Statistical Science 1994

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Statistical Science
1994, Vol. 9, No. 3, 429-438 (abridged)
Equidistant Letter Sequences in the Book of Genesis
Doron Witztum, Eliyahu Rips and Yoav Rosenberg

Abstract. It has been noted that when the Book of Genesis is written as two-dimensional arrays, equidistant letter sequences spelling words with related meanings often appear in close proximity. Quantitative tools for measuring this phenomenon are developed. Randomization analysis shows that the effect is significant at the level of 0.00002.
Key words and phrases: Genesis, equidistant letter sequences, cylindrical representations, statistical analysis.

1. INTRODUCTION

The phenomenon discussed in this paper was first discovered several decades ago by Rabbi Weissmandel [7]. He found some interesting patterns in the Hebrew Pentateuch (the Five Books of Moses), consisting of words or phrases expressed in the form of equidistant letter sequences (ELS's)--that is, by selecting sequences of equally spaced letters in the text.
As impressive as these seemed, there was no rigorous way of determining if these occurrences were not merely due to the enormous quantity of combinations of words and expressions that can be constructed by searching out arithmetic progressions in the text. The purpose of the research reported here is to study the phenomenon systematically. The goal is to clarify whether the phenomenon in question is a real one, that is, whether it can or cannot be explained purely on the basis of fortuitous combinations.
The approach we have taken in this research can be illustrated by the following example. Suppose we have a text written in a foreign language that we do not understand. We are asked whether the text is meaningful (in that foreign language) or meaningless. Of course, it is very difficult to decide between these possibilities, since we do not understand the language. Suppose now that we are equipped with a very partial dictionary, which enables us to recognise a small portion of the words in the text: "hammer" here and "chair" there, and maybe even "umbrella" elsewhere. Can we now decide between the two possibilities?
Not yet. But suppose now that, aided with the partial dictionary, we can recognise in the text a pair of conceptually related words, like "hammer" and "anvil." We check if there is a tendency of their appearances in the text to be in "close proximity." If the text is meaningless, we do not expect to see such a tendency, since there is no reason for it to occur. Next, we widen our check; we may identify some other pairs of conceptually related words: like "chair" and "table," or "rain" and "umbrella." Thus we have a sample of such pairs, and we check the tendency of each pair to appear in close proximity in the text. If the text is meaningless, there is no reason to expect such a tendency. However, a strong tendency of such pairs to appear in close proximity indicates that the text might be meaningful.
Note that even in an absolutely meaningful text we do not expect that, deterministically, every such pair will show such tendency. Note also, that we did not decode the foreign language of the text yet: we do not recognise its syntax and we cannot read the text.
This is our approach in the research described in the paper. To test whether the ELS's in a given text may contain "hidden information," we write the text in the form of two-dimensional arrays, and define the distance between ELS's according to the ordinary two-dimensional Euclidean metric. Then we check whether ELS's representing conceptually related words tend to appear in "close proximity."
Suppose we are given a text, such as Genesis (G). Define an equidistant letter sequence (ELS) as a sequence of letters in the text whose positions, not counting spaces, form an arithmetic progression; that is, the letters are found at the positions

n, n+d, n+2d, ... , n+(k-1)d.

We call d the skip, n the start and k the length of the ELS. These three parameters uniquely identify the ELS, which is denoted (n,d,k).
Let us write the text as a two-dimensional array--that is, on a single large page--with rows of equal length, except perhaps for the last row. Usually, then, an ELS appears as a set of points on a straight line. The exceptional cases are those where the ELS "crosses" one of the vertical edges of the array and reappears on the opposite edge. To include these cases in our framework, we may think of the two vertical edges of the array as pasted together, with the end of the first line pasted to the beginning of the second, the end of the second to the beginning of the third and so on. We thus get a cylinder on which the text spirals down in one long line.

