¿c nation of duration T. In other words, chroniclers calculate small, "short" reign durations better, and in doing so, make insignificant mistakes. On the contrary, a chronicler would calculate long reign durations, those encountered rather rarely, with a significant error. The longer the reign, the greater the possible error.

The errors function h(T) for indicated probability density of a random value (reign duration) was determined experimentally ([884], p. 115). Let us divide the segment [0, 100] of integer axis T into ten segments of identical length, namely:

[0,9], [10,19], [20,29], [30,39],... [90,99]. Then it appears that: h(T) = 2, if T varies from 0 to 19, h(T) = 3, if T varies from 20 to 29, h(T) = 5 ([7710]- 1), if T varies from 30 to 100. The integer part of number s is designated as [s], fig. 5.34.

Let us now consider the errors of chroniclers in plotting the "environment" for point a. For this end, we expand the parallelepiped P'(ay b)y making it a larger parallelepiped P(ay b)y where point a is again its centre, and segments with the ends

[a{ - |a{ - bi\ - ft(fli), a{ + |a{ - b{\ + fc(af-)]

are orthogonal projections thereof on the coordinate axes.

It is clear that the parallelepiped P'(ay b) lies entirely within the large parallelepiped P(ay b)y q.v. in fig. 5.33. Vector a-b + h(a) is the diagonal of this large parallelepiped, where vector h (a) is h(a) = (h(a1)y...yh(ak)).

It is possible to name it the vector of chroniclers' errors.

Thus, we simulated all three basic errors that the chroniclers would make while calculating reign durations. As the final coefficient c(ay b) measuring the proximity or distance from each other of two dynasties a and by we assume the following number:

the number of points from the set vir(D)

falling in P(ay b) the total number of points in the set vir(D) '

Fig. 5.34. A "scribe error function" calculated experimentally.

It is clear that the number c(ay b) is the integral of density function z(x) along the parallelepiped P(ay b). In fig. 5.35, the number c(ay b) is symbolically pre

Fig. 5.35. Coefficient c(a, b) presented as the volume of a prism, or an integral of the function z(x) along the parallelepiped P(a, b).

Fig. 5.35. Coefficient c(a, b) presented as the volume of a prism, or an integral of the function z(x) along the parallelepiped P(a, b).

sented as the volume of a prism with parallelepiped P(a, b) as its base, and limited on top by the graph of the function z. Number c(a, b) may, if desired, be interpreted as the probability that a random "dynastic vector" distributed in space Rk with density function z proves to be at a distance from point a, keeping within the distance between points a and b> with the error h(a) taken into account. In other words, the random "dynastic" vector distributed with the density function z falls into the environment P(a, b) of point a with the "radius" a-b + h(a).

It is evident from the above that the role of dynasties a and b in the calculation of the coefficient c(a, b) is not the same. Dynasty a was placed into the centre of parallelepiped P(a, b), and dynasty b determined its diagonal. Certainly, it was possible "to grant equal rights" to dynasties a and b> likewise the preceding coefficient p(X, Y). In other words, it is possible to change the positions of dynasties a and b, calculate coefficient c(b, a), and then obtain the arithmetic mean value of numbers c(a, b) and c(b, a). We refrained from this for two reasons. Firstly, as certain experiments have shown, replacement of coefficient c(a> b) by its "symmetric analogue" does not actually change the obtained results. Secondly, in certain cases dynasties a and b may actually have unequal rights in the sense that one of them may be the original, and the second merely its duplicate, a phantom reflection. In this case it is natural to place dynasty a, which claims to be the original, in the centre of the parallelepiped, and consider the "phantom reflection" b a "disturbance" of dynasty a. The resulting differences between coefficients c(a, b) and c(b, a), albeit small, may serve as useful material for further, more complex research, which has not been performed yet.

4.3. Refinement of the model and the computation experiment

The small distortions principle as formulated above was checked on the basis of coefficient c(a, b).

1) For verification purpose we used Chronological Tables by J. Blair ( [76] ) containing virtually all basic chronological data from the Scaligerian version of the history of Europe, the Mediterranean, the Middle East, Egypt, and Asia allegedly from 4000 b.c. to 1800 a.d. This data was then complemented with lists of rulers and their reign durations taken from other tables and monographs, both mediaeval and contemporary. Let us mention the following books here, for example: C. Bemont, G. Monod ( [64] ), E. Bickerman ([72]), H. Brugsch ([99]), A. A. Vasilyev ([120]), F. Gregorovius ([195] and [196]), J. Assad ([240]), C. Diehl ( [247] ), F. Kohlrausch ([415] ), S. G. Lozinsky ([492]), B. Niese ([579]), V. S. Sergeyev ([766] and ( [767]), Chronologie égiptienne ([1069] ), F. K. Ginzel ( [ 1155] ), L. Ideler ( [1205] ), L'art de vérifier les dates des faits historiques ( [ 1236] ), T. Mommsen ( [ 1275] ), Isaac Newton ([1298]), D. Petavius ([1337]), J. Sca-liger ([1387]).

2) As we have already noted, by dynasty we understand a sequence of actual rulers of the country, irrespectively of their titles and kinship. Subsequently, we will sometimes refer to them as kings for the sake of brevity.

3) The existence of co-rulers sometimes makes it difficult to arrange dynasties into a sequence. We accepted the simplest principle of ordering - by the average reign durations.

