DNA-Proven Fathers of “Babylonian” Astronomy:

Aristarchos of Samos & Hipparchos of Nicaea

** Summary**:

Years ago (2002-2003), the empirical sources of the three highly accurate “Babylonian” lunar motions (eqs.1-3, below) were discovered and fully comprehended for the first time since antiquity. Long accepted as of Babylonian origin, all three of these motions turned out to be derivable by simple mathematics from the work of Greek scientists, who had skillfully applied reasonable, attested, standard, and accurate methods to Babylonian and Greek celestial data. Which is why each of the three motions is accurate to ordmag 1 part in at least a million, a consistency in naked-eye astronomical success which alone confirms — to an astronomer, at any rate — that their empirical foundation must have been lunar eclipse cycles.

Introductory Context:

Though there are several cases in which the same astronomical numbers are found in both Babylonian and Greek records, there was never proof of direction of transmission in any given case. Until, that is, DR's 1982 discovery (

[1991] ‡6 §A [pp.49-51]) — now immortalized at the British Museum — that the yearlength on the central System B Babylonian text BM55555 (ACT 210) is based upon*DIO 1.1*of known date and nation: famous solstices of 432BC and 135BC, both unquestionably*observations*, specifically Meton of Athens and Hipparchos of Nicaea (later of Rhodos).*Greek*

*As of the present, this remains the only case of proven direction for*

Greek↔Babylonian transmission of an astronomical parameter.

(I.e., while we have this case of a Greek parameter adopted by Babylonian astronomy, there is not a single proven case of a Babylonian astronomical parameter adopted by Greek astronomers.) About the time of this discovery, several scholars openly spoke our realization that many of us had been too unthinkingly herded into accepting that all of Babylonian astronomy grew independently of Greek. Among the now-vindicated dissenters: David Dicks

[1994] ‡1 [pp.3f], and Hugh Thurston*DIO 4.1*:58-69 (2002) p.58.*Isis 93*By the 3rd century BC era (which is the earliest period of direct attestation of any of the “Babylonian” equations in question), Babylon was a gradually dying city, ruled by Greeks. (Since eqs.1-3 below are alleged to derive from Seleukid Babylonia, one ought to keep in mind that Seleukos was a Greek general who had inherited the eastern sector of Alexander's empire.) It had always seemed a funny-coincidence that all the “Babylonian” lunar motions of high accuracy only appeared on cuneiform texts written after the Greek conquest, which is why DR has asked (

[2002] ‡1 n.2 [p.5]): “By the time [the ‘Babylonian’ month] is known to have been used, Babylon had been ruled by Greece for many decades. [Several useful sources here cited in original.] Are Babylonianists not implicitly contending that subjugation somehow made Babylonian astronomers more original & accurate than ever?” Greece had for some time been technologically superior, which is why it was occupying Babylon instead of the other way about. It is not unreasonable to suggest that such superiority might relate to astronomical sophistication.*DIO 11.1*

[Note that none of this affects our gratitude for the data Babylon preserved — a preservation which DR is here contending (for eq.3) was a half-millennium greater than Babylonianists believe. Not only did these data make possible the Greek triumphs here laid out, but even today the work of F.R.Stephenson & colleagues with Babylonian data give us our best modern estimate of the rate of secular deceleration of the Earth's rotation.]So it's no surprise to find that Babylonian astronomy is inferior to Greek astronomy: no trig, no vertical observations, no solstice or equinox observations (Neugebauer

p.366), no awareness of lunar parallax, or precession, observations of needlessly rough accuracy (see F.R.Stephenson's consistent findings or*HAMA*[1991] n.223 [p.152]), no knowledge of (or even evident interest in) Babylon's geographical latitude, planets taken in good-to-bad astrological order instead of the Greeks' physical order. (Summation:*DIO 1.3*[1991] §E3 [p.112].) Obvious question here:*DIO 1.2*?*how could Babylon's crude astrologers have discovered eqs.1-3 below, each accurate to at least 6 places*

[The unadorned saros ratio has merely 1/30000 accuracy, though its (exclusively Greek) 10°2/3 remainder (in eq.0 below) represents about the same 1-in-millions accuracy as eq.1 — because it tightly inter-relates with eq.1, as we will see.]

