemy to have stolen all of his data from Hipparchus as indirect evidence of the number of stars that would have been included in Hipparchus’ catalog indicate that Ptolemy’s catalog had around $200$ stars that Hipparchus’ presumptive catalog did not. Furthermore, we cited Dreyer and Fotheringham who both showed that errors in the determination of the position of equinoxes and solstices would have resulted in the $1º$ error at the heart of the accusation against Ptolemy, eliminating the need for Ptolemy to have used Hipparchus’ catalog.
Thus, while it’s not necessary that Ptolemy took all of his data from Hipparchus, the possibility remains that he took some. But to determine that, we’d need more information about Hipparchus’ presumed catalog which is what we’ll explore in this post looking at an important 1925 paper by Vogt.
Recovering the Hipparchan Star Catalog
As discussed previously, the only work from Hipparchus to survive in its entirety is Hipparchi in Arati et Eudoxi Phaenomena Commentariorum, (Hipparchus’ Commentary on the Phenomena of Aratus and Eudoxus)1. In this work Hipparchus is criticizing the work of these previous astronomers. In particular, Aratus and Eudoxus give descriptions of the rising/setting of constellations from Athens. Hipparchus first presents the rising/setting of these constellations as described by Aratus and Eudoxus and contrasts them with the “true” phenomena.
However, in a second part, Hipparchus presents the same work entirely without the context of Aratus and Eudoxus, giving descriptions of the heavens as constellations rise for Rhodes. Notably, he describes what stars rise and set first and last in various constellations and then gives the point on the ecliptic which is culminating at those moments. He also gives the point on the ecliptic culminating for other stars within the constellations.
This gives us a method of determining the celestial coordinates of the stars listed as we can determine their ecliptic longitudes by knowing what point of the ecliptic is culminating when they are on the horizon. And, if Hipparchus drew on his presumed star catalog as the basis for this, then we have uncovered an entry from that catalog. Thus, it should be possible to attempt to reconstruct the Hipparchan catalog by this method and compare the ecliptic longitude for each of these stars to those given by Ptolemy. If each star has a discrepancy of exactly $2;39º$ in its ecliptic longitude then this would suggest that Ptolemy used this data and simply added that constant. Similarly, if Ptolemy simply updated the ecliptic longitude, then all ecliptic latitudes should be identical. However, if there is a range of values, then this suggests that Ptolemy observed independently and his systematic errors just happened to produce the cumulative error of $1º$ in ecliptic longitude.
The first person to successfully apply this strategy and reconstruct a portion of the Hipparchan catalog was Heinrich Vogt in 1925. In this work, he was able to determine the position of $122$ stars without the need for additional hypotheses out of the $374$ stars listed. Instead of comparing stars head-to-head, Vogt compared both Hipparchus and Ptolemy to the true positions from modern astronomical calculations, and compared the number of stars for various ranges of error. Here is the results:
Since the table in in German, here’s a bit of explanation: The first column is the error range. The second is the number of stars with errors from the expected values (calculated using modern astronomy) for Hipparchus. The third for Ptolemy. The fourth is the number of individual stars in that error bracket that had the same error between each catalog.
Vogt also repeated this procedure for the ecliptic longitude:
So how to interpret this?
Well, immediately we see that the number of stars in each bracket are not identical, indicating that Ptolemy couldn’t have simply copied all of the data even though it was apparently available. However, the third column indicates stars that do have the same errors. And if the star has the same error in both ecliptic latitude and longitude, it makes it a prime candidate for having potentially been copied. Ultimately Vogt only concludes that only $5$ stars are similar enough that they were potentially copied: τ Ari, α Car, ν Boo, π Hya, θ Eri.
Criticism of Delambre
In this same publication, Vogt also tackles Delambre’s criticism that Ptolemy cherry-picked the observations to fit the theory, using only six of the $18$ observations available to him. The discussion is a bit confusing in Grasshoff’s book2, but based on my reading, Vogt first notes that Delambre attempted to recalculate the constant of precession using all of Ptolemy’s values3 and by doing so, Delambre demonstrated that Ptolemy could have arrived at a much better value.
However, Vogt finds two issues with his method. The first is that Delambre didn’t truly have the Hipparchan coordinates for all of the stars as Ptolemy didn’t cite them. Rather, Delambre evidently simply subtracted $2;40º$ because he assumed that Ptolemy had lifted Hipparchus’ values and added this. In addition, since the values in Ptolemy’s time were shy by $1º$ Delambre evidently added $1º$ to each of the values from his time. The effect of this was simply to artificially correct all of the values, which Vogt claims boiled down to mathematical circular reasoning:
It is unhappily the case, however, that in the results it is not so much the truth that emerges, but rather the hypothesis which had already been introduce as its very premise. In logic, this is known as petitio principii.
