4.1. The formulation of the small dynastic distortions principle

The small dynastic distortions principle, and a method based thereupon, was proposed and developed by the author in [884], [885], [888], [1129], [895] and [1130].

Let us assume a historical text to be found, describing a dynasty of rulers unknown to us, indicating the duration of their rule. The question arises whether this dynasty is a new one, unknown to us, and therefore requiring dating, or is it one of the dynasties we know, but described in the terms we are not used to - for example, the names of rulers are altered, etc.? The answer is in the procedure below ([904], [908], [1137], [885] and [886]).

Let us examine the k value of any successive actual rulers or kings in the history of some state or region. We shall agree to name this sequence an actual dynasty; its members by no means have to be related, though. Frequently, the same actual dynasty is described in different documents, by different chroniclers, and from different points of view - for example, the activity of rulers, their significance, personal qualities, and so forth, evaluated in a different way. Nevertheless, there are the "invariant" facts, the description of which is less dependent on sympathies or antipathies of chroniclers. These more or less "invariant facts" include, for example, the duration of the rule of a king. Usually there are no special reasons for a chronicler to significantly or intentionally distort this figure. However, chroniclers would frequently encounter natural difficulties in calculating reign duration for this or that king.

These natural difficulties are as follows: incompleteness of information, distortion in documents, etc. They sometimes resulted in the fact that chronicles or tables by different chroniclers would report different numbers, which to them seemed to be the reign duration of the same king. Such divergences, sometimes significant, are characteristic, for example, for the pharaohs in the tables by H. Brugsch ( [99] ) and in the Chronological Tables by J. Blair ([76]). For example, the tables by J. Blair, going as far as the beginning of the XIX century, collected all basic historical dynasties, with dates of rule, the information about which is available to us. The value of the tables by J. Blair for us lies in the fact that they were compiled in an epoch sufficiently close to the time of the creation of the Scaligerian chronology. Therefore, they contain clearer imprints of the "Scaligerian activity" which were subsequently shaded and plastered by the historians of the XIX-XX century.

Thus, each chronicler describing an actual dynasty M calculates the reign duration of its kings in his own way, to the best of his abilities and possibilities. As a result, he obtains a certain sequence of numbers a = (aj, a2,..., ak)y where number a, shows, possibly with an error, the actual reign duration for a king with the value i. Let us recall that the value k represents the total number of kings in the dynasty. We agreed to call this sequence of values extracted from the chronicle, a dynasty of annals, convenient to be represented as vector a in Euclidean space Rk.

Another chronicler describing the same real dynasty M may assign somewhat different reign durations to the same kings. As a result, another dynasty of annals b = (bly b2, ... ,bk) will appear. Thus, the same actual dynasty M, described in different chronicles, may be depicted therein as different dynasties of annals a and b. The question is that of how great resulting distortions are? In this case, errors and objective difficulties impeding precise determination of the actual duration of rule play a significant part. We describe the basic types of errors below.

Let us formulate a statistical model, or a hypothesis, which we agree to call the small distortions principle.

The small distortions principle for the reign durations.

If the two dynasties of annals a and b are "slightly" different, they refer to the same actual dynasty M, i.e., these are two versions of its descriptions in different chronicles. We call such dynasties of annals dependent.

On the contrary, if the two dynasties of annals a and b refer to two different actual dynasties M and N, they differ "considerably". We call them independent.

We shall call the remaining pairs of dynasties neutral

In other words, according to this hypothetical model, different chroniclers would distort the same actual dynasty "slightly" when writing their chronicles. In any case, the resulting differences proved to be smaller "on the average" than those existing between evidently different, or independent, actual dynasties.

The hypothesis or the model formulated above requires an experimental verification. In case of its validity, an important and by no means obvious quality is revealed, one that characterizes the activity of ancient chroniclers. Namely, the dynasties of annals that appeared in the description of the same actual dynasty differ from one another and from their prototype less than truly different actual dynasties do.

Is there a natural numerical coefficient, or a measure c(a, b), computed for each pair of dynasties of annals a and b and possessing the quality of being "small" for dependent dynasties and, on the contrary, "large" for independent ones? In other words, this coefficient must distinguish between the dependent and independent dynasties. We have discovered such coefficient.

