History Online - Ancient Dynasties

Dynasties of rulers. The real dynasty and the numerical dynasty. Dependent and independent dynasties. The small-distortion principle

We shall now describe a concrete problem for whose solution we introduce the measure A. Let a historical text describing a previously unknown dynasty of rulers be discovered with an indication of the duration of their reigns. The question arises whether this historical dynasty is new and not mentioned in the known documents or whether it is one of the rulers' dynasties already known to us but described in a text in unusual terms (with the rulers' names distorted, etc.).

Consider n consecutive authentic rulers (kings). Let the true rule durations of these kings be pi,p2,... ,pn, respectively. We call this sequence a real dynasty. Note that the same real dynasty of rulers is often described in the primary sources from different standpoints by different chroniclers. But there exist more or less 'invariant facts' concerning these rules, and their description depends little on the tastes of the author of a primary source (chronicler). Such facts include, for example, the duration of a king's rule, since there are usually no special reasons for which the chronicler should considerably and intentionally distort it. Nevertheless, chroniclers often encounter serious difficulties in calculating the regal rule duration, which leads to giving different values to the duration of the rule of the same historical character in different historical documents.

Thus, each author (chronicler), while describing a real dynasty p = (pi,p2, ... ,?"), calculates the duration a, of a king's rule and obtains a certain sequence of numbers a = (ai, 03,... , a'). This sequence of numbers represented as an integral vector a in the space R' will be called a numerical dynasty. Another chronicler, while describing the same real dynasty of kings, will possibly obtain another vector b from R', i.e., another numerical dynasty. Thus, one and the same real dynasty can be represented as different numerical dynasties in different documents.

As the set D described in 5.1, we take a sufficiently large set of real dynasties of length n, i.e., D = {p = (pi,... ,?")}. We formulate the following theoretical model (statistical hypothesis).

The small-distortion principle.
If two numerical dynasties are sufficiently close (in the sense of the measure \), then they indeed represent the same real dynasty of kings, i.e., they are merely two different versions of its description.

Such numerical dynasties will be called dependent. On the contrary, if two numerical dynasties represent two real dynasties of kings, known a priori as different, then the numerical dynasties are much different from one another (in the sense of the measure X). Such numerical dynasties will be called independent.

Later in Section 5.4, before verifying this model experimentally, we shall give an exact description of the measure A. Meanwhile, we identify the set of all numerical dynasties describing real historical dynasties from the set D in the space R' (see above) with the set V(D).

Basic errors leading to controversy among chroniclers as to the duration of kings' rules

We now point out concrete errors most often leading to the controversy among chroniclers as to the duration of the rules of kings.

(a) Permutation of the names of (or confusion between) two neighbouring rulers.
(b) Replacing two neighbouring rulers by one, the duration of whose rule was assumed to be equal to the sum of the rule durations of both.
(c) Computational error by a chronicler. The longer the duration of a king's rule, the greater the error that arises in its computation.

It turns out that these three basic types of concrete errors made by chroniclers can be sufficiently simply described by means of a suitable mapping, V : D -' .R'. Let p be a certain real dynasty in the set D. We call the dynasty (vector) c a virtual variation (virtual vector) of the dynasty p and write c = v(p) if the following conditions are fulfilled, namely, each coordinate c; of c coincides with one of the three coordinates of the original vector p, i.e.,
P.-li Pi, P'+i, or with pi +P.+I.

It is clear that each of such virtual vectors, or virtual dynasties, can be regarded as a numerical dynasty and be obtained from a real dynasty p, because of chroniclers' errors of type (a) and (b).
Eventually, we take as V(D) the union of all virtual vectors (virtual dynasties) c = v(p), where p ranges over all the real dynasties of D. It remains to model an error of type (c).
Let a piecewise smooth, nonnegative function a(t) be given on the positive half-axis t > 0. In our case, the role of a(t) will be played by the probability density of a random variable i] to be specified below. Put h(t) = f(a(t)), where f(s) is a certain monotonically decreasing function of a parameter s, given on the half-axis s > 0, and such that lim/(s) = +00 when s -r +0. For example, as f(s), we can take the function ^. If r/ is a discrete random variable with probability density oc(t), then the quantity h(t) becomes greater as r] assumes the value t with lesser probability. In our problem, we take as T] the duration of a king's rule in a dynasty. Let t range over all positive integers (i.e., possible values of the rule duration). If t is a certain fixed rule duration, or the value of i], then a(t) will mean the number of historical characters ruling for t years. We call h(t) the error amplitude in measuring a rule t years long. The graph a(t) in Fig. 12 shows that short rules are most frequent, and, conversely, long rules are rare.