Grasshoff takes issue with Vogt’s criticisms of Delambre. In particular, Delambre did not simply calculate the rate of precession by taking the change in ecliptic longitude over the period, but rather, employed a more complex formula4:
$$p = \frac{(\delta_2 – \delta_1) cos(\frac{\delta_2 + \delta_1}{2})}{n \; sin \epsilon \; cos \beta \; cos (\frac{\lambda_2 + \lambda_1}{2})}$$
where $n$ is the time difference between the periods 5, ε is the obliquity of the ecliptic6, $\delta_1$ is the declination for a given star in Hipparchus’ time and $\delta_2$ in Ptolemy’s. Similarly, $\lambda_1$ is the ecliptic longitude in Hipparchus’ time and $\lambda_2$ in Ptolemys. $\beta$ is the ecliptic latitude7. Here, the only place that Delambre used the ecliptic longitude from Hipparchus’ time ($\lambda_1$) which wasn’t really known, was in the denominator.
Grasshoff explores the propagation of errors for $\lambda_1$. In math terms, this means that he took the partial differential of the above equation with respect to $\lambda_1$, $\frac{\delta p}{\delta \lambda_1}$, and then divided it by $p$ to determine the percent error based on a variance in $\lambda_1$ and shows that, for longitude close to the solstices, even a fairly substantial error in guessing at the Hipparchan latitude, would not result in a large error in the value for precession so long as the stars chosen were near the solstices, as these stars were. As an example, Grasshoff considers the case of η UMa in which the Hipparchan latitude was a full degree off. This would only result in an error of $0.5%$. Thus, even if Delambre were wrong about the Hipparchan longitude, it wouldn’t make a significant difference in the value of precession he calculates.
The more important is the error that would result if there was an error in $\delta$. However, there can be no error here because we have the declinations given directly from Hipparchus’ time. Thus, Delambre’s method is more robust than Vogt gives him credit for.
Regardless, Vogt attempts the calculations himself using the values from his derived Hipparchan star catalog. However, Grasshoff describes his method of calculation as “unintelligible” and that it is “entirely unclear how Vogt calculates his precession constants.” While Vogt manages to come up with an average precession of all $18$ stars that matches Ptolemy, Grasshoff is unable to explain how and instead notes that there was a much simpler way to arrive at the precession constant8 but, had Vogt done so, it would have affirmed what Delambre had said all along: That the selection of the $6$ stars Ptolemy chose is notably lower than a better average and thus do appear chosen to try to conform to theory.
Scale and Gradation of Ptolemy’s Instrument(s)
Next, Vogt takes a look at the stars Dreyer had been interested in that stood out by being recorded in increments of $\frac{1}{4}º$. While Dreyer had argued this was an indication of two different instruments (and likely two different observers), Vogt disagrees giving two reasons.
First he compares the number of these $\frac{1}{4}º$ stars in the Almagest with those from his reconstructed star catalog. He notes that there are $95$ stars with latitudes ending in $\frac{1}{4}º$ and $47$ with latitudes ending in $\frac{3}{4}º$ for a total of $142$ stars that fit this description. Vogt notes that $47$ of these stars are also in the reconstructed Hipparchan catalog. This begs the question of why Ptolemy would have observed them if they were so readily available from Hipparchus?
Second, Vogt argues that Dreyer significantly underestimated the number of stars measured in $\frac{1}{4}º$ increments but were not immediately recognizable as such (i.e., any that were a whole number or were a half since those could be either $\frac{1}{4}º$ increments or $\frac{1}{6}º$).
As noted above, Vogt notes that there are $142$ immediately identifiable $\frac{1}{4}º$ stars. Similarly, there are $441$ stars that are immediately identifiable $\frac{1}{6}º$ stars. Since there are $1,025$ distinct stars in Ptolemy’s catalog that leaves $442$ stars that that could either have been measured by a $\frac{1}{4}º$ instrument or $\frac{1}{6}º$. Vogt goes through some math to conclude that the number of $\frac{1}{4}º$ stars should be somewhere around $315$ stars which is well more than the $170$ that Dreyer had estimated and only serves to further reinforce the first point.
Still, why were there two different precisions listed?
Vogt notes that this problem was not simply limited to the equatorial latitude and longitude and that there there is a variety of $\frac{1}{4}º$ and $\frac{1}{6}º$ measurements for the various declinations given in the Almagest. He personally finds it unlikely that Ptolemy used two different instruments and instead seeks an explanation that places both scales on a single instrument.
As we have seen, Ptolemy claims to have used and armillary sphere (which he calls an astrolabe) to have observed the stars. Vogt considers what size it could have been in order to make gradations of both of the scales that would be discernable.
For no particular reason, he starts with one that has a radius of one cubit. However, that begs the question of what definition of cubit as the Egyptian cubit was $0.525 m$ ($1.72ft$) while the Roman cubit was $0.4436m$ ($1.46ft$). For the former, each whole degree on the ecliptic latitude and longitude rings would be separated by $4.28mm$ while for the latter it would be $3.61mm$9. If we divide that into fourths and sixths, that would imply that the $\frac{1}{4}º$ and $\frac{1}{6}º$ marks would each be approximately $1mm$ and $\frac{2}{3}mm$ apart respectively. Quite small indeed. But is it plausible?
