Having reviewed Vogt’s work, Grasshoff now turns to dissecting Gundel’s. As a reminder, Gundel discovered a $15^{th}$ century set of hermetic writings that contained a list of stars that was dated to the time of Hipparchus, potentially making it the first direct evidence of a Hipparchan star catalog.
But what Gundel’s analysis lacked is any exploration of how this related to whether or not Ptolemy stole Hipparchus’ data. In this post, we’ll explore Grasshoff’s analysis of that topic. However, before doing so, Grasshoff works to ensure that we have the best possible understanding of the hermetic catalog before comparing the hermetic catalog to Ptolemy’s.
Grasshoff first explores Gundel’s claim that the hermetic catalog contained stars from both the era of Hipparchus as well as an even earlier one. In the previous post in which we discussed Gundel, I noted that this notion was dispelled by Neugebauer. Grasshoff now explains why. Neugebauer finds two faults with Gundel’s reasoning.
The first is that the coordinates in the hermetic catalog are all rounded to whole degrees. This introduces significant error that Gundel did not account for.
Second, Gundel’s procedure was sloppy, simply comparing to the Almagest and assuming any stars with a difference of $\approx 2;40$ was in Hipparchus’ era and stars with a larger difference were from an earlier one. A better analysis would have been to compare against the calculated positions using modern astronomical techniques to determine the best epoch for each.
Grasshoff states that Neugebauer corrected these issues and repeats the calculations to find that the data has a good fit to the year $128$ BCE.
In the above image, Grasshoff has made a histogram of the number of stars in various bins and their discrepancy in degrees from their calculated position for an epoch of $128$ BCE. We can clearly see that there is a decent fit to a normal distribution indicating that the stars were all from a single epoch – not two, as Gundel had proposed.
Grasshoff next dives a bit further into an exploration of the text, noting that the stars weren’t uniquely given in a single coordinate system. Four stars are given in the
number of degrees along the ecliptic which simultaneously rise or culminate with the star.
Sound familiar? It should as this is the same method in which Hipparchus described the positions of stars in the Aratus Commentary. Indeed, the figures given these four stars are identical with those in the Aratus Commentary1, further establishing the connection between this catalog and Hipparchus.
Grasshoff speculates at the reason that these stars were expressed differently than the others. He proposes that the author of the hermetic text (or someone previously) had converted to ecliptic coordinates for most stars, possibly by use of a globe as discussed in the last post, but failed to do so for these four.
Furthermore, Neugebauer explored the coordinates and found that the coordinates of three of the stars matched another hermetic text, albeit rounded. However, the other text included arcminutes and analyzing them with the additional precision gives an even better fit to the time of Hipparchus.
Next, Grasshoff explores the question of dating further as there are different methods by which the authors may have rounded to a whole degree. The first is the method with which we are most familiar: rounding up if the value was more than a half degree and down if less. If this method is assumed, it should not change the position of the mean, and the result would be a date slightly earlier than that of the Aratus Commentary. However, this does not fit well with the terminology used which Grasshoff describes as “unambiguously late-Hipparchan”.
However, there is another method by which this could have been done. This is, perhaps, better described as truncation in which the arcminutes are simply dropped. In effect, this is always rounding down. This method was proposed by Neugebauer in his analysis and gives a better fit to a later Hipparchan period.
Unfortunately, this makes trying to compare this to the Ptolemaic catalog more difficult. It means that, in addition to the errors inherent in the original catalog that we want to compare against the errors of the Almagest, there would be further error introduced by the truncation that is difficult to account for. In addition, there may be error introduced by imprecise conversions on a physical globe instead of mathematical methods.
Still, using the same analysis Grasshoff used for Vogt2 can again be applied. The additional errors above will only increase the scatter on the axis for the hermetic catalog.
In doing so, Grasshoff again finds that there is a strong correlation between the two sources, indicating that there is a common source for both. Since we know that the hermetic catalog is based on Hipparchan data, this indicates that Ptolemy’s data was as well. That is, unless there was a common systemic error to both…
This closes out Grasshoff’s commentary on previous authors. In upcoming posts, we’ll follow along as he dives into the Almagest itself.