In the last post, we followed along as Grasshoff explored the distribution of errors within Ptolemy catalog as compared to those of the reconstructed Hipparchan one. He demonstrated that the errors are highly correlated indicating that Ptolemy likely did use Hipparchan data, unless there was a systematic error common to both.
To build on that concept, Grasshoff begins exploring structures in the catalog by mapping the position of the stars in Ptolemy’s catalog as compared to the true positions as calculated by modern astronomy for the year $137$ CE1. Mapping things in this manner allows for several patterns to become apparent.
Grasshoff produces two maps – one in ecliptic coordinates and the other in equatorial.
Ecliptic Graph
Equatorial Graph
We can see immediately from the ecliptic graph the consistent horizontal shift in ecliptic longitude by about $1º$. This image doesn’t show which end of each of the lines is the position as Ptolemy gave it and which is the true position as calculated using modern methods, but Ptolemy’s positions were averaged $1º$ low in ecliptic longitude. Thus, the true position is the left end of each line and Ptolemy’s is the right.
This same structure is apparent in the equatorial graph, but since the shift is along the ecliptic, is has the characteristic sinusoidal curve when plotted in equatorial coordinates.
What we can also quickly see is that there are certain groups of stars that show a consistent deviation in addition to this horizontal shift. There’s several examples of this. The most notable is the cluster of stars around $\lambda = 252$, $\beta = -18$ which is the constellation of Corona Australis.
Conversely, we can also see some examples where regions of the sky have significantly less error than others such as in Lyra which is around $\lambda = 160$, $\beta = 54$.
This becomes especially obvious when Grasshoff plots the deviation in ecliptic latitude as a function of catalog number
Since the catalog is ordered by constellation, we can clearly see that there are small groups with similar deviations in latitude that correspond to the various constellations.
This can be explained by how Ptolemy states positions were determined. To quote Grasshoff:
According to Ptolemy, the positions of the individual reference stars were determined with the aid of the moon and the position of the sun. Then the positions of the other stars were measured relative to the places of the reference stars. Such a procedure implies that a positional error of the reference star is transferred to all the stars whose positions are measured relative to it.
In short, Ptolemy didn’t determine the position of every star independently. He collected a number of reference stars for which he believed he accurately determined the celestial coordinates. Then he could measure the position of other stars from it. Thus, if his reference star had a notable error, stars measured from it would inherit the error.
We can also notice that along the ecliptic, there doesn’t seem to be much in the way of error in ecliptic latitude. However, further from the ecliptic, there seems to be more stars that show an error in latitude2.
Subgroup Analysis
Grasshoff then returns to the propositions raised by Bjornbo, Dreyer, and Vogt that Ptolemy’s catalog may have been based on Hipparchus’ but with additions by Ptolemy or even potentially other sources as Borjnbo proposed. As discussed previously, Dreyer proposed that perhaps the different precisions in which stars are recorded ($\frac{1}{4}º$ and $\frac{1}{6}º$) were indicative of different observers.
To analyze this Grasshoff looks at the distribution of errors compared to an epoch of Hipparchus’ time for the $\frac{1}{4}º$ stars as compared to those of the entire dataset. He finds that there is little difference in the distribution of errors between sets which does not support the hypothesis of different origins.
Entire Dataset
$\frac{1}{4}º$ Stars
Here we can see that, if both of these distributions were fitted to a normal distribution, they have very similar means and standard deviations.
The only difference that Grasshoff does note is that, on average, the $\frac{1}{4}º$ stars tend to be somewhat fainter, including no first magnitude stars and only one second magnitude. This too speaks against the hypothesis that the $\frac{1}{4}º$ stars have a different origin since it would make little sense for that observer to skip such prominent stars.
But despite the $\frac{1}{4}º$ star idea not panning out, Grasshoff notes that the general principle may still be sound – finding subgroups of stars that have significantly different distributions of errors.
The first group of stars Grasshoff considers is “southern stars”. These are defined as stars that never rise more than $15º$ above the horizon at Rhodes. At this declination, they would be too southern to ever have their positions determined through a planetary or lunar conjunction3.
When analyzing just these stars, they do not show the same distribution as before:
Southern Stars
In this case, there is no obvious peak and the distribution is quite wide having a number of stars that have very little error. Grasshoff interprets this as potentially a group of stars which whose position was determined prior to Ptolemy since their positions more accurately match the Hipparchan era (i.e., they have an error in this diagram near $0$.
