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Evokation
 
 
Index
 

 

 

 

 

NUCLEAR FAMILY 19769

 

 

 

 

 

 

THE

MAGICALALPHABET

 

..................

 

 

-
-
-
-
-
THE RAINBOW LIGHT
-
-
-
T
=
2
-
3
THE
33
15
6
R
=
9
-
7
RAINBOW
82
37
1
L
=
3
-
5
LIGHT
56
29
2
-
-
14
-
15
THE RAINBOW LIGHT
171
81
9
-
-
1+4
-
1+5
-
1+7+1
8+1
-
Q
-
5
-
6
THE RAINBOW LIGHT
9
9
9

 

 

 

 

 

 

 

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 DIVINE COMEDY

OF

DANTE ALIGHIERI (1265-1321)

THE FLORENTINE

CANTICA I

HELL

(L'INFERNO)

INTRODUCTION

Page 9

"Midway this way of life we're bound upon

I woke to find myself in a dark wood,

Where the right road was wholly lost and gone."

 

 

M
=
4
-
6
MIDWAY
75
30
3
T
=
2
-
4
THIS
56
20
2
W
=
5
-
3
WAY
49
13
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
4
LIFE
32
23
5
W
=
5
-
4
WE'RE
51
24
6
B
=
2
-
5
BOUND
56
20
2
U
=
3
-
4
UPON
66
21
3
-
-
30
-
32
-
406
163
28
-
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
W
=
5
-
4
WOKE
54
18
9
T
=
2
-
2
TO
35
8
8
F
=
6
-
4
FIND
33
24
6
M
=
4
-
6
MYSELF
80
26
8
I
=
9
-
2
IN
23
14
5
A
=
1
-
1
A
1
1
1
D
=
4
-
4
DARK
34
16
7
W
=
5
-
4
WOOD
57
21
3
-
-
45
-
28
-
326
137
56
-
-
-
-
-
-
-
-
-
W
=
5
-
5
WHERE
59
32
5
T
=
2
-
3
THE
33
15
6
R
=
9
-
5
RIGHT
62
35
8
R
=
9
-
4
ROAD
38
20
2
W
=
5
-
3
WAS
43
7
7
W
=
5
-
6
WHOLLY
95
32
5
L
=
3
-
4
LOST
66
12
3
A
=
1
-
3
AND
19
10
1
G
=
7
-
4
GONE
41
23
5
-
-
46
-
37
-
456
186
42
-
-
-
-
-
-
-
-
-
-
-
121
-
97
First Total
1188
486
126
-
-
1+2+1
-
9+7
Add to Reduce
1+1+8+8
4+8+6
1+2+6
Q
-
4
-
16
Second Total
18
18
9
-
-
-
-
1+6
Reduce to Deduce
1+8
1+8
-
-
-
4
-
7
Essence of Number
9
9
9

 

 

THE DIVINE COMEDY

OF

DANTE ALIGHIERI (1265-1321)

THE FLORENTINE

CANTICA I

HELL

(L'INFERNO)

INTRODUCTION

Page 9

"Power failed high fantasy here; yet, swift to move

Even as a wheel moves equal, free from jars,

Already my heart and will were wheeled by love,

The Love that moves the sun and other stars."

 

P
=
7
-
5
POWER
77
32
5
F
=
6
-
6
FAILED
37
28
1
H
=
8
-
4
HIGH
32
32
5
F
=
6
-
7
FANTASY
86
23
5
H
=
8
-
4
HERE
36
27
9
Y
=
7
-
3
YET
50
14
5
S
=
1
-
5
SWIFT
77
23
5
T
=
2
-
2
TO
35
8
8
M
=
4
-
4
MOVE
55
19
1
-
-
49
-
40
First Total
485
206
44
-
-
4+9
-
4+0
Add to Reduce
4+8+5
2+0+6
4+4
Q
-
13
-
4
Second Total
17
8
8
-
-
1+3
-
-
Reduce to Deduce
1+7
-
-
-
-
4
-
4
Essence of Number
8
8
8

 

 

E
=
5
-
4
EVEN
46
19
1
A
=
1
-
2
AS
20
2
2
A
=
1
-
1
A
1
1
1
W
=
5
-
5
WHEEL
53
26
8
M
=
4
-
5
MOVES
74
20
2
E
=
5
-
5
EQUAL
56
20
2
F
=
6
-
4
FREE
34
25
7
F
=
6
-
4
FROM
52
25
7
J
=
1
-
4
JARS
48
12
3
-
-
34
-
34
First Total
384
150
33
-
-
3+4
-
3+4
Add to Reduce
3+8+4
1+5+0
3+3
Q
-
7
-
7
Second Total
15
6
6
-
-
-
-
-
Reduce to Deduce
1+5
-
-
-
-
7
-
7
Essence of Number
6
6
6

 

 

A
=
1
-
7
ALREADY
66
30
3
M
=
4
-
2
MY
38
11
2
H
=
8
-
5
HEART
52
25
7
A
=
1
-
3
AND
19
10
1
W
=
5
-
4
WILL
56
20
2
W
=
5
-
4
WERE
51
24
7
W
=
5
-
7
WHEELED
62
35
8
B
=
2
-
2
BY
27
9
9
L
=
3
-
4
LOVE
54
18
9
-
-
34
-
38
First Total
425
182
47
-
-
3+4
-
3+8
Add to Reduce
4+2+5
1+8+2
4+7
Q
-
7
-
11
Second Total
11
11
11
-
-
-
-
1+1
Reduce to Deduce
1+1
1+1
1+1
-
-
7
-
2
Essence of Number
2
2
2

 

 

T
=
2
-
3
THE
33
15
6
L
=
3
-
4
LOVE
54
18
9
T
=
2
-
4
THAT
49
13
4
M
=
4
-
5
MOVES
74
20
2
T
=
2
-
3
THE
33
15
6
S
=
1
-
3
SUN
54
9
9
A
=
1
-
3
AND
19
10
1
O
=
6
-
5
OTHER
66
30
3
S
=
1
-
5
STARS
77
14
5
-
-
22
-
35
First Total
459
144
45
-
-
2+2
-
3+5
Add to Reduce
4+5+9
1+4+4
4+5
Q
-
4
-
8
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
-
8
Essence of Number
9
9
9

 

 

3
THE
33
15
6
4
HOLY
60
24
6
7
MESSAGE
-
-
-
2
M+E
18
9
9
1
S
19
10
1
3
S+A+G
27
18
9
1
E
5
5
5
7
MESSAGE
69
42
24
14
Add
162
81
36
1+4
Reduce
1+6+2
8+1
3+6
5
Deduce
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 NUMBER WHEN IN CONJUNCTION SET

THE

FAR YONDER SCRIBE

MADE RECORD OF THEIR FALL

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
ADDED
18
18
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
2
TO
35
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
3
ALL
25
7
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
5
MINUS
76
22
4
-
-
-
-
4
-
-
-
-
-
N
=
5
-
4
NONE
48
21
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
6
SHARED
55
28
1
-
1
-
-
-
-
-
-
-
-
B
=
2
-
2
BY
27
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
10
EVERYTHING
133
61
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
10
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
58
First Total
626
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+8
Add to Reduce
6+2+6
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
13
Second Total
14
14
10
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+3
Reduce to Deduce
1+4
1+4
1+0
-
-
-
-
-
-
-
-
-
-
-
-
8
-
4
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
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

 

 

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

 

15
ANUBIS A NUMBER IS
-
-
-
1
ANUBIS
66
30
3
-
A
1
1
1
1
NUMBER
73
28
1
2
IS
28
19
1
15
ANUBIS A NUMBER IS
168
78
6
1+5
1+6+8
7+8
-
6
ANUBIS A NUMBER IS
15
15
6
-
1+5
1+5
-
6
ANUBIS A NUMBER IS
6
6
6

 

 

12
INTELLIGENCE
-
-
-
-
I
9
9
9
-
N+T+E+L+L
63
18
9
-
I
9
9
9
-
G
7
7
7
-
E+N+C+E
27
18
9
12
INTELLIGENCE
115
61
43
1+2
-
1+1+5
6+1
4+3
3
INTELLIGENCE
7
7
7

 

 

