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Evokation
  
 
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THE ACT OF CREATION

 

 

THE NUCLEAR FAMILY 1969

 

 

 

 

EHT NAMUH 1973

 

 

 

 

AFRICAN NIGHTMARE SPECTRE OF FAMINE 1975

 

 

 

 

EHT NAMUH 1977

 

 

 

 

THE JOURNEYMAN 1977

 

 

Letter frequency
From Wikipedia, the free encyclopediaJump to: navigation, search
The frequency of letters in text has often been studied for use in cryptography, and frequency analysis in particular. No exact letter frequency distribution underlies a given language, since all writers write slightly differently. Linotype machines sorted the letters' frequencies as etaoin shrdlu cmfwyp vbgkqj xz based on the experience and custom of manual compositors. Likewise, Modern International Morse code encodes the most frequent letters with the shortest symbols; arranging the Morse alphabet into groups of letters that require equal amounts of time to transmit, and then sorting these groups in increasing order, yields e it san hurdm wgvlfbk opjxcz yq. Similar ideas are used in modern data-compression techniques such as Huffman coding.

More recent analyses show that letter frequencies, like word frequencies, tend to vary, both by writer and by subject. One cannot write an essay about x-rays without using frequent Xs, and the essay will have an especially strange letter frequency if the essay is about the frequent use of x-rays to treat zebras in Qatar. Different authors have habits which can be reflected in their use of letters. Hemingway's writing style, for example, is visibly different from Faulkner's. Letter, bigram, trigram, word frequencies, word length, and sentence length can be calculated for specific authors, and used to prove or disprove authorship of texts, even for authors whose styles aren't so divergent.

Accurate average letter frequencies can only be gleaned by analyzing a large amount of representative text. With the availability of modern computing and collections of large text corpora, such calculations are easily made. This Deafandblind link details examples from a variety of sources, (press reporting, religious text, scientific text and general fiction) and there are differences especially for general fiction with the position of 'h' and 'i'. The example differs from the linotype 'etaoin shrdlu' to come out as 'etaoHn Isrdlu'. There is an unproven statement[by whom?] that conversation is similar in frequency to general fiction.

Herbert S. Zim, in his classic introductory cryptography text "Codes and Secret Writing", gives the English letter frequency sequence as "ETAON RISHD LFCMU GYPWB VKXJQ Z", the most common letter pairs as "TH HE AN RE ER IN ON AT ND ST ES EN OF TE ED OR TI HI AS TO", and the most common doubled letters as "LL EE SS OO TT FF RR NN PP CC".[1]

The 'top twelve' letters comprise about 80% of the total usage. The 'top eight" letters comprise about 65% of the total usage. A spy using the VIC cipher or some other cipher based on a straddling checkerboard typically uses a mnemonic such as "a sin to err" (dropping the second "r") to remember the top 8 characters.

The use of letter frequencies and frequency analysis plays a fundamental role in several games, including hangman, Scrabble, Wheel of Fortune, Definition, Bananagrams, and cryptograms.

Letter frequencies had a strong effect on the design of some keyboard layouts. The most-frequent letters are on the bottom row of the Blickensderfer typewriter. The most-frequent letters are on the home row of the Dvorak Simplified Keyboard.

Contents [hide]
1 Relative frequencies of letters in the English language
2 Relative frequencies of the first letters of a word in the English language
3 Relative frequencies of letters in other languages
4 See also
5 References
6 External links

[edit] Relative frequencies of letters in the English language

Relative frequencies of letters in text.
Relative frequencies ordered by frequency.The letter frequencies for English are listed below.[2] However, this table differs slightly from others, such as Cornell University Math Explorer's Project, which produced this table after measuring over 40,000 words.

Letter Frequency
a 8.167%
b 1.492%
c 2.782%
d 4.253%
e 12.702%
f 2.228%
g 2.015%
h 6.094%
i 6.966%
j 0.153%
k 0.772%
l 4.025%
m 2.406%
n 6.749%
o 7.507%
p 1.929%
q 0.095%
r 5.987%
s 6.327%
t 9.056%
u 2.758%
v 0.978%
w 2.360%
x 0.150%
y 1.974%
z 0.074%

In English, the space is slightly more frequent than the top letter (7% more frequent than, or 107% as frequent as, e), and the non-alphabetic characters (digits, punctuation, etc.) occupy the fourth position, between t and a.[3]

[edit] Relative frequencies of the first letters of a word in the English language
First Letter of a word frequencies:[4]

Letter Frequency
a 25.602%
b 4.702%
c 3.511%
d 2.670%
e 2.000%
f 3.779%
g 1.950%
h 7.232%
i 6.286%
j 0.631%
k 0.690%
l 2.705%
m 4.374%
n 2.365%
o 6.264%
p 2.545%
q 0.173%
r 1.653%
s 7.755%
t 16.671%
u 1.487%
v 0.619%
w 6.661%
x 0.005%
y 1.620%
z 0.050%

