BACK TO ARISTOTLE? 1. ON URANOCHEMISTRY

BACK TO ARISTOTLE? 3. ON THE PIONEER ANOMALY

B. Lukács

CRIP RMKI, H-1525 Bp. 114. Pf. 49., Budapest, Hungary

lukacs@rmki.kfki.hu

ABSTRACT

This is the third study of a sequence discussing as if Aristotelian science is finally, eternally and fully surpassed. There is presently a controversy between some dating methods by eclipses in History and the present state of Celestial Mechanics; it seems that the older retrocalculations fit better to History. The phenomenon and possible consequences are discussed.

0. FOREWORD TO THE SERIES

Aristotle of Stageira is the Prügelknabe of History of Science since the French Enlightment, c. 1750. As you know, the Prügelknabe is cca. scapegoat; but in a more direct way. Originally the Prügelknabe was a substitute boy beaten if the young son of the Fürst gave a wrong answer to the tutor.

For 250 years Aristotle is beaten instead of (or: together with) the Catholic Church by Historians of Science (h. of sc.). Any ancient Greek scientist other that Aristotle may have told anything: historians of science are still enthusiastic telling that it was originally a congenial idea. Platon may have discussed the Soul which grows wings in ten thousand years and then flees above the sky; h. of sc. tell what a nice allegory. His nephew Speusippus may have written a morally terrible letter to Philip of Macedon; h. of sc. do not discuss this letter telling that it may not be genuine. Speusippus' son, Eurymedon, may have accused Aristotle for asebeia (roughly: a mortal sin against religion), h. of sc. elegantly ignore the event. Only Aristotle has no right to err.

No doubt, this aversion originated from the close ties between Aristotle and the Catholic Church. Since St. Thomas Aquinas, the teachings of the Philosopher were in the fundaments of Church Science. For some centuries almost anything which he had taught was regarded as Truth.

There are old stories, e.g. when a faculty colleague of Albertus Magnus was confronted with a fly. He was told to count the legs. His answer was: if the Philosopher's text were not clear, I should accept that a fly has six legs. (The story is too nice. The Philosopher's texts are not unequivocal about the numbers of legs of the fly. Some years ago my colleague and Aristotle expert, K. Martinás tried with a hypothesis that Aristotle counted the spider's legs and applied the result on the fly by analogy. It is possible: a fly is not only smaller than a spider, but also its legs move much faster. So it is easier to perform the counting on a spider. Remember that the number of human chromosomes was also miscounted up to the 1950's as 24 pairs, which is the correct number for our closest living relatives but not for us.)

St. Thomas Aquinas listed a very few points where he (and so later the Church) did not agree with Aristotle; they were mainly in connection with Man's Immortal Soul. Well, he could not use the gifts of Divine Revelation (albeit Sir Thomas Browne cites an antique author of Jewish religion in Alexandria who claimed Aristotle to have converted). Anyway, Aristotle is an authority in Science for the Church, and continuously ascends in authority. The (posthumous) heresy trial of Wyclif ends in 1415 with the sentence that Wyclif was indeed a heretic; some 260 points of his heresy/errors are condemned and one of them is that he contradicted Aristotle's De lineis insecabilibus. And Pierre Chaunu [0] tells us that Giordano Bruno was burned because he had refused Aristotle's Cosmology. (Indeed, the multiplicity of Suns and Earths is blatantly contrary to Aristotle.)

So, when Enlightened Catholic French freethinkers and anti-Roman Protestants find the Roman Church as common adversary, they attack Aristotle as a symbolic figure. Platon is harmless; he was simply a pagan.

Now, surely our present science is better than the Philosopher's 2350 years ago. Also, the paradigm of the recent physics is different from the Aristotelian one, except in Thermodynamics. (For other sciences, and for social disciplines the differences in paradigms are not so clear.) However, we are just learning that Nature can be described in more than one way. And paradigms cannot be disproven in another paradigm.

Galileo believed he could prove the Copernican system Now we know this would have been an impossible task. In General Relativity you may choose any coordinate system whose origo you consider at rest. In the Ptolemaic ~ Aristotelian description this is the center of Earth, and the (x,y,z) axes are fixed within Earth. With this choice some "inertial forces" appear in the laboratory; the most spectacular continuously changes the plane of the Laplace pendulum ("so proving Earth's rotation", but this is not a proof in the sense of Mathematical Logic). On the other hand, in the Copernican description the center of Sun is the origin. Then the inertial forces in the terrestrial laboratory are much more moderate; however not zero. Two old examples are the tides, well known from the time of Alexander the Great, and the aberration of starlight, observed first by Bradley in XVII^th century.

If you want to use astronomy and physics simultaneously, then you can do it for first approximation more easily in the Copernican paradigm. Without physics if you want to describe the motions of planets, it is simpler in a heliocentric coordinate system, while for the lunar motion a geocentric system is more convenient. But neither of them is "exact, so the true one".

But then, sometimes the Aristotelian paradigm may be useful even in 2011 AD. My colleague, K. Martinás succintly called my attention to this fact some twenty years ago.

1. INTRODUCTION

This study is mainly about controversies between historical dating by old eclipse retrocalculations and modern celestial mechanics. Lots of datings seem to fit into the network of History, but now we, physicists & astronomers, are doubtful about the old dating, or even not doubtful but sure that that eclipse could not have been seen thence. This is a problem, worthy for discussion. And in the same time there is the famous Pioneer Anomaly...

Chapter 2 poses the problem of the Pioneer Anomaly. Chapter 3 gives a bird's eye view of the Universe. Then Chap. 4 discusses the problems of dating history with eclipses.

The main problem is tidal friction. It surely exists but until c. 1900 was overlooked. It makes the rotation of Earth slowing down. Previously calculated eclipses still happen, but the visibility zone is different. In spite of this it seems that the old calculations are "good for history". This is a problem, if we believe in One Truth, so now we start with the discussion. (Maybe Pioneer Anomaly and this anomaly have some common root.) Chap. 5 is about the fundaments of Earth's rotation, the ΔT of eclipses and possible changes between Classical Ancient Times and Present. Chap. 7 discusses briefly the reliability of old eclipse observations. Then in Sect. 7 we discuss in details the influence of Earth's changing angular velocity on the visibility of eclipses. Chap. 8 is a short overview of eclipses connected somehow to the 11 Aug., 1999 solar eclipse, which was the first solar eclipse total in Central Europe for a long time.

Having the scene now prepared, Sect. 9 is the detailed analysis if the discrepancy between History and Physic exists or not. The discussion is not exclusively but mainly about "King Šulgi's Solar Eclipse", calculated about 1880 to 2050 BC. If so, then Babylon was destroyed by Mursilis I of Hattusas vvery probably in 1595.

Lunar eclipses are much less sensitive on tidal friction than solar ones because the totality strip is roughly half of Earth's circumference. Still in 1998 Gurzadyan recalculated the 2050 eclipse, with the result that it could not be Šulgi's Eclipse. He suggested another; and then the Fall of Babylon came up to 1499. Argumentations are going.

As it turns out, this would need a real interdisciplinary work. Changing the date of the Fall of Babylon a lot of things change on direct & indirect ways sometimes thousand miles abroad, so it should have been an easy task to check. Unfortunately at that time there is just a lacuna in the list of the Kaššú kings, earlier but not much earlier we know the names of some Assyrian kings but not the years, and there is not yet a reliable Assyrian-type King List for the Hittites. So we end up with the Scottish verdict as for Discrepancies Between History and Physics; as for Šulgi's Eclipse, that was very probable not the 1954 one. For Thucydides' First Solar Eclipse, believed excellent for dating the Peloponnesian War and so Classical Antiquity now it seems that either there are disjoint Terrestrial and Heavenly Laws (believed until Galileo), or the Peloponnesian War's dating is off by decades, or neither Thucydides' nor possible informants did not observe the eclipse and reported it quite falsely. Any of the 3 alternatives is serious enough.

Sect. 10 is the Conclusions. Appendix A gives a few data about transplutonians, while the two others give historical and statistical details about the list of the Kaššú Kings.

2. WHAT IS THE PIONEER ANOMALY?

Probes Pioneer 10 & 11 are leaving the Solar System. While astronomy does not define the "border of the Solar System", now the probes are well beyond the farthest major planet Neptune (≈30 AU), the "ninth planet" Pluto (»39 AU) or even the bigger known members of the "second asteroid belt" unofficially baptized as Quouar, Xena &c.). Pioneer 10 started in 1972, 11 in 1973.

The two probes continuously were showing (I have here troubles with Indo-European tenses) some anomaly since 1980: as time goes by a small but unexplained excess deceleration have been proved more and more significant. Of course, the probes are decelerating, since the fuel in the (main) drives has been exhausted and solar gravity is still the biggest factor influencing the route. However, the deceleration is greater than it should be according to our best theories; recently long fits gave the excess over 3 decades as

Δa_an = -(8.74±1.33)*10^-8 cm/s² (1.1)

so significantly nonzero, where the sign shows that the excess is Sunwards.

This is a disturbing situation. The excess acceleration, ~10^-7 cm/s², is moderate, but significant. In App. A we show that such an extra acceleration causes roughly 0.1 % change in orbital period; only generally the distances of Pluto and trans-Plutonians are not measured for 3 digits. With the Pioneers the long string of times, Doppler shifts &c. it seems the problem has become explicit.

Of course, it is possible that some ignored effect is not negligible and this is the reason for the discrepancy. Some such effects have been suggested and discussed (as e.g. gas leakage, thermal radiation &c.) without yielding any explanation. Fortunately, NASA still keeps a third copy of the probe, so such suggestions can be confronted with the apparatus, even if the originals are beyond 80 AU.

The problem is extensively discussed in the literature (I give here two references [1], [2], but the second has a very extensive list of literature), with lots of suggestions but no accepted explanation. For any case the possible explanations can be roughly classified into four big groups: i) the outer Solar System is unsimilar to our expectations; ii) the theory of gravity should be modified; iii) we see the effect of Final Unification; and iv) "miracle" or "new physics is needed". In this Chapter we discuss very briefly the first three possibilities; of course miracles’ discussion would not be so straightforward.

For the most conservative Group i) of course nobody would be too surprised if the outer Solar System were different from today's best picture. However the two probes are receeding in quite different paths. So local irregularities would not help. Of course, a general, spherically symmetric peculiarity is possible, however the simplest thing would be the presence of some unexpected matter. Now, it may result in extra drag Sunwards via extra mass or friction. But spherical extra mass distributions outside Pluto could be easily detected just from the flight data and sufficiently high friction would have been removed the small bodies from the outer solar system in a few million years.

As for Goup ii), the modification of Newton's Law, we can, of course, try with a deviation from the inverse square law. Now, an r^-(2-ε) form, with the fact that the r^-2 law was conjectured from data in ~1 AU distances, could produce the measured extra acceleration with cca. r^-1.9999 law. However this deviation would result in an "anomalous" ten-fold increase of gravity acceleration from Sun at the nearest stars, transforming the "invisible matter" problems of galaxies into the much worse "nongravitating bright matter" one.

As for Group iii), lots of suggestions exist for unifying all three fundamental phenomena. However simple inclusions of QFT terms do not help, as I will show. Namely, consider the much wanted Triune Unification (i.e. Relativity + Gravity + Quantumness). We do not yet know the form of the Unified theory, but do know the constants of it: c, G and h. If there were other fundamental constants in it, then it would be either not the Triune Unification or it would not yet be fundamental. Now these constants uniquely define the fundamental ("Planck") scales of the (still unknown) Unified Theory, namely

λ_Pl	(hG/c³)^1/2	1.61*10^-33 cm
τ_Pl	(hG/c⁵)^1/2	5.37*10^-44 s
M_Pl	(hc/G)^1/2	2.17*10^-5 g

Now, the first two scales are now beyond observation limits, but the third is not. Still, we do not see elementary objects of Planck scale. Maybe such objects (or what) played their roles when the Universe was still Planck time old with Planck length curvature radius. Maybe the infinities of our present theories mean really Planck data, so some 13 billion years ago the Universe started with the energy density c⁷/hG² = 4.68*10¹¹⁴ erg/cm³, and similarly great but finite results would appear at zero-point energy, singularities & such.

Maybe, but observe that M_Pl is greater by 19 orders of magnitude than our known elementary particles. This may result in problems of Particle Physics in Triune Unification; but let us pass for now. We are interested if Triune Unification will be able to produce Δa_an of (1.1) as a fundamental correction to Gravity. Now, in the next Chapter we are going to see that a Hubble-law type correction would give good numerical order of magnitude for Δa_an (with opposite sign, to be sure), but the present Hubble constant does not belong to Laws of Physics but to the Initial Condition for the Whole Universe. Of course, the Triune Unification might impose a Hubble Constant on the Universe. However the Planck scales listed above would result in a constant again 10¹⁹ too great. So by large probability Triune Unification will not explain the anomalous acceleration of the Pioneer anomaly.

As I told earlier, unknown miracles are hard to discuss. So I will not discuss miracles and Quite New Physics here; but a few words will be told in the next Chapter, and I return to the problem in the Conclusion.

