Report on new ELS tests of Torah
================================
Dror Bar-Natan, Alec Gindis, Aryeh Levitan, Brendan McKay
29 May 1997
==== SUMMARY ====
We have performed two series of experiments similar to that
published by Witztum, Rips, and Rosenberg. One matches the
appellations of famous rabbis against the names of the books
they wrote. The other matches their appellations against the
years of their birth or death.
In each case, the result was unambiguously negative.
No indication of any extraordinary phenomenon was found.
==== PROTOCOLS ====
The following experimental protocol was published on 17 Apr 1997.
1. Statement of Purpose
Our aim is to further test the hypotheses made by Witztum, Rips,
and Rosenberg in [WRR]. Several new lists of word pairs will
be tested against the Koren edition of Genesis by two methods:
A. A program identical in behaviour to Mr Rosenberg's program
ELS2.C, with a permutation test equivalent to [WRR] for the
statistics P1 and P2.
B. The following method suggested by Persi Diaconis. For each
pair of persons p,p', compute one distance t(p,p') by averaging
the defined values c(w,w') where w is in the first word-set
of p and w' is in the second word-set of p'. If there are no
such values defined, t(p,p') is undefined. For a permutation
pi of the persons, define T(pi) to be the average over all
p of the defined values t[p,pi(p)]. If there are no such
defined values, T(pi) is undefined. The result will be the
rank position of T(id) amongst all defined T(pi) for a large
set of random permutations pi.
2. Principles
Our subjects are chosen to permit as little subjective choice
as possible in the data preparation.
We will use the reference encyclopaedia [EH] as our primary source
of data, with the less authoritative work [M] as a secondary
source. In all cases we will use the data in [EH] unless it is
obviously wrong, in which case we will use [M] to resolve the
error. For other decisions we will follow the precedents set
by [WRR] wherever possible.
The data for each experiment will be made available for
challenge, and the experiments will be run again if any error
is demonstrated.
3. Experiments E1.1 and E1.2
Experiment E1.1 will use the list of 34 rabbis and their
appellations exactly as in Table 1 of [WRR]. The second
word-set for each rabbi will be generated from the year of
birth and the year of death, according to these rules:
R1. Years will be taken from [EH]. If there is an obvious
error in [EH], we will use [M] to resolve it. We will
also use [M] to assist when [EH] gives a year in the
Western calendar but insufficient additional information
to determine which of the two possible years of the
Hebrew calendar is correct. However, following the
precedent of [WRR], we will not use any year which is
indicated by [EH] as being uncertain.
R2. According to the precedent established in [WRR], the
numbers 15 and 16 appearing as years within a century
will be expressed in two ways.
R3. Subject to the rules above, the list will comprise
those words of 5-8 letters (precedent of [WRR]) formed
from the year in each of these ways:
Let yyy be the year within the millennium, and let myyy
be the same with the millennium indicated. The following
eight forms were approved by the linguist Professor
Michael Sokolov of Bar-Ilan University:
F1: yyy
F2: Byyy ('in yyy')
F3: $NTyyy ('the year yyy')
F4: B$NTyyy ('in the year yyy')
F5-F8: The same as F1-F4 with myyy in place of yyy.
Experiment E1.2 will be the same except that it uses the list
of 32 rabbis and their appellations given in Table 2 of [WRR].
4. Experiments E2.1 and E2.2
Experiment E2.1 will use the list of 34 rabbis and their
appellations exactly as in Table 1 of [WRR]. The second
word-set for each rabbi will contain the titles of his most
notable written works.
R4. Our definition of 'most notable written work' will be
that the work is mentioned in both [M] and [EH] in the
entry for that rabbi.
R5. The exact title as given in [EH] will be used unless
there is a very clear error. In the latter case, [M]
will be used to correct the error.
R6. Subject to the rules above, the list will comprise those
titles containing 5-8 letters (precedent of [WRR]).
Experiment E2.2 will be the same except that it uses the list
of 32 rabbis and their appellations given in Table 2 of [WRR].
On April 21, the following addition to the protocols was made:
As separate experiments, we will also apply the same tests to
each of the other four books of the Torah.
On May 1, the following request was received from Professor E. Rips,
and accepted as an addition to the experiment:
"I would like to suggest (in addition to the procedure R3) to
consider the forms {F1,F2,F5,F6} (i.e. without $NT) separately
and to consider the forms (F3,F4,F7,F8) (i.e. with $NT) separately."
