Did Greek Astronomers Have Access to Such?

at Late Ancient Use of That Century's Eclipse Records

The Five Lunar Eclipses: 1292BC, 1274BC, 1245BC, 1201BC, 1190BC

Single Fruitful Theory Is Consistent With Integral Solution of Last 3

Hitherto-Mysterious Ancient Period-Relations. Each On-the-Nose.

** Summary**
(See also tight precis of entire case at inside cover of

At the end of the 20th century, only three precise ancient lunar period-relations yet remained unsolved.

[“Period-relations” are integral ratios of celestial speeds, the central ancient one being the very accurate equation of 251 synodic months with 269 anomalistic months. The better known Meton→Easter relation, equating 19 years with 235 months, is far less accurate.

(Neither of these are among the triple-quarry sought below.)]

Testimony on these 3 relations had come down to us from sources spread across nearly half a millennium: from the 3rd century BC to the 2nd century AD.

Yet in 2002-2003, it was discovered by ** DIO** that
all
three period-relations were precisely consistent with
common-prime-defactored integral eclipse-cycles and thereby explainable
by the theory that the period-relations' classical-era discoverers had,

Thus, one single

There are no less than ** six** obvious
evidences recommending the theory. These are summarized at

[See also

When this case is looked back upon someday, it will be apparent that a prime reason that this approach was not considered was that scholars (including DR) had uniformly discounted the very idea of such early eclipse records surviving for a millennium. (Note: DR was not smarter than others here. He was simply

Our time-bases' vastness — over 1000y in each case — can also account for the previously-inexplicable astonishing accuracy of all three of these period-relations.

[The

Each's accuracy is ordmag one part in a million or better — ten million in the draconitic case. (So since there are three equations & three super-accuracies, we are actually solving not 3 but 6 mysteries at one Occamite swoop.)

[Note, also, that other anciently attested lunar period-relations (also founded on ordmag 1000y time-bases) are equally accurate, e.g., 251 syn.mo. = 269 anom.mo., 1979 syn.mo. = 160 sid.yr. (Both discussed at

DR's theory is indeed shocking: classical-era access to accurate
eclipse-times over half a millennium older than the previously-accepted limit
of eclipse-data availability (more reliable than the Ammizaduga data)
— so shocking, that some historians reject
the idea purely on the basis of the enormity of the remoteness.

[Ptolemy refers (** Almajest** 4.11)
to a series of eclipses brought over from Babylon,
but does not give the duration of it.
Some, e.g., Toomer 1984 p.166 n.59, point to
the

However, doubters of the eclipse solutions ought to recognize that not one but several possibilities inherently reside in the remoteness-enormity of DR's proposed Babylonian eclipse-records:

The very idea is crackpot.

[Easy to undo that complaint, since older data are known, and the method proposed here to explain the three integral equations is common-sense, highly accurate, and very naturally emits integral relations — self-evidently apt for solving the sources of equations which are entirely expressed in integral form. Our method is also.]*attested and standard for the era, since the best ancient scientists understood its advantages*The proposal is possible but not strongly so.

Gauging probability-inherence of our proposed matches occurring by chance is discussed below. Several soft & hard probability-tests have already been applied:[2002] ‡3 §E [pp.24-25]; and*DIO 11.1*[2003] ‡2 §C [p.13]. Results: odds against chance are statistically significant, if not lock-certain. Below, we have chain-linked these various surprisingly confirmatory indicia, four such in all:*DIO 13.1*

just keep on clicking upon each's glowing link, starting with this link, to go right on to the next (ultimately returning here).The very enormity of astronomical-historian-clique shock tells us that

*if the proposal is true, it represents a clarifying advance*in our perception of the devoted preservation (and possibly precision) of early Assyrian-Babylonian astronomical records — long before the Persian and Greek conquests of Babylonia.If the theory is ultimately accepted, it also may be of some ancillary use — as a non-extrapolated snapshot (

[2002] ‡3 §D5 [p.23], for one felicitously remote century — to expert extrapolative researches on Δ*DIO 11.1**T*behavior, possibly assisting in suggesting an upper bound on Δ*T*for that century.

Keep in mind:
if the hypothesized 13th century data had survived to our day,
there would be no inductive achievement here.
I.e.,
there is no sense in attacking an unexpected induction of lost science
— on the grounds that it's lost!
Inductions of lost data obviously must be judged on merits other
than the self-evident fact of lostness.

[** DIO 11.1** [2002]
‡2 n.7 [p.12]: “If we cannot accept any finding
in ancient science without direct attestation, then: should we all park
our brains at the entrance to the ancient science field?
Is it forbidden to induce beyond the texts?”

Note that there are several pieces of ancient astronomy which are not mentioned in extant classical-era materials, yet which we know existed: e.g., a valid Alexandria winter solstice solar altitude observation (DR

** Integral Relation→Integral Relation**:

— From Visible Empirical ECLIPSE-CYCLE to Tabular PERIOD-RELATION

We know that all pre-Ptolemy lunar speeds were based upon period relations found from

[For independent evidence that accurate lunar-orbit elements were based upon eclipse-observations, see elsewhere.]