9
LANGUAGES
87
33
6
3
AND
19
10
1
7
NUMBERS
92
29
2
19
-
198
72
9
1+9
-
1+9+8
7+2
-
10
-
18
9
9
1+0
-
1+8
-
-
1
-
9
9
9

 

 

-
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
5
-
-
-
-
-
1
-
-
5
-
-
5
-
-
-
-
-
1
+
=
17
1+7
=
8
=
8
=
8
-
-
-
-
14
-
-
-
-
-
19
-
-
14
-
-
14
-
-
-
-
-
19
+
=
80
8+0
=
8
=
8
=
8
-
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
--
-
-
-
-
-
-
3
1
-
7
3
1
7
5
-
-
1
-
4
-
-
3
4
2
5
9
-
+
=
55
5+5
=
10
1+0
1
=
1
-
-
12
1
-
7
21
1
7
5
-
-
1
-
4
-
-
21
13
2
5
18
-
+
=
118
1+1+8
=
10
1+0
1
=
1
-
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
--
-
-
-
-
-
-
12
1
14
7
21
1
7
5
19
-
1
14
4
-
14
21
13
2
5
18
19
+
=
198
1+9+8
=
18
1+8
9
=
9
-
-
3
1
5
7
3
1
7
5
1
-
1
5
4
-
5
3
4
2
5
9
1
+
=
72
7+2
=
9
=
9
=
9
-
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
-
1
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
5
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
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
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
7
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
14
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
31
-
-
19
-
72
-
45
1+4
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
3+1
-
-
1+9
-
7+2
-
4+5
5
10
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
4
-
-
10
-
9
-
9
-
1+0
3
1
5
7
3
1
7
5
1
-
1
5
4
-
5
3
4
2
5
9
1
-
-
-
-
-
1+0
-
-
-
-
5
1
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
4
-
-
1
-
9
-
9

 

 

19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
-
-
-
1
-
-
5
-
-
5
-
-
-
-
-
1
+
=
17
1+7
=
8
=
8
=
8
-
-
-
14
-
-
-
-
-
19
-
-
14
-
-
14
-
-
-
-
-
19
+
=
80
8+0
=
8
=
8
=
8
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
--
-
-
-
-
-
3
1
-
7
3
1
7
5
-
-
1
-
4
-
-
3
4
2
5
9
-
+
=
55
5+5
=
10
1+0
1
=
1
-
12
1
-
7
21
1
7
5
-
-
1
-
4
-
-
21
13
2
5
18
-
+
=
118
1+1+8
=
10
1+0
1
=
1
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
--
-
-
-
-
-
12
1
14
7
21
1
7
5
19
-
1
14
4
-
14
21
13
2
5
18
19
+
=
198
1+9+8
=
18
1+8
9
=
9
-
3
1
5
7
3
1
7
5
1
-
1
5
4
-
5
3
4
2
5
9
1
+
=
72
7+2
=
9
=
9
=
9
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
-
1
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
5
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
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
occurs
x
5
=
25
2+5
7
-
-
-
-
7
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
19
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
31
-
-
19
-
72
-
45
1+9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
3+1
-
-
1+9
-
7+2
-
4+5
10
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
4
-
-
10
-
9
-
9
1+0
3
1
5
7
3
1
7
5
1
-
1
5
4
-
5
3
4
2
5
9
1
-
-
-
-
-
1+0
-
-
-
-
1
L
A
N
G
U
A
G
E
S
-
A
N
D
-
N
U
M
B
E
R
S
-
-
4
-
-
1
-
9
-
9

 

 

18
THE ENGLISH ALPHABET
-
-
-
1
THE
33
15
6
-
ENGLISH
74
38
2
1
ALPHABET
65
29
2
18
THE ENGLISH ALPHABET
172
82
10
1+8
-
1+7+2
8+2
1+0
9
THE ENGLISH ALPHABET
10
10
10
-
-
1+0
1+0
1+0
9
THE ENGLISH ALPHABET
1
1
1

 

 

-
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
5
-
-
9
1
8
-
-
-
-
8
-
-
-
-
+
=
39
3+9
=
12
1+2
3
=
3
-
-
-
8
-
-
-
14
-
-
9
19
8
-
-
-
-
8
-
-
-
-
+
=
66
6+6
=
12
1+2
3
=
3
-
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
5
-
7
3
-
-
-
-
1
3
7
-
1
2
5
2
+
=
43
4+3
=
7
=
7
=
7
-
-
20
-
5
-
5
-
7
12
-
-
-
-
1
12
16
-
1
2
5
20
+
=
106
1+0+6
=
7
=
7
=
7
-
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
5
14
7
12
9
19
8
-
1
12
16
8
1
2
5
20
+
=
172
1+7+2
=
10
1+0
1
=
1
-
-
2
8
5
-
5
5
7
3
9
1
8
-
1
3
7
8
1
2
5
2
+
=
82
8+2
=
10
1+0
1
=
1
-
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
-
--
-
-
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
5
-
5
5
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
10
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
35
-
-
18
-
82
-
37
1+0
1+8
-
-
-
-
--
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
3+5
-
-
1+8
-
8+2
-
3+7
1
9
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
8
-
-
9
-
10
-
10
-
-
2
8
5
-
5
5
7
3
9
1
8
-
1
3
7
8
1
2
5
2
-
-
-
-
-
-
-
1+0
-
1+0
1
9
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
8
-
-
9
-
1
-
1

 

 

18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
5
-
-
9
1
8
-
-
-
-
8
-
-
-
-
+
=
39
3+9
=
12
1+2
3
=
3
-
-
8
-
-
-
14
-
-
9
19
8
-
-
-
-
8
-
-
-
-
+
=
66
6+6
=
12
1+2
3
=
3
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
5
-
7
3
-
-
-
-
1
3
7
-
1
2
5
2
+
=
43
4+3
=
7
=
7
=
7
-
20
-
5
-
5
-
7
12
-
-
-
-
1
12
16
-
1
2
5
20
+
=
106
1+0+6
=
7
=
7
=
7
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
5
14
7
12
9
19
8
-
1
12
16
8
1
2
5
20
+
=
172
1+7+2
=
10
1+0
1
=
1
-
2
8
5
-
5
5
7
3
9
1
8
-
1
3
7
8
1
2
5
2
+
=
82
8+2
=
10
1+0
1
=
1
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
--
-
-
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
5
-
5
5
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
18
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
35
-
-
18
-
82
-
37
1+8
-
-
-
-
--
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
3+5
-
-
1+8
-
8+2
-
3+7
9
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
8
-
-
9
-
10
-
10
-
2
8
5
-
5
5
7
3
9
1
8
-
1
3
7
8
1
2
5
2
-
-
-
-
-
-
-
1+0
-
1+0
9
T
H
E
-
E
N
G
L
I
S
H
-
A
L
P
H
A
B
E
T
-
-
8
-
-
9
-
1
-
1

 

 

18
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
5
-
-
9
1
8
-
-
-
8
-
-
-
-
+
=
39
3+9
=
12
1+2
3
=
3
-
-
8
-
-
14
-
-
9
19
8
-
-
-
8
-
-
-
-
+
=
66
6+6
=
12
1+2
3
=
3
18
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
2
-
5
5
-
7
3
-
-
-
1
3
7
-
1
2
5
2
+
=
43
4+3
=
7
=
7
=
7
-
20
-
5
5
-
7
12
-
-
-
1
12
16
-
1
2
5
20
+
=
106
1+0+6
=
7
=
7
=
7
18
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
5
14
7
12
9
19
8
1
12
16
8
1
2
5
20
+
=
172
1+7+2
=
10
1+0
1
=
1
-
2
8
5
5
5
7
3
9
1
8
1
3
7
8
1
2
5
2
+
=
82
8+2
=
10
1+0
1
=
1
18
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
--
-
-
3
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
5
5
5
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
8
-
-
-
8
-
-
-
-
-
-
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
18
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
35
-
-
18
-
82
-
37
1+8
-
-
-
--
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
3+5
-
-
1+8
-
8+2
-
3+7
9
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
8
-
-
9
-
10
-
10
-
2
8
5
5
5
7
3
9
1
8
1
3
7
8
1
2
5
2
-
-
-
-
-
-
-
1+0
-
1+0
9
T
H
E
E
N
G
L
I
S
H
A
L
P
H
A
B
E
T
-
-
8
-
-
9
-
1
-
1