3. ON THE NUMERICAL VALUE OF THE HUBBLE PARAMETER

Observational cosmology knows the Hubble parameter H (or H₀) since 1924, as the proportionality factor between galaxy velocities (measured via Lorentz redshift) and distances (measured e.g. via RR Lyrae brightness-period time calibration curves):

v » Hr (3.1)

In GR Robertson-Walker cosmologies, where there is full spatial Killing symmetry, so the global line element must be

ds² = dt² - R²(t){dx²+f²(x)dΩ²} (3.2)

f(x) = (sh x, x or sin x) (3.3)

eq. (2.1) automatically follows for not too far objects, and

H₀ = R^-1(dR/dt) at t₀ (3.4)

So in GR the Hubble parameter belongs to the global, actual Universe. According to recent observations H is somewhere in the range

H = (75-100) km/sMpc (3.5)

where the units are somewhat hybrid, but practical. Now, [2] notes that for numerical value you could get the anomalous acceleration as

|a_an| = Hr (3.6)

with H=82 km/sMpc.

This H value is quite compatible with the cosmologic observations, still GR cosmology in itself cannot explain (1.1), since there the effect would be rather anti-Sunward. Note that in GR cosmologies the Hubble constant cannot be derived from fundamental constants because primarily it depends on the "initial conditions" in the Big Bang. However, of course, if Cosmos is a coherent entity even now (which is not so in General Relativity or even in suggested forms of the still nonexistent Full Unification), then the Theory of the Cosmos may give a (3.6)-type formula, only this theory of Cosmos is not yet suggested/accepted.

By other words, if now the probe does not feel "the whole Universe" when flying, and also the photons fly between the probe and Earth freely, then a form (3.6) is hopeless.

4. NEWTON AND FOMENKO

This Chapter starts quite independently of Chaps. 2 & 3; if there is any connection, that will be seen only later. Note that this Chapter contains some History as well, even if I have relegated some arguments to the Appendix. If you do not like History and believe me, jump to Chap. 5 where some conclusion of this Chapter will be recapitulated.

According to academic folklore Astronomy (and especially Chronology of Eclipses) corroborates some results of History; and in some points where History in herself cannot decide the question, gives an answer which is conform to our knowledge. Some fellows at the periphery of History do challenge the Big Picture of Chronology, but only because they are ignorant both in History and in Astronomy. My point (and I am a physicist) is that this statement surely holds upward from High Middle Ages, but cannot yet be verified beyond doubt for earlier times, and especially not to Early Antiquity.

Observe that verification/falsification of a really interdisciplinary study is rather difficult; the first is even more than the second. There is a nice example mentioned by Sagan [3] who wanted to organise a discussion about Velikovsky's famous work "Worlds in Collision" [4]. As we should know, Velikovsky's theory is based on an assumption that Planet Venus was gorged out by Planet Jupiter in historical times as a comet; of course then historical records should exist for the process, and Velikovsky really enumerated lots of such records. Of course the arguments must then include at least a silence about Venus (the third brightest object of the skies) in the oldest historical times and then some transient behaviour of Venus' orbit for a time. And indeed, Velikovsky mentions King Amissaduga's Venus tablets as well as Joshua's story from the Bible about the anomalous heavenly revolution of Sun in the time of the Battle of Gilead; and many more.

I can use just myself for one horn of the argumentation: a physicist/astronomer generally states that comets are smaller by at least 10 orders of magnitude than Venus; that very probably Jupiter does not have volcanoes but even if he had, the escape velocity on the surface is >50 km/s so Jupiter could have not disgorged Venus; that a temporary stopping of Earth's rotation 3200 years ago would have caused lots of megacatastrophes among the archaeological finds; and so on. But on the other hand, Velikovsky's Bible analysis and historical arguments are at least disturbing for me.

Fortunately, Sagan tells us the other horn of the argumentation. He asked a Hebraist about Worlds in Collision". (I seem to remember that the expert was Patai and Sagan asked him to be an opponent on a discussion; but just now I cannot find the reference.) Now, the Hebraist answered [3]: "The Assyriology ... and all of that Talmudic and Midrashic pilpul is, of course, nonsense; but I was impressed by the astronomy".

Now, the usual argumentation for the corroboration & finishing of ancient history by astronomy of eclipses goes somewhat as follows. Thales of Miletus florebat c. 585 (or 609) BC because ancients mention a total solar eclipse during a battle of Lydians and Medes (Alyattes vs. Kyaxares) which he either predicted or explained on the spot; this eclipse happened according to Astronomy in 585 BC (or 603, or 609) [5], [6].

The III^rd Dynasty of Ur ended in 2007 BC (or 2012/2017) because Archaeologist Rawlinson found the record of a total lunar eclipse [7] and Oppolzer himself in 1880 AD calculated that eclipse to 2050 BC.

Our Classical Greek chronology must be more or less good, because we put its breakout to 431 BC and i) Thucydides in the history of the Peloponnesian War mentions a sequence of 3 eclipses (2 solar, 1 lunar) in definite seasons with given time gaps between them; and ii) astronomers determined the first eclipse of this rather rare sequence indeed as 431 BC.

Neo-Assyrian chronology is absolutely fixed by a (partial but big and well-recorded) solar eclipse which himself Oppolzer dated to June 15, 763 BC.

And so on.

And indeed, the differential equations of Celestial Mechanics are believed to be reliable to calculate orbits and revolutions anywhere outside of the Asteroid Belt for times much longer than 5000 years, back or forth. So there is no problem, is it?

However, the Sun-Earth-Moon system is not a 3-body system in celestial mechanical sense. Therefore we cannot calculate it from first principles! This should have been known from the time of I. Newton, or, at least from Laplace, but it was not emphasized until the end of the XIX^th century and even now it is generally not known by the historians. For briefly explaining this statement before fully understanding the problem: the reason is the tidal friction. Earth is oblate, but until the oblateness were fixed, this would not be a problem. However the liquid hydrosphere is moving, which means that the number of degrees of freedom is in principle infinite, moreover during the relative motion viscous forces work, so the problem is nonconservative. (To be sure, it is also true for completely gaseous Sun, still there the phenomenon is rather negligible.)

But first we must see the sketchy history of "the lunar motion" because the timetable will be important.

Of course both for prediction and for retrocalculation of the lunar motion (and so of eclipses) the knowledge of the whole Solar System would be needed; but since the 1840's, the discovery of Neptune, we practically knew everything. Pluto and the asteroids are quite negligible and the relativistic corrections, of course unknown until 1905, are also.

Indeed, the formalism seemed to be ready at least with the work of Delaunay, which was definitely ready and published in 1867 [8]. Even a decade earlier only the computational technique differed. Hansen in 1857 gave a system of equations and tables calculated by them [9]. Comparing the calculations and the observations it turned out that the equations described the lunar motion correctly (up to observational errors) in the century 1750-1850. Then surely we can calculate forward.

However soon it turned out that the calculations did not agree with the observations after 1857! What is more, in 1878 Newcomb recognised that the calculations yielded incorrect results also before 1750.

To defend Astronomy's honour in front of historians, the discrepancies were moderate. In 1890 the observation showed 18" error in the lunar longitude, so along her orbit. This was just the hundredth of the width of the lunar disc; but that means that if we have bad luck, there could be a problem with a solar eclipse in 3300 years or even somewhat earlier.

The work to find the error started along two quite different ways. One was that of G. H. Darwin, but we will return to this approach (successful in 1879 but rather ignored for a while) in due course. The more conservative effort was taken by Newcomb and Cowell. They checked Hansen's equations. You can imagine e.g. possible pen errors!

For Moon the leading terms formally come from Sun's gravity, but the acceleration is almost as that of Earth, so in a geocentric system terrestrial terms are big, the solar ones small and those from the planets are even smaller. The checks verified the first two groups, but they found two errors in the planetary perturbations. First, Hansen miscalculated the secular acceleration of Moon (we do not know why). Second, there was a term on the right hand side, describing a perturbing force of 239 year periodicity. By calculating correctly, the amplitude (on the rhs.) is 0".27, but in Hansen's book the amplitude was written 21",47! That is a real difference.

OK, they corrected the errors. The discrepancy changed its sign, but for absolute value it did not decrease. Here Newcomb gave up and henceforth tried to get only equations whence he could calculate at least between 1625 and 1875. Somewhat surprisingly, the new totally empirical term ("large empirical term") was almost Hansen's second error: a "force" of 257 year periodicity, with 10."71 amplitude.

Hansen's erroneous term was not a force, but some complicated combination from all the planets. But Newcomb included that combination with the correct but rather small amplitude, so the extra term empirical term formally corresponded an extra planet, with rather substantial mass and 257 year revolution.

Could it have been Pluto, decades before its discovery? Surely not. The period time would be almost correct (Pluto has the period time 248 years), but Pluto, because of its small mass, would make a much smaller perturbation. There is no such planet!

Darwin had an interesting idea. Take a moment when Moon is over mid-Pacific. Then the tide pulls up the water just below Moon (and also in the antipodal point). But Earth's angular velocity is much higher than that of Moon (in her orbit), so by viscosity of seawater and by friction to ocean bed Earth somewhat brings forward the "bulge". This search of gravity accelerates Moon forward, while somewhat "lagging" Moon decelerates the bulge, so, via viscosity & friction, the rotation is braked [10].

It is not easy to calculate this effect. It is definitely hopeless to calculate it from first principles; but we know everything except a strength or coupling. Leaving this factor free we can then fit it from the lunar motion itself.

Darwin indeed estimated the effect, but for first approach now ignore the tidal effects of Sun. This is not an absurd approximation; every sailor knows that solar tides are less than half of the lunar ones.

In the Earth-Moon system still the total angular momentum is conserved. So the phenomenon practically adds the same orbital angular momentum to Moon's revolution which is taken away from Earth's rotation. (Moon's rotation is negligible.) But this means that Moon's orbital motion is decelerating (the greater orbital angular momentum means greater orbital radius, where the circular velocity is smaller). So Moon's longitude decelerates from double reasons: the rotation happens on higher orbit which in itself would mean lesser angular velocity, plus the orbital velocity is slowing down as well.

Since the effect surely must exist and everybody saw an extra longitudinal deceleration, one would expect that Darwin's idea would have been incorporated into the lunar tables; but no: Newcomb's "large empirical term" won and remained in the canonical equations until 1954; Brown's "empirical term" [11] even remained the same as of Newcomb. (However, Darwin's idea explicitly appears in the classical sci-fi or H. G. Wells’ The Time Machine.)

To the 50's atomic clocks directly proved the slowing rotation of Earth: then finally the experts got rid of the "large empirical term" and instead they declared that the "Mean Sun" on large scale slows down. By other words there is still a time variable where the equations of the mechanics are true, but this is not the time what is measured via Earth's rotation (it would be simple to say: but the time measured by atomic clocks; but the choice of the time standard still took more than one decade). In this time variable the lunar longitude gets an extra term, which even for moderately long times can be approximated as

Δλ = -8".72 - 26".75t_c - 11".22t_c² (4.1)

where t_c is the time (after 1850.0 AD) [12].

By other words, the best lunar tables contain an extra angular acceleration

λ^·· = -22".44/cy² (4.2)

So going futureward in local time (which is not the atomic time but rather is shown by Sun as pointer) eclipses happen later and later. In a nonrotating geocentric frame slowing down Moon catches Sun later and later. In pastward the effect is opposite.

It is not easy to determine the lengthening of Terrestrial day. Still, observations suggest cca. 20 s in a million years. This is more or less conform with palaeontologic observations (some annual and diurnal rings can be seen on very ancient "coral" fossils), but the "measurements" are not sufficiently reliable).

But then historical eclipse calculations performed in XIX^th century must be all wrong and even most XX^th century ones up to 1954, since the until that time astronomical tables the lunar tables contained not a constant lunar deceleration but a periodic superplutonian perturbation. The term was fitted in a 250 year interval centered around 1750, but for times longer than 257 years its effect would average out, so the term could not imitate the slowing down at millennial ranges. For lunar eclipses the difference is "only" that the phenomenon happened at different local times; however in an unlucky case this may mean e.g. that an eclipse really started when Sun & Moon still were belong the horizon. But for total solar eclipses, the difference will be dramatic. The total eclipse happened, but at a different geographic narrow strip. For partial solar eclipses the phenomenon might be seen even from the incorrect place, but the extent of the occultation might be anything.

And now note that such heroic astronomer helpers of History as e.g. Stockwell & Oppolzer worked in XIX^th century and used, instead the tidal friction term, either Newcomb's "empirical term" from a nonexistent very massive planet at the place of small, unknown Pluto, or the pen error of Hansen! If miraculous cancellations of errors did not happen, then for solar eclipses of antiquity the calculations of the classical heroes are totally irrelevant (except as History of Science/Scholarship); for lunar ones they may or may not be first approximations.

Obviously one could, with great work, to repeat all classical calculations without any "empirical term" except (4.1) on the basis that maybe the tidal deceleration was constant in a few millennia range. However there appears 2 problems.

1) With a few exceptions (I will mention soon one) the recalculations did not yet happen.

2) (R. R.) Newton in 1972 tried to check if the deceleration was constant in the past or not [13], [14]. His result was very unexpected.

