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Evokation
 
 
Index
 

 

 

 

 




 

 

 

 

 

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

 

 

Nature's Numbers
Ian Stewart 1995

Numerology is the easiest-and consequently the most dangerous-method for finding patterns. It is easy because anybody can do it and dangerous for the same reason. The difficulty lies in distinguishing significant numerical patterns from accidental ones. Here's a case in point. Kepler was fascinated with patterns in nature, and he devoted much of his life to looking for them in the behaviour of the planets. He devised a simple and tidy theory for the existence of precisely six planets (in his time only Mercury, Venus, Earth, Mars, Jupiter, and Saturn were known). He also discovered a very strange pattern relating the orbital period of a / planet- the time it takes to go once around the Sun-to its distance from the Sun. Recall that the square of a number is what you get when you multiply it by itself: for example, the square of 4 is 4 x 4 = 16. Similarly, the cube is what you get when you multiply it by itself twice: for example, the cube of 4 is 4 x 4 x 4 = 64. Kepler found that if you take the cube of the distance of any planet from the Sun and divide it by the square of its orbital period, you always get the same number. It was not an especially elegant number, but it was the same for all six planets.

Which of these numerological observations is the more significant? The verdict of posterity is that it is the second one, the complicated and rather arbitrary calculation with squares and cubes. This numerical pattern was one of the key steps towards Isaac Newton's theory of gravity, which has explained all sorts of puzzles about the motion of stars and planets. In contrast, Kepler's neat, tidy theory for the number of planets has been buried without trace. For a start it must have been wrong, because we now know of nine planets, not six. There could be even more, farther out from the Sun, and small enough to be undetectable But more important, we no longer expect to find a neat, tidy theory for the number of planets. We think that the Solar System condensed from a cloud of gas surrounding the Sun, and the number of planets presumably depended on the amount of matter in the gas cloud, how it was distributed, and how fast and in what directions it was moving. An equally plausible gas cloud could have given us eight planets, or eleven; the number is accidental, depending on the initial conditions of the gas cloud, rather than universal, reflecting a general law of nature"

Page 6

" The big problem with numerological pattern-seeking is that it generates millions of accidentals for each universal. Nor is it always obvious which is which. For example, there are three stars, roughly equally spaced and in a straight line, in the belt of the constellation Orion. Is that a clue to a significant law of nature?
Here's a similar question. Io, Europa, and Ganymede are three of Jupiter's larger satellites. They orbit the planet in , respectively, 1.77, 3.55, and 7.16 days. Each of these numbers is almost exactly twice the previous one. Is that a significant pattern? Three stars in a row, in terms of orbital period. Which pattern if either, is an important clue..."
    "… In addition to numerical patterns there are geometric ones…"
    "… Until recently the main shapes that appealed to mathematicians were very simple ones: triangles, squares, pen / Page 7 /tagons, hexagons, circles, ellipses, spirals, cubes, spheres, cones, and so on. All of these shapes can be found in nature, although some are far more common, or more evident, than others. The rainbow, for example, is a collection of circles, one for each colour. We don't normally see the entire circle just an arc; but rainbows seen from the air can be complete circles. You also see circles in the ripples on a pond, in the human eye, and on butterflies wings.
         Talking of ripples, the flow of fluids provides an inexhaustible supply of nature's patterns. There are waves of many different kinds-surging toward a beach in parallel ranks, spreading in a V-shape behind a moving boat, radiating outward from an underwater earthquake…"
"…There are swirling spiral whirlpools and tiny vortices. And there is the apparently structureless, random frothing of turbulent flow, one of the great enigmas of mathematics and physics. There are similar patterns in the atmosphere, too, the most dramatic being the vast spiral of a hurricane…"
    "…There are also wave patterns on land. The most strikingly mathematical landscapes on Earth are to be found in the great ergs, or sand oceans, of the Arabian and Sahara deserts. Even when the wind blows steadily in a fixed direction, sand dunes form. The simplest pattern is that of transverse dunes, which-just like ocean waves-line up in parallel straight rows at right angles to the prevailing wind direction. Sometimes the rows themselves become wavy in which case they are called barchanoid ridges; sometimes they break up into / Page 8 / innumerable shield-shaped barchan dunes. If the sand is slightly moist, and there is a little vegetation to bind it together, then you may find parabolic dunes-shaped like a U, with the rounded end pointing in the direction of the wind. These sometimes occur in clusters, and they resemble the teeth of a rake. If the wind direction is variable, other forms become possible. For example, clusters of sand shaped dunes can form, each having several irregular arms radiating from a central peak. They arrange themselves in a random pattern of spots.