To preserve the a-priori nature of the experiment, no further
requests for additions or changes were accepted.
==== COLLECTION OF THE DATA ====
Collection of the data posed no special problems. The following
is a summary of all cases where some unusual action was required.
1. Rabbi Benvenisti: [EH] gives the birth year as 5363 and the
death year as 5333, which is impossible. We corrected this
error from [Mar], which gives the death year as 5433.
2. Rabbi Margalit: His date of death is given as 12 Tevet and the
year as 1780 (Gregorian calendar). However, there was no
12 Tevet appearing in 1780. We corrected this error from [Mar],
which gives the year of death as 5541 (so he died on 9 Jan 1781).
The actual data collected is given at the end of this document.
As noted in the protocols, we will rerun the computations if
any errors are demonstrated in the data AS DEFINED BY THE
PROTOCOLS. Piece-meal correction of errors using outside sources
will not be accepted, because non-systematic investigation is
known to be a fertile source a-posteriori bias.
==== DISCUSSION OF THE METHOD ====
We have previously expressed criticism of Experiment A on various
mathematical grounds. However, since it was the method used in
[WRR] (other than minor changes), we included it in order to make
the present experiment independent of that debate.
Experiment B has been severely criticised by E. Rips on the
grounds that it does not satisfactorily measure the phenomenon
he believes to occur in Genesis. Essentially, he is concerned
that the exceptionally small distances which occur occasionally
may be masked by averaging them with a larger number of ordinary
distances.
==== THE RESULTS ====
We computed ranks out of one million permutations by calculating
the statistics for each of five million random permutations and
dividing the rank by five. "B" refers to the statistic T(pi)
defined in Experiment B.
Books All year Without With
forms $NT $NT
Genesis
Table 1
P1 946597 343991 268265 518287
P2 897962 288110 079486 683097
B 804395 063461 036526 232923
Table 2
P1 227835 417339 834959 105783
P2 268628 201746 720029 041576
B 713015 244322 442305 033625
Exodus
Table 1
P1 520579 708113 525718 740976
P2 264919 701410 496007 747468
B 212843 529410 753949 012059
Table 2
P1 732340 906005 685038 916417
P2 666454 537493 581435 458208
B 553204 697032 421129 383316
Leviticus
Table 1
P1 288488 929194 845731 847117
P2 689177 945211 742176 915300
B 440922 792107 986974 551347
Table 2
P1 191194 856053 778712 750315
P2 073466 640612 653329 527906
B 494075 935923 815783 802847
Numbers
Table 1
P1 761420 390130 394191 444578
P2 412348 353797 351793 442018
B 569661 822768 474874 485556
Table 2
P1 305941 085703 188226 145228
P2 467604 467653 631375 335369
B 422428 711498 796720 584605
Deuteronomy
Table 1
P1 612340 488540 752630 221857
P2 770759 543284 738072 297303
B 627681 677182 912213 604377
Table 2
P1 437418 473301 427928 526740
P2 334035 422429 338638 546324
B 192979 111412 554526 054611
It is seen that the lowest value is 1.2%, produced by Experiment
B for the Book of Exodus. Considering the number of experiments
performed, this value is not small.
==== FURTHER COMPUTATIONS ====
It must be stressed that none of the additional computations
described in this section represent a-priori experiments.
We will consider three matters.
1. The boundary between the two lists is an artifact of the
history of [WRR]. Therefore it makes sense to consider
the effect of using both lists together.
2. The strongest result in [WRR] was obtained after the
removal of appellations starting with the word "Rabbi".
Therefore it makes sense to try that here also.
3. On May 15, E. Rips requested that we use only the years
of death, not the years of birth and death together.
We did not agree to that change, but in any case we will
present the results of that experiment here also.
The results from the original experiment are included to make
comparisons easier. Each rank is given like mmmmmm/nnnnnn,
where mmmmmm includes the appellations starting with "Rabbi",
and nnnnnn does not. In the case of the years, we give two pairs.
The upper pair is the rank for both birth and death years
together, and the lower is the rank for the death year alone.
We will only present the results for the Book of Genesis.
The results for years of death in the other books are even
less interesting.