 

 

-
-
-
-
THE MAGICAL ALPHABET
-
-
-
T
=
2
3
THE
33
15
6
M
=
4
7
MAGICAL
46
28
1
A
=
1
8
ALPHABET
65
29
2
-
-
7
18
THE MAGICAL ALPHABET
144
72
9
-
-
-
1+8
-
1+4+4
7+2
-
-
-
7
9
THE MAGICAL ALPHABET
9
9
9

 

 

-
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
--
-
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
8
-
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
8
-
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
4
1
7
-
3
1
3
-
1
3
7
-
1
2
5
2
+
=
47
4+7
=
11
1+1
2
=
2
-
-
20
-
5
-
13
1
7
-
3
1
12
-
1
12
16
-
1
2
5
20
+
=
119
1+1+9
=
11
1+1
2
=
2
-
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
13
1
7
9
3
1
12
-
1
12
16
8
1
2
5
20
+
=
144
1+4+4
=
9
=
9
=
9
-
-
2
8
5
-
4
1
7
9
3
1
3
-
1
3
7
8
1
2
5
2
+
=
72
7+2
=
9
=
9
=
9
-
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
4
=
4
=
4
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
-
--
-
-
-
3
-
3
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
6
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
39
-
-
18
-
72
-
45
-
1+8
-
-
-
-
--
1
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
3+9
-
-
1+8
-
7+2
-
4+5
6
9
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
12
-
-
9
-
9
-
9
-
-
2
8
5
-
4
1
7
9
3
1
3
-
1
3
7
8
1
2
5
2
-
-
1+2
-
-
--
-
-
-
-
6
9
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
3
-
-
9
-
9
-
9

 

 

18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
8
-
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
8
-
-
-
-
+
=
25
2+5
=
7
=
7
=
7
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
4
1
7
-
3
1
3
-
1
3
7
-
1
2
5
2
+
=
47
4+7
=
11
1+1
2
=
2
-
20
-
5
-
13
1
7
-
3
1
12
-
1
12
16
-
1
2
5
20
+
=
119
1+1+9
=
11
1+1
2
=
2
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
13
1
7
9
3
1
12
-
1
12
16
8
1
2
5
20
+
=
144
1+4+4
=
9
=
9
=
9
-
2
8
5
-
4
1
7
9
3
1
3
-
1
3
7
8
1
2
5
2
+
=
72
7+2
=
9
=
9
=
9
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
4
=
4
=
4
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
-
--
-
-
-
3
-
3
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
18
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
39
-
-
18
-
72
-
45
1+8
-
-
-
-
--
1
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
3+9
-
-
1+8
-
7+2
-
4+5
9
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
12
-
-
9
-
9
-
9
-
2
8
5
-
4
1
7
9
3
1
3
-
1
3
7
8
1
2
5
2
-
-
1+2
-
-
--
-
-
-
-
9
T
H
E
-
M
A
G
I
C
A
L
-
A
L
P
H
A
B
E
T
-
-
3
-
-
9
-
9
-
9

 

 

18
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
9
-
-
-
-
-
-
8
-
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
-
8
-
-
-
-
9
-
-
-
-
-
-
8
-
-
-
-
+
=
25
2+5
=
7
=
7
=
7
18
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
2
-
5
4
1
7
-
3
1
3
1
3
7
-
1
2
5
2
+
=
47
4+7
=
11
1+1
2
=
2
-
20
-
5
13
1
7
-
3
1
12
1
12
16
-
1
2
5
20
+
=
119
1+1+9
=
11
1+1
2
=
2
18
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
-
-
-
--
-
-
-
-
-
20
8
5
13
1
7
9
3
1
12
1
12
16
8
1
2
5
20
+
=
144
1+4+4
=
9
=
9
=
9
-
2
8
5
4
1
7
9
3
1
3
1
3
7
8
1
2
5
2
+
=
72
7+2
=
9
=
9
=
9
18
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
1
-
-
-
1
-
-
-
-
-
1
occurs
x
4
=
4
=
4
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
2
occurs
x
3
=
6
=
6
-
-
-
-
--
-
-
-
3
-
3
-
3
-
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
18
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
39
-
-
18
-
72
-
45
1+8
-
-
-
--
1
-
-
-
1
-
1
-
-
-
1
-
-
-
-
-
3+9
-
-
1+8
-
7+2
-
4+5
9
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
12
-
-
9
-
9
-
9
-
2
8
5
4
1
7
9
3
1
3
1
3
7
8
1
2
5
2
-
-
1+2
-
-
--
-
-
-
-
9
T
H
E
M
A
G
I
C
A
L
A
L
P
H
A
B
E
T
-
-
3
-
-
9
-
9
-
9

 

 

5
LIGHT
56
29
2
5
SIGHT
63
18
9
4
EYES
54
18
9

 

 

ADVENT 2205 ADVENT

AVATAR 2 TWO 2 AVATAR

THE

TEARS

OF

QUETZALCOATL

 

 

3
THE
33
15
6
5
TEARS
63
36
9
2
OF
21
12
3
12
QUETZALCOATL
153
45
9
22
-
270
108
27
2+2
-
2+7+0
1+0+8
2+7
4
-
9
9
9

 

 

FINGER PRINTS OF THE GODS

Graham Hancock 1995

Page 153

Adventures in the Underworld, Journeys to the Stars

"In Egypt's early dynastic period more than 4500 years ago, an 'Ennead' of nine omnipotent deities was particularly adored by the priesthood at Heliopolis.5 Likewise in Central America, both the Aztecs and the Mayas believed in an all-powerful system of nine deities"

"The majority of the traditions of the God-King Quetzalcoatl, as we have seen, focus on his deeds and teaching as a civilizer. His followers in ancient Mexico, however, also believed that his human manifestation had experienced death and that afterwards he was reborn as a star.9"

 

INTERNET SITE

http://theosophy.org/tlodocs/teachers/Quetzalcoatl.htm

QUETZALCOATL

"What does your mind seek?
Where is your heart?
If you give your heart to each and every thing,
You lead it nowhere: you destroy your heart.
Can anything he found on earth?

Beyond is the place where one lives.
I would he lying to myself were Ito say:
"Perhaps everything ends on this earth;
Here do our lives end."

No, O Lord of the Close Vicinity,
It is beyond, with those who dwell in Your house,
That I will sing songs to You, in the innermost of heaven.
My heart rises;
I fix my eyes upon You,
Next to You, beside You,
O Giver of Life.

Cantares Mexicanos

"No one outside the adyta of initiation can know the ultimate origins of American Indian spirituality, and only a few have penetrated the veil of metaphor and symbol constituting the core of Nahuatl literature. No one knows how many, if any, of those who still speak the Nahuatl tongue, spoken by the Aztecs before them and already richly developed in the time of the earliest Toltecs, really fathom the inner meaning of its startling profusion of juxtaposed images, symbolic descriptions and ethereal allusions. The Nahuatl mind found truth only in "flowers and songs", in intuitive apprehension, and entirely dispensed with delusive dichotomies and mechanistic categories. Only the barest lineaments of the history of the Aztecs, latest of the major pre-Columbian civilizations in Meso-America, are known. The peoples before them are immersed in an obscurity dimly illuminated here and there by legend and archaeological discovery. Ancient Mexico and the lands immediately south of it are, as H.P. Blavatsky said, "a land of mystery". Yet within that lost continent is to be found Quetzalcoatl, one of the iridescent spiritual impulses of poorly recorded history. Quetzalcoatl emblazoned a trail through human thought and culture that could not be effaced by the indifference of the rapacious conquistador and the ruthless zeal of the Inquisition."

 

12
QUETZALCOATL
153
45
9
7
CHOLULA
72
27
9
11
TEOTIHUACAN
117
63
9
-
-
-
-
-
-
-
-
-
-
11
TEOTIHUACAN
     
3
T+E+O+T
60
15
6
5
I
9
9
9
2
H+U+A
30
30
3
11
C+A+N
18
9
9
11
TEOTIHUACAN
117
63
27
2+2
-
1+1+7
6+3
2+7
4
9
9
9

 

"The earliest American high culture known to history was that developed by the Toltecs, whose name in Nahuatl means 'master craftsmen'. They built the great city of Tollan, a sacred precinct laid out to mirror and intimate the mysteries of existence. Tollan, literally 'metropolis', became the prototype of later cities which bore its name as well as specific names of their own. Tollan was the magnificent Teotihuacan which was recognized as the source of Nahuatl civilization, and Quetzalcoatl was the spiritual source of the earthly Tollan. Like Osiris in Egypt, Quetzalcoatl was a divine king who taught all the arts and sciences. Like Prometheus, he gave mankind sacred keys to wisdom. He is the spiritual progenitor of the tlamatinime, the wise men who were the priests and preservers of divine knowledge.

