Ask a Catalogue What Year It Is
Six star catalogues, from AD 137 to 1690, still carry the date they were made. Getting it out of them is another matter. One way of asking answers to the year and should not be believed for a moment. The other is honest and very nearly blind. Five of the six catalogues have dates on the public record, which is the only reason we can tell which is which.
A star catalogue is a list of positions, and positions go stale. The equinox drifts, so every longitude in an old catalogue is wrong by an amount that grows with age. Stars also move on their own, each in its own direction, at its own speed. Both are clocks. Both are running inside the numbers Ptolemy wrote down.
The trouble is that the loud clock lies. Precession shifts every longitude in a catalogue by almost exactly the same amount, about 1.4 degrees per century, so a thousand stars all vote the same way and the fit comes out to within a year. But a single global shift in longitude is also precisely what an astronomer produces by misjudging where the equinox is, and precisely what a compiler produces by reducing his stars to a chosen date. Nothing in the arithmetic can tell those three apart.
The quiet clock is honest. Stars move individually, so no global correction can imitate them: Arcturus goes one way at 2.28 arcseconds a year while Sirius goes another at 1.34, and no amount of rotating the sky will produce that. The catch is the size of it. Over the 265 years between Hipparchus and Ptolemy, the fastest star in the Almagest moves about ten arcminutes, and Ptolemy's typical error per star is three times larger than that.
So here are both clocks, running on all six catalogues, in your browser.
The instrument
Pick a catalogue. The curve shows how badly the whole catalogue disagrees with the real sky, as a function of what year you suppose it was observed. The bottom of each curve is that clock's answer. What matters is not where the bottom sits but how sharp it is.
Loading the stars.
The red curve has nothing free: precession alone sets the date. It is a needle. The blue curve lets the entire sphere rotate, solving for the best rigid rotation of the catalogue onto the sky exactly rather than approximately. Precession is a rigid rotation, so this removes it completely, along with any error in the observer's equinox, obliquity or ecliptic node. What is left is the one thing no rotation can imitate: stars moving individually. It is a shallow bowl.
That difference in shape is the entire argument. The needle is not more accurate than the bowl. It is more confident, about a quantity it cannot see.
What the five known dates say
Only one of these six catalogues has a contested date. The other five are a matter of record, and they are the only reason any of this is checkable.
| catalogue | stars | scatter | on the record | nothing free | equinox free | frame free |
|---|
The middle clock was never reading the stars
Between the two extremes sits the obvious compromise: let the catalogue's zero point of longitude float, which is exactly a rotation about the pole of the ecliptic. This is the alternative explanation in the Ptolemy dispute turned into a parameter, and if his equinox was a degree out of place, this clock forgives it. It looks unimpeachable. No single offset can imitate stars moving in different directions, so whatever dates the catalogue now must be proper motion.
It is not, and the test is one line: set every proper motion to zero and run it again. If the clock still dates the catalogue when the stars are forbidden to move, it was never reading the stars. Tick freeze the stars above and watch the orange curve keep its minimum while the blue curve wanders off by millennia.
Getting that asymmetry to hold took two corrections, and the verifier found both of them, not the reasoning that preceded it. The first: an approximate rotation is not good enough. Fitted as a small-angle rotation, the frame leaves a second-order remainder at trial epochs centuries away, and on de Houtman's catalogue that remainder dated the stars all by itself while they were standing still. The rotation is now solved exactly. The second: a constant error in latitude is not a rotation at all, because no rigid motion moves every star the same distance towards one pole. Left in place it traded against the epoch and pulled the three precise catalogues 30 to 50 years early. It is now measured once per catalogue, at that catalogue's own stated equinox, and removed. That measurement is checkable against work done by other people, because those offsets are published:
| catalogue | equinox free | frozen | frame free | frozen |
|---|
One star out of nine hundred and ninety three
When Dennis Duke reanalysed the first serious proper-motion dating of the Almagest in 2002, his objection was not that the idea was wrong. It was that a handful of stars, Arcturus above all, carried the whole fit while appearing to be one vote among a thousand, and that the published confidence interval had not noticed.
That is measurable. Drop each of the fastest stars in turn and see how far the answer moves.
| catalogue | star | proper motion | share of signal | epoch without it | shift |
|---|
Are the error bars honest?
A bootstrap tells you how much an answer wobbles when you resample the stars you already have. It cannot tell you whether the estimator is biased, and it cannot tell you whether that wobble is the real spread. Only a catalogue whose date you already know can do that, and there are five of those.