Books All year Without With
forms $NT $NT
Table 1
P1 946597/786917 343991/417444 268265/309097 518287/567437
107933/188022 040073/059591 548529/622865
P2 897962/657465 288110/261657 079486/156629 683097/511968
025506/032244 004150/020425 488451/310752
B 804395/558328 063461/046783 036526/026419 232923/127553
010521/013124 010639/017908 400176/307306
Table 2
P1 227835/100008 417339/414092 834959/778628 105783/149229
677190/786334 819212/799814 346106/557166
P2 268628/194173 201746/274949 720029/722988 041576/075552
434804/679920 626708/715350 268936/495786
B 713015/220562 244322/105014 442305/404187 033625/019805
379753/319462 375565/410462 302149/145498
Tables 1 and 2 together
P1 823043/463562 337511/362989 601181/536583 208691/289420
301703/449190 322849/315978 402008/621510
P2 753366/437302 179017/190047 321917/389296 214528/186159
100186/189912 093055/191680 344982/382331
B 848098/383244 079685/019538 108264/060895 038674/011613
050206/026570 051700/049921 291890/116418
Here again we see no reason to claim other than chance behaviour.
Removing of years of birth sometimes improves the result and
sometimes worsens it. Similarly for removing the names starting
with "Rabbi".
The smallest value 0.4% is not very small considering the large
number of computations we have performed. In fact, a close look
shows just how weak it is. There are 72 defined values c(w,w')
for which w and w' belong to the same rabbi. If they were
independent random variables with uniform distribution on (0,1),
the expected values of the smallest two would be 0.0137 and 0.0274.
The actual smallest two values are larger: 0.0172 and 0.0320.
Hence, this example certainly does not support the hypothesis that
very small distances are unusually common. This conclusion is even
more inescapable if we remember that the appellations in this list
tend to produce below-random c(w,w') values even for random words w'.
Looking at c(w,w') values where w and w' belong to different rabbis,
we find 22 values better than 0.0172, including 9 perfect scores of
1/125. It is hard to reconcile these facts with the score of 0.4%,
but it seems to be due to the small number of smallish distances
(8 at most 0.05) being unevenly distributed: there are 3 for rabbi
#22 and 2 for rabbi #5. Removing rabbi #22 alone is enough to raise
the P2-rank by a factor of more than 8.
==== SOME OBSERVATIONS ====
We begin with an observation that serves as a warning for
future experiment design:
Rank orders out of 1 million.
Genesis: 004311
Exodus: 004948
Numbers: 004071
These consistently low values come from the famous books of the
rabbis in Table 1. What they measure is the rank order of the
number of defined c(w,w') values, with large values taken as
better than small values. This statistic depends only slightly
on the exact text, but more on its length and letter frequencies.
Most of all, these results are due to a built-in correlation
within the list of appellations and books. Probably what is
occurring is merely that the rabbis with more books to their
credit have been written about more and hence have more
appellations on average.
Lest it be suspected that these results represent the discovery
of a new phenomenon, we hasten to add that the same thing happens
with randomly permuted Genesis texts. Out of 45 random permutations,
the best three scores (ranks out of a million) were 236, 762, 775.
In order for c(w,w') to be defined, it is usually enough that some
ELS for each word exists, irrespective of how many ELSs exist or
where they are placed. This is a very crude measure, in which
the Genesis text is not known to be special in any way.
In the case of the years of death, the strongest correlation of
this form (98%) occurs just at the place were the P2-rank is least.
It might be a coincidence, but the mathematics of the P2 statistic
is far too complicated to permit a satisfactory analysis.
- - - -
In experiments like this, it is essential to follow the rules
exactly, as otherwise the results are rendered meaningless.
It has been thoroughly established that even a small amount of
freedom in constructing the data can be exploited to obtain a
result much better or much worse than it should be. A case in
point occurs for The Ramhal (#28 in first list). We have omitted
his book DRKH$M because it does not appear in either [EH] or [Mar],
thus failing our criterion. However, it is one of his most
famous books. Such apparent anomolies cannot be predicted in
advance and cannot be corrected a-posteriori without introducing
an undesirable subjectiveness. (As a matter of interest,
neither DRKH$M nor the alternative spelling DRKYHWH make any
significant difference.)
- - - -
It is instructive to examine one of the perfect scores obtained
for the experiment on books. For the Book of Exodus we find
c(HLBW$,LBW$YM)=1/125. The reason is the one suggested by the
words themselves: there is an ELS for HLBW$YM, which includes
both HLBW$ and LBW$YM as substrings. The chance of this
happening is obviously much greater than 1/125.
