The Historic Discussion of Ptolemy’s Star Catalog

From the time of its writing in the 2nd century CE, Claudius Ptolemy’s Almagest stood at the forefront of mathematical astronomy for nearly 1,500 years. This work included a catalog of 1,025 stars, listing their coordinates (in ecliptic longitude and latitude) and brightnesses. While astronomers within a few centuries realized that the models for the sun, moon, and planets all had issues (which we today recognize as being a result of them being incorrect, geocentric models relying on circles and epicycles instead of a heliocentric model with elliptical orbits), the catalog of stars was generally believed to be correct.

That was, until the end of the 16th century, when the renowned observational astronomer Tycho Brahe realized that there was a fundamental flaw with the catalog: the ecliptic longitudes were low by an average of 1 degree.

What’s more, Brahe proposed an explanation for why. He suggested that Ptolemy had stolen the data from the astronomer Hipparchus some 250 years earlier, and then incorrectly updated the coordinates.

The question of whether this was a cosmic coincidence or the oldest case of scientific plagiarism is a question that historians of astronomy have argued for over 400 years.

The Accusation

To understand why Brahe made this accusation, we must first understand the preferred coordinate system of astronomers at the time: the ecliptic coordinate system.

In general, this system functions exactly like coordinates on earth, with a longitude and latitude. The difference is these are applied to the celestial sphere with the sun’s path (the ecliptic) replacing the equator, and the position of the sun on the vernal equinox replacing Greenwich.

However, a phenomenon known as the precession of the equinoxes means that the position of the vernal equinox drifts slowly west, dragging the entire coordinate system with it. It does this at a rate of 1 degree every 72 years, or roughly 0.014 degrees per year.

In the Almagest, Ptolemy gives the time between Hipparchus and the year for which his star catalog was created as a period of 265 years. In that amount of time, the coordinate system would have drifted by about 3 2/3 degrees.

However, while Ptolemy was aware of precession, he had an incorrect value for its rate. He estimated it at 1 degree per century. So by his estimation, the coordinate system would have only moved 2 2/3 degrees, leaving it short by exactly the amount by which Brahe discovered them to be. To astronomers of the time, this seemed to condemn Ptolemy as the observer for the catalog within the Almagest.

The Hipparchan Catalog

Before we get any further, it is a fair question to ask whether there is any evidence that Hipparchus had a star catalog which Ptolemy could have used. And indeed there is.

Pliny the Elder, circa 1st C. CE, records that Hipparchus did indeed create a star catalog because he had witnessed the birth of a new star, which was thought to be impossible. In response, Hipparchus allegedly set out to create a catalog of stars against which future astronomers could compare to determine how the heavens changed. But if Hipparchus did have a catalog, why did historians have no other record of it?

The answer is the Almagest itself. Ptolemy’s work was so comprehensive that astronomers were no longer inclined to commission the tedious copying of other works. The Almagest was all they needed. Most prior astronomical works have been lost to time.

Indeed, the only surviving work of Hipparchus is a commentary he wrote criticizing two other natural philosophers, Aratus and Eudoxus.

Counting Stars

Towards the end of the 19th C., many historians were invigorated by the discovery and subsequent reinterpretation of medieval and ancient astrological manuscripts.

In 1892, Ernst Maas was put in charge of editing a codex from the 18th C. and discovered it contained a list of constellations. Based on the use of language, the text was attributed to Hipparchus, further supporting the notion that he may have had a star catalog.

Shortly thereafter, two more texts were discovered. The first was by Alessandro Olivieri in 1898 and the second by Franz Boll in 1901. These were also attributed to Hipparchus and, in addition to the names of the constellations, they also contained a count of stars in each constellation.

However, the number of stars given was inconsistent with the number that Ptolemy provides in the Almagest. If these texts summarized a lost Hipparchan catalog, it would indicate that Ptolemy must have observed at least some of the stars in his catalog. But this did not free Ptolemy from the accusation that he stole some, if not most, of the data.

Dreyer’s 1/4 Degree Stars

One of the oddities in Ptolemy’s catalog surrounds the precision to which the coordinates are given. For the majority of stars, these seem to be given in increments of 1/6 of a degree. However, for a subset of stars, we find that they are given to a precision of 1/4 of a degree.

