Having discussed the authoritative version of the text and the stellar identifications, Grasshoff now turns towards reviewing and checking the work of previous authors, starting with Vogt.
Further Commentary on Vogt/Delambre/Newton
He first returns to the discussion between Vogt and Delambre1. From the post on that topic, you’ll recall that Vogt criticized Delambre’s claim that Ptolemy selected which stars he used for demonstration/calculation of the precession to arrive at the conclusion he wanted, but did so in a way that ultimately made no sense.
Grasshoff explores this further, making a histogram of the precession constants from all $18$ stars that Ptolemy mentions in his discussion of precession:
Here, we can clearly see there are two clusters of values: One around $40$ arc seconds per year, and one in the low $50$’s of arc seconds per year. This is obviously not a normal2 distribution, (although we shouldn’t really expect it to be given the small sample size). Grasshoff’s interpretation of this is that it could indicate that Ptolemy did indeed have a biased selection of data, as the six stars Ptolemy selected as examples all came from the left cluster.
In short, Grasshoff agrees that had Ptolemy done a more thorough examination of the stars he listed, he would have come to a better conclusion. However, taking averages of all the values was a statistical technique not known in Ptolemy’s time.
Regardless, Grasshoff points out that this in no way supports the claim from Newton that Ptolemy fabricated the data. Rather, Grasshoff notes that the stars outside the six Ptolemy used as examples was likely “chosen in order to get a sample of observations distributed over the whole celestial sphere” such that they were used to discuss the poles of precession, but not considered “suitable for the demonstration of the precession constant”.
Commentary on Vogt’s Reconstructed Coordinates
Diving further into Vogt, Grasshoff critiques the methods by which Vogt reconstructed portions of the Hipparchan catalogue. Grasshoff first challenges Vogt’s claim that there were no “supplementary hypotheses” he needed. Specifically,
[t]he geographical latitude of the observation site as well as the time of the observations are, among other things, incorporated in the equations.
The first of these pertains to the dates for which the coordinates were being calculated. This becomes a rather complicated subject because historians have suggested that different parts of the Aratus Commentary were composed at different times. For Vogt’s part, he broke his analysis of the Aratus Commentary into four sections for which he find different dates by finding when the values best fit the data. The divisions and conclusions are as follows:
1) The actual Aratus Commentary: $150 \pm 3.77$ BCE;
2) The $18$ right ascensions of the star clock: $130 \pm 7.45$ BCE;
3) The $18$ declinations taken from the actual Commentary: $156 \pm 10$ BCE; and
4) The $16$ Hipparchan declinations handed down from Ptolemy and Strabo: $130 \pm 6.46$ BCE.
Grasshoff notes that what Vogt means by the first of these is not well defined. Grasshoff states, “[i]t still remains unclear which distinct sections of the Aratus Commentary Vogt relied on in the dating “.
After some analysis, Grasshoff concludes that “the first and second parts [of the Aratus Commentary] were dated together [for $1$ above].” This is an odd choice given the aforementioned differences in latitude.
Furthermore, for many of the stars Vogt calculated, there is more information than necessary, resulting in overdetermination. While having more information than necessary isn’t inherently a bad thing, it should be incorporated carefully to avoid putting too much emphasis on any individual components of the calculations. Evidently, Vogt combines the extra data in manners that aren’t explained. While they may not be wrong, it does make it hard to critique the methodology.
As an additional reference, Grasshoff points to another analysis done by Y. Maeyama, which broke up the Aratus Commentary into more sensible and well-defined chunks. There, Maeyama finds that the first part of the Aratus Commentary was written somewhere between $156$ to $140$ BCE, and the second portion around $130$ BCE.
Aside from these attempts at dating, Grasshoff notes that the longitudes Vogt derived can also be used to determine a date. Doing so, he arrives at a mean epoch of $131$ BCE but with such a large standard deviation that he concludes “the scattering of these data does not allow an exact dating”, thus underscoring the complication of Vogt’s claim that there are no “supplementary hypotheses.”
With regards to the latitude, Vogt again is painfully vague in some of his methodology. As described in the post on Vogt’s work, there were two parts to the Aratus Commentary. The first is a discussion by Hipparchus on Aratus’ work. This portion is based on a latitude of $37º$ N from which Aratus was working. In the second half, in which Hipparchus gave his own values, this was done for his home in Athens of $36º$ N. Vogt gives an example of his conversions, but only mentions the $36º$ latitude. Thus, it is not clear whether Vogt performed the transformations correctly for the ones at $37º$ .
Another point of interpretation that Vogt would have to contend with is conflicting information from Hipparchus’ Aratus commentary in the first place. In particular, “Hipparchus assigns slightly differing values for simultaneous culminations to the same star at different places in the Commentary.”
Grasshoff gives the example of β Cnc, which has three different longitudes given, differing by as much as $1.5º$ from each other. How Vogt treated this is not clear either.
Grasshoff also considers the implications of the differing numbers on the source material itself and concludes,
the Hipparchan values cannot possibly be the product of a rigorous derivation from a fixed star register. One is compelled to the view that Hipparchus had determined the data in the Commentary with the help of a glob, in the parts where he disputes against Aratus and Eudoxus, with no high standards of accuracy.
This throws the reliability of Vogt’s calculations into question as they may well have been based on a faulty assumption: that the figures presented in the Commentary were based on a star catalog in the first place.