Ptolemy explains the centrality of
just this process at ** Almajest** 4.2,
when he discusses the most important
(

which was repeatedly verified empirically as of a duration (126007d01h)
virtually independent of position
(proving
its anomalistic cyclicity).
Thus, division by common-prime 17 provided the canonical
and extremely accurate *period-relation*:

** Inverting the Foregoing Process — to Untransform**:

Three Long-Mysterious PERIOD-RELATIONS→ECLIPSE-CYCLES

Three famous but hitherto-unsolved integral lunisolar period-relations are found on texts of the classical period. (No selectivity here: as of the start of 2002, these were the only outstanding ancient lunisolar period-relations remaining to be solved.)

[1] 6247 synodic months = 6695 anomalistic months (System A, 3rd century BC)

[2] 5458 synodic months = 5923 draconitic months (Hipparchos, 2nd century BC)

[3] 3277 synodic months = 3512 anomalistic months (

[Note that in what follows we will often use abbreviations for lunar periods: u for synodic, v for anomalistic, w for draconitic.

Thus, the foregoing will henceforth below often be written like: [1] 6247u = 6695v, [2] 5458u = 5923w, [3] 3277u = 3512v.

(Or as ratios: [1] 6247u/6695v, [2] 5458u/5923w, [3] 3277u/3512v.)]

** Conservative Math Automatically Yields Way-Unconservative Triple Discovery**:

What could possibly be more obvious and conservative than to use ancients'

Therefore, DR just probed (2002/3/18-2003/1/26) via
the simplest, most astronomically-sound, and
only-attested technique:
Ptolemy's common-sense method — which we just invert for each equation.
For all three *variously disparate* cases, the unambiguous recovery-math
was ** mere
multiplication by an integer** (or half-integer:

The three DR inductions' directness — leading (unambiguously in all three cases) to data in the range 1292BC-1190BC is what others must verify before they either accept or reject the theory. After DR's exhaustive 2001-2003 investigations, he has consistently contended that no post-1500 BC data, other than the c.13th century BC eclipse records cited, will [a] directly (by attested ancient procedure) and [b]

(

In the years since 2002-2003 publication, not one alternate eclipse has been found by any of the several committed cultist knee-jerk disbelievers in the DR theory. This, despite fervent desire among them (particularly AJ, PH, &JB) to find some way — any way — to kill off DR's eclipse theory. Note that those (esp. AJ) who reject the theory because of the proposed eclipses' great antiquity are engaging in literally preposterous logic. Such mental inversion is typical of math-challenged cultists who (selectively) balk at going beyond the directly attested.

[Yes, deductive logic indeed tells us that only flawed premises can lead to a false conclusion, but it is the inverse of reason to Jonesely attack an evidential argument-towards-a-conclusion by rejecting the conclusion

To the contrary, the very import of the ** DIO** triple-discovery

In each case, DR's solution remarkably LED RIGHT TO the close vicinity
of the 13th century BC.

The underlying eclipse-relations were
(length in years, respectively, about 1010y, 1103y, 1325y):

[1] 12494 synodic months = 13390 anomalistic months
(System A, 3rd century BC)

[2] 13645 synodic months = 14807 1/2 drac months
(Hipparchos, 2nd century BC)

[3] 16385 synodic months = 17560 anom months
(** PlanetaryHyp**, 2nd century AD)

To check the connexions, divide each of these 3 hitherto-unknown eclipse-cycles by its corresponding period-relation to see how simple the ratios are:

[1] two;

[2] five halves;

[3] five;

again: how hard was that?To supersimply verify the 13th century BC clustering, just subtract the relations' durations from the approximate dates of their attestation, for century-rounded estimates:

c.300 BC − 1010y ≈ 1300 BC;

c.100 BC − 1103y ≈ 1200 BC;

c.100 AD − 1325y ≈ 1200 BC;

again, this isn't higher math. For most of us.

That, in skeletal outline, is what has been uncovered. And probably
no set of discoveries in ancient astronomy has been more ordinary
(in method or math-execution) or more unexpected (in result) by all
— certainly including DR.
Since history of astronomy archons are reluctant learners,
there will be disbelief in officialdom
at how easily and directly such treasure was captured.

[Upon hearing of DR's eclipse-solutions to
the three famous hitherto-intractable period-relations,
Morphiosi doubtless immediately suspected the matches were rigged.

(After all, that's how they get some of their own matches:

** DIO 16** [2009]
‡3 eqs.7\&14 & n.38 [pp.27\&30] and §J5 [p.36];

Years later, the seethers are still at the eternally frustrating task of finding the slightest evidence that the DR solutions are anything but what fell naturally out of an unprejudiced, non-manipulated investigation via the only anciently attested method for finding lunar period-relations.]

It is true that the the implications here may subtract from the rep of
decaying post-Arbela Babylon. But we should have been realistic all along.
There is, e.g., no sign in late Babylon of the sophistication we find
in contemporary Hellenistic math —
i.e., there's no known Seleukid Euklid.