Anyone who doesn't realize that this triply-consistent high accuracy is based upon eclipses is only revealing his own astronomical innocence.As David Dicks (University of London) and even ultimo-Babylonianist F.X.Kugler have noted (

[1994] ‡1 §C4 [p.10]), what we have from Babylon are strictly tabular computing schemes, without any extant explanation of their empirical sources.*DIO 4.1*

[Don't miss[2002] p.26.]*DIO 11.1*While Babylonianists hotly resent the present thesis, note: they themselves have no coherent explanation of how Babylonians arrived at such highly accurate equations, while the present analysis will produce both the exact numbers and the means by which such accuracy was achievable.

For the last century, the three
lunar mean motions — synodic (eq.1),
anomalistic (eq.2), and draconitic
(eq.3) — have been routinely ascribed to Babylon.
(See, e.g., O.Neugebauer ** HAMA** 1975 p.310, or
the Babylonianist who wrote Wikipedia's founding entry on Babylonian
astronomy.) The three equations follow,
starting with the saros as “eq.0” — since, though important,
it does not give a highly accurate lunar motion
(good, as already noted, to only about 1 part in 30000 without
its Greek remainder, which we include here for reasons that will be apparent).

[The superscripts: u = mean synodic (civil) months (return of mean Moon to mean Sun); v = mean anomalistic months (return of mean Moon to apogee); w = mean draconitic (eclipse) months (return of mean Moon to a node); g = mean anomalistic year (return of Sun to apogee [Earth to aphelion]); K = Kallippic years of 365 1/4 days (mean Sun's return to a tropic, by Kallippos' 330BC estimate); d = days; h = hours; m = time-minutes.]

All of these equations are presented by Ptolemy at

[0] saros = 223

[1] 1

[2] 251

[3] 5458

We will not here argue the origin of the saros itself,
which was presumably recognized in deepest antiquity
(6th century BC Thales could be but one of many who knew of it),
since eclipses recurred at any given place every 19756.0 days.

[That is, an eclipse would be followed by another
— of similar magnitude and at a similar time of day —
after 54 years and slightly more than a month. This is 3 saros,
which Greeks called the “exeligmos”.]

What will be examined here is the peculiar non-Babylonian 10°2/3,
which is attested
(at ** Almajest** 4.2) as much older than Hipparchos.

** The Greek DNA-Clue — That Glaring 10°2/3 Remainder**:

It has been known since Paul Tannery (1888) and Thomas Heath (1913 pp.314-315) that the 10°2/3 relates to two well-established astronomical quantities, the Kallippic year and Aristarchos' “Great Year”. Note that

DR went beyond this (British Museum 2001/6/27,
** DIO 11.1** [2002]
‡1), to establish the empirical origin of
the 10°2/3 in above eq.0 and thereby the precise empirical
ancestry of
the “Babylonian” month

To probe the situation, we test by substituting eq.1 into eq.0, producing a saros equal to:

The spectacular closeness of agreement in this preliminary investigation
tells us that:

[a] whoever had
the 10°2/3 remainder had *M* — and

[b] *M* was father to that remainder.

(If there was realization
of items [a]&[b] previous to DR, no record survives of it.)
Even Neugebauer (** HAMA** p.603) agrees that Tannery had
convincingly related this remaindered version of the saros to Aristarchos.

[No one previously discovered where the 10°2/3 came from in the 1st place. It is mentioned at

DR is amazed that no one previously appears to have sensed an accurate month in these. Even the rougher and more derivative 6585d1/3 (divided by 223) leads to a month-length merely 4 time-seconds high (5 times more accurate than Kallippos' month-length). Since Heath (1913 p.315) knew that the 18 years were Kallippic, the exact saros time-span would be easy to find: 365d1/4 times (18 + 10°2/3) = 6585d.32222, which when divided by 223 produced a month-length only 1/10 of a time-sec less than “Babylonian”

Since Neugebauer read (

(More generally: the Muffia-H.A.D.(A.A.S.)-

Babylonianists' inability to reasonably explain the empirical origin of any
of their three lunar motions leaves us with the challenge of doing so.
Let us start with our above eq.1 and
the 10°2/3 remainder of our eq.0:
from what outdoor data could they have originated?
This has already been explained in general by Ptolemy
(** Almajest** 4.2) and in precise detail by DR
(

Ptolemy cites the 4267^{u} eclipse cycle (nearly 345^{y} long)
as the empirical basis of *M*, noting that division by the integer 17
produces our eq.2, which is barely 20 years long — *but is attested
earlier than the bigger cycle that is 17 times longer*. Here we find
the source and crux
of the gulf between astronomer DR and Babylonianists.