He calculated then historical eclipses in such a way that the extra acceleration in Moon's longitude (proportional to the strength of the tidal friction) was a constant (this assumption seemed harmful enough: at least the shorelines have not been changed too much in neothermal times), but a free constant. Now, you take the historical eclipses of a given century. You calculate the free constant of tidal accelerations necessary for the individual eclipses needed to be the eclipse "good". Then you average the constant, called D, now = -22".44/century². In such a way you reconstructed

D = D(t) (4.3)

for the past, as far as (solar) eclipse records are available.

And then he got Fig. 1 of [13]. The present D is near to -20"/century². That value is more or less good (small deviations are arguable) back to AD 1300. There is a "transition time" between 600 and 1300 AD. And before 600 AD the proper D is near to 0 and positive.

Note that by definition D(t) is an average value between t and present. So an almost 0 (but positive!) D from classical antiquity to present means either a very definitely accelerating Moon (in longitude) in antiquity, which would contradict to tidal friction, or some transitional acceleration from an unknown external cause starting after antiquity and ending before pendulum clock observations (say, between 600 and 1300 AD, as Newton's Figures in [13] suggest and what is the least absurd assumption because of the state of astronomy in Medieval Europe). But the second choice is only slightly less problematic than the first one

Let us take briefly first Thales' Eclipse as an example: Thales' Eclipse is mentioned, e.g., in Herodotus. Alyattes, King of Lydia is fighting with Kyaxares, King of Media at R. Halys. Just before sunset there is an eclipse, and there is a danger of panic in the Lydian host, but Thales, advisor of Alyattes, explains what is happening. No month or day is given; however local time is then given. Stockwell concludes that 3 years are possible, 609, 602 or 584 BC. E.g. the totality line of the 609 crossed R. Halys at the Gordion-Boghazköy line at 16h 28m (although he likes better the 602 eclipse, I do not know why). Recently the 584 eclipse is more accepted. But remember that Stockwell's Moon equations were wrong!

Now, somebody could calculate with D=-22".44/century² total solar eclipses near Halys just before sunset and select the most proper. Then Ancient History changes somewhat. But, as an alternative, one may believe in the usual chronology; then Thales' Eclipse is an evidence for acceleration of terrestrial rotation in ancient times, which is quite surprising. Or one may say that there is some problem somewhere, and now we cannot say what Earth's rotation did in Classical Antiquity, which is harmless but not constructive at all.

A. T. Fomenko, excellent differential geometer, chose a third way of approach [15]. As a mathematician, he tends to believe in equations. Anyways, our best equations for the lunar motion are given in [12], and the essential change of D would need essential changes of maritime geography; and History tells us that e.g. the shorelines in the Aegean region were essentially the present ones back to at least 700 BC. But Fig. 1 of [13] shows great changes in D before 1300 AD. On the other hand, History cannot be checked by measurement while we know lots of historical falsifications; while the present D is measurable and great changes in a few millennia are improbable. Then Fomenko reinterprets Fig. 1 of [13].

After 1300 AD there is no problem, while before 1900 even falsifiers could not falsify a nonexistent eclipse with the present D because then everybody used the wrong equations. So these eclipses are genuine even if political lies may contaminate the chronicles.

Between 700 & 1300 AD the smoothened D(t) curve goes from the neighbourhood of 0 to the present, so correct value. However Fomenko's figure [15] suggests that this transitional D is an average of eclipses with the present D and of ones with 0 values. Then some eclipses together with their usual dating are genuine, some ones happened, but were included in falsified chronicles with falsified dating, and finally some eclipses were construed from thin air.

Before 700 AD with the usual historical datings all eclipses would need the impossible/improbable tidal acceleration. So no traditional dating can be genuine; a few eclipses happened later (say, around 1200 AD), and the majority is mere fantasy.

Now let us see the example of the 3 eclipses of the Peloponnesian War according Thucydides [16]. According to the historians' consensus Thucydides was contemporary of the War, in Classical Antiquity and his book is essentially genuine albeit sporadic falsifications may contaminate it. The text mentions 3 eclipses, in the 1st, 8th and 19th year, we know the seasons, once the very month, and of course we know the days because the Athenian calendar was lunar. So the triad is:

Bk 2, Sect. 28, solar. Year 1, summer, total, in Greece, after noon.

Bk 4, Sect. 52, solar. Year 8, very beginning of summer, partial, in Greece.

Bk 7, Section 50, lunar. Year 19, summer, seen in Sicily.

Such a sequence is rare, so let us see 3 different approaches.

1) I. Newton (1728): The War started in 431 BC, so the triad must be: {-430, -423, -412} [17]. Very probably Newton did not calculate the eclipses but used the Parian Marble [18] + some Humanist timetables.

2) J. Stockwell (1890): The triad is: {Aug. 3, -429, March 21, -422, (-411)} [6]. One year difference to Newton, but exact datings; however from wrong equations.

3) A. T. Fomenko (1981): Two and only two solutions for constant present D, as follows. {Aug. 22, +1039, Apr. 9, +1046, Sep. 15, +1057} and {Aug. 2, +1133; Mar. 20, 1140, Aug. 28, 1151}; no solution before +1000 from the correct equations.

I stop here. Obviously either we must choose between A) traditional history and great, improbable changes in tidal friction in mere 2500 years, including ancient acceleration of rotation; and B) highly nontraditional history but no great changes in tidal friction; or we may choose the harmless but completely defeatist C) we do not know the lunar equations (and so cannot retrocalculate eclipses) before 1300 AD!

5. WHAT DO WE SEE FROM THE ROTATION OF EARTH IF WE BELIEVE EVERYTHING?

According to present best physical cosmology, geophysics & astronomy we learn that

1) in the Universe there is no preferred point (since even our Earth is not such); by the best possible measurements we observe a Doppler shift of cca.

v = Hr (5.1)

H = (75-100)*km/sMpc (5.2)

which is quite well explained by a solution without preferred points of the Einstein equation of GR;

2) the Earth is rotating with the angular inertia now as

L=αMR²ω (5.3)

where α < 0.4 characteristic for the internal shape of density and surely ω<0.07;

3) History gives us solar & lunar eclipses to retrocalculate/check.

Now, assuming all 1)-3) problems arise. First let us ignore completely Point 1) on the ground that maybe the cosmologic expansion disturbs infinitesimally the rotation of Earth, which seems indeed so in our theories. Now, let us assume that α, R & M are constant in 2) and do the retrocalculations. Then we run into the problem of Newton in [13]: the present deceleration may be wrong before 1300 AD, and surely before 700 AD. Assuming the constancies the only ways out are i) exotic forces in the Solar System at 700 AD [13]; or ii) falsifications in mss. believed more than a millennium old [15].

Well; we can drop Point 3). Then we do not believe in History before Middle Ages, but there is no problem with Celestial Mechanics (except for the Pioneer Anomaly which will be solved somehow in due course). This is a possible if somewhat purist standpoint; of course then we cannot give dating to Pericles or Jesus Christ until somebody elaborates New Chronology. This is the Fomenko Project [15].

Of course, we can drop, alternatively, the constancies of data in Point 2). This is a less radical way out; but as we going to see it, hardly less hanging in mid-air.

The Mean Day cycle is slowing compared to precision pendulum clocks or modern atomic clocks. The difference of mean astronomic time and atomic time is called Delta-T, and one would expect a parabolic ΔT(t) because of the roughly constant deceleration. This parabolic shape is not seen too well (explanation will come), but of course we can make a parabolic fit. Using the last 4 centuries for deceleration we would get c. 1-3 hours back to 1^st c. AD. (ΔT tables depend on authors; in the NASA tables ΔT was 0 in 1870 (!; a calibration) and was cca. 3 hours in 1 AD [19]. As for more exact data: at the beginning of 1st c. ΔT is: 10520 s [19], 9462 s [20], 9848 s [21], 10600 s [22].) But then there will be problems before 600 AD!

Now let us forget totally about the tidal friction (as it was usual till Darwin). A similar decrease of angular velocity could be caused by some increase of the terrestrial radius R. If the day is lengthening as

T_D = T₀(1+t/t₀) (5.4)

then from conservation of angular momentum

R = R₀(1+t/2t₀) (5.5)

while

ΔT = -t²/2t₀ (5.6)

if the 0 point is that of the calibration. Now, let us take 2 hours in 2000 years, for simplicity. Then

t₀ = 1.4*10¹⁷ s (5.7)

or 4.4 billion years. While this is almost exactly the age of Earth, the value is aphysical, because the tidal friction does exist. However, if a part of ΔT is not explained by the tidal friction, then there will be an anomaly, and a bigger t₀.

Now, with the above t₀ R would increase in 2000 years by 1.5 meters. Past terrestrial radii are unknown for such precision, I may imagine (but am not sure) that such differences in sea level could be detected in the 1^st c. geography of the Aegeic, and uplift of similar range is reported from Scandinavia. And surely, human building activity is the main cause behind the bad parabolic shape of ΔT in the past centuries: high buildings increase the angular inertia so decrease the angular velocity; more or less in the same order of magnitude as tidal friction does.

6. ON THE RELIABILITY OF OBSERVING ANCIENT ECLIPSES

Here "ancient" stands for cca. "before telescopes". Not as if the observation of an eclipse needed a telescope; but after Galileo the number of observers went up. In Hellenistic ages there are eclipses about which a single record is extant, and then one may assume that the report is not a true observation; after Galileo lots of astronomers & dilettantes observed the eclipses.

Of course heavily overcast sky made impossible the observation, and we generally cannot know the ancient meteorology. So let us ask a conditional probability: what is the probability that if the observer had believed to observe an eclipse then it was really an eclipse; and if that was, how reliable was the time of the eclipse.

Practically nothing mimics an eclipse. Of course, peripheric partial eclipses may have escaped observation, especially solar ones. One does not look into the solar orb without filters if it can be avoided, and a minimal decrease of the daylight brightness may be overlooked. However it is difficult to imagine any celestial event which could have been confused with a, say, 50% eclipse by professional astronomers since Old Babylonian and Old Egyptian times.

The solar orb never shows phases so any truncation of Sun must be an eclipse. Moon have phases; but lunar eclipses happen at full Moon, and the truncation is transient, for a few hours at most, so an astronomer generally followed te whole process during the night.

Especially total eclipses are interesting & dramatic. The Sun cannot be seen, the daylight illumination goes down to an early evening dimness, something strange can be seen at the edge of the blackness substituting the solar orb, and a few stars come out to the sky for minutes. Total lunar eclipse is interesting too, e.g. because the lunar orb generally remains on the sky as a dim, coppery ghost of Moon.

As for the time data, any astronomer from Sumerians upwards was familiar with the calendar, so day, month & year should be correct. The same is true for medieval monks. For other observers, it depends on the circumstances. As for the hour & minute, it seems to be a thumb rule that reliable ancient astronomers could guess the time for 1/3 hour; even nonprofessionals must have observed "early morning", "not too much before noon" and such, so say they should have reported a correct 3 hour interval.

Third hand information, hearsays and literary inventions do not fulfil the above accuracies. Lots of reasons can be imagined why a historian/essayist invented an eclipse from thin air. As for a semirecent example look at Mark Twain's A Connecticut Yankee... where somebody somehow gets back to VIth c. AD, in the time of King Arthur & Mage Merlin, and in a tight situation survives by predicting an eclipse. Now, in the novel the Yankee remembers the eclipse listed in an astronomical table. However it is generally told that Mark Twain did not consult astronomical tables and there was no retrocalculated eclipse handy at the end of the XIXth c. for the author's purposes. He simply declared it.

As for ancient & medieval purposes observe that an eclipse was rather mysterious & supernatural for almost everybody outside astronomy, and it was a commonplace to regard them as "some signals of something". So surely it was a part of chroniclers' techniques to record some eclipses before key events. This might have been done even if the chronicler was not sure that the eclipse happened; much easier if there was a tradition of the eclipse, "only" its exact time was not preserved.

Obviously such "eclipsewrongs" should be excluded from discussions. The problem is that this task cannot be done within Astronomy. E.g. only historians may know if a chronicler is unreliable, if the chronicle is a forgery &c. And in cases even historians are not sure.

7. THE EFFECT OF CHANGES OF EARTH'S ROTATION ON ECLIPSES

This Chapter will be a commonplace for astronomers. However, according to my first-hand experience, not for historians.

Atomic clock observations indicate some changes of the length of day, there is a fluctuation from year to year c. 0.1 ms and the tendency was a lengthening until 1973; then there was a shortening until 1987, and again a lengthening until 1994.

As for longer periods the tendency of lengthening seems to be almost sure. Delta-T tables indicate that the day now is c. 2 ms longer than at the middle of XIXth century and c. 13 ms longer than at the beginning of telescopic & pendulum clock observations. Of course such 4 century tables are composed mainly from eclipse observations.

This lengthening is expected from tidal frictions, and, independently of the friction from the eclipse data as well. Morrison & Stephenson have been working on this from the 80's, and report a rate +(17-18)*10^-6 s/y [23], [24], [25]. They also observe a medium-range oscillation of c. 5*10^-6 s/y, and, what is really surprising, state that pure tidal friction would cause 23*10^-6 s/y lengthening rate. No doubt, in principle the tidal friction part of the decay of Earth's spin can be calculated from lunar observations, because in an Earth-Moon system far from any other gravitation the total angular momentum would be conserved, so the tidal friction caused by Luna converts Earth's spin into Luna's orbital angular momentum; I simply note that the -5*10^-6s/y "residual lengthening" is not a straightforward datum.