[DR's bluntness about
Babylonian astrology's mathematical inferiority
to Hellenstic astronomy (after 300BC) has not won him any archonal friends.
Babylon: no trig, no transit circle, no vertical observations, no solstice or
equinox observations (Neugebauer ** HAMA** p.366; Jones 2005),
no awareness of lunar parallax, or precession, observations of needlessly
rough accuracy (see
F.R.Stephenson's consistent findings or

But DR's discovery that Greeks used Babylonian 13th century BC eclipse records adds to the glory of the height of early Babylonian civilization — a thousand years before Greece became serious about astronomy — a revelation that ought to appeal to cliques who have long affected a love of Babylonian astronomy — a passion which evidently becomes somewhat “confused” when Babylon is exalted from Unapproved quarters:

[There is little doubt that 13th century BC Babylonia was more than advanced enough to care to record simple eclipse-data. E.g., the Hilprecht Collection in Jena possesses a strikingly precise clay tablet map of Nippur, Babylonia, from c.1300 BC.]

To assist predictably glacial archonal realization of what has been found, the following less condensed discussions may be of enlightening value.

It is not at all controversial that Greek astronomy had access to Babylonian records of eclipses and planetary data. And Babylonian celestial data much older than the 13th century BC have survived far longer (c.1500BC→present for the Ammizaduga Venus tablets: 3500y) than the data-survivals proposed here (1010y, 1103y, 1325y).

[And over 99% of classical-era science mss have been lost. (Less than 10% of even Hipparchos' writings survive. Under 1% [perhaps 0% of primary writings] for Aristarchos.) Which makes it hard to be confident — as confident as certain moderns, anyway — that we would know of classical-era references to 13th century data if astronomers of that time had used them. After all, there are no such references to the earlier Ammizaduga data —*which we know existed*.]All pre-Ptolemy lunar speeds are attested by Ptolemy as having been based upon period-relations obtained from long eclipse cycles. (

4.2 & 6.9; Toomer 1984*Almajest*pp.176 & 309.)*Almajest*

[Note that Ptolemy's top modern defenders reject his direct testimony here on method. See, e.g., O.Neugebauerpp.310-311 & 391, or Toomer*HAMA*p.176 n.10.]*1984 Almajest*

Take, e.g., 251 syn mos = 269 anom mos: its accuracy — 1 part in*millions*— was based upon the 345y eclipse-cycle (which is 17 times the 251-month relation), just as Ptolemy says at4.2. This is obvious to any astronomer familiar with how astronomical periods are determined accurately: record a huge time base*Almajest**T*for a known number*N*of cycle-repeats, then the mean-motion's period*P*is found via*P = T/N*.The idea is that the observational errors at each end of the span are automatically reduced to relative insignificance after division by the large number

*N*.

E.g., when errors of ordmag 1h (3600s) at each end of a 345y cycle are divided by 4267, the period (month*M*) is obviously determined to ordmag 1 time-second.]All 3 ancient month-estimates are accurate to ordmag 1 in (at least) a million, not accomplishable (in a naked-eye era) without vast time-bases.

All 3 are anciently reported in integral-ratio form, exactly the form which empirical eclipse cycles would emit, simply & naturally — merely through dividing by an integer (or half-integer).

When we attempt to reconstruct the ancients' three hitherto-unsolved ratios (from System A, Hipparchos, & Ptolemy) by

*undoing such division*(i.e., multiplying by an integer), we find: all three reconstructions lead into the same narrow 10-decade temporal window, the 13th century BC,—*and all three do so unambiguously**despite the fact that they come from eras centuries apart*(spread over nearly 1/2 a millennium). I.e., the front-end eclipses are conspicuouslyas tightly bunched (into the same 13th century BC) as the back-end eclipses which are spread throughout the classical era. [I.e., the three results all zero-in on an early*four times**theoretical*span (c.100y: 1292 BC to 1190 BC) which is about*four times narrower*than the later*empirical*base: 3 different observers, spanning over 400y: mid-3rd century BC to late 2nd century AD.]

This provocative over-arching statistical point is one which the history-of-astronomy community has yet to take in (a point independent of whether it will be persuasive, even if&when finally understood). And said general point is merely prelude to presentation of some specific matches yet to come (below: as the earlier-cited chain-links continue), which multiplicatively shrink even further the statistical likelihood of chance-explanation here.Starting in 1980, historians refused for nearly 1/4 century to accept that all Greek planet speeds were also based upon cycles, even though DR had shown that three planets' mean motions (Mercury, Venus, & Saturn) positively were — and that the numbers generating their speeds were right in the

9.3 preface to their tables. Then (*Almajest*[2003]) Alex Jones showed (while correcting a DR misjudgement) that the mean motions of Mars & Jupiter were also based upon cycles cited in the same preface — ending that controversy.*DIO 11.2*DR has traced (

*idem*) all five planets'mean motions to integral sidereal cycles of ordmag 1000y. (Note the Jupiter case's glaring indication that the*Almajest*'s tropical or Metonic planet relations must have come from larger sidereal relations:*Almajest*[2003] ‡4 §H4 [p.45].) And we note that Ptolemy's*DIO 11.2*explicitly provides such huge sidereal cycles for the planets (Neugebauer*Planetary Hypotheses*p.906),*HAMA**sidereal-integral being the sort of speeds which ancient observers would find from centuries of raw stationary-point planet data*(*ibid*p.390) — just as*synodic-integral are the sorts of lunar speeds one would find from eclipse-cycles*, and are precisely the format of the lunar speeds we find in pre-Ptolemy records.