The wise maintain a light, a torch, a stout torch
that does not smoke;
A perforated mirror, a mirror pierced on both sides.
His are the black and red ink, his are the illustrated
manuscripts. . . .
He himself is writing and wisdom.
He is the path, the true way for others.

Codice Matritense

The priest who through purity and insight emerged foremost amongst his peers was given the name Quetzalcoatl, reminiscent of the Egyptian Initiate who earned the epithet Hermes Trismegistus. He is a mirror of the world and of the Divine, "pierced on both sides", so that the transcendent shines forth upon the world and man sees beyond the immanent through the wise man. The tlamatini mirrors the primordial and aeviternal activity of Quetzalcoatl, the divine sage, high priest and inner being of humanity.

Quetzalcoatl's ineluctably numinous nature, a mystery impenetrable to theological and mythological analysis, abides in his role as the bridge between ontological levels and between pairs of opposites within each level. The duality essential to manifestation is constrained by and unified in Quetzalcoatl. Thus, to explain the intertwined and enigmatic functions of this man-god, mythographers have been compelled to import the Sanskrit concept of avatara. Surviving fragments of myth, legend and history provide a tantalizingly incomplete mosaic of a priest-king and spiritual principle omnipresent in Nahuatl thought and life. Accounts of his functions and activities seem confused and contradictory, most likely because the keys to levels of interpretation perished with the silent tlamatinime. Nonetheless, Quetzalcoatl, believed to have incarnated as a righteous priest-king in Tollan, was first a metaphysical principle involved in the primordial creative emanation of the world. Ometeotl is the great god who abides forever in the twelfth and thirteenth heavens. In the highest realm, he alone is unaffected by the emergence and dissolution of the cosmos. In the twelfth heaven, Omeyocan (the Realm of Duality), he is "Our Mother and Our Father, Ometeotl-Omecihuatl, who is Dual Lord and Dual Lady", the first cause.

Our Lord, Lord of the Ring,
Self-engendered, self-willing, self-enjoying;
Even as He wills, so shall He desire that
it shall be.

The appearance and passing of worlds is the work of hierarchies of divine beings who operate in strict obedience to the universal law, the will of Ometeotl. Yet he also dwells in the centre of the cosmos and on every plane of being as Xiuhtecuhtli, the Lord of Fire and Time.

As Mother-Father, the Dual Lord confirms the connection of a soul to the body engendered by conception. In this role, he sits on high with his consort, his feminine self, and Quetzalcoatl sits between them, for Quetzalcoatl creates the connection willed by Ometeotl. As Ometeotl is the heart of manifestation, Quetzalcoatl is the heart of the dual Ometeotl. Within the vast unfoldment of cosmos, the world has emerged five times through five rebirths of the sun. Whilst some say the fifth sun is the last, others suggest that there will be seven suns, and still others hint at the possibility of twelve suns. Each sun has come into existence through the sacrifice of a god, just as Ometeotl must sacrifice his utter transcendence to become the dual first cause. Tezcatlipoca, son of Ometeotl, sacrificed himself in the cosmic fire so that the Sun of Night and of Earth might arise. Represented by the jaguar or tiger, the raw forces of this world were sterile from an evolutionary standpoint and therefore perished. Quetzalcoatl sacrificed himself to produce the second world, the Sun of Air, but its purely spiritual powers could not sustain form. Creatures of this world who corresponded to human beings in the world of the fifth sun became monkeys. Tlaloc, Lord of Rain, immolated himself in the cosmic fire to give birth to the Sun of Rain and Fire, but the volcanic intensity of this world allowed only birds to survive, though during its existence the prototype of maize was grown. Chalchiuhtlicue, life-giving goddess of waters, offered herself so that the fourth Sun of Waters might appear. Whilst men consumed the acicintli seed, it could not grow in water alone, and the world perished in a universal deluge.

Two gods volunteered themselves in the fiery sacrifice to create the fifth sun. The divine hearth was constructed at Teotihuacan, the centre of what would be the fifth world. After suitable ritual preparations were made and the gods had purified themselves, the moment came to approach the fire. Tecuciztecatl, Lord of the Snails, who had arrogantly claimed primacy, could not muster the courage to enter the cosmic fire. Nanahuatzin or Nanahuatl, the god whose form is diseased, who therefore understood the pain of limitation and imperfection, stepped forward and threw himself on the pyre. Shamed by such detachment, Tecuciztecatl followed him as the moon (which Tecuciztecatl became) follows the sun. The sun did not rise immediately, however, and the gods became anxious in the oppressive darkness. Quetzalcoatl, however, divined the locus of the sunrise and proceeded to the east. There he welcomed the rising sun as Lord of the Dawn and, when the sun wobbled uncertainly on its rising course, steadied it as god of wind. Thus, the fifth sun is called Nahui Ollin, Four Movement, Naollin, the synthesis of the four elements through dynamic interaction, the Sun of Quetzalcoatl, who as movement is the active ingredient of the ever-changing balance which sustains – and is – life. Its symbol is the human face, the countenance signifying life and intelligence, self-conscious will or choice in the service of unalterable cosmic law, that mystic promise of immortality within necessary dissolution that alone can mirror unmanifest eternity. Its glyph includes the four transient elements and three aspects of divine creativity, arranged as a quincunx that points to both the Fourth Round and Fifth Root Race.

In his Promethean aspect Quetzalcoatl is involved in the creation of human beings and in inspiring them with intelligence. Before Naollin's roseate splendour had burst into full day and brought the present world to light, Quetzalcoatl had to descend into the realm of the dead, Mictlan, to secure the precious bones of man so that humans might again inhabit the earth. In Mictlan, the realm of the fleshless, he confronted Mictlantecuhtli and Mictlancihuatl, Lord and Lady of the Land of the Dead, the 'masks' or reflections of Ometeotl and Omecihuatl in the lowest sphere of duality, beyond which is unknowable darkness, just as there is the Unknown above Omeyocan, the highest heaven. When Quetzalcoatl demanded the bones, Mictlantecuhtli offered them on condition that Quetzalcoatl sound the conch-shell and circle the kingdom four times. Whilst this seemed to be a genuine challenge, the shell had no sounding-hole and was ever mute. Quetzalcoatl called upon the worms to pierce the shell, and bees entered through the hole and made it sound. Whilst appearing to yield possession of the bones, Mictlantecuhtli called upon the forces of the underworld to prevent Quetzalcoatl from fulfilling his charge. Mirroring this deception, Quetzalcoatl sent his double, nahualli, who is Xolotl, his twin and another aspect of himself, to inform the Lord of the Dead that the bones would be left in Mictlan. Even whilst this message was being delivered, Quetzalcoatl gathered the bones of Man and Woman and fled.

The forces of the underworld did not pursue Quetzalcoatl directly; they had prepared a trap. Quetzalcoatl fell into the trap and lost consciousness for a time. When he recovered, he found the bones damaged and in disarray. Crying out to his nahualli, he asked, "What shall I do now?" His twin gave the pre-ordained response: "Since things have turned out badly, let them turn out as they may."

And as soon as he arrived, the woman called Quilaztli, who is Cihuacoatl, took them to grind and put them in a precious vessel of clay. Upon them Quetzalcoatl bled his member. The other gods and Quetzalcoatl himself did penance. And they said, "People have been born, O gods, the macehuales – those 'deserved' into life through penance." Because for our sake, the gods did penance!

Manuscript of 1558

Within this mysterious allegory one can see the failure of nature alone to produce intelligent men, the gathering of the lower vestures and their animation with the breath of life within the body of clay, as well as the penitential self-sacrifice of the gods, represented by the seminal blood and signifying the incarnation of the spiritual and divine within the prepared living human form. Once this complex process was completed, Quetzalcoatl stole maize, the proper food of self-consciously intelligent beings, and gave it to humanity.