Chapter 6

Page 81

"Nature's symmetries can be found on every scale, from the structure of subatomic particles to that of the entire universe. Many chemical molecules are symmetric. The methane molecule is a tetrahedron - a triangular-sided pyramid - with one carbon atom at its center and four hydrogen atoms at its corners Benzene has the sixfold symmetry of a regular hexagon. The fashionable molecule buckminsterfullerene is a truncated icosahedral cage of sixty carbon atoms. (An icosahedron is a regular solid with twenty triangular faces;
"truncated" means that the corners are cut off.) Its symmetry lends it a remarkable stability, which has opened up new possibilities for organic chemistry.
    On a slightly larger scale than molecules, we find symmetries in cellular structure; at the heart of cellular replication lies a tiny piece of mechanical engineering. Deep within each / Page 82  / living cell, there is a rather shapeless structure known as the centrosome, which sprouts long thin microtubules, basic components of the cell's internal "skeleton", like a diminutive sea urchin. Centrsomes were first discovered in 1887 and play an important role in organizing cell division. How-ever in one respect the structure of the centresome is astonishingly symmetric. Inside it has two structures, known as centrioles, positioned at right angles to each other. Each centriole is cylindrical, made from twenty-seven microtubules fused together along their lengths in threes, and arranged with perfect ninefold symmetry. The microtubules themselves also have an astonishingley regular symmetric form. They are hollow tubes, made from a perfect regular checkerboard pattern of units that contain two distinct proteins, alpha- and betatubulin. One day, perhaps, we will understand why nature chose these symmetric forms. But it is amazing to see symmetric structures at the core of a living cell. "

 

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Preface to the New Edition

Page 5
Twelve years ago a little boy entered my imagination as he hopped across the centuries and played with numbers. I began to see how the simple architecture of our decimal system could be constructed in secret ways — not a building project this time but an abstract one. On the surface of our arithmetic countless combinations of numbers take part in tedious and exacting calculations but underneath it all there is pattern, governed by a repeating code of integers. The Sigma Code reduces numbers to a single digit and the illusion of the many is seen to be but the reflection of a few. This is not a book on maths: this is a book for anyone who can carry out simple sums in their heads, and who won't be short-changed knowingly.
When Number 9 first came out I received mail from many who played with numbers. They chased patterns; some had special numbers and even mystical systems. I was tempted to write about numerology but resisted. I wanted to write about the intricacy of what the.. numbers actually do and leave the reader to wonder about the larger irrational that seems to hover around such constructions.
If I were writing this book today the numbers would have featured in a wider context of structuring nature's patterns, and also playing the role of animator in algorithms that create unique architectural forms and shapes. I would also include my previous research into other base systems. But this book was a first step which came from a child-like urge, like playing with building blocks, to build out of our numbers — just the simple 1, 2, 3, up to number 9.

 

RESEARCH R E SEARCH ER RESEARCH

 

 

0
-
Z
=
8
-
4
ZERO
64
28
1
1
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
56
29
2
3
-
T
=
2
-
5
THREE
52
16
7
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
65
20
2
6
-
S
=
1
-
3
SIX
42
24
6
7
-
S
=
1
-
5
SEVEN
58
13
4
8
-
E
=
5
-
5
EIGHT
49
31
4
9
-
N
=
5
-
4
NINE
42
24
6
45
-
-
-
42
-
40
Add
522
225
45
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
9
-
-
-
6
-
4
Deduce
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
-
-
7
-
-
2
-
T
=
2
-
3
TWO
56
29
2
-
-
2
-
-
-
-
-
-
-
3
-
T
=
2
-
5
THREE
52
16
7
-
-
-
-
-
-
-
7
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
65
20
2
-
-
2
-
-
-
-
-
-
-
6
-
S
=
1
-
3
SIX
42
24
6
-
-
-
-
-
-
6
-
-
-
7
-
S
=
1
-
5
SEVEN
58
13
4
-
-
-
-
4
-
-
-
-
-
8
-
E
=
5
-
5
EIGHT
49
31
4
-
-
-
-
4
-
-
-
-
-
9
-
N
=
5
-
4
NINE
42
24
6
-
-
-
-
-
-
6
-
-
-
45
-
-
-
42
-
40
Add
522
225
45
-
1
4
3
8
5
18
14
8
9
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
-
-
-
-
-
-
1+8
1+4
-
-
9
-
-
-
6
-
4
Deduce
9
9
9
-
1
4
3
8
5
9
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
4
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
7
2
-
T
=
2
-
3
TWO
56
29
2
-
-
2
-
-
-
3
-
T
=
2
-
5
THREE
52
16
7
-
-
-
-
-
7
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
6
-
5
-
F
=
6
-
4
FIVE
65
20
2
-
-
2
-
-
-
6
-
S
=
1
-
3
SIX
42
24
6
-
-
-
-
6
-
7
-
S
=
1
-
5
SEVEN
58
13
4
-
-
-
4
-
-
8
-
E
=
5
-
5
EIGHT
49
31
4
-
-
-
4
-
-
9
-
N
=
5
-
4
NINE
42
24
6
-
-
-
-
6
-
45
-
-
-
42
-
40
Add
522
225
45
-
1
4
8
18
14
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
-
-
-
-
1+8
1+4
9
-
-
-
6
-
4
Deduce
9
9
9
-
1
4
8
9
5