However it would not be proper to go into the details of this lengthening in this Chapter. Instead, let us accept here +18*10^-6 s/y, and let us see the consequences for ancient total eclipses.

The clock of Celestial Mechanics of Newton & Laplace ticks uniformly, the final equations being those of Newton's Dynamics & Gravity. However Earth's rotation is not uniform; therefore i) Sun rose later and later in the time of Celestial Mechanics as we go farther to the past, and ii) Moon's orbital angular velocity was greater and greater.

The two effects are not the same, even if Earth's spin is transferred to Luna's revolution. For example, Sun also causes tides even if smaller ones. However for first approach, let us consider only the consequences of the lengthening of the day. That will be enough for a while.

In 2000 years the lengthening have gone up to c. 3.6*10^-2 s, but ΔT is accumulating, so the Solar Day Time was more by 1320 s, c. 3.5 hours than the Celestial Mechanics time. We have seen that detailed calculations gave rather 10600 s [22], but I told that in this Chapter we remain at first approximations.

Now, 3.5 hours difference must be recognised in the record of any firsthand observer. For total lunar eclipses this 3.5 hours is cca. the duration of totality. What is more, an eclipse we retrocalculate without tidal friction may have happened at the definite longitude below the horizon, so unseen.

Now, the tidal friction was first seriously and quantitatively suggested in 1879 [9], and not used in eclipse retrocalculations for some more years. I would try with the assumption that all classical retrocalculations of the XIXth century happened without tidal friction. I will return to this problem soon; but let us see total solar eclipses.

Total solar eclipses are brief, a few minutes. This means a c. 50-100 km strip at a fixed latitude. As a thumb rule we may tell that if a total solar eclipse is miscalculated by 10 minutes then it did not happen at the fixed geographic position. The exception is when the totality strip is (almost) W-E; then the strip ius sifted in itself and only the local time changes.

Of course it did happen at some other locality, displaced to East or West. Namely, Luna got between Sun and Earth at the calculated Celestial Mechanics time, the shadow cone reached Earth's surface, but at another longitude. (Again: in this Chapter I ignore the changes in Luna's orbital revolution and the solar tides). About 1550, when Nostradamus gave his dated prophecy about the 1999 total solar eclipse the effect of tidal friction may have been 9 minutes in the present approximation, so for a definite position it might or might not be observable; but Nostradamus did not give definite localities. For older eclipses, XIXth century retrocalculations from Newton's equations must have given totality strips not even overlapping with the true ones.

I will immediately return to the phrase in italics; but first let us see if the historical error from this fact is moderate or not.

Consider the situation when we have an ancient event which we want to date from eclipse retrocalculation. E.g. this is the acme of Thales: roughly a century later Herodotus recorded the tradition that Thales was present at a Lydian-Median battle at River Halys in the HQ of King of Lydia Alyattes, when a solar eclipse happened and Thales, giving an explanation, prevented the panic. From the recorded reactions everybody guesses total eclipse, so Oppolzer could give a date and this 585 BC is conform with our historical knowledge as well as with the chronological Tables of Alexandrine historians.

Now assume that this eclipse is miscalculated. How much time is needed for a real total solar eclipse to happen at the given place?

The question is not quite definite: what is exactly the same place? However after long periods of observations we know, as an observational fact, that

1) at an exactly defined geographical point about 45° latitude you can expect at least 2 solar eclipses per year, but the probability is high that they will not be total:

2) the 1999 summer eclipse was total at a part of Hungary, but the previous such was in 1846;

3) for a European country of moderate size total solar eclipses are separated by 2-3 centuries.

So a misretrocalculated total solar eclipse means that there was no total eclipse in the neighbourhood at that time at all, and a not calculated one might happen at any time but the chances are that they happened a century earlier or later.

The discrepancy will be even bigger if we have extra information as e.g. the season of the eclipse or the eclipse was a member of a sequence. An excellent example is Thukydides' 3 eclipses during the Peloponnesian War [16] discussed in an earlier Chapter. I cannot calculate how frequently such a sequence repeats itself; but it must be mightily infrequent. And the classical retrocalculations are conform with classical history!

There is some mystery here. Fomenko believes that the "ancient documents" were written in Rinascimento and that the Peloponnesian War is a fiction written from a war within Greece in the XIVth century. However I would like to be cautious. From Classical Antiquity there are ways to calculate (more or less) eclipses from eclipses; and it is possible that our classical XIXth century astronomers (as e.g. Oppolzer & Stockwell) used them too. Of course then you should be sure about the previous eclipse and so on ad infinitum; but let us see here the old method. I follow a classical XIXth century astronomer here, Houzeau [26].

The first phenomenologically discovered cycle of eclipse repetition was the Saros, 18 years and 11 days. We have several incommensurable periods, (in ancient language) the orbiting of Moon around Earth which is the synodic month, the motion of Sun on the sky, i.e. the year, the orbit of Moon’s perigeum (the anomalistic month), and return of Moon to the same “knot” or intersection of orbit of Moon with the ecliptic, the draconic month. For a complete repetition of eclipses one should find the common multiple of all. According to modern measurements the period is as follows:

Day	1 day = 86400 s
Synodic month	29.530589 days
Anomalistic month	27.554551 days
Draconic month	27.212220 days
Year	365.242199 days

The values now are reliable to all digits given here, although they are known to change very slowly. In Classical Antiquity the precision was somewhat lower, but enough to conclude.

If two eclipses are separated simultaneously by integer synodic and anomalistic months and at least a half-integer number of anomalistic ones (being two “knots”), then the whole processes of the two eclipses are very similar; if, in addition there is integer number of days in it, then even they happen at similar local times. Of course, it would take a tremendously long cycle to satisfy this.

What is absolutely necessary to an eclipse is being at a knot at “new moon”; then the detaist may change, but there will be another eclipse (except it happens below the horizon). Now, 223 synodic months are almost exactly 242 draconic ones, so there will be a new eclipse after 223 synodic months. This is Saros.

Unfortunately, the 223 lunar months mean 6585.32 days, so in Mean Day Time there is cca. 8 hours difference. So the eclipse happens, but maybe already under the horizon.

To correct this, in Hellenistic times the Exeligmos was proposed: 3 times the Saros. It is almost good, a 19,756 day cycle (roughly 54 years 1 month). In the next Chapter I will demonstrate the quality of the repetition. Here I note only that even in exeligmos there will be an hour difference. So after 54 years the totality zone is still away for 15°.

Now, Saros & Exeligmos are good, but not too good for the anomalistic month. So Moon’s distance is not the same in a Saros series, and then a total eclipse may be followed by an annular one or vice versa.

For synodic, draconic & anomalistic months the shortest useful “multiple” is the Tritos, 135 synodic months ≈ 146.5 draconic ones. The anomalistic months are not integer, so the actual process of the eclipses may be different.

Hipparchus on the other side in IInd c. BC suggested a cycle of 126,007 days, almost common multiple of the synodic & anomalistic months and not too wrong for half draconic ones. The clock error is c. 1 hour, so only a minority of eclipses go below the horizon. The 126,007 days is 345 years.

There is one important cycle discovered after Antiquity: the Inex. 358 synodic months are almost equal to 388.5 draconic ones.

On a conference I heard about a c. 145 year cycle as well, but I cannot now find the reference.

The above cycles are purely phenomenologic, coming from long observations. So they are independent even of Physics. However with lengthening days the point where the lunar umbra reaches Earth’s surface shifts West, so still the longitude will not be the predicted one.

Astronomers use these cycles, so it is nontrivial to tell that the Peloponnesian War could not start in 431 (or 430 BC because XIXth century retrocalculations gave that date). However, minimally, we can tell that each date conform with retrocalculations without tidal friction should be suspect. And, as seen, the true eclipse may have happened a century away. But you should completely rewrite Ancient History with Pericles in 540 or 340 BC.

No surprise that the lunar acceleration parameter is calculated almost exactly 0 from eclipses of Classical Antiquity [13], [14], [15]. And then there is some discrepancy between Best History and Best Astronomy, not easy to remedy without very serious consequences.

8. EXAMPLES FOR SAROS, EXELIGMOS & HIPPARCHUS’ CYCLE

NASA maintains a site for eclipses [27]. Here you can find time data for eclipses from 2000 BC to 3000 AD, and in many cases some information about the observed eclipse. I hope the tidal friction is taken into account in an efficient way, bur for that I do not know the details. For the last 400 years the retrocalculated eclipses seem to fit.

Let us take the solar eclipse of 11th Aug. 1999 as starting point. For many decades this was the first solar eclipse whose totality zone traversed Central Europe. The umbra reached the terrestrial surface South of Nova Scotia, traversed the Atlantic Ocean going to ESE., the Isle of Scilly, Southwestern England and Normandy, then some other parts of France, Bavaria, Austria, Hungary, Rumania, the Black Sea, Turkey, peripheral Syria, Iran, Pakistan & India, and left Earth somewhere in the Bay of Bengal. The maximal duration of totality was just under 2.5 minutes.

The previous Tritos pair of this eclipse happened on 11th Sep. 1988. The shape of the path was similar as for the 1999 eclipse, but shifted much to SE. The eclipse started at the shores of Somalia, for a while it followed more or less the Equator, then went to SE, but always remained over the Indian Ocean.There was just time to observe the subsequent member of the Tritos sequence too. On July 11, 2010 there was an annular eclipse at the Pacific. At sunset it ended in Patagonia.

The previous Saros pair happened on 31st July 1981. While the general shape of the path was not too unsimilar to that in 1999, it was much shifted to E, as expected. The maximal length of totality happened about the Lake Baykal, then to ESE the umbra traversed Sakhalin, and continued towards Hawaii.

The last Inex pair happened on 31th Aug. 1970. (The Inex sequence was the only one of the sequences mentioned in this Chapter not known by Clasical Antiquity.) While Inex pairs are geographically more similar to each other than Saros ones, the umbra of this eclipse completely missed Central Europe. Indeed the eclipse was visible in the Southern Pacific.

The last Exeligmos counterpart was on 9th July 1945. Being Exeligmos pairs much more similar than simple Saros ones, the path of the umbra was quite similar than in 1999, but shifted northward by some c. 20°.

The last Hipparchus pair happened on 12th Aug. 1654. Indeed, the umbra followed almost the path of the 1999. The northward shift was moderate. But even this moderate shift means that the umbra traversed Scotland instead of SW England, or Poland instead of Hungary.

All of these eclipses, including even the 1654 one, were observed and recorded. For ΔT this fact gives measured values as

Year	ΔT, s
1999	61.7
1988	56.1
1981	51.9
1970	40.9
1945	27.1
1654	42.4

Before 11th Aug. 1999 the last solar eclipse traversing present Hungary (just under 100,000 km²) was on 8th July 1842. The two eclipses did not belong to the same sequence in any senses told in this Chapter, even if the path of the umbra in Hungary was similar; so the similarity is "accidental". The time gap is 157 years, only marginally smaller than the estimate of "200 years between total eclipses at the same pont". This is so because the total linear size of present Hungary is only slightly larger than 264 km, the theoretical upper limit of the width of the totality strip.

9. SCOTTISH VERDICT IN ORTHODOX HISTORY VS. ASTRONOMY

We must stop sooner or later with the eclipses, because the topic is only one trace toward the goal of the present study while the historical background is almost an infinity. So in this Chapter I argue about postponing the decision whether tidal friction & al. are disproving the history of the last half millennium or not. That history was claimed to be in accordance with the lunar equations of XIXth century; and then, as I argued, it is hard to get now an accordance between modern lunar equations and orthodox history.

Obviously there would be ample reason to supervise all the "historical" solar eclipses of Antiquity; but it seems it will not be done: nobody is interested, mainly because of very weak coupling between historians and physicists. It is a pity; Gurzadyan's recalculation of Šulgi's Lunar Eclipse [28], [29], [30] demonstrates that the method is effective (here exceptionally even for a lunar eclipse).

First I deal with only 3 total solar eclipses: all very important. They are:

A) the eclipse on which the Neo-Assyrian and Neo-Babylonian chronology is based;

B) Thales' eclipse (according to Herodotus); and

C) Thukydides' first eclipse.

Assyrian tablets speak about a solar eclipse, not total but substantial at Babylon and at Niniveh, and if we can date that eclipse, and believe the Assyrian King Lists, then Assyrian and Babylonian chronology is fixed at least back to the beginning of the Kaššu dynasty in Babylon (the reign of Agum II), when Babylon recovers from the catastrophe of the occupation and demolition of Mursilis I. In the most accepted chronology the Mursilis attack is 1595 BC (Middle Chronology); by recalculating Šulgi's lunar eclipse Gurzadyan suggested 1499 instead, but we are not yet interested in Šulgi's eclipse. The records for Babylon are without gap from the recover, and so the data from Agum II depend only on the well-documented solar eclipse many centuries later.

