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Evokation
  
 
Index
  

 

A

MAZE

IN

ZAZAZA ENTER AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

WORK DAYS OF GOD

Herbert W Morris D.D.circa 1883

Page 22

"As all the words in the English language are composed out of the twenty-six letters of the alphabet,.."

 

 

LIGHT AND LIFE

Lars Olof Bjorn 1976

Page 197

"By writing the 26 letters of the alphabet in a certain order one may put down almost any message (this book 'is written with the same letters' as the Encyclopaedia Britannica and Winnie the Pooh, only the order of the letters differs). In the same way Nature is able to convey with her language how a cell and a whole organism is to be constructed and how it is to function. Nature has succeeded better than we humans; for the genetic code there is only one universal language which is the same in a man, a bean plant and a bacterium."

"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
J
K
L
M
N
O
P
Q
R
10
11
12
13
14
15
16
17
18
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
S
T
U
V
W
X
Y
Z
I
19
20
21
22
23
24
25
26
9
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
ME
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
I
ME
I
ME
I
ME
I
ME
I
9
18
9
18
9
18
9
18
9
=
1+8
=
1+8
=
1+8
=
1+8
=
=
9
=
9
=
9
=
9
=
I
ME
I
ME
I
ME
I
ME
1
9
9
9
9
9
9
9
9
9
I
ME
I
ME
I
ME
I
ME
1

 

 

"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
 -
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically; the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by combining the letters with the sacred numbers, rearranging them in endless configurations, the mystic weaned his mind away from the normal connotations of words."

 

 

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

THE ACCOUNT IS SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL CONNOTATIONS OF WORDS

 

THIS IS THE SCENE OF THE SCENE UNSEEN

THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE

 

 

3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
18
First Total
189
90
18
1+8
Add to Reduce
1+8+9
9+0
1+8
9
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
9
Essence of Number
9
9
9

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

IN SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

AT THE THROW OF THE NINTH RAM WHEN IN CONJUNCTION SET

THE

FAR YONDER SCRIBE

MADE RECORD OF THEIR FALL

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

FINGERPRINTS OF THE GODS

A QUEST FOR THE BEGINNING AND THE END

Graham Hancock 1995

Chapter 32

Speaking to the Unborn

Page 285

"It is understandable that a huge range of myths from all over the ancient world should describe geological catastrophes in graphic detail. Mankind survived the horror of the last Ice Age, and the most plausible source for our enduring traditions of flooding and freezing, massive volcanism and devastating earthquakes is in the tumultuous upheavals unleashed during the great meltdown of 15,000 to 8000 BC. The final retreat of the ice sheets, and the consequent 300-400 foot rise in global sea levels, took place only a few thousand years before the beginning of the historical period. It is therefore not surprising that all our early civilizations should have retained vivid memories of the vast cataclysms that had terrified their forefathers.
Much harder to explain is the peculiar but distinctive way the myths of cataclysm seem to bear the intelligent imprint of a guiding hand.l Indeed the degree of convergence between such ancient stories is frequently remarkable enough to raise the suspicion that they must all have been 'written' by the same 'author'.
Could that author have had anything to do with the wondrous deity, or superhuman, spoken of in so many of the myths we have reviewed, who appears immediately after the world has been shattered by a horrifying geological catastrophe and brings comfort and the gifts of civilization to the shocked and demoralized survivors?
White and bearded, Osiris is the Egyptian manifestation of this / Page 286 / universal figure, and it may not be an accident that one of the first acts he is remembered for in myth is the abolition of cannibalism among the primitive inhabitants of the Nile Valley.2 Viracocha, in South America, was said to have begun his civilizing mission immediately after a great flood; Quetzalcoatl, the discoverer of maize, brought the benefits of crops, mathematics, astronomy and a refined culture to Mexico after the Fourth Sun had been overwhelmed by a destroying deluge.
Could these strange myths contain a record of encounters between scattered palaeolithic tribes which survived the last Ice Age and an as yet unidentified high civilization which passed through the same epoch?
And could the myths be attempts to communicate?

A message in the bottle of time"

'Of all the other stupendous inventions,' Galileo once remarked,

what sublimity of mind must have been his who conceived how to communicate his most secret thoughts to any other person, though very distant either in time or place, speaking with those who are in the Indies, speaking to those who are not yet born, nor shall be this thousand or ten thousand years? And with no greater difficulty than the various arrangements of two dozen little signs on paper? Let this be the seal of all the admirable inventions of men.3

If the 'precessional message' identified by scholars like Santillana, von Dechend and Jane Sellers is indeed a deliberate attempt at communication by some lost civilization of antiquity, how come it wasn't just written down and left for us to find? Wouldn't that have been easier than encoding it in myths? Perhaps.
Nevertheless, suppose that whatever the message was written on got destroyed or worn away after many thousands of years? Or suppose that the language in which it was inscribed was later forgotten utterly (like the enigmatic Indus Valley script, which has been studied closely for more than half a century but has so far resisted all attempts at decoding)? It must be obvious that in such circumstances a written / Page 287 / legacy to the future would be of no value at all, because nobody would be able to make sense of it.
What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them - and the city of Teotihuacan may be the calling-card of a lost civilization written in the eternal language of mathematics.
Geodetic data, related to the exact positioning of fixed geographical points and to the shape and size of the earth, would also remain valid and recognizable for tens of thousands of years, and might be most conveniently expressed by means of cartography (or in the construction of giant geodetic monuments like the Great Pyramid of Egypt, as we shall see).
Another 'constant' in our solar system is the language of time: the great but regular intervals of time calibrated by the inch-worm creep of precessional motion. Now, or ten thousand years in the future, a message that prints out numbers like 72 or 2160 or 4320 or 25,920 should be instantly intelligible to any civilization that has evolved a modest talent for mathematics and the ability to detect and measure the almost imperceptible reverse wobble that the sun appears to make along the ecliptic against the background of the fixed stars..."

"What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them"

"WRITTEN IN THE ETERNAL LANGUAGE OF MATHEMATICS"

 

 

-
THE RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
9

 

 

15
THE RAINBOW LIGHT
-
-
-
-
THE
33
15
6
-
R
18
9
9
-
A
1
1
1
-
I
9
9
9
-
N+B+O+W
54
18
9
-
L
12
3
3
-
I
9
9
9
-
G+H+T
35
17
8
15
THE RAINBOW LIGHT
171
81
54
1+5
-
1+7+1
8+1
5+4
6
THE RAINBOW LIGHT
9
9
9

 

 

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-

 

 

9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
+
=
28
2+8
=
10
1+0
1
-
1
2
3
4
5
6
7
-
-
+
=
28
2+8
=
10
1+0
1
9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
+
=
45
4+5
=
9
=
9
-
1
2
3
4
5
6
7
8
9
+
=
45
4+5
=
9
=
9
9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
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
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
-
+
=
6
occurs
x
1
=
6
-
-
-
-
-
-
-
7
-
-
+
=
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
8
-
+
=
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
9
+
=
9
occurs
x
1
=
9
9
A
B
C
D
E
F
G
H
I
-
-
45
-
-
9
-
45
-
1
2
3
4
5
6
7
8
9
-
-
4+5
-
-
-
-
4+5
9
A
B
C
D
E
F
G
H
I
-
-
9
-
-
9
-
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
9
A
B
C
D
E
F
G
H
I
-
-
9
-
-
9
-
9

 

 

9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
-
-
-
-
5
6
-
-
-
+
=
11
1+1
=
2
=
2
-
-
-
-
-
14
15
-
-
-
+
=
29
2+9
=
11
1+1
2
9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
1
2
3
4
-
-
7
8
9
+
=
34
3+4
=
7
=
7
-
10
11
12
13
-
-
16
17
18
+
=
97
9+7
=
16
1+6
7
9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
10
11
12
13
14
15
16
17
18
+
=
126
1+2+6
=
9
=
9
-
1
2
3
4
5
6
7
8
9
+
=
45
4+5
=
9
=
9
9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
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
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
-
+
=
6
occurs
x
1
=
6
-
-
-
-
-
-
-
7
-
-
+
=
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
8
-
+
=
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
9
+
=
9
occurs
x
1
=
9
9
J
K
L
M
N
O
P
Q
R
-
-
45
-
-
9
-
45
-
10
11
12
13
14
15
16
17
18
-
-
4+5
-
-
-
-
4+5
9
J
K
L
M
N
O
P
Q
R
-
-
9
-
-
9
-
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
9
J
K
L
M
N
O
P
Q
R
-
-
9
-
-
9
-
9

 

 

8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
1
-
-
-
-
6
-
8
+
=
15
1+5
=
6
=
6
-
19
-
-
-
-
24
-
26
+
=
69
6+9
=
15
1+5
6
8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
2
3
4
5
-
7
-
+
=
21
2+1
=
3
=
3
-
-
20
21
22
23
-
25
-
+
=
111
1+1+1
=
3
=
3
8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
19
20
21
22
23
24
25
26
+
=
 180
1+8+0
=
9
=
9
-
1
2
3
4
5
6
7
8
+
=
36
3+6
=
9
=
9
8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
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
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
1
=
6
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
1
=
8
8
S
T
U
V
W
X
Y
Z
-
-
36
-
-
8
-
36
-
1
2
3
4
5
6
7
8
-
-
3+6
-
-
-
-
3+6
8
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
8
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
=
7
=
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
=
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
=
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
=
9
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
=
9
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

 

 

 

 

THE DEATH OF GODS IN ANCIENT EGYPT

Jane B. Sellars 1992

Page 204

"The overwhelming awe that accompanies the realization, of the measurable orderliness of the universe strikes modern man as well. Admiral Weiland E. Byrd, alone In the Antarctic for five months of polar darkness, wrote these phrases of intense feeling:

Here were the imponderable processes and forces of the cosmos, harmonious and soundless. Harmony, that was it! I could feel no doubt of oneness with the universe. The conviction came that the rhythm was too orderly. too harmonious, too perfect to be a product of blind chance - that, therefore there must be purpose in the whole and that man was part of that whole and not an accidental offshoot. It was a feeling that transcended reason; that went to the heart of man's despair and found it groundless. The universe was a cosmos, not a chaos; man was as rightfully a part of that cosmos as were the day and night.10

Returning to the account of the story of Osiris, son of Cronos god of' Measurable Time, Plutarch takes, pains to remind the reader of the original Egyptian year consisting of 360 days.

Phrases are used that prompt simple mental. calculations and an attention to numbers, for example, the 360-day year is described as being '12 months of 30 days each'. Then we are told that, Osiris leaves on a long journey, during which Seth, his evil brother, plots with 72 companions to slay Osiris: He also secretly obtained the measure of Osiris and made ready a chest in which to entrap him.

The, interesting thing about this part of the-account is that nowhere in the original texts of the Egyptians are we told that Seth, has 72 companions. We have already been encouraged to equate Osiris with the concept of measured time; his father being Cronos. It is also an observable fact that Cronos-Saturn has the longest sidereal period of the known planets at that time, an orbit. of 30 years. Saturn is absent from a specific constellation for that length of time.

A simple mathematical fact has been revealed to any that are even remotely sensitive to numbers: if you multiply 72 by 30, the years of Saturn's absence (and the mention of Osiris's absence prompts one to recall this other), the resulting product is 2,160: the number of years required, for one 30° shift, or a shift: through one complete sign of the zodiac. This number multplied by the /Page205 / 12 signs also gives 25,920. (And Plutarch has reminded us of 12)

If you multiply the unusual number 72 by 360, a number that Plutarch mentions several times, the product will be 25,920, again the number of years symbolizing the ultimate rebirth.

This 'Eternal Return' is the return of, say, Taurus to the position of marking the vernal equinox by 'riding in the solar bark with. Re' after having relinquished this honoured position to Aries, and subsequently to the to other zodiacal constellations.

Such a return after 25,920 years is indeed a revisit to a Golden Age, golden not only because of a remarkable symmetry In the heavens, but golden because it existed before the Egyptians experienced heaven's changeability.

But now to inform the reader of a fact he or she may already know. Hipparaus did: not really have the exact figures: he was a trifle off in his observations and calculations. In his published work, On the Displacement of the Solstitial and Equinoctial Signs, he gave figures of 45" to 46" a year, while the truer precessional lag along the ecliptic is about 50 seconds. The exact measurement for the lag, based on the correct annual lag of 50'274" is 1° in 71.6 years, or 360° in 25,776 years, only 144 years less than the figure of 25,920.