==== REFERENCES ====
[WRR] D. Witztum, E. Rips, and Y. Rosenberg, Equidistant
Letter Sequences in the Book of Genesis, Statistical
Sciences Vol 9 (1994) 429-438.
[M] M. Margaliot (ed.), Encyclopaedia of Great Men of Israel.
[EH] Encyclopaedia Hebraica.
==== APPENDIX - The Data ====
We will give the data here using the Michigan-Clairmont
transliteration scheme. The Hebrew alphabet in this scheme
is )BGDHWZX+YKLMNS(PCQR$T. Postscript files containing the
data in Hebrew can be fetched from directory
http://cs.anu.edu.au/~bdm/ELS. The file names are books1.ps,
books2.ps, years1.ps, and years2.ps.
With the data for years, if only a single year is given it is
the year of death. If two years are given, the first is the
year of death and the second is the year of birth.
---------------------------------------------------------------
Books for Table 1.
#1 has 2+2 words
RBY)BRHM HR)BD )SWRM$HW B(LYHNP$
#2 has 1+1 words
RBY)BRHM M($HNSYM
#3 has 4+13 words
RBY)BRHM )BN(ZR) BN(ZR) HR)B(
SPRH(BWR SPRHMSPR $PHBRWRH SPRH$M SPRYHWH SPRH(WLM SPRH)XD
YSWDMWR) )GRTH$BT SPRHYSWD $PTYTR SPRDQDWQ SPRHCXWT
#4 has 3+4 words
RBY)LYHW HBXWR B(LHBXWR HHRQBH SPRHBXWR +WB+(M MTWRGMN
#5 has 2+1 words
RBY)LYHW HG)WN )YLM$WL$
#6 has 2+0 words
RBYGR$WN HGR$NY
#7 has 4+0 words
RBYDWD DWDGNZ DWDG)NZ CMXDWD
#8 has 3+4 words
RBYDWD DWDHLWY B(LH+Z +WRYZHB MGNDWD ZHBMZWQQ DBRYDWD
#9 has 4+3 words
RBYXYYM BN(+R )BN(+R )WRHXYYM XPCH$M XPCYHWH PRYT)R
#10 has 1+0 words
RBYYHWDH
#11 has 1+1 words
RBYYHWDH SPRHKBWD
#12 has 4+6 words
RBYYHWDH RBYLYW) HMHRL MHRLMPRG
GWR)RYH NCXY$R)L DRKXYYM B)RHGWLH )WRXD$ NRMCWH
#13 has 3+2 words
RBYYWNTN )YB$YC B(LHTMYM Y(RTDB$ $M(WLM
#14 has 2+0 words
RBYYHW$( RBYH($YL
#15 has 2+1 words
RBYYHW$( B(LHSM( BYTY$R)L
#16 has 3+3 words
RBYYW)L SYRQ$ B(LHBX BYTXD$ M$YBNP$ $WTHBX
#17 has 0+3 words
+WB+(M TWRTH)$M PR$THXD$
#18 has 2+0 words
RBYYWNH RBNWYWNH
#19 has 7+3 words
RBYYWSP YWSPQRW YWSPQ)RW MHRYQRW MHRYQ)RW BYTYWSP HMXBR
$LXN(RWK BYTYWSP KSPM$NH
#20 has 1+0 words
B(LHCLX
#21 has 1+0 words
PNYYHW$(
#22 has 2+1 words
RBYY(QB RBNWTM SPRHY$R
#23 has 3+1 words
RBYYCXQ )LPSY RB)LPS TLMWDQ+N
#24 has 3+0 words
RBYY$R)L B(L$M+WB HB($+
#25 has 2+0 words
RBYM)YR HMHRM
#26 has 4+1 words
RBYMRDKY MRDKYYPH HLBW$ B(LHLBW$ LBW$YM
#27 has 2+4 words
RBYM$H )YSRL$ DRKYM$H TWRTX+)T MXYRYYN $WTHRM)
#28 has 3+0 words
LWC+W LWC)+W HRMXL
#29 has 2+3 words
RBYM$H HRMBM SPRHMCWT YDXZQH M$NHTWRH
#30 has 2+0 words
RBYCBY XKMCBY
#31 has 4+5 words
RBY$BTY $BTYKHN $BTYHKHN B(LH$K
$PTYKHN H)RWK TQPWKHN PW(LCDQ MGYLT(PH
#32 has 1+2 words
RBY$LMH SDWRR$Y $WTR$Y
#33 has 4+4 words
RBY$LMH LWRY) MHR$L HMHR$L
YM$L$LMH XKMT$LMH (+RT$LMH $WTMHR$L
#34 has 3+0 words
)YDL$ MHR$) HMHR$)
---------------------------------------------------------------
Books for Table 2.
