So far in Grasshoff’s analysis, he’s come to the conclusion that the distribution of errors (as well as some of the specific ones) point to Ptolemy having used Hipparchus’ data. The only possibility to rescue Ptolemy as an observer is if there is some sort of underlying bias in both of their observations that led to a similar set of errors.
The primary suspect for this would be errors in the solar theory because Ptolemy’s model was essentially the same as that of Hipparchus’1. So, at long last, Grasshoff is ready to explore the errors in the solar theory.
One of the first statements that I quite like is that Grasshoff gives a nod towards the complexity of observation:
Every single measurement of a stellar position is an evaluation process far more complex that just reading off numbers from an instrument: The coordinate system has to be defined, the instrument must be properly adjusted, the mathematical transformations suitable, and the auxiliary astronomical hypotheses should be correct. Every one of these steps in the construction of the final catalogue data can introduce a deviation from the accurate value that, retrospectively, is summed into the total error.
As someone trying to compile a star catalog using period methods, this is a good reminder of the challenge I face.
The reason Grasshoff bring it up, however, is to remind us that the previously discussed cluster analysis was based in the notion that the errors analyzed was a normal distribution, or perhaps multiple normal distributions. However, if the error is not a normal distribution, then the cluster analysis isn’t a suitable test of much of anything. Thus, we need to figure out what sort of distribution errors in the solar theory might play.
To explore this, Grasshoff first divides the zodiac into $30$3:
Here, we can easily see that the error is highly dependent on ecliptic longitude and Grasshoff concludes that this is due to a “variable systematic error in the measurement method.” The question is whether this pattern of error matches the error in the stellar model.
The first error of the solar theory to be recognized, the error of the mean position of the sun, is due to inaccuracies in determining the solar year. The solar theory which Ptolemy took over from Hipparchus estimates the length of the tropical year as $365 \frac{1}{4} – \frac{1}{300}$ days, which is longer than the accurate $365 \frac{1}{4} – \frac{1}{128}$ days. The theoretical motion of the mean sun is therefore too slow.
The cumulative error of this is that, if you apply the two different rates from the time of Hipparchus to Ptolemy, right about $1º$
This is the error which, according to one possible historical interpretation, caused the systematic error of the Ptolemaic star coordinates.
However, while this does align well with the mean error of $1º$, this does not address the distribution of errors we’ve been discussing, and why they’re so similar between the reconstructed Hipparchan catalog and the Ptolemaic one.
For that, we need to look at the errors in the eccentric motion which would be imprinted on the distribution of errors. And for that discussion, Grasshoff turns to the $1967$ PhD thesis of John Britton which, in part, explored the error caused by the fact that the earth orbits the sun, not in the oval shape described the the solar model, but by an ellipse.
Doing so reveals a pattern of errors very similar in amplitude and period to the graph above, but the phase4 doesn’t line up – It is off by about half of the phase as shown in the figure below5:
For Grasshoff, the similarity in amplitude6 and period of these is enough to state that the solar theory was obviously used to derive stellar positions7, but the question then becomes why is there a phase shift?
Grasshoff explains that this was likely a result of the technique in measurement. Obviously, Ptolemy could not observe stars with the same ecliptic longitude as the sun as they would be up during the day and thus, not visible. Rather, he finds it more likely that Ptolemy attempted to determine the positions of stars from the sun by seeing which bright stars were at opposition – i.e., ones that were just rising at sun set and thus, would be culminating at midnight. This means that the error in the solar position would be imprinted by stars $180º$ away from the sun.
If that were the case and the phase of the stellar errors adjusted by that amount, the graph lines up much better:
It’s still not perfect, but it’s not a bad fit.
From this, Grasshoff concludes a few things. First, that whoever did the original observing likely only used a small number of techniques – i.e., they consistently observed stars the same time after sunset consistently since, if they didn’t, the distribution would not mirror the error in the solar model so well.
Second, the best fit is a phase shift of $250º$ which is more akin to stars that were rising about four hours after sunset. Why the observer would do this isn’t clear, but the data does suggest it nonetheless.
Third, the mean of the errors of the solar theory ($2.55º$) is slightly different than the offset we would expect from the time between Hipparchus and Ptolemy if he used the incorrect constant of precession ($2.67º$). However, the value of error in the solar theory was based on Hipparchus having observed in $128$ BCE which isn’t firmly established. Rather, that time period was simply the best fit for the data from the reconstructed catalog that minimized the errors in Grasshoff’s analysis.
If we instead consider what epoch best matches the mean error, it’s roughly year $137$ BCE which better matches some other potential dates8.
So what does all of this mean in terms of whether or not Ptolemy stole the data from Hipparchus?
In short, it offers absolution for Ptolemy and offers an explanation for how the distribution of errors would plausibly match those from a presumptive Hipparchan catalog. As Grasshoff puts it:
The very good correspondence of the error in the solar theory … shows that Ptolemy could very well have compared the adjusted position of the Hipparchan star register with actual observations made by him without obtaining significant differences. In the end, the false positions of the stars must have been confirmed by the observations.
The plausibility that Ptolemy could have observed independently and arrived at similar incorrect results as he would have arrived at from an incorrect adjustment of the Hipparchan catalog doesn’t mean that he did.
To further investigate, Grasshoff promises to return to the “distribution of the fractions of the degrees”, which we will explore in the next post.
- You’ll recall that the basic parameters on which the model depends were the same as Hipparchus’ and, while Ptolemy checked them, he did not change them. Thus, Ptolemy’s solar model is likely identical to Hipparchus’.
- In the text, Grasshoff states that he divided the zodiac into $20$ segments. However, this is clearly contradicted by the fact that the graph we’re about to present has $15$ segments in half of the ecliptic. Thus, I suspect the $20$ was a typo.[\note] segments and then looks at the average error for each as well as the standard deviations in error about that mean for each of the stars in that bin. Here’s the graph for the first half of the zodiac2This only included stars within $\pm 20º$ of the ecliptic in latitude and the error is relative to an epoch of $128$ BCE (i.e., the time of Hipparchus).
- I.e., the position of the peaks and troughs.
- Grasshoff does not make it clear which is which author.
- Which he notes is slightly smaller than the actual amplitude for the stars.
- Something that was never really in question since Ptolemy told us as much.
- Grasshoff mentions Vogt‘s analysis as one of them. However, my previous post recorded that Vogt had a dating of $130$ BCE $\pm6.46$ years, which is actually closer to the $128$ BCE. So I’m not sure where Grasshoff is getting at here.