Oppolzer dated this solar eclipse to June 15, -762, and this date is not influenced by tidal friction. Namely, the retrocalculated path of totality was almost exactly W-E, and it crossed Northern Mesopotamia. So the local time when the eclipse was observed is not that of calculated by Oppolzer and still that eclipse was seen in Mesopotamia.

For Thales' total solar eclipse somewhere on the Western bank of R. Halys, recorded by Herodotus, earlier various data were given between 609 & 585 BC, but for historians all were acceptable. Now it seems that only the latest one survives: the eclipse on May 28, -584 was total on a strip between Central America and Anatolia; the trip was almost W-E and ended somewhere in Asia Minor at sunset. Minor geographic details may depend on the exact ΔT; but this eclipse is marginally possible still, no doubt because of the almost W-E path.

But Thucydides' eclipse is not clear. The author mentions 2 solar eclipses, the first total, the second partial in Greece, one in summer, the other "at the beginning of summer", in the years 1 & 8 of the War. From the Parian Marble Newton himself [17] fixed them to -430 & -423, even if Stockwell shifted them 1 year upward.

Thucydides, however, mentions a “crescent” when speaking about a total eclipse. In Warner’s translation [16]: “The sun took the appearance of a crescent and some of the stars became visible before it returned to its normal shape.” But stars do not become visible at a partial eclipse.

There was no total solar eclipse at all in -430 (so “stars … visible”), and there was one in -429, but the totality strip touched the Cape, Southern Madagascar, and then crossed Indonesia (and in January). The centrality strip of the -430 annular eclipse was rather far north from Greece. In March 21, -423 a nice partial eclipse was seen from Greece; in -422 an eclipse on March 10 was just marginally partial in Southernmost Greece. So the -430 eclipse would be possible except that Thucydides speaks about a total one and neither he nor anybody he knew could observe totality.

Now, we may believe in Thucydides about visible stars, but then the eclipse could not have been even near to -430. The next eclipse total anywhere near to Greece is Jan. 18, -401, and even this is not too hopeful, because January is not the beginning of summer. Also, there is no good eclipse 7 years later.

So Thucydides seems unreliable. Maybe he wrote historical facts, but he did observe a partial eclipse, or somebody other observed, and then he wrote sentences about total eclipses read somewhere. Maybe. Or the problem is even bigger, and then we are back to R. R. Newton & Fomenko.

I am a physicist; therefore I, contrary to many (but not all) historians, do not believe that our main goal would be to formulate nice, round and soothing stories. I think there was a Past, very probably it was unique, and one of History's aim is to know this unique Past. Eclipse retrocalculations, C¹⁴ or isotope palaeothermometers suggested ways to get data from Past directly.

Now, look at [31] in 1969. It speaks about a temperature maximum bw. 5500 & 3000 BC, 2.5 centigrade above the present; other textbooks mention even 3-5 centigrades. This phenomenon is called "Atlantic thermal optimum" or the "altithermal age" or such. But today we hear that a 2.5 centigrade warming up would cause the breakdown of human civilisation, massive extinction & such, so now we do not believe the past altithermal optimum. OK, maybe the warming up was smaller; but we believed to use objective palaeothermometers 40 years ago.

As for C¹⁴, now we do know that precise nuclear measurements are not enough to get reliable data. A process called calibration is needed as well. The clearest calibration is taking the palaeotemperature into account: the C¹⁴ concentration in the atmosphere is temperature-dependent. But there are more tricky deviations as well. When trying to date Greenland Viking bones, first the results were contrary to any history, but soon somebody recognised that the Greenlander lived mainly from the sea, and the C¹⁴ concentration of seawater is different from that of the atmosphere. So Fomenko simply regards calibration as a way reconciling observations with orthodox, and according to him, falsified, history. We do not have to take sides here; only note that calibrated C¹⁴ ages are too indirect tools to decide.

And now the eclipse retrocalculations. Historians might have believed in eclipse retrocalculations in XIXth century: they did not know the problems with Moon, but they did know that astronomers had found Neptune by means of Celestial Mechanics. So for them Celestial Mechanics was a Magic Wand handled by mages. Of course, the astronomers did know the problems and "calibrated".

The inclusion of tidal friction seemed to eliminate the "calibration", but no; the problem remained even if for somewhat smaller degree. If tidal friction would be constant and there were no crustal motions, building activities & such, then there would be a constant deceleration of Earth's rotation and a constant transfer of angular momentum to Moon's orbiting. Then we could calculate the constant from the observations of the telescopic age and then we could extrapolate back for many millennia. However, we do observe the non-parabolic shape of ΔT(t) in the telescopic age, we have R. R. Newton's ingenious demonstration about the problem, we have Morrison & Stephenson's calculation showing a quasioscillatory part of ΔT(t) smaller but still 30 % of the "leading, friction" term, and know that such things as changes of Earth's radius by 1.5 m (improbable but not impossible), shoreline changes changing water-land ratios (specific gravity 1 instead of 2.5) or motions towards isostacy (well documented at least in Scandinavia) might change the length of day throughout the historic ages in the needed order of magnitude.

The present Canon of Eclipses produced by NASA is the present peak of state of art; but it is a compromise. The compilers do not believe in all chronicles, but they believe in some records. ΔT is calculated from the better documented records, then smoothened, interpolated &c. What is unexplained, remains unexplained.

If we choose to believe History from Antiquity upwards totally, then lots of problems arise from the Canon. If we believe that History more or less, then everybody can try to produce a New Synthesis; until the New Synthesis there is no final answer. If we like Illig's idea, we can excise sections from History telling that that special period is a mere invention; then maybe the corrected historical year numbers become more conform with the Canon. Or we may accept Fomenko's radical suggestion: everything is very seriously suspect in Russia before 1682 AD, in Western Europe before 1300 and in Byzantium before 1050. Then all the problems demonstrated first very succinctly by R. R. Newton have been heaped to the tables of historians and Physics is not disturbed at all. This is a possible and convenient approach, even if I feel it minimalist. And this is possible to be extended back to 763 BC, from that anchor even to somewhat more back we may believe in history; but as we are just going to see, the previous anchors are even more seriously suspect.

Namely for Early Antiquity additional problems arise: and this is the proper pace for Šulgi's Eclipse, c. 4 millennia ago. Of course, as you go farther and farther into Past, everything is more and more hazy; but Šulgi's Eclipse is a good example when brand new Astronomy collides with orthodox but modern history. As for the first, see [32], as the best for astronomical argumentation, but of course [29] and [30] give some details as well. As for the orthodox but modern history, see the King Lists in an Appendix of [33]. Alternative readings of the Lists vary, but generally only a few years from Middle Babylonian/Assyrian times, and there are also good list for Sumerian & Old Babylonian times. After the astronomical reanalisation [28] was written as synthesis; however [28] and [33] contradict each other in a tricky way not revised in [28]. Let us see.

Mesopotamian chronology is based on many sources, but mostly on the King Lists and short Chronicles written on indestructible bricks, and on 2 eclipses. The latter of the eclipses is the 763 one discussed above. The earlier one is Šulgi's Lunar Eclipse c. 4000 years ago. Lunar eclipses are repeated quite frequently to be not too useful for dating, but Šulgi's Eclipse is very well documented. Namely, already just after the eclipse it was regarded as a bad omen, King Šulgi died next year, and then the description went into the omen thesaurus, copied repeatedly. The eclipse happened in the 47th year of Šulgi, History preferred the middle of XXIth century, and astronomers calculated one at 2048 (or 2050, according to other sources) in the good month, so [33] gives the reign of Šulgi as 2094-2047. This is a solid anchor, and then Mesopotamian chronology can be calculated until the Fall of Babylon in 1595, when the Hittite King Mursilis I occupied and razed the city and the Hamurabbi dynasty vanished from History. (We shall soon see how this can be calculated.)

After Muršiliš' raid Babylon is empty and in ruins for some time, but then becomes a big city again, but now under the dominance of the Kaššú Dynasty. The Kaššús are a mysterious enough people, neither Semitic nor Indo-European, originally maybe hill people at Northeast (maybe in Mt. Zagros; a people of similar name is mentioned even in Classical Antiquity somewhere there), and for a time some pastoralists in Mesopotamia far from big cities. Surely the Hittite campaign left them undisturbed, and so the Kaššú tribal confederation (or anything) remained the strongest power in Akkad, so able to start Consolidation. (The Southern part of Old Sumer seems to be at most a loose dependence for a while. Assur seems to be independent from the beginning.)

The Kaššú Dynasty starts with a "Dark Age". This is caused partly by the fact that the first several Kings of the King List were sheiks of the countryside, far from centers of literacy. But also, observe that in the time of the ascent of Cuneiform Studies, in the second half of XIXth century, in Europe races were in fashion. For Europe of that time there were 3 "important races", which we now regard rather linguistic families. In the terms of that age they were the Aryans, Semites and Turanians. The first meant more or less the present Indo-European family (of which Aryan now is the Indo-Iranic subbranch), the second term is used even now more or less in the same way as in 1850, and the third now forms the Uralic and Altaic families. In addition, in the lack of clear evolutionary thinking (we are just in the time when Darwin is finishing his first great book), people loosely believed that there are older and newer languages/races/peoples amongst the extant ones. So the par excellence Aryan is Sanskrit, and texts similar to Sanskrit must belong to Mainstream.

To the set table came then the ideologists. Since Europe is predominantly Indo-European, they generally believed in Aryan supremacy; that of course for ideologists meant that Semites come second, and Turanic horse-breeder barbarians not at all.

Then European intelligentsia started to argument. And Ancient Mesopotamia was a good place to yield examples for a completely absurd argumentation.

At the Beginning there was Sumer. They were neither Semites, nor Aryans; Oppert as early as 1869 guessed them to have been Turanians. For this he was liked in Hungary. But after the Sumerians came Semites: Amorites, Babylonians, Assyrians, Aramaeans. Aryan supremacists generally told: wait for the Persians & Greeks eating up these peoples from the middle of 1st millennium BC; but ancient Aryans were also detected soon: the upper crust of Mitannians in Northern Assyria spoke a Sanskrit-like (but older) language and honoured gods with names Indara & Mitara (Indra & Mitra in Sanskrit). So people, for more arguments, were interested in Semitic & Indo-European ethnics in Mesopotamia.

Now, the Kaššú were neither; not even Uralic or Altaic. So nobody was really interested. History books read by me generally make the impression that the Kassú era was a stagnation, and for half a millennium Akkad (called Kar-Duniaš by the Kassú) was a backwater.

But the big Niniveh library of the Assyrian King Assur-ban-apli, excavated early, has a copy of an inscription of the Kaššú King Agum-kakrime, from the Dark Ages. The inscription states that Agum-kakrime is the Lord of various geographic entities, including Akkad, the Kaššú Land, Kar-Duniaš and "the plains of Babylon". Experts generally agree that the Kaššú entered Mesopotamia from the Zagros, so maybe the Kaššú Land is there; that in later times Kar-Duniaš meant the Mesopotamian Kaššú dominions, so that may be Northern Sumer or Middle Mesopotamia or something; and the text states that Agum-kakrime controls Babylon, although the city may be even depopulated. Then Agum-kakrime tells that the statues of Marduk & Sarpanitum were carried away from Babylon, but recently he has found them in the country of Hani, sent an embassy and has got back the statues now restored in their temple.

OK, so maybe Agum-kakrime is the first Kaššú King controlling Babylon, either a live city of any size or a field of honourable ruins, and obviously the Marduk temple is just restored. When did this happen?

Unfortunately Agum-kakrime's name now cannot be read in the King List, whence the Kaššú Dynasty's data are taken. This List is called King List A, for details see e.g. [35] and references therein. If we believe everything mainstream Assyrologists read from that List, we can restore the year with moderate mean deviation. For the mathematics see Appendix B, for not offending scholars. In the raw data [33] and [34] agree, so I worked from [33]. That study accepts Middle Chronology, but this simply means that we are in the convention where the 18th Kaššú King, Kadašman-Enlil I, starts in 1374. If you accept anything else for this King, every other one has to be shifted by the same time.

Now I tell the story in two steps. First from the viewpoint of a naive physicist reading one book about Agum-kakrime and believing everything told by the particular historian; and secondly at least recognising that there are real problems in History even if Mesopotamian clay bricks keep a lot of writings for us.

Agum-kakrime is generally translated as "Agum II", even if our Kaššú is rather weak. (It is not sure at all that "kakrime" is "second". Namely the 2nd King is Agum too, this Agum cannot be our Agum (being too old), so he is Agum I, and then Agum-kakrime is not the first. (I did not see any serious suggestion for the affiliation of Kaššú, so maybe it is Na-Dene, but that is a big and diverse macrofamily with such members as Basque, Burushaski, Chinese & Navajo.)

Brinkman reads the 6th King as Urzigurumaš, where the “Ur” may even be a Sumerogram. About the previous turn of centuries it was read "Tašigurumaš"; but in cuneiform not each sign has a single reading. Very probably the two readings belong to the same person. Also a similar name is read in the copy in the Niniveh library as the father of Agum-kakrime. So the natural idea would be that Agum-kakrime is the 7th King, and this was the opinion decades ago. However now the first half of the name of the 7th King is read and it is Harbe-.... The names of the 8th and 9th kings are still unread (see [33] & [35]), the 10th is Burnaburiaš I, and again there are lacunae at the 11th & 12th. Now the experts generally guess that Agum II is the 9th, and we may accept this. Namely in ancient times the names generally repeated with at least 1 generation gap; and Agum II does not declare his father King. Maybe Agum-kakrime is the son of a younger son.