With Hipparchus's incorrect figures a 'Great Year' takes from 28,173.9 to 28,800 years, Incorrect by a difference of from 2,397.9 years to 3,024.

Since Nicholas Copernicus (AD 1473-1543) has always been credited with giving the correct numbers (although Arabic astronomer Nasir al-Din Tusi,11 born AD 1201, is known to have fixed the Precession at 50°), we may correctly ask, and with justifiable astonishment 'Just whose information was Plutarch transmitting'

AN IMPORTANT POSTSCRIPT

Of course, using our own notational system, all the important numbers have digits that reduce to that amazing number 9 a number that has always delighted budding mathematician.

Page 206

Somewhere along the way, according to Robert Graves, 9 became the number of lunar wisdom.12

This number is found often in the mythologies of the world. the Viking god Odin hung for nine days and nights on the World Tree in order to acquire the secret of the runes, those magic symbols out of which writing and numbers grew. Only a terrible sacrifice would give away this secret, which conveyed upon its owner power and dominion over all, so Odin hung from his neck those long 9 days and nights over the 'bottomless abyss'. In the tree were 9 worlds, and another god was said to have been born of 9 mothers.

Robert Graves, in his White Goddess, Is intrigued by the seemingly recurring quality of the number 72 in early myth and ritual. Graves tells his reader that 72 is always connected with the number 5, which reflects, among other things, the five Celtic dialects that he was investigating. Of course, 5 x 72= 360, 360 x 72= 25,920. Five is also the number of the planets known to the ancient world, that is, Saturn, Jupiter, Mars, Venus Mercury.

Graves suggests a religious mystery bound up with two ancient Celtic 'Tree Alphabets' or cipher alphabets, which as genuine articles of Druidism were orally preserved and transmitted for centuries. He argues convincingly that the ancient poetry of Europe was ultimately based on what its composers believed to be magical principles, the rudiments of which formed a close religious secret for centuries. In time these were-garbled, discredited and forgotten.

Among the many signs of the transmission of special numbers he points out that the aggregate number of letter strokes for the complete 22-letter Ogham alphabet that he is studying is 72 and that this number is the multiple of 9, 'the number of lunar wisdom'. . . . he then mentions something about 'the seventy day season during which Venus moves successively from. maximum eastern elongation 'to inferior conjunction and maximum western elongation'.13

Page 207

"...Feniusa Farsa, Graves equates this hero with Dionysus Farsa has 72 assistants who helped him master the 72 languages created at the confusion of Babel, the tower of which is said to be built of 9 different materials

We are also reminded of the miraculous translation into Greek of the Five Books of Moses that was done by 72 scholars working for 72 days, Although the symbol for the Septuagint is LXX, legend, according to the fictional letter of Aristeas, records 72. The translation was done for Ptolemy Philadelphus (c.250 BC), by Hellenistic Jews, possibly from Alexandra.14

Graves did not know why this number was necessary, but he points out that he understands Frazer's Golden Bough to be a a book hinting that 'the secret involves the truth that the Christian dogma, and rituals, are the refinement of a great body of primitive beliefs, and that the only original element in Christianity- is the personality of Christ.15

Frances A. Yates, historian of Renaissance hermetisma tells, us the cabala had 72 angels through which the sephiroth (the powers of God) are believed to be approached, and further, she supplies the information that although the Cabala supplied a set of 48 conclusions purporting to confirm the Christian religion from the foundation of ancient wisdom, Pico Della Mirandola, a Renaissance magus, introduced instead 72, which were his 'own opinion' of the correct number. Yates writes, 'It is no accident there are seventy-two of Pico's Cabalist conclusions, for the conclusion shows that he knew something of the mystery of the Name of God with seventy-two letters.'16

In Hamlet's Mill de Santillarta adds the facts that 432,000 is the number of syllables in the Rig-Veda, which when multiplied by the soss (60) gives 25,920" (The reader is forgiven for a bit of laughter at this point)

Thee Bible has not escaped his pursuit. A prominent Assyriologist of the last century insisted that the total of the years recounted
mounted in Genesis for the lifetimes of patriarchs from the Flood also contained the needed secret numbers. (He showed that in the 1,656.years recounted in the Bible there are 86,400 7 day weeks, and dividing this number yields / Page 208 / 43,200.) In Indian yogic schools it is held that all living beings exhale and inhale 21,600 times a day, .multiply this by 2 and again we have.the necessary 432 digits.

Joseph Campbell discerns the secret in the date set for the coming of Patrick to Ireland. Myth-gives this date-as.- the interest-
ing number of AD.432.18

Whatever one may think-of some of these number coincidences, it becomes. difficult to escape the suspicion that many signs (number and otherwise) -indicate that early man observed the results.. of the movement of Precession . and that the-.transmission of this information was .considered of prime importance.

'With the awareness of the phenomenon, observers would certainly have tried for its measure, and such an endeavour would
have constituted the construction-of a 'Unified Field Theory' for nothing .less than Creation itself. Once determined, it would have been information worthy of secrecy and worthy of the passing on to future adepts.

But one last word about mankind's romance with number coincidences.The antagonist in John Updike's novel, Roger's Version, is a computer hacker, who, convinced.,that scientific evidence of God's existence is accumulating, endeavours to prove it by feeding -all the available scientific information. into a comuter. In his search for God 'breaking, through', he has become fascinated by certain numbers that have continually been cropping up. He explains them excitedly as 'the terms of Creation':

"...after a while I noticed that all over the sheet there seemed to hit these twenty-fours Jumping out at me. Two four; two,four.Planck time, for instance, divided by the radiation constant yields a figure near eight times ten again to the negative twenty-fourth, and the permittivity of free space, or electric constant, into the Bohr radiusekla almost exactly six times ten to the negative twenty-fourth. On positive side, the electromagnetic line-structure constant times Hubble radius - that is, the size of the universe as we now perceive it gives us something quite close to ten to the twenty-fourth, and the strong-force constant times the charge on the proton produces two point four times ten to the negative eighteenth, for another I began to circle twenty-four wherever it appeared on the Printout here' - he held it up. his piece of striped and striped wallpaper, decorated / Page 209 / with a number of scarlet circles - 'you can see it's more than random.'19
This inhabitant of the twentieth century is convinced that the striking occurrences of 2 and 4 reveal the sacred numbers by which God is speaking to us.

So much for any scorn directed to ancient man's fascination with number coincidences. That fascination is alive and well, Just a bit more incomprehensible"

 

 

WE ARE THE DEAD SHORT TIME AGO WE LIVED FELT DAWN SAW SUNSET GLOW AND NOW

?

 

 

-
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
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
26
6
A
T
U
M
-
R
A
-
-
19
-
-
6
-
20
2+6
-
-
-
-
-
-
9
-
-
-
1+9
-
-
-
-
2+0
8
6
A
T
U
M
-
R
A
-
-
10
-
-
6
-
2
-
-
-
-
-
-
-
-
-
-
-
1+0
-
-
-
-
-
8
6
A
T
U
M
-
R
A
-
-
1
-
-
6
-
2

 

 

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
-
-
-
1+9
-
-
-
-
2+0
6
A
T
U
M
-
R
A
-
-
10
-
-
6
-
2
-
1
2
3
4
-
9
1
-
-
1+0
-
-
-
-
-
6
A
T
U
M
-
R
A
-
-
1
-
-
6
-
2

 

 

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
-
-
-
1+9
-
-
-
Q
2+0
6
A
T
U
M
R
A
-
-
10
-
-
6
-
2
-
1
2
3
4
9
1
-T
-
1+0
-
-
-
-
-
6
A
T
U
M
R
A
-
-
1
-
-
6
-
2

 

 

-
4
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
4
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
-
-
-
-
-
6
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
35
4
A
T
U
M
-
-
10
-
-
4
-
10
3+5
-
-
-
-
-
-
-
1+0
-
-
-
Q
1+0
8
4
A
T
U
M
-
-
1
-
-
4
-
2

 

 

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

 

 

OF TIME AND STARS

Arthur C. Clarke 1972

FOREWORD

"'Into the Comet' and 'The Nine Billion Names of God' both involve computers and the troubles they may cause us. While writing this preface, I had occasion to call upon my own HP 9100A computer, Hal Junior, to answer an interesting question. Looking at my records, I find that I have now written just about one hundred short stories. This volume contains eighteen of them: therefore, how many possible 18-story collections will I be able to put together? The answer ­as I am sure will be instantly obvious to you - is 100 x 99. . . x 84 x 83 divided by 18 x 17 x 16 ... x .2 x 1. This is an impressive number - Hal Junior tells me that it is approximately 20,772,733,124,605,000,000.

 

3
THE
33
15
6
4
NINE
42
24
6
7
BILLION
73
37
1
5
NAMES
52
16
7
2
OF
21
12
3
3
GOD
26
17
8
24
-
247
121
31
2+4
-
2+4+7
1+2+1
3+1
6
-
13
4
4
-
-
1+3
-
-
6
-
4
4
4

Page 15

The Nine Billion Names of God

 


'This is a slightly unusual request,' said Dr Wagner, with what he hoped was commendable restraint. 'As far as I know, it's the first time anyone's been asked to supply a Tibetan monastery with an Automatic Sequence Computer. I don't wish to be inquisitive, but I should hardly have thought that your - ah - establishment had much use for such a machine. Could you explain just what you intend to do with it?'
'Gladly,' replied the lama, readjusting his silk robes and carefully putting away the slide rule he had been using far currency conversions. 'Your Mark V Computer can carry out any routine mathematical operation involving up to ten digits. However, for our work we are interested in letters, not numbers. As we wish you to modify the output circuits, the machine will be printing words, not columns of figures.'
'I don't quite understand. . .'
'This is a project on which we have been working for the last three centuries - since the lamasery was founded, in fact. It is somewhat alien to your way of thought, so I hope you will listen with an open mind while I explain it.'
'Naturally.'
'It is really quite simple. We have been compiling a list which shall contain all the possible names of God.'
'I beg your pardon?'

Page16

'We have reason to believe,' continued the lama imperturbably, 'that all such names can be written with not more than nine letters in an alphabet we have devised.'
'And you have been doing this for three centuries?'
'Yes: we expected it would take us about fifteen thousand years to complete the task.'
'Oh,' Dr Wagner looked a little dazed. 'Now I see why you wanted to hire one of our machines. But what exactly is the purpose of this project?'
The lama hesitated for a fraction of a second, and Wagner wondered if he had offended him. If so, there was no trace of annoyance in the reply.
'Call it ritual, if you like, but it's a fundamental part of our belief. All the many names of the Supreme Being - God Jehova, Allah, and so on - they are only man-made labels. There is a philosophical problem of some difficulty here, which I do not propose to discuss, but somewhere among all the possible combinations of letters that can occur are what one may call the real names of God. By systematic permutation of letters, we have been trying to list them all.'
'I see. You've been starting at AAAAAAA . . . and working up to ZZZZZZZZ . . .'
'Exactly - though we use a special alphabet of our own. Modifying the electromatic typewriters to deal with this is, of course, trivial. A rather more interesting problem is that of devising suitable circuits to eliminate ridiculous combinations. For example, no letter must occur more than three times in succession.'
,'Three? Surely you mean two.'
'Three is correct: I am afraid it would take too long to explain why, even if you understood our language.' "

Page 68

Into the Comet


"Pickett's fingers danced over the beads, sliding them up and down the wires with lightning speed. There were twelve wires in all, so that the abacus could handle numbers up to 999,999,999,999 - or could be divided into separate sections where several independent calculations could be carried out simultaneously.
'374072,' said Pickett, after an incredibly brief interval of time. 'Now see how long you take to do it, with pencil and paper.'
There was a much longer delay before Martens, who like most mathematicians was poor at arithmetic, called out '375072'. A hasty check soon confirmed that Martens had taken at least three times as long as Pickett to arrive at the wrong answer.
The atronomer's face was a study in mingled chagrin, astonishment, and curiosity.
'Where did you learn that trick?' he asked. 'I thought those things could only add and subtract.'
'Well - multiplication's only repeated addition, isn't it? All I did was to add 856 seven times in the unit column, three times in the tens column, and four times in the hundreds column. You do the same thing when you use pencil and paper. Of course, there are some short cuts, but if you think I'm fast, you should have seen my granduncle. He used to work in a Yokohama bank, and you couldn't see his fingers / Page 69 / when he was going at speed"

 

 

I

SAY

THREAD THAT THREAD

THREAD READ DEATH DEATH READ THREAD

THREAD R DEATH DEATH R THREAD

THREAD READ DEAR THREAD

 

 

THE NEW ELIZABETHAN

REFERENCE DICTIONARY

An up-to-date vocabulary of the living English language

Circa 1900

FOURTH EDITION

Page 1472

thread (thred) [A.-S. thraed, from thrawan, to THROW (cp. Dut. draad, G. draht, Icel. thrathr)], n. A slender cord consisting of two or more yarns doubled or twisted ; a single filament of cotton, silk, wool, etc., esp. Lisle thread ; anything resembling this ; a fine line of colour etc. ; a thin seam or vein ; the spiral on a screw ; (fig.) a continuous course (of life etc.). v.t. To pass a thread through the eye or aperture of ; to string (beads etc.) on a thread ; (fig.) to pick (one's way) or to go through an intricate or crowded place, etc. ; to streak (the hair) with grey etc. ; to cut a thread on (a screw). thread and thrum : Good and bad together, all alike. threadbare, a. Worn so that the thread is visible, having the nap worn off ; (fig.) worn, trite, hackneyed. threadbareness, n. thread-mark, n. A mark produced by coloured silk fibres in banknotes to prevent counterfeiting. thread-paper, n. Soft paper for wrapping up thread, thread-worm, n. A thread-like nematode worm, esp. one infesting the rectum of children. threader, n. threadlike, a. and adv. thready, a. threadiness, n.