#1 has 5+1 words
RBY)BRHM HR)BY HRB)BD HR)BD H)$KWL )$KWL
#2 has 3+0 words
RBY)BRHM YCXQY ZR()BRHM
#3 has 2+0 words
RBY)BRHM HML)K
#4 has 3+0 words
RBY)BRHM )BRHMSB( CRWRHMR
#5 has 1+0 words
RBY)HRN
#6 has 2+1 words
M($YH$M M($YYHWH YWSPLQX
#7 has 2+0 words
RBYDWD )WPNHYM
#8 has 2+0 words
RBYDWD DWDHNGYD
#9 has 2+2 words
RBYDWD DWDNY+W M+HDN KWZRY$NY
#10 has 1+6 words
RBYXYYM
(CHXYYM MQR)YQD$ YWSPLQX Y$R$Y(QB $BWTY(QB XNN)LHYM
#11 has 2+3 words
RBYXYYM BNBN$T DYN)DXYY B(YXYY XMR)WXYY
#12 has 4+0 words
RBYXYYM KPWSY B(LNS B(LHNS
#13 has 4+2 words
RBYXYYM XYYM$BTY MHRX$ HMHRX$
$WTSHRX$ TWRTXYYM
#14 has 1+1 words
XWTY)YR XW+H$NY
#15 has 1+0 words
RBYYHWDH
#16 has 2+5 words
RBYYHWDH MHRY(Y)$
LXMYHWDH BYTYHWDH BNYYHWDH M+HYHWDH $B+YHWDH
#17 has 1+0 words
RBYYHWSP
#18 has 2+2 words
RBYYHW$( MGNY$LMH MGNY$LMH PNYYHW$(
#19 has 9+2 words
RBYYWSP M+RNY YWSP+RNY +R)NY M+R)NY MHRYM+ HMHRYM+ MHRY+ HMHRY+
CPNTP(NX $WTMHRY+
#20 has 3+3 words
RBYYWSP T)WMYM PRYMGDYM PWRTYWSP GNTWRDYM R)$YWSP
#21 has 4+0 words
RBYY(QB Y(QBBYRB MHRYBYRB HRYBR
#22 has 2+3 words
X)GYZ B(LHLQ+ (CHXYYM TXLTXKMH PTYLTKLT
#23 has 8+1 words
RBYY(QB MWLYN Y(QBSGL Y(QBHLWY MHRYSGL MHRYHLWY MHRYL HMHRYL
$WTMHRYL
#24 has 5+4 words
HY(BC HRY(BC (MDYN HRY(MDN HRY(MDYN
$)LTY(BC LXM$MYM MRWQCY(H MGYLTSPR
#25 has 3+0 words
RBYYCXQ HWRWWYC YCXQHLWY
#26 has 4+0 words
RBYMNXM QRWKML RBYM(NDL CMXCDQ
#27 has 11+2 words
RBYM$H ZKWT) ZKWTW M$HZKWT M$HZKWT) M$HZKWTW MHRMZKWT MHRMZ
HMHRMZ HMZLN QWLHRMZ
$WTHRMZ TPTH(RWK
#28 has 3+0 words
RBYM$H MRGLYT PNYM$H
#29 has 1+0 words
RBY(ZRYH
#30 has 2+2 words
)XH(R Y$RLBB M($HXW$B HWN($YR
#31 has 6+2 words
RBY$LWM MZRXY $R(BY $R$LWM MHR$$ HMHR$$
)MTW$LWM NHR$LWM
#32 has 1+1 words
RBY$LMH LB$LMH
---------------------------------------------------------------
Years for Table 1.