In 1917, the astronomer John Dreyer published a paper highlighting this fact and suggesting that this may be indicative of two instruments or two observers.

Statistical tests were done on this hypothesis by Gerd Grasshoff in his 1990 book The History of Ptolemy’s Star Catalogue. There, he compared the error in the positions of each star as compared to their true positions as calculated by modern astronomical means. He separated the stars with 1/4 and 1/6 degree accuracy and examined the distributions of errors of each.

A notably different distribution would support Dreyer’s hypothesis, but Grasshoff found that they were virtually indistinguishable making the test inconclusive.

Here Comes the Sun

One of the largest challenges to Brahe’s assertion came from the famous French mathematician Pierre-Simon Laplace. Examining Ptolemy’s work, he noticed that there was a minute error in the value Ptolemy used for the length of a year. It was slightly too long – by about 6.5 minutes (a 0.0012% error).

This becomes important because Ptolemy’s solar model is, in fact, Hipparchus’ solar model. While Ptolemy checked Hipparchus’ values, he found them to be accurate and thus made no changes. Thus, the date for which the solar model was calibrated was actually during the epoch of Hipparchus.

If the error in the length of the year is compounded over the 265 years between the two astronomers, it results in the sun’s average position lagging behind where it should have been by roughly 1 degree in the time of Ptolemy.

This matters because to determine the position of a star, historical astronomers would first have to determine the position of an object with known coordinates and measure from there.

For historic astronomers, this meant calculating the position of the sun (which would then be “known”) and then observing the difference between the position of the sun just as it set, and the position of a bright star that was visible in the fading daylight. That star could then be used as a reference point from which to measure other stars. But ultimately, any error in the solar position would be imprinted on the coordinates of the stars.

Further evidence for this explanation is given in by Grasshoff. The Greeks knew that the sun’s apparent motion sped up and slowed down throughout the year. Today, we recognize this as being due to our elliptical orbit but Ptolemy modeled it by placing the sphere on which the sun traveled slightly off of the earth. When the sun was closer, it would appear to move faster and vice-versa when it was further. However, this period motion would also be captured in the solar model and would affect the average error Laplace described by similarly increasing and decreasing the error throughout the year.

Grasshoff examined the error in the stellar positions as a function of ecliptic longitude and found that they had a sinusoidal pattern that matched the one from the solar model. Thus, the error in the solar model was an entirely plausible explanation for the discrepancy.

However, it did not fully absolve Ptolemy because the very same pattern would have been present if Hipparchus had been the original observer, but without the 1 degree error since the solar model was calibrated for his time.

Proper Motions

An interesting method to determine the era the star catalog was created involved the positions of the stars relative to one another. Although ancient astronomers believed the positions of the stars were fixed, they do move very slowly – a phenomenon known as proper motion.

In the late 1980’s the astronomers Efremov and Pavlovskaya attempted to use this to date the star catalog. Their method was to consider the fastest moving stars in the catalog and determine what year the position of these fast moving stars were best described relative to the other stars in the same constellation.

Initially, they had trouble as the year varied wildly depending on which other stars they included to compare against. However, in 2000, Efremov and Dambis published a second paper claiming success and dating the star catalog to the era of Hipparchus.

This conclusion was swiftly rejected by Dennis Duke from the University of Florida who noted that the amount of proper motion over the 265 years between Hipparchus and Ptolemy was below the 1/6 degree precision for nearly every star. Only three stars moved more than 1/6 of a degree in that time period.

While Dambris and Efremov claimed to have sufficiently accounted for the inherent uncertainty from the instrumentation, Duke refuted this stating that, when properly accounted for, this method would not be able to distinguish between the era of Hipparchus and Ptolemy at all.

The Southern Limit

Another attempt to answer the question of authorship first arose in 1982 when Dennis Rawlins considered stars near the southern horizon.

While Ptolemy claims that the catalog should be a complete accounting of all stars brighter than 6th magnitude, this is clearly false as there are numerous stars that meet this criteria but aren’t included. However, the fainter a star is, the more likely it is to be excluded.