Under Naollin, the fifth sun, Quetzalcoatl is the dynamic order of Nature, the homoeostasis in which humanity can flourish. Celestially, he guards the Milky Way, 'the Luminous Petticoat of Stars'. Tezcatlipoca, son of Ometeotl, became the four Tezcatlipocas who guard the four quarters of the world. In the west this fourfold hypostasis is Quetzalcoatl, whilst in the east he is the red Tezcatlipoca, the two constituting the tension between birth and death, which is also death in this world of change and birth into the Divine Darkness. The red Tezcatlipoca is also Xolotl, the twin of Quetzalcoatl, the other half of one ceaseless activity. In the atmosphere which blankets the fertile earth, Quetzalcoatl is the wind and the water it bears in the air. He is also lightning, sudden illuminator of darkness, who, like an ambassador, precedes Tlaloc, the god of rain. His multivalent functions are intimated in the deliberate ambiguity of his name: Quetzalcoatl is derived from quetzal, 'feathered' or 'precious', and coati, 'serpent' or 'twin'. Thus he is both the Plumed Serpent and the Precious Twin.

In tlalticpac, the dream world which is earth, Quetzalcoatl is the divine king who, like Osiris, the second divine pharaoh of Egypt, brought civilization to humanity. As the divine ruler in Tollan, he taught all the arts and sciences, from cultivation of maize to metallurgy and from astrology to poetry, as well as the sacred tlilli tlapalli, red and black ink, that is, writing and, by extension, wisdom. During the golden age he dwelt in his invisible form, guiding and governing in a kingdom of innocent joy. Yet the forces of limitation, shadows in this realm of light, plotted Quetzalcoatl's downfall. Tezcatlipoca took a mirror and invited Quetzalcoatl to gaze into it. To his horror, he thereby gained a body, rather like Anthropos, and seeing himself reflected in the mirror of inchoate Nature, became one with it, according to the Hermetic tradition. In his confusion he allowed a mask and feathered head-dress to be made for him, so that people might look upon him without fear. Whilst he was disoriented, demons made pulque, a fermented drink from sap of the maguey, and gave it to Quetzalcoatl. Thus intoxicated, he took Quetzalpetatl, his feminine aspect from which he now felt alienated, and slept with her, falling afterwards into a stupor. As the archetype of humanity, his deeds brought pain and suffering to humanity – the pain of having a body, the suffering of loneliness, the disharmonies of striving, contention, fear and guilt, which pit person against person and turn the powers of human consciousness into instruments of selfishness and its inevitable offspring, conflict and greed.

In the morning Quetzalcoatl awoke filled with grief and remorse. As god, he knew the unavoidable problems of incarnation, but as king, he saw the massive failure of civilization. Between potentiality and actualization fell the dread shadow of self-induced ignorance. Within the architectonics of human life, the problems of creating man had been wholly reflected, and thus Quetzalcoatl's earthly work was completed. He resolved to leave his beautiful Tollan and set out with his closest devotees. He journeyed throughout his kingdom, leaving at different sites marks of his presence – a sacred footprint here, a raised stone there – and stripped himself of his arts and powers as he went so that these might remain with humanity in his absence. He ordered a stone casket to be made, and when it was finished he lay in it for four days so that his most precious secrets might be absorbed into it. When he was ready, he ordered the stone box sealed up to prevent theft or contamination of its contents. Only those who have redeemed Quetzalcoatl's wisdom through severe penance and self-sacrifice can hope to know the contents of that mystic sarcophagus now secreted in the human breast, in the place of purity where Quetzalcoatl was accustomed to bathing.

All work finished, Quetzalcoatl went to the sea.

When he reached the holy sea
And the shore of the luminous ocean,
He stayed his steps and wept.
He gathered his garments together
And dressed himself again
In his robes of quetzal feathers
And his turquoise mask.

Annals of Cuauhtitlan

There at Tlillin Tlapallin, the place of burning, he built a huge pyre, mounted it and set it aflame. His ashes rose into the air and the rarest birds of the earth appeared. As the red flames lit up the celestial vault, Quetzalcoatl became again the Lord of the Dawn.

When the ashes had ceased to burn,
Quetzalcoatl's heart rose up.
They say it was raised to heaven
And entered there.
Wise men say it became
The morning star, appearing at dawn,
And they add that it was not seen
For four days after his death,
Whilst he had sojourned in the Kingdom of Death.
And in these four days, he gathered arrows,
And eight days later, he appeared again
As the great star.
Since then he has sat enthroned.

"The heart of Quetzalcoatl became Venus, the morning star which promises first the dawn, then the rising sun itself."

 

12
QUETZALCOATL
153
45
9
5
VENUS
81
27
9
2
V+E
27
9
9
3
S+U+N
54
18
9
5
VENUS
81
27
18
-
-
8+1
2+7
1+8
5
VENUS
9
9
9

 

 

10
OSIRISISIS
145
55
1
12
HORUS
81
27
9
5
VENUS
81
18
9
2
SUN
54
9
9
3
JUPITER
99
36
9
 
ORIONIS
99
45
9
5
SAGITTARIUS
144
45
9

 

GOD OF NAMES 99 NAMES OF GOD

 

12
QUETZALCOATL
153
45
9
13
SETI-MER-EN-PTAH
153
72
9
14
PHARAOH + PYRAMID
153
81
9
14
ALBERT + EINSTEIN
153
63
9

 

 

HOW MANY FISH RISHI RISHI HOW MANY FISH

?

153

ALL

12

INCHES LONG SAID RISHI CLAPPING A HAND

ONE FOR EACH

DISCIPLE

 

 

THE EGYPTIAN HEAVEN AND HELL

E. E. Wallis Budge 1857-1934

Page 59

"CHAPTER OF COMING FORTH BY DAY AND OF MAKING A WAY THROUGH THE AMMEHET."

"SETI MER EN PTAH"

13
SETI-MER-EN-PTAH
153
72
9
9
ALIVE+DEAD
63
36
9
11
DYING+RISING
135
72
9
11
LIGHT+DARK
90
45
9
16
POSITIVE+NEGATIVE
198
81
9
10
MIND+MATTER
117
45
9
13
MAGNETIC+FIELD
1O8
45
9
9
ATOMS +MIND
108
36
9
4
KNOW
63
18
9
4
REAL
36
18
9
7
REALITY
90
36
9
4
GODS
45
18
9
4
LOVE
54
18
9
6
DIVINE
63
36
9
7
THOUGHT
99
36
9

 

GODS JUDGEMENT GODS

DIVINE LAW ISISISISISIS LAW DIVINE

 

5
THE HOLY WORD
81
27
9
3
THE
33
15
6
4
HOLY
60
24
6
4
WORD
60
24
6
11
HOLY THE WORD
153
63
9
1+1
-
1+5+3
6+3
-
2
THE HOLY WORD
9
9
9

 

WHATEVER THOU SOWETH THOU REAPEST

 

THE TIMES

Monday September 12 2005

Advert opposite Page 16

"GLOBAL WARNING"

 

GLOBALWARNINGGLOBALWARMINGGLOBALWARMINGGLOBALWARNING

 

 

DAILY STAR

Front Page Headlines

Monday, September 12, 2005

"999 PANIC"

 

 

-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
6
-
-
5
+
=
19
1+9
=
10
1+0
1
=
1
-
-
8
-
-
-
15
-
-
14
+
=
37
3+7
=
10
1+0
1
=
1
-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
-
7
4
9
-
7
5
-
+
=
32
3+2
=
5
-
5
=
5
-
-
-
25
4
18
-
7
5
-
+
=
59
5+9
=
14
1+4
5
=
5
-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
8
25
4
18
15
7
5
14
+
=
96
9+6
=
15
1+5
6
=
6
-
-
8
7
3
9
6
7
5
5
+
=
51
5+1
=
6
-
6
=
6
-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
7
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
6
8
H
Y
D
R
O
G
E
N
-
-
39
-
-
8
-
51
-
33
-
-
-
-
-
9
-
-
-
-
-
-
3+9
-
-
-
-
5+1
-
3+3
6
8
H
Y
D
R
O
G
E
N
-
-
12
-
-
8
-
6
-
6
-
-
8
7
3
9
6
7
5
5
-
-
1+2
-
-
-
-
-
-
-
6
8
H
Y
D
R
O
G
E
N
-
-
3
-
-
8
-
6
-
6