So DR has proposed his General Theory of Ancients' Cyclicities ([2002] ‡2 §H [p.19]): all ancient celestial mean motions were ultimately based upon integral cycles — the Moon, the 5 planets, even the Sun's accurate sidereal motion (*DIO 11.1*[1996] ‡1 eqs.23&31 [pp.22&24]), and Aristachos' durable inaccurate precession (arising out of his 4868y-cycle Great Year:*DIO 6*[2002] ‡1 eq.11 [p.8];*DIO 11.1*[1999] ‡3 eq.16 [p.37]). Since there is no longer any question that ancients' planet motions were based upon cycles, isn't the analogy obvious regarding ancient scientists' likely approach to determining lunar motions?*DIO 9.1*

*What Led DR to His Eclipse-Cycle Triple?*

We repeat the surprisingly-ultra-conservative answer: DR simply took
ancient scientists' STANDARD METHOD
— that is, their use of very long eclipse cycles —
to search out the sources of three previously unsolved problems.

*In no case was there serious ambiguity*: ALL 3 arithmetically-indicated
early eclipses directly plopped down into the 13th century BC or barely after.
We examine these shockers one-by-one,
in order of date and (equivalently, as it happens) of DR discovery.

[According to the 1992 canons of Meeus-Mücke and Liu-Fiala
(both works based upon F.R.Stephenson's pioneering researches),
each of the ten eclipses used here were above the appropriate horizon
for at least part of the event's duration.

(The relation of visibility and Δ*T* questions
are touched on in a 2008 note appended to the online pdf
of ** DIO 13.1** ‡2 §G [p.17].)

In any case, the σ in Δ

** Solution [1]: System A**:

The characteristic integral equation defining System A (found only on Babylonian tablets) is a period-relation of length c.505y (Neugebauer

[See, e.g., J.Britton
** Arch Hist Exact Sci 61**:83-145
[2007] p.124. (The relation is mis-typed there.) There is no citation of DR's
proposal that the 6247u relation was obtained, not by Britton's wildly
speculative theory of massive averaging
of elaborately-elicited Lunar-Four-based anomalistic estimates
(there's not-a-jot of ancient testimony that anyone computed lunar speeds so),
but simply through dividing an eclipse-cycle by a small integer:
below; or DR

Well, given [a] Ptolemy's ascription of all pre-Ptolemy lunar speeds to integral eclipse-cycle bases; [b] his account (based upon division by an integer) of the ancestry (above) of the 251-month period-relation; [c] the fact that eclipses are the most obvious,

But when (2002/3/18) DR tried multiplication by 2, he was delighted to find that (against the

The 22° remainder is so near
the limit
that allows eclipse-pairs to occur at all, that such pairs are rare —
fortunate in helping delimit possibilities for those which underlay
period-relation [1].

[Only three post-400BC pairs could have been used. These are listed at
** DIO 11.1** [2002]
‡2 §B4 [p.13], (Due to a DR book-keeping error,
a half-invisible 1418BC-408BC pair was accidentally listed there.
However, the pair was not used in any analysis originally or latterly.
The online version of

The search for pairs that are suspect of thus triggering the invention of System A find (between 336BC and 67AD) merely two pair:

1274BC/12/05→263BC/1/26

(** DIO 11.1** [2002]
‡2 §B4 [p.13].) Presumably the to-the-hour agreement between
the two pairs' time-gaps suggested an anomalistic return.
The possibility that
13th century BC eclipse reports specified the time (not just date)
of the eclipse could be a hint of early precision-astronomy;
however, the fact that most (all but the 1st) of the five older eclipses
(underlying the present triple-analysis) were near the horizon
suggests the possibility that near-horizon eclipse reports were
preferentially selected in order to narrow the event's time-uncertainty
(a technique

** Solution [2]: Hipparchos' Draconitic Month**:

At

(For simplicity in this equation, we use the remainders implied by
his parameters, which are close to reality.)

[The “draconitic month” or “eclipse month”
is the time the mean Moon takes to return to a node.
A visible (umbral) eclipse can
only occur within about a dozen degrees of a node.]
Though this is not an eclipse cycle as its stands,
we could (if this is all we had testimony for) recover the eclipse cycle
*empirically underlying the equation* merely by finding
when a multiple of the equation produces regularly-repeating eclipses.
Here, multiplying by 10 does it, producing the following eclipse cycle:

(Accurate to about 1 part in 2 million.)
The 1st of these equations (716 mos) is of course an obvious consequent
of the 2nd, via simple ** division by common primes** —
2 and 5 in this case (thus the 1st equation is 1/10th the 2nd equation).

Now comes the disjunct-impediment (which
** DIO 11.1** [2002]
‡3 finally [2002/4/3-4] penetrated past): Ptolemy says
(

But in fact Ptolemy is (partly) wrong:
this equation positively does not follow
from the above 7160 month eclipse cycle.

