Eudoxus' Star Catalogue - how old is it really?

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Berlin · 2010  Uwe Topper topper


There is an ever so often reappearing puzzle of time shift when ancient astronomical dates are confronted to modern retrocalculations. Ancient dates are far to low, and modern calcualtions for the described phenomena yield dates in the distant past where they historically cannot be located.

A very striking example for this experience has recently been published in DIO 15, Dec 2008 by Dennis Duke, “Statistical Dating of the Phenomena of Eudoxus”. The professional and academic mathematician offered a purely mathematical analysis of the chances that the coordinates indirectly contained in the star atlas of Eudoxus (preserved in Aratus and Hipparchos) could be pinned down to a narrow date in Greek pre-history by using the well known precession model. By thoroughly examining the data of five determining celestial circles following the description of fixed stars, all of them giving very near and consistant dates in time, Duke found as average result for the time of Eudoxus’ star catalog 1130 plus/minus 90 BC. The date is off Eudoxus‘ supposed lifetime (370 BC) by 760 years.

Moreover, Duke  (p.12) says that the result is in good agreement with previous results of similar calculations quoting in fn.5  Bradley E. Schaefer (in JHA, 2004) and in fn.9 German scientist R. Böker (1952) who reached nearly the same chronological borderline as Duke himself.

Duke made considerations about inaccuracy of the identified stars and found that the inaccuracy can be judged very small, since around a dozen star groups for each colure and on-circle are at hand. In general the given dates are coherent and consistent with each other. Though high probability of correctness of those dates is diminished as Duke‘s statistical research shows, the outcome nevertheless is so outcrying far from historian‘s thoughts about the dates of the mentioned originators of the star atlas that Duke sees no other way to console them but by proposing (p. 16):
“Therefore, it appears that if we invoke the knowledge we have independent of Eudoxus, we might tolerate a uniform prior for perhaps a century before Eudoxus ...“
Surely a century is not enough as there are more than seven centuries at stake.

Now Duke admits in continuing at the same place:
„The implementation of the prior knowledge of t in a statistical analysis is unavoidably somewhat subjective ...“
which is the least a mathematician should say.

To avoid misunderstandings: I do not criticise Duke‘s highly professional mathematics but try to make use of his remarkable results proposing a solution to the problem.

In fact there is quite a common nasty feature of retrocalculations of ancient coordinates using modern precession rate, as not only early chronologists like Scaliger, Calvisius and Petavius and their opponent Isaac Newton had to tackle, but scholars of the 19th century alike of whom I shall only quote Alexander v. Humboldt, Kosmos (1845, III,149):
The Ptolemaic star atlas “gives, because of mistaken precession reduction, star positions as if they were defined in 63 CE“ (instead of 137 CE) [my translation from the German version]

Induced by a number of similar indications we (i.e. my son Ilya Topper and the undersigned) suggest that a possible jump or jolt in the movement of precession might have occasioned the erroneous calculations noticed by many scientists and which are the more obvious the older the investigated traditions are, possibly on account of accumulation of deviation by distinct precession rates.

Having in mind that such an enlightened researcher as Nic. Copernicus did acknowledge the Arabic rate of precession (around 66 years per 1 degree) which practically all Arab/Persian astronomers used for many centuries based on very neat observations, I cannot but accept that this was the actual rate at their time. The switch to about 72 y/1° by Ulugh Beg ocurred in the 15th century on account of observations. Since the 16th century this rate has been stable until today.

Part 1: Proposal of a new term: precession jolt

The new term we have coined for the lapse of virtual time which might be falsifying any retrocalculations was first named “cosmic jolt“ or “leap of the earth”, now we call it “precession jolt”. It implies that the axis of the  earth jumps with regard to the background of the fixed stars for a tiny fraction in its sun-attached orbit, a fraction that can be measured afterwards by newly calculating the tropical year length and thus the speed of the actual precession. After a short lived instability the earth regains its stable precession movement which differs slightly from the previous one. The amount of the drift in reference to the stars can be ascertained from earlier historic documents of astronomical and calendaric nature. The type of jump the earth undergoes is essentially not different from its usual precessional movement with the only exception that it is very quick, nearly instant. Only the length of the tropical year changes slightly while other parameters like the sidereal year or the rotation speed or the inclination of the Earth’s axis (ε) remain essentially the same.

