

Statistical Science
1994, Vol. 9, No. 3, 429438 (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 twodimensional 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
twodimensional arrays, and define the distance between ELS's
according to the ordinary twodimensional 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+(k1)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 twodimensional arraythat is, on a single
large pagewith 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 'patishùéèô'
(hammer) and 'sadanðãñ'
(anvil); in Figure 2, 'Zidkiyahuåäé÷ãö'
(Zedekia) and 'Matanyaäéðúî'
(Matanya), which was the original name of King Zedekia (Kings II,
24:17). In Figure 3 we see yet another example of 'hachanukaäëåðçä'
(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 'patishùéèô'
(hammer) and 'sadanðãñ'
(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 'patishùéèô'
(hammer), there is no ELS with a smaller skip than that of Figure
1 in all of Genesis; for 'sadanðãñ'
(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 pvalue, that is, we find the results
highly statistically significant.
Up to Section 1
Down to Section 3 Down to Appendix



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 hammeranvil and ZedekiaMatanya). 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
twodimensional 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 P_{1} and P_{2} ,
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 listTable
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 listTable
2). The significance test was carried out on the second sample
only.
Note
that personalitydate 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 personalitydate 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 P_{1} and P_{2} .
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 P_{1}
and P_{2} , when applied to this smaller
sample, by P_{3} and P_{4} .
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 P_{1}^{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 P_{1}^{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 P_{1}
occupies any one of the 32! places in this order as any other.
Similarly for P_{2}, P_{3} and
P_{4}. 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 P_{1}^{p};
together with P_{1}, we have thus 1,000,000 numbers.
Define the rank order of P_{1} among these
1,000,000 numbers as the number of P_{1}^{p}
not exceeding P_{1}; if P_{1} is
tied with other P_{1}^{p},
half of these others are considered to "exceed" P_{1}.
Let r_{1} be the rank order of P_{1},
divided by 1,000,000; under the null hypothesis, r_{1}
is the probability that P_{1} would rank as low as
it does. Define r_{2}, r_{3}
and r_{4} similarly (using the
same 999,999 permutations in each case).
After
calculating the probabilities r_{1}
through r_{4}, 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 r_{3} =
0.01, the other r_{i}
being higher. There is then the temptation to consider r_{3}
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 E_{3}, but of "E_{1} or
E_{2} or E_{3} or E_{4}."
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 pvalue),
using all four statistics, is r_{0}
:= 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 P_{4} means that for
precisely 3 out of the 999,999 random permutations p,
the statistic P_{4}^{p}
was smaller than P_{4} (none was equal). It
follows that min r_{i} =
0.000004 so r_{0} = 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 r_{0}
= 4 min r_{i} exceeds 1,
and in the remaining case r_{0}
= 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}
___ 
P_{1} 
P_{2} 
P_{3} 
P_{4} 
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 P_{1},
P_{2}, P_{3} and P_{4}.
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') := f^{2}
+ f'^{2 }+ l^{2}. 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 = h_{1},h_{2},...,
where hi is the integer nearest to d/i (1/2
is rounded up). Thus when h = h_{1} = d,
then e appears as a column of adjacent letters (as in
Figure 1); and when h = h_{2}, 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') :=



(e, e') + 


(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
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
Genesisthe 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 (D1)(2L(k1)(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 + (k4)d,
n + (k3)d + x, n + (k2)d
+ x + y, n + (k1)d +x +y
+ z,
instead of the positions n, n +d, n
+ 2d,..., n + (k1)d. Note that in a
word of length k, k2 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 EvenShoshan'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
"alefà"
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 58. 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 P_{1}, P_{2}, P_{3}
and P_{4}
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 


Nj 
p_{1} := 

(


) 
(


) 
(


) 
To understand this definition, note that if the c(w,
w') were independent random variables that are uniformly
distributed over [0,1], then P_{1} 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 P_{1},
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 P_{1} 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
P_{2} := F^{N} := (P
c(w, w')),
with N as above, and
F^{N} (X) := X 
(

1  ln X + 

+ . . . + 
(ln X) ^{N1} 
 
(N  1)! 

). 
To understand this definition, note first that if x_{1},x_{2},...,x_{N}
are independent random variables that are uiformly distributed
over [0,1], then the distribution of their product X := x_{1}x_{2}
... x_{N} is given by Prob(X <= X_{0})
=F^{N} (X_{0}); 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 P_{2}
is then analogous to that for P_{1}: If the c(w,w')
were independent random variables that are uniformly distributed
over [0,1], then P_{2} 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 P_{1},
the statistic P_{2} 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, P_{3} and P_{4}.
They are defined like P_{1} and P_{2},
except that for each personality, all appellations starting with
the title "Rabbi" are omitted. The reason for
considering P_{3} and P_{4} 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 P_{1}^{p}
and P_{2}^{p}
for the permuted samples, so in misleadingly low significance
levels for P_{1} and P_{2} and so,
conceivably, to an unjustified rejection of the research
hypothesis. Note that this effect is "oneway"; 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 TurboPascal 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 2^{32}. 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] EVENSHOSHAN, 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. AddisonWesley, 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
14. 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.
Up to Section 1 Up to Section
2 Up to Section 3 Up to Appendix

Go
Back to Home Page