However, Grasshoff is of the opinion that, even if the Aratus Commentary was not directly based on the star catalog, the globe in question likely was. Therefore, it’s not completely unrelated, but the additional layer in transmission introduces further errors which should be investigated.
The result of all of this is that there are significant sources of error which Vogt did not properly analyze that need to be conducted before any declaration of independence between Hipparchus and Ptolemy can be made. As Grasshoff puts it,
Assuming that Ptolemy did take over the coordinates from Hipparchus, it still would not be assured that Vogt’s reconstructions would coincide with the values of the Almagest [due to these additional sources of error].
This is somewhat mitigated by Vogt’s analysis that tries to look at the distribution of errors instead of comparing stars head-to-head. However, even then, Grasshoff notes that the analysis Vogt employs is flawed.
As a reminder, Vogt binned the data into errors vs the true positions of the stars computed using modern methods3, and then counted how many stars in each bin were shared by both the Hipparchus and Ptolemy. We saw this in the table below from the post on Vogt’s work:
However, the fuzziness of the data caused by the above sources of error can easily push stars between bins in which case Vogt would no longer have considered them potentially related even if they actually were. Thus,
[n]ot only is Vogt’s classification procedure untrustworthy, even his estimation of the number of independent coordinates in relation to the genetically identical stars is false.
So how can we contend with these sources of error to determine whether or not Ptolemy’s catalog is truly independent?
A New Test – Direct Correlation Analysis
Grasshoff suggests employing a different statistical method: comparing stars head-to-head on a graph. The graphs would have the error in ecliptic longitude for Hipparchus on one axis and for Ptolemy on the other. If there is a correlation between them and the data were directly copied (without error) then it should form a well defined, diagonal line described by $x = y$ – a diagonal line from bottom left to upper right.
While the above sources of error will cause scatter, a statistical test of the points vs the best fit line should give some indication as to the relationship. In doing so, Grasshoff finds that there is a strong correlation between the two sets of errors both in the errors in latitude and longitude.
The above graph presents the errors in longitude for Vogt’s reconstructed Hipparchan catalog on the x-axis vs Ptolemy’s on the y-axis. Immediately we can see several stars that fall along the diagonal line. While there is a clump in the center in which the random error dominates, the extreme outliers on this line were almost certainly copied.
However, there are also immediate counter-examples. The two stars at far left which have errors of $5-6º$ in Hipparchus’ catalog but $1-2º$ in the Almagest would be ones in which it seems unlikely Ptolemy could have copied the data.
Here’s the same graph broken down by errors in latitude:
Here, the outliers that make the line clear are much rarer, but there are still some stars which clearly fall along the line. Similarly, there are also counter-examples again.
Grasshoff further explores this, breaking down this data further looking at it for the $\frac{1}{4}º$ and $\frac{1}{6}º$ degree stars independently. For the $\frac{1}{4}º$ stars, Grasshoff finds no significant correlation in either the latitude or longitude4 as we can see in this graph:
Thus, the notion that the $\frac{1}{4}º$ stars could be ones from the Hipparchan catalog is unlikely.
In contrast, if the $\frac{1}{6}º$ stars are singled out, the correlation becomes more pronounced. Here’s a graph of them in latitude:
And in longitude:
Grasshoff considers this “strong evidence that [Ptolemy’s values] do indeed stem from the same source [as those in the Aratus Commentary].”
We might ask whether, as discussed previously, this could be the result of the systematic error be from the solar model?
Grasshoff dismisses this, stating
[n]or can a periodic error of the solar theory, which could lead to systematic errors especially in the longitudes, explain the similarly large differences.
In short, while the errors in the solar model can explain away the mean error, it cannot account for the correlation of errors evidenced here.
Closing Remarks
Grasshoff demonstrated that Vogt’s conclusion (that Ptolemy’s stars could not possibly have been taken from Hipparchus) is without merit. Rather, a more thorough statistical analysis indicates that Ptolemy likely took at least some of the Hipparchan values, but we’ll need to explore other sources of systematic bias before making any final conclusion.
Grasshoff closes out the chapter stating that, as a result of this,
[i]t remains unclear whether a systematic register of all stars with ecliptical coordinates like those in the Almagest existed. Notwithstanding that it is certain that Hipparchan coordinates were indeed taken over by Ptolemy.
This seemingly contradicts Grasshoff’s earlier pronunciation that we must examine those other sources of systematic error, so we’ll have to see where he goes with this.
In the next post, we’ll turn to Grasshoff’s analysis of the partial star catalog found by Gundel, which Grasshoff previews, stating it
goes beyond Vogt’s reconstructions in showing a correlation with the errors of the Ptolemaic coordinates. Besides the genetic identity of the Hipparchan coordinates with their counterparts in the Almagest, it also demonstrates that the Hipparchan coordinates had already been noted down and possibly, did not have to be converted in the first place by Ptolemy from other Hipparchan values.
- Some of this discussion I’ve gone back and edited into the previous post on Vogt as it seemed to fit better there.
- I.e., bell curve or Gaussian.
- Well, modern to $1925$. I’d be interested to see some analysis of how Vogt’s derived positions align with ones derived using modern proper motion data.
- You’ll undoubtedly notice that the number of stars in this graph is quite small. This is because the $\frac{1}{4}º$ stars are uncommon to start with and the number that overlap with Vogt’s reconstructed catalog even smaller. Thus, the lack of correlation may well be due to lack of data.