 

 

THE NEW ELIZABETHAN

REFERENCE DICTIONARY

An up-to-date vocabulary of the living English language

FOURTH EDITION

Circa 1900

Page 1472

thread (thred) [A.-S. thraed, from thrawan, to THROW (cp. Dut. draad, G. draht, Icel. thrathr)], n. A slender cord consisting of two or more yarns doubled or twisted ; a single filament of cotton, silk, wool, etc., esp. Lisle thread ; anything resembling this ; a fine line of colour etc. ; a thin seam or vein ; the spiral on a screw ; (fig.) a continuous course (of life etc.). v.t. To pass a thread through the eye or aperture of ; to string (beads etc.) on a thread ; (fig.) to pick (one's way) or to go through an intricate or crowded place, etc. ; to streak (the hair) with grey etc. ; to cut a thread on (a screw). thread and thrum : Good and bad together, all alike. threadbare, a. Worn so that the thread is visible, having the nap worn off ; (fig.) worn, trite, hackneyed. threadbareness, n. thread-mark, n. A mark produced by coloured silk fibres in banknotes to prevent counterfeiting. thread-paper, n. Soft paper for wrapping up thread, thread-worm, n. A thread-like nematode worm, esp. one infesting the rectum of children. threader, n. threadlike, a. and adv. thready, a. threadiness, n.

 

 

lisle thread: lisle thread

A strong tightly twisted cotton thread (usually made of long-staple cotton) - lisle. Derived forms: lisle threads. Type of: cotton. Nearest ... www.wordwebonline.com/en/LISLETHREAD

 

 

Definition - of Lisle from Dictionary.net

Lisle thread, a hard twisted cotton thread, originally produced at Lisle. Source: Webster's Revised Unabridged Dictionary (1913) ... www.dictionary.net/lisle - 9k

 

 

CASSELL'S ENGLISH DICTIONARY

1974

Lisle thread (lil thred) [ town in France, now Lille], n, A fine, hard thread orig. made at Lille.

 

 

L
=
3
4
LIFE
32
23
5
H
=
8
7
HANGING
60
42
6
B
=
2
2
BY
27
9
9
A
=
1
1
A
1
1
1
T
=
2
6
THREAD
56
29
2
-
-
16
20
First Total
176
104
23
-
-
1+6
2+0
Add to Reduce
1+7+6
1+0+4
2+3
Q
-
7
2
Second Total
14
5
5
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
7
2
Essence of Number
5
5
5

 

 

-
THREAD
-
-
-
1
R
18
9
9
5
DEATH
38
20
2
6
THREAD
56
29
2

 

LIFE HANGING BY A THREAD

 

-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
44
-
-
9
-
-
-
8
-
5
-
9
5
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
44
4+4
=
8
=
8
=
8
62
-
-
9
-
-
-
8
-
14
-
9
14
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
62
6+2
=
8
=
8
=
8
-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
60
-
3
-
6
5
-
-
1
-
7
-
-
7
-
2
7
-
1
-
2
-
9
5
1
4
+
=
60
6+0
=
6
=
6
=
6
114
-
12
-
6
5
-
-
1
-
7
-
-
7
-
2
25
-
1
-
20
-
18
5
1
4
+
=
114
1+1+4
=
6
=
6
=
6
-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
176
-
12
9
6
5
-
8
1
14
7
9
14
7
-
2
25
-
1
-
20
8
18
5
1
4
+
=
176
1+7+6
=
14
1+4
5
=
5
104
-
3
9
6
5
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
+
=
104
1+0+4
=
5
=
5
=
5
-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
1
occurs
x
3
=
3
=
3
2
``-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
5
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
7
-
-
7
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
45
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
45
-
-
20
-
104
-
41
4+5
2+0
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
2+0
-
1+0+4
-
4+1
9
2
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
2
-
5
-
5
-
-
3
9
6
5
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
-
-
-
-
-
-
-
-
-
-
9
2
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
2
-
5
-
5

 

 

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

2
IS
28
10
1
9
UNIVERSAL
121
40
4
4
MIND
40
22
4
3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
33
First Total
378
162
27
3+3
Add to Reduce
3+7+8
1+6+2
2+7
6
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
6
Essence of Number
9
9
9

 

 

9
UNIVERSAL
121
40
4
4
MIND
40
22
4
2
IS
28
10
1
3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
33
First Total
378
162
27
3+3
Add to Reduce
3+7+8
1+6+2
2+7
6
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
6
Essence of Number
9
9
9

 

 

E
=
5
-
2
EX
11
2
2
U
=
3
-
6
UMBRIS
82
28
1
E
=
5
-
2
ET
25
7
7
I
=
9
-
10
IMAGINIBUS
104
50
5
I
=
9
-
2
IN
23
14
5
V
=
4
-
9
VERITATEM
113
41
5
-
-
35
-
31
First Total
358
142
25
-
-
3+5
-
3+1
Add to Reduce
3+5+8
1+4+2
2+5
-
-
8
-
4
Second Total
16
7
7
-
-
-
-
-
Reduce to Deduce
1+6
-
-
-
-
8
-
4
Essence of Number
7
7
7

 

 

O
=
6
-
3
OUT
56
11
2
O
=
6
-
2
OF
21
12
3
S
=
1
-
7
SHADOWS
89
26
8
A
=
1
-
3
AND
82
28
1
P
=
7
-
9
PHANTASMS
111
30
3
I
=
9
-
4
INTO
58
22
4
T
=
2
-
5
TRUTH
87
24
6
-
-
32
-
33
Add to Reduce
441
135
27
-
-
3+2
-
3+3
Reduce to Deduce
4+4+1
1+3+5
2+7
-
-
5
-
6
Essence of Number
9
9
9

 

 

ADVENT 895 ADVENT

 

LIFE HANGING BY A THREAD
I
ME
EGO
OGRE
EGOCENTRIC
CONSCIENCE
DIVINE CONSCIENCE
CONSCIOUSNESS
SOURCE
ONENESS
COMPLETENESS
WHOLE
CIRCLE
CIRCULAR
CIRCULAR MOTION
THE CYCLE OF THE CIRCLE
THE HE AS IN SHE THAT IS ME

 

 

1
I
11
2
9
2
ME
18
9
9
3
EGO
27
18
9
4
OGRE
45
27
9
10
EGOCENTRIC
99
54
9
10
CONSCIENCE
90
45
9

 

 

-
1
I
-
-
-
I
-
--
-
-
-
-
9
+
=
9
-
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
3
I
-
-
-
I
-
-
-
-
-
-
9
+
=
9
-
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
1
I
-
-
-
I
-
-
-
-
-
-
9
+
=
9
-
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
1
I
T
-
0
ZERO
-
-
-
0
1
-
-
-
-
1
ONE
-
-
-
1
2
-
-
-
-
2
TWO
-
-
-
2
3
-
-
-
-
3
THREE
-
-
-
3
4
-
-
-
-
4
FOUR
-
-
-
4
5
--
-
-
-
5
FIVE
-
-
-
5
6
-
-
-
-
6
SIX
-
-
-
6
7
-
-
-
-
7
SEVEN
-
-
-
7
8
-
-
-
-
8
EIGHT
-
-
-
8
-
-
9
-
-
9
NINE
x
1
=
9
36
1
I
-
-
36
-
-
1
-
9
3+6
-
9
-
-
3+6
-
-
-
-
-
9
1
I
-
-
9
-
-
1
-
9
-
-
9
-
-
-
-
-
-
-
-
9
1
I
-
-
9
-
-
1
-
9

 

 

-
1
I
-
-
-
-
-
--
-
-
-
-
9
+
=
9
1+8
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
3
I
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+8
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
1
I
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+8
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
1
I
T
-
-
-
-
-
-
-
1
-
-
-
-
1
--
-
-
-
1
2
-
-
-
-
2
--
-
-
-
2
3
-
-
-
-
3
-
-
-
-
3
4
-
-
-
-
4
--
-
-
-
4
5
--
-
-
-
5
--
-
-
-
5
6
-
-
-
-
6
--
-
-
-
6
7
-
-
-
-
7
--
-
-
-
7
8
-
-
-
-
8
--
-
-
-
8
-
-
9
-
-
9
occurs
x
1
=
9
36
1
I
-
-
36
-
-
1
-
9
3+6
-
9
-
-
3+6
-
-
-
-
-
9
1
I
-
-
9
-
-
1
-
9
-
-
9
-
-
-
-
-
-
-
-
9
1
I
-
-
9
-
-
1
-
9

 

 

-
1
I
-
-
-
-
-
--
-
-
-
-
9
+
=
9
1+8
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
3
I
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+8
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
1
I
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+8
=
9
=
9
-
-
9
+
=
9
-
=
9
=
9
-
1
I
T
-
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
36
1
I
-
-
36
-
-
1
-
9
3+6
-
9
-
-
3+6
-
-
-
-
-
9
1
I
-
-
9
-
-
1
-
9
-
-
9
-
-
-
-
-
-
-
-
9
1
I
-
-
9
-
-
1
-
9

 

 

-
2
M
E
-
-
-
-
-
--
-
-
-
-
13
5
+
=
18
1+8
=
9
=
9
-
-
4
5
+
=
9
-
=
9
=
9
-
2
M
E
-
-
-
-
-
--
-
-
-
-
13
5
+
=
18
1+8
=
9
=
9
-
-
4
5
+
=
9
-
=
9
=
9
-
2
M
E
-
-
-
-
-
--
-
-
-
-
13
5
+
=
18
1+8
=
9
=
9
-
-
4
5
+
=
9
-
=
9
=
9
-
2
M
E
-
-
-
-
-
-
-
-
1
-
--
--
--
--
1
--
--
--
--
--
2
-
--
--
--
--
2
--
--
--
--
--
3
-
--
--
--
--
3
--
--
--
--
--
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
--
--
--
--
6
--
--
--
--
--
7
-
--
--
--
--
7
--
--
--
--
--
8
-
--
--
--
--
8
--
--
--
--
--
9
-
--
--
--
--
9
--
--
--
--
--
36
2
M
E
-
-
9
-
-
2
-
9
3+6
-
4
5
-
-
-
-
-
-
-
-
9
2
M
E
-
-
9
-
-
2
-
9
-
-
4
5
-
-
-
-
-
-
-
-
9
2
M
E
-
-
9
-
-
2
-
9

 

 

2
M
E
-
-
-
-
-
--
-
-
-
13
5
+
=
18
1+8
=
9
=
9
-
4
5
+
=
9
-
=
9
=
9
2
M
E
-
-
-
-
-
--
-
-
-
13
5
+
=
18
1+8
=
9
=
9
-
4
5
+
=
9
-
=
9
=
9
2
M
E
-
-
-
-
-
--
-
-
-
13
5
+
=
18
1+8
=
9
=
9
-
4
5
+
=
9
-
=
9
=
9
2
M
E
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
5
-
-
5
occurs
x
1
=
5
2
M
E
-
-
9
-
-
2
-
9
-
4
5
-
-
-
-
-
-
-
-
2
M
E
-
-
9
-
-
2
-
9
-
4
5
-
-
-
-
-
-
-
-
2
M
E
-
-
9
-
-
2
-
9

