Previously, we discussed Vogt’s attempt to reconstruct the Hipparchan catalog by reverse calculating its coordinates from Hipparchus’ Commentary on Aratus. This Commentary was Hipparchus’ response to a poem by Aratus entitled the Phaenomena. Grasshoff ultimately took issue with Vogt’s methods, finding them insufficiently explained given the number of assumptions required to perform the transformation, to put too much stock in. Although not overtly stated, the fact that no one else has attempted to reproduce Vogt’s methods with better explanations, including Grasshoff himself, implies that the uncertainty surrounding such assumptions are considered sufficiently prohibitive that it is not worth attempting to refine Vogt’s methods.
However, Grasshoff isn’t finished with the Aratus Commentary just yet. While the issues with the dates and longitudes may make the Aratus Commentary too messy to use to reverse calculate Hipparchus’ catalog from, Grasshoff instead proposes going the other way around – using Ptolemy’s catalog to calculate same rising/culminating/setting descriptions given in the Aratus Commentary. These can then be compared to those in the Aratus Commentary without needing to worry about recovering Hipparchus’ catalog.
Before beginning, Grasshoff takes a bit of time to explain how Hipparchus’ commentary is laid out. Hipparchus begins with a narrative criticism of Aratus’ work but quickly launches into a clear and formulaic description of various constellations. Grasshoff gives the example of Bootes:
Bootes rises simultaneously with the zodiac from the beginning of Virgo until the $27^{th}$ degree1 of Virgo. While Bootes is rising, the zodiac culminates from the middle of the $27^{th}$ degree of Taurus until to the $27^{th}$ degree of Pegasus. The first star of Bootes to rise is the one in the head2, and the last is the one in the right foot3.
The passage continues, listing each of the stars culminating at the moments of the star and finish of the rising of the constellation, but because Hipparchus has given the point of the zodiac culminating at those times, this is effectively telling us the ecliptic longitude of the degree on the zodiac culminating simultaneously with those stars.
Hipparchus also gives the amount of time taken for each constellation to cross the horizon (i.e., rise).
From this, Grasshoff distills five types of phenomena being described:
- The degree of the ecliptic rising simultaneously with a star.
- The degree of the ecliptic culminating when a particular star rises.
- The degree of the ecliptic setting simultaneously with a star.
- The degree of the ecliptic culminating when a particular star sets.
- The degree of the ecliptic culminating together with a particular star.
In total, the second portion of Hipparchus’ work contains $619$ such descriptions which give information on $292$ stars, most frequently in the fifth of these forms.
Grasshoff notes that there is a bit of variation on this such that, for the fifth type, Hipparchus will occasionally describe the position, not as the co-culmination, but describe how far a star is from culminating at the same time as something else. For example, he might say “$1$ diameter of the moon west of the meridian”. Other examples are less specific as vague statements like “nearly” and “a little” are also evidently employed.
Other issues crop up. For example, Hipparchus’ descriptions may not be sufficient for modern astronomers to determine the star in question, the text is corrupt, or the description is incorrect4. Excluding the stars that have such problems, Grasshoff is left with $271$ stars to consider.
Since Grasshoff is not attempting to recover coordinates for each description, he instead considers the phenomena itself and compares the description of the phenomena to how it should have appeared according to modern astronomical theory.
This is not something we’ve done before, so I’ll pause a moment and explain using an example of a phenomena type $1$ from above.
Consider this situation in which we have the star Debnola just rising and on the horizon5.
Here, Hipparchus would describe the point on the ecliptic6 which is on the horizon at the same time as the star. This is not the same as its ecliptic longitude because the horizon is likely tilted with respect to the ecliptic.
The ecliptic longitude could be found by taking a line from the star that is perpendicular to the ecliptic which I’ve drawn in here going beneath the ground. However, determining that would require all the assumptions Grasshoff found problematic previously.
Thus, Grasshoff instead considers the phenomena itself – the point of the ecliptic co-rising in this case. He then uses modern astronomical theory to determine what it should have been and calculates the error. For example, if Hipparchus said that the degree of the ecliptic rising at the same time as Debnola was $172;00$ and it was really $168;45$7, then there would be an error of $3;15º$ in the phenomena Hipparchus described.
Next, he takes Ptolemy’s star catalog and calculates the same phenomena as Hipparchus described. In this case, what degree of the ecliptic would be rising at the same time as Debnola? Based on that, Grasshoff then compares the calculated Ptolemaic phenomena against the modern astronomical value and again considers the error.