We do know a bit more about one of Ptolemy’s other reported instruments: the Parallactic Instrument. Ptolemy gives a minimum length of $4$ cubits for the measuring rod on this instruments. This is divided into $60$ parts. Because his observations include measurements that had an angle of $\frac{7}{12}º$, this indicates that the instrument he used had each degree divided into twelfths in which case each $\frac{1}{12}º$ would be separated by $2.5 – 2.9mm$, notably larger than the separations Vogt proposes. Thus, if this is the limit of how fine the gradations could be made, then the armillary sphere Vogt proposes could be no more finely divided than $\frac{1}{2}º$.
Vogt digs into this a bit deeper taking all the stars that are in any increment of $\frac{1}{6}º$. He then counts how many there are for each sixth and discovers the number is notably uneven. In total, there are $1,017$ stars that are given in increments of $\frac{1}{6}º$. If they were distributed evenly, there would then be $\approx 170$ for each measurement. But they’re not. The only sixth that comes close to this is $\frac{1}{6}º$ whereas $\frac{3}{6}º$ and $\frac{5}{6}º$ are notably lower and $\frac{2}{6}º$ and $\frac{4}{6}º$ notably higher. Vogt takes this as an indication that the instrument didn’t have gradations of sixths, but of thirds, allowing the observer to infer sixths if the star appeared between two, but supposing there might be a bias for an observer to place it on the gradation which would be the $\frac{1}{3}º = \frac{2}{6}º$ and $\frac{2}{3}º = \frac{4}{6}º$ marks explaining why they are higher.
This is quite familiar to me as the divisions on my quadrant are in $\frac{1}{10}º$ gradations. And in the overwhelming number of cases, I read the measurement as on those gradations. It’s only in very rare cases that I’ll suppose that the reading is between two leading to a reporting of the measurement to a recorded precision of $\frac{1}{20}º$.
So what of the $\frac{1}{4}º$ increments?
As previously noted, these were quite rare on the ecliptic longitude scale, but common on the ecliptic latitude. Was this potentially the result of the longitude being divided into sixths and the latitude being in fourths? Vogt concludes that this doesn’t really make any sense since the rings for both ecliptic latitude and longitude are the same size, so why not use the same gradations? Furthermore, there are still many observations that are in $\frac{1}{6}º$ increments. Vogt has no explanation for this and ultimately cannot discount the possibility of two instruments – or two observers.
The Epoch of Hipparchan Observations
In the last post, I noted that the sole piece of direct evidence there was a Hipparchan star catalog was that Pliny tells us there was. However, he indicates that it was created after Hipparchus’ Commentary on Aratus. If true, then this calls into question the legitimacy of using the Commentary on Aratus to attempt to infer the catalog since the catalog would not have been written yet. So how to rectify this discrepancy?
Vogt first assumes that there was a star catalog prior to the Commentary on Aratus and then tries to determine an epoch for this – an approximate date when the catalog would have been compiled. This is done by comparing the positions in the hypothetical catalog with calculated positions using modern methods and determining when the best fit was. Vogt finds it to be approximately $150$ BCE $\pm3.77$ years using only the $77$ stars in the zodiac near the celestial equator. He also does this using the declinations given in the Commentary and comes up with an epoch of $156$ BCE $\pm 10$ years – consistent with the previous test.
He then tries to do the same with the positions recorded in the Almagest and finds an epoch of $130$ BCE $\pm 6.46$ years.
This discrepancy in dates paints a possible solution to the Paradox implied by Pliny: Hipparchus had a star catalog, but updated it after writing the Commentary. It is possible that this was done, as Pliny stated, because Hipparchus observed a “new star” and wanted to improve the measurements but there is another possibility: It could have been during this time that Hipparchus discovered the precession of the equinoxes and revised his catalog – converting it from equatorial coordinates (RA/Dec) to ecliptic (lat/long) as managing precession is far easier in the latter system.
Conclusion
Vogt’s work in a major step towards and exoneration of Ptolemy. While we had previously established that it was likely impossible for Ptolemy to have lifted all of his catalog from Hipparchus, Vogt indicates that only a very minimal number of stars share a common error and thus, only $5$ stars are appear to have been copied.
There’s still lots more to go in the scholarly history of the catalog so we’ll continue exploring in the next post!
- If you’d like to see a $15^{th}$ century copy of this text, it is viewable here. It is part of a larger work that appears to be a copy of Aratus’ Phaenomena bundled with commentary on it.
- I’ve been slowly working on translating Vogt’s work from the German, but it’s taking too long and I don’t want to get lost in that rabbit hole.
- By which I mean the $18$ stars Ptolemy’s discussion of the poles of precession.
- This equation comes from differentiation of the transformation between ecliptic coordinates and equatorial:
$$sin \delta = cos \epsilon \; sin \beta + sin \epsilon \; cos \beta \; sin \lambda$$
- A period of $265$ years between Hipparchus and Ptolemy.
- $23.86º$ according to Hipparchus and Ptolemy.
- Which did not change between the time of Hipparchus and Ptolemy, hence why it is held as a constant and does not have a Δ symbol.
- Which I won’t get into here.
- This calculation also takes into consideration the fact that each ring in the sphere must be smaller than the previous.