#1 has 2+10 words
RBY)BRHM HR)BD
TTQN+ BTTQN+ $NTTTQN+ DTTQN+ BDTTQN+ $NTTTP B$NTTTP BDTTP
$NTDTTP B$NTDTTP
#2 has 1+11 words
RBY)BRHM
BTTCX $NTTTCX B$NTTTCX DTTCX BDTTCX $NTDTTCX TTQMW BTTQMW
$NTTTQMW DTTQMW BDTTQMW
#3 has 4+5 words
RBY)BRHM )BN(ZR) BN(ZR) HR)B(
TTQKD BTTQKD $NTTTQKD DTTQKD BDTTQKD
#4 has 3+4 words
RBY)LYHW HBXWR B(LHBXWR
$NT$X B$NT$X $NTH$X B$NTH$X
#5 has 2+10 words
RBY)LYHW HG)WN
BTQNX $NTTQNX B$NTTQNX HTQNX BHTQNX $NTHTQNX $NTTP B$NTTP
$NTHTP B$NTHTP
#6 has 2+5 words
RBYGR$WN HGR$NY
$NTTNG B$NTTNG BHTNG $NTHTNG B$NTHTNG
#7 has 4+9 words
RBYDWD DWDGNZ DWDG)NZ CMXDWD
$NT$(G B$NT$(G BH$(G $NTH$(G B$NTH$(G $NT$) B$NT$) $NTH$) B$NTH$)
#8 has 3+10 words
RBYDWD DWDHLWY B(LH+Z
$NTTKZ B$NTTKZ BHTKZ $NTHTKZ B$NTHTKZ $NT$MW B$NT$MW BH$MW
$NTH$MW B$NTH$MW
#9 has 4+10 words
RBYXYYM BN(+R )BN(+R )WRHXYYM
$NTTQG B$NTTQG BHTQG $NTHTQG B$NTHTQG $NTTNW B$NTTNW BHTNW
$NTHTNW B$NTHTNW
#10 has 1+7 words
RBYYHWDH $NTQ+ B$NTQ+ $NTHQ+ B$NTHQ+ B$NTL $NTHL B$NTHL
#11 has 1+5 words
RBYYHWDH TTQ(Z BTTQ(Z $NTTTQ(Z DTTQ(Z BDTTQ(Z
#12 has 4+5 words
RBYYHWDH RBYLYW) HMHRL MHRLMPRG
$NT$S+ B$NT$S+ BH$S+ $NTH$S+ B$NTH$S+
#13 has 3+6 words
RBYYWNTN )YB$YC B(LHTMYM
BTQKD $NTTQKD B$NTTQKD HTQKD BHTQKD $NTHTQKD
#14 has 2+5 words
RBYYHW$( RBYH($YL $NTTKD B$NTTKD BHTKD $NTHTKD B$NTHTKD
#15 has 2+5 words
RBYYHW$( B(LHSM( $NT$(D B$NT$(D BH$(D $NTH$(D B$NTH$(D
#16 has 3+3 words
RBYYW)L SYRQ$ B(LHBX B$NTT $NTHT B$NTHT
#17 has 0+10 words
$NTTYD B$NTTYD BHTYD $NTHTYD B$NTHTYD $NT$L+ B$NT$L+ BH$L+
$NTH$L+ B$NTH$L+
#18 has 2+4 words
RBYYWNH RBNWYWNH $NTKD B$NTKD $NTHKD B$NTHKD
#19 has 7+10 words
RBYYWSP YWSPQRW YWSPQ)RW MHRYQRW MHRYQ)RW BYTYWSP HMXBR
$NT$LH B$NT$LH BH$LH $NTH$LH B$NTH$LH $NTRMX B$NTRMX BHRMX
$NTHRMX B$NTHRMX
#20 has 1+11 words
B(LHCLX
BTQNG $NTTQNG B$NTTQNG HTQNG BHTQNG $NTHTQNG $NTT(D B$NTT(D
BHT(D $NTHT(D B$NTHT(D
#21 has 1+17 words
PNYYHW$(
BTQ+Z $NTTQ+Z B$NTTQ+Z HTQ+Z BHTQ+Z $NTHTQ+Z BTQYW $NTTQYW
B$NTTQYW HTQYW BHTQYW $NTHTQYW $NTTM) B$NTTM) BHTM)
$NTHTM) B$NTHTM)
#22 has 2+5 words
RBYY(QB RBNWTM TTQL) BTTQL) $NTTTQL) DTTQL) BDTTQL)
#23 has 3+12 words
RBYYCXQ )LPSY RB)LPS
BTTSG $NTTTSG B$NTTTSG DTTSG BDTTSG $NTDTTSG BT$(G $NTT$(G