[The disagreement is about 1 part in 2 million.
Small, yes. But the two ratios are definitely incompatible,
so 5458u = 5923w cannot have arisen from the 7160u = 7770w cycle.
Neugebauer *HAMA* p.314 argues that the discrepancy is due
to slight deviations (which Ptolemy notes) from the exact apsidal line.
The problem no pre-** DIO** investigator faced is that such
a consideration would've ruined the purely integral ratio
Hipparchos concluded for. Had he been concerned with the trivial effect
of the imperfections so blinding to previous modern investigators, he would
at the very least have appended remainders to the integers of his ratio
(as for the adjusted period-relations of

DR found that the above 5458 month cycle follows instead from
cycle-pairing an earlier eclipse
*with the very same −140/1/27 eclipse
(Almajest 6.5)
which Hipparchos had initially used (Almajest 6.9)
to find the 7160 month cycle,*

[Hipparchos perhaps made a similar decision (much less successfully, given the less accurate data he inherited for prior generations' solar places) after recording his 135BC Summer Solstice, when he used it to gauge the length of the tropical year by comparing to an earlier solstice. He evidently switched from using Meton's 432BC solstice as his older datum (

The search for a parent eclipse-cycle that is proportional to
this period-relation is again narrowly restricted, since the latter is already
long: 441y. But, again, a little testing elicits the solution
(** DIO 11.1** [2002])
‡3 §§B2-B3 [p.21]): multiplication by 5/2 discovers
an eclipse-cycle ([2])
about 1103y long:

The half-integral anomalistic term requires: only apogee-perigee pairs
need apply.

[Any pair (of the type specified by Ptolemy)
with identical anomaly would have sufficed;
but Hipparchos apparently preferred to try eliminating (as completely
as possible) differences between true and mean longitude,
by using events near the lunar orbit's major axis. (With analogous cleverness,
he usually observed the Moon when its longitudinal parallax
was near null: ** Almajest** 5.3&5.)]

It happens that the nearest-to-perigee eclipse during Hipparchos' lifetime that works with the 13645-month cycle is that of 141BC/1/27 (anomaly −1°), which

— and we find that modern theory indicates that
the end of the 1245BC eclipse was
probably just above the horizon in
Babylon and was near-apogee as required (anomaly = 171°).

[** DIO 11.1** [2002]
‡3 §C2 [p.21] suggests the 1239BC/7/12-136BC/9/24 pair,
alternatively.]

The fact that the astronomy of the above 13645-month eclipse-cycle
requires a perigee-apogee pair is striking because Hipparchos is
the only astronomer in history
who is known to have tried such a ploy (** Almajest** 6.9).
That is, the 13645-month cycle's half-integral anomaly
shouts Hipparchos.
And, though Ptolemy is not always reliable, it is worth noting that
he (

[Firmly establishing the 4267-month eclipse-cycle or cycle [1] or cycle [3] requires multiple eclipse-pairs. But once the synodic and anomalistic motions are nailed down (preferably by the 251-month period-relation, or cycle [3]), one needs but a single well-chosen equal-magnitude eclipse-pair to establish draconitic-return cycle [2]. This is why

** Solution [3]: Ptolemy's Anomalistic Equation **:

At c.160AD, Ptolemy in the

Here there are two possible integers (which will by multiplication
produce an eclipse cycle not too outlandishly long): 3 and 5. But the former
produces no eclipse pairs after 36BC until the mid-3rd century AD, while
the latter connects to two eclipses reported in the ** Almajest**,
compiled by the very author (Ptolemy) whose

Again against chance expectation
(continuing the train of indicia
suggesting we are on the right track here throughout), Ptolemy reports
eclipses in 125/4/5 and 136/3/6 *that work with the above eclipse cycle*.
(The odds are 20-to-1 against his selection occurring by chance: computed at
** DIO 13.1** [2003]
‡2 §C1 [p.13].) Thus, applying the 13645-month eclipse-cycle to
those, we find two pairs that could have handed him that cycle's
durations (which produce exactly
period-relation [[3]):

1190BC/6/12→136/3/6

Note that it makes sense that Ptolemy would not attempt
to find a period-relation better than the famous one of
251 months unless he were using a huge time-base
— which he appears also to do elsewhere
(** DIO 11.2** [2003]
§L [pp.49-50]) in the same

** Temporal Coherence**:

The earlier eclipses just derived are only about a century later than the earliest we adduced (above) for System A — emphasizing the tightness of the temporal window here, where all five of our remote eclipses fall in a

System A 1010y 281BC&263BC↔1292BC&1274BC

Hipp 1103y 141BC↔1245BC

Ptol 1325y 125AD&136AD↔1201BC&1190BC

Ancients may, if several relations seemed comparably useful, have chosen
the one where both sides of the equation contained a common prime,
which rendered the final relation less cumbersome.

[Hipparchos' alternate draconitic eclipse-cycle,
7160 synodic months = 7770 draconitic months was divisible on both sides
by 10, leading to the {\ti Almajest} 6.9 period-relation
716 syn mo = 777 drac mos. If there were no common prime factor,
then the period-relation was identical to the eclipse-cycle
and thus directly obvious —
e.g., 781 sidereal years = 9660 synodic months, which was used to develop
Ptolemy's final luni-solar equation
(** DIO 6** [1996]
‡1 eqs.20-31 [pp.21-24]).
This and other period-relations were not anomalistically integral
but were used anyway — another neat example being
800 sid yrs = 9895 syn mos, was divided by 5
to give Geminos' 160 sidereal years = 1979 syn mos.]