 

 

8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
6
-
-
5
+
=
19
1+9
=
10
1+0
1
=
1
-
8
-
-
-
15
-
-
14
+
=
37
3+7
=
10
1+0
1
=
1
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
7
4
9
-
7
5
-
+
=
32
3+2
=
5
-
5
=
5
-
-
25
4
18
-
7
5
-
+
=
59
5+9
=
14
1+4
5
=
5
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
8
25
4
18
15
7
5
14
+
=
96
9+6
=
15
1+5
6
=
6
-
8
7
3
9
6
7
5
5
+
=
51
5+1
=
6
-
6
=
6
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
8
H
Y
D
R
O
G
E
N
-
-
39
-
-
8
-
51
-
33
-
-
-
-
9
-
-
-
-
-
-
3+9
-
-
-
-
5+1
-
3+3
8
H
Y
D
R
O
G
E
N
-
-
12
-
-
8
-
6
-
6
-
8
7
3
9
6
7
5
5
-
-
1+2
-
-
-
-
-
-
-
8
H
Y
D
R
O
G
E
N
-
-
3
-
-
8
-
6
-
6

 

The hydrogen line, 21-centimeter line, is the electromagnetic radiation spectral line

 

-
-
-
-
-
-
-
-
-
T
=
2
-
3
THE
33
15
6
H
=
8
-
8
HYDROGEN
96
51
6
L
=
3
-
4
LINE
40
22
4
T
=
2
-
9
TWENTYONE
141
42
6
C
=
3
-
10
CENTIMETER
112
49
4
L
=
3
-
4
LINE
40
22
4
I
=
9
-
2
IS
28
10
1
T
=
2
-
3
THE
33
15
6
E
=
5
-
15
ELECTROMAGNETIC
150
69
6
R
=
9
-
9
RADIATION
91
46
1
S
=
1
-
8
SPECTRAL
94
31
4
L
=
3
-
4
LINE
40
22
4
-
-
50
-
79
First Total
898
394
52
-
-
5+0
-
7+9
Add to Reduce
8+9+9
3+9+4
5+2
-
-
5
-
16
Second Total
25
16
7
-
-
-
-
1+6
Reduce to Deduce
2+5
1+6
-
-
-
5
-
7
Essence of Number
7
7
7

 

 

Hydrogen line - Wikipedia

https://en.wikipedia.org › wiki › Hydrogen_line
The hydrogen line, 21-centimeter line, or H I line is the electromagnetic radiation spectral line ... This wavelength falls within the microwave region of the electromagnetic spectrum, and it is ... Despite these problems, 21 cm observations, along with space-based gravitational wave observations, are generally viewed as the ...

 

21-centimeter line, or H I line is the electromagnetic radiation spectral line

 

-
-
-
-
-
-
-
-
-
T
=
2
-
9
TWENTYONE
141
42
6
C
=
3
-
10
CENTIMETER
112
49
4
L
=
3
-
4
LINE
40
22
4
-
-
8
-
23
First Total
293
113
14
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
8
-
5
Second Total
14
5
5
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
8
-
5
Essence of Number
5
5
5

 

The hydrogen line, 21-centimeter line, or H I line[1] is the electromagnetic radiation spectral line that is created by a change in the energy state of neutral hydrogen atoms. This electromagnetic radiation is at the precise frequency of 1,420,405,751.7667±0.0009 Hz,[2][3] which is equivalent to the vacuum wavelength of 21.1061140542 cm in free space. This wavelength falls within the microwave region of the electromagnetic spectrum, and it is observed frequently in radio astronomy because those radio waves can penetrate the large clouds of interstellar cosmic dust that are opaque to visible light. This line is also the theoretical basis of hydrogen maser.

The microwaves of the hydrogen line come from the atomic transition of an electron between the two hyperfine levels of the hydrogen 1 s ground state that have an energy difference of ˜ 5.87433 µeV [9.411708×10-25 J].[4] It is called the spin-flip transition. The frequency, ?, of the quanta that are emitted by this transition between two different energy levels is given by the Planck–Einstein relation E = h?. According to that relation, the photon energy of a 1,420,405,751.7667 Hz photon is ˜ 5.87433 µeV [9.411708×10-25 J]. The constant of proportionality, h, is known as the Planck constant.
Further information: Hyperfine structure

The ground state of neutral hydrogen consists of an electron bound to a proton. Both the electron and the proton have intrinsic magnetic dipole moments ascribed to their spin, whose interaction results in a slight increase in energy when the spins are parallel, and a decrease when antiparallel. The fact that only parallel and antiparallel states are allowed is a result of the quantum mechanical discretization of the total angular momentum of the system. When the spins are parallel, the magnetic dipole moments are antiparallel (because the electron and proton have opposite charge), thus one would expect this configuration to actually have lower energy just as two magnets will align so that the north pole of one is closest to the south pole of the other. This logic fails here because the wave functions of the electron and the proton overlap; that is, the electron is not spatially displaced from the proton, but encompasses it. The magnetic dipole moments are therefore best thought of as tiny current loops. As parallel currents attract, the parallel magnetic dipole moments (i.e., antiparallel spins) have lower energy.[5] The transition has an energy difference of 5.87433 µeV that when applied in the Planck equation gives:
? = 1 ? · c = h E · c = 4.1357 · 10 - 15 eV s 5.874 33 · 10 - 6 eV · 2.9979 · 10 8 m s - 1 = 0.211 06 m = 21.106 cm {\displaystyle \lambda ={\frac {1}{\nu }}\cdot c={\frac {h}{E}}\cdot c={\frac {4.1357\cdot 10^{-15}\;{\text{eV}}\;{\text{s}}}{5.874\,33\cdot 10^{-6}\;{\text{eV}}}}\cdot 2.9979\cdot 10^{8}\;{\text{m}}\;{\text{s}}^{-1}=0.211\,06\;{\text{m}}=21.106\;{\text{cm}}} {\displaystyle \lambda ={\frac {1}{\nu }}\cdot c={\frac {h}{E}}\cdot c={\frac {4.1357\cdot 10^{-15}\;{\text{eV}}\;{\text{s}}}{5.874\,33\cdot 10^{-6}\;{\text{eV}}}}\cdot 2.9979\cdot 10^{8}\;{\text{m}}\;{\text{s}}^{-1}=0.211\,06\;{\text{m}}=21.106\;{\text{cm}}}
where h is the Planck constant and c is the speed of light.

This transition is highly forbidden with an extremely small transition rate of 2.9×10-15 s-1,[6] and a mean lifetime of the excited state of around 10 million years. A spontaneous occurrence of the transition is unlikely to be seen in a laboratory on Earth, but it can be artificially induced using a hydrogen maser. It is commonly observed in astronomical settings such as hydrogen clouds in our galaxy and others. Owing to its long lifetime, the line has an extremely small natural width, so most broadening is due to Doppler shifts caused by bulk motion or nonzero temperature of the emitting regions.

Discovery?[edit]

During the 1930s, it was noticed that there was a radio 'hiss' that varied on a daily cycle and appeared to be extraterrestrial in origin. After initial suggestions that this was due to the Sun, it was observed that the radio waves seemed to propagate from the centre of the Galaxy. These discoveries were published in 1940 and were noted by Jan Oort who knew that significant advances could be made in astronomy if there were emission lines in the radio part of the spectrum. He referred this to Hendrik van de Hulst who, in 1944, predicted that neutral hydrogen could produce radiation at a frequency of 1420.4058 MHz due to two closely spaced energy levels in the ground state of the hydrogen atom.

The 21 cm line (1420.4 MHz) was first detected in 1951 by Ewen and Purcell at Harvard University,[7] and published after their data was corroborated by Dutch astronomers Muller and Oort,[8] and by Christiansen and Hindman in Australia. After 1952 the first maps of the neutral hydrogen in the Galaxy were made, and revealed for the first time the spiral structure of the Milky Way.
Uses?[edit]

In radio astronomy?[edit]

The 21 cm spectral line appears within the radio spectrum (in the L band of the UHF band of the microwave window to be exact). Electromagnetic energy in this range can easily pass through the Earth's atmosphere and be observed from the Earth with little interference.