Now, the List states that it contains 36 Kings with 576 years and 9 months total. The months may indicate the rule of the very last king; but they may indicate as well the interregnum when Tukulti-Ninurta carried Kaštilias IV to Aššur (as an Assyrian chronicle states it). In Mesopotamia the general usage was that the year belongs to the king who performed the New Year rites; if he dies later, that year is still his. So: 577 years for 36 kings. Calculated back, we get that the first King of the List, Gandaš, started in 1732 (±1-2 years for rounding, enthroning conventions &c., ignored).

[33] takes the years of 22 Kings known. Most data for lengths of rule come from King List A, some from Assyrian chronicles &c. Of them 2 is 0, 3 is 1 and 1 is 3 years, the others ruled substantial years. This altogether is 290 years. So 287 fall to the 14 unreconstructible ones. However the unreadable ones in average ruled only marginally longer than the 16 readable ones with substantial years. Maybe the List simply did not contain counter-Kings, pretenders &c. in its partially damaged first half, already centuries from the scribes.

From these data we know the average time + mean deviation for the undamaged Kings, and also the average for the partially/totally damaged ones. In the lack of any other information we may assume the same mean deviation; and then the midpoint of Agum II's rule is 1550±23.

For anything more definite extra historical information would be needed. Of course, this is the result of absolute belief in the Brinkman Appendix of [33].

This picture is self-consistent. Muršiliš I destroys the City in 1595; the survivors flee. However Muršiliš immediately leaves the City, going home with treasures, and some survivors return. After 45±23 years (1-2 generations) there already was some population and much Tradition, so Agum-kakrime declared himself Lord of the City. So far, so good.

However, according to Modern Astronomy and Tidal Friction, Earth revolved faster in 2000 BC, and Gurzadyan estimated the Delta-T [32]. So the 2050 eclipse started below the horizon of Babylon, contrary to the records. Then this was the 1954 eclipse of the same month.

Now comes the Second Wave. Observe first that there are four widely accepted Chronologies for the First Dynasty of Babylon and for the Dark Age afterward.

The Old Babylonian chronology is anchored down to this eclipse. Let us see, how.

1) Šulgi's 47th year is the Eclipse. (Eclipse record.)

2) After Šulgi there are 3 more kings of the Ur III dynasty, and the last one, Ibbi-Sín, starts in the 19th year after Šulgi. (Lists.)

3) Isbí-erra, officer of King Ibbi-sín, revolts in Isin, according to the best historians, in the 13th year of Ibbi-sín. (Records.)

4) The total of Isin I is 224/223 years until the death or dethronement of Dámiq-ilíšu. (List.)

5) Larsa was another city revolting against Ibbi-sín; the first king was Naplánum, the last Rím-Sín, 263 years total. (List.)

6) Both Isin and Larsa was defeated by Hammurabbi of Babylon; he occupied Isin in his 7th year and Larsa in his 31st, according to the recorded year names of his reign. (Records, mainly Hamurabbi's Year Names.)

7) The Babylon I dynasty starts with Sumuabum and ends with Samsuditana, 299 years altogether. Hammurabbi is the 6th king, from the 103th year of the dynasty. (Lists.)

Conclusion: Hammurabbi's victory on Isin must be the 261st year after Šulgi's Eclipse. You may forget about Long Chronology, now it is not in favour; but older texts use it, and also see recent [36]. There it is 1842. The Middle Chronology (preferred by Brinkman [33] and incidentally by me) is anchorred by Šulgi's Eclipse, so if that was in 2050, then the date is 1786. For the Short Chronology wait a minute, but the Ultra-Low simply shifts the Middle upwards 96 years, so in it the Conquest of Isin must be 1690. You can check this by Isin-Larsa synchronisms.

As for the Short Chronology, there are the Venus Tablets of Amissaduga (the penultime King of Babylon I). Velikovsky believed that Venus appeared just then, but it is not important for now. The Tablets give when Venus becomes visible and when she ceases to be. Compared the data with modern retrocalculations, we can at least check the above mentioned years.

Alas, the check is not really powerful, because of an interesting and unexplained resonance in the Solar System. While 1 terrestrial calendar year is 365.2422 days, 1 Venus year is 224.70 days. So 8 calendar years is 13.004 Venus revolution, and with a similar very small error this is 5 synodic Venus years. So the Tables must be very similar in 8 years periods.

Anyways, the Venus Tablets suggested a 64=8*8 years shift from Middle Chronology, and this is Short. I use Middle, because I like it, and because Brinkman's data for the Kaššú Dynasty are given in Middle.

OK, historians know this and maybe astronomers lost the track 29 kings ago. But this is History. Astronomers then may tell that these are 3500 year old records, possible objects of falsifications &c.; but then I will tell that Fomenko or Illig argue similarly and they regarded as odious by historians.

But then let us go one step further (say I, not Gasche, Gurzadyan & al.). Let us first return to the Kaššú Dynasty.

King List A is very precious, but is not in good state; and it seems that the knowledge about i is in even worse. It is generally told that there is a lacuna for Kings 4-17. However early scholars, at the beginning of XXth century, reported readings for some kings "in the lacuna". For the 4th they gave Ušši or Duši, for the 5th Abirattaš, son of the 3rd or Adumetaš, for the 6th Tašigurumaš or Tazzigurumaš, son of Abirattaš/Adumetaš, and for the 7th Agum-kakrime.

Brinkman in 1974 is not so optimistic. The names of the 4th and 5th are quite illegible; the 6th is legible, Urzigurumaš (so he is the earlier Tazzigurumaš; better reading), and the first half of the 7th is still legible: Harbe-. Being Harbe the Kaššú Sky God, equivalent of Enlil, this king cannot be Agum-kakrime.

And now, before turning to Astour, let us see the text found in Assurbanapli's library. A king, calling himself Agum-kakrime, uses ambitious royal titles which are, however, not exactly the same as earlier or later (e.g. he is King of Kaššú & Akkad, not Sumer & Akkad, or king of the land of Babylon, but not of the city). He tells his descent. He is son of Taššigurumaš (surely Urzigurumaš), descendant of Abirumaš (another reading of the 5th King?), and direct descendant of Agum (surely the 2nd King).

Now, the problem is that he does not declare his father, Urzigurumaš, King. So maybe his father was not the 6th King, but a later younger son in the same family.

No problem at all; but then he cannot be the 7th King. And in present decades he is rather believed to be the 9th. This was originally the idea of Weidner in 1926 reading A-gu-uX, who, however, retracted the statement in 1960 [35]. Now, Astour in 1986 claims that he can read in the ninth column Kak-ri-iX-e (or something such) even if he cannot read the first half seen by Weidner in 1926, Agum [35]. Incidentally he explains "kakrime" as "Thunder weapon" or such; there are parallels from almost synchronous analogies from Mitanni about Indara or Teshub, fighting with thunders. Note that Astour somewhat corrects the readings of the 4th and 5th Kings too: Ušši may be on the King List A, but Ušši Ašú means simply "His son", not a name. As for the 5th he reads Abirattaš, but calls the attention for another List giving A-bi-rX-taš for the 4th, and Kaš-til-a-šú for the 5th. Since the royal name Kaštiliaš is recorded both for the 3rd and for the 28th King, we may remain with this. More is rather hopeless until better grasp on the Kaššú language (even if "uz" seems to be the root "protect" [37]).

So Agum-kakrime is the 9th King. If the length of rule cannot be read (majority opinion), then from statistics he ruled 20.5 years, however the mean deviation is 9.8 years. Or we may believe in a single claim that he ruled 22 years; not a big difference. However I repeat that the midpoint of his rule was in 1550±23. And this date, even if it depends on many details of scholarship, as readings and so, is completely independent of the concurrence of Long, Middle, Short and Ultra-Short Chronologies, being independent of eclipses of Šulgi or Ur-Nammu, and Venus Tablets of Amissaduga. It is anchored to the 763 eclipse, through Assyrian-Babylonian synchronisms an via the statistics of the Kaššú Dynasty.

On the other hand we have 4 date for the Fall of Babylon:

Chronology	Long	Middle	Short	Ultra-Short
Fall of Babylon	1651	1595	1531	1499

In the Long Chronology Mursilis' raid and the Fall of Babylon happened in the time of the 4th King, Abirattaš, and the Marduk-Zarpanitum statues are recovered only by Agum-kakrime, the 9th, c. a century later. It is possible, it belongs to historians, but it is a long time. However, in Middle Chronology the Fall is during c. in the time of the 7th, Harbe-X, and Agum-kakrime restores the Marduk temple 45±23 years later. This gap seems quite nice. In the Short Chronology the Fall of Babylon happens, in the best case and stretching the deviations, under the rule of Agum-kakrime, in the second half, so Agum started towards Babylon just as Mursilis left it. And Ultra-Short Chronology is impossible if we accept the Kaššú King List.

Now, there are Assyrian lists as well, and by using them we can count again downward from the Assyrian Partial Solar Eclipse in 763, as well. What is more, for Babylonian Kings after the Fall of Babylon there is no direct way to utilize Šulgi's Eclipse, because the continuity is broken.

The Assyrian List is very, very long. In the present elaborated form it contains 120 kings [33], of which maybe 3-5 are counter-kings, parallel to somebody. The first 38 do not have duration data, and this sporadically do happen upwards too. In total 38+16=54 are without duration. However we may forget about the first 38, and then we have 82 kings, 64 with durations and only 16 without. This gives some numbers for statistics. And the 39th king, first of the present attention, Šamši-Adad I, was synchronous with Hammurabbi, so old enough. By this synchronicity in Middle Chronology he rules between 1813 & 1781.

Here it will be unnecessary to make a full statistical analysis, because the numbers of the list directly collide with the Gasche & al. chronology. Namely in the List after Šamši-Adad there is one more king with recorded duration, Išmé-Dagan, so he dies in 1741 in standard Middle Chronology. After him 3 king have no durations, then one, Aššur-dugul, has 6 years, 6 kings without duration, then 17 kings uninterrupted with duration from Bélu-báni to Aššur-šaduni. Then 2 kings without durations, but even the previous Aššur-šaduni ruled a mere 1 month, so the next 2 could not reign too much. And then an uninterrupted sequence of 16 kings, then 2 kings without reigns (Ninurta-tukulti-Aššur & Mutakkil-Nusku), which, however Standard Middle Chronology believes to be of negligible reigns, and then an uninterrupted sequence of reign data until the Assyrian Solar Eclipse under Aššur-dán II, and beyond. From Enlil-násir II (starting in 1430 in Middle Chronology) everybody is anchored to the Assyrian Solar Eclipse, so recalculation of Šulgi's Lunar Eclipse do not influence them. From the enthroning of Bélu-báni to the Assyrian Solar Eclipse 906 or 907 years + the unknown years of 4 kings, but that is probably only 1-2 years.

But we have the List backwards to Hammurabbi's coeval, Šamši-Addad I. That is 3 more kings with a total reign of 79 years, and 9 more without reign data. In the Middle Chronology from the beginning of Šamši-Adad I to the Assyrian Solar Eclipse is 985+x years, where x is the total of 13 kings without data. Because Middle Chronology puts Šamši-Adad I at 1813, x=65 years. So we get 5 ys/king for the ones with missing data; short, but possible enough.

In the Long Chronology it is 1105+x years, so x=185 years, 14 years/king, quite possible but maybe too long for practically unknown kings. (But this is again the business of historians). For the Short Chronology x=1 year and we are at the edge of possibility. (All of them counter-Kings?)

Now, shift upwards Šulgi by 96 years as suggested in [28] & [32]! Then x<0! There is no time for 13 kings in the new chronology!

Hittite Kings are much less known than ones of Mesopotamia. But there seems to be 15 kings between Mursilis I and 2 [34]; and Morby refers a Ph.D. dissertation claiming astronomical dating for Mursilis II. That would mean c. 270+x years for the 15 kings + for a gap between Old and New Empires in Middle Chronology, but only c. 170 in the Ultra-Short one. While the second is not impossible, it is rather short. Shifting to Mursilis I's campaign against Babylon to 1499 there remains no gap at all. In the traditional Hittite chronology the New Empire starts with Tudhalyas I c. 1430, and then mere 69 years would remain for 9 kings and the gap.

No doubt, the phenomenon can be explained away. You can reclassify some Assyrian kings (the less known ones), e.g., as counter-kings. Also, Huzziyas I & II, Zidantas I & II or Hantilis I & II may be identical. But such revisions violate the Sacredness of the Historical Methods on the same way as Fomenko or Illig does it. In the best case, Everything would become again liquid.

And now comes the real surprise. The most recent NASA calculations for lunar eclipses give an 1954 total eclipse which must be the eclipse of the Ultra-Short Chronology, but it was seen (or at least seems to have been seen) mainly in South America. It just touched the Westernmost Africa and the Iberian Peninsula, but surely not the Middle East. NASA's ΔT now seriously differs from Gasche & al's in 1998. The Short Chronology would need a total eclipse in summer 1986, but there was no total lunar eclipse at all in 1986! As for Middle and Long Chronologies, NASA did not publish lunar eclipses for 3rd millennium BC. Of course they can be individually calculated.