** Responses to Objections**:

As with so many of DR's discoveries, at first only the most enlightened are taking this seriously. Centrists imagine (hope?) that there is some impediment that all-by-itself makes the hypothesis flat-impossible. After a century of Babylonian-astronomy-exaltation, the findings here are obviously an unwelcome shock.

[A dozen years after publication, not one professional historian has yet reported checking even the easy arithmetic, much less the more laborious task (which may be beyond the abilities of some among the set) of confirming the delicate behavior of the eclipse-cycles we discuss. I.e.,

(But, then, there's the old saying that the one thing we learn from history is that most folks learn nothing else from it.)]

So let us look critically at the most predictable objections:

[a] Data so remote couldn't have survived.

Yet the Ammizaduga Venus data have come through to us, and they are quite pre-13th century. Recall too that the availability of pre-747BC lunar records is indicated by Geminos' reportage of the 800y sidereal lunar cycle:

[b] Thousand-year cycles aren't in the

No, but they're in Ptolemy's

[c] The early eclipse-dates DR is proposing could not have been ascertained reliably enough before the Babylonian calendar became regular.

Yet Ptolemy's eclipse trio (721BC-720BC) is correctly dated, for events well before said calendar became regular. Why assume that early Babylonian astronomers were not up to Ptolemy's smart habit of dating all by the unambiguously rigid Egyptian calendar?

[d] There are cuneiform records using the 5458-month cycle c.200BC, well before Hipparchos.

But this objection falls, upon detection of a flaw in tablet-dating firmness (noted elsewhere here and at

[It turns out that the only explicitly

** Creativity vs Creationism**:

For now, this paper may prove a vain exercise.

In dealing with those who already know the wrongness of alternate theories even before looking at the evidence (in the present case, a seething clique, which has spent years straining to draw high-precision astronomy out of Babylonian not Greek records), DR feels a kinship with those scientists who (like himself) have debated creationists. The cementality is the same: no interest in what is the most reasonable explanation of the admittedly fragmentary data (though the present theory's mechanism is [unlike Darwin's] anciently attested), but an obsession instead with rejecting anything but a pre-arrived-at sacred conclusion (in this case: non-scientific Babylonian astrologer-mystics were the ultimate pre-Ptolemy astronomical genii) for which there is no evidence whatever on method — fragmentary or otherwise.

** Mental Archaeology and the Future**:

Throughout his career, DR has enjoyed numerous vindications by emergent new evidence. The above theory will probably not be one of them, since 13th century BC cuneiform eclipse records are unlikely to turn up at this late date. But the case as it stands is sufficiently coherent, direct, and simple that (even if the theory is never 100% established) its merits will be gingerly appreciated by balanced scholars in the not-too-remote future.

A theorist specializes in envisioning beyond available data, so the creator's vision of that future will for-now suffice as vindication in this matter — which is in any case of lesser importance, compared to DR's 2002-2003 combination of delight and initial-disbelief when experiencing the three discoveries themselves.

Appendix 1. On Method: Standard Greek vs Double-Secret Babylonian.

DR does not at all object to any scientist who wishes to question
the foregoing on logical or empirical grounds.
But it is disappointing to observe the unscientifically passionate
*resentment* of the theory, by a clique that has long been committed
(professionally & emotionally) to ascribing the discovery of high-accuracy
ancient lunar speeds to Babylonia, though:

[1] The only proven relevant inter-cultural transmission of
a mathematical parameter based upon known dated observations
was in the direction Greece→Babylonia:
the sole Babylonia-attested yearlength, found
upon the key System B cuneiform tablet BM55555 (a DR discovery,
now honored by the tablet's permanent display at the British Museum),
based upon Summer Solstices by Meton (432BC) and Hipparchos (135BC).

[B.L. van der Waerden's
evaluation
began the realization of this finding's import, and the discovery is
now so universally accepted that it has long since entered
the ** Encyclopedia of Astronomy and Astrophysics**
(2000, Hipparchos entry).
See also A.Jones' generous paper in the

[2] Even DR-critics' most adventurous unattested (purely hypothetical) Lunar-4or6 schemes would not automatically produce the integral-ratio format of all the high-accuracy extant pre-Ptolemy ancient lunar speeds. The cultists' pretzel-processes have never been able convincingly to account for the extant lunar speeds' precise integral ratios (which automatically issue from eclipse-cycles

Babylonianist attempts to solve the three ratios' sources are based on crude horizon data that are less steady and less directly related to anomaly than eclipses' neat periodic returns, so these Muffia guesses must posit MASSIVE use of averaging, thus negating the use of merely two bounding events separated by a huge time-base — which all astronomers know is the preferred method for accurately determining steady temporal periods. (Not just for eclipses, but also for, e.g., rotations of planets or pulsars. Note that some cultists [e.g., P.Huber] propose explaining ancient period-relations by combining arbitrarily and flexibly-

DR has long contended that the “Babylonian” month's 1-in-millions accuracy was based simply upon the 4267 syn.mo. eclipse cycle's empirical invariance within under 1h (

[a] JB does not cite DR's confirmation here, though his calculations find and use the very same range which DR was (2002 ‡A3 [p.6]) perhaps first to quantify (in hours and in equations), tabulate, and use in this connexion (see above

[Nor does JB cite DR elsewhere except at his n.66, where he cavils about periphery, not noting online