|3||16.924%

[edit] Relative frequencies of letters in other languages

Spanish letter frequenciesLetter French [5] German [6] Spanish [7] Portuguese [8] Esperanto [9] Italian[10] Turkish Swedish[11] Polish[12] Toki Pona [13] Dutch [14]
a 7.636% 6.51% 12.53% 14.63% 12.12% 11.74% 11.68% 9.3% 8.0% 17.2% 7.49%
b 0.901% 1.89% 1.42% 1.04% 0.98% 0.92% 2.95% 1.3% 1.3% 0.0% 1.58%
c 3.260% 3.06% 4.68% 3.88% 0.78% 4.5% 0.97% 1.3% 3.8% 0.0% 1.24%
d 3.669% 5.08% 5.86% 4.99% 3.04% 3.73% 4.87% 4.5% 3.0% 0.0% 5.93%
e 14.715% 17.40% 13.68% 12.57% 8.99% 11.79% 9.01% 9.9% 6.9% 7.4% 18.91%
f 1.066% 1.66% 0.69% 1.02% 1.03% 0.95% 0.44% 2.0% 0.1% 0.0% 0.81%
g 0.866% 3.01% 1.01% 1.30% 1.17% 1.64% 1.34% 3.3% 1.0% 0.0% 3.40%
h 0.737% 4.76% 0.70% 1.28% 0.38% 1.54% 1.14% 2.1% 1.0% 0.0% 2.38%
i 7.529% 7.55% 6.25% 6.18% 10.01% 11.28% 8.27%* 5.1% 7.0% 14.8% 6.50%
j 0.545% 0.27% 0.44% 0.40% 3.50% 0.00% 0.01% 0.7% 1.9% 3.0% 1.46%
k 0.049% 1.21% 0.01% 0.02% 4.16% 0.00% 4.71% 3.2% 2.7% 5.1% 2.25%
l 5.456% 3.44% 4.97% 2.78% 6.14% 6.51% 5.75% 5.2% 3.1% 10.2% 3.57%
m 2.968% 2.53% 3.15% 4.74% 2.99% 2.51% 3.74% 3.5% 2.4% 4.4% 2.21%
n 7.095% 9.78% 6.71% 5.05% 7.96% 6.88% 7.23% 8.8% 4.7% 11.6% 10.03%
o 5.378% 2.51% 8.68% 10.73% 8.78% 9.83% 2.45% 4.1% 7.1% 7.7% 6.06%
p 3.021% 0.79% 2.51% 2.52% 2.74% 3.05% 0.79% 1.7% 2.4% 3.7% 1.57%
q 1.362% 0.02% 0.88% 1.20% 0.00% 0.51% 0 0.007% - 0.0% 0.009%
r 6.553% 7.00% 6.87% 6.53% 5.91% 6.37% 6.95% 8.3% 3.5% 0.0% 6.41%
s 7.948% 7.27% 7.98% 7.81% 6.09% 4.98% 2.95% 6.3% 3.8% 4.1% 3.73%
t 7.244% 6.15% 4.63% 4.74% 5.27% 5.62% 3.09% 8.7% 2.4% 4.6% 6.79%
u 6.311% 4.35% 3.93% 4.63% 3.18% 3.01% 3.43% 1.8% 1.8% 3.2% 1.99%
v 1.628% 0.67% 0.90% 1.67% 1.90% 2.10% 0.98% 2.4% - 0.0% 2.85%
w 0.114% 1.89% 0.02% 0.01% 0.00% 0.00% 0 0.03% 3.6% 2.8% 1.52%
x 0.387% 0.03% 0.22% 0.21% 0.00% 0.00% 0 0.1% - 0.0% 0.04%
y 0.308% 0.04% 0.90% 0.01% 0.00% 0.00% 3.37% 0.6% 3.2% 0.0% 0.035%
z 0.136% 1.13% 0.52% 0.47% 0.50% 0.49% 1.50% 0.02% 5.1% 0.0% 1.39%
à 0.486% 0 0 see a 0 see a 0 0.0% 0 - see a
å 0 0 0 0 0 0 0 1.6% 0 - -
ä 0 - 0 0 0 0 0 2.1% 0 - see a
ą 0 - 0 0 0 0 0 0 see a - -
œ 0.018% 0 0 0 0 0 0 0 0 - -
ç 0.085% 0 0 see c 0 0 1.26% 0 0 - -
ĉ 0 0 0 0 0.66% 0 0 0 0 - -
ć 0 - 0 0 0 0 0 0 see c - -
è 0.271% 0 0 0 0 see e 0 0.0% 0 - see e
é 1.904% 0 0 see e 0 see e 0 0.0% 0 - see e
ê 0.225% 0 0 see e 0 0 0 0 0 - -
ë 0.001% 0 0 0 0 0 0 0 0 - see e
ę 0 - 0 0 0 0 0 0 see e - -
ĝ 0 0 0 0 0.69% 0 0 0 0 - -
ğ 0 0 0 0 0 0 1.13% 0 0 - -
ĥ 0 0 0 0 0.02% 0 0 0 0 - -
î 0.045% 0 0 0 0 0 0 0 0 0 -
ì 0 0 0 0 0 see i 0 0 0 - see i
ï 0.005% 0 0 0 0 0 0 0 0 - see i
ı 0 0 0 0 0 0 5.20%* 0 0 - -
ĵ 0 0 0 0 0.12% 0 0 0 0 - -
ł 0 - 0 0 0 0 0 0 see l - -
ñ 0 0 0.31% 0 0 0 0 0 0 - -
ń 0 - 0 0 0 0 0 0 see n - -
ò 0 0 0 0 0 see o 0 0 0 - see o
ö 0 - 0 0 0 0 0.87% 1.5% 0 - see o
ó 0 - 0 see o 0 0 0 0 see o - see o
ŝ 0 0 0 0 0.38% 0 0 0 0 - -
ş 0 0 0 0 0 0 1.94% 0 0 - -
ś 0 - 0 0 0 0 0 0 see s - -
ß 0 0.31% 0 0 0 0 0 0 0 - -
ù 0.058% 0 0 0 0 see u 0 0 0 - see u
ŭ 0 0 0 0 0.52% 0 0 0 0 - -
ü 0 - 0 0 0 0 1.99% 0 0 - see u
ź 0 - 0 0 0 0 0 0 see z - -
ż 0 - 0 0 0 0 0 0 0.7% - -

*See Turkish dotted and dotless I

The figure below illustrates the frequency distributions of the 26 most common Latin letters across some languages.

