In the last post, we explored a few potential explanations for the distributions of the increment for latitude. In this post, we’ll explore the various explanations for the distribution in longitude.
What Grasshoff is really doing in this section is exploring various scenarios and asking which one best gives the reason for the distribution of increments in longitude. So let’s take a look at the different scenarios.
1. The coordinates are based on Ptolemy’s readings
Despite Grasshoff opening this section stating, “no explanation considers the ecliptical longitudes of Ptolemy’s star catalogue as direct recordings of the measurements”, that’s precisely what this first scenario does.
However, this is quickly dismissed by Grasshoff stating that there is effectively no evidence for it. Rather, the evidence of some stars that show a strong relationship with the Aratus Commentary are evidence against this.
1.1 A considerable proportion of the coordinates in the Almagest were measured by Ptolemy
Grasshoff quotes Vogt who proposed a rounding scenario in which Ptolemy used an astrolabe with increments of $\frac{1}{3}º$ and he estimated the sixths between them. As shown in the previous post, this leads to an undercounting of the estimates between increments on the device and thus neatly explains why the number of stars with increments of $\frac{3}{6} = \frac{1}{2}º$ are lacking. This proposition would then expect that the whole degree increments should be over-represented which is also true.
But contradicting the expectations of this proposition are the stars ending with $\frac{1}{6}º$. As this would require an estimation, the should be less common, yet are more common than the stars ending in $\frac{2}{6}º = \frac{1}{3}º$.
Ultimately, Grasshoff concludes that this explanation does not do a good job of explaining the distribution and states that this implies that there must be “an intermediate step of converting the longitudes to the finally documented data” which imprints itself on the distribution. This is Grasshoff’s next scenario.
1.1.11 A considerable portion of the coordinates in the Almagest were measured by Ptolemy but the longitudes were not directly read off the graduated ring but rather converted to their final form
Grasshoff considers what this conversion process could look like. He describes a process in which the observer aligns the armillary sphere to reference stars using the coordinates of the star from Hipparchus’ time, knowing that he will then need to update the coordinates later to account for precession.
In other words, Ptolemy did make the observations, but when aligning on the reference stars, set their coordinates to the values from the epoch of Hipparchus. Ptolemy would then have longitudes for the stars he observed in Hipparchus’ epoch2 and need to adjust the observations to his epoch with what he incorrectly believed to be the amount of precession ($2;40º$).
So how does this address the issue of distribution of the increments? To explain, I’ll quote from this paper3 by James Evans who gives some insight stating,
In a raw distribution of measured longitudes, one would expect the fraction $0’$ (i.e., a raw whole number of degrees) to occur more often than any other. In fact, in the star catalogue of the Almagest, the fraction $40’$ occurs more often than any other. Thus it appears that Ptolemy added $40’$ (plus a whole number of degrees) to the original longitudes. This is consistent with Ptolemy’s supposed expropriation of Hipparchus’s catalogue by the addition of $2º40’$ to the longitudes. Supporting evidence for Newton’s theory has been offered by Dennis Rawlins, who argues from the absence of a particular kind of periodic error in Ptolemy’s star catalogue that Newton must be correct.
In other words, if we take a standard distribution of the increments and add $n + 0;40$ to them, it would align fairly well with the distributions we see in longitude.
Grasshoff notes that doing this would introduce “a small periodical error into the final coordinates” but does not explain what that is.
Instead, he cites an series of articles which boils down to a published exchange between R. R. Newton and Owen Gingerich between $1979-1982$ with a few others weighing in. Unfortunately, there’s lots of topics raised in these articles that have nothing to do with the above issue thus obscuring it, but the topic first seems to come up in this $1979$ article from Newton. Newton doesn’t do a great job of explaining it there, but points out that the argument was brought to him by Dennis Rawlins who explains it much better in a paper from $1982$. So I’ll explain based on that paper.