 

 

-
3
E
G
O
-
-
-
-
-
--
-
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
-
15
+
=
15
1+5
=
6
=
6
-
3
E
G
O
-
-
-
-
-
--
-
-
-
-
5
7
-
+
=
12
1+2
=
3
=
3
-
3
E
G
O
-
-
-
-
-
--
-
-
-
-
5
7
15
+
=
27
2+7
=
9
=
9
-
-
5
7
6
+
=
18
1+8
=
9
=
9
-
3
E
G
O
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
2
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
3
-
-
-
-
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
6
-
-
6
occurs
x
1
=
6
-
-
-
7
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
27
3
E
G
O
-
-
18
-
-
3
-
18
2+7
-
-
-
-
-
-
1+8
-
-
-
-
1+8
9
3
E
G
O
-
-
9
-
-
3
-
9
-
-
5
7
6
-
-
-
-
-
-
-
-
9
3
E
G
O
-
-
9
-
-
3
-
9

 

 

3
E
G
O
-
-
-
-
-
--
-
-
-
-
-
6
+
=
6
-
=
6
=
6
-
-
-
15
+
=
15
1+5
=
6
=
6
3
E
G
O
-
-
-
-
-
--
-
-
-
5
7
-
+
=
12
1+2
=
3
=
3
3
E
G
O
-
-
-
-
-
--
-
-
-
5
7
15
+
=
27
2+7
=
9
=
9
-
5
7
6
+
=
18
1+8
=
9
=
9
3
E
G
O
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
-
-
6
-
-
6
occurs
x
1
=
6
-
-
7
-
-
-
7
occurs
x
1
=
7
3
E
G
O
-
-
18
-
-
3
-
18
-
-
-
-
-
-
1+8
-
-
-
-
1+8
3
E
G
O
-
-
9
-
-
3
-
9
-
5
7
6
-
-
-
-
-
-
-
-
3
E
G
O
-
-
9
-
-
3
-
9

 

 

11
4
O
G
R
E
-
-
-
-
--
-
-
-
-
6
-
-
-
+
=
6
-
=
6
=
6
-
-
15
-
-
-
+
=
15
1+5
=
6
=
6
11
4
O
G
R
E
-
-
-
-
-
--
-
-
-
-
-
7
9
5
+
=
21
2+1
=
3
=
3
-
-
-
7
18
5
+
=
30
3+0
=
3
=
3
11
4
O
G
R
E
-
-
-
-
-
--
-
-
-
-
15
7
18
5
+
=
45
4+5
=
9
=
9
-
-
6
7
9
5
+
=
27
2+7
=
9
=
9
11
4
O
G
R
E
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
--
-
-
-
-
-
3
-
-
-
-
-
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
18
4
O
G
R
E
-
-
27
-
-
4
-
27
1+8
-
-
-
9
-
-
-
2+7
-
-
-
-
2+7
9
4
O
G
R
E
-
-
9
-
-
4
-
9
-
-
6
7
9
5
-
-
-
-
-
-
-
-
9
4
O
G
R
E
-
-
9
-
-
4
-
9

 

 

4
O
G
R
E
-
-
-
-
--
-
-
-
6
-
-
-
+
=
6
-
=
6
=
6
-
15
-
-
-
+
=
15
1+5
=
6
=
6
4
O
G
R
E
-
-
-
-
-
--
-
-
-
-
7
9
5
+
=
21
2+1
=
3
=
3
-
-
7
18
5
+
=
30
3+0
=
3
=
3
4
O
G
R
E
-
-
-
-
-
--
-
-
-
15
7
18
5
+
=
45
4+5
=
9
=
9
-
6
7
9
5
+
=
27
2+7
=
9
=
9
4
O
G
R
E
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
-
7
-
-
-
-
7
occurs
x
1
=
7
-
-
-
9
-
-
-
9
occurs
x
1
=
9
4
O
G
R
E
-
-
27
-
-
4
-
27
-
-
-
9
-
-
-
2+7
-
-
-
-
2+7
4
O
G
R
E
-
-
9
-
-
4
-
9
-
6
7
9
5
-
-
-
-
-
-
-
-
4
O
G
R
E
-
-
9
-
-
4
-
9

 

 

--
EGO CENTRIC
-
-
-
3
EGO
27
18
9
7
CENTRIC
72
36
9
10
EGO CENTRIC
99
54
18
1+0
-
9+9
5+4
1+8
1
EGO CENTRIC
18
9
9
-
-
1+8
-
-
1
EGO CENTRIC
9
9
9

 

 

11
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
-
5
-
-
9
-
+
=
20
2+0
=
2
-
2
-
2
-
-
-
-
15
-
-
-
14
-
-
9
-
+
=
38
3+8
=
11
1+1
2
-
2
11
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
5
7
-
-
3
5
-
2
9
-
3
+
=
34
3+4
=
7
-
7
-
7
-
-
5
7
-
-
3
5
-
20
18
-
3
+
=
61
6+1
=
7
1+4
7
1+4
7
11
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
5
7
15
-
3
5
14
20
18
9
3
+
=
99
9+9
=
18
1+8
9
-
9
-
-
5
7
6
-
3
5
5
2
9
9
3
+
=
54
5+4
=
9
-
9
-
9
11
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
-
--
-
3
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
-
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
5
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
13
10
E
G
O
-
C
E
N
T
R
I
C
-
-
32
-
-
10
-
54
-
36
1+3
1+0
-
-
-
-
-
-
-
-
9
9
-
-
-
3+2
-
-
1+0
-
5+4
-
3+6
4
1
E
G
O
-
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9
-
-
5
7
6
-
3
5
5
2
9
9
3
-
-
-
-
-
-
-
-
-
-
4
1
E
G
O
-
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9

 

 

10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
5
-
-
9
-
+
=
20
2+0
=
2
-
2
-
2
-
-
-
15
-
-
-
14
-
-
9
-
+
=
38
3+8
=
11
1+1
2
-
2
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
5
7
-
-
3
5
-
2
9
-
3
+
=
34
3+4
=
7
-
7
-
7
-
5
7
-
-
3
5
-
20
18
-
3
+
=
61
6+1
=
7
1+4
7
1+4
7
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
--
-
-
-
-
-
5
7
15
-
3
5
14
20
18
9
3
+
=
99
9+9
=
18
1+8
9
-
9
-
5
7
6
-
3
5
5
2
9
9
3
+
=
54
5+4
=
9
-
9
-
9
10
E
G
O
-
C
E
N
T
R
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
--
-
3
-
-
-
-
-
3
-
-
3
occurs
x
2
=
6
-
6
-
5
-
-
-
-
5
5
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
2
=
18
1+8
9
10
E
G
O
-
C
E
N
T
R
I
C
-
-
32
-
-
10
-
54
-
36
1+0
-
-
-
-
-
-
-
-
9
9
-
-
-
3+2
-
-
1+0
-
5+4
-
3+6
1
E
G
O
-
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9
-
5
7
6
-
3
5
5
2
9
9
3
-
-
-
-
-
-
-
-
-
-
1
E
G
O
-
C
E
N
T
R
I
C
-
-
5
-
-
1
-
9
-
9

 

 

-
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
-
6
5
1
-
9
-
5
-
-
+
=
26
2+6
=
8
-
8
-
8
-
`-
-
15
14
19
-
9
-
14
-
-
+
=
71
7+1
=
8
-
8
-
8
-
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
3
-
-
-
3
-
5
-
3
5
+
=
19
1+9
=
10
1+0
1
-
1
-
-
3
-
-
-
3
-
5
-
3
5
+
=
19
1+9
=
10
1+0
1
-
1
-
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
`-
3
15
14
19
3
9
5
14
3
5
+
=
90
9+0
=
9
-
9
-
9
-
-
3
6
5
1
3
9
5
5
3
5
+
=
45
4+5
=
9
-
9
-
9
-
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
-
-
-
3
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
5
5
5
-
-
5
occurs
x
4
=
20
=
2
-
-
-
6
-
-
-
-
-
3
-
2
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
9
-
-
9
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
21
10
C
O
N
S
C
I
E
N
C
E
-
-
24
-
-
10
-
45
-
27
2+1
1+0
-
-
5
-
-
-
5
5
5
5
-
-
2+4
-
-
1+0
-
4+5
-
2+7
3
1
C
O
N
S
C
I
E
N
C
E
-
-
6
-
-
1
-
9
-
9
-
-
3
6
5
1
3
9
5
5
3
5
-
-
-
-
-
-
-
-
-
-
3
3
C
O
N
S
C
I
E
N
C
E
-
-
6
-
-
1
-
9
-
9

 

 

10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
6
5
1
-
9
-
5
-
-
+
=
26
2+6
=
8
-
8
-
8
`-
-
15
14
19
-
9
-
14
-
-
+
=
71
7+1
=
8
-
8
-
8
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
3
-
-
-
3
-
5
-
3
5
+
=
19
1+9
=
10
1+0
1
-
1
-
3
-
-
-
3
-
5
-
3
5
+
=
19
1+9
=
10
1+0
1
-
1
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
`-
3
15
14
19
3
9
5
14
3
5
+
=
90
9+0
=
9
-
9
-
9
-
3
6
5
1
3
9
5
5
3
5
+
=
45
4+5
=
9
-
9
-
9
10
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
3
-
-
-
3
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
5
-
-
-
5
5
5
5
-
-
5
occurs
x
4
=
20
=
2
-
-
6
-
-
-
-
-
3
-
2
-
-
6
occurs
x
1
=
6
=
6
-
9
-
-
9
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
10
C
O
N
S
C
I
E
N
C
E
-
-
24
-
-
10
-
45
-
27
1+0
-
-
-
-
-
9
-
-
-
-
-
-
2+4
-
-
1+0
-
4+5
-
2+7
1
C
O
N
S
C
I
E
N
C
E
-
-
6
-
-
1
-
9
-
9
-
3
6
5
1
3
9
5
5
3
5
-
-
-
-
-
-
-
-
-
-
3
C
O
N
S
C
I
E
N
C
E
-
-
6
-
-
1
-
9
-
9

 

 

-
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
9
5
-
-
-
6
5
1
-
9
-
5
-
-
+
=
49
4+9
=
13
1+3
4
=
4
-
-
-
9
-
9
14
-
-
-
15
14
19
-
9
-
14
-
-
+
=
103
1+0+3
=
4
=
4
=
4
-
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
4
-
4
-
-
5
-
3
-
-
-
3
-
5
-
3
5
+
=
32
3+2
=
5
=
5
=
5
-
-
4
-
22
-
-
5
-
3
-
-
-
3
-
5
-
3
5
+
=
50
5+0
=
5
=
5
=
5
-
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
4
9
22
9
14
5
-
3
15
14
19
3
9
5
14
3
5
+
=
153
1+5+3
=
9
=
9
=
9
-
-
4
9
4
9
5
5
-
3
6
5
1
3
9
5
5
3
5
+
=
81
8+1
=
9
=
9
=
9
-
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
5
-
-
-
5
-
-
-
5
5
-
5
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
17
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
28
-
-
16
-
81
-
36
1+6
1+6
-
-
-
-
5
5
-
-
-
5
-
-
-
5
5
-
5
-
-
2+8
-
-
1+6
-
8+1
-
3+6
8
7
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
10
-
-
7
-
9
-
9
-
-
4
9
4
9
5
5
-
3
6
5
1
3
9
5
5
3
5
-
-
1+0
-
-
-
-
-
-
-
8
7
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
1
-
-
7
-
9
-
9

 

 

16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
-
9
-
9
5
-
-
-
6
5
1
-
9
-
5
-
-
+
=
49
4+9
=
13
1+3
4
=
4
-
-
9
-
9
14
-
-
-
15
14
19
-
9
-
14
-
-
+
=
103
1+0+3
=
4
=
4
=
4
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
4
-
4
-
-
5
-
3
-
-
-
3
-
5
-
3
5
+
=
32
3+2
=
5
=
5
=
5
-
4
-
22
-
-
5
-
3
-
-
-
3
-
5
-
3
5
+
=
50
5+0
=
5
=
5
=
5
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
--
-
-
-
-
-
4
9
22
9
14
5
-
3
15
14
19
3
9
5
14
3
5
+
=
153
1+5+3
=
9
=
9
=
9
-
4
9
4
9
5
5
-
3
6
5
1
3
9
5
5
3
5
+
=
81
8+1
=
9
=
9
=
9
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
3
-
-
-
3
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
5
-
-
-
5
-
-
-
5
5
-
5
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
9
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
16
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
28
-
-
16
-
81
-
36
1+6
-
-
-
-
5
5
-
-
-
5
-
-
-
5
5
-
5
-
-
2+8
-
-
1+6
-
8+1
-
3+6
7
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
10
-
-
7
-
9
-
9
-
4
9
4
9
5
5
-
3
6
5
1
3
9
5
5
3
5
-
-
1+0
-
-
-
-
-
-
-
7
D
I
V
I
N
E
-
C
O
N
S
C
I
E
N
C
E
-
-
1
-
-
7
-
9
-
9