Assuming that the hydrogen atoms are uniformly distributed throughout the galaxy, each line of sight through the galaxy will reveal a hydrogen line. The only difference between each of these lines is the doppler shift that each of these lines has. Hence, one can calculate the relative speed of each arm of our galaxy. The rotation curve of our galaxy has been calculated using the 21 cm hydrogen line. It is then possible to use the plot of the rotation curve and the velocity to determine the distance to a certain point within the galaxy.

Hydrogen line observations have also been used indirectly to calculate the mass of galaxies, to put limits on any changes over time of the universal gravitational constant and to study the dynamics of individual galaxies.

In cosmology?[edit]

The line is of great interest in Big Bang cosmology because it is the only known way to probe the "dark ages" from recombination to reionization. Including the redshift, this line will be observed at frequencies from 200 MHz to about 9 MHz on Earth. It potentially has two applications. First, by mapping the intensity of redshifted 21 centimeter radiation it can, in principle, provide a very precise picture of the matter power spectrum in the period after recombination. Second, it can provide a picture of how the universe was reionized, as neutral hydrogen which has been ionized by radiation from stars or quasars will appear as holes in the 21 cm background.

However, 21 cm observations are very difficult to make. Ground-based experiments to observe the faint signal are plagued by interference from television transmitters and the ionosphere, so they must be made from very secluded sites with care taken to eliminate interference. Space based experiments, even on the far side of the Moon (where they would be sheltered from interference from terrestrial radio signals), have been proposed to compensate for this. Little is known about other effects, such as synchrotron emission and free–free emission on the galaxy. Despite these problems, 21 cm observations, along with space-based gravitational wave observations, are generally viewed as the next great frontier in observational cosmology, after the cosmic microwave background polarization.

Relevance to the search for non-human intelligent life?[edit]


The hyperfine transition of hydrogen, as depicted on the Pioneer and Voyager spacecraft.
The Pioneer plaque, attached to the Pioneer 10 and Pioneer 11 spacecraft, portrays the hyperfine transition of neutral hydrogen and used the wavelength as a standard scale of measurement. For example, the height of the woman in the image is displayed as eight times 21 cm, or 168 cm. Similarly the frequency of the hydrogen spin-flip transition was used for a unit of time in a map to Earth included on the Pioneer plaques and also the Voyager 1 and Voyager 2 probes. On this map, the position of the Sun is portrayed relative to 14 pulsars whose rotation period circa 1977 is given as a multiple of the frequency of the hydrogen spin-flip transition. It is theorized by the plaque's creators that an advanced civilization would then be able to use the locations of these pulsars to locate the Solar System at the time the spacecraft were launched.

The 21 cm hydrogen line is considered a favorable frequency by the SETI program in their search for signals from potential extraterrestrial civilizations. In 1959, Italian physicist Giuseppe Cocconi and American physicist Philip Morrison published "Searching for Interstellar Communications", a paper proposing the 21 cm hydrogen line and the potential of microwaves in the search for interstellar communications. According to George Basalla, the paper by Cocconi and Morrison "provided a reasonable theoretical basis" for the then-nascent SETI program.[9] Similarly, Pyotr Makovetsky proposed SETI use a frequency which is equal to either
p × 1420.40575177 MHz = 4.46233627 GHz
or
2p × 1420.40575177 MHz = 8.92467255 GHz
Since p is an irrational number, such a frequency could not possibly be produced in a natural way as a harmonic, and would clearly signify its artificial origin. Such a signal would not be overwhelmed by the H I line itself, or by any of its harmonics.[10]

A suggested modification to the mentioned idea is to search on currently unused Sky or other satellite LNB frequencies, as it is possible that with enough synchronized receivers AI inference techniques could be used. In this case, the target frequency would be in the 13 GHz band (13.37–13.39 GHz), derived from 3pH, but there may be merit in searching on other unused bands within the transition areas not normally used, with the PCB also including an onboard reference clock derived from Droitwich or MSF.[citation needed]

See also?[edit]
Chronology of the universe
Dark Ages Radio Explorer
H-alpha, the visible red spectral line with wavelength of 6562.8 ångstroms
Hydrogen spectral series
Rydberg formula
Balmer series
Schelling point
Timeline of the Big Bang

https://en.wikipedia.org › wiki › Hydrogen_line
The hydrogen line, 21-centimeter line, or H I line is the electromagnetic radiation spectral line ... This wavelength falls within the microwave region of the electromagnetic spectrum, and it is ... Despite these problems, 21 cm observations, along with space-based gravitational wave observations, are generally viewed as the ...

21-cm Line Radiation
In 1944 Hendrik van de Hulst predicted that the cold atomic hydrogen (H I) gas should emit a particular wavelength of radio energy from a slight energy change in the hydrogen atoms. The wavelength is 21.1 centimeters (frequency = 1420.4 MHz) so this radiation is called 21-cm line radiation.28 May 2019

 

T
=
2
-
9
TWENTYONE
141
42
6
C
=
3
-
10
CENTIMETER
112
49
4
L
=
3
-
4
LINE
40
22
4
I
=
9
-
2
IS
28
10
1
T
=
2
-
3
THE
33
15
6
E
=
5
-
15
ELECTROMAGNETIC
150
69
6
R
=
9
-
9
RADIATION
91
46
1
S
=
1
-
8
SPECTRAL
94
31
4
L
=
3
-
4
LINE
40
22
4
-
-
37
-
64
First Total
729
306
36
-
-
3+7
-
6+4
Add to Reduce
7+2+9
3+0+6
3+6
-
-
10
-
10
Second Total
18
9
9
-
-
1+0
-
1+0
Reduce to Deduce
1+8
-
-
-
-
1
-
1
Essence of Number
9
9
9

 

 

TWENTY ONE CENTIMETER LINE

21-centimeter line, or H I line is the electromagnetic radiation spectral line

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
W
=
5
2
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
Y
=
7
6
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
42
-
9
-
141
42
42
-
-
-
-
-
-
-
-
-
-
C
=
3
10
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
12
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
13
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
15
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
16
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
17
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
18
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
19
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
40
-
10
-
112
49
40
-
-
-
-
-
-
-
-
-
-
L
=
3
10
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
12
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
18
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
22
-
4
-
40
22
22
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

TWENTY ONE CENTIMETER LINE

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
W
=
5
2
1
W
23
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
T
=
2
5
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
Y
=
7
6
1
Y
25
7
7
-
1
-
-
-
-
-
7
8
-
O
=
6
7
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
C
=
3
10
1
C
3
3
3
-
1
-
3
-
-
-
-
8
-
E
=
5
11
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
T
=
2
13
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
M
=
4
15
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
E
=
5
16
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
T
=
2
17
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
R
=
9
19
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
L
=
3
10
1
L
12
3
3
-
1
-
3
-
-
-
-
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
13
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
17
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
C
=
3
10
1
C
3
3
3
-
1
-
3
-
-
-
-
8
-
L
=
3
10
1
L
12
3
3
-
1
-
3
-
-
-
-
8
-
M
=
4
15
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
W
=
5
2
1
W
23
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
11
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
16
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
Y
=
7
6
1
Y
25
7
7
-
1
-
-
-
-
-
7
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
R
=
9
19
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

TWENTY ONE CENTIMETER LINE

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
13
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
17
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
C
=
3
10
1
C
3
3
3
-
1
-
3
-
-
-
-
8
-
L
=
3
10
1
L
12
3
3
-
1
-
3
-
-
-
-
8
-
M
=
4
15
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
W
=
5
2
1
W
23
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
11
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
16
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
Y
=
7
6
1
Y
25
7
7
-
1
-
-
-
-
-
7
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
R
=
9
19
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

TWENTY ONE CENTIMETER LINE

 

I SAY HAVE I MENTIONED GODS DIVINE THOUGHT

?