[** Almajest** 4.2 has the empirical ancestry right, but
ascribes to Hipparchos what we now find
actually goes back at least to Aristarchos.

One must blame this mis-chronology for much of Babylonianists' misunderstandings here, though they ought to have noticed that Ptolemy rightly reports that the key 10°2/3 remainder is pre-Hipparchos. Note that Hipparchos occasionally adopted Aristarchan material without attribution: e.g., precession; also orbital parameters. (

[The important exception appears to be John Britton, who after thinking the matter over, allows (2001/9/1) that DR may well be right in his contra-Neugebauer contention (central to the present analyses) that the 126007d01h interval was the origin of the 251-month relation (instead of the other way about, as Neugebauer&co had thought) — and adds that Aristarchos indeed was probably the author of the exeligmos' 32° remainder (1/3 of the 10°2/3). However, Britton strongly disagrees that the round 32° could be the source of anything as exact as

In response, DR points out the following:

[i]

[ii] Multiplying 126007d01h/4267 or

[iii] The remainder could have been approximated via unit-fractions, day-fractions, or whatever — before Aristarchos stumbled upon the realization that degree-division of the year (not likely one's first resort) clicked with a neat sexagesimal expression: 10°2/3.

[iv] No scientist would expand this into a Great Year unless he knew that remainder was taken as

It is hard politely to express an astronomer's appropriate reaction to
the stubbornness of some on the point that the 4267-month eclipse-cycle
bred
the 251-month relation (and not vice-versa). That is, to volk who cementally
and epithetically mutter that dissenters (from the traditional understanding
of *M*'s ancestry) are fools or nuts —
and refuse — out of a haughty sense of expertise —
** even to consider** new views.

[Note recent

Scholars who greet heresy thusly are inviting future scholars' ironic snickering in this instance, since

[And probably to any other sort of scientist. The need for huge time-bases was obvious even to Ptolemy for

Additionally, note: the 251-month period-relation is not directly
visible (no eclipse-pair),
while the 4267-month eclipse-cycle was repeatedly visible.

[Further,
no matter the cleverness of Babylonianist speculation,
it is obvious that eclipse-cycles [a] provide the best accuracy
and [b] directly emit integral results.]

Even without further indicia, the *accuracy* of ALL the above eqs.1-3
** alone** tells us that they were based upon centuries-long cycles,
for there is no other way to ensure that naked-eye-observational error
(at either end
of the measured duration of a cycle) will only trivially corrupt
estimation of a mean period, than through

[Note: Babylonianists' dependence upon the fact that cuneiform records pre-date Greek ones counts on a situation which may reflect nothing more than that clay outlasts papyrus. (Classic case of: absence-of-evidence-is-not-evidence-of-absence.) See, e.g.,

For years, Babylonianists (rejecting ** Almajest** 4.2) have
tried to establish the 251-month period-relation by non-eclipse means,
refusing to realize
that eclipses not only automatically provide
the integers we find in eqs.1-3 but
provide the highest accuracy possible in antiquity
since mid-eclipses can be naked-eye timed to ordmag 1m.
(Though, even the best extant ancient records do not report better
than an ordmag less precise than this.)

Consider:
how, other than by high-precision means, could a putative ancient
Babylonian analyst — whose work is of course totally unattested! —
distinguish between eq.2 and a relation 14 synodic months higher or lower?
E.g., 237 synodic months ≈ 254 anomalistic months, or
265 synodic months ≈ 284 anomalistic months —
ratios differing by less than 1 part in 60000 from that of eq.2.
There is obviously no way that such distinctions could have been made,
given that (see
the excellent and non-worshipping papers of Stephenson et al:
** DIO 4.1** [1994]
‡1 n.46 [p.12]) Babylonian mean time-reportorial accuracy was about
a half-hour — and such errors would have vitiated any determination
of lunar periods not founded upon an extremely long time-base.

For folks who normally require attestation, Babylonianists are curiously
quick to reject *the only attested method*, the one used by DR, which is
clearly set out by Ptolemy at ** Almajest** 4.2, namely:
long eclipse cycles, the sole method Ptolemy cites anywhere for determination
of lunar speeds before his own work. He specifically (

[Babylonianists just won't believe it, while treating those who even THINK it could be true as inferior. How can one get depressed at the folly, while such ironic theatre keeps ever-bubbling?]