 

 

 

N
U
M
B
E
R
S
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ZERO
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
Z
=
8
-
-
Z
26
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
O
=
5
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
0
-
-
-
27
-
4
ZERO
64
28
2+8
=
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ONE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
1
-
-
-
16
-
3
ONE
34
16
1+6
=
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
TWO
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
W
=
5
5
-
W
23
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
2
-
-
-
13
-
3
TWO
58
13
1+3
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THREE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
3
-
-
-
29
-
5
THREE
56
29
2+9
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FOUR
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
U
=
3
-
-
U
21
3
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
4
-
-
-
24
-
4
FOUR
60
24
2+4
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
FIVE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
24
-
4
FIVE
42
24
2+4
=
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SIX
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
X
=
6
-
-
X
24
6
6
-
-
-
-
-
-
-
-
6
-
-
-
6
-
-
-
16
-
3
SIX
52
16
1+6
=
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SEVEN
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
7
-
-
-
20
-
5
SEVEN
65
20
2+0
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
G
=
7
-
-
G
7
7
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
T
=
2
-
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
8
-
-
-
31
-
5
EIGHT
49
31
3+1
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
NINE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
9
-
-
-
24
-
4
NINE
42
24
2+4
=
6
-
-
-
-
-
-
-
-
-
-
45
-
-
-
-
70
40
Add
522
225
45
-
45
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
-
7+0
4+0
Reduce
5+2+2
2+2+5
4+5
-
4+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
-
7
4
Deduce
9
9
9
-
9
-
2
6
3
8
7
6
7
6
9

 

 

N
U
M
B
E
R
S
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
-
-
-
-
4
ZERO
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
Z
=
8
-
-
Z
26
8
8
-
-
-
-
-
-
-
-
8
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
O
=
5
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
1
-
-
-
-
-
3
ONE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
2
-
-
-
-
-
3
TWO
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
W
=
5
5
-
W
23
5
5
-
-
-
-
-
5
-
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
3
-
-
-
-
-
5
THREE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
-
-
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
8
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
4
-
-
-
-
-
4
FOUR
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
-
-
O
15
6
6
-
-
-
-
-
-
6
-
-
-
-
-
U
=
3
-
-
U
21
3
3
-
-
-
3
-
-
-
-
-
-
-
-
R
=
9
-
-
R
18
9
9
-
-
-
-
-
-
-
-
-
9
5
-
-
-
-
-
4
FIVE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
F
=
6
-
-
F
6
6
6
-
-
-
-
-
-
6
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
6
-
-
-
-
-
3
SIX
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
1
-
-
-
-
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
X
=
6
-
-
X
24
6
6
-
-
-
-
-
-
6
-
-
-
7
-
-
-
-
-
5
SEVEN
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
-
S
19
1
1
-
1
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
V
=
4
-
-
V
22
4
4
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
8
-
-
-
-
-
5
EIGHT
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
G
=
7
-
-
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
H
=
8
-
-
H
8
8
8
-
-
-
-
-
-
-
-
8
-
-
-
T
=
2
-
-
T
20
2
2
-
-
2
-
-
-
-
-
-
-
9
-
-
-
-
-
4
NINE
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
-
-
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
N
=
5
5
-
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
5
-
E
5
5
5
-
-
-
-
-
5
-
-
-
-
45
-
-
-
-
70
40
ADD
522
225
45
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
-
7+0
4+0
REDUCE
5+2+2
2+2+5
4+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
-
7
4
DEDUCE
9
9
9
-
2
6
3
8
7
6
7
6
9