NASA claims that in the time of the 1954 eclipse ΔT was 45404±3548 s. It is an interesting question, what is this standard error; but clearly Šulgi's Eclipse did not happen in the summer of 1954. We do not yet know when it happened.

And there is Beckman [39]. He methodically counted the generations of Hittite Kings using the available historical information about descents and concluded that the Ultra-Short Chronology would lead to impossibly short Hittite generations.

I close this Chapter concluding that

1) You may save Textbook History's picture back to somewhat before 763 AD, because the Assyrian Eclipse as anchor still seems to work. But for this you must be negligent about the statements of such giants as Thukydides or Plutarch believing them writing down hearsays or inventing eclipses.

2) But in 2nd millennium BC you are meeting self-contradictions anyways.

I personally believe that the dating confusions in Egyptian XVIIIth Dynasty have something to do with this; and definitely some morasses about the chronology of Moses may originate hence as well. Simply: we cannot "predict" Delta-T data for 1500 BC, and for that time eclipse observations cannot be selected from rubbish.

10. CONCLUSIONS

We saw here that problems do arise between History and Celestial Mechanics would be to declare that they have nothing to do with each other. This might happen in different ways. E.g. if Historians reach the ears of politicians, a legislation may declare that certain statements of Present History (e.g. Thucydides on the eclipse in the first year of the Peloponnesian War) are Final Truths and they cannot be in doubt. You can imagine the subsequent problems.

The opposite solution is not too good either. Physicists tend to believe that they should not enter historical discussions which are "not exact". But then historians will manufacture their own eclipse calculations; or, more probably, would cite the calculations of Freiherr von Oppolzer without tidal friction ad infinitum.

Another possibility is duplex veritas. As you can remember, this was a legitimate viewpoint between 1250 and the Galileo Controversy; but generally does not result in anything.

And then, the desired goal can also be reached via Back to Aristotle. There are two Physics. One in Heaven, one Below Moon (more or less). Eclipses belong to Celestial Mechanics so to Heaven; atomic clocks belong to Terrestrial Physics. Then there is no problem with ancient eclipses and records; Fomenko may go back to Differential Geometry, historians may use the Oppolzer Canon, Babylon was raided by Mursilis I in 1595, and still the Standard people can sometimes insert a leap second on December 31. Also, there is a difference in the motion of probes inside and outside the Solar System: good, let us discover the differences between/among different Physics and the real border between Terra and Heaven.

I am not advising this; but surely, the monist solution is in greater doubt now than in any time since Galileo. History and Celestial Mechanics do not just now cohabitate too well. Of course, another possible reason is that there is some problem either with History or with Physics.

Well, historians do not like to revise. But if they do not do anything, then discrepancies accumulate, as has happened recently, provoking New Chronology into existence.

But, although I am a physicist, I can also imagine that the Pioneer Anomaly (not the eclipse controversies!) originates in the use of an unsatisfactory Physics. I do not state this, but can imagine. Namely, as I told in Chap. 3, writing

Δa_an = H'r (10.1)

where H' is a signal that it is not the Hubble constant but some "related" quantity, the numerical value is near to the Hubble constant, even if that quantity appears in a phenomenon of opposite direction.

Our present best cosmological theory is based on General Relativity; and in it the Hubble constant comes from the really initial initial condition: the Universe appeared with some initial expansion and for now it produces the Hubble constant. If things had started otherwise, the Hubble constant would have been double or half as well. As I mentioned in Chap. 2, the order of magnitude of our Hubble constant cannot be derived from Triune Unifications either.

Of course, it is not too logical to impose arbitrary initial conditions on the Universe, which exists in one, unique "copy". However our present best theory needs an initial constant. The problem is that this quantity does not appear in the free motions of probes, so does not explain the Pioneer Anomaly.

And now we can discuss very briefly the Group iv) of possible explanations in Chap. 2: Really New Physics. If the Universe is really a Unity even now, then probes at 70 AU may be influenced by Something also behind Hubble constant and such. I mentioned this already at the end of Chap. 3. And look, in Teleparallelism the photons flying from the probe to Earth are not flying really freely in between.

I will not discuss here the various suggestions of Type iv); only will use one New Physics as a model for all Unity-type theories. The model is Barbour & Bertotti's Machian Universe [45].

The idea grew out of course the philosophical ideas of Berkeley and the physical ones of Mach. Physicists heard a lot about Mach, but much less about Bishop Berkeley.

Bishop Berkeley was a younger contemporary and concurrent of Newton. Also, he was the Anglican Bishop of Dublin. Of course, this Bishopry was a sinecure, given under the influence of a friend high in Government. Namely, the Irish of Dublin were of course Catholic. Only Government officials belonged to Bishop Berkeley, not many. So he could write treatises.

Now, he introduced more or less the same idealised "observers" as Einstein, but then observed that an observer alone in the Universe could not measure distances and times, because it would not have rods and clocks. Even for that several other entities would be needed in the Universe. So -he concluded- such a theory in which even a lone mass point in the entire Universe would move on straight lines without acceleration cannot be correct. Now, the Newtonian and Einsteinian theories are such.

Ernst Mach, a hairbreadth before Einstein, suggested that Inertia may be not an inherent property of the body but the consequence of interaction with all the other bodies of the Universe. This may be so: but then we are at a physics in which the body feels the entire Universe, so such theories belong to Group iv) of Chap. 2. Of course, Mach's and Berkeley's ideas can fit together, and Barbour and Bertotti made such a Physics, and in a simple model Universe for local motions in first approximation they got back Newton's Equations of Motion, and in second approximation they got something resembling relativistic corrections and connections with Hubble Law.

If somebody likes it, the Machian Universe is a good model to work out explanations for the Pioneer Anomaly. I will not do exactly this, but return to something mentioned in Chapter 5.

At various subregions of Physics & Astronomy appears a quantity in the order of 1-10 billion years I called in Chap. 5 t₀. Earth's rotation is slowing on this scale, Moon is receding on this scale, the age of the Solar System is on this scale, and the Age of the Universe differs only marginally, being 13.6 almost 10.

The present best canonical age of the Universe is (now in physical units) 4.28*10¹⁷ s. On the other hand, in spite of the seemingly complicated dimension of the Hubble constant, in physical units it is 1/time, so H^-1 is seconds. Slightly greater than the Age of the Universe. And for the H' of Pioneer Anomaly in eq. (10.1)

H'^-1 = 6.0*10¹⁷ s (10.2)

I stop here. A quantity ~10¹⁷ s seems to appear in many disciplines, sometimes we understood wy, sometimes we do it only partly and then it appears in the discrepancy, sometimes it seems to be an accidental value and in the Pioneer Anomaly it should not appear but does (and so it is an anomaly). Interesting, ne c'est pas?

Fomenko looks for plots which falsified History; and tells that he has found some. Then the Professional Historians, not following his statistical arguments about falsifications and completely ignoring his note about R. R. Newton's observations, became angry and started to tell that he is a "revisionist" (meaning that he revised important statements of History, odious because Actual Politics but called Copernican Revolution in Astronomy), and "Russian fundamentalist" (which is no surprise if we know that he regards Ivan the Terrible as a good ruler; or rather 4 ones). However you do not have to accept Fomenko's Paradigm in face value to explain mysterious repetitions in History.

By statistical analysis Fomenko showed out strong (auto)correlations in History. E.g. the ruling length sequences in one dynasty significantly resemble another dynasty centuries (and maybe geographically) apart. (In Statistics there are objective criteria when the similarity is very probably not accidental.) Then he considers the last dynasty real and the previous one(s) with similar numbers "invented" or "clones". By this way Byzantine Emperor Alexios Comnenus is real, but Constantine I the Great, Alexander III of Macedon and Joshua, son of Nun are his clones.

Now, this radical viewpoint is not necessary. We do know that in Medieval and Early Modern times dynasties employed scholars to write the early history, descent &c. of the dynasties. The Early Medieval Ages were indeed dark and the dynasties of Western Europe descended from illiterate barbarians a few centuries back "in the backwoods". E.g. according to Frankish tradition, the first known ancestor of the Merovingians, so of Louis (Chlodwig) I of France was a sea monster from the North Sea. The ancestors of many Western Emperors, the Habsburgs, were simply the Counts of Habichtsburg in Schwitzerland about 1200. And so on.

Now, imagine that a rich ruler tells a scholar: "Write the family tree & history of my glorious family, and, if the work is of quality, you get a bag of gold!"

Then maybe the correct answer would be: "Sire, the history of that age is virtually unknown and so I cannot produce the study"; but such an answer is contrary to both the mores of that age and the human nature. More probably, the scholar would have answered: "Glorious Majesty! The historical period is virtually unknown, only I could do this study and even I with great difficulty. So 4-5 years will be needed." Then the scholar lived in opulence for 6 years and at the end he compiled a history. If he found real data, he used those, if not, he invented ancestors with names in use in the family and ruling times & such taken from thin air. Except that nobody is a good random generator; consciously or unconsciously he uses memories of other dynasties, so this may be the explanation of statistical similarities.

This mechanism did work; in lots of cases we recognised the fallacy later. I mention here a few without proper references. Saxo Grammaticus, the Danish chronicler of XIIth century (the source of the Hamlet) traced back the descent of the Danish Kings to a bear carrying away a girl of good family. Thence came a son called Björn (bear) who became a chieftain. The honourable scholar Antoine du Pinet (1515-1584) compiled a history for the Counts d’Agoult. The family arms show a wolf. Now the story. In the far past in Pomerania there was a princess Valdugue and a young guy Hugo. A son was born and given away. The baby was stolen by a she-wolf, the she-wolf was then shot by the king, the baby recovered. Later the royal grandson married the daughter of the Byzantian Emperor & so on.

Prudencio de Sandoval compiled the descent of Charles V of Spain. After 120 generations he arrived at Adam.

And in the early XVIth century the descendents of the Counts von Habichtsburg were already glorious Habsburgs, rich enough for Johannes Stabius, court historian of Maximilian I and poeta laureatus to compile the family tree well before Charlemagne. Recently historians may or may not accept the descent back to a court official of Charlemagne, but Stabius gave the ancestors back to Noah through Cham; the work had been approved by the theologic faculty of the Vienna University.

We have found lots of mystifications in our past. This does not mean that we have already found all. The unfound ones still pester the concord of History & Astronomy; but also History in herself.

ACKNOWLEDGEMENTS

Now there are none. My colleagues definitely refuse to discuss problems arising between Physics and History telling "who believes in histories?"; they listen when I am speaking about Triune Unification and Particle Masses, but they are much more optimistic than I am. Historians do not want to discuss the details of eclipses with me, even if they are polite.

APPENDIX A: ANOMALIC ACCELERATION AND TRANSPLUTONIANS

Since we do not know the reason of the anomalic acceleration (1.1) yet, of course we do not know its dependence on distance. However as first approximation, let us consider (1.1) an average on Pioneer's travel. The part of the travels whence data were sent traverses almost 80 AU, so the midpoints were roughly at Pluto's distance. So let us write Δa=9*10^-8 cm/s² at R=6*10¹⁴ cm.

On a circular orbit, without anomaly

a = v²/R = GM/R² (A.1)

where M is the solar mass. Then

2δv/v = δa/a (A.2)

So with our data

δ/v ≈ 10^-4

This would lead to a c. 10 days difference in Pluto's time of revolution. The effect would not be small; except for 2 difficulties. First, Pluto's revolution will not be very well known until finishing the first orbit after discovery, and, second, Pluto's distance has not been measured until now by ways independent of the orthodox Law of Gravitation.

APPENDIX B: ON THE STATISTICS OF THE KAŠŠÚ KING LIST IN BRINKMAN'S RECONSTRUCTION

Since King List A has lacunae and at places it is illegible or at least some scholars believe it to be illegible, reconstructions were made by using inscriptions, parallel lists &c. The depths of such reconstructions belong to the discipline of Assyrology and will not be discussed in this Appendix, except for the very fundamental facts. Here I follow Brinkman [33], the leading Kaššú expert of the last generation.

The original King List A at the time of the scribe had 36 kings, with 36 lengths of rule; 576 years 9 months altogether. I take here integer years, so 577.

Kings 4-21 are mostly regarded now as illegible, at least partly. Kings 1-3 and 22-36 are unanimously regarded legible. After lots of thinking, reading & filling crossword puzzles you can get a small amount of extra information. So did Brinkman. He got the string of ruling years as

{26,22,22,?,?,?,?,?,?,?,?,?,?,?,?,?,?,15,27,0,0,25,26,18,9,9,13,8,1,1,6,30,15,13,1,3}

and here I accept his numbers in face value. He believes that the names of Kings 6 & 10 are certain enough (but not the years) and also the first half of Name 7. He gives the names of Kings 15-21, but then he gives a footnote telling that his 21st King may even not be on King List A and then Kings 15-20 must be shifted by 1 position. So it seems he considered Names 15-21 illegible. Remember this in App. C. However here names will be unimportant.

Now we have the above string of numbers. The unknowns are in one block from 4 to 17.