These errors are then plotted against one another. As with before, if Ptolemy’s data was based on Hipparchus’, we should see a diagonal line from bottom left to upper right:
Here, there is clear evidence of just such a correlation as many stars fall on the line $x = y$ and have some rather extreme errors. This would be quite difficult to explain without Ptolemy having based these stars on Hipparchus’ data.
Grasshoff gives a list of some of the largest offenders8: ζ Cas, β Sge, θ Gem*, i Cnc, β Vir, ε Vir, β Sgr, χ Psc, ρ Cet, θ Eri*, α Car*, π Hya*, and α Cen.
However, there’s also some stars that very clearly do not fall on this line such as the one9 that is $8º$ off on Ptolemy’s (x-axis) but shows basically no error from Hipparchus (y-axis). This would be strong evidence that this star is independent of Hipparchus.
But in the end, the majority of stars falls into the mass in the middle which is difficult to draw any conclusions from because random error will dominate here10.
However, there is a large caveat that needs to be made here11. Just as there are assumptions that would be made from converting from the Aratus Commentary to ecliptic coordinates, there would have been assumptions made by Hipparchus in creating the Aratus Commentary in the first place.
In short, the comparison Grasshoff does here is a bit uneven. The Almagest to phenomena conversion is done using modern astronomical techniques and thus free of error. However, we do not know how Hipparchus would have performed his conversion. Thus, the two methods may not produce identical results if there were hidden errors in Hipparchus’.
Grasshoff does take some time to explore how Hipparchus might have done his transformations and settles on the idea that he used a globe (based on coordinates from Hipparchus’ catalog) and simply measured the distances directly instead of performing tedious calculations. His reason for coming to this conclusion is rather weak. It relies entirely on the fact that some pairs of stars are described twice and the values given vary, sometimes by a degree and a half! Grasshoff finds it unlikely that a strict mathematical transformation would result in such discrepancies since they would be done in the same manner each time (and thus should produce identical errors). However, if these phenomena were based on a measuring tool with graduations of a half a degree, such errors would be produced reasonably easy.
Still, based on the large common errors, I agree that Grasshoff makes a strong case that it is virtually impossible that some of the data in Ptolemy’s star catalog did not have an origin in Hipparchus’.
- Grasshoff pauses to explain a bit about how Ptolemy describes the position on the zodiac. Specifically, Hipparchus is using ordinal numbers. In other words $0-1$ is the “first” degree of a constellation. Thus, when Hipparchus describes the the “first” degree, he’s really meaning the portion starting with $0º$ in much the same way the “first” century was from the year $0$ to $99$ and does not, in fact, start with a $1$ at all. To avoid confusion, Grasshoff declares that he will instead use cardinal numbering.
- Star $93$ in Ptolemy’s catalog and $6$ in the constellation of Bootes – β Boo.
- Star $106$ in Ptolemy’s catalog and $19$ in the constellation of Bootes – ζ Boo.
- Grasshoff gives the example of two or more stars jointly mentioned in reference to one numerical value when the stars could not rise, set, or culminate at the same time thereby making the meaning of the text unclear.
- We’re going to pretend atmospheric refraction isn’t an issue here. And in general, I believe this is a good assumption. After all, observers likely weren’t trying to observe stars on the horizon. Instead, they observed them when they were high in the sky. This “on the horizon” position was not given based on observations of stars actually on the horizon, but rather, calculated from knowing the configuration of the heavens from the presumptive catalog.
- The ecliptic is the red line.
- This value is roughly the correct modern value from St. Louis and should not be taken as indicative of something Hipparchus would have said. After all, the precession of the equinoxes has significantly changed the positioning of the ecliptic in the past $2,200$ years.
- Ones I’ve marked with a * are ones that Vogt had also identified based on his analysis.
- It’s not clear which star this is. Perhaps it’s θ Crt which Grasshoff states has “a large Hipparchan error for the setting phenomena of about $10$ and $7$ degrees while the position in the Almagest is rather precise.” There doesn’t seem to be a star on the chart that matches this description but this description would describe this star if reversed.
- I do find it interesting that this clump is drawn out more along the x-axis (Ptolemy) than it is from the y-axis (Hipparchus). I’m honestly a little surprised Grasshoff didn’t explore this as comparing the distribution of errors is something that Grasshoff has done a lot of in this book.
- And if Grasshoff makes it, I have lost it in the noise.