It turns out that, in order to evaluate the "proximity" of the two dynasties a and b, it is possible to introduce the numerical coefficient c(a, b), similar to the coefficient PACY = p(X, Y) as described above. This coefficient c(a, b) also stands for probability. Let us first describe a rough idea of determining the coefficient c(ay b). The dynasty of annals may be conveniently presented in the form of a graph, with the number of kings on the horizontal axis, and the duration of their reigns on the vertical axis. We will say that dynasty q "is similar" to the two dynasties a and by if the graph of dynasty q differs from the graph of dynasty a no more than the graph of dynasty b differs from the graph of dynasty a. See details below in [904], [1137], [885], [886]and [884].

The part that dynasties "similar" to dynasties a and b constitute in the set of all dynasties is assumed as c(a, b). In other words, we calculate the ratio:

quantity of dynasties "similar" to a and b total quantity of dynasties described in the chronicles

Chroniclers may determine the reign durations of kings with an error. We actually extract only their ap proximate values from the chronicles. It is possible to describe the mechanisms of probability resulting in such errors mathematically. Furthermore, we considered two additional errors that the chroniclers might have possibly made: the permutation of two successive kings and the replacement of these two successive kings by one "king" with a summary duration of rule.

The coefficient c(a, b) may be called PACD, i.e., Probability of Accidental Coincidence of Dynasties a and b.

4.2. The statistical model

Let us now provide a formal definition of the coefficient c(ay b)y designating the set of all actual dynasties with the length fc, i.e., consisting of k sequential kings, as D. We will actually have to denote as set D those historical dynasties the information about which is available to us from the preserved historical chronicles. We have compiled an almost complete list of all such dynasties based on a large number of different chronological tables listed below. On the basis of these tables, we composed a list of all groups of 15 successive kings, who, according to the Scaligerian chronology, had ruled within the range of 4000 b.c. -1900 a.d. in Europe, the Mediterranean, the Middle East, Egypt, and Asia.

Each dynasty of annals may be depicted as a vector in fc-dimensional Euclidean space Rk. In our specific experiment we assumed k- 15, q.v. above. We consider two dynasties essentially different if the number of kings, or actual rulers simultaneously listed in both dynasties does not exceed kl2, or a half of the members of the entire dynasty. Two randomly chosen real dynasties may intersect, have common members, since we may declare, at our own discretion, one or another king as "the progenitor of a dynasty". Along with dependent and independent dynasties, there also exist "intermediate" or "neutral" pairs of dynasties, in which the number of common kings, or actual rulers, exceeds kl2 "(although the dynasties aren t dependent). It is clear that if the total number of dynasties in question is large, the quantity of intermediate or neutral pairs of dynasties is relatively small. Therefore, primary attention should be paid to dependent and independent pairs of dynasties.

The small distortions principle as formulated above means that in practice, "on the average", chroniclers made insignificant mistakes, which means that they would not distort actual numerical data greatly.

Let us now discuss the errors most frequently made by chroniclers in calculating the reign durations of ancient kings. We found these three types of errors while working on a large number of actual historical texts. These particular errors proved to most frequently result in the distortion of actual durations of rules of kings.

Error one. The permutation or confusion of two adjacent kings.

Error two. The replacement of two kings by one, whose duration of rule equals the sum of durations of both rules.

Error three. Inaccuracy in calculating the very reign duration per se. The longer the duration, the greater error the chronicler would usually make in its determination.

These three types of errors may be described and simulated mathematically. Let us begin with errors (1) and (2). We shall examine a dynasty p = (ply p2>..., pk) from the set D. We shall call vector q = (qly q2,..., qk) a virtual variation of vector (dynasty) py and designate it as q=vir(p)y if each coordinate qt of vector q is derived from coordinates of vector p in one of the two following procedures (1) and (2).

(1) Either q{ -p{ (the coordinate does not change), or pi coincides withp,^, orp, coincides with pi+1, i.e., with one of the "adjacent coordinates" of vector p.

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