 

 

-
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
6
5
1
8
9
6
-
1
5
-
1
1
+
=
35
3+5
=
8
=
8
=
8
-
--
-
15
14
19
-
9
15
-
19
14
-
19
19
+
=
143
1+4+3
=
8
=
8
=
8
-
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
3
-
-
-
3
-
-
3
-
-
5
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
3
-
-
-
3
-
-
21
-
-
5
-
-
+
=
32
3+2
=
5
=
5
=
5
-
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
3
15
14
19
3
9
15
21
19
14
5
19
19
+
=
175
1+7+5
=
13
1+3
4
=
4
-
-
3
6
5
1
3
9
6
3
1
5
5
1
1
+
=
49
4+9
=
13
1+3
4
=
4
-
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
1
-
-
1
occurs
x
4
=
4
=
4
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
-
-
-
-
-
-
-
-
3
-
-
-
3
-
-
3
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
5
-
-
-
-
5
occurs
x
3
=
15
=
6
-
-
-
6
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
=
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
21
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
24
-
-
13
-
49
-
-
2+1
1+3
-
-
-
1
-
-
-
-
1
-
-
1
1
-
-
2+1
-
-
1+3
-
4+9
-
-
3
4
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
3
-
-
4
-
13
-
-
-
--
3
6
5
1
3
9
6
3
1
5
5
1
1
-
-
-
-
-
--
-
-
-
-
3
4
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
3
-
-
4
-
13
-
-

 

 

13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
6
5
1
8
9
6
-
1
5
-
1
1
+
=
35
3+5
=
8
=
8
=
8
--
-
15
14
19
-
9
15
-
19
14
-
19
19
+
=
143
1+4+3
=
8
=
8
=
8
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
3
-
-
-
3
-
-
3
-
-
5
-
-
+
=
14
1+4
=
5
=
5
=
5
-
3
-
-
-
3
-
-
21
-
-
5
-
-
+
=
32
3+2
=
5
=
5
=
5
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
3
15
14
19
3
9
15
21
19
14
5
19
19
+
=
175
1+7+5
=
13
1+3
4
=
4
-
3
6
5
1
3
9
6
3
1
5
5
1
1
+
=
49
4+9
=
13
1+3
4
=
4
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
1
-
-
1
occurs
x
4
=
4
=
4
-
3
-
-
-
3
-
-
3
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
5
-
-
-
-
-
-
5
5
-
-
-
-
5
occurs
x
3
=
15
=
6
-
-
6
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
=
3
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
13
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
24
-
-
13
-
49
-
-
1+3
-
-
-
1
-
-
-
-
1
-
-
1
1
-
-
2+1
-
-
1+3
-
4+9
-
-
4
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
3
-
-
4
-
13
-
-
--
3
6
5
1
3
9
6
3
1
5
5
1
1
-
-
-
-
-
--
-
-
-
-
4
C
O
N
S
C
I
O
U
S
N
E
S
S
-
-
3
-
-
4
-
13
-
-

 

 

 

-
5
W
H
O
L
E
-
-
--
-
-
-
-
-
-
-
-
-
-
8
6
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
8
15
-
-
+
=
23
2+3
=
5
=
5
=
5
-
5
W
H
O
L
E
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
3
5
+
=
13
1+3
=
4
=
4
=
4
-
-
23
-
-
3
5
+
=
40
4+0
=
4
=
4
=
4
-
5
W
H
O
L
E
-
-
--
-
-
-
-
-
-
-
-
-
23
8
15
12
5
+
=
63
6+3
=
9
=
9
=
9
-
-
5
8
6
3
5
+
=
27
2+7
=
9
=
9
=
9
-
5
W
H
O
L
E
-
-
--
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
23
5
W
H
O
L
E
-
-
22
-
-
5
-
27
-
18
2+3
-
-
-
-
-
-
-
-
2+2
-
-
-
-
2+7
-
1+8
5
5
W
H
O
L
E
-
-
4
-
-
5
-
9
-
9
-
-
5
8
6
3
5
-
-
--
-
-
-
-
-
-
-
5
5
W
H
O
L
E
-
-
4
-
-
5
-
9
-
9

 

 

-
6
S
O
U
R
C
E
-
-
-
-
-
--
-
-
-
-
1
6
-
-
-
-
+
=
7
-
=
7
=
7
-
-
19
15
-
-
-
-
+
=
34
3+4
=
7
=
7
-
6
S
O
U
R
C
E
-
-
-
-
-
--
-
-
-
-
-
-
3
9
3
5
+
=
20
2+0
=
2
=
2
-
-
-
-
21
18
3
5
+
=
47
4+7
=
11
1+1
2
-
6
S
O
U
R
C
E
-
-
-
-
-
--
-
-
-
-
19
15
21
18
3
5
+
=
81
8+1
=
9
=
9
-
-
1
6
3
9
3
5
+
=
27
2+7
=
9
=
9
-
6
S
O
U
R
C
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
-
3
-
-
-
3
occurs
x
2
=
6
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
-
6
-
-
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
21
6
S
O
U
R
C
E
-
-
27
-
-
6
-
27
2+1
-
--
-
-
9
-
-
-
-
2+7
-
-
-
-
2+7
3
6
S
O
U
R
C
E
-
-
9
-
-
6
-
9
-
-
1
6
3
9
3
5
+
=
27
2+7
=
9
=
9
3
6
S
O
U
R
C
E
-
-
9
-
-
6
-
9

 

 

`-
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
6
5
-
5
-
1
1
+
=
18
1+8
=
9
=
9
-
9
-
-
15
14
-
14
-
19
19
+
=
81
8+1
=
9
=
9
-
9
-
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
5
-
-
+
=
10
1+0
=
1
=
1
-
1
-
7
O
N
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
15
14
5
14
5
19
19
+
=
91
9+1
=
10
1+0
1
-
1
-
-
6
5
5
5
5
1
1
+
=
28
2+8
=
10
1+0
1
-
1
-
7
O
N
E
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
5
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
33
7
O
N
E
N
E
S
S
-
-
12
-
-
7
-
28
-
10
3+3
-
-
5
5
5
5
-
-
-
-
1+2
-
-
-
-
2+8
-
1+0
6
7
O
N
E
N
E
S
S
-
-
3
-
-
7
-
10
-
1
-
-
6
5
5
5
5
1
1
-
-
-
-
-
--
-
1+0
-
-
6
7
O
N
E
N
E
S
S
-
-
3
-
-
7
-
1
-
1

 

 

-
10
C
O
M
P
L
E
T
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
-
-
-
5
-
1
1
+
=
13
1+3
=
4
=
4
=
4
-
-
-
15
-
-
-
-
-
-
14
-
19
19
+
=
67
6+7
=
13
1+3
4
=
4
-
10
C
O
M
P
L
E
T
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
3
-
4
7
3
5
2
5
-
5
-
-
+
=
34
3+4
=
7
=
7
=
7
-
-
3
-
13
16
12
5
20
5
-
5
-
-
+
=
79
7+9
=
16
1+6
7
=
7
-
10
C
O
M
P
L
E
T
E
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
3
15
13
16
12
5
20
5
14
5
19
19
+
=
146
1+4+6
=
11
1+1
2
=
2
-
-
3
6
4
7
3
5
2
5
5
5
1
1
+
=
47
4+7
=
11
1+1
2
=
2
-
10
C
O
M
P
L
E
T
E
N
E
S
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
occurs
x
4
=
20
2+0
2
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-`
6
occurs
x
1
=
6
=
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
17
10
C
O
M
P
L
E
T
E
N
E
S
S
-
-
28
-
1
12
-
47
-
29
1+7
1+0
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+8
-
-
1+2
-
4+7
-
2+9
8
1
C
O
M
P
L
E
T
E
N
E
S
S
-
-
10
-
-
3
-
11
-
11
-
-
3
6
4
7
3
5
2
5
5
5
1
1
-
-
1+0
-
-
-
-
1+1
-
1+1
8
1
C
O
M
P
L
E
T
E
N
E
S
S
-
-
1
-
-
3
-
2
-
2

 

 

-
-
-
-
CIRCLE
-
-
-
-
-
-
1
C
3
3
3
-
-
-
1
I
9
9
9
-
-
-
1
R
18
9
9
-
-
-
1
C
3
3
3
-
-
-
1
L
12
3
3
-
-
-
1
E
5
5
5
C
=
3
6
CIRCLE
50
32
32
-
-
-
-
-
5+0
3+2
3+2
C
=
3
6
CIRCLE
5
5
5

 

 

R
=
9
-
ROUND
-
-
-
-
-
-
1
R
18
9
9
-
-
-
1
O
15
6
6
-
-
-
1
U
21
12
3
-
-
-
1
N
14
5
5
-
-
-
1
D
4
4
4
R
=
9
5
ROUND
72
36
27
-
-
-
-
-
7+2
3+6
2+7
R
=
9
5
ROUND
9
9
9

 

 

R
=
9
-
ROUND
-
-
-
-
-
-
1
R
18
9
9
-
-
-
2
O+U
36
9
9
-
-
-
2
N+D
18
9
9
R
=
9
5
ROUND
72
36
27
-
-
-
-
-
7+2
3+6
2+7
R
=
9
5
ROUND
9
9
9

 

 

R
=
9
-
ROUNDED
-
-
-
-
-
-
1
R
18
9
9
-
-
-
1
O
15
6
6
-
-
-
1
U
21
12
3
-
-
-
1
N
14
5
5
-
-
-
1
D
4
4
4
-
-
-
1
E
5
5
5
-
-
-
1
D
4
4
4
R
=
9
7
ROUNDED
81
45
45
-
-
-
-
-
8+1
4+5
4+5
R
=
9
7
ROUNDED
9
9
9

 

 

R
=
9
-
ROUNDED
-
-
-
-
-
-
1
R
18
9
9
-
-
-
2
O+U
36
9
9
-
-
-
2
N+D
18
9
9
-
-
-
1
E+D
5
5
5
R
=
9
7
ROUNDED
81
45
45
-
-
-
-
-
8+1
4+5
4+5
R
=
9
7
ROUNDED
9
9
9

 

 

-
PHI
-
-
-
-
P
16
7
7
-
H
8
8
8
-
I
9
9
9
3
PHI
33
24
24
-
-
3+3
2+4
2+4
3
PHI
6
6
6

 

►Golden ratio - Wikipedia, the free encyclopedia Mark Barr (20th century) suggests the Greek letter phi (φ), the initial ...
en.wikipedia.org/wiki/Golden_ratio -

Show more results from wikipedia.orgPhi: 1.618. The best source to the golden section, golden mean ... The best information on the Golden Section, Divine Proportion, Fibonacci series and phi. Explore its application to art, design, life, beauty, mathematics, ...
www.goldennumber.net/ -

the golden ratio = 1.61803399
More about calculator.

Search ResultsPhi - Wikipedia, the free encyclopediaPhi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced /ˈfaɪ/ or sometimes /ˈfiː/ in English, and [ˈfi] in modern Greek, is the 21st letter of the ...
en.wikipedia.org/wiki/Phi -

PhiFrom Wikipedia, the free encyclopediaJump to: navigation, search
For other uses, see Phi (disambiguation).

Greek alphabet
Αα Alpha Νν Nu
Ββ Beta Ξξ Xi
Γγ Gamma Οο Omicron
Δδ Delta Ππ Pi
Εε Epsilon Ρρ Rho
Ζζ Zeta Σσς Sigma
Ηη Eta Ττ Tau
Θθ Theta Υυ Upsilon
Ιι Iota Φφ Phi
Κκ Kappa Χχ Chi
Λλ Lambda Ψψ Psi
Μμ Mu Ωω Omega
Other characters
Digamma Stigma
Heta San
Qoppa Sampi
Greek diacritics

Phi (uppercase Φ, lowercase φ or math symbol ϕ), pronounced /ˈfaɪ/ or sometimes /ˈfiː/ in English,[1] and [ˈfi] in modern Greek, is the 21st letter of the Greek alphabet. In modern Greek, it represents [f], a voiceless labiodental fricative. In Ancient Greek it represented [pʰ], an aspirated voiceless bilabial plosive (from which English ultimately inherits the spelling "ph" in words derived from Greek). In the system of Greek numerals it has a value of 500 (φʹ) or 500,000 (͵φ). The Cyrillic letter Ef (Ф, ф) arose from Φ.