Meanwhile other stars have better match the correct positions for Ptolemy’s time – i.e., they have an error around $3;40º$. Among the latter of these include the constellations of Lepus and Eridanus.
Distribution for Lepus and Eridanus
Oddly, Grasshoff declines to consider this possibility much further stating:
For those stars [ones with high northern and southern ecliptic latitudes], though, substantial errors are easily explicable by the difficult measurement conditions. Consequently, the peculiar errors do not necessarily point to the existence of another observer of a different epoch.
In short, Grasshoff decides he is not going to try to determine if the flattened distribution of stars here is really the superposition of two distributions (one from Hipparchus and one from Ptolemy) and simply tosses the observation to the side without further comment.
Cluster Analysis
Instead, Grasshoff returns to the entire dataset and attempts to use a $\chi^2$ test to explore the fits of various superpositions of hypothetical subsets of distributions with varying parameters. Specifically, he iterates through a number of normal distributions between one and four, varying the mean and standard deviation of each and then analyzing the fit of their superpositions4.
I’ll skip the case for only one distribution as there’s not really any interesting discussion to be had there. We already know that the peak therein lies at $\approx 2.4º$ error in ecliptic longitude as compared to the time of Hipparchus. This was the original observation that started this whole discussion and is consistent with the notion that Ptolemy’s data was based on Hipparchus’.
Only slightly more interesting is the best fit when two distributions are used:
In this case, both distributions have nearly an identical peak5. Since both peaks are near $2.4º$ this is still aligned with the hypothesis that Ptolemy used Hipparchus’ data but could potentially indicate that Hipparchus used two different instruments6.
Where things start to get interesting is when three curves are fit:
In this case, we see one distribution near the expected $2.4º$, but smaller ones near $3.89º$ and also one at $1.37º$. We could explain the small group on the right as observations of Ptolemy’s as they’re roughly aligned with the error we should expect of such observations. However, even then it’s rather high as we should expect a peak of $3.4º$.
Meanwhile, the small distribution on the left with the lower mean would potentially be a small number of positions from an astronomer prior to Hipparchus. Grasshoff suggests Timocharis as Ptolemy was obviously familiar with work from him.
If observations from Timocharis were used and Ptolemy likewise applied an incorrect rate of precession, the error for this group should have had a mean of $1.9º$. Again, this doesn’t fit the data particularly well.
Thus, Grasshoff states that, while this configuration does provide a good fit, it doesn’t make any historical sense and thus, the possibility of three observational groups be discarded. In the same manner, Grasshoff also rejects the possibility of four observational groups.
Ultimately, the only one that makes sense is that of one observational epoch, again, consistent with it being from the time of Hipparchus.
But as usual, Grasshoff closes out this chapter stating:
Still, the method of cluster analysis cannot distinguish between two measurement procedures with systematic errors, both finally leading to the same error in the catalogue. Therefore, it remains an open question whether the stars of the Almagest are of Hipparchan origin, or a large section of several hundred stars was in fact observed by other astronomers.
In the opening of the next chapter, Grasshoff immediately notes that this cluster analysis is likely highly flawed as one of the underlying assumptions is that there was no source of error beyond that of normal instrumental error. If we consider other sources of error, it throws enough of a wrench in the works that trying to analyze things with this method is not particularly useful.
In the next chapter, we’ll finally explore the errors in the solar theory and how they play into the potential errors we’ve been discussing and whether or not they can save Ptolemy as an observer.
- This is a deviation from previous analysis in which an epoch of the time of Hipparchus was used.
- See especially the cluster of stars near $\lambda = 195$, $\beta = -50$.
- Grasshoff could also consider northern stars which are too far from the ecliptic to measure in this manner, but dismisses such a group before going further because the proximity to the poles makes accurate measurements are difficult.
- This methodology is dangerously close to p-hacking. Grasshoff doesn’t really have a strong hypothesis here. Iterating through all possible distributions and increasing the number of them is naturally going to produce well fit results. Fortunately, I believe Grasshoff stays on the right side of scientific interpretation here because the mean of each distribution would be indicative of the epoch in which they were observed and if those don’t line up with known astronomers, the results can be discarded. This is ultimately what Grasshoff does so I think the analysis is fair.
- At $2.46º$ for the narrow one and $2.51º$ for the wide one.
- This is my interpretation. Not Grasshoff’s.