[b] We now toss at Muffiosi a question its Lunar-4&6 wing has never faced — if this cult has even thought of it: can one discover (except by the 345y eclipse cycle) how ancients pinned-down and settled-upon the wonderfully accurate period-relation 251u = 269v,

[There's no record that any ancient realized that 355/113 is a better approximation to π (good to 1 part in better than 10 million) than

JB (still p.124) ascribes this scalpel-like precision to sinuous but (to quote

[c] Thus, taking his grouplet's dreamed-up non-eclipse methods as an Expert conclusion (though no fundamental astronomer's support is cited), JB continues privately to replace demonstration with scorn, since he can find no mathematical fault with DR's 2002-2003 eclipse-cycle solutions to the parallel problem of the other period-relations considered here. Where is the detailed math of Britton's alternate explanation for

And (Muffiosi, are you sitting down?):

[Lunar-four data of course have nothing whatever to do with eclipses. This point parallels R.Newton's comment on laborious Muffia attempts to explain-away Ptolemy's solar fakes by assuming (against the testimony of

The rock-base mental blocks preventing Muffia&co
acceptance of DR's solution are pretty obvious.

[a] We note first that the three-eclipse-cycles solution
(to the mystery of super-accurate ancient lunar speeds) discussed here
was not thought-of earlier by Muffiosi and rejected for cause.
No, these sadly limited volk (while proposing wildly
unsupported speculations of their own) never even ** thought**
of an eclipse-cycle explanation for system A,
or the Hipparchos draconitic cycle, or Ptolemy's final equation
— and never checked to find out whether it would work.
(Since DR's 2002-3 publication of such solutions, no one has found
any eclipse cycles other than those he cited, which would explain
the lunar relations they

[b] And, beyond cultism: once a problem is solved, it's no longer available as a money-cow. So there is a simple economic motive for not admitting that anything in the field has been solved. (Even one of DR's associates has explicitly claimed that nothing is ever proved.) But as DR has repeatedly asked (

Appendix 2. Table-Flip's Elementary Consistency with Hipparchan Use of 13th Century BC Eclipses

As noted at
** DIO 1.3** [1991]
n.211, Neugebauer & Toomer rightly regard Pliny's comment on Hipparchos'
600y of eclipse “predictions” as Pliny's misunderstanding of
Hipparchos eclipse calculations going not forward but backward —
to Nabonassar 1 (−746/2/26 Alexandria Apparent Noon).
Pliny's testimony is valuable as indicating that
the

It is obvious from ** Almajest** 3.1 that young Hipparchos used
as epoch “the death of Alexander” (likely adopted from Kallippos),
Philip 1 Thoth 1 Alexandria Apparent Noon (−323/11/12 Julian).
(Notice the alternate interpretation at Neugebauer

But: WHY Nabonassar? Some have assumed
(*HAMA* p.608) that the choice was related to
the antiquity of surviving eclipse records. But there is
no surviving hint that any important astronomical event occurred
during Nabonassar's reign. The oldest record Ptolemy cites
(** Almajest** 4.6&9 and 6.9) is from the reign of
the 4th king down from Nabonassar in the Canon of Kings
(5th king listed at Toomer 1984 p.11).

Since ancient scientists were addicted to round numbers
(** DIO 1.3** [1991]
§ [p.];

** Giving Kept Rabbots a New Destructive Opportunity**:

A reasonable theory. But should our investigation end with this? Does the theory account for all mysteries here? E.g., we know (above) that planetary observations from the 9th century BC survived at least as late as the 2nd century AD. How does a theory of 600y-tables back only to 747BC comport with that reality? Or with our triple-finding that classical-era astronomical period-relations were found via use of 13th century BC eclipse observations? Those Muffia-troughing rabbots who earn their kepthood by nay-jerk-pulling the automatic-eject lever for such obviously valid discoveries from outlanders are bound to ask: if Hipparchos really computed his draconitic period-relation using a 13th century BC eclipse report, why didn't he create tables 1200y long instead of just 600y?

HE DID.

This reality has lain-plain before our eyes for thousands of years:
Hipparchos' Pliny-attested 600y table was *actually good for 1200y*
— by the elementary expedient of ** using the 600y table
in both directions**.

This suggests a (very tentative) theory: that Nab 1 was chosen as Hipparchos' tabular zero-year because

[The fact that the king-list (Toomer 1984 p.11) starts with Nabonassar (mentioning no rulers prior to him) suggests that either Hipparchos never calculated anything prior to Nab 1 or that it originated with or passed through a party that did not know of his pre-747 BC research and-or the Hipparchos tables' reflectivity. (Or the pre-Nabonassar king-list was too spotty for cataloging.)]

Let's test-explore this ultra-elementary revelation of table-inversion by
using Hipparchos' original mean solar motion 600y table,
exactly preserved at ** Almajest** 3.2
(with its later expansion [

The table provides spans of 18 Egyptian years
(6570d for each such span), single Egyptian years (365d each),
Egyptian months (30d each), as well as days & hours.
Alexandria local time differs less than 2 time-minutes from
local time at Lindos, Hipparchos' observational home-base
(** DIO 4.1** [1994]
‡3 §G3 [p.46]),
which he based his tables on.