Based on these tables, the 'etaoin shrdlu'-equivalent results for each language is as follows:

French: 'esait nrulo'; (Indo-European: Romance; traditionally, 'esartinulop' is used, in part for its ease of pronunciation[15])
Spanish: 'eaosr nidlc'; (Indo-European: Romance)
Portuguese: 'aeosr indmt' (Indo-European: Romance)
Italian: 'eaion lrtsc'; (Indo-European: Romance)
Esperanto: 'aieon lsrtk' (artificial language – influenced by Indo-European languages, Romance, Germanic mostly)
German: 'enisr atdhu'; (Indo-European: Germanic)
Swedish: 'eantr slido'; (Indo-European: Germanic)
Turkish: 'aeinr ldkmu'; (Turkic: a non Indo-European language)
Dutch: 'enati rodsl'; (Indo-European: Germanic)[16]
Polish: 'aoiez nscwr'; (Indo-European: Slavic)
All these languages use a basically similar 25+ character alphabet.

[edit] See also
Corpus linguistics
ETAOIN SHRDLU
RSTLNE
Frequency analysis (cryptanalysis)
Linotype machine
Most common words in English
Scrabble
[edit] References
1.^ Zim, Herbert Spencer. Codes and secret writing (abridged edition). Scholastic Book Services, fourth printing, 1962. Copyright 1948 Herbert S. Zim. Originally published by William Morrow.
2.^ Beker, Henry; Piper, Fred (1982). Cipher Systems: The Protection of Communications. Wiley-Interscience. p. 397. Table also available from Lewand, Robert (2000). Cryptological Mathematics. The Mathematical Association of America. p. 36. ISBN 978-0883857199. http://books.google.com/books?id=CyCcRAm7eQMC&pg=PA36. and [1]
3.^ Lee, E. Stewart; Essays about Computer Security; University of Cambridge Computer Laboratory, p. 181
4.^ Calculated from "Project Gutenberg Selections" available from the NLTK Corpora
5.^ "CorpusDeThomasTempé". http://gpl.insa-lyon.fr/Dvorak-Fr/CorpusDeThomasTemp%C3%A9. Retrieved 2007-06-15.
6.^ Albrecht Beutelspacher, Kryptologie, 7. Aufl., Wiesbaden: Vieweg Verlagsgesellschaft, 2005, ISBN 3-8348-0014-7, p.10
7.^ Fletcher Pratt, Secret and Urgent: the Story of Codes and Ciphers Blue Ribbon Books, 1939, pp. 254-255.
8.^ "Frequência da ocorrência de letras no Português". http://www.numaboa.com/criptografia/criptoanalise/310-Frequencia-no-Portugues. Retrieved 2009-06-16.
9.^ "La Oftecoj de la Esperantaj Literoj". http://lingvakritiko.com/2007/09/13/literoftecoj-kaj-tabelvortoftecoj/. Retrieved 2007-09-14.
10.^ Simon Singh, Codici e Segreti, 1999, RCS, ISBN 88-17-12539-3
11.^ Simon Singh, Kodboken, 1999, Norstedts, ISBN 91-1-1300708-4
12.^ Wstęp do kryptologii, counting [space] 17.2%, [dot point] 0.9%, [comma] 0.9% and [semicolon] 0.5%
13.^ "lipu pi jan Jakopo pi toki pona". http://bellsouthpwp.net/j/i/jimhenry1973/conlang/tokipona/tokipona.htm. Retrieved 2007-09-14.
14.^ "Letterfrequenties". Genootschap OnzeTaal. http://www.onzetaal.nl/advies/letterfreq.php. Retrieved 2009-05-17.
15.^ Perec, Georges; Alphabets; Éditions Galilée, 1976
16.^ "Letterfrequenties". Genootschap OnzeTaal. http://www.onzetaal.nl/advies/letterfreq.php. Retrieved 2008-12-26.
[edit] External links
A site with content of Cryptographical Mathematics by Robert Edward Lewand
Some examples of letter frequency rankings in some common languages
Java-Application for building letter frequencies out of a text file
Some useful tables for single letter, digram, trigram, tetragram, and pentagram frequencies based on 20,000 words that take into account word-length and letter-position combinations for words 3 to 7 letters in length. The references are as follows: 1. Mayzner, M.S. & Tresselt, M.E. Tables of single-letter and digram frequency counts for various word-length and letter-position combinations.Psychonomic Monograph Supplements, 1965, 1, #2, 13-32. 2. Mayzner, M.S., Tresselt, M.E. & Wolin, B.R. Tables of trigram frequency counts for various word-length and letter-position combinations.Psychonomic Monograph Supplements, 1965, 1, #3, 33-78. 3. Mayzner, M.S., Tresselt, M.E. & Wolin, B.R. Tables of tetragram frequency counts for various word-length and letter-position combinations. Psychonomic Monograph Supplements, 1965, 1, #4, 79-143. 4. Mayzner, M.S., Tresselt, M.E. & Wolin, B.R. Tables of pentagram frequency counts for various word-length and letter-position combinations. Psychonomic Monograph Supplements, 1965, 1, #5, 144-190.