HAVE I MENTIONED THAT YET

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
1
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
2
3
SAY
45
18
9
-
-
-
-
-
-
-
-
-
9
H
=
8
3
4
HAVE
36
18
9
-
-
-
-
-
-
-
-
-
9
I
=
9
4
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
5
9
MENTIONED
99
45
9
-
-
-
-
-
-
-
-
-
9
G
=
7
6
4
GODS
45
18
9
-
-
-
-
-
-
-
-
-
9
D
=
4
7
6
DIVINE
63
36
9
-
-
-
-
-
-
-
-
-
9
T
=
2
8
7
THOUGHT
99
36
9
-
-
-
-
-
-
-
-
-
9
H
=
8
9
4
HAVE
36
18
9
-
-
-
-
-
-
-
-
-
9
I
=
9
10
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
11
9
MENTIONED
99
45
9
-
-
-
-
-
-
-
-
-
9
T
=
2
12
4
THAT
49
13
4
-
-
-
-
4
-
-
-
-
-
Y
=
7
13
3
YET
50
14
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
74
-
56
First Total
648
288
108
-
1
2
3
4
5
6
7
8
99
-
-
7+4
-
5+6
Add to Reduce
6+4+8
2+8+8
1+0+8
-
-
-
-
-
-
-
-
-
9+9
-
-
11
-
11
Second Total
18
18
9
-
1
2
3
4
5
6
7
8
18
-
-
1+1
-
1+1
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
1+8
-
-
2
-
2
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
I
=
9
1
1
I
9
9
9
-
1
2
3
-
-
6
7
8
9
S
=
1
2
3
SAY
45
18
9
-
1
2
3
-
-
6
7
8
9
H
=
8
3
4
HAVE
36
18
9
-
1
2
3
-
-
6
7
8
9
I
=
9
4
1
I
9
9
9
-
1
2
3
-
-
6
7
8
9
M
=
4
5
9
MENTIONED
99
45
9
-
1
2
3
-
-
6
7
8
9
G
=
7
6
4
GODS
45
18
9
-
1
2
3
-
-
6
7
8
9
D
=
4
7
6
DIVINE
63
36
9
-
1
2
3
-
-
6
7
8
9
T
=
2
8
7
THOUGHT
99
36
9
-
1
2
3
-
-
6
7
8
9
H
=
8
9
4
HAVE
36
18
9
-
1
2
3
-
-
6
7
8
9
I
=
9
10
1
I
9
9
9
-
1
2
3
-
-
6
7
8
9
M
=
4
11
9
MENTIONED
99
45
9
-
1
2
3
-
-
6
7
8
9
T
=
2
12
4
THAT
49
13
4
-
1
2
3
4
-
6
7
8
-
Y
=
7
13
3
YET
50
14
5
-
1
2
3
-
5
6
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
74
-
56
First Total
648
288
108
-
1
2
3
4
5
6
7
8
99
-
-
7+4
-
5+6
Add to Reduce
6+4+8
2+8+8
1+0+8
-
-
-
-
-
-
-
-
-
9+9
-
-
11
-
11
Second Total
18
18
9
-
1
2
3
4
5
6
7
8
18
-
-
1+1
-
1+1
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
1+8
-
-
2
-
2
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

LOGARITHM


Logarithm - Wikipedia
en.wikipedia.org › wiki › Logarithm

In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed ...
Logarithmic scale · ?Natural logarithm · ?Common logarithm · ?Binary logarithm

 

Introduction to Logarithms - Math is Fun
www.mathsisfun.com › algebra › logarithms
Introduction to Logarithms. In its simplest form, a logarithm answers the question: How many of one number do we multiply to get another number?

?Working with Exponents · ?Logarithms Can Have Decimals · ?Exponents, Roots

 

-
-
-
-
9
ALGORITHM
103
49
49
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
LOGARITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
-
-
49
-
9
LOGARITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
9
LOGARITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
9
LOGARITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
10
ALGORITHM
103
49
49
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
10
LOGARITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

LOGARITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
10
ALGORITHM
103
49
49
-
1
2
3
4
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
10
LOGARITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
LOGARITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
LOGARITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
LOGARITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

MINUS 5

 

LOGARITHM

ALGORITHM

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
4
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
4
4
4

 

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

ALGORITHM

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL MORDER

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
5
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
5
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
5
6
7
8
9

 

MINUS THE 5

 

-
-
-
-
-
ALGORITHM
-
-
-
-
1
2
3
4
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
49
-
10
ALGORITHM
103
49
49
-
2
2
3
4
6
7
8
18
-
-
4+9
-
1+0
-
1+0+3
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
1
ALGORITHM
4
13
13
-
2
2
3
4
6
7
8
9
-
-
1+3
-
1+0
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
1
ALGORITHM
4
4
4
-
2
2
3
4
6
7
8
9

 

Algorithm - Wikipedia, the free encyclopedia en.wikipedia.org/wiki/Algorithm

In mathematics and computer science, an algorithm is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and ...

 

A
=
1
-
9
ALGORITHM
103
49
4
A
=
1
-
10
ALGORITHMS
122
59
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
-
ALGORITHMS
-
-
-
-
-
-
-
1
A
1
1
1
-
-
-
-
1
L
12
3
3
-
-
-
-
1
G
7
7
7
-
-
-
-
1
O
15
6
6
-
-
-
-
1
R
18
9
9
-
-
-
-
1
I
9
9
9
-
-
-
-
1
T
20
2
2
-
-
-
-
1
H
8
8
8
-
-
-
-
1
M+S
32
14
5
A
=
1
-
10
ALGORITHMS
122
59
50
-
-
-
-
1+0
-
1+2+2
5+9
5+0
A
=
1
-
1
ALGORITHMS
5
14
5
-
-
-
-
-
-
-
1+4
-
A
=
1
-
1
ALGORITHMS
5
5
5

 

 

-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
9
-
8
-
1
+
=
24
2+4
=
6
=
6
-
-
-
-
-
15
-
9
-
8
-
19
+
=
51
5+1
=
6
=
6
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
3
7
-
9
-
2
-
4
-
+
=
26
2+6
=
8
=
8
-
-
1
12
7
-
18
-
20
-
13
-
+
=
71
7+1
=
8
=
8
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
1
12
7
15
18
9
20
8
13
19
+
=
122
1+2+2
=
5
1+0
5
-
-
1
3
7
6
9
9
2
8
4
1
+
=
50
5+0
=
5
1+0
5
-
10
A
L
G
O
R
I
T
H
M
S
-
-
-
-
-
-
-
-
-
-
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
FIVE
5
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
9
-
-
-
-
-
-
9
occurs
x
1
=
9
5
10
A
L
G
O
R
I
T
H
M
S
-
-
40
-
-
10
-
41
-
1+0
-
-
-
-
9
9
-
-
-
-
-
-
4+0
-
-
1+0
-
4+1
5
1
A
L
G
O
R
I
T
H
M
S
-
-
4
-
-
1
-
5
-
-
1
3
7
6
9
9
2
8
4
1
-
-
-
-
-
-
-
-
5
1
A
L
G
O
R
I
T
H
M
S
-
-
4
-
-
1
-
5

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
T
=
2
-
1
T
20
2
2
L
=
3
-
1
L
12
3
3
M
=
4
-
1
M
13
4
4
5
-
5
-
-
5
-
-
5
O
=
6
-
1
O
15
6
6
G
=
7
-
1
G
7
7
7
H
=
8
-
1
H
8
8
8
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

A
=
1
-
10
ALGORITHMS
122
59
5
A
=
1
-
9
ALGORITHM
103
49
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ALGORITHM
-
-
-
A
=
1
-
1
A
1
1
1
L
=
3
-
1
L
12
3
3
G
=
7
-
1
G
7
7
7
O
=
6
-
1
O
15
6
6
R
=
9
-
1
R
18
9
9
I
=
9
-
1
I
9
9
9
T
=
2
-
1
T
20
2
2
H
=
8
-
1
H
8
8
8
M
=
4
-
1
M
13
4
4
-
-
49
-
9
ALGORITHM
103
49
49
-
-
4+9
-
-
-
1+0+3
4+9
4+9
-
-
13
-
9
ALGORITHM
103
13
13
-
-
1+3
-
-
-
-
1+3
1+3
-
-
4
-
9
ALGORITHM
103
4
4

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBERS REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
5
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
ALGORITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
-
S
=
1
10
1
S
1
1
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
4
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
-
-
50
-
10
ALGORITHMS
122
50
50
-
2
2
3
4
6
7
8
18
-
-
5+0
-
1+0
-
1+2+2
5+5
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
1
ALGORITHMS
5
5
5
-
2
2
3
4
6
7
8
9

 

 

 
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