B$NTT$(G DT$(G BDT$(G $NTDT$(G
#24 has 3+5 words
RBYY$R)L B(L$M+WB HB($+
$NTTQK B$NTTQK BHTQK $NTHTQK B$NTHTQK
#25 has 2+4 words
RBYM)YR HMHRM $NTNG B$NTNG $NTHNG B$NTHNG
#26 has 4+5 words
RBYMRDKY MRDKYYPH HLBW$ B(LHLBW$
$NT$(B B$NT$(B BH$(B $NTH$(B B$NTH$(B
#27 has 2+5 words
RBYM$H )YSRL$ $NT$LB B$NT$LB BH$LB $NTH$LB B$NTH$LB
#28 has 3+10 words
LWC+W LWC)+W HRMXL
$NTTQZ B$NTTQZ BHTQZ $NTHTQZ B$NTHTQZ $NTTSZ B$NTTSZ BHTSZ
$NTHTSZ B$NTHTSZ
#29 has 2+11 words
RBYM$H HRMBM
TTQSH BTTQSH $NTTTQSH DTTQSH BDTTQSH BTTCX $NTTTCX B$NTTTCX
DTTCX BDTTCX $NTDTTCX
#30 has 2+9 words
RBYCBY XKMCBY
$NTT(X B$NTT(X BHT(X $NTHT(X B$NTHT(X $NTTK B$NTTK $NTHTK B$NTHTK
#31 has 4+10 words
RBY$BTY $BTYKHN $BTYHKHN B(LH$K
$NTTKB B$NTTKB BHTKB $NTHTKB B$NTHTKB $NT$PB B$NT$PB BH$PB
$NTH$PB B$NTH$PB
#32 has 1+6 words
RBY$LMH BTTSH $NTTTSH B$NTTTSH DTTSH BDTTSH $NTDTTSH
#33 has 4+5 words
RBY$LMH LWRY) MHR$L HMHR$L
$NT$LH B$NT$LH BH$LH $NTH$LH B$NTH$LH
#34 has 3+15 words
)YDL$ MHR$) HMHR$)
$NT$CB B$NT$CB BH$CB $NTH$CB B$NTH$CB $NT$+W B$NT$+W BH$+W
$NTH$+W B$NTH$+W $NT$YH B$NT$YH BH$YH $NTH$YH B$NTH$YH
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Years for Table 2.
#1 has 5+10 words
RBY)BRHM HR)BY HRB)BD HR)BD H)$KWL
TTQL+ BTTQL+ $NTTTQL+ DTTQL+ BDTTQL+ $NTTT( B$NTTT( BDTT(
$NTDTT( B$NTDTT(
#2 has 3+10 words
RBY)BRHM YCXQY ZR()BRHM
$NTTP+ B$NTTP+ BHTP+ $NTHTP+ B$NTHTP+ $NTTK) B$NTTK) BHTK)
$NTHTK) B$NTHTK)
#3 has 2+11 words
RBY)BRHM HML)K
BTQLD $NTTQLD B$NTTQLD HTQLD BHTQLD $NTHTQLD $NTTQ) B$NTTQ)
BHTQ) $NTHTQ) B$NTHTQ)
#4 has 3+0 words
RBY)BRHM )BRHMSB( CRWRHMR
#5 has 1+11 words
RBY)HRN
BTQLB $NTTQLB B$NTTQLB HTQLB BHTQLB $NTHTQLB $NTTCW B$NTTCW
BHTCW $NTHTCW B$NTHTCW
#6 has 2+10 words
M($YH$M M($YYHWH
$NT$MW B$NT$MW BH$MW $NTH$MW B$NTH$MW $NTR(G B$NTR(G BHR(G
$NTHR(G B$NTHR(G
#7 has 2+10 words
RBYDWD )WPNHYM
$NTTCZ B$NTTCZ BHTCZ $NTHTCZ B$NTHTCZ $NTTKD B$NTTKD BHTKD
$NTHTKD B$NTHTKD
#8 has 2+0 words
RBYDWD DWDHNGYD
#9 has 2+5 words
RBYDWD DWDNY+W $NTTPX B$NTTPX BHTPX $NTHTPX B$NTHTPX
#10 has 1+9 words
RBYXYYM
$NTTQD B$NTTQD BHTQD $NTHTQD B$NTHTQD $NTTK B$NTTK $NTHTK B$NTHTK
#11 has 2+10 words
RBYXYYM BNBN$T