First, I should note that Rawlins is arguing against the possibility that the solar model could be the culprit for the $1º$ error in longitude by getting into the nitty-gritty of how the armillary sphere4 works. He asks the reader to imagine the simplest situation in which the star which is to be recorded directly on the vernal equinox.
How would this actually play out?
First, the observer would align the armillary sphere such that the equatorial ring is aligned with the ecliptic. However, to do so, the observer would align based on the position of the sun, setting the dial on this ring to match the calculated coordinates of the sun. But if the solar model was in error, then it would cause the ring to be incorrectly set by a little over the degree by $1;06º$5.
This sets up the following scenario:
This takes a moment to understand, but first recall that the definition of the equinox is the intersection of the ecliptic and equator. Thus, the position for which the armillary sphere is set (the Astrolabe Vernal Equinox here), must be the intersection of the plane for which the armillary sphere’s ecliptic intersects the celestial equator. But if this is off by $1;06º$, then it will actually change the ecliptic longitude as well by $0;29º$.
This would mean that if the observer then tried to measure a star that was on the vernal equinox, they would have an error of $0;29º$ in ecliptic latitude – i.e., they would read the star as having an ecliptic latitude of $+0;29º$ instead of $0;00º$.
Flipping this around, the opposite would happen at the autumnal equinox but the observer would record the ecliptic latitude as $-0;29º$. Obviously intermediate values would happen between these two extremes.
If the hypothesis that the solar model led to incorrect solar positions, causing the observer to incorrectly align the instrument, were true, then we could look for this sinusoidal pattern reaching its maximums at the equinoxes as it is a fairly substantial swing of nearly a full degree.
Rawlins examines the data and finds no such periodicity thus, in his opinion, discredits the possibility that an observer in Ptolemy’s time could have misaligned the instrument.
So that’s the argument6.
Oddly, Grasshoff dismisses Rawlins’ claims, stating that the periodic variation would be too small to be tested. No support is offered for this claim7.
Grasshoff then proposes a second iteration of this situation but I can’t particularly tell what the functional difference is.
Ultimately, Grasshoff dismisses both possibilities because they cannot account for the fact that some stars clearly have common origin as shown by the Aratus Commentary. I’m not sure why Grasshoff thinks that matters since we’re in the subset of scenarios in which Ptolemy didn’t observe all the stars. Thus, there is no contradiction.
Moving on, Grasshoff next considers the scenarios in which the coordinates in the Almagest did originally come from Hipparchus.
His next scenario is as follows:
2.1: The star coordinates were originally measured by Hipparchus. Ptolemy converted the Hipparchan data by adding $2;40º$ to the longitudes and by rounding the original $15’$ fractions to $0’$ (i.e., $15′ + 40’$) and the $45’$ fractions to $20’$ (of $45′ + 40’$).
This method was evidently proposed by Newton. In that hypothesis,
Newton begins with the distribution for the increments in latitude, assuming this to be the normal distribution before transformation, and then puts it through the rounding method he proposed. Indeed, it does return something vaguely similar to the observed distribution for longitude.
Grasshoff, however, immediately pans it, stating it
does not grow out of an historical examination of Greek mathematical and astronomical practice, but was proposed with the sole intention of constructing the best approximation to the frequency distribution of the fractions in longitude. This strategy can prove neither the correctness of the initial hypothesis nor the origin of the star catalogue. It only shows that one set of hypothetical set of assumptions can derive the documented properties of the data8.
Furthermore, Grasshoff notes that there is no basis for the assumption that the distribution in longitude should have mirrored the one in latitude.
Instead, Newton claims that “all evidence” that the $\frac{1}{4}º$ stars were observed with a different instrument. This seems to contradict his own subgroup analysis in which we noted that these stars shared a very similar distribution to the rest of the dataset.
2.2 A large proportion of the star coordinates were originally observed by Hipparchus who also used them for writing the second part of his Commentary on Aratus. Ptolemy transformed the Hipparchan data to ecliptic coordinates referred to the epoch $+137$, using the precession constant of $2;40º$.