It has been noted that when Genesis is written in this way, ELS's spelling out words with related meanings often appear in close proximity. In Figure 1 we see the example of 'patish-ùéèô' (hammer) and 'sadan-ðãñ' (anvil); in Figure 2, 'Zidkiyahu-åäé÷ãö' (Zedekia) and 'Matanya-äéðúî' (Matanya), which was the original name of King Zedekia (Kings II, 24:17). In Figure 3 we see yet another example of 'hachanuka-äëåðçä' (the Chanuka) and 'chashmonaee -éàðåîùç' (Hasmonean), recalling that the Hasmoneans were the priestly family that led the revolt against the Syrians whose successful conclusion the Chanuka feast celebrates.
Indeed, ELS's for short words, like those for 'patish-ùéèô' (hammer) and 'sadan-ðãñ' (anvil), may be expected on general probability grounds to appear close to each other quite often, in any text. In Genesis, though, the phenomenon persists when one confines attention to the more "noteworthy" ELS's, that is, those in which the skip |d| is minimal over the whole text or over large parts of it. Thus for 'patish-ùéèô' (hammer), there is no ELS with a smaller skip than that of Figure 1 in all of Genesis; for 'sadan-ðãñ' (anvil), there is none in a section of text comprising 71% of G; the other four words are minimal over the whole text of G. On the face of it, it is not clear whether or not this can be attributed to chance. Here we develop a method for testing the significance of the phenomenon according to accepted statistical principles. After making certain choices of words to compare and ways to measure proximity, we perform a randomization test and obtain a very small p-value, that is, we find the results highly statistically significant.

Up to Section 1 Down to Section 3 Down to Appendix

Figures 2 and 3 showing the codes Matanya, Zidkiyahu, Hachanuka, and Chashmonaee

2. OUTLINE OF THE PROCEDURE

In this section we describe the test in outline. In the Appendix, sufficient details are provided to enable the reader to repeat the computations precisely, and so to verify their correctness. The authors will provide, upon request, at cost, diskettes containing the program used and the texts G, I, R, T, U, V and W (see Section 3).
We test the significance of the phenomenon on samples of pairs of related words (such as hammer-anvil and Zedekia-Matanya). To do this we must do the following:

(i) define the notion of "distance" between any two words, so as to lend meaning to the idea of words in "close proximity";
(ii) define statistics that express how close, "on the whole," the words making up the sample pairs are to each other (some kind of average over the whole sample);
(iii) choose a sample of pairs of related words on which to run the test;
(iv) determine whether the statistics defined in (ii) are "unusually small" for the chosen sample.