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1 Relative frequencies of letters in the English language... Jump to text »
the frequency distributions of the 26 most common Latin letters across some languages. Jump to text »

 

 

 

 

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U
M
-
M
U
T
A

 

 

6
A
T
U
M
-
R
A
-
-
-
-
-
--
-
-
`-
1
20
21
13
-
18
1
+
=
74
7+4
=
11
1+1
2
-
1
2
3
4
-
9
1
+
=
20
2+0
=
2
-
2
6
A
T
U
M
-
R
A
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
6
A
T
U
M
-
R
A
-
-
19
-
-
6
-
20
-
-
-
-
-
-
9
-
-
-
10
-
-
-
Q
2+0
6
A
T
U
M
-
R
A
-
-
1
-
-
6
-
2

 

 

4
PTAH
45
18
9
-
-
-
-
-
-
-
-
-
-
7
SEKHMET
81
36
9
6
SPHINX
90
36
9
13
Add Reduce Deduce
171
72
18
1+3
-
1+7+1
7+2
-
4
Essence of Number
9
9
9

 

 

4
ATUM
55
10
1
6
OSIRIS
89
35
8
10
Add Reduce Deduce
144
45
9
1+0
-
1+4+4
4+5
-
1
Essence of Number
9
9
9

 

 

-
8
R
E
D
E
E
M
E
R
-
-
-
-
-
--
-
-
-
-
-
-
18
5
4
5
5
13
5
18
+
=
73
7+3
=
10
1+0
1
=
1
-
-
9
5
4
5
5
4
5
9
+
=
46
4+6
=
10
1+0
1
=
1
-
8
R
E
D
E
E
M
E
R
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
=
-
5
-
5
5
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
27
8
R
E
D
E
E
M
E
R
-
-
18
-
-
8
-
46
-
19
2+7
-
9
-
-
-
-
-
-
9
-
-
1+8
-
-
-
-
4+6
-
1+9
9
8
R
E
D
E
E
M
E
R
-
-
9
-
-
8
-
10
-
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+0
-
1+0
9
8
R
E
D
E
E
M
E
R
-
-
9
-
-
8
-
1
-
1

 

 

8
R
E
D
E
E
M
E
R
-
-
-
-
-
--
-
-
-
-
-
18
5
4
5
5
13
5
18
+
=
73
7+3
=
10
1+0
1
=
1
-
9
5
4
5
5
4
5
9
+
=
46
4+6
=
10
1+0
1
=
1
8
R
E
D
E
E
M
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
=
-
5
-
5
5
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
9
-
-
-
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
8
R
E
D
E
E
M
E
R
-
-
18
-
-
8
-
46
-
19
-
9
-
-
-
-
-
-
9
-
-
1+8
-
-
-
-
4+6
-
1+9
8
R
E
D
E
E
M
E
R
-
-
9
-
-
8
-
10
-
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+0
-
1+0
8
R
E
D
E
E
M
E
R
-
-
9
-
-
8
-
1
-
1

 

 

8
REDEEMER
-
-
-
1
R
18
9
9
2
E+D
9
9
9
-
E
5
5
5
2
M+E
18
9
9
1
R
18
9
9
8
REDEEMER
68
41
41
-
-
6+8
4+1
4+1
8
REDEEMER
14
5
5
-
-
1+4
-
-
8
REDEEMER
5
5
5

 

 

-
8
R
E
D
E
E
M
E
D
-
-
-
-
-
--
-
-
-
-
-
-
18
5
4
5
5
13
5
4
+
=
59
5+9
=
14
1+4
5
=
5
Q
-
9
5
4
5
5
4
5
4
+
=
41
4+1
=
5
=
5
=
5
-
8
R
E
D
E
E
M
E
D
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
-
4
-
-
4
occurs
x
3
=
12
1+2
3
-
=
-
5
-
5
5
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
27
8
R
E
D
E
E
M
E
R
-
-
18
-
-
8
-
41
-
14
2+7
-
9
-
-
-
-
-
-
-
-
-
1+8
-
-
-
-
4+1
-
1+4
9
8
R
E
D
E
E
M
E
D
-
-
9
-
-
8
-
5
-
5

 

FOLLOW

THE

PATH OF PTAH

 

 

-
-
-
-
A
-
-
-
-
-
-
-
-
T
-
-
-
-
-
-
-
-
O
-
-
-
-
-
-
-
-
N
-
-
-
-
A
T
O
N
E
M
E
N
T
-
-
-
-
M
-
-
-
-
-
-
-
-
E
-
-
-
-
-
-
-
-
N
-
-
-
-
-
-
-
-
T
-
-
-
-

 

 

-
-
-
-
-
A
-
-
-
-
-
-
-
-
-
-
B
-
-
-
-
-
-
-
-
-
-
R
-
-
-
-
-
-
-
-
-
-
A
-
-
-
-
-
-
-
-
-
-
C
-
-
-
-
-
A
B
R
A
C
A
D
A
B
R
A
-
-
-
-
-
D
-
-
-
-
-
-
-
-
-
-
A
-
-
-
-
-
-
-
-
-
-
B
-
-
-
-
-
-
-
-
-
-
R
-
-
-
-
-
-
-
-
-
-
A
-
-
-
-
-

 

 