 

 

N
U
M
B
E
R
S
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
ZERO
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Z
=
8
1
1
Z
26
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
E
=
5
2
1
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
R
=
9
3
1
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
0
-
O
=
6
4
4
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
5
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
N
=
5
6
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
1
-
E
=
5
7
3
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
T
=
2
8
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
W
=
5
9
3
W
23
5
5
-
-
-
-
-
-
-
5
-
-
-
-
2
-
O
=
6
10
3
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
T
=
2
11
-
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
H
=
8
12
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
R
=
9
13
-
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
E
=
5
14
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
3
-
E
=
5
15
5
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
F
=
6
16
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
O
=
6
17
-
O
15
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
U
=
3
18
-
U
21
3
3
-
-
-
-
-
3
-
-
-
-
-
-
4
-
R
=
9
19
4
R
18
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
F
=
6
20
-
F
6
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
I
=
9
21
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
V
=
4
22
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
5
-
E
=
5
23
4
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
S
=
1
24
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
I
=
9
25
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
6
-
X
=
6
26
3
X
24
6
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
S
=
1
27
-
S
19
1
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
E
=
5
28
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
V
=
4
29
-
V
22
4
4
-
-
-
-
-
-
4
-
-
-
-
-
-
-
E
=
5
30
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
7
-
N
=
5
31
7
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
E
=
5
32
-
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
33
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
G
=
7
34
-
G
7
7
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
H
=
8
35
-
H
8
8
8
-
-
-
-
-
-
-
-
-
-
8
-
8
-
T
=
2
36
5
T
20
2
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
N
=
5
37
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
-
-
I
=
9
38
-
I
9
9
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
N
=
5
39
-
N
14
5
5
-
-
-
-
-
-
-
5
-
-
-
-
9
-
E
=
5
40
4
E
5
5
5
-
-
-
-
-
-
-
5
-
-
-
-
45
-
-
-
42
-
40
Add
522
225
45
-
45
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
-
4+5
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
6
-
4
Deduce
9
9
9
-
9
-
2
6
3
8
7
6
7
6
9

 

 

1
occurs
x
2
=
2
=
2
2
occurs
x
3
=
6
=
6
3
occurs
x
1
=
3
=
3
4
occurs
x
2
=
8
=
8
5
occurs
x
14
=
70
7+0
7
6
occurs
x
7
=
42
4+2
6
7
occurs
x
1
=
7
=
7
8
occurs
x
3
=
24
2+4
6
9
occurs
x
7
=
63
6+3
9
45
-
-
40
-
225
-
54
4+5
-
-
4+0
-
2+2+5
-
5+4
9
-
-
4
-
9
-
9

 

 