Task (i) has several components. First, we must define the notion of "distance" between two given ELS's in a given array; for this we use a convenient variant of the ordinary Euclidean distance. Second, there are many ways of writing a text as a two-dimensional array, depending on the row length; we must select one or more of these arrays and somehow amalgamate the results (of course, the selection and/or amalgamation must be carried out according to clearly stated, systematic rules). Third, a given word may occur many times as an ELS in a text; here again, a selection and amalgamation process is called for. Fourth, we must correct for factors such as word length and composition. All this is done in detail in Sections A.1 and A.2 of the Appendix.
We stress that our definition of distance is not unique. Although there are certain general principles (like minimizing the skip d) some of the details can be carried out in other ways. We feel that varying these details is unlikely to affect the results substantially. Be that as it may, we chose one particular definition, and have, throughout, used only it, that is, the function c(w,w') described in Section A.2 of the Appendix had been defined before any sample was chosen, and it underwent no changes. [Similar remarks apply to choices made in carrying out task (ii).]
Next, we have task (ii), measuring the overall proximity of pairs of words in the sample as a whole. For this, we used two different statistics P₁ and P₂ , which are defined and motivated in the Appendix (Section A.5). Intuitively, each measures overall proximity in a different way. In each case, a small value of P_i indicates that the words in the sample pairs are, on the whole, close to each other. No other statistics were ever calculated for the first, second or indeed any sample.
In task (iii), identifying an appropriate sample of word pairs, we strove for uniformity and objectivity with regard to the choice of pairs and to the relation between their elements. Accordingly, our sample was built from a list of personalities (p) and the dates (Hebrew day and month) (p') of their death or birth. The personalities were taken from the Encyclopedia of Great Men in Israel [5].
At first, the criterion for inclusion of a personality in the sample was simply that his entry contain at least three columns of text and that a date of birth or death be specified. This yielded 34 personalities (the first list--Table 1). In order to avoid any conceivable appearance of having fitted the tests to the data, it was later decided to use a fresh sample, without changing anything else. This was done by considering all personalities whose entries contain between 1.5 and 3 columns of text in the Encyclopedia; it yielded 32 personalities (the second list--Table 2). The significance test was carried out on the second sample only.
Note that personality-date pairs (p,p') are not word pairs. The personalities each have several appellations, there are variations in spelling and there are different ways of designating dates. Thus each personality-date pair (p,p') corresponds to several word pairs (w,w'). The precise method used to generate a sample of word pairs from a list of personalities is explained in the Appendix (Section A.3).
The measures of proximity of word pairs (w,w') result in statistics P₁ and P₂ . As explained in the Appendix (Section A.5), we also used a variant of this method, which generates a smaller sample of word pairs from the same list of personalities. We denote the statistics P₁ and P₂ , when applied to this smaller sample, by P₃ and P₄ .
Finally, we come to task (iv), the significance test itself. It is so simple and straightforward that we describe it in full immediately.
The second list contains of 32 personalities. For each of the 32! permutations p of these personalities, we define the statistic P₁^p obtained by permuting the personalities in accordance with p, so that Personality i is matched with the dates of Personality p(i). The 32! numbers P₁^p are ordered, with possible ties, according to the usual order of the real numbers. If the phenomenon under study were due to chance, it would be just as likely that P₁ occupies any one of the 32! places in this order as any other. Similarly for P₂, P₃ and P₄. This is our null hypothesis.
To calculate significance levels, we chose 999,999 random permutations p of the 32 personalities; the precise way in which this was done is explained in the Appendix (Section A.6). Each of these permutations p determines a statistic P₁^p; together with P₁, we have thus 1,000,000 numbers. Define the rank order of P₁ among these 1,000,000 numbers as the number of P₁^p not exceeding P₁; if P₁ is tied with other P₁^p, half of these others are considered to "exceed" P₁. Let r₁ be the rank order of P₁, divided by 1,000,000; under the null hypothesis, r₁ is the probability that P₁ would rank as low as it does. Define r₂, r₃ and r₄ similarly (using the same 999,999 permutations in each case).
After calculating the probabilities r₁ through r₄, we must make an overall decision to accept or reject the research hypothesis. In doing this, we should avoid selecting favorable evidence only. For example, suppose that r₃ = 0.01, the other r_i being higher. There is then the temptation to consider r₃ only, and so to reject the null hypothesis at the level of 0.01. But this would be a mistake; with enough sufficiently diverse statistics, it is quite likely that just by chance, some one of them will be low. The correct question is, "Under the null hypothesis, what is the probability that at least one of the four r_i would be less than or equal to 0.01?" Thus denoting the event "r_i <= 0.01" by E_i, we must find the probability not of E₃, but of "E₁ or E₂ or E₃ or E₄." If the E_i were mutually exclusive, this probability would be 0.04; overlaps only decrease the total probability, so that it is in any case less than or equal to 0.04. Thus we can reject the null hypothesis at the level of 0.04, but not 0.01.
More generally, for any given d, the probability that at least one of the four numbers r_i is less than or equal to d is at most 4 d. This is known as the Bonferroni inequality. Thus the overall significance level (or p-value), using all four statistics, is r₀ := 4 min r_i.

Up to Section 1 Up to Section 2 Down to Appendix

3. RESULTS AND CONCLUSIONS

In Table 3, we list the rank order of each of the four P_i among the 1,000,000 corresponding P_i^p. Thus the entry 4 for P₄ means that for precisely 3 out of the 999,999 random permutations p, the statistic P₄^p was smaller than P₄ (none was equal). It follows that min r_i = 0.000004 so r₀ = 4 min r_i = 0.000016. The same calculations, using the same 999,999 random permutations, were performed for control texts. Our first control text, R, was obtained by permuting the letters of G randomly (for details, see Section A.6 of the Appendix). After an earlier version of this paper was distributed, one of the readers, a prominent scientist, suggested to use as a control text Tolstoy's War and Peace. So we used text T consisting of the initial segment of the Hebrew translation of Tolstoy's War and Peace [6]--of the same length of G. Then we were asked by a referee to perform a control experiment on some early Hebrew text. He also suggested to use randomization on words in two forms: on the whole text and within each verse. In accordance, we checked texts I, U and W: text I is the Book of Isaiah [2]; W was obtained by permuting the words of G randomly; U was obtained from G by permuting randomly words within each verse. In addition, we produced also text V by permuting the verses of G randomly. (For details, see Section A.6 of the Appendix.) Table 3 gives the results of these calculations, too. In the case of I, min r_i is approximately 0.900; in the case of R it is 0.365; in the case of T it is 0.277; in the case of U it is 0.276; in the case of V it is 0.212; and in the case of W it is 0.516. So in five cases r₀ = 4 min r_i exceeds 1, and in the remaining case r₀ = 0.847; that is, the result is totally nonsignificant, as one would expect for control texts.
We conclude that the proximity of ELS's with related meanings in the Book of Genesis is not due to chance.