A
-
-
-
A
-
-
-
-
A
-
B
-
-
-
B
-
-
-
R
-
-
R
-
-
R
-
-
B
-
-
-
-
A
-
A
-
A
-
-
-
-
-
-
C
C
D
-
-
-
-
A
B
R
A
C
A
D
A
B
R
A
-
-
-
C
D
D
-
-
-
-
-
-
A
-
A
-
A
-
-
-
-
R
-
-
B
-
-
B
-
-
-
B
-
-
-
R
-
-
-
R
-
A
-
-
-
A
-
-
-
-
A

 

 

5
PEACE
30
21
3
2
BE
7
7
7
4
UPON
66
21
3
3
YOU
61
16
7
-
-
-
-
-
-
-
-
-
-
14
PEACE BE UNTO YOU
168
60
6
-
-
-
-
-
-
-
-
-
-
5
PEACE
30
21
3
4
LOVE
54
18
9
5
LIGHT
56
29
2
14
First Total
140
68
14
1+4
Add to Reduce
1+4+0
6+8
1+4
5
Second Total
5
14
5
-
Reduce to Deduce
-
1+4
-
5
Essence of Number
5
5
5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
35
6
A
T
U
M
-
-
10
-
-
4
-
10
3+5
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
8
6
A
T
U
M
-
-
1
-
-
4
-
1

 

 

6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
=
1
6
A
T
U
M
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
4
occurs
x
1
=
4
6
A
T
U
M
-
-
10
-
-
4
-
10
-
1
2
3
4
-
-
1+0
-
-
-
Q
1+0
6
A
T
U
M
-
-
1
-
-
4
-
1

 

AUTUMN ATUM AUTUMN

 

-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
-
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
-
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
30
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
3+0
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
3
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
3
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

 

6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

 

6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
+
=
5
-
=
5
=
5
-
-
-
-
-
-
14
+
=
14
1+4
=
5
=
5
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
3
2
3
4
-
+
=
13
1+3
=
4
=
4
-
1
21
20
21
13
-
+
=
76
7+6
=
13
1+3
4
6
A
U
T
U
M
N
-
-
-
-
-
--
-
-
-
1
21
20
21
13
14
+
=
90
9+0
=
9
=
9
-
1
3
2
3
4
5
+
=
18
1+8
=
9
=
9
6
A
U
T
U
M
N
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
A
U
T
U
M
N
-
-
15
-
-
6
-
18
-
-
-
-
--
-
-
-
-
1+5
-
-
-
-
1+8
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9
-
1
3
2
3
4
5
-
-
-
-
-
--
-
-
6
A
U
T
U
M
N
-
-
6
-
-
6
-
9

 

AUTUMN ATUM AUTUMN

QUANTUM ATUM QUANTUM

 

 

-
QUANTUM
-
-
-
4
.......A...TUM.
55
10
1
7
QUANTUM
107
26
8
4
.......A...TUM.
55
10
1
7
QUANTUM
107
26
8
3
.....QU...N.............
52
16
7
7
QUANTUM
107
26
8

 

 

-
QUANTUM
-
-
-
1
Q
17
8
8
3
UAN
35
9
9
3
TUM
54
9
9
7
QUANTUM
107
26
26
-
-
1+0+7
2+6
2+6
7
QUANTUM
8
8
8

 

 

-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
 1+2
3
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
-
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
-
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
6
SIX
-
-
-
-
7
``-
-
-
-
-
-
-
-
-
-
7
SEVEN
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
9
-
-
-
-
-
-
-
-
-
-
9
NINE
-
-
-
-
22
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
2+2
-
-
-
-
-
-
-
-``
-
-
2+3
-
-
-
-
2+6
4
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
4
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

 

7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
 1+2
3
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
-
-
-``
-
-
-
-``
-``
-
-
2+3
-
-
-
-
2+6
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

 

7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
+
=
5
-
=
5
=
5
-
-
-
-
14
-
-
-
+
=
14
1+4
=
5
=
5
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
8
3
1
-
2
3
4
+
=
21
2+1
=
3
=
3
-
17
21
1
-
20
21
13
+
=
93
9+3
=
12
 1+2
3
7
Q
U
A
N
T
U
M
-
-
-
-
-
--
-
-
-
17
21
1
14
20
21
13
+
=
107
1+0+7
=
8
=
8
-
8
3
1
5
2
3
4
+
=
26
2+6
=
8
=
8
7
Q
U
A
N
T
U
M
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
``-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
``-
-
3
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
Q
U
A
N
T
U
M
-
-
23
-
-
7
-
26
-
-
-
1
-
2
3
4
-
-
2+3
-
-
-
-
2+6
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8
-
8
3
1
5
2
3
4
-
-
-
-
-
-
-
-
7
Q
U
A
N
T
U
M
-
-
5
-
-
7
-
8

 

RA ATUM ATUM RA

 

 

Q
=
8
-
7
QUANTUM
107
26
8
E
=
5
-
8
ENERGIES
82
46
1
-
-
13
-
27
First Total
189
72
9
-
-
1+3
-
2+7
Add to Reduce
1+8+9
7+2
-
Q
-
4
-
9
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
5
9
Essence of Number
9
9
9

 

 