So the study manufactures more. For each real catalogue it builds synthetic ones from the same stars at a randomly chosen known year, with the same per-star scatter, the same recording quantum (Ptolemy wrote sixths of a degree; Hevelius wrote arcseconds), and a random rigid error in the observer's frame. Then it runs the same clocks and asks whether they come back with the year it chose.
| catalogue | nothing free: bias | true | quoted | ratio | frame free: bias | true | quoted | ratio |
|---|
And so, Ptolemy
The Almagest gives positions for a stated epoch, the start of the reign of Antoninus, AD 137. Its longitudes do not fit AD 137. They are about a degree short, and they fit the era of Hipparchus, around 129 BC, far better. That much is not in dispute and has not been since Tycho Brahe noticed it.
What it means has been in dispute ever since, because the one-degree offset is exactly as consistent with two very different stories. Ptolemy may have taken Hipparchus's catalogue and moved it forward using his own precession constant of one degree per century, when the true rate in his era was nearer one degree per 72 years. Or he may have observed the stars himself with an equinox that was a degree out of place, which would come to the same numbers. Verbunt and van Gent put it plainly in 2012: opinion "has oscillated ever since Tycho Brahe".
The longitudes cannot settle it, because a wrong precession constant and a wrong equinox are the same arithmetic. That leaves proper motion, and proper motion has been tried, repeatedly, since 1987. This page's contribution is not the idea. It is the ladder of controls underneath it, and the ladder says: do not believe the answer.
What this page does not claim
- Not a new method. Dating a star catalogue by proper motion goes back at least to Efremov and Pavlovskaya (1987), and runs through Dambis and Efremov (2000), Duke's reanalysis (2002), and Baiget Orts (2026). The degeneracy between precession and a zero-point error was written out formally by Duke in 2008. None of that is original here.
- Not a verdict on Ptolemy. This page finds the proper-motion clock too weak to separate Hipparchus from Ptolemy, which is what Duke concluded in 2002. It adds controls and a calibration; it does not add an answer.
- Not a claim that Ptolemy did or did not observe. The scholarly picture since 2020 has moved toward a composite catalogue: Marx (2021) and the 2022 multispectral recovery of part of Hipparchus's own catalogue from the Codex Climaci Rescriptus both point away from either extreme.
- Not a homogeneous re-derivation of catalogue accuracy. The per-star scatter printed here is a robust spread computed one way for all six, not the Gaussian widths Verbunt and van Gent fitted, and the two are not interchangeable. Theirs are cited below.
The check
- Precession model checked against an independent implementation.
- Reproduces the published offsets that ship with the data.
- Recovers the editors' documented correction from the data alone.
- Every clock ablated by freezing proper motion.
- Error bars calibrated against synthetic catalogues of known date.
- Every figure on this page is computed in your browser, by the same modules the study runs, copied here byte for byte. The verifier fails if the copies drift.
Sources
- Verbunt, F. & van Gent, R. H., The star catalogues of Ptolemaios and Ulugh Beg, Astronomy & Astrophysics 544, A31 (2012). doi:10.1051/0004-6361/201219596. Machine-readable catalogues at CDS J/A+A/544/A31.
- Verbunt, F. & van Gent, R. H., The star catalogue of Tycho Brahe, A&A 516, A28 (2010); The star catalogue of Hevelius, A&A 516, A29 (2010); Early star catalogues of the southern sky, A&A 530, A93 (2011).
- Duke, D. W., Dating the Almagest star catalogue using proper motions: a reconsideration, Journal for the History of Astronomy 33, 45 (2002). Full text.
- Duke, D. W., Statistical dating of the phenomena of Eudoxus, DIO 15, 7 (2008). Full text.
- Dambis, A. K. & Efremov, Yu. N., Dating Ptolemy's star catalogue through proper motions: the Hipparchan epoch, JHA 31, 115 (2000).
- Baiget Orts, C., Speed-error cross-correlation dating of ancient star catalogues, arXiv:2604.02521 (2026). A preprint, not peer reviewed.
- Gysembergh, V., Williams, P. J. & Zingg, E., New evidence for Hipparchus' star catalogue revealed by multispectral imaging, JHA 53, 383 (2022).
- Capitaine, N., Wallace, P. T. & Chapront, J., Expressions for IAU 2000 precession quantities, A&A 412, 567 (2003). Coefficients as given in the IERS Conventions (2010), Technical Note 36.
- ESA, The Hipparcos and Tycho Catalogues, ESA SP-1200 (1997), CDS I/239; Anderson, E. & Francis, C., XHIP: an extended Hipparcos compilation (2012), CDS V/137D, used for radial velocities.
The lab notebook, the verifier and every script are in research/fifteen-centuries-of-eyes/ in this project's repository. The page's numbers come from data.json, which is generated, never edited.