Claudius Ptolemy's great work, the Almagest, contains in books 7 and 8 a catalog of over a thousand stars, each given with its position in ecliptic latitude and ecliptic longitude for his epoch of 137 AD.
As one might expect from such an early work, these positions contain sizable errors. Most obviously, there is a systematic error in the longitudes: the cataloged values average 1.2 degrees too low for Ptolemyís epoch.
The Danish astronomer Tycho Brahe, who compiled his own star catalog, noted these errors. He suggested that Hipparchos had really observed the catalog in the second century BC; and that Ptolemy had plagiarized the catalog of Hipparchos, by precessing all of the longitudes by 2 2/3 degrees while leaving the latitudes unchanged. This would explain the 1-degree longitude error, because Ptolemy had adopted a precessional constant of one degree per century, which is too low. The actual precession between Hipparchosí time and Ptolemy's was closer to 3 2/3 degrees; so the one-degree difference nicely accounts for the systematic longitude error.
In the 19th century, Pierre-Simon Laplace suggested that the 1-degree longitude error could be explained in a less sinister way. Ptolemy had also adopted his theory of solar motion from Hipparchos. Although the errors in the solar theory were small in Hipparchosí day, by Ptolemy's day these solar position errors had risen to about a degree, in the same direction as the longitude errors of the catalog. Since the longitudes of the stars are ultimately determined by reference to the longitude of the Sun, Laplace proposed to acquit Ptolemy of theft on this basis.
Even more recently, James Evans has suggested that Ptolemy used the fundamental reference stars of Hipparchos, unprecessed, as his own reference points, then observed all the other stars himself, and later added 2 2/3 degrees to all 1025 stars in the catalog.
We now have three hypotheses that explain the one-degree longitude error. Fortunately, there is a way to test them, proposed by Dennis Rawlins in 1982.
When measuring a star's position with the astrolabe, the observer's first task is to rotate the astrolabe around the equatorial axis until the ecliptic ring is directly aligned with the ecliptic of the sky. It is this critical first step which, it is alleged, Ptolemy botched.
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| Astrolabe Misrotation |
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Here, the inner red globe is the ecliptical reference frame of the astrolabe, and the outer blue globe is the reference frame of the sky. On the left, the astrolabe has been rotated correctly; on the right, the it has been rotated incorrectly.
As the astrolabe rotates around the equatorial axis, the longitudes diverge, as we expect. But note that the latitudes diverge too: the entire reference frame wobbles out of kilter. This error in latitudes is positive at the spring equinox, negative at the autumn equinox, and near zero at the solstices. In other words, misrotation of the astrolabe will cause a wave of errors to sweep through the ecliptic latitudes of the stars.
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| Expected Latitude Errors by Longitude |
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This graph shows the expected latitude error wave under the various hypotheses.
Note the phase of the expected error wave: itís a cosine wave, with a peak at zero and a valley at 180 degrees. This phase is required by the geometry of the situation. Now there is another possible source of periodic latitude errors, and that is if the observer did not know the correct obliquity of the ecliptic. That would also show up as an error wave in the latitudes; but the obliquity error is a sine wave, 90 degrees out of phase with the rotation error.
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| Actual Latitude Errors by Longitude |
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When we look at the actual latitude errors, it is clear that this wave is mostly an obliquity error and that very little is a cosine wave caused by astrolabe misrotation. When we fit a sinusoid curve through this data, we see that the rotation error component has an amplitude of only 8 arcminutes, less than a third of the amplitude that the Laplace hypothesis requires, and about a twelfth the amplitude that the Evans hypothesis requires.
This is critical to the entire case. The key point is this:
The latitude errors show that observer of the catalog must have aligned his astrolabe with the ecliptic correctly to within a third of a degree in longitude. That means that the observer of the catalog knew the longitudes of his fundamental stars correctly to within a third of a degree; and by extension, knew the longitude of the Sun correctly to within a third of a degree. Ptolemy did not know these data nearly that well; therefore Ptolemy cannot have been the catalog's observer.
So the 1-degree longitude error cannot be due to Ptolemy's error in the solar theory. But the maximum error in the solar theory at the time of Hipparchos was about a third of a degree.
Note also that the overall mean error in the latitudes is near zero. This cannot happen unless the observer has aligned his astrolabe correctly with the celestial pole (to within two minutes of arc for these data). This implies that the observer knew his geographic latitude correctly to within two arcminutes. Ptolemyís latitude error was 14 arcminutes.
About ten years ago, Evans suggested that Rawlinsí analysis was not conclusive, because of the possibility of other causes of periodic errors in the catalog data. In particular, he pointed to the possibility of errors caused by eccentricity within the astrolabe itself, and other kinds of astrolabe mis-manufacture and misalignment.
However, the astrolabe is a machine with internal rotating parts. Any sizable eccentricity in such a machine will not allow the machine to rotate at all; and this is especially true if the machine is manufactured to tight tolerances, as seems likely for a scientific instrument. Therefore, any eccentricities in the astrolabe were probably small.
Now let's look at the southern limit of the catalog. From anywhere in the Northern Hemisphere, there is a zone of the sky that is permanently below the southern horizon. Obviously the size of this zone depends on your latitude. Less obviously, the position of this zone on the celestial sphere depends on your epoch, because the zone is centered on the South Celestial Pole, and the position of the pole moves as precession advances.
Rawlins analyzed the bright stars of the southern sky, both in the catalog, and missing from the it. Using a very transparent model of atmospheric extinction, he found that the catalog's latitude fit the latitude of Hipparchos, and also that the catalogís epoch fit the epoch of Hipparchos.
