Ancient Plagiarism? An Analysis of Claudius Ptolemy's Star Catalog

September, 2023

Claudius Ptolemy (100-170 CE) was a Greek scientist who lived in Roman Egypt. The Almagest (his work on astronomy) was the definitive treatise on astronomy from its completion in the 2nd century CE until Kepler about 1500 years later, making it likely the longest-lived detailed scientific model in history. The importance of the Almagest makes the accusations of plagiarism from over a thousand years later all the more surprising.

In Book VII of the Almagest, Ptolemy provides a star catalog listing the positions and magnitudes of over 1000 stars. Below is a visualization of the catalog in the Almagest:

The visualization is oval shaped because objects in the sky effectively lie on a sphere around the Earth, and so in order to visualize them on a flat screen they need to be projected (just like a map of the Earth).

As a disclaimer, there are several potential sources of error in this data. There are no surviving "first editions" of the Almagest, and what we have today are copies of copies of copies. These ancient copies contain numerous mistakes and often disagree with each other. For example, Λ (which represented 30°) was sometimes miscopied as Δ (which represented 4°), which was sometimes miscopied as Α (which represented 1°), and so on (Grasshoff 94).

There is also some difficulty in matching Ptolemy's descriptions with modern stars. For example, Ptolemy's description of what is probably Polaris is "The star on the end of the tail [of Ursa Minor]", along with its position in latitude and longitude relative to the ecliptic, and its magnitude (a measure of brightness). Researchers like Gerald Toomer (the primary source for the data used in this project) have matched these descriptions to star in modern catalogs, but there are always a handful of ambiguous stars.

The star chart below compares Ptolemy's data with modern calculations. Each star is represented by a line from Ptolemy's observed position to the computed true position, with color representing the distance between observed and computed positions:

Most of the lines are short, indicating that Ptolemy's catalog roughly matches the modern calculations. However, from the colors you can see that most of the positions given in Ptolemy's catalog are off by about a degree. As with any real-world data, some inaccuracy is expected, but these errors in Ptolemy's star catalog have fueled accusations of plagiarism for the last few hundred years.

To demonstrate why, the scatter plot below shows the differences between the latitudes and longitudes given in the Almagest, and the true latitudes and longitudes as computed with modern ephemerides:

This is unexpected! Usually errors cluster around 0, as can be seen with the errors in latitude. However, the median error of longitudes in Ptolemy's data is closer to 1. This means that the errors in the longitudes are not just noise, and they likely have a systematic cause.

"Plagiarism"?

The systematic errors in Ptolemy's star catalog can be explained if he "stole" data from an earlier astronomer.

Before Ptolemy, the dominant figure in Greek astronomy was Hipparchus (190-120 BCE) who lived about 300 years earlier. Though he was generally skeptical of other astronomers, Ptolemy made frequent reference to Hipparchus and openly adopted some of his models (though often with modifications).

Hipparchus is generally credited with discovering the precession of the equinoxes, a component of the Earth's motion which causes the fixed stars to appear to move very slowly, circling the Earth every 26,000 years or so. Precession causes the longitudes of the fixed stars to appear to change by about 1 degree every 72 years. By comparing even more ancient observations with his own, Hipparchus estimated the rate of precession at 1 degree for every 100 years, which is a pretty good estimate of the true value.

This plot shows the positions of stars in the constellation Eridanus in the year 130 BCE (blue) and 2023 CE (red). The stars are all shifted horizontally as a result of precession:

Hipparchus is also credited with creating a star catalog in 129 BCE, but its contents have been (largely) lost. If Ptolemy wanted to pass off Hipparchus's catalog as his own, he would need to correct for precession. In particular, because Hipparchus's star catalog was created 266 years before Ptolemy's, the effects of precession would shifted the fixed stars by about \(266/72 \approx 3.7\) degrees longitude. However, because Ptolemy only had access to Hipparchus's inaccurate estimate of the rate of precession (1 degree per 100 years instead of 1 degree per 72 years), he would have computed that precession shifted the fixed stars by about \(266/100 = 2.66 \approx 2+2/3\) degrees longitude. This means Ptolemy's corrected longitudes would be off by about \(266/72 - 266/100 = 1.034\) degrees, which is close to what is observed in the data!

To summarize, if Ptolemy had corrected Hipparchus's star catalog using his inaccurate value for the rate of precession, the results would be shifted by about 1 degree longitude, which is what we see in the actual data.

Ever since this systematic error was discovered (probably by Tycho Brahe in the 16th century) there has been considerable debate over its origin and interpretation. Given that Ptolemy credits Hipparchus with so many other developments, why would he not also credit him with some of the observations in the star catalog? Why explain the methods of observation in great detail if these methods were not used? Wouldn't any plagiarism be obvious to astronomers from Ptolemy's time who also had access to the original works of Hipparchus? Could there be other causes for the systematic errors (such as inaccuracy in the solar theory)?

These questions have led to controversy for the last several hundred years. I'm going to focus on analysis of the data in the Almagest, and avoid too much discussion of blame, the social context, possible motivations, etc.

Dating Through Proper Motion

There are several components to the apparent movement of stars, including the daily rotation of the Earth and precession of the equinoxes (discussed above). Proper motion is the component caused by the actual movement of stars through space relative to the Sun. The rate and direction of proper motion varies from star to star, and so it can cause stars to move relative to one another. This raises the possibility that the catalog in the Almagest could be dated by looking at the relative positions of stars.