TABLE 3

*Rank order of P_i among one million P_i^p*
___	P₁	P₂	P₃	P₄
G	453	5	570	4
R	619,140	681,451	364,859	573,861
T	748,183	363,481	580,307	277,103
I	899,830	932,868	929,840	946,261
W	883,770	516,098	900,642	630,269
U	321,071	275,741	488,949	491,116
V	211,777	519,115	410,746	591,503

APPENDIX: DETAILS OF THE PROCEDURE
In this Appendix we describe the procedure in sufficient detail to enable the reader to repeat the computations precisely. Some motivation for the various definitions is also provided.
In Section A.1, a "raw" measure of distance between words is defined. Section A.2 explains how we normalize this raw measure to correct for factors like the length of a word and its composition (the relative frequency of the letters occurring in it). Section A.3 provides the list of personalities p with their dates p' and explains how the sample of word pairs (w, w') is constructed from this list. Section A.4 identifies the precise text of Genesis that we used. In Section A.5, we define and motivate the four summary statistics P₁, P₂, P₃ and P₄. Finally, Section A.6 provides the details of the randomization.
Sections A.1 and A.3 are relatively technical; to gain an understanding of the process, it is perhaps best to read the other parts first.

A.1 The Distance between Words
To define the "distance" between words, we must first define the distance between ELS's representing those words; before we can do that, we must define the distance between ELS's in a given array; and before we can do that, we must define the distance between individual letters in the array.
As indicated in Section 1, we think of an array as one long line that spirals down on a cylinder; its row length h is the number of vertical columns. To define the distance between two letters x and x', cut the cylinder along a vertical line between two columns. In the resulting plane each of x and x' has two integer coordinates, and we compute the distance between them as usual, using these coordinates. In general, there are two possible values for this distance, depending on the vertical line that was chosen for cutting the cylinder; if the two values are different, we use the smaller one.
Next, we define the distance between fixed ELS's e and e' in a fixed cylindrical array. Set
f := the distance between consecutive letters of e,
f' := the distance between consecutive letters of e',
l := the minimal distance between a letter of e and one of e',
and define d(e, e') := f² + f'²+ l². We call d(e, e') the distance between the ELS's e and e' in the given array; it is small if both fit into a relatively compact area. For example, in Figure 3 we have f = 1, f' =Ö5, l = Ö34 and d = 40.
Now there are many ways of writing Genesis as a cylindrical array, depending on the row length h. Denote by d_h (e, e') the distance d(e, e') in the array determined by h, and set m_h (e, e') := 1/d_h (e, e'); the larger m_h (e, e') is, the more compact is the configuration consisting of e and e' in the array with row length h. Set e = (n,d,k) (recall that d is the skip) and e' = (n',d',k'). Of particular interest are the row lengths h = h₁,h₂,..., where hi is the integer nearest to |d|/i (1/2 is rounded up). Thus when h = h₁ = |d|, then e appears as a column of adjacent letters (as in Figure 1); and when h = h₂, then e appears either as a column that skip alternate rows (as in Figure 2) or as a straight line of knight's moves (as in Figure 3). In general, the arrays in which e appears relatively compactly are those with row length h_i with i "not too large."
Define h_i' analogously to h_i. The above discussion indicates that if there is an array in which the configuration (e,e') is unusually compact, it is likely to be among those whose row length is one of the first 10 h_i or one of the first 10 h_i'. (Here and in the sequel 10 is an arbitrarily selected "moderate" number.) So setting

s (e, e') :=


10

S

i=1



m_h

i

   (e, e') +

10

S

i=1



m_h

i

   (e, e'),

we conclude that s (e, e') is a reasonable measure of the maximal "compactness" of the configuration (e, e') in any array. Equivalently, it is an inverse measure of the minimum distance between e and e'.
Next, given a word w, we look for the most "noteworthy" occurrence or occurrences of w as an ELS in G. For this, we chose those ELS's e = (n,d,k) with |d| >= 2 that spell out w for which |d| is minimal over all of G, or at least over large portions of it. Specifically, define the domain of minimality of e as the maximal segment T_e of G that includes e and does not include any other

^ ^ ^ ^

ELS e =( n, d, k )

for w with

^

d < |d|.