For the moment and for the sake of the mathematical argument of this paper I suggest  that there is no need of knowing or discussing the cause that unleashes such a jolt of the earth. Each event must have had far reaching consequences for life and environment: Sea coasts and islands changed places, mountains were rising or sinking, volcanoes erupted and ravines opened for new riverbeds, the atmosphere will have been darkened for many years, in certain regions higher life may have ceased altogether, cultural achievements had to be regained by hard efforts for generations. Denying those devastations would give a wrong picture of human cultural evolution. But this is not the topic of the present paper which is only concerned with the mathematical problem: Retrocalculated chronological distance for certain ancient astronomical observations might be wrong if only based on stable precession rates.

Is there astronomical evidence for such jolts?

There are many to this effect, but they have not been taken seriously into account or not been related to each other to give a clear sight. Let us regard some astronomical traditions proving those jolts.

Part 2: Four points are to be considered:

A: Although north remains north on earth after a jolt, the celestial north pole changes its place for the observer.

B: By a jolt the speed of the precession is altered because the length of tropical year has changed a bit. This can be ascertained after some lapse of time, say after one or two generations. Also, the current year of the jolt will be shortened by a number of days once. By this leap the cardinal points of the year jump abruptly. This has been marked in calendaric records such as almanachs.

C: Retrocalculations using modern values cannot give real historic dates when refering to a time before a jolt.

D: Trepidation is a temporary irregular movement of the observable star positions right after a jolt.

Let us consider those four points closely.

Point A: The movement of the Celestial North Pole.

The term ‘precession’ is described as the circling of the earth like a spinning top by which its axis moves in relation to the celestial pole, and thus the equinox wanders backwards.
A simple way to understand this as a layman is to compare positions of the polar star during the centuries. Today Alpha Ursae minoris (our polar star) is less than 1° distant from the North pole and therefore quite apt to be used as guidance, while Columbus when crossing the Atlantic Ocean had difficulties in determining true north because the deviation of the same star amounted to more than 3° from celestial pole.

The slow dislocation of the polar star in the course of centuries is an effect of precession and its most obvious example. Joseph Scaliger who – like later Isaac Newton – secured his chronological work by taking into account precession, used an ancient tradition which said that a star in the constellation of Draco, Thuban by name (meaning ‘the serpent’), had been the polar star in those times, supposedly the times of the Phoenicians. Scaliger selected the star Thuban, the one we today call by that same name (Alpha Draconis) because it would lay on the precession circle as retrocalculated from data available to Scaliger (roughly the same we use today). Only this star could have been the polar star of the ancients, argued Scaliger, and this is taught until now in popular books. Yet the ‘historical’ moment for the position of Alpha Draconis as polar star would be chronologically far off: 2800 B.C., i.e. traditionally about two thousand years too early for the Phoenician culture.

(Note: Kunitzsch, Almagest 1974, p.172 relates this exceptional mistake of Scaliger – the wrong attribution of Thuban to α DRA – to John of London, 1246, and repeats this p.224. See: Kunitzsch, Paul (1986): “John of London and his Unknown Arabic Source” in JHA 17, pp. 51-57)

A similar observation can be made concerning the polar star of the Arabs. They are reported to have chosen Kochab (which equals Beta Ursae minoris) and called it al-Kawkab al-Shamali, the north star, from which it took its modern name Kochab. Yet if we retrocalculate using modern values the Arabs would have lived about three thousand years ago or they would have retained a corresponding old term which in their own time was useless. Those reputedly excellent Arab astronomers deserve a better judgement.

Aristotle (Peri Kosmou) described the Earth as a globe floating motionless in the centre of the universe and around which are circling the sun, the moon, and all the stars except one star: the polar star. This one is an unchangeable point which sailors use to determine their course.

Now, which star did Aristotle refer to? His thought is ideally correct at least from our actual point of view. But in his time – say around 300 BC – there was no star that could have served as polar star if we apply our data for retrocalculation. The earth‘s axis would have pointed to a section of the northern sky without any bright stars in the (present-day) constellation of Camelopardalis. Not even approximately would there have been a star to serve as guide for sailors. Then what is Aristotle talking about? We don’t know but we can deduce that the sky at the time of Aristotle looked different from what it would have looked like if modern retrocalculation is applied.