But Evans criticized this analysis on the basis of Rawlins' model for atmospheric extinction. Using the more conventional King model for extinction, Evans used Ptolemy's reported magnitudes for various southern stars to show that the catalog fit Ptolemy's latitude well, and was probably too far south for Hipparchos. The question therefore is one of atmospheric extinction, and particularly its most variable component, the amount of dust and other aerosol in ancient skies.
Perhaps the best measure of ancient atmospheric transparency can be found in Ptolemy's own reports of what he calls ëphasesí of the planets, as explained in Book 13 of the Almagest. Here Ptolemy describes the limits of visibility of the planets under the most difficult conditions possible: right on the horizon, and during twilight. Ptolemy says that the critical parameter for visibility is the angle of the Sun below the horizon; and he gives this angle for each of the planets near solar conjunction. In his shorter works Planetary Hypotheses and Phaseis, Ptolemy gives more of these values, although they differ somewhat from those of the Almagest.
This gives us a lot of data to work with. We plot these values here; the x-values are given by Ptolemy, and the y-values, the magnitudes, are computed modernly.
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| Limits of visibility for planets near solar conjunction |
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On the same graph, we plot the modern computed limits of visibility for the same conditions: on the horizon, at twilight. The lowest thick lines are computed with a standard dust atmosphere at two different elongations. Above these we have medium weight lines computed the same way, except for 50% of standard dust. And the narrow lines at the top are computed for a null-dust atmosphere. As you can see, the ancient sky looks a lot like a very transparent null-dust sky.
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| Limits of visibility for planets near opposition |
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We also compute the values for the outer planets near opposition. Once again, the ancient sky looks a lot like a null-dust sky. Therefore Rawlins' analysis, which was based on unusually transparent skies, and which supported both the latitude and the epoch of Hipparchos for the catalogís origin, remains firm.
Here are the southern stars found and not found in the catalog, along with the visibility limit computed for null dust. It is clear that the actual southern limit of the catalog closely follows the theoretical limit for Hipparchos, especially for the brighter stars. By contrast, there are many stars, even as bright as second magnitude, that Ptolemy missed if he was the catalog's observer.
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| Southern Limit of the Catalog: Theoretical vs. Actual |
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Our final main point concerns fractional endings in the catalog, an argument first made by R.R. Newton in 1977. These fractional endings are not uniform. If we examine the fractional endings in the latitudes, we see that the cataloger used fractions of 1/2, 1/3, 1/4, and 1/6 (and combinations thereof.) Because these fractions are not evenly spaced, we would expect more integers and halves than other fractions.
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| Latitude Fractions in the Catalog |
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Here we put each latitude in the northern part of the catalog into a separate bucket according to its fractional ending. Note especially that there are more integers than we expect from chance. This is because the astrolabe was graduated in integers, and the eye tends to gravitate toward the graduation marks on the scale.
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| Longitude Fractions in the Catalog |
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But the longitudes are a completely different story. Here we find that the 1/4 and 3/4 fractions are almost entirely missing. Further, the integer fractional bucket is underfilled from what we would expect based on the latitudes, and the 2/3 degree bucket is overfilled.
This distribution of the longitude fractions is clearly and obviously non-random. The longitudes preserved in the Almagest simply cannot be the same longitudes actually observed at the astrolabe; they have been fiddled with after the fact. Newton realized that an upward shift of 40 arcminutes could explain the observed distribution of longitudes.
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| Longitude Fractions: Newton's Hypothesis |
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Under this scenario, the longitudes are originally observed with about the same distribution of fractional endings as the latitudes, including the 1/4 and 3/4 endings. Then Ptolemy adds 40 arcminutes, or any integer plus 40 arcminutes, to the observed longitudes. The 1/4 bucket would then become 11/12, a fraction that does not occur elsewhere in the catalog, so to avoid this dead-giveaway Ptolemy rounds these numbers up to integers. Similarly, the 3/4 bucket would become 5/12, which is also not present in the catalog, so Ptolemy rounds these fractions down to 1/3. If Ptolemy followed this procedure, the observed distribution of longitude fractions can be recreated quite nicely, including the missing 1/4 and 3/4 fractional buckets.
Now if we examine the 14th century star catalog of Ulug Beg, we find that the integer bucket is also underfilled. This also shows that the epoch of his catalog is different than the epoch of observation. But in that case, the overfull bucket is 55í, which implies a difference of only six years in these epochs. Therefore, it is clear that Ulug Beg observed the catalog himself, and adjusted his own longitudes for six years of precession that occurred between his adopted epoch and his actual observation.
In Ptolemy's case, the overfull bucket is 40í. If Ptolemy observed the catalog himself, and adjusted his own observations for precession, as did Ulug Beg, then that would mean that the catalog was observed in 170 AD -- long after the Almagest was written, and probably after Ptolemy was dead. Or going the other way, it might mean that the catalog was observed in 70 AD, before Ptolemy was born; or, subtracting whole integers from the longitudes, the epoch of observation could also be 29 BC; or 129 BC, the epoch of Hipparchos. Thus while the fact that the overfull bucket is non-integer does not in itself convict Ptolemy of fraud, the amount of the shift not only convicts him, it point directly to Hipparchos as the catalog's observer.
Here then is a summary of the evidence we've seen. Today or in the future, you may hear alternate explanations for some of this evidence. If you do, I ask you to bear in mind Occam's Razor: what is the simplest theory that explains all of this evidence? The simplest theory is this: Hipparchos observed the catalog, and Ptolemy stole it.
| Summary of Evidence |
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