We can then distinguish 3 Blocks:

Block	Kings	Total length	N
A	1-3	70	3
B	4-17	577-290=287	15
C	18-36	220	18

For strict statements we would need distributions with N>>1; we cannot have them. For first approach we can use formulae valid only in N>>1 for Gaussian or normal distributions [40], but in many cases they gave at least no distortion for expectation values. So, <x> being here the expectation value of ruling time, δ its measurement error and σ its standard deviation or variance we can take (as first approximation)

<x> ≈ ∑_i x_i

σ² ≈ [N∑_ix_i² - (∑_ix_i)²]/N(N-1) (B.1)

δ ≈ σ/(N-2)^1/2

Hence we get:

Block	<x>	σ	Δ	N
A	23.33	2.31	2.31	3
B	20.50	?	?	14
C	12.22	9.91	2.40	19

Now, the distributions of Blocks A & C do differ. The difference of the block expectation values is 11.11±3.33, which means that the null hypothesis that they originate from identical distribution but with sampling fluctuation has a chance well below 0.5 % (of course, for Gaussian distribution, but we cannot use any else because we do not know the distribution). As for Block B, σ and δ cannot be calculated, but for <x> the difference from A is 2.83±X where X>2.31, that is not significantly different from 0.

So Blocks A & B may be similar, Blocks A & C are surely not, and B & C are probably not in either.

However in Block C there are probably two distinct kinds of kings. To see this, let us put them into 6 year bins, and then we get the distribution of Fig. 1.

Obviously there is a bunch of kings of very short rule. We can guess that they were generals catching the throne with force and such. Let us collect all from Bin 1, transfer them into Block C2, and the remainder of C (there are no such kings in Block A) into C1. Then we get

Block	<x>	σ	Δ	N
A	23.33	2.31	2.31	3
B	20.50	?	?	14
C1	16.46	8.07	2.36	13
C2	1.00	1.10	0.55	6

The details of this way of correction must remain to App. C. However the last Table demonstrates that the distribution of the damaged ruling years may be not too different from the undamaged ones, so for Block B we may substitute the mean deviation or variance (σ) with that of the undamaged ones (the unification of Blocks A & C), namely 8.94 ys. Then we get years & mean errors for Kings 4-17 and midpoint for King 9 (maybe Agum II) as told in Chap. 9. (And observe that still <x> for Block C2 is somewhat smaller than for Blocks A & B, but lots of historical explanations could be invented for this.)

APPENDIX C: RECONSTRUCTING THE KASSITE KING LIST (KING LIST A)

This Appendix is not to be taken too seriously. And its content was not used for any dating for Šulgi’s Eclipse. I give it for 3 reasons:

1) It is unaesthetic to leave gaps in a List.

2) As Galileo’s Lorini told: I have other things to do but I would like to show that I too exist.

3) It is difficult to speak of the “17^th King”.

If there is any definite statement of his, I accept Brinkman’s data, all numbers, all names, only I shifted back a block of 3 kings by 1 position; see there. If he does not read anything, I take the date from elsewhere and I tell whence.

Now, the scribe composing King List A added up the data and wrote: 36 kings, 576 years, 9 months. While 36 and 576 might be artificial, for some number magic, being 6² and 4²*6², the 9 months is not a nice number. So as first approach, we may accept the numbers.

It is nontrivial how the “9 months” can be interpreted. The Babylonian way of recording ruling years was that the King was reinstated on some New Year ritual. Until he was able to attend to the ritual, a new year of his started. So his first official year started on the New Year Holiday after his succession, and his last year was finished by his successor, but still belonged to him. No possibility to fragments of years in a well-kept King List.

However this was known better by the scribe than me and still there is the 9 months. There are lots of possibilities, of which I mention 4 (but the first 2 will be ruled out immediately):

1) the first King, Gandaš, was enthroned definitely not in Babylon, so he started not on Babylonian New Year, so maybe the scribe remembered the different calendar. However his ruling years are legible and integer.

2) The last King, Enlil-nadin-ahé, surely was not dethroned on New Year. But again, the number is legible and integer.

3) 2 Kings, Kara-hardaš & Nazibugaš, have not too unequivocal ruling years. Brinkman reads “0 year” for both; older scholars read some fragment years. So maybe Burnaburiaš II died 9 months before the New Year Ceremony, came Kara-hardaš, ruled a few months, then came Nazibugaš, also a few months, and at New Year Kurigalzu II performed the ritual. But centuries later a scribe knew something about the early deaths. Well, maybe; but maybe not.

4) A King, very probably Agum-kakrime transferred his seat into Babylon. Maybe he performed 2 ceremonies with 9 months difference in a year, the second in Babylon. For any case, Agum-kakrime’s ruling years are illegible, so we cannot yet see if they are integer or not.

Now, for simplicity, I accept the 9 extra months, but round it to 1 year. So the total length is 577 years.

The last year of the 36^th King, Enlil-nadin-ahé, seems to be 1155. Then Gandaš started in 1732±1, where the uncertainty comes from our ignorance about the pre-Babylonian Kaššú ceremonies, ignored henceforth.

Here I standardized the orthography of names to something near to international Assyrologic ways. It seems that -aš is a Kaššú formative ending, so maybe the closing vowels as -ašú are pen errors of Akkadian scribes.

Brinkman reads/reconstructs 28 names and 22 ruling years. In the Table I give them New Times Roman Bold. For names read/reconstructed by anybody else I use Lucida Calligraphy; as for years, Arial, and I will give some explanation for the source. For the numbers still unrecontruable I give later the average and mean deviation recalculated because the statistics here is different from that of App. B.

N°	King	Years	Note
1	Gandaš	26
2	Agum I	22
3	Kaštiliaš I	22
4	Abirattaš	9	Old sources as e.g. [41]-[43] generally read 9 years, they know why. They read an impossible name. This name is the suggestion of Astour [35].
5	Kaštiliaš II	?	Name is suggested by Astour
6	Urzigurumaš	?
7	Harbe-šipak	?	The second half of the name is uncertain, and being both Harbe and Šipak Kaššú gods, an unusual compound. Another reading is -šihu.
8	Tiptakzi	?	Tiptakzi is certainly a Kaššú King, cca. in this position.
9	Agum-kakrime	?	Weidner saw Agum, but not kakrime [44]. Astour sees kakrime, although not Agum. Both agree in the position.
10	Burnaburiaš I	22	Brinkman gives the name and position. [41] and [42] give the years, and another source gives the 22 years to Agum-kakrime.
11	Kaštiliaš III	?	Many early sources (e,.g. [41] and [42] mention a Kaštiliaš as a son of a Burnaburiaš
12	Ulamburiaš	?	A Kaštiliaš, an Ulamburiaš and an Agum is often suggested for positions after Burnaburiaš
13	Agum III	?	See previous Note
14	Karaindaš	?	Shifted back with 1
15	Kadašman-Harbe I	?	Shifted back with 1
16	Kurigalzu I	?	Shifted back with 1
17	Meli-Šipak I	?	A ghost of an early Meli-Šipak is haunting the literature, even if [42] was the only source found by me which explicitly gave him. However some modern secondary sources give a Meli-Šipak II, even without a I. As for the previous 3 Kings, they are often mentioned as Kings to be inserted c. to this position, and close kins. The Burnaburiaš-Kaštiliaš III and Kadašman-Harbe-Kurigalzu-Meli-Šipak descents are given by [42]. I have shifted the block Karaindaš-Kadašman-harbe-Kurigalzu I back from the Brinkman position by 1 place to make place for a Meli-Šipak after Kurigalzu; but they are in the lacuna anyway
18	Kadašman-Enlil I	15	Brinkman tells “at least” 15 years.
19	Burna-buriaš II	27
20	Kara-hardaš	0	Or fragmentary length
21	Nazibugaš	0	Or fragmentary length
22	Kurigalzu II	25
23	Nazi-Maruttaš	26
24	Kadašman-turgu	18
25	Kadašman-Enlil II	9
26	Kudur-Enlil	9
27	Šagarakti-Šuriaš	13
28	Kaštiliaš IV	8	Tukuli-ninurta I defeats him, and carries to Aššur.
29	Enlil-nadin-šumi	1	Maybe Assyrian client
30	Kadašman-Harbe II	1	Maybe Assyrian client, but observe the genuine Kaššú name
31	Adad-šuma-iddina	6	Henceforth rather Akkadian names
32	Adad-šuma-usur	30
33	Meli-Šipak II	15	Šipak is a Kaššú god.
34	Marduk-apla-iddin	13
35	Zababa-šumu-iddin	1
36	Enlil-nadin-ahé	3	The last Kaššú King, defeated by Elam, kingdom going to Isin

Now the statistics is as follows. We have for existing ruling years

N = 24

<x> = 13.37 ys

σ_x = 9.9 ys

As for the others, then

N = 12

<x> = 21.33 ys

but we, of course, cannot directly determine σ_x for them.

We might substitute the value of the previous group of 24; but the difference in the expectation values warn us that the distributions of the 24 and the 12 differ. This question (for 22 and 14) was, however, investigated in App. B, with the result that by removing the “nonsubstantial” kings removes the significance of the difference of expectation values. Let us do that again in the same way as in App. B. Removing everybody with no longer than 3 ys we get <x>=17.5 ys, σ_x = 7.6 ys. Now Block A is Kings 1-4, B is Kings 5-9 & 11-17, C1 is the “significants” of Kings 10 & 18-36, or Kings of 6 ys or more, and C2 is the others in 18-36. The statistics is:

Group	<x>	Σ	Δ	N
A	19.75	7.41	5.24	4
B	21.33	?	?	12
C1	16.86	7.89	2.28	14
C2	1.00	1.10	0.55	6

Now the difference of the expectation values <x> is then 3.83 ys, while the statistical error of the “substantial readable” subgroup is 1.9 years. Since the average expectation value of the unreadables would also have an error, only we cannot calculate it, let us take as null hypothesis that the two distributions have the same parameters. Then the error of the average is 2.41 ys. So the measured value of the difference is Δ = 3.83±3.06 ys, not too significantly nonzero. For such an accidental fluctuation the chance is still 42 % (for Gaussians, at least). So you may indeed does not have statistical evidences against putting <x>=21.33 ys, σ_x=7.67 ys for the 12 unreconstruables, and then δ=2.43 ys. Consequently Gandaš starts of course in 1732 and Agum-kakrime rules c. 21 ys with the midpoint at 1557±16, and finally we get a Table.

Times New Roman Bold names and/or years are from Brinkman, even if he puts the names in Positions 14-16 one place upwards. Lucida Calligraphy names were read by somebody else; generally the previous Table gave some information, who. As for years not given by Brinkman, the code is as follows. The beginning of the Table is as sure as Brinkman’s years, because the total length of King List A is known and Brinkman gives years for the end. Similarly he gives names and years for the first 3 kings. Still, he did not give chronology for them in [33], he knows why not; this is done here and the years are in CG Times Bold. Years calculated from numbers read by anybody else than Brinkman are in Arial; you can consult with the previous Table. For all others the statistical interpolation is what was told in the previous paragraphs, and the numbers are in Baskerville Old Face.

The last column is the statistical error of the calendar chronology of the king at midpoint of rule. For start/endpoints the value is (δ_MP²+29.41)^1/2. Naturally lots of other sources of errors are possible, but here we give only the consequence of the present method.

N°	King	Starting year	Last year	Error at midpoint
1	Gandaš	1732	1707	0
2	Agum I	1706	1685	0
3	Kaštiliaš I	1684	1663	0
4	Abirattaš	1662	1654	0
5	Kaštiliaš II	1653	1633	5
6	Urzigurumaš	1632	1611	8
7	Harbe-šipak	1610	1590	12
8	Tiptakzi	1589	1569	14
9	Agum-kakrime	1568	1547	16
10	Burnaburiaš I	1546	1525	17
11	Kaštiliaš III	1524	1503	18
12	Ulamburiaš	1502	1482	18
13	Agum III	1481	1460	16
14	Karaindaš	1459	1439	14
15	Kadašman-Harbe I	1438	1418	12
16	Kurigalzu I	1417	1396	8
17	Meli-Šipak I	1395	1375	5
18	Kadašman-Enlil I	1374	1360	0
19	Burna-buriaš II	1359	1333	0
20	Kara-hardaš	1333	1333	0
21	Nazibugaš	1333	1333	0
22	Kurigalzu II	1332	1308	0
23	Nazi-Maruttaš	1307	1282	0
24	Kadašman-turgu	1281	1264	0
25	Kadašman-Enlil II	1263	1255	0
26	Kudur-Enlil	1254	1246	0
27	Šagarakti-Šuriaš	1245	1233	0
28	Kaštiliaš IV	1232	1225	0
29	Enlil-nadin-šumi	1224	1224	0
30	Kadašman-Harbe II	1223	1223	0
31	Adad-šuma-iddina	1222	1217	0
32	Adad-šuma-usur	1216	1187	0
33	Meli-Šipak II	1186	1172	0
34	Marduk-apla-iddin	1171	1159	0
35	Zababa-šumu-iddin	1158	1158	0
36	Enlil-nadin-ahé	1157	1155	0

For Kings 4-5b & 8-17 the number-year connections are primary, not name-year ones.

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