GREEKS 3 Babylon 0

Greek Priority With Babylon's Three Lunar Equations
DNA-Proven Fathers of “Babylonian” Astronomy:
Aristarchos of Samos & Hipparchos of Nicaea

Satanic Heresy in Ancient-Astronomy-World's Church

Dennis Rawlins           2007 December 18

  • 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:

    1. 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 (DIO 1.1 [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 observations of known date and nation: famous solstices of 432BC and 135BC, both unquestionably Greek, specifically Meton of Athens and Hipparchos of Nicaea (later of Rhodos).

      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 DIO 4.1 [1994] ‡1 [pp.3f], and Hugh Thurston Isis 93:58-69 (2002) p.58.

    2. 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 (DIO 11.1 [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.
      [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.]

    3. 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 HAMA p.366), no awareness of lunar parallax, or precession, observations of needlessly rough accuracy (see F.R.Stephenson's consistent findings or DIO 1.3 [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 1.2 [1991] §E3 [p.112].) Obvious question here: 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.

    4. As David Dicks (University of London) and even ultimo-Babylonianist F.X.Kugler have noted (DIO 4.1 [1994] ‡1 §C4 [p.10]), what we have from Babylon are strictly tabular computing schemes, without any extant explanation of their empirical sources.
      [Don't miss DIO 11.1 [2002] p.26.]

    5. 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 Almajest 4.2:

    [0] saros = 223u = 239v = 242w = 18K + 10°2/3 = 6585d1/3 days

    [1] 1u = 29d31'50''08'''20'''' = M (“Babylonian” System B month; primes = 1/60ths)

    [2] 251u = 269v (Babylon System B)

    [3] 5458u = 5923w (Hipparchos c.130BC)

  • 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 both Kallippus of Kyzicos (c.330BC) and Aristarchos of Samos (c.280BC) were Greek astronomers, who preceded System B's debut (Neugebauer Astronomical Cuneiform Texts 1955 1:xvi) by decades.

  • 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 M (eq.1), which scholars are unanimous in regarding as ancient astronomy's central parameter.

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

    223M ≈ 18K + 10°40'01" ≈ 18K + 10°2/3

    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 Almajest 4.2 as long before Hipparchos and prior to any math-development of the saros.
    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” M. How could this not have been noticed?
    Since Neugebauer read (HAMA p.603 n.18) Heath's Kallippic analysis of our eqs.0-2 and also knew that eq.3 was not nailed-down as Babylonian, one cannot help but wonder about the origins of Neugebauer-Muffia Babylonianism salesmanship's history of ugly, aggressive attacks on the competence of all those perceived as encroaching upon the Muffia's preserves or questioning its infallibility: is this behavior related to an inner awareness that the Babylonian case really didn't hold water as well as repeatedly proclaimed? Every iota of the unpleasantness of the decades of war between the Muffia and DIO grew out of the gang's buzz-off nastiness. (See DIO 1.1 [1991] ‡1 §C7 [p.] & ‡3 §§D2-D3 [p.20].) As DR has repeatedly inquired (e.g., DIO 4.3 [1994] ‡15 §§D5-D6 [pp.126-127]): do scholars with confidence in their views act this way? I.e., one can make a psychological argument (quite aside from scientific exposures of the obviously non-Babylonian origins of eqs.1-3) that the Muffia not only had a weak case but itself sensed that fact to some extent all along.
    (More generally: the Muffia-H.A.D.(A.A.S.)-JHA history-of-astronomy “centrist” clique has consistently aimed at mutter-slandering, exiling, or killing-off dissenters, by contrast to DIO's equally consistent reaching-out to communicate.) Along the same line: is the reader aware that none of the vicious combatants we refer to discovered any of eqs.1-3? These all go back to F.X.Kugler, a century ago. Question: what has the Neugebauer Muffia's 3/4 century crusade-for-Babylonianism actually added to serious history of science? How important is it to figure out the details of the computational shortcuts of Babylonian astrologers, whose astronomy was almost entirely leeched off the real science of the Greeks? See DIO 1.3 [1991] n.266 [p.167].]

  • 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 (DIO 11.1 [2002] ‡1 §A [p.6]), though Babylonianists (who normally regard Ptolemy as a giant) won't believe either of us. The DR reconstruction will follow a brief but instructive diversion.

  • Ptolemy cites the 4267u eclipse cycle (nearly 345y 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. (DIO 1.3 [1991] §N4, eq.8, and nn.235 & 237 [pp.150-151 & 157]. John Britton & Hugh Thurston both verified this math in detail and deserve our thanks for a laborious kindness.)]

    Earlier Babylonian citation of the 251-month period relation definitively settles the issue for Babylonialists: just forget all that astronomical and statistical stuff — one single textual point proves that Babylon's 251-month relation MUST have been the ancestor of the 4267-month cycle. (Even though there's no cuneiform attestation of said cycle.) This attitude has convinced just about the lot of them that there is no need even to look at evidence against their most-sacred of sacred tenets — said tenet being that the shorter 251-month relation was discovered first in Babylon, and subsequent Greek multiplication of it by 17 was the origin of the 4267-month eclipse cycle found at Almajest 4.2.
    [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 M. (Though Britton remains a DR-critic, it is important to recognize here that he has understood the point that — despite the 251-month relation's earlier citation — eclipses have an advantage in accounting for the accuracy of the 251-month period-relation. The reason he is sympathetic to reading beyond the texts to the science behind is that he, Alex Jones, and DIO — whatever their sometimes huge disagreements — are not only scholars but adventurers.)
    In response, DR points out the following:
    [i] Almajest 4.2 finds the 32° from the 10°2/3 by tripling it, not the reverse.
    [ii] Multiplying 126007d01h/4267 or M by 223 and dividing by 365d1/4 yields 18K + 1/33.7497 or 1/33.7492, resp. Either way, formal continued-fraction-analysis would hardly be required to conclude that the ending is best approximated by 4/135, a deliciously sexagesimal gift felicitously handed to Aristarchus' rounding.
    [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 extremely precise. (DIO 11.1 [2002] ‡1 §A11 [p.7].) Which it was. So Aristarchos jumped on it as the basis of a durable 4868y calendar.]

