## 18 Mathematical and Statistical Methods for Dating Events of Ancient History

There is a need for new and independent scientific methods to investigate the correctness of Scaliger's chronology. In this section we would like to mention some of those new mathematical and statistical methods that were developed by A.T. Fomenko with the purpose of examining dependences between historical texts. Due to its complexity and large amount of material that was analyzed, it is not possible to discuss here this topic in details. We will only present a brief discussion of these methods and describe the most significant results that were obtained by A.T. Fomenko and G.V. Nosovskiy. We refer all the interested readers to the Fomenko's monograph  for more details and more information.

Let us point out that the problem of recognizing dependences and dependent texts (for example the texts with the same primary source) arise in many branches of applied statistics, linguistics, physics, genetics, forensics, etc. There are many various methods used for finding dependences of this sort. In particular, they are useful in criminal investigations, where they can provide tangible proofs, based on the collected physical evidence, that a person was linked to a crime scene. Identifying fingerprints, bloodstains, fibers, markings on bullets, footprints, various marks on the body and all kinds of even microscopic traces, require more than just an impression that two patterns are similar. Figure 1.21: Medieval drawing of Ptolemy

Maximum 3

Maximum 4

Historical Text 1

Historical Text 2

Historical Text 3

Historical Text 4

Historical Text 5 Historical Text 2

Historical Text 3

Historical Text 4

### Historical Text 5

Figure 1.22: Comparison of the volume functions of dependent historical texts: 1) Vremennik Timo-feeva, 2) Piskarevskiy letopisets, 3) Skazaniye o Fyodore, 4) Novyi letopisets, 5) The 1617 Chronograph

Determination with high confidence that two patterns match is a serious mathematical problem requiring precise calculations of probabilities.

The existence of strange "repetitions" in chronology was already known for a long time, since N.A. Morozov identified several similarities between the so-called dynasty functions. In order to be able to present a solid proof that these similarities are not coincidental and that they are in fact indicate a historical "mistake" or misinterpretation of historical documents, it is necessary to analyze them statistically exactly in the same way as it is done regarding the physical evidence in a criminal investigation. The method suggested by A.T. Fomenko is based on empirico-statistical procedures that can be extremely useful not only in analyzing narrative texts such as historical chronicles, but in other areas of sciences, for example in studying biological codes.

Let us explain more clearly how the historical material, such as chronicles, can be analyzed using mathematical methods? For example, it is possible to extract all kinds of numerical information from these documents. Suppose we are analyzing a text X — a historical chronicle. We can create a sequence of integers representing the number of words used in the chronicle to describe events in subsequent years. If T denotes the year T, then the value X (T)) is the number of words in the chapter describing the year T, which corresponds to the volume of a fragment for the year T. Such a function X (T) is called by Fomenko the volume function for X. Of course, it is easy to define many other similar numerical functions that could be used as "identifiers" of the text X. For instance, the frequency an year T is mentioned in the subsequent chapters of X, the number of all the names of historical personae listed in the text, or how many times these names were mentioned in the whole text. In his monograph , A.T. Fomenko used several such functions to analyze similarities and differences between various historical documents, for which we carried appropriate statistical calculations and precise evaluations of probabilities. The main problem was to identify which of the considered documents referred to the same epoch or two different epochs. In the case of two volume functions, this could be done as follows: It is clear that for two different documents X and Y their volume functions X (T) and Y(T) can be completely different even if they refer to the same epoch. However, if X and Y describe the same sequence of events, then it is most probable that they will have the same distribution of local maxima. It is quite obvious, that important events in both documents would be presented in more elaborated forms. That means for those "important years" T the volume functions X(T) and Y(T) would have local maxima. Consequently, the both functions X(T) and Y(T) would have similar distribution of their maxima or the "important years." A question arises, how to determine the probability that two volume functions with similar maxima distribution are referring to the same sequence of events? A.T. Fomenko did the probability calculation and verify their accuracy on the chronicles related to the well documented epochs. On Figure 1.22 we show the distribution of local maxima for five different chronicles describing the same sequence of events. It is clear that, indeed, their maxima are located almost in the same positions.

On the other hand, the coincidence of local maxima of two volume functions can be used to identify, with very high probability, that two historical texts describe the same epoch and the same sequence of events, even if they were mistakenly associated with different epochs. This method of matching dependent texts is called by A.T. Fomenko the principle of maximal correlation. This

Dynasty Function 