-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
9
-
1
+
=
20
2+0
=
2
-
2
=
2
-
-
-
-
-
14
-
-
-
-
-
14
-
-
-
9
-
19
+
=
56
5+6
=
11
1+1
2
=
2
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
8
3
1
-
2
3
4
-
5
-
5
9
7
-
5
-
+
=
52
5+2
=
7
-
7
=
7
-
-
17
21
1
-
20
21
13
-
5
-
5
18
7
-
5
-
+
=
133
1+3+3
=
7
-
7
=
7
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
--
-
-
-
-
-
-
17
21
1
14
20
21
13
-
5
14
5
18
7
9
5
19
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
+
=
72
7+2
=
9
-
9
=
9
-
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
``-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
``-
-
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
-
-
-
-
5
5
5
-
-
-
5
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
-
-
-
-
-
-
-
``-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
6
15
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
39
-
-
15
-
72
-
45
=
1+5
-
-
-
-
-
-
-``
-
-
-
-
9
-
9
-
-
-
-
3+9
-
-
1+5
-
7+2
-
4+5
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
12
-
-
6
-
9
-
9
-
-
8
3
1
5
2
3
4
-
5
5
5
9
7
9
5
1
-
-
1+2
-
-
-
-
-
-
-
6
6
Q
U
A
N
T
U
M
-
E
N
E
R
G
I
E
S
-
-
3
-
-
6
-
9
-
9

 

QUANTUM THOUGHTS GODS THOUGHTS QUANTUM

 

A
=
1
-
3
AND
19
10
1
Q
=
8
-
7
QUANTUM
107
26
8
I
=
9
-
2
IS
28
10
1
A
=
1
-
2
AS
20
2
2
Q
=
8
-
7
QUANTUM
107
26
8
D
=
4
-
4
DOES
43
16
7
-
-
31
-
25
First Total
324
90
27
-
-
3+1
-
2+5
Add to Reduce
3+2+4
9+0
2+7
Q
-
4
-
7
Second Total
9
9
9
-
-
-
-
-
Reduce to Deduce
-
-
-
-
-
4
5
7
Essence of Number
9
9
9

 

 

8
EXOTERIC
99
45
9
8
ESOTERIC
94
40
4
16
First Total
193
85
13
1+6
Add to Reduce
1+9+3
8+5
1+3
7
Second Total
13
13
4
-
Add to Reduce
1+3
1+3
-
7
Essence of Number
4
4
4

 

 

-
ESOTERIC
-
-
-
1
E
5
5
5
1
S
19
10
1
1
O
15
6
6
1
T
20
2
2
1
E
5
5
5
1
R
18
9
9
1
I
9
9
9
1
C
3
3
3
8
ESOTERIC
94
49
40
-
-
9+4
4+9
4+0
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

-
ESOTERIC
-
-
-
1
E
5
5
5
3
SOT
54
18
9
1
E
5
5
5
1
R
18
9
9
1
I
9
9
9
1
C
3
3
3
8
ESOTERIC
94
49
40
-
-
9+4
4+9
4+0
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

-
ESOTERIC
-
-
-
5
ESOTE
64
28
1
1
R
18
9
9
1
I
9
9
9
1
C
3
3
3
8
ESOTERIC
94
49
40
-
-
9+4
4+9
4+0
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

-
ESOTERIC
-
-
-
1
I
9
9
9
6
SECRET
70
34
7
1
O
15
6
6
8
ESOTERIC
94
49
22
-
-
9+4
4+9
2+2
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

-
ESOTERIC
-
-
-
1
O
15
6
6
6
SECRET
70
34
7
1
I
9
9
9
8
ESOTERIC
94
49
22
-
-
9+4
4+9
2+2
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

-
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
--
-
-
-
-
-
-
6
-
1
-
-
-
-
-
-
9
+
=
16
1+6
=
7
=
7
=
7
-
-
15
-
19
-
-
-
-
-
-
9
+
=
43
4+3
=
7
=
7
=
7
-
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
3
9
5
2
-
-
+
=
24
2+4
=
6
=
6
=
6
-
-
-
-
-
5
3
18
5
20
-
-
+
=
51
5+1
=
6
=
6
=
6
-
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
--
-
-
-
-
-
-
15
-
19
5
3
18
5
20
-
9
+
=
94
9+4
=
13
1+3
4
=
4
-
-
6
-
1
5
3
9
5
2
-
9
+
=
40
4+0
=
4
=
4
=
4
-
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
-
-
-
-
-
``-
-
-
--
1
-
-
-
-
-
--
-
-
-
1
occurs
x
1
=
1
=
1
-
``-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
19
16
O
-
S
E
C
R
E
T
-
I
-
-
26
-
-
8
-
40
-
22
1+9
1+6
-
---
-
-
-
9
-
-
---
9
-
-
2+6
-
-
-
-
4+0
-
2+2
10
7
O
-
S
E
C
R
E
T
-
I
-
-
8
-
-
8
-
4
-
4
1+0
-
6
-
1
5
3
9
5
2
-
9
-
-
-
1
-
-
-
-
-
-
1
7
O
-
S
E
C
R
E
T
-
I
-
-
8
-
-
8
-
4
-
4

 

 

16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
--
-
-
-
-
-
6
-
1
-
-
-
-
-
-
9
+
=
16
1+6
=
7
=
7
=
7
-
15
-
19
-
-
-
-
-
-
9
+
=
43
4+3
=
7
=
7
=
7
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
3
9
5
2
-
-
+
=
24
2+4
=
6
=
6
=
6
-
-
-
-
5
3
18
5
20
-
-
+
=
51
5+1
=
6
=
6
=
6
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
--
-
-
-
-
-
15
-
19
5
3
18
5
20
-
9
+
=
94
9+4
=
13
1+3
4
=
4
-
6
-
1
5
3
9
5
2
-
9
+
=
40
4+0
=
4
=
4
=
4
16
O
-
S
E
C
R
E
T
-
I
-
-
-
-
-
-
-
-
-
-
-
-
--
1
-
-
-
-
-
--
-
-
-
1
occurs
x
1
=
1
=
1
``-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
5
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
9
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
16
O
-
S
E
C
R
E
T
-
I
-
-
26
-
-
8
-
40
-
22
1+6
-
---
-
-
-
9
-
-
---
9
-
-
2+6
-
-
-
-
4+0
-
2+2
7
O
-
S
E
C
R
E
T
-
I
-
-
8
-
-
8
-
4
-
4
-
6
-
1
5
3
9
5
2
-
9
-
-
-
1
-
-
-
-
-
-
7
O
-
S
E
C
R
E
T
-
I
-
-
8
-
-
8
-
4
-
4