For an example of proper motion, this plot shows the positions of stars in the constellation Eridanus in the year 1000 CE (blue) and 2021 CE (red), but corrected for precession. Note how the stars do not quite line up, and that some stars moved more than others (such as the one near the top-left). This movement that varies from star to star is proper motion:

There have been several attempts to date Ptolemy's star catalog by proper motion, but it is unclear to what extent these attempts have been successful. The most cited source is probably this paper by Dambis and Efremov (Dambis) which claims to use the effects of proper motion to date Ptolemy's catalog to time of Hipparcus. However, others (who seem to be cited less) have argued that there are statistical faults in that paper (Duke), such as its central reliance on this dubious linear regression (with \(R^2=0.000009\)):

Worse yet, Dobler claims that Dambis and Efremov made the embarrassing mistake of computing \(x^1/2 = x/2\) instead of \(x^{1/2} = \sqrt{x}\) when deriving the standard error, resulting in a margin of error of about ±50 years when the true margin of error is between ±500 years and ±1000 years, which does now allow for definitive dating between Hipparcus and Ptolemy (whose star catalogs were created 266 years apart).

To understand why dating the star catalog with proper motion is difficult, it is useful to get a sense of scale. This plot shows the distance each cataloged star moved due to proper motion between 129 BCE (the time of Hipparchus) and 137 CE (the time of Ptolemy):

The coordinates given in Ptolemy's star catalog are usually rounded to the nearest 10 arcminutes (a sixth of a degree). Only three stars moved further than that between Hipparchus and Ptolemy, and most stars moved an infinitesimal amount. This means that the effects of proper motion would be largely drowned out by the effects of Ptolemy's rounding, and that isn't even including the approximately 1° standard deviation of random observation error in the longitudes.

I have made several attempts to date the diaries through proper motion. Every method has given different and unreliable results. Proper motion is probably too slight and the noise too large for the star catalog to be dated in this way.

Local Error

Ptolemy gives a detailed description of how he (supposedly) constructed his star catalog. He describes some kind of mechanical device (referred to as an "astrolabe" in most translations) that was used to measure the latitudes and longitudes of the stars. In particular, he says:

We always arranged the first of the above-mentioned astrolabe rings [to sight] one of the bright stars whose position we had previously determined by means of the moon [...] For when these conditions were met, we could readily obtain both coordinates of the required star at the same time by means of its astrolabe ring ... (Toomer 339)

There is a bit of technical detail in that description, but the important claim is that Ptolemy first measured the positions of a few "bright stars" relative to the Moon, and then measured the positions of all other stars relative to the bright stars. This means that errors in the positions of the bright stars would have carried over into positions of all other nearby stars, so neighboring stars should have correlated errors.

The star chart below shows how errors in the star catalog vary across the sky. Each star is colored according to its error in latitude and longitude (after correcting for the ~1 degree systematic error):

Though it isn't exact, there appear to be patches of sky with similar colors. For example, most stars in the upper right are red, while most in the lower middle are blue. This indicates that nearby stars have correlated errors, which is what would be expected if the catalog were observed in the way that Ptolemy claims.

Constellations

There is also some evidence that there is a correlation between the ordering of the star catalog and the errors. This could happen if observations were made in groups based on the bright reference star. The following plot (inspired by one in (Grasshoff 135)) shows error in latitude as a function of the ordering of the star catalog:

This plot shows the same for longitudes (corrected for the ~1° systematic error):

In both plots, errors seem to be correlated in stars that appear in nearby rows of the catalog. Errors also appear to be especially correlated within constellations (colored bands in the plots), though more statistics would be needed to show this conclusively.

It isn't clear whether correlated errors are caused by the proximity of stars in the sky or by their ordering in the catalog because the catalog is ordered so that nearby stars are often listed sequentially. At this point it is useful to look at how the catalog is ordered.

Ordering of the Catalog

This chart shows the ordering of stars in Ptolemy's star catalog:

Stars appear to have been listed from north to south, and from east to west. This can be seen even better in these plots of latitude and longitude as a function of position in the catalog:

There is a constant downward trend in latitude, and a constant upward trend in longitude (though it wraps around).

This can also be seen in this animation that shows the order of stars in the catalog:

It isn't entirely clear whether the ordering or the proximity of stars is causing correlated errors. Looking at some of the star charts above, it appears that proximity is more important than ordering, but more statistics would be needed to confirm this.

Code

A notebook with all code needed to reproduce this project is available here.

Works Cited

Grasshoff, G. (1990). The History of Ptolemy's Star Catalogue (Vol. 14, Ser. Studies in the History of Mathematics and Physical Sciences). Springer-Verlag.

Ptolemy's Almagest (G. J. Toomer, Trans.). (1984). Duckworth.

Wright, E. (n.d.). Seeing Ancient Stars: Visualization of the Almagest Catalog. http://www.etwright.org/astro/almagest.html

Dambis, A. K., & Efremov, Yu. N. (2000). Dating Ptolemy’s Star Catalogue through Proper Motions: The Hipparchan Epoch. Journal for the History of Astronomy, 31(2), 115–134. https://doi.org/10.1177/002182860003100202

Duke, D. W. (2002). Dating the Almagest Star Catalogue Using Proper Motions: A Reconsideration. Journal for the History of Astronomy, 33(1), 45–55. https://doi.org/10.1177/002182860203300106

Dobler, H. R. (2002). The Dating of Ptolemy’s Star Catalogue. Journal for the History of Astronomy, 33(3), 265–277. https://doi.org/10.1177/002182860203300305