  • 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 DIOs' respect for (and welcome to the criticism of) those who differ, e.g., DIO 11.1 [2002] p.2 n.1.]
    Scholars who greet heresy thusly are inviting future scholars' ironic snickering in this instance, since
    Preference for long periods is chapter-one obvious to any astronomer .
    [And probably to any other sort of scientist. The need for huge time-bases was obvious even to Ptolemy for all his treatments of mean motions (lunar or otherwise) — despite his Almajest's failure to understand his predecessors' preference for integral cycles, a miscomprehension which Ptolemy appears creditably to have overcome by the time he compiled his Planetary Hypotheses, which gives period-relations for solar, lunar, and planetary motions.] Yet the Babylonian tradition does not cite long time-periods in connexion with founding the 251-month period-relation.

  • 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 division by a huge number of periodic repeats. Superior airs affected by those who cannot follow science this simple (or won't even take the trouble to compute-out a few examples for themselves) verge on mirrorlessly-projective buffoonery.
    [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., DIO 1.3 [1991] n.266 [p.167]. Even so, the writing-materials-inequity is trumped by our realization that System B is attested only after Aristarchos' 10°2/3, a datum which we have found tells us that he possessed all of eqs.0-2.]

  • 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 (idem) states (contra modern Babylonianists) that the 251-month period-relation (our eq.2) came from dividing the 4267-month eclipse-cycle by 17. The obvious-anyway descent is thus anciently attested, and anyone with an astronomical sense can see that it alone can explain M's accuracy.
    [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 idem) proving (idem) a near-perfect return in lunar anomaly (and only a 7°1/2 shortfall in solar anomaly):

    4267u = 4573v = 345g − 7°1/2 = 4630w1/2 + 11° = 126007d01h = 3024169h

    Substituting this into eq.0, we find:

    saros = 223u = 223×126007d01h/4267μ(1K/365d1/4) = 18K + 10°40'00''…

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

    saros = 18K + 10°2/3 = 18K + 4/135 = 4868K/270 = 1778037d/270

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

    GY = 1778037d = 270 saros = 270×223M = 60210u = 4868K

    Aristarchos' month M must therefore be:

    M = 1778037d/223/270 ≈ 7973d06m15m/270 = 29d31'50''08'''20''''

    [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 DIO 11.1 [2002] ‡1 §A11 [p.7]) unless he knew that the basic 10°2/3 fits astonishingly exactly to the empirical 345y cycle, as shown just above.
    So the longest ancient calendar was established by the ambitiously visionary astronomer who was also proposing the largest universe.

  • 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' M was not merely close to but exactly equal to canonical eq.1. Details in Appendix here.

  • 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 on the nose, keep in mind that cavilling Babylonianists do not even claim to have a way of explaining eq.3's two numbers, 5458 and 5923.

  • Since Ptolemy (Almajest 4.2) attributes eq.3 to Hipparchos, we go to Almajest 6.9, where Ptolemy reports that eq.3 is based upon comparison of Hipparchos' observation of a 141BC perigee eclipse (1/27; anomaly 359°) with a midnight apogee eclipse of 720BC (3/8) fortunately preserved by Babylon. Since the two eclipses are separated by 7160 months, we have a draconitic equation:

    7160u = 7770w = 7673v1/2 −12° ≈ 211439d ≈ 579y

    — 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 (DIO 4.1 [1994] ‡3 n.7 [p.21].) And eq.3 is indeed about this accurate — perhaps the highest achievement to come out of the Hipparchos school, sadly mis-attributed these many decades by Babylonianist cultism.
    [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]):

    13645u = 14807w1/2 = 14623v1/2 + 7° = 1103g + 63° = 402945d

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

    5458u = 5923w

    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. (DIO 4.1 [1994] ‡3 sect;§D1-D2 [pp.22-23].)

  • 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 exact Hipparchan draconitic period-relation (eq.3), by using a highly peculiar perigee-apogee eclipse-pair — while the only astronomer anciently cited for such a bizarrity is Hipparchos.
    [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 Precise “Babylonian” Month (Eq.1)

    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:

    YAt = 365d1/[4 + 1/(20 + 2/60)] = 365 1/4 − 15/4868 days

    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) 4868×235M/19 ≈ 1778022d — thus his Metonic-tropical year is

    YAt = 1778022d/4868 = 365d1/4 − 15/4868

    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 M), this would have led to a non-matching remainder of 16/4868 rather than 15/4868 (ibid §B6 [p.8]) — a discriminator-test that confirms Aristarchos' pre-System B possession of the exact “Babylonian” month M.

    1st posted 2007/12