If e' is an ELS for another word w', then T_e Ç T_e_' is called the domain of simultaneous minimality of e and e'; the length of this domain, relative to the whole of G, is the "weight" we assign to the pair (e,e'). Thus we define w(e,e') := l(e, e')/l(G), where l(e, e') is the length of T_e Ç T_e_', and l(G) is the length of G. For any two words w and w', we set

W (w, w') := S w (e, e') s (e, e'),

where the sum is over all ELS's e and e' spelling out w and w', respectively. Very roughly, W (w, w') measures the maximum closeness of the more noteworthy appearances of w and w' as ELS's in Genesis--the closer they are, the larger is W (w, w').
When actually computing W (w, w'), the sizes of the lists of ELS's for w and w' may be impractically large (especially for short words). It is clear from the definition of the domain of minimality that ELS's for w and w' with relatively large skips will contribute very little to the value of W (w, w') due to their small weight. Hence, in order to cut the amount of computation we restrict beforehand the range of the skip |d| <= D(w) for w so that the expected number of ELS's for w will be 10. This expected number equals the product of the relative frequencies (within Genesis) of the letters constituting w multiplied by the total number of all equidistant letter sequences with 2 <= |d| <= D. [The latter is given by the formula (D-1)(2L-(k-1)(D+2)), where L is the length of the text and k is the number of letters in w.] The same restriction applies also to w' with a corresponding bound D(w'). Abusing our notation somewhat, we continue to denote this modified function by W (w, w').

A.2 The Corrected Distance

In the previous section we defined a measure W(w,w') of proximity between two words w and w' -- an inverse measure of the distance between them. We are, however, interested less in the absolute distance between two words than in whether this distance is larger or smaller than "expected." In this section, we define a "relative distance" c (w,w'), which is small when w is "unusually close" to w', and is 1, or almost 1, when w is "unusually far" from w'.

The idea is to use perturbations of the arithmetic progressions that define the notion of an ELS. Specifically, start by fixing a triple (x,y,z) of integers in the range {-2,-1,0,1,2}; there are 125 such triples. Next, rather than looking for ordinary ELS's (n,d,k), look for "(x,y,z)- perturbed ELS's" (n,d,k)^(x,y,z), obtained by taking the positions

n, n + d,..., n + (k-4)d, n + (k-3)d + x, n + (k-2)d + x + y, n + (k-1)d +x +y + z,

instead of the positions n, n +d, n + 2d,..., n + (k-1)d. Note that in a word of length k, k-2 intervals could be perturbed. However, we preferred to perturb only the three last ones, for technical programming reasons.

The distance between two (x,y,z)-perturbed ELS's (n,d,k)^(x,y,z) and (n',d',k')^(x,y,z) is defined as the distance between the ordinary (unperturbed) ELS's (n,d,k) and (n',d',k').

We may now calculate the "(x,y,z)-proximity" of two words w and w' in a manner exactly analogous to that used for calculating the "ordinary" proximity W (w,w'). This yields 125 numbers W ^(x,y,z) (w,w'), of which W (w,w') = W^(0,0,0) (w,w') is one. We are interested in only some of these 125 numbers; namely, those corresponding to triples (x,y,z) for which there actually exist some (x,y,z)-perturbed ELS's in Genesis for w, and some for w' [the other W^(x,y,z) (w,w') vanish]. Denote by M(w,w') the set of all such triples, and by m(w,w') the number of its elements.

Suppose (0,0,0) is in M (w,w'), that is, both w and w' actually appear as ordinary ELS's (i.e., with x = y = z = 0) in the text. Denote by v (w,w') the number of triples (x,y,z) in M (w,w') for which W^(x,y,z) (w,w') >= W (w,w'). If m (w,w') >= 10 (again, 10 is an arbitrarily selected "moderate" number),

c (w,w') := v (w,w') / m (w,w').

If (0,0,0) is not in M (w,w'), or if m (w,w') < 10 (in which case we consider the accuracy of the method as insufficient), we do not define c (w,w').

In words, the corrected distance c (w,w') is simply the rank order of the proximity W (w,w') among all the "perturbed proximities" W^(x,y,z) (w,w'); we normalize it so that the maximum distance is 1. A large corrected distance means that ELS's representing w are far away from those representing w', on a scale determined by how far the perturbed ELS's for w are from those for w'.