 

 

-
16
O
S
E
C
R
E
T
-
-
-
-
-
--
-
-
-
-
-
-
6
1
-
-
-
-
-
9
+
=
16
1+6
=
7
=
7
=
7
-
-
15
19
-
-
-
-
-
9
+
=
43
4+3
=
7
=
7
=
7
-
16
O
S
E
C
R
E
T
I
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
3
9
5
2
-
+
=
24
2+4
=
6
=
6
=
6
-
-
-
-
5
3
18
5
20
-
+
=
51
5+1
=
6
=
6
=
6
-
16
O
S
E
C
R
E
T
I
-
-
-
-
-
--
-
-
-
-
-
-
15
19
5
3
18
5
20
9
+
=
94
9+4
=
13
1+3
4
=
4
-
-
6
1
5
3
9
5
2
9
+
=
40
4+0
=
4
=
4
=
4
-
16
O
S
E
C
R
E
T
I
-
-
-
-
-
-
-
-
-
-
``-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
``-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
5
-
-
5
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
19
16
O
S
E
C
R
E
T
I
-
-
26
-
-
8
-
40
-
22
1+9
1+6
-
-
-
-
9
-
-
9
-
-
2+6
-
-
-
-
4+0
-
2+2
10
7
O
S
E
C
R
E
T
I
-
-
8
-
-
8
-
4
-
4
1+0
-
6
1
5
3
9
5
2
9
-
-
-
1
-
-
-
-
-
-
1
7
O
S
E
C
R
E
T
I
-
-
8
-
-
8
-
4
-
4

 

 

16
O
S
E
C
R
E
T
-
-
-
-
-
--
-
-
-
-
-
6
1
-
-
-
-
-
9
+
=
16
1+6
=
7
=
7
=
7
-
15
19
-
-
-
-
-
9
+
=
43
4+3
=
7
=
7
=
7
16
O
S
E
C
R
E
T
I
-
-
-
-
-
--
-
-
-
-
-
-
-
5
3
9
5
2
-
+
=
24
2+4
=
6
=
6
=
6
-
-
-
5
3
18
5
20
-
+
=
51
5+1
=
6
=
6
=
6
16
O
S
E
C
R
E
T
I
-
-
-
-
-
--
-
-
-
-
-
15
19
5
3
18
5
20
9
+
=
94
9+4
=
13
1+3
4
=
4
-
6
1
5
3
9
5
2
9
+
=
40
4+0
=
4
=
4
=
4
16
O
S
E
C
R
E
T
I
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
``-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
5
-
-
5
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
9
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
16
O
S
E
C
R
E
T
I
-
-
26
-
-
8
-
40
-
22
1+6
-
-
-
-
9
-
-
9
-
-
2+6
-
-
-
-
4+0
-
2+2
7
O
S
E
C
R
E
T
I
-
-
8
-
-
8
-
4
-
4
-
6
1
5
3
9
5
2
9
-
-
-
1
-
-
-
-
-
-
7
O
S
E
C
R
E
T
I
-
-
8
-
-
8
-
4
-
4

 

 

-
ESOTERIC
-
-
-
1
O
15
6
6
6
SECRET
70
34
7
1
I
9
9
9
8
ESOTERIC
94
49
22
-
-
9+4
4+9
2+2
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

-
8
E
S
O
T
E
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
1
6
-
-
-
9
-
+
=
16
1+6
=
7
=
7
=
7
-
-
-
19
15
-
-
-
9
-
+
=
43
4+3
=
7
=
7
=
7
-
8
E
S
O
T
E
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
2
5
9
-
3
+
=
24
2+4
=
6
=
6
=
6
-
-
5
-
-
20
5
18
-
3
+
=
51
5+1
=
6
=
6
=
6
-
8
E
S
O
T
E
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
5
19
15
20
5
18
9
3
+
=
94
9+4
=
13
1+3
4
=
4
-
-
5
1
6
2
5
9
9
3
+
=
40
4+0
=
4
=
4
=
4
-
8
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
``-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
3
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
19
8
E
S
O
T
E
R
I
C
-
-
26
-
-
8
-
40
-
22
1+9
-
-
-
-
-
-
9
9
-
-
-
2+6
-
-
-
-
4+0
-
2+2
10
8
E
S
O
T
E
R
I
C
-
-
8
-
-
8
-
4
-
4
1+0
-
5
1
6
2
5
9
9
3
-
-
-
1
-
-
-
-
-
-
1
8
E
S
O
T
E
R
I
C
-
-
8
-
-
8
-
4
-
4

 

 

-
ESOTERIC
-
-
-
1
O
15
6
6
6
SECRET
70
34
7
1
I
9
9
9
8
ESOTERIC
94
49
22
-
-
9+4
4+9
2+2
8
ESOTERIC
13
13
4
-
-
1+3
1+3
-
8
ESOTERIC
4
4
4

 

 