Enter atmospheric extinction. This is a phenomenon in which stars are made fainter as their light is scattered depending on how much atmosphere is being looked through. Near the zenith, the stars magnitude is reduced by about a third of a magnitude. But near the horizon, this increases exponentially.

Since Hipparchus is known to observe at a higher latitude than Ptolemy, this means that southern stars would appear closer to the horizon and thus, be dimmed more than for Ptolemy. Rawlings wondered, for which observer did the number of stars of each magnitude after extinction match the predicted amount for that magnitude given the number of stars available?

Based on his estimations, Rawlins concluded it was Hipparchus.

However, in 2001, this claim was challenged by Bradley Schaefer. In this paper, Schaefer contended that Rawlins had vastly underestimated the amount of extinction, stating that Rawlins assumed clearer skies than even the best observatories in the world have today. He also criticized Rawlins selection of stars to use as the basis for the probability of a star of given magnitude being observed.

Attempting to correct these errors, Schaefer recalculated and claimed that Hipparchus being the author was overwhelmingly unlikely. Furthermore, he claims that the magnitudes reported in the Almagest do not fall off in brightness due to extinction the further south the stars were. Thus, he concludes that Ptolemy must have known about and attempted to correct for extinction despite no mention of if being known about, let alone a method for accounting for it, ever mentioned in antiquity.

In 2002, a swift response to Schaefer was released by Keith Pickering. In it, Pickering challenged the notion that, if Schaefer’s model was to be believed, then it would rule out Hipparchus as the author of the only extant work we have of his: the Aratus Commentary. Thus, Schaefer’s results could not pass the sniff test.

Pickering attributes the error mostly to the amount of extinction and argued that the skies in antiquity were far more pristine than Schaefer estimated, likely due to industrialization. After attempting to correct this as well as derive a better function for the probability of stars of a given magnitude being included, Pickering ultimately decides that the parameters are not sufficiently well determined to allow this methodology to be useful since wildly different results can be achieved depending on what values are selected.

Recovering Hipparchus

With all of the tests that astronomers devised unable to answer the question, what would really be needed was a copy of Hipparchus’ lost catalog. Sadly, the only text historians had available was the Aratus Commentary. This text was Hipparchus’ response to a poem by Aratus entitled The Phaenomena, in which Aratus attempted to describe the stars that were rising, setting, or culminating (i.e., at their highest point) at the same time that other stars or points on the ecliptic were also rising, setting, or culminating.

Hipparchus criticized Aratus’ descriptions and, first corrected them for the latitude of Aratus, and then provided pairings for his own longitude of Athens in an appendix. Starting in the early 1900’s, historians considered that, if these pairings had been calculated based on the lost Hipparchan catalog, it might be possible to reverse-calculate the coordinates from the catalog.

In 1925 Heinrich Vogt claimed to have successfully recovered the coordinates of 122 stars without “auxiliary hypotheses.” Vogt first compared the positions for these stars to the ones given by Ptolemy. If the coordinates were the same, with the exception of a 2 2/3 offset, then it would support the hypothesis that Ptolemy used Hipparchus’ data.

Vogt found that, by and large, the coordinates did not match. He went further, by comparing the error against the modern calculated position for both Hipparchus and Ptolemy and analyzed the distribution of errors. Again, he found that they were notably different. Only for a handful of stars did Vogt find that the coordinates for both the Almagest (after accounting for the 1 degree error) and the recovered Hipparchan catalog were off by similar amounts in the same direction. Theta Eridani, for example, was off by more than three degrees from its true position in both catalogs, again indicating that Ptolemy used at least some of Hipparchus’ data.

However, Vogt’s findings were not free from criticism. When Grasshoff reviewed Vogt’s analysis, he questioned many of the underlying assumptions Vogt used. In particular, historians have recognized that different portions of the Aratus Commentary were written at different times during Hipparchus’ life.

While Vogt says he accounted for this, he is unclear on which calculations were for which era. Similarly, the two portions of the text were given for different latitudes. Vogt’s methods only ever mention Aratus’ latitude raising doubt over whether or not he accounted for this at all.

Perhaps worst, some stars are given as part of multiple pairings allowing for their positions to be calculated in multiple ways. If the assumptions and methodology are sound they should match, but often they did not.