N
U
M
B
E
R
S
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
N
U
M
B
E
R
S
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
Z
-
S
=
1
-
1
S
19
1
1
-
-
1
-
-
-
-
-
-
-
-
E
-
S
=
1
-
2
S
19
1
1
-
-
1
-
-
-
-
-
-
-
-
R
-
T
=
2
-
3
T
20
2
2
-
-
-
2
-
-
-
-
-
-
-
O
-
T
=
2
-
4
T
20
2
2
-
-
-
2
-
-
-
-
-
-
-
O
-
T
=
2
-
5
T
20
2
2
-
-
-
2
-
-
-
-
-
-
-
N
-
U
=
3
-
6
U
21
3
3
-
-
-
-
3
-
-
-
-
-
-
E
-
V
=
4
-
7
V
22
4
4
-
-
-
-
-
4
-
-
-
-
-
T
-
V
=
4
-
8
V
22
4
4
-
-
-
-
-
4
-
-
-
-
-
W
-
E
=
5
-
9
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
O
-
N
=
5
-
10
N
14
5
5
-
-
-
-
-
-
5
-
-
-
-
T
-
E
=
5
-
11
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
H
-
W
=
5
-
12
W
23
5
5
-
-
-
-
-
-
5
-
-
-
-
R
-
E
=
5
-
13
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
E
-
E
=
5
-
14
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
E
-
E
=
5
-
15
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
F
-
E
=
5
-
16
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
O
-
N
=
5
-
17
N
14
5
5
-
-
-
-
-
-
5
-
-
-
-
U
-
E
=
5
-
18
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
R
-
E
=
5
-
19
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
F
-
N
=
5
-
20
N
14
5
5
-
-
-
-
-
-
5
-
-
-
-
I
-
N
=
5
-
21
N
14
5
5
-
-
-
-
-
-
5
-
-
-
-
V
-
E
=
5
-
22
E
5
5
5
-
-
-
-
-
-
5
-
-
-
-
E
-
O
=
6
-
23
O
15
6
6
-
-
-
-
-
-
-
6
-
-
-
S
-
O
=
6
-
24
O
15
6
6
-
-
-
-
-
-
-
6
-
-
-
I
-
O
=
6
-
25
O
15
6
6
-
-
-
-
-
-
-
6
-
-
-
X
-
F
=
6
-
26
F
6
6
6
-
-
-
-
-
-
-
6
-
-
-
S
-
O
=
6
-
27
O
15
6
6
-
-
-
-
-
-
-
6
-
-
-
E
-
F
=
6
-
28
F
6
6
6
-
-
-
-
-
-
-
6
-
-
-
V
-
X
=
6
-
29
X
24
6
6
-
-
-
-
-
-
-
6
-
-
-
E
-
G
=
7
-
30
G
7
7
7
-
-
-
-
-
-
-
-
7
-
-
N
-
Z
=
8
-
31
Z
26
8
8
-
-
-
-
-
-
-
-
-
8
-
E
-
H
=
8
-
32
H
8
8
8
-
-
-
-
-
-
-
-
-
8
-
I
-
H
=
8
-
33
H
8
8
8
-
-
-
-
-
-
-
-
-
8
-
G
-
R
=
9
-
34
R
18
9
9
-
-
-
-
-
-
-
-
-
-
9
H
-
R
=
9
-
35
R
18
9
9
-
-
-
-
-
-
-
-
-
-
9
T
-
R
=
9
-
36
R
18
9
9
-
-
-
-
-
-
-
-
-
-
9
N
-
I
=
9
-
37
I
9
9
9
-
-
-
-
-
-
-
-
-
-
9
I
-
I
=
9
-
38
I
9
9
9
-
-
-
-
-
-
-
-
-
-
9
N
-
I
=
9
-
39
I
9
9
9
-
-
-
-
-
-
-
-
-
-
9
E
-
I
=
9
-
40
I
9
9
9
-
-
-
-
-
-
-
-
-
-
9
45
-
-
-
42
-
-
Add
522
225
45
-
-
2
6
3
8
70
42
7
24
63
4+5
-
-
-
4+2
-
-
Reduce
5+2+2
2+2+5
4+5
-
-
-
-
-
-
7+0
4+2
-
2+4
6+3
9
-
-
-
6
-
-
Deduce
9
9
9
-
-
2
6
3
8
7
6
7
6
9

 

 

 

Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
-
26
5
18
15
-
15
14
5
-
20
23
15
-
20
8
18
5
5
-
6
15
21
18
-
6
9
22
5
-
19
9
24
-
19
5
22
5
14
-
5
9
7
8
20
-
14
9
14
5
522
8
5
9
6
-
6
5
5
-
2
5
6
-
2
8
9
5
5
-
6
6
3
9
-
6
9
4
5
-
1
9
6
-
1
5
4
5
5
-
5
9
7
8
2
-
5
9
5
5
225
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
5
-
-
-
--
5
5
-
-
5
-
-
-
-
-
5
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
5
5
-
5
-
-
5
5
-
-
-
-
5
5
-
-
-
6
-
6
-
-
-
-
--
6
-
-
-
-
-
-
-
6
6
-
-
-
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
9
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
45
-
5
-
-
-
--
5
5
-
-
5
-
-
-
-
-
5
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
5
5
-
5
-
-
5
5
-
-
-
-
5
4+5
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
9
8
5
9
6
-
6
5
5
-
2
5
6
-
2
8
9
5
5
-
6
6
3
9
-
6
9
4
5
-
1
9
6
-
1
5
4
5
5
-
5
9
7
8
2
-
5
9
5
5
-
Z
E
R
O
-
O
N
E
-
T
W
O
-
T
H
R
E
E
-
F
O
U
R
-
F
I
V
E
-
S
I
X
-
S
E
V
E
N
-
E
I
G
H
T
-
N
I
N
E
9

 

 