8
EXOTERIC
99
45
9
8
ESOTERIC
94
40
4
16
First Total
193
85
13
1+6
Add to Reduce
1+9+3
8+5
1+3
7
Second Total
13
13
4
-
Reduce to Deduce
1+3
1+3
-
7
Essence of Number
4
4
4

 

 

-
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
9
-
-
-
1
6
-
-
-
9
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
-
24
15
-
-
-
9
-
-
-
19
15
-
-
-
9
-
+
=
91
9+1
=
10
1+0
1
=
1
-
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
2
5
9
-
3
-
5
-
-
2
5
9
-
3
+
=
48
4+8
=
12
1+2
3
=
3
-
-
5
-
-
20
5
18
-
3
-
5
-
-
20
5
18
-
3
+
=
102
1+0+2
=
3
=
3
=
3
-
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
5
24
15
20
5
18
9
3
-
5
19
15
20
5
18
9
3
+
=
193
1+9+3
=
13
1+3
4
=
4
-
-
5
6
6
2
5
9
9
3
-
5
1
6
2
5
9
9
3
+
=
85
8+5
=
13
1+3
4
=
4
-
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
``-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
4
=
36
3+6
9
19
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
26
-
-
16
-
85
-
31
1+9
1+6
-
-
-
-
-
9
9
-
---
-
-
-
-
-
9
9
-
-
-
2+6
-
-
1+6
-
8+5
-
3+1
10
7
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
8
-
-
7
-
13
-
4
1+0
-
5
6
6
2
5
9
9
3
-
5
1
6
2
5
9
9
3
-
-
-
-
-
-
-
1+3
-
-
1
7
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
8
-
-
7
-
4
-
4

 

 

16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
9
-
-
-
1
6
-
-
-
9
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
24
15
-
-
-
9
-
-
-
19
15
-
-
-
9
-
+
=
91
9+1
=
10
1+0
1
=
1
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
5
-
-
2
5
9
-
3
-
5
-
-
2
5
9
-
3
+
=
48
4+8
=
12
1+2
3
=
3
-
5
-
-
20
5
18
-
3
-
5
-
-
20
5
18
-
3
+
=
102
1+0+2
=
3
=
3
=
3
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
5
24
15
20
5
18
9
3
-
5
19
15
20
5
18
9
3
+
=
193
1+9+3
=
13
1+3
4
=
4
-
5
6
6
2
5
9
9
3
-
5
1
6
2
5
9
9
3
+
=
85
8+5
=
13
1+3
4
=
4
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
``-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
-
5
-
-
-
5
-
-
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
4
=
36
3+6
9
16
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
26
-
-
16
-
85
-
31
1+6
-
-
-
-
-
9
9
-
---
-
-
-
-
-
9
9
-
-
-
2+6
-
-
1+6
-
8+5
-
3+1
7
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
8
-
-
7
-
13
-
4
-
5
6
6
2
5
9
9
3
-
5
1
6
2
5
9
9
3
-
-
-
-
-
-
-
1+3
-
-
7
E
X
O
T
E
R
I
C
-
E
S
O
T
E
R
I
C
-
-
8
-
-
7
-
4
-
4

 

 

-
19
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
9
-
-
-
5
-
-
-
1
6
-
-
-
9
-
+
=
42
4+2
=
6
=
6
=
6
-
-
-
24
15
-
-
-
9
-
-
-
14
-
-
-
19
15
-
-
-
9
-
+
=
105
1+0+5
=
6
=
6
=
6
-
19
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
2
5
9
-
3
-
1
-
4
-
5
-
-
2
5
9
-
3
+
=
53
5+3
=
8
=
8
=
8
-
-
5
-
-
20
5
18
-
3
-
1
-
4
-
5
-
-
20
5
18
-
3
+
=
107
1+0+7
=
8
=
8
=
8
-
19
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
5
24
15
20
5
18
9
3
-
1
14
4
-
5
19
15
20
5
18
9
3
+
=
212
2+1+2
=
5
=
5
=
5
-
-
5
6
6
2
5
9
9
3
-
1
5
4
-
5
1
6
2
5
9
9
3
+
=
95
9+5
=
14
1+4
5
=
5
-
19
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
1
-
-
--
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
``-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
-
5
-
-
-
-
-
5
-
-
5
-
-
-
5
-
-
-
-
-
5
occurs
x
5
=
25
2+5
7
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
4
=
36
3+6
9
15
19
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
30
-
-
19
-
95
-
41
1+5
1+9
-
-
-
-
-
9
9
-
---
-
-
-
---
-
-
-
-
-
9
9
-
-
-
3+0
-
-
1+9
-
9+5
-
4+1
6
10
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
3
-
-
10
-
14
-
5
-
1+0
5
6
6
2
5
9
9
3
-
1
5
4
-
5
1
6
2
5
9
9
3
-
-
-
-
-
1+0
-
1+4
-
-
6
1
E
X
O
T
E
R
I
C
-
A
N
D
-
E
S
O
T
E
R
I
C
-
-
3
-
-
1
-
5
-
5

H
=
8
-
4
HOLY
60
24
6
C
=
3
-
10
CONSCIENCE
90
45
9
T
=
2
-
3
THE
33
15
6
I
=
9
-
5
INNER
60
33
6
V
=
4
-
5
VOICE
54
27
9
-
-
26
-
27
First Total
297
144
27
-
-
2+6
-
2+7
Add to Reduce
2+9+7
1+4+4
2+7
Q
-
8
-
9
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
8
5
9
Essence of Number
9
9
9

 

 

 

THE SCULPTURE OF VIBRATIONS 1971
 
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