This scenario only seems to substantively differ from the previous in that no rounding is implied. Furthermore, it makes explicit the assumption that the Aratus Commentary was derived from a Hipparchan catalog instead of leaving that as an auxiliary hypothesis.
This seems to be the favored scenario of Grasshoff who states that, looking at the previous evidence, “a consistent picture emerges”.
First, he considers the reconstructed Vogt catalog. As we saw in this post, Grasshoff did criticize Vogt’s methods of interpretation. Specifically, the statistical test Vogt used to compare the error distributions was flawed. However, while there would certainly be some scatter introduced by the errors in Vogt’s methods, the general positions of the stars was still likely to be correct and suggests some stars were indeed taken from Hipparchus.
However, I think Grasshoff vastly overstates the conclusions here as he claims it indicates “all stars with an accuracy of $\frac{1}{6}º$ in latitude were originally observed by Hipparchus.”
I think a very simple reading of the graphs in the post in which Grasshoff does his analysis easily disproves this as there are notable outliers where the differences in error were significant. Grasshoff offers no explanation as to why these are discounted.
Next, Grasshoff recapitulates the solar model. He reminds us that, the errors here suggest that, if Ptolemy had used Hipparchus’ catalog, the errors in the solar model in Hipparchus’ time would have been best fit to a time of $137$ BCE which is the same conclusion that Vogt came to independently.
Lastly, Grasshoff cites the case of θ Eridani. Beyond just the significant error of over $3º$ from both sources, Grasshoff notes that Ptolemy describes it as a magnitude $1$ star despite its true magnitude being $3.42$. No explanation is ever given by Ptolemy on how he arrived at the magnitudes, but if we assume his information was taken from Hipparchus, it immediately becomes obvious as Hipparchus described this star as “brilliant9“.
Conclusions
While this section is Grasshoff coming to his conclusion that Ptolemy likely did take the majority of his data from Hipparchus, I think his conclusions here are overstated for the reasons I gave above.
Regardless, I find it rather odd that Grasshoff engaged in his summary at this point because there’s still a major section left to go that has direct implications to this question.
So in the next post, we’ll explore Grasshoff’s own analysis of the Aratus Commentary.
- I’m deviating from Grasshoff’s numbering here as he called this 1.11 which doesn’t make sense and I suspect was a typo.
- This may seem a bit odd, but it does make a certain amount of sense in my mind. Doing so would allow Ptolemy to directly check the accuracy of values for the stars in Hipparchus’ catalog without having to do conversions.
- This is not one that Grasshoff cites but is I stumbled across that I think does a particularly good job of explaining.
- Rawlins refers to the instrument as an astrolabe as this was the term used in period. However, it is more commonly referred to as an armillary sphere today and astrolabe tends to refer to the flat instrument.
- This is the average amount by which Ptolemy’s star catalog is off. I have generally rounded this to $1º$ for simplicity.
- Grasshoff cites several more articles in a frustrating, out of order and confused manner. Doing my best to put things in order, the next is this one by Owen Gingerich which comments heavily on Newton, but I can’t find anywhere that Gingerich actually addresses this issue. Newton then replied to Gingerich and again, I find nowhere he actually addresses the periodic variation although he does reiterate that the distribution of errors can be best explained by a shift of $n + 0;40º$. Gingerich then replies to Newton, and admits as much and goes on to agree that Rawlins’ argument seems to disprove the notion that Ptolemy could have observed since the tell-tale curve is not found.
- This is where Grasshoff includes his muddled list of papers that I mention in the previous footnote. None of them appear to support his argument that Rawlins’ variation is too small to test. Indeed, they state quite the opposite.
- I feel the same argument could be made for much of what Grasshoff has done previously.
- The Aratus Commentary doesn’t use magnitudes, but rather, describes stars using descriptive words such as this or, “glaring,” “bright,” or “dark”.