All three eclipses Hipparchus used
for finding his two ratios (716u/777w & 5458u/5923w)
for the draconitic motion of the Moon were near-apsidal,
and were of course partial (with semi-duration ordmag 1 hour).
Rounding to integral hours and following Hipparchos
(** DIO 16** [2009]
‡1 n.10 [p.6]) in ignoring the Equation of Time —
whose effect on solar places is trivial anyway — these are the eclipses:

−140/1/27 22h near-perigee

−719/3/8 23h near-apogee

−1244/11/13 16h near-apogee.

The −140/1/27 eclipse observed by Hipparchos at Lindos
was 221311^{d}10^{h} after Nab 1.
To conveniently enter and extract data from
the Hipparchos mean Sun tables,
we (using superscript E for Egyptian years) break this interval
*T*_{H} into sub-intervals of time for which the tables
explicitly supply solar longitudinal arcs:

T_{H} = 594^{E} + 12^{E} + 120^{d}
+ 1^{d} + 10^{h}

We then simply (and we mean *simply* — this is
astrologer-level math) read directly in the table (where we round to 1'),
for each of these sub-intervals, the corresponding longitude arcs.
These we merely add, in the following equation
(where the corresponding time-sub-intervals are in parentheses),
to the mean-longitude-at-epoch 330°45'
(** Almajest** 3.2), to learn the Sun's mean longitude

f_{H} = 330°45' +
215°35' (594^{E}) + 357°05' (12^{E})
+ 118°17' (120^{d}) + 0°59' (1^{d})
+ 0°25' (10^{h}) = 303°06'

We again emphasize the ultra-convenience that every component of
the foregoing equation is provided explicitly in Hipparchos' table, as those
who consult it at ** Almajest** 3.2 will see for themselves.
(I.e., no need for interpolation unless computing with

The Babylon −719/3/8 eclipse was 9872^{d}11^{h}
after Nab 1, an interval *T*_{F} which we break into:

T_{F} = 18^{E} + 9^{E} +
17^{d} + 11^{h}

Proceeding as before, we have:

f_{F} = 330°45' +
355°37' (18^{E}) + 357°49' (9^{E}) +
16°45' (17^{d}) +27' (11^{h}) = 341°23'

Given Hipparchos' +2°22' equation of center here,
we have his longitude for mid-eclipse: 343°45'.
** Almajest** 4.6
renders the same result as 343°3/4.

The Babylon moonrise −1244/11/13 eclipse was
181633^{d}20^{h} before Nab 1, or equivalently
−181634^{d} + 4^{h}
“after” Nab 1, an interval we render:

T_{D} = −504^{E} + 6^{E} +
120^{d} + 16^{d} + 4^{h}

Computing as before:

f_{D} = 330°45'
−237°28' (−504^{E}) + 358°32' (6^{E})
+ 118°17' (120^{d}) + 15°46' (16^{d})
+ 0°10' (4^{h}) = 226°02'.

[If one just eyeballs +122°32' from the 504^{E} entry,
the above calculation can then be done purely additively,
like the previous two.]

Hipparchos' −50' equation of center then provides his longitude for mid-eclipse: 225°12'. This completes our illustration of tabular-reflection's simplicity: the Hipparchos tables can be used going backwards from Nab 1 just as easily as forwards from there.

The −1244 eclipse may have been chosen by Hipparchos partly because,
presuming the Babylonian record mentioned that the eclipse
ended near moonrise, this gave a time of day
that was accurate enough for his purposes. (An ordmag 1 hour error
would have minuscule effect on the deduced mean lunar motion.)
If the reality of the Babylon −1244 eclipse record becomes understood
in academe, it will contribute to modern knowledge by
setting an upper limit on Δ*T*
for that remote date.

Appendix 3. Summary of Exploratory Research Results.

Summarizing what we have available supporting the reality of pre-Nab 1
Babylonian celestial observations:

[1] Sidereal period-relations back to at least to
850 BC.

[2] Geminos' knowledge in the 1st century BC
of the 800y eclipse cycle.

[3] Use of 13th century BC eclipses to solve all three of
the only hitherto-mysterious four-digit numerator&denominator
lunar period-relations — not one of which
had ever previously (nor subsequently)
been mathematically traced by modern academe —
before ** DIO** (2002-2003) produced

[By unique and non-arbitrary, we mean: in the dozen years since publication, none of

[4] Long before

[5] This figure is consistent with a reasonable interpretation of Pliny's testimony connecting Hipparchos to a 600y span, presumably that of the original 18-year-interval table for his eclipse research, a table of which our reflective flip showed was actually good for 1200y years back from 145 BC or 600y back from Nab 1 — i.e., c.1350 BC.

[6] Recall also our novel if explicitly speculative hypothesis that Hipparchos' choice of Nab 1 as his tables' zero-epoch might have occurred because

The foregoing items vary in strength but they are consistent with the
** DIO** hypothesis that usable Babylonian astronomical observations
from as early as the 14th-13th century BC were made & preserved
(probably for superstitious reasons, ironically!) —
and were much later known to and profitably exploited by
mathematically able classical-era Greek scientists,
to establish astonishingly accurate
pioneering estimates of celestial motions —
the earliest instance in all history of precise predictivity and technology,
the hallmark of high science.

[Intermittent revisions 2007/12/23-2015/1/1.]