
History Online  Chronology and Eclipses
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Eclipses
What is to be done with the moon's elongation?
Let us return to the beginning of the chapter, to the moon's elongation and its second derivative. The computation of D' was based on the data of ancient eclipses adopted by traditional chronology. The attempts to explain the surprising square wave in the graph of D' do not touch at all upon the question whether the data of the eclipses were determined correctly. We will assume that an eclipse has been dated correctly if its characteristics exactly described in a historical source coincide with the parameters of the real eclipse offered by chronology.
Morozov suggested a method of 'impartial' dating, namely, the comparison of the characteristics of an eclipse given in a primary source with those from astronomical tables. Analysis demonstrates that, while not questioning the chronology of ancient events and a priori regarding it as true, the astronomers often could not find a suitable eclipse in the 'desired' century and thus resorted to strained interpretations. For example, in the History of the Peloponnesian War by Thucydides, three eclipses were described, traditionally dated as belonging to the 5th century B.C. However, even in the last century, a discussion around this triad started, being caused by the fact that there were no eclipses with suitable characteristics in the assumed epoch. Still, an exact solution can be found if we extend the interval of the search. One solution is the 12th century A.D., and the second one the llth century A.D. There are no other solutions.
A similar effect of 'shifting the dates forwards' can be extended to those eclipses which are traditionally dated in the interval from A.D. 400 to 900. It is only after A.D. 900 that the traditional dates are satisfactorily consistent with the precise datings given by astronomy, and undoubtedly after A.D. 1300.
But why, in fact, speak of it here? Because such a shift of dates is completely consistent with the GCD being glued together from four identical chronicles. If an earlier and traditional date for an eclipse was assigned to an epoch, say, labelled by C on the GCD, then its precise astronomical date lies much farther to the right on the time axis. It occurs in the period of history denoted on the diagram by the same letter. In particular, the date shift just described is reduced to advancing certain groups of eclipses up by about 333 years, others by 1,053 years, and so on. In such a time advance, the mutual occurrence of dates inside each of these groups is practically unaltered, and the group is advanced as a block.
But what's to be done with D'7 Its recalculation on the basis of the reconsidered dates of ancient eclipses showed that the graph (Fig. 5) is qualitatively altered. It cannot now be moved reliably to the left earlier than the 10th century A.D., while in the later period, it almost coincides with the curve already found and is represented by an almost horizontal line. No square wave is found in the second derivative, and no mysterious nongravitational theories should be invented. ...
It goes without saying that the work discussed here cannot claim to be the basis for any final conclusion, the more so as the most complicated, multifarious and often subjectively interpreted historical data are analyzed here by strictly mathematical methods. To process the material will certainly require a large variety of methods, purely historical, archaeological, philological, physical and chemical, and, inter alia, mathematical, which as the reader can see, will permit us to look at the problems of chronology from a new angle.
Computation of the Second Derivative of the Moon's Elongation and Statistical Regularities in the Distribution of the Records of Ancient Eclipses
Parameter D' and R. Newton's paper 'Astronomical evidence concerning nongravitational forces in the EarthMoon system
It is known that, for certain problems of computational astronomy, the behaviour of the socalled second derivative of the moon's elongation D'(t) as a function of time t should have been known for large time intervals in the past . Let OM be the acceleration of the moon with respect to ephemeris time, and WE that of the earth. The quantity D' = huf  0.03S862wE, which is the second derivative of the moon's elongation, is called an acceleration parameter . D' is normally measured in arc seconds per century squared. The dependence of the parameter D'(t) on time has been established in a series of remarkable works by the American astronomer R. Newton who calculated 12 values of the parameter D' on the basis of the investigation of 370 observations of ancient and medieval eclipses, extracted from historical sources. In computing the date fed. of the observation of a particular concrete eclipse, the parameter D' can be neglected. Therefore, it can, in turn, be found from the distribution of ancient eclipse dates [ecl., which is a priori regarded as known. In R. Newton's papers, the computation ofD' was based on the dates of ancient eclipses contained in the chronological canons of F. Ginzel and T. Oppolzer . They are generally accepted in the contemporary literature. The results of Newton, related to those of Martin, who studied about 2,000 telescopic observations of the moon from 1627 to 1860, allowed him to construct an experimental curve for D'(t) in the interval from 900 B.C. to A.D. 1900. In the following, we will sometimes designate A.D. by '+', and B.C. by ''. In Fig. 6, the symbol o indicates the values of the parameter D' calculated by means of solar eclipse data, while 0 denotes those of D' which were computed from the lunar eclipse durations fixed in the documents. The sign A implies the values of D' calculated on the basis of information regarding the duration of solar eclipses. Finally, V indicates the values of D' computed from the phases of solar eclipses. Commenting upon the graph of D' obtained, Newton wrote:
'D' has had surprisingly large values and ... it has undergone large and sudden changes within the past 2000 years ...'.
Newton's paper 'Astronomical evidence concerning nongravitational forces in the EarthMoon system' was also devoted to the attempts to explain this strange gap (oneorder jump) in the parameter D'.
Thus, on the basis of Newton's works we can make the following conclusions.
(1) In the interval A.D. 400600, the parameter D' starts falling sharply (oneorder jump).
(2) Before this interval, until A.D. 300400, the values of D' do not deviate much from zero.
(3) Starting with about A.D. 1000, the values of D' are close to those of today; in particular, they are practically constant.
(4) In the interval 6000 B.C. to A.D. 1000, the parameter D' undergoes considerable variance, with the oscillation amplitude reaching up to 60'/century.
Hereafter, the bounds of the time intervals indicated are approximate. Newton writes that D' 'has even changed its sign near about A.D. 800' ([II], p. 115).
In the following, we shall point out two bounds in the behaviour of the graph D', the first of them being about A.D. 500 (the beginning of the square wave on the graph), and the other one about A.D. 1000 (the end of the square wave).
In the present section, we give the results of a new interpretation and calculations of the graph of D', based on the dates of astronomical observational data made precise, which form the basis for computing the parameter D'. The curve of D' which we obtained has qualitatively different character. In particular, the incomprehensible oneorder gap of the graph completely vanishes. As it turns out, the new graph of D' is, in reality, oscillating around a constant numerical value which coincides with the modern one. As a corollary, the necessity to invent 'nongravitational forces' for the explanation of the 'gap' in the graph becomes unnecessary.