The combined result of these potential errors is that the coordinates in Vogt’s reconstructed Hipparchan catalog could have been scrambled, making them not match when indeed they should have. And unfortunately, due to the unresolved questions about when various portions were written, it seemed unlikely this approach could ever be successful.

The Phaenomena of Ptolemy

But if attempts to reconstruct Hipparchus’ catalog from the Aratus Commentary were unlikely to work, what would happen if we went the other way around, recreating the rising/setting/culminating descriptions of the Aratus Commentary using the data from the Almagest? This would remove the questionable assumptions in converting the phenomena described in the Aratus Commentary to standardized coordinates.

Grasshoff performed exactly these calculations and compared the phenomena described in the Aratus Commentary to what should be expected using modern astronomical theory to determine the error in the description for each star. He then did the same for the ones calculated from the Almagest. The error for each could then be plotted against one another. If both had the same, large error, it would be a strong indication of common origin.

When he did so, Grasshoff found that there were, indeed, a number of stars that showed a strong correlation in their errors (including Theta Eridani and three other stars that Vogt had also flagged as suspicious).

However, there were some that had notably different errors. This view supports the hypothesis that Ptolemy’s catalog was only partially based on Hipparchus’.

Aratus Latinus

In the 8th C., this same Aratus Commentary had grown popular among scholars studying astronomy and it was translated into Latin. These texts became known collectively as Aratus Latinus. To supplement the work, they were often compiled with additional astronomical material.

Among the works that became included in some, were passages containing descriptions of the boundaries of three constellations (Ursa Major, Ursa Minor, and Draco) by giving the distances of the bounding stars from the north celestial pole and their ecliptic longitude.

While these portions were originally attributed to Aratus, in the late 19th C., historians realized their descriptions and language better matched those of Hipparchus. It was further realized that the coordinates for Polaris exactly matched the value Hipparchus used according to Ptolemy in another of his works, Geographia.

Thus, historians now accept Hipparchus as the author of these portions, and these passages offered another way to recover the positions of some stars from the Hipparchan catalog.

Codex Climaci Rescriptus

In 2012, biblical scholar Peter Williams, gave his students several pages from a text found at a Greek Orthodox monastery in Egypt. These pages contained Christian texts, but were known to have been scraped clean before the Christian text was added, removing a previous text (a phenomenon known as palimpsest).

The students imaged the pages under numerous colors of light and used computer algorithms to attempt to recover the latent text beginning in 2017. They quickly realized that the texts were astronomical, containing star-origin mythos from Eratosthenes, but also segments of the Aratus Latinus and associated works. Among those associated works was the Hipparchan work, but this time included a new constellation: Corona Borealis.

The researchers examined the coordinates for the stars of all four constellations and compared them with those of Ptolemy. The team found that they did not match. What’s more, they compared the distribution of errors and found that Hipparchus’ catalog was likely more accurate based on the limited selection of stars available.

Conclusions

When reviewing the 400+ year debate over the origin of Ptolemy’s star catalog, the historian Noel Swerdlow remarked, “I am only too happy that I have never written anything on the subject myself that I might wish to defend.”

Here, Swerdlow summarizes the ferocity of the debate on this subject, noting that there are convincing arguments in both directions. The startlingly similar errors for stars in both Ptolemy’s catalog as in the Aratus Commentary seem to indicate that some data was most likely taken. But the independence of other stars in the Aratus Latinus collections suggests that much was not.

Thus, the general consensus among historians seems to be crystalizing around the notion that there is almost certainly a portion of the Almagest’s star catalog that is based on Hipparchus’ data. In response, historians have been asking: How much?

Estimates have ranged wildly. Swerdlow concludes that the amount of overlap should be no more than 10%. Duke confidently declares it to be over 80%, and more likely over 90%.

But while this debate has raged for over four centuries, Swerdlow considers the question of whether it has been worth it:

“The labour and ingenuity that have been spent on this question must far exceed the labour spent by Ptolemy in compiling the catalogue in the first place, however he may have done it, and perhaps it is time for a moratorium on the subject on the ground that life is too short to waste on questions that cannot be answered.”

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A more thorough discussion of each of the arguments here and more is available in Jon’s blog.