A.3 The Sample of Word Pairs

The reader is referred to Section 2, task (iii), for a general description of the two samples. As mentioned there, the significance test was carried out only for the second list, set forth in Table 2. Note that the personalities each may have several appelations (names), and there are different ways of designating dates. The sample of word pairs (w, w') was constructed by taking each name of each personality and pairing it with each designation of that personality's date. Thus when the dates are permuted, the total number of word pairs in the sample may (and usually will) vary.
We have used the following rules with regard to Hebrew spelling:

1. For words in Hebrew, we always chose what is called the grammatical orthography--"ktiv dikduki." See the entry "ktiv" in Even-Shoshan's dictionary [1].
2. Names and designations taken from the Pentateuch are spelled as in the original.
3. Yiddish is written using Hebrew letters; thus, there was no need to transliterate Yiddish names.
4. In transliterating foreign names into Hebrew, the letter "alef-à" is often used as a mater lectionis; for example, "Luzzatto" may be written "åèöåì" or "åèàöåì." In such cases we used both forms.

In designating dates, we used three fixed variations of the format of the Hebrew date. For example, for the 19th of Tishri, we used éøùú è'é, éøùú è'éá and éøùúá è'é. The 15th and 16th of any Hebrew month can be denoted as ä'é or å'è and å'é or æ'è, respectively. We used both alternatives.
The list of appellations for each personality was provided by Professor S. Z. Havlin, of the Department of Bibliography and Librarianship at Bar Ilan University, on the basis of a computer search of the "Responsa" database at that university.
Our method of rank ordering of ELS's based on (x, y, z)-perturbations requires that words have at least five letters to apply the perturbations. In addition, we found that for words with more than eight letters, the number of (x, y, z)-perturbed ELS's which actually exist for such words was too small to satisfy our criteria for applying the corrected distance. Thus the words in our list are restricted in length to the range 5-8. The resulting sample consists of 298 word pairs (see Table 2).

A.4 The Text

We used the standard, generally accepted text of Genesis known as the Textus Receptus. One widely available edition is that of the Koren Publishing Company in Jerusalem. The Koren text is precisely the same as that used by us.

A.5 The Overall Proximity Measures P₁, P₂, P₃ and P₄

Let N be the number of word pairs (w, w') in the sample for which the corrected distance c(w, w') is defined (see Sections A.2 and A.3). Let k be the number of such word pairs (w, w') for which c(w, w') <= 1/5.
Define

j N-j

p₁ :=

N

S

j=k

(

N

j

)
(

1

---

5

)
(

4

---

5

)

To understand this definition, note that if the c(w, w') were independent random variables that are uniformly distributed over [0,1], then P₁ would be the probability that at least k out of N of them are less than are equal to 0.2. However, we do not make or use any such assumptions about uniformity and independence. Thus P₁, though calibrated in probability terms, is simply an ordinal index that measures the number of word pairs in a given sample whose words are "pretty close" to each other [i.e., c(w, w') <= 1/5], taking into account the size of the whole sample. It enables us to compare the overall proximity of the word pairs in different samples; specifically, in the samples arising from the different permutations of the 32 personalities.
The statistic P₁ ignores all distances c(w,w') greater than 0.2, and gives equal weight to all distances less than 0.2. For a measure that is sensitive to the actual size of the distances, we calculate the product Pc(w,w') over all word pairs (w,w') in the sample. We then define
P₂ := F^N := (P c(w, w')),
with N as above, and

F^N (X) := X
(
1 - ln X +

(-ln X)²

-------

2!

+ . . . +

(-ln X) ^N-1

-----

(N - 1)!

).