$NTTLG B$NTTLG BHTLG $NTHTLG B$NTHTLG $NT$SG B$NT$SG BH$SG
$NTH$SG B$NTH$SG
#12 has 4+0 words
RBYXYYM KPWSY B(LNS B(LHNS
#13 has 4+4 words
RBYXYYM XYYM$BTY MHRX$ HMHRX$ $NTTZ B$NTTZ $NTHTZ B$NTHTZ
#14 has 1+10 words
XWTY)YR
$NTTSG B$NTTSG BHTSG $NTHTSG B$NTHTSG $NT$CX B$NT$CX BH$CX
$NTH$CX B$NTH$CX
#15 has 1+6 words
RBYYHWDH BTQL) $NTTQL) B$NTTQL) HTQL) BHTQL) $NTHTQL)
#16 has 2+6 words
RBYYHWDH MHRY(Y)$
BTQK) $NTTQK) B$NTTQK) HTQK) BHTQK) $NTHTQK)
#17 has 1+12 words
RBYYHWSP
BTTKZ $NTTTKZ B$NTTTKZ DTTKZ BDTTKZ $NTDTTKZ BT$CW $NTT$CW
B$NTT$CW DT$CW BDT$CW $NTDT$CW
#18 has 2+4 words
RBYYHW$( MGNY$LMH $NTTX B$NTTX $NTHTX B$NTHTX
#19 has 9+10 words
RBYYWSP M+RNY YWSP+RNY +R)NY M+R)NY MHRYM+ HMHRYM+ MHRY+ HMHRY+
$NT$C+ B$NT$C+ BH$C+ $NTH$C+ B$NTH$C+ $NT$K+ B$NT$K+ BH$K+
$NTH$K+ B$NTH$K+
#20 has 3+11 words
RBYYWSP T)WMYM PRYMGDYM
BTQNB $NTTQNB B$NTTQNB HTQNB BHTQNB $NTHTQNB $NTTPZ B$NTTPZ
BHTPZ $NTHTPZ B$NTHTPZ
#21 has 4+4 words
RBYY(QB Y(QBBYRB MHRYBYRB HRYBR $NT$) B$NT$) $NTH$) B$NTH$)
#22 has 2+9 words
X)GYZ B(LHLQ+
$NTTLD B$NTTLD BHTLD $NTHTLD B$NTHTLD $NT$P B$NT$P $NTH$P B$NTH$P
#23 has 8+5 words
RBYY(QB MWLYN Y(QBSGL Y(QBHLWY MHRYSGL MHRYHLWY MHRYL HMHRYL
$NTQPZ B$NTQPZ BHQPZ $NTHQPZ B$NTHQPZ
#24 has 5+6 words
HY(BC HRY(BC (MDYN HRY(MDN HRY(MDYN
BTQLW $NTTQLW B$NTTQLW HTQLW BHTQLW $NTHTQLW
#25 has 3+6 words
RBYYCXQ HWRWWYC YCXQHLWY
BTQKZ $NTTQKZ B$NTTQKZ HTQKZ BHTQKZ $NTHTQKZ
#26 has 4+0 words
RBYMNXM QRWKML RBYM(NDL CMXCDQ
#27 has 11+5 words
RBYM$H ZKWT) ZKWTW M$HZKWT M$HZKWT) M$HZKWTW MHRMZKWT MHRMZ
HMHRMZ HMZLN QWLHRMZ
$NTTNX B$NTTNX BHTNX $NTHTNX B$NTHTNX
#28 has 3+6 words
RBYM$H MRGLYT PNYM$H BTQM) $NTTQM) B$NTTQM) HTQM) BHTQM) $NTHTQM)
#29 has 1+9 words
RBY(ZRYH
$NTTZ B$NTTZ $NTHTZ B$NTHTZ $NT$L+ B$NT$L+ BH$L+ $NTH$L+ B$NTH$L+
#30 has 2+10 words
)XH(R Y$RLBB
$NTTQG B$NTTQG BHTQG $NTHTQG B$NTHTQG $NTTMX B$NTTMX BHTMX
$NTHTMX B$NTHTMX
#31 has 6+6 words
RBY$LWM MZRXY $R(BY $R$LWM MHR$$ HMHR$$
BTQLZ $NTTQLZ B$NTTQLZ HTQLZ BHTQLZ $NTHTQLZ
#32 has 1+6 words
RBY$LMH
BTQM) $NTTQM) B$NTTQM) HTQM) BHTQM) $NTHTQM)
Overview on numerical features in different scriptures
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