Contents [hide]
1 Use as a symbol
2 Computing
3 See also
4 References

Use as a symbolThe lower-case letter (or often its variant, ) is often used to represent the following:

The golden ratio in mathematics, art, and architecture.
Euler's totient function φ(n) in number theory; also called Euler's phi function.
In algebra, group or ring homomorphisms
In probability theory, ϕ(x) = (2π)−½e−x2/2 is the probability density function of the normal distribution.
In probability theory, φX(t) = E[ eitX ] is the characteristic function of a random variable X.
An angle, typically the second angle mentioned, after θ (theta). Especially:
The argument of a complex number.
The phase of a wave in signal processing.
In spherical coordinates, mathematicians usually refer to phi as the polar angle (from the z-axis). The convention in physics is to use phi as the azimuthal angle (from the x-axis).
One of the dihedral angles in the backbones of proteins.
Internal or effective angle of friction.
Electric potential in physics.
The work function in electronics.
A shorthand representation for an aromatic functional group in organic chemistry.
The fugacity coefficient in thermodynamics.
The ratio of free energy destabilizations of protein mutants in phi value analysis.
In cartography and navigation, latitude.
In combustion engineering, equivalence ratio. The ratio between the actual fuel air ratio to the stoichiometric fuel air ratio.
A sentence in first-order logic.
Porosity in geology and hydrology.
Strength (or resistance) reduction factor in structural engineering, used to account for statistical variabilities in materials and construction methods.
The symbol for a voiceless bilabial fricative in the International Phonetic Alphabet (using the straight line variant character)
In economics, this is usually an additive term.[clarification needed]
In flight dynamics, the roll angle.
The upper-case letter Φ is used as a symbol for:

The golden ratio conjugate −0.618... in mathematics.
The magnetic flux and electric flux in physics, with subscripts distinguishing the two.
The cumulative distribution function of the normal distribution in mathematics and statistics.
Philosophy.[why?]
The number of phases in a power system in electrical engineering, for example 1Φ for single phase, 3Φ for three phase.
A common symbol for the parametrization of a surface in vector calculus.
In Lacanian algebra, Φ stands for the imaginary phallus and also represents phallic signification. -Φ stands in for castration.[2]
The diameter symbol in engineering, ⌀, is often incorrectly referred to as "phi". This symbol is used to indicate the diameter of a circular section, for example "⌀14", means the diameter of the circle is 14 units.

 

 

-
PHI
-
-
-
-
P
16
7
7
-
H
8
8
8
-
I
9
9
9
3
PHI
33
24
24
-
-
3+3
2+4
2+4
3
PHI
6
6
6

 

 

-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
7
-
-
+
=
7
-
=
7
=
7
-
`-
16
-
-
+
=
16
1+6
=
7
=
7
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
16
8
9
+
=
33
3+3
=
6
=
6
-
-
7
8
9
+
=
24
2+4
=
6
=
6
-
3
P
H
I
-T
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
-
4
FOUR
4
-
-
-
5
-
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
-
6
SIX
6
-
-
-
-
--
7
-
-
-
-
7
occurs
x
1
=
7
-
-
-
8
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
9
occurs
x
1
=
9
21
3
P
H
I
-
-
24
-
-
3
-
24
2+1
-
-
-
9
-
-
2+4
-
-
-
-
2+4
3
3
P
H
I
-
-
6
-
-
3
-
6
-
-
7
8
9
-T
-
-
-
-
-
-
-
3
3
P
H
I
-
-
6
-
-
3
-
6

 

 

-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
7
-
-
+
=
7
-
=
7
=
7
-
`-
16
-
-
+
=
16
1+6
=
7
=
7
-
3
P
H
1
-
-
-
-
-
-
-
-
-
-
16
8
9
+
=
33
3+3
=
6
=
6
-
-
7
8
9
+
=
24
2+4
=
6
=
6
-
3
P
H
I
-T
-
-
-
-
-
-
-
-
--
7
-
-
-
-
7
occurs
x
1
=
7
-
-
-
8
-
-
-
8
occurs
x
1
=
8
-
-
-
-
9
-
-
9
occurs
x
1
=
9
21
3
P
H
I
-
-
24
-
-
3
-
24
2+1
-
-
-
9
-
-
2+4
-
-
-
-
2+4
3
3
P
H
I
-
-
6
-
-
3
-
6
-
-
7
8
9
-T
-
-
-
-
-
-
-
3
3
P
H
I
-
-
6
-
-
3
-
6

 

 

-
PHI
-
-
-
-
P
16
7
7
-
H
8
8
8
-
I
9
9
9
3
PHI
33
24
24
-
-
3+3
2+4
2+4
3
PHI
6
6
6

 

 

Pi - Wikipedia, the free encyclopediaπ (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in the Euclidean plane; ...

Fundamentals - History - Open questions - Use in mathematics and science
en.wikipedia.org/wiki/Pi -

 

PiFrom Wikipedia, the free encyclopediaJump to: navigation, search
This article is about the number. For the Greek letter, see Pi (letter). For other uses, see Pi (disambiguation).

Mosaic depicting π at the entrance to the math building at Technische Universität BerlinPart of a series of articles on
the mathematical constant π

Uses
Area of disk · Circumference
Use in other formulae
Properties
Irrationality · Transcendence
Less than 22/7
Value
Approximations · Memorization
People
Archimedes · Zu Chongzhi
Madhava of Sangamagrama
William Jones · John Machin
John Wrench
History
Chronology · Book
In culture
Legislation · Holiday
Related topics
Squaring the circle · Basel problem
Other topics related to π
v · d · e

π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in the Euclidean plane; this is the same value as the ratio of a circle's area to the square of its radius. It is approximately equal to 3.14159265 in the usual decimal notation. Many formulae from mathematics, science, and engineering involve π, which makes it one of the most important mathematical constants.[1]

π is an irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers. Consequently, its decimal representation never ends or repeats. It is also a transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics. Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.

Probably because of the simplicity of its definition, the concept of π has become entrenched in popular culture to a degree far greater than almost any other mathematical construct.[2] It is, perhaps, the most common ground between mathematicians and non-mathematicians.[3] Reports on the latest, most-precise calculation of π (and related stunts) are common news items.[4][5][6] The current record for the decimal expansion of π, if verified, stands at 5 trillion digits.[7]

The Greek letter π, often spelled out pi in text, was first adopted for the number as an abbreviation of the Greek word for perimeter "περίμετρος" (or as an abbreviation for "perimeter/diameter") by William Jones in 1706. The constant is also known as Archimedes' Constant, after Archimedes of Syracuse who provided an approximation of the number, although this name for the constant is uncommon in modern English-speaking contexts.

Contents [hide]
1 Fundamentals
1.1 The letter π
1.2 Geometric definition
1.3 Irrationality and transcendence
1.4 Decimal representation
1.5 Estimating π
2 History
2.1 Antiquity
2.2 Second millennium AD
2.3 Computation in the computer age
2.4 Pi and continued fraction
2.5 Memorizing digits
3 Open questions
4 Use in mathematics and science
4.1 Geometry and trigonometry
4.2 Complex numbers and calculus
4.3 Physics
4.4 Probability and statistics
4.5 Geomorphology and chaos theory
5 In popular culture
6 See also
7 References
8 External links

FundamentalsThe letter πMain article: pi (letter)

Lower-case π is used to symbolize the constantThe name of the Greek letter π is pi.[8] The name pi is commonly used as an alternative to using the Greek letter. As a mathematical symbol, the Greek letter is not capitalized (Π) even at the beginning of a sentence, and instead the lower case (π) is used at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced "pie" in English, which is the conventional English pronunciation of the Greek letter. The constant is named "π" because "π" is the first letter of the Greek word περίμετρος (perimeter), probably referring to its use in the formula perimeter/diameter which is constant for all circles, the word "perimeter" being synonymous here with "circumference."[9] William Jones was the first to use the Greek letter in this way, in 1706,[10] and it was later popularized by Leonhard Euler in 1737.[11][12] William Jones wrote:

There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to ... 3.14159, &c. = π[13]

The capital letter pi (Π) has a completely different mathematical meaning; it is used for expressing products (notice that the word "product" begins with the letter "p" just like "perimeter/diameter" does). It can also refer to the osmotic pressure of a solution.

Geometric definitionIn Euclidean plane geometry, π is defined as the ratio of a circle's circumference C to its diameter d:[9]

The ratio C/d is constant, regardless of a circle's size. For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.

Alternatively π can be defined as the ratio of a circle's area A to the area of a square whose side is equal to the radius r of the circle:[9][14]

These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar, and the fact that the right-hand-sides of these two equations are equal to each other (i.e. the area of a disk is Cr/2). These two geometric definitions can be considered a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its analytic properties as a definition. A common choice is to define π as twice the smallest positive x for which the trigonometric function cos(x) equals zero.[15

(Diagrams omitted)

Irrationality and transcendenceMain article: Proof that π is irrational
π is an irrational number, meaning that it cannot be written as the ratio of two integers. π is also a transcendental number, meaning that there is no polynomial with rational coefficients for which π is a root.[16] An important consequence of the transcendence of π is the fact that it is not constructible. Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.[17] This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.[18][19]

Decimal representationSee also: Numerical approximations of π
The decimal representation of π truncated to 50 decimal places is:[20]

π = 3.14159265358979323846264338327950288419716939937510...
Various online web sites provide π to many more digits.[21] While the decimal representation of π has been computed to more than a trillion (1012) digits,[22] elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places. For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the Earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.[23][24]

Because π is an irrational number, its decimal representation does not repeat, and therefore does not terminate. This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating π's properties.[25] Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of the decimal representation of π, no simple base-10 pattern in the digits has ever been found.[26] Digits of the decimal representation of π are available on many web pages, and there is software for calculating the decimal representation of π to billions of digits on any personal computer.

Estimating πMain article: Numerical approximations of π

Number system Approximation of π
Binary 11.00100100001111110110...[27]
Octal 3.11037552421026430215...
Decimal 3.14159265358979323846264338327950288...
Hexadecimal 3.243F6A8885A308D31319...[28]
Rational approximations 3, 22⁄7, 333⁄106, 355⁄113, 103993/33102, ...[29]
(listed in order of increasing accuracy)

Continued fraction [3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1...][30]
(This fraction is not periodic. Shown in linear notation)

3.1415926535 8979323846 2643383279 5028841971 6939937510
5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959

An estimate of π accurate to 1120 decimal digits was obtained using a gear-driven calculator in 1948, by John Wrench and Levi Smith. This was the most accurate estimate of π before electronic computers came into use.[31]

The earliest numerical approximation of π is almost certainly the value 3.[32] In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the perimeter of an inscribed regular hexagon to the diameter of the circle.

π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometry-based approach, attributed to Archimedes,[33] is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then compute the limit of a sequence as n increases to infinity:

This limit converges because the more sides the polygon has, the closer the ratio approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:[34]

π can also be calculated using purely mathematical methods. Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions.[16] Formulas for calculating π using elementary arithmetic typically include series or summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π.[35] The more terms included in a calculation, the closer to π the result will get.

Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in trigonometry and calculus. However, some are quite simple, such as this form of the Gregory-Leibniz series:[36]

 

 

-
PI
-
-
-
-
P
16
7
7
-
I
9
9
9
2
PI
25
16
16
-
-
2+5
1+6
1+6
2
PI
7
7
7

 

 

-
2
P
1
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+7
=
8
=
8
-
-
-
9
+
=
9
1+7
=
8
=
8
-
2
P
1
-
-
-
-
-
-
-
-
-
-
7
-
+
=
7
-
=
7
=
7
-
-
16
-
+
=
16
1+6
=
7
=
7
-
2
P
1
-
-
-
-
-
-
-
-
-
-
16
9
+
=
25
2+5
=
7
=
7
-
-
7
9
+
=
16
1+6
=
7
=
7
-
2
P
I
-T
-
-
-
-
-
-
-
1
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
4
FOUR
4
-
-
-
5
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
29
2
P
I
-
-
16
-
-
2
-
16
2+9
-
-
9
-
-
2+4
-
-
-
-
1+6
11
2
P
I
-
-
6
-
-
2
-
7
1+1
-
7
9
-T
-
-
-
-
-
-
-
2
2
P
I
-
-
6
-
-
2
-
7

 

 

-
2
P
1
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+7
=
8
=
8
-
-
-
9
+
=
9
1+7
=
8
=
8
-
2
P
1
-
-
-
-
-
-
-
-
-
-
7
-
+
=
7
-
=
7
=
7
-
-
16
-
+
=
16
1+6
=
7
=
7
-
2
P
1
-
-
-
-
-
-
-
-
-
-
16
9
+
=
25
2+5
=
7
=
7
-
-
7
9
+
=
16
1+6
=
7
=
7
-
2
P
I
-T
-
-
-
-
-
-
-
1
-
-
-
-
-
1
ONE
1
-
-
-
2
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
4
FOUR
4
-
-
-
5
-
-
-
-
-
5
FIVE
5
-
-
-
6
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
29
2
P
I
-
-
16
-
-
2
-
16
2+9
-
-
9
-
-
2+4
-
-
-
-
1+6
11
2
P
I
-
-
6
-
-
2
-
7
1+1
-
7
9
-T
-
-
-
-
-
-
-
2
2
P
I
-
-
6
-
-
2
-
7

 

 

-
2
P
1
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
1+7
=
8
=
8
-
-
-
9
+
=
9
1+7
=
8
=
8
-
2
P
1
-
-
-
-
-
-
-
-
-
-
7
-
+
=
7
-
=
7
=
7
-
-
16
-
+
=
16
1+6
=
7
=
7
-
2
P
1
-
-
-
-
-
-
-
-
-
-
16
9
+
=
25
2+5
=
7
=
7
-
-
7
9
+
=
16
1+6
=
7
=
7
-
2
P
I
-T
-
-
-
-
-
-
-
-
-
7
-
-
-
7
occurs
x
1
=
7
-
-
-
9
-
-
9
occurs
x
1
=
9
29
2
P
I
-
-
16
-
-
2
-
16
2+9
-
-
9
-
-
2+4
-
-
-
-
1+6
11
2
P
I
-
-
6
-
-
2
-
7
1+1
-
7
9
-T
-
-
-
-
-
-
-
2
2
P
I
-
-
6
-
-
2
-
7

 

 

-
-
-
-
CIRCLE
-
-
-
-
-
-
1
C
3
3
3
-
-
-
1
I
9
9
9
-
-
-
1
R
18
9
9
-
-
-
1
C
3
3
3
-
-
-
1
L
12
3
3
-
-
-
1
E
5
5
5
C
=
3
6
CIRCLE
50
32
32
-
-
-
-
-
5+0
3+2
3+2
C
=
3
6
CIRCLE
5
5
5

 

 

-
-
-
-
CIRCLE
-
-
-
-
-
-
5
CIRCL
45
27
9
-
-
-
1
E
5
5
5
C
=
3
6
CIRCLE
50
32
32
-
-
-
-
-
5+0
3+2
3+2
C
=
3
6
CIRCLE
5
5
5

 

 

-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
+
=
9
-
=
9
=
9
=
9
-
-
-
9
-
-
-
-
+
=
9
-
=
9
=
9
=
9
-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
5
+
=
23
2+3
=
5
=
5
=
5
-
-
3
-
18
3
12
5
+
=
32
3+2
=
5
=
5
=
5
-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
12
5
+
=
50
5+0
=
5
=
5
=
5
-
-
3
9
9
3
3
5
+
=
32
3+2
=
5
=
5
=
5
-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
28
6
C
I
R
C
L
E
-
-
17
-
-
6
-
32
-
23
2+8
-
-
9
9
-
-
-
-
-
1+7
-
-
-
-
3+2
-
2+3
10
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5
1+0
-
3
9
9
3
3
5
-
-
-
-
-
-
-
-
-
-
1
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5

 

 

6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
+
=
9
-
=
9
=
9
=
9
-
-
9
-
-
-
-
+
=
9
-
=
9
=
9
=
9
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
5
+
=
23
2+3
=
5
=
5
=
5
-
3
-
18
3
12
5
+
=
32
3+2
=
5
=
5
=
5
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
12
5
+
=
50
5+0
=
5
=
5
=
5
-
3
9
9
3
3
5
+
=
32
3+2
=
5
=
5
=
5
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
9
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
6
C
I
R
C
L
E
-
-
17
-
-
6
-
32
-
23
-
-
9
9
-
-
-
-
-
1+7
-
-
-
-
3+2
-
2+3
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5
-
3
9
9
3
3
5
-
-
-
-
-
-
-
-
-
-
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5

 

 

-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
+
=
9
=
9
-
=
8
=
8
-
-
-
9
-
-
-
-
-
-
+
=
9
=
9
-
=
8
=
8
-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
+
=
31
3+1
4
=
=
4
=
4
-
-
3
-
18
3
21
12
1
18
+
=
76
7+6
13
1+3
=
4
=
4
-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
+
=
85
8+5
13
1+3
=
4
=
4
-
-
3
9
9
3
3
3
1
9
+
=
40
4+0
4
-
=
4
=
4
-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
-
-
3
3
3
-
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
-
-
-
-
-
1
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
9
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
32
8
C
I
R
C
U
L
A
R
-
-
13
-
-
8
-
40
-
13
3+2
-
3
-
-
3
3
3
-
-
-
-
3+0
-
-
-
-
4+0
-
3+0
5
8
C
I
R
C
U
L
A
R
-
-
3
-
-
8
-
4
-
3
-
-
3
9
9
3
3
3
6
9
-
-
-
-
-
-
-
-
-
-
5
8
C
I
R
C
U
L
A
R
-
-
3
-
-
8
-
4
-
3

 

 

-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
+
=
9
=
9
-
=
8
=
8
-
-
-
9
-
-
-
-
-
-
+
=
9
=
9
-
=
8
=
8
-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
+
=
31
3+1
4
=
=
4
=
4
-
-
3
-
18
3
21
12
1
18
+
=
76
7+6
13
1+3
=
4
=
4
-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
+
=
85
8+5
13
1+3
=
4
=
4
-
-
3
9
9
3
3
3
1
9
+
=
40
4+0
4
-
=
4
=
4
-
8
C
I
R
C
U
L
A
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
=
1
-
-
3
-
-
3
3
3
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
9
9
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
32
8
C
I
R
C
U
L
A
R
-
-
13
-
-
8
-
40
-
13
3+2
-
3
-
-
3
3
3
-
-
-
-
3+0
-
-
-
-
4+0
-
3+0
5
8
C
I
R
C
U
L
A
R
-
-
3
-
-
8
-
4
-
3
-
-
3
9
9
3
3
3
6
9
-
-
-
-
-
-
-
-
-
-
5
8
C
I
R
C
U
L
A
R
-
-
3
-
-
8
-
4
-
3

 

 

-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
6
-
9
6
5
+
=
35
3+5
8
-
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
15
-
9
15
14
+
=
62
6+2
8
-
=
8
=
8
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
-
4
-
2
-
-
-
+
=
37
3+7
10
1+0
=
1
=
1
-
-
3
-
18
3
21
12
1
18
-
13
-
20
-
-
-
+
=
109
1+0+9
10
1+0
=
1
=
1
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
-
13
15
20
9
15
14
+
=
171
1+7+1
9
-
=
9
=
9
-
-
3
9
9
3
3
3
1
9
-
4
6
2
9
6
5
+
=
72
7+2
9
-
=
9
=
9
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
3
3
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
1
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
15
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
30
-
-
14
-
72
-
27
1+5
1+4
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
3+0
-
-
1+4
-
7+2
-
2+7
6
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9
-
-
3
9
9
3
3
3
6
9
-
4
6
6
9
6
5
-
-
-
-
-
-
-
-
-
-
6
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9

 

 

14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
6
-
9
6
5
+
=
35
3+5
8
-
=
8
=
8
-
-
9
-
-
-
-
-
-
-
-
15
-
9
15
14
+
=
62
6+2
8
-
=
8
=
8
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
-
4
-
2
-
-
-
+
=
37
3+7
10
1+0
=
1
=
1
-
3
-
18
3
21
12
1
18
-
13
-
20
-
-
-
+
=
109
1+0+9
10
1+0
=
1
=
1
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
-
13
15
20
9
15
14
+
=
171
1+7+1
9
-
=
9
=
9
-
3
9
9
3
3
3
1
9
-
4
6
2
9
6
5
+
=
72
7+2
9
-
=
9
=
9
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
3
3
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
1
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
30
-
-
14
-
72
-
27
1+4
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
3+0
-
-
1+4
-
7+2
-
2+7
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9
-
3
9
9
3
3
3
6
9
-
4
6
6
9
6
5
-
-
-
-
-
-
-
-
-
-
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9

 

 

4
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
6
-
9
6
5
+
=
35
3+5
8
-
=
8
=
8
-
-
9
-
-
-
-
-
-
-
15
-
9
15
14
+
=
62
6+2
8
-
=
8
=
8
14
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
4
-
2
-
-
-
+
=
37
3+7
10
1+0
=
1
=
1
-
3
-
18
3
21
12
1
18
13
-
20
-
-
-
+
=
109
1+0+9
10
1+0
=
1
=
1
14
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
13
15
20
9
15
14
+
=
171
1+7+1
9
-
=
9
=
9
-
3
9
9
3
3
3
1
9
4
6
2
9
6
5
+
=
72
7+2
9
-
=
9
=
9
14
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
3
3
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
1
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
9
9
-
-
-
-
9
-
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
14
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
30
-
-
14
-
72
-
27
1+4
-
9
9
-
-
-
-
9
-
-
-
9
-
-
-
-
3+0
-
-
1+4
-
7+2
-
2+7
5
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9
-
3
9
9
3
3
3
6
9
4
6
6
9
6
5
-
-
-
-
-
-
-
-
-
-
5
C
I
R
C
U
L
A
R
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9

 

 

T
=
2
-
3
THE
33
15
6
C
=
3
-
5
CYCLE
48
21
3
O
=
6
-
2
OF
21
12
3
T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CIRCLE
50
32
5
-
-
16
-
19
First Total
185
95
23
-
-
1+6
-
1+9
Add to Reduce
1+8+5
9+5
2+3
-
-
7
-
10
Second Total
14
14
5
-
-
-
-
1+0
Reduce to Deduce
1+4
1+4
-
-
-
7
-
1
Essence of Number
5
5
5

 

 

-
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
9
-
-
-
-
+
=
31
3+1
=
4
-
4
=
4
-
-
-
8
-
-
-
-
-
-
-
-
15
-
-
2
-
-
-
-
9
2
-
-
-
+
=
40
4+0
=
4
=
4
=
4
-
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
3
7
3
3
5
-
-
6
-
2
-
5
-
3
-
9
3
3
5
+
=
64
6+4
=
10
1+0
1
=
1
-
-
20
-
5
-
3
25
3
12
5
-
-
6
-
20
-
5
-
3
-
18
3
12
5
+
=
145
1+4+5
=
10
1+0
1
=
1
-
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
3
25
3
12
5
-
15
6
-
20
8
5
-
3
9
18
3
12
5
+
=
185
1+8+5
=
14
=
5
=
5
-
-
2
8
5
-
3
7
3
3
5
-
6
6
-
2
8
5
-
3
9
9
3
3
5
+
=
95
9+5
=
14
1+4
5
=
5
-
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
-
3
3
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
2
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
3
=
18
1+8
9
5
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
40
-
-
19
-
95
-
41
-
1+9
-
-
-
-
3
-
3
3
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
4+0
-
-
1+9
-
9+5
-
4+1
5
10
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
4
-
-
10
-
14
-
5
-
1+0
2
8
5
-
3
7
3
3
5
-
6
6
-
2
8
5
-
3
9
9
3
3
5
-
-
-
-
-
1+1
-
1+4
-
-
5
1
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
4
-
-
1
-
5
-
5

 

 