4) We will call the sequence of numbers showing the reign durations of all rulers over the course of the entire history of a certain state (where the length of a sequence is not limited a priori) a dynastic current. Sub-sequences obtained by neglecting some of co-rulers will be called dynastic jets. Each jet is to be even, which means that middles of periods of rule must increase monotonically. A dynastic jet must also be complete, or cover the entire historical period included in the given flow without gaps or lapses; reign period superpositions are in order here.

5) In actual situations the above requirements may be somewhat disrupted for natural reasons, - for example, one or several years of interregnum may be missing in a chroniclers story, - therefore insignificant gaps have to be acceptable. We only allowed gaps with durations not exceeding one year. Furthermore, while analysing dynastic currents and jets, the possibility of authentic picture distortion as a result of abovementioned errors (l)y (2)y and (3)> made by chroniclers - must be constantly kept in mind.

6) Another reason for the distortion of a clear formal picture lies in the fact that the beginning of a king's reign is sometimes hard to determine for certain. For example, should we start counting from the moment of actual accession, or from the moment of formal inauguration? Different tables give diverse variants of the beginning of rule of Friedrich II: 1196, 1212,1215, and 1220 a.d. At the same time, usually there is no problem to determine the end of a rule -in most cases, the death of a king. Thus, a need arises for the "bifurcation", or even a review of the three versions of a king. Fortunately, in practice larger numbers of versions are exceptionally rare. All these versions were included in a general dynastic current. In doing so, not one single jet under research should have contained two different versions of the same reign.

7) A complete list D of all dynasties of annals with the length of 15 - i.e., a list of all dynasties of 15 successive kings - was made for all states of the above-indicated geographical regions on the basis of chronological data that we collected from the Scaligerian version. Moreover, every king could appear in several 15-member dynasties, i. e., dynasties may "overlap". Let us enumerate the basic dynastic currents that underwent statistical analysis. They are: the bishops and popes in Rome, patriarchs of Byzantium, Saracens, high priests in Judah, Greek-Bactrians, exarchs in Ravenne, pharaoh dynasties of Egypt, the mediaeval dynasties of Egypt, dynasties of Byzantium, the Roman empire, Spain, Russia, France, Italy, Ottoman = Ataman empire, Scotland, Lacedaemon, Germany, Sweden, Denmark, Israel, ludah, Babylon, Syria, Portugal, Parthia, the kingdom of Bosporus, Macedonia, Poland, England.

8) Having applied disturbances of types (1) and (2), see above, to list D of dynasties of annals, we turned out to have obtained approximately 15 X1011 virtual dynasties, i.e., the set vir(D) appears to contain approximately 15 X1011 points.

4.4. Result of the experiment: coefficient c(a, b)

positively distinguishes between the dependent and independent dynasties of kings

Computation experiment performed in 1977-1979 that M. Zamaletdinov, P. Puchkov, and yours truly performed together confirmed the small distortions principle. Namely, the number PACD=c(ay b) turned out to never exceed 10"8, and usually vary from 10~12 to 10-10, for a priori dependent dynasties of annals a and b. In probabilistic interpretation, it means that if we examine the observed proximity of two dependent dynasties of annals as a random event, then its probability is small, such event is exceptionally rare, since only one of hundred billion chances occurs.

It further appeared that if two dynasties of annals a and b refer to two a priori different real dynasties, coefficient PACD = c(ay b) "is substantially larger" -namely, never less than 10"3, or "large". Likewise, in the case of coefficient p(X, Y)y we are certainly not interested absolute values of PACD = c(ay b) but, rather, the difference of several orders of magnitude between the "dependent zone" and the "independent zone", q.v. in fig. 5.36.

Thus, with the aid of coefficient PACD it was possible to discover the essential difference between a priori dependent and a priori independent dynasties of annals.

4.5. The method of dating the royal dynasties and the method detecting the phantom dynastic duplicates

And so, the coefficient c(ay b) helps us to distinguish between dependent and independent pairs of dynasties of annals with reasonable certainty. The important experimental condition is that the mistakes of chroniclers are not "too grave". In any case, their errors are substantially less than the value distinguishing between independent dynasties.

This makes it possible to propose a new method of recognizing dependent dynasties of annals and a dating procedure for unknown dynasties within the framework of the experiment performed. Likewise in the paragraph above, for an unknown dynasty d we calculate the coefficient c(a, d)y where a denotes known and already dated dynasties of annals. Let us assume that we have discovered dynasty ay for which the coefficient c(ay d) is small, that is to say, it does not exceed 10 8. This allows us to say that dynasties a and d are dependent with the probability of 1 - c(a, d)y - i.e., dynasties of annals a and d obviously correspond to one actual dynasty M, the dating of which is already known to us. Thus, we can date the dynasty of annals d.

This procedure was tested on mediaeval dynasties with a known dating. The efficiency of the procedure was completely confirmed ([904] and [908]).

The same method makes it possible to reveal phantom duplicates in the "Scaligerian textbook on history". Namely, if we find two dynasties of annals a and by for which coefficient c(ay b) does not exceed 10~8, this allows us to assume having just seen two copies, or two versions describing the same actual dynasty M multiplied on the pages of different chronicles, and then placed into different parts of the "Scaligerian textbook".

Let us reiterate that any conclusions or hypotheses appealing to "similarities" or on the contrary, "dissimilarities" of dynasties may be considered sensible only when based on extensive numeric experiments, similar to the ones performed by us. Otherwise, vague subjective considerations hardly worthy of being discussed may surface.

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