** Aristarchos' Great Year as the “Babylonian” Month's
Double-Helix**:

Once we get past the impediment of cultist pathology, we can go straight to the DR 2001 British Museum reconstruction of the ancestry of the “Babylonian” month (eq.1). Ancient scientists chose the 4267-month eclipse cycle because an ancient with a Halleyesque eye (for pattern in centuries-long records) noticed that eclipse-pairs separated by 4267 months occurred after virtually the same time, 126007d01h (± less than an hour), no matter where a pair occurred in the lunar orbit — thus (as Ptolemy explains at

Substituting this into eq.0, we find:

by accident a near-perfectly-round remainder, a building-block which Aristarchos evidently exploited by rendering the equation exact:

which in turn sets up his 4868^{K} double-cycle
— the double-helix of early high ancient astronomy —
his saros-cycle-embedded-in-solar-cycle “Great Year”:

Aristarchos' month *M* must therefore be:

[Note the neat alternate way J.Britton prefers to derive *M*,
using Babylonian day-division by degrees:
126007d01h/4267 ≈ 29d191'00''49''' which (by normal Babylonian rounding)
equals 29d191°00'50'' and that is the same as *M*.
(However, there is no cuneiform attestation of 126007d01h.
Nor does this route tell us anything about the 10°2/3 remainder which
pre-dates cuneiform attestation of the *M* it produces.)]

The result is the famous “Babylonian” month of above
eq.1, thus illustrating its parentage
out of Aristarchos' well-grounded calendar (itself effectively parented
by the saros' 10°2/3 remainder) — a Great-Year calendar which he
would hardly have ** founded for millennia** (see

The foregoing evidence for Aristarchos' possession of *M*
is confirmed by independent analysis at
** DIO 11.1** [2002]
‡1 §B's eqs.12&13 [pp.7-8], showing that Aristarchos'

The foregoing has now Hellenistically accounted for above eqs.0-2, showing that they are intimately inter-related.

Eq.3 is a separate case. It is of later origin and is appreciably more accurate — good to one part in ordmag 10 million. Its Greek origin is our next subject.

Babylonianists are particularly upset at what we are about to trace.
So it is best to start with this comment: as in the earlier cases (eqs.1-2),
the critics themselves have no convincing explanation of eq.3's origin —
** much less its astonishing accuracy**.
Since you're about to experience
a simple & direct
derivation which produces both ordmag-10000 (!) values of
eq.3

Since Ptolemy (** Almajest** 4.2) attributes eq.3
to Hipparchos, we go to

— good to 1 part in c.2 million.

Problem: this is not the equation promised (eq.3).
So, what went wrong? Answer: DR discovered
(** DIO 11.1** [2002])
‡3 eq.3 [p.21]) that Hipparchos had pulled a switch
and compared his 141BC eclipse not to the cited 720BC one but instead
to a 13th century BC apogee eclipse, a magnificent pairing which left
only about half the anomalistic disjunct in about twice the span: obviously
a 4-times-better choice — and so remote that even an hour's inaccuracy
in the earlier eclipse would (after
division by
14807 1/2 draconitic months) cause only a 1-part-in-ten-million error
in eq.3's estimate of the length of the draconitic month
(

[Note that the foregoing tells us not only that Hipparchos' draconitic equation is explained by his switch to a much earlier eclipse but that additionally he had good cause to switch.]

We can tell from eq.3 that, rather than the 720BC apogee eclipse, Hipparchos
instead chose the Babylonian apogee eclipse of 1245BC (11/13; anomaly
171°). The Moon's position and anomaly were already computable for
Hipparchos from eqs.1-2, so he had just checked that the magnitude (39%)
roughly matched that of his own 141BC eclipse (26%). The resulting equation
(** DIO 4.1** [1994]
‡3 eq.3 [p.21]):

Division by 5/2 then produced
(** DIO 4.1** [1994]
‡3 eq.1 [p.20])

Thus, we have found the exact long-mysterious source of Hipparchos' draconitic
period-relation, our above eq.3.
Of course, a certain cult has no interest in any Greek solution,
because it positively ** knows** that the answer can't be Greek.