19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
9
-
-
-
-
+
=
31
3+1
=
4
-
4
=
4
-
-
8
-
-
-
-
-
-
-
-
15
-
-
2
-
-
-
-
9
2
-
-
-
+
=
40
4+0
=
4
=
4
=
4
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
3
7
3
3
5
-
-
6
-
2
-
5
-
3
-
9
3
3
5
+
=
64
6+4
=
10
1+0
1
=
1
-
20
-
5
-
3
25
3
12
5
-
-
6
-
20
-
5
-
3
-
18
3
12
5
+
=
145
1+4+5
=
10
1+0
1
=
1
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
3
25
3
12
5
-
15
6
-
20
8
5
-
3
9
18
3
12
5
+
=
185
1+8+5
=
14
=
5
=
5
-
2
8
5
-
3
7
3
3
5
-
6
6
-
2
8
5
-
3
9
9
3
3
5
+
=
95
9+5
=
14
1+4
5
=
5
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
3
-
3
3
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
5
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
3
=
18
1+8
9
19
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
40
-
-
19
-
95
-
41
1+9
-
-
-
-
3
-
3
3
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
4+0
-
-
1+9
-
9+5
-
4+1
10
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
4
-
-
10
-
14
-
5
1+0
2
8
5
-
3
7
3
3
5
-
6
6
-
2
8
5
-
3
9
9
3
3
5
-
-
-
-
-
1+1
-
1+4
-
-
1
T
H
E
-
C
Y
C
L
E
-
O
F
-
T
H
E
-
C
I
R
C
L
E
-
-
4
-
-
1
-
5
-
5

 

 

19
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
2
8
-
2
-
-
-
2
6
-
-
2
-
-
9
-
-
2
-
+
=
31
3+1
=
4
-
4
=
4
-
-
8
-
-
-
-
-
-
15
-
-
-
-
-
9
-
-
-
-
+
=
40
4+0
=
4
=
4
=
4
19
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
2
-
5
3
7
3
3
5
-
6
2
-
5
3
-
9
3
3
5
+
=
64
6+4
=
10
1+0
1
=
1
-
20
-
5
3
25
3
12
5
-
6
20
-
5
3
-
18
3
12
5
+
=
145
1+4+5
=
10
1+0
1
=
1
19
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
20
8
5
3
25
3
12
5
15
6
20
8
5
3
9
18
3
12
5
+
=
185
1+8+5
=
14
=
5
=
5
-
2
8
5
3
7
3
3
5
6
6
2
8
5
3
9
9
3
3
5
+
=
95
9+5
=
14
1+4
5
=
5
19
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
3
-
3
3
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
-
-
-
5
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
2
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
3
=
18
1+8
9
19
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
40
-
-
19
-
95
-
41
1+9
-
-
-
3
-
3
3
-
-
-
-
-
-
3
-
-
3
3
-
-
-
4+0
-
-
1+9
-
9+5
-
4+1
10
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
4
-
-
10
-
14
-
5
1+0
2
8
5
3
7
3
3
5
6
6
2
8
5
3
9
9
3
3
5
-
-
-
-
-
1+1
-
1+4
-
-
1
T
H
E
C
Y
C
L
E
O
F
T
H
E
C
I
R
C
L
E
-
-
4
-
-
1
-
5
-
5

 

 

T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CIRCLE
50
32
5
O
=
6
-
2
OF
21
12
3
T
=
2
-
3
THE
33
15
6
C
=
3
-
5
CYCLE
48
21
3
-
-
16
-
19
First Total
185
95
23
-
-
1+6
-
1+9
Add to Reduce
1+8+5
9+5
2+3
-
-
7
-
10
Second Total
14
14
5
-
-
-
-
1+0
Reduce to Deduce
1+4
1+4
-
-
-
7
-
1
Essence of Number
5
5
5

 

 

-
-
-
-
CIRCLE
-
-
-
-
-
-
1
C
3
3
3
-
-
-
1
I
9
9
9
-
-
-
1
R
18
9
9
-
-
-
1
C
3
3
3
-
-
-
1
L
12
3
3
-
-
-
1
E
5
5
5
C
=
3
6
CIRCLE
50
32
32
-
-
-
-
-
5+0
3+2
3+2
C
=
3
6
CIRCLE
5
5
5

 

 

T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CIRCLE
50
32
5
I
=
9
-
2
IS
28
10
1
P
=
7
-
7
PERFECT
73
37
1
I
=
9
-
2
IS
28
10
1
P
=
7
-
7
PERFECT
73
37
1
I
=
9
-
2
IS
28
10
1
T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CIRCLE
50
32
5
-
-
51
-
38
First Total
396
198
27
-
-
5+1
-
3+8
Add to Reduce
3+9+6
1+9+8
2+7
-
-
6
-
11
Second Total
18
18
9
-
-
-
-
1+1
Reduce to Deduce
1+8
1+8
-
-
-
6
-
2
Essence of Number
9
9
9

 

 

-
-
-
-
CIRCLING
-
-
-
-
-
-
1
C
3
3
3
-
-
-
1
I
9
9
9
-
-
-
1
R
18
9
9
-
-
-
1
C
3
3
3
-
-
-
1
L
12
3
3
-
-
-
1
I
9
9
9
-
-
-
1
N
14
5
5
-
-
-
1
G
7
7
7
C
=
3
6
CIRCLING
75
48
48
-
-
-
-
-
7+5
4+8
4+8
C
=
3
6
CIRCLING
12
12
12
-
-
-
-
-
1+2
1+2
1+2
C
=
3
6
CIRCLING
3
3
3

 

 

-
-
-
-
CIRCLING
-
-
-
-
-
-
1
C
3
3
3
-
-
-
1
I
9
9
9
-
-
-
1
R
18
9
9
-
-
-
1
C
3
3
3
-
-
-
1
L
12
3
3
-
-
-
1
I
9
9
9
-
-
-
2
N+G
21
12
3
C
=
3
8
CIRCLING
75
48
39
-
-
-
-
-
7+5
4+8
3+9
C
=
3
8
CIRCLING
12
12
12
-
-
-
-
-
1+2
1+2
1+2
C
=
3
8
CIRCLING
3
3
3

 

 

-
-
-
-
Q
CONCENTRIC
7
7
7
-
1
2
3
4
5
6
7
8
9
-
-
-
-
Q
-
Q
Q
Q
-
-
-
-
-
-
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
-
1
R
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
C
=
3
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
Q
-
Q
Q
Q
-
-
-
-
-
-
-
-
-
-
-
-
50
4
10
CONCENTRIC
104
50
50
-
1
2
9
4
15
6
7
8
18
-
-
5+0
Q
1+0
Q
1+0+4
5+0
5+0
-
-
-
-
-
1+5
-
-
-
1+8
Q
-
5
-
1
CONCENTRIC
5
5
5
-
1
2
9
4
6
6
7
8
9

 

 

-
-
-
-
Q
CONCENTRIC
7
7
7
-
2
3
5
6
9
-
-
-
-
Q
-
Q
Q
Q
-
-
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
3
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
6
-
N
=
5
-
1
N
14
5
5
-
-
-
5
-
-
C
=
3
-
1
C
3
3
3
-
-
3
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
5
-
-
N
=
5
-
1
N
14
5
5
-
-
-
5
-
-
T
=
2
-
1
T
20
2
2
-
2
-
-
-
-
R
=
9
-
1
R
9
9
9
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
9
C
=
3
-
1
C
3
3
3
-
-
3
-
-
-
-
-
-
-
Q
-
Q
Q
Q
-
-
-
-
-
-
-
-
50
4
10
CONCENTRIC
104
50
50
-
2
9
15
6
18
-
-
5+0
Q
1+0
Q
1+0+4
5+0
5+0
-
-
-
1+5
-
1+8
Q
-
5
-
1
CONCENTRIC
5
5
5
-
2
9
6
6
9

 

 

-
-
-
-
Q
CONCENTRIC
7
7
7
-
2
3
5
6
9
-
-
-
-
Q
-
Q
Q
Q
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
2
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
3
-
-
-
C
=
3
-
1
C
3
3
3
-
-
3
-
-
-
C
=
3
-
1
C
3
3
3
-
-
3
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
5
-
-
E
=
5
-
1
E
5
5
5
-
-
-
5
-
-
N
=
5
-
1
N
14
5
5
-
-
-
5
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
6
-
R
=
9
-
1
R
9
9
9
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
9
-
-
-
-
Q
-
Q
Q
Q
-
-
-
-
-
-
-
-
50
4
10
CONCENTRIC
104
50
50
-
2
9
15
6
18
-
-
5+0
Q
1+0
Q
1+0+4
5+0
5+0
-
-
-
1+5
-
1+8
Q
-
5
-
1
CONCENTRIC
5
5
5
-
2
9
6
6
9

 

 

T
=
2
3
THE
33
15
6
H
=
8
2
HE
13
13
4
A
=
1
2
AS
20
2
2
I
=
9
2
IN
23
14
5
S
=
1
3
SHE
32
14
5
T
=
2
4
THAT
49
13
4
I
=
9
2
IS
28
10
1
M
=
4
2
ME
18
9
9
-
-
36
20
-
216
90
36
-
=
3+6
2+0
-
2+1+6
9+0
3+6
-
-
9
2
=
9
9
9

 

 

-
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
8
-
-
-
1
-
9
5
-
1
8
-
-
-
8
-
-
-
9
1
-
-
-
+
=
58
5+8
=
13
1+3
4
=
4
-
-
-
8
-
-
8
-
-
-
19
-
9
14
-
19
8
-
-
-
8
-
-
-
9
19
-
-
-
+
=
121
1+2+1
=
4
-
4
=
4
-
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
-
5
-
1
-
-
-
-
-
-
-
5
-
2
-
1
2
-
-
-
-
4
5
+
=
32
3+2
=
5
-
5
-
5
-
-`
20
-
5
-
-
5
-
1
-
-
-
-
-
-
-
5
-
20
-
1
20
-
-
-
-
13
5
+
=
95
9+5
=
14
1+4
5
=
5
-
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
8
5
-
1
19
-
9
14
-
19
8
5
-
20
8
1
20
-
9
19
-
13
5
+
=
216
2+1+6
=
9
-
9
=
9
-
-
2
8
5
-
8
5
-
1
1
-
9
5
-
1
8
5
-
2
8
1
2
-
9
1
-
4
5
+
=
90
9+0
=
9
-
9
=
9
-
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
1
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
5
=
5
-
5
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
3
=
6
-
6
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
5
-
-
5
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
--
--
-
8
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
4
=
32
3+2
5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
16
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
29
-
-
20
-
90
-
36
1+6
2+0
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
2+9
-
-
2+0
-
9+0
-
3+6
7
2
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
11
-
-
2
-
9
-
9
-
-
2
8
5
-
8
5
-
1
1
-
9
5
-
1
8
5
-
2
8
1
2
-
9
1
-
6
5
-
-
1+1
-
-
-
-
--
=
-
7
2
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
11
-
-
2
-
9
-
9

 

 

20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
8
-
-
-
1
-
9
5
-
1
8
-
-
-
8
-
-
-
9
1
-
-
-
+
=
58
5+8
=
13
1+3
4
=
4
-
-
8
-
-
8
-
-
-
19
-
9
14
-
19
8
-
-
-
8
-
-
-
9
19
-
-
-
+
=
121
1+2+1
=
4
-
4
=
4
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
-
5
-
1
-
-
-
-
-
-
-
5
-
2
-
1
2
-
-
-
-
4
5
+
=
32
3+2
=
5
-
5
-
5
-`
20
-
5
-
-
5
-
1
-
-
-
-
-
-
-
5
-
20
-
1
20
-
-
-
-
13
5
+
=
95
9+5
=
14
1+4
5
=
5
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
8
5
-
1
19
-
9
14
-
19
8
5
-
20
8
1
20
-
9
19
-
13
5
+
=
216
2+1+6
=
9
-
9
=
9
-
2
8
5
-
8
5
-
1
1
-
9
5
-
1
8
5
-
2
8
1
2
-
9
1
-
4
5
+
=
90
9+0
=
9
-
9
=
9
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
1
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
5
=
5
-
5
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
3
=
6
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
5
-
-
5
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
5
=
25
2+5
7
-
-
8
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
4
=
32
3+2
5
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
20
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
29
-
-
20
-
90
-
36
2+0
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
2+9
-
-
2+0
-
9+0
-
3+6
2
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
11
-
-
2
-
9
-
9
-
2
8
5
-
8
5
-
1
1
-
9
5
-
1
8
5
-
2
8
1
2
-
9
1
-
6
5
-
-
1+1
-
-
-
-
--
=
-
2
T
H
E
-
H
E
-
A
S
-
I
N
-
S
H
E
-
T
H
A
T
-
I
S
-
M
E
-
-
11
-
-
2
-
9
-
9

 
 
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