To understand this definition, note first that if x₁,x₂,...,x_N are independent random variables that are uiformly distributed over [0,1], then the distribution of their product X := x₁x₂ ... x_N is given by Prob(X <= X₀) =F^N (X₀); this follows from (3.5) in [3], since the -ln x_i are distributed exponentially, and -ln X = Si (ln x_i). The intuition for P₂ is then analogous to that for P₁: If the c(w,w') were independent random variables that are uniformly distributed over [0,1], then P₂ would be the probability that the product P c(w,w') is as small as it is, or smaller. But as before, we do not use any such uniformity or independence assumptions. Like P₁, the statistic P₂ is calibrated in probability terms; but rather than thinking of it as a probability, one should think of it simply as an ordinal index that enables us to compare the proximity of the words in word pairs arising from different permutations of the personalities.
We also used two other statistics, P₃ and P₄. They are defined like P₁ and P₂, except that for each personality, all appellations starting with the title "Rabbi" are omitted. The reason for considering P₃ and P₄ is that appellations starting with "Rabbi" often use only the given names of the personality in question. Certain given names are popular and often used (like "John" in English or "Avraham" in Hebrew); thus several different personalities were called Rabbi Avraham. If the phenomenon we are investigating is real, then allowing such appellations might have led to misleadingly low values for c(w,w') when p matches one "Rabbi Avraham" to the dates of another "Rabbi Avraham." This might have resulted in misleadingly low values P₁^p and P₂^p for the permuted samples, so in misleadingly low significance levels for P₁ and P₂ and so, conceivably, to an unjustified rejection of the research hypothesis. Note that this effect is "one-way"; it could not have led to unjustified acceptance of the research hypothesis, since under the null hypothesis the number of P_i^p exceeding P_i is in any case uniformly distributed. In fact, omitting appellations starting with "Rabbi" did not affect the results substantially (see Table 3); but we could not know this before perfoming the calculations.
An intuitive feel for the corrected distances (in the original, unpermuted samples) may be gained from Figure 4. Note that in both the first and second samples, the distribution for R looks quite random, whereas for G it is heavily concentrated near 0. It is this concentration that we quantify with the statistics P_i.

A.6 The Randomizations

The 999,999 random permutations of the 32 personalities were chosen in accordance with Algorithm P of Knuth [4], page 125. The pseudorandom generator required as input to this algorithm was that provided by Turbo-Pascal 5.0 of Borland Inter Inc. This, in turn, requires a seed consisting of 32 binary bits; that is, an integer with 32 digits when written to the base 2. To generate this seed, each of three prominent scientists was asked to provide such an integer, just before the calculation was carried out. The first of the three tossed a coin 32 times; the other two used the parities of the digits in widely separated blocks in the decimal expansion of p. The three resulting integers were added modulo 2³². The resulting seed was 01001 10000 10011 11100 00101 00111 11.
The control text R was constructed by permuting the 78,064 letters of G with a single random permutation, generated as in the previous paragraph. In this case, the seed was picked arbitrarily to be the decimal integer 10 (i.e., the binary integer 1010). The control text W was constructed by permuting the words of G in exactly the same way and with the same seed, while leaving the letters within each word unpermuted. The control text V was constructed by permuting the verses of G in the same way and with the same seed, while leaving the letters within each verse unpermuted.
The control text U was constructed by permuting the words within each verse of G in the same way and with the same seed, while leaving unpermuted the letters within each word, as well as the verses. More precisely, the Algorithm P of Knuth [4] that we used requires n - 1 random numbers to produce a random permutation of n items. The pseudorandom generator of Borland that we used produces, for each seed, a long string of random numbers. Using the binary seed 1010, we produced such a long string. The first six numbers in this string were used to produce a random permutation of the seven words constituting the first verse of Genesis. The next 13 numbers (i.e., the 7th through the 19th random numbers in the string produced by Borland) were used to produce a random permutation of the 14 words constituting the second verse of Genesis, and so on.

REFERENCES

[1] EVEN-SHOSHAN, A. (1989). A New Dictionary of the Hebrew Language. Kiriath Sefer, Jerusalem.

[2] FCAT (1986). The Book of Isaiah, file ISAIAH.MT. Facility for Computer Analysis of Texts (FCAT) and Tools for Septuagint Studies (CATSS), Univ. Pennsylvania, Philadelphia. (April 1986.)

[3] FELLER, W. (1966). An Introduction to Probability Theory and Its Applications 2. Wiley, New York.

[4] KNUTH, D. E. (1969). The Art of Computer Programming 2. Addison-Wesley, Reading, MA.

[5] MARGALIOTH, M., ed. (1961). Encyclopedia of Great Men in Israel; a Bibliographical Dictionary of Jewish Sages and Scholars from the 9th to the End of the 18th Century 1-4. Joshua Chachik, Tel Aviv.

[6] TOLSTOY, L. N. (1953) War and Peace. Hebrew translation by L. Goldberg, Sifriat Poalim, Merhavia.

[7] WEISSMANDEL, H. M. D. (1958). Torath Hemed. Yeshivath Mt. Kisco, Mt. Kisco.

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