As far as I can see, modern retrocalculations going back more than six centuries very often do not give meaningful results when compared to old traditions. There is one way of solving the problem: Transposing those traditions of polar stars like Kochab (as well as Thuban) to older cultures to make the retrocalculation fit. Kochab could go back to the Phoenicians (or some other sailing people talking Semitic) because the CH instead of K in the word Kochab might betray Aramaic or similar origin.  And the Dragon with Thuban could stem from Mesopotamian roots, as cuneiform seals vaguely suggest. Even the Egyptian Old Kingdom was considered to comply with such a suggestion when traditional chronology is applied but there the constellation Dragon has no warrant in art or writing anywhere.

Point B: Variation of the speed of precession

Well known is the conclusion that nearly all data and descriptions in the Almagest of Ptolemy are somehow wrong; they do not fit his time if retrocalculated; some do fit an earlier time (e.g. Hipparchos’). This had been noted by critical Arab astronomers when they compared the Almagest to their own observations. A possible solution that makes sense to me are the jolts that have taken place in the meantime.

Precession jolts can be tracked back in history for quite a distance.

Our actual value of the speed of precession (ca. 72 years for 1° displacement of the spring equinox) is known since the Renaissance. In the Middle Ages the speed was noted to be a bit quicker. Arab astronomers ascertained the precession speed at their time as around 65 to 66 years per degree, starting from al-Battani (around 880 AD) passing to Kushayr and as-Sufi until Haraqi (1112 AD) and their followers. An anonymous contemporary of Zarqallu (11th century) noted likewise 66, castilian Canones de Albateni, 1250, cap. 52 (ed. Bossong) has neat 66, and the Latin book of Alfonso X, the Wise (13th century) gives the same value. There is at least another handful of testimonies to this speed, here a selection of Arab precession rates (extracted from Kunitzsch Ibn Salah 1975):

al-Battani (880 AD) = 66,4
Kuschayr (933 AD) = 66,25
as-Sufi (964 AD)   = 65,4
Haraqi (1112 AD)  = 65,77

We can retain: All values for some centuries oscillate around 66. The astonishing uniformity of the value cannot be explained by copying because the values differ very slightly and are often declared to be the result of painstaking and individual calculations of which basic elements are also given. (Although the correct dating of those texts is open to discussion, yet it seems clear to me that they pertain to the time before the Renaissance.)

Let us step back into classical time:
Ptolemy gave the speed of precession as exactly 100 years per degree (and Arab authors like Thabit ibn Qurra confirmed this for Ptolemy). It might seem this is just a round number but the mistake can only add up to a few years, as this value had been handed down for Hipparchos as well.

Even older dates have been discovered: Dennis Rawlins (1981, published only in DIO 9.1, 1999, p. 37) shows that Aristarchos 130 years before Hipparchos confered the same value of 100.
This means once again that an original value for the precession speed had been handed down (in this case by classical Greeks) over a time span of several centuries. Again I have to admit that we now don’t know when this really was and for how long this precession speed was uniform. But since the Arabs knew this value it must have been calculated some time before them.
Stepping further back until Babylonian time we are confronted with values near 50 (Schnabel 1927) but they are highly contested and so I will not insist on.

To call these figures rounded to fit magical purposes is not logical. I regard these different values as calculated by astronomers on the basis of real observations in their own time. The calculation can depart from the difference between sidereal and tropical year or using information of generations past about the location of stars on the ecliptic. Both methods had been used.
It strikes me that any intermediate values are missing.

There is a curious letter of Thabit ibn Qurra to his colleague Ishaq ibn Hunayn in Baghdad (only preserved in the star tables of Ibn Yunus, see Morelon 1987, p. XXI ; 1994 ; and Ragep p. 274, note 18) in which he asks whether Ishaq could supply him with some intermediate values of the tropical year (or precession rate) between Ptolemy and the ‘House of Knowledge’ (of Khalif Mu’min in Baghdad, 830 AD) because this would help him to understand the great difference in Ptolemy’s and his own observations. The answer is not known, but to my knowledge there are no intermediate values extant either.