The readiest objection to the foregoing equation is that 1245BC is too old
for data to survive a few hundred years. Yet the Ammizaduga data are older
and have lasted over 3 millennia, so 1245BC can hardly be ruled out.
A more interesting objection is the only one that doubters have
taken
the trouble to lodge face-to-face (2002) against the 13th century proposal:
there are 200BC cuneiform tablets using eq.3, six decades before the 141BC
Hipparchos eclipse. This would be a killer. If true.
But after investigating and reading carefully the little-known Neugebauer
fine print (which his gang never spoke, while for decades pretending
its equations were Babylonian *beyond the slightest doubt*), DR instead
found that of Neugebauer's seven draconitic-computed lunar latitude cuneiform
texts, the six that were computed for 200BC displayed a provocative split:
the three using the Hipparchan period-relation (our eq.3) bear no
date-of-writing, while the three clay tablets that *are* marked with
the date on which they were written (c.200BC) ** do not use eq.3**.
(And Neugebauer knew this.)
The 7th tablet is the only one of the set of 7 tablets
which satisfies the requirements of proving unambiguously that
eq.3 was used on its date, but its date is 103BC, well after Hipparchos.
(

Even so, since the foregoing point is more indicative than probative, one might understand the cult clinging to its earlier view — if that were all the evidence we have bearing upon the matter. But it's not.

Just put the Neugebauer cult's sole mushy point (undated 200BC clay)
up against the points supportive of Hipparchos' authorship of eq.3
(** DIO 11.1** [2002]
‡3 §C [pp.21-22]):

[a] Ptolemy attests that eclipses were the basis of eq.3 (which is reasonable since how else will one find such perfect integers), and no other eclipse-cycle under 2000y will produce it.

[b] The above equation yields the

[c] The 2nd-eclipse cited for this very bizarrity is 141BC, just our choice above. [d] The nearest-to-perigee of all 5458-month-cycle contemporary-to-H nominees is that of 141BC. [e] The Hipparchos draconitic equation is one of 3 previously unsolved equations — all of which in 2002-2003 were explained (on the nose in every case) by just the same means as the above draconitic case: millennial eclipse-cycles, with common prime factors removed.

** Conclusion of Study and of Purported Experts' Relevance**:

In sum, all three of the equation we started out to examine are now solved. And all have been traced to purely Greek sources: Aristarchos and Hipparchos.

Don't expect Babylonianist-catechismed cultists' enlightenment anytime soon. Few if any among them have the open-mindedness (even assuming the math talent were there) to check whether the eclipse-pairs cited here and in other DIO 13th century BC reconstructions are indeed as rare as DR claims — a point that adds heavily to the odds in favor of his hypotheses.

But there's no need for fancy explanations of non-appreciation by folks who for a century had the job to solve the foregoing but couldn't. When a self-styled Expert clique condemns a party and journal for decades as crazy and incompetent, and then (oops) the banishee solves the gang's “owned” mysteries, to perfect precision in all cases (and the Experts can offer nothing more than an empty cupboard in that regard), well, the enormity of the irony (and the folly of repeatedly greeting amiability with denigration) as well as entertaining all expert onlookers: it's all just too face-loss-humbling for the losers.

Aristarchos' Possession of the

One of the Vatican collection's two invaluable ancient lists
of yearlengths (Neugebauer ** HAMA** p.601) gives for Aristarchos:
365 1/4 20' 60 2'. Since the whole list turns out to be continued-fractions,
and since Neugebauer suggests a sexagesimal interpretation for the 60,
we may write and develop the Aristarchos entry as follows:

The appearance of 4868 and the nearness of the result
(remainder −15/4868 ≈ −1/316)
to the Hipparchos-Ptolemy Metonic —
supposedly “tropical” — year (negative remainder 1/300)
nudge us to ask: what is the length of the Metonic year necessitated
by Aristarchos' Great Year? The Greek Metonic cycle
(still used for the Easter calendar) equates 235 months to 19 years,
so Aristarchos' Metonic Great Year will be (from eq.1)
4868×235*M*/19 ≈ 1778022d —
thus his Metonic-tropical year is

The match with the previous equation is striking
(** DIO 11.1** [2002]
‡1 §§B3-B4 [p.8]:
eq.13 vs eq.12), especially when we find that if the above
substitution had used the 126007d01h equation's
monthlength (instead of eq.1's “Babylonian” month

1st posted 2007/12