The change of the precession speed was not slow and surreptitious but abrupt. At certain moments the speed changed suddenly, that is what I call a jolt.

Point C: Chronology by means of retrocalculation
Archaeo-astronomers researching megalithic monuments or medievalists analyzing astrolabes always use a simple method to find out the age of the object in question: They look for an indication of the spring equinox (or any other similar date) given by their object and then just apply mathematics using the 72 y/1° precession rate like a ruler because this speed is regarded a natural constant that holds good for millenniums and even millions of years. The law of uniformity is fully put into action. The result is wrong.

How wrong can be shown by an example: Alfonsine castilian “books of wisdom” (said to be written about 1270 AD) always give star positions with 17;08° longuitude distance to the positions of Ptolemy. The traditional chronologic distance between the two authors would amount to 1130 years. Applying the modern rate of 72 the distance would be wrong by hundred years more. Taking into account Ptolemy’s rate the distance would be completely out of order: 1700 years. Only with the Arab (as well as Alfonsine) value of 66 we arrive at 1130 years. This looks well arranged. The conclusion should be that the speed of precession between Ptolemy and Alfonso was regarded stable at 66 y/1°. Our chronology is based on particular precession rates. As long as we take a wrong precession speed for retrocalculation the result must be erroneous.

But what if a jolt had happened in-between? Then the time lapse between the two proposed persons is wrong. If jolts have taken place, all chronography before the last jolt based on the actual precession rate is wrong. This would resolve the problem stated by Duke and others as mentioned above.

The model of a Platonic Year (of roughly 25.900 years) as revolution of the spring equinox by precession is unreal. Not even an ellipse or oval form would help to improve the dates, because in this case Kochab could never have been the polar star of the Arabs. It stands too far off in time when calculated using actual values. For the last millennium or two we can only pretend to know that precession has always been retrograde (but not always steady).

Point D: Trepidation

There is another hint to the correctness of the above thoughts: the notion of trepidation of the Arab and early Renaissance writers.

The Latin term trepidatio, also called accessio and recessio, or in Arabic iqbal and idbar, had been regarded as fact by certain astronomers of the Middle Ages. It was already mentioned by Theon of Alexandria before 400 AD, and shortly by Proklos (5th cent. CE). Trepidation is described as to-and-fro movement of the position of the equinox (and corresponding data). It could not be brought into accordance to the known mechanisms and never found a satisfying theoretical explanation. It was observed only during some short decades and then vanished, at least twice in recorded history: before Theon of Alexandria (otherwise he could not have recorded it) and in the time around Thabit ben Qurra and Alfonso X.

From the Greeks the notion of trepidation had reached the Arabs who handed it on to Andalusian and Christian scientists who proved it by using slightly differing data. They held fierce discussions about this subject. King Alfonso X is on record having considered trepidation as a fact when he started to reign (around 1260 AD), but towards the end of his long rule he disregarded it (about 1290 A.D.). The disturbance had ceased, the values had gone back to constancy.

French astronomers in Paris had noted the same: Campanus de Novare (1261-64) used compromise values, but from 1290 onwards the Toledan tables give stable data. Morelon (1997) names five French astronomers around 1300 who agreed on this.

Some time later John of Königsberg (Regiomontanus, end of 15th century) wrote a pamphlet refuting vehemently this notion of irregular movements. I suppose that in his time trepidation had definitely ceased.

Trepidation is thus revealed to be an indication of a short lived unstable movement of the earth after a jolt.

When confronting historical models with astronomical calculations one is prone to exclaim: There is something wrong here! just as Duke found out.
And such was Isaac Newton’s expression when he checked the chronology of Scaliger and Petavius by precessional mathematics.

Result: Retrocalculation of bygone years by using an unchanging value of precession (taken from actual observation) cannot give true historic years if jolts of the earth’ axis and an abrupt change of precession speed are not taken into account. Best results may be obtained when traditions of earlier astronomers and calendar usage are considered valuable..


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DIO: The International Journal of Scientific History, Baltimore, USA, can be found via internet at and loaded down free of charge.
JHA: Journal for the History of Astronom

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