1
occurs
x
2
=
2
=
2
2
occurs
x
3
=
6
=
6
3
occurs
x
1
=
3
=
3
4
occurs
x
2
=
8
=
8
5
occurs
x
14
=
70
7+0
7
6
occurs
x
7
=
42
4+2
6
7
occurs
x
1
=
7
=
7
8
occurs
x
3
=
24
2+4
6
9
occurs
x
7
=
63
6+3
9
45
-
-
40
-
225
-
54
4+5
-
-
4+0
-
2+2+5
-
5+4
9
-
-
4
-
9
-
9

 

 

0
-
4
ZERO
8
5
9
6
-
=
28
2+8
=
10
1+0
1
1
-
3
ONE
6
5
5
-
-
=
16
1+6
=
7
-
7
2
-
3
TWO
2
5
6
-
-
=
13
1+3
=
4
-
4
3
-
5
THREE
2
8
9
5
5
=
29
2+9
=
11
1+1
2
4
-
4
FOUR
6
6
3
9
-
=
24
2+4
=
6
-
6
5
-
4
FIVE
6
9
4
5
-
=
24
2+4
=
6
-
6
6
-
3
SIX
1
9
6
-
-
=
16
1+6
=
7
-
7
7
-
5
SEVEN
1
5
4
5
5
=
20
2+0
=
2
-
2
8
-
5
EIGHT
5
9
7
8
2
=
31
3+1
=
4
-
4
9
-
4
NINE
5
9
5
5
-
=
24
2+4
=
6
-
6
45
-
40
Add
42
70
58
43
12
-
225
-
-
63
-
45
4+5
-
4+0
-
4+2
7+0
5+8
4+3
1+2
-
2+2+5
-
-
6+3
-
4+5
9
-
4
Reduce
6
7
13
7
3
-
9
-
-
9
-
9
-
-
-
-
-
-
1+3
-
-
-
-
-
-
-
-
-
9
-
4
Deduce
6
7
4
7
3
-
9
-
-
9
-
9

 

 

O
=
6
-
3
ONE
34
16
7
T
=
2
-
3
TWO
56
29
2
T
=
2
-
5
THREE
52
16
7
F
=
6
-
4
FOUR
60
24
6
F
=
6
-
4
FIVE
65
20
2
S
=
1
-
3
SIX
42
24
6
S
=
1
-
5
SEVEN
58
13
4
E
=
5
-
5
EIGHT
49
31
4
N
=
5
-
4
NINE
42
24
6
-
-
34
-
36
-
458
197
44
-
-
3+4
-
3+6
-
4+5+8
1+9+7
4+4
-
-
7
4
9
-
17
17
8
-
-
-
-
-
-
1+7
1+7
-
-
-
7
-
9
-
8
8
8

 

 

O
=
6
3
ONE
34
16
7
T
=
2
3
TWO
58
13
4
T
=
2
5
THREE
56
29
2
F
=
6
4
FOUR
60
24
6
F
+
6
4
FIVE
42
24
6
S
=
1
3
SIX
52
16
7
S
=
1
5
SEVEN
65
20
2
E
=
5
5
EIGHT
49
31
4
N
=
5
4
NINE
42
24
6
-
-
39
36
-
458
197
44
-
-
3+9
3+6
-
4+5+8
1+9+7
4+4
-
-
12
9
-
17
17
8
-
-
1+2
-
-
-
-
-
-
-
3
9
-
8
8
8

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Page 32

5


To Sorcerers and Magicians number FIVEis the most powerful - five is the mark of the pentacle, a five pointed star drawn by extending the sides of a Pentagon. Five surely is in the possession of the occult. And the Pentagon is the geometric figure in which the golden ratio of classical art and architecture is found most.

 

 

THE

BALANCING

ONE TWO THREE FOUR

FIVE

NINE EIGHT SEVEN SIX

 

 

O
=
15
-
3
ONE
34
16
7
-
1
T
=
20
-
3
TWO
58
13
4
-
2
T
=
20
-
5
THREE
56
29
2
-
3
F
=
6
-
4
FOUR
60
24
6
-
4
-
-
61
-
15
Add
208
82
19
-
10
-
-
6+1
-
1+5
Reduce
2+0+8
8+2
1+9
-
1+0
-
-
7
-
6
Deduce
10
10
10
-
1
-
-
-
-
-
Produce
1+0
1+0
1+0
-
-
-
-
7
-
6
Essence
1
1
1
-
1

 

 

N
=
14
-
4
NINE
42
24
6
-
9
E
=
5
-
5
EIGHT
49
31
4
-
8
S
=
19
-
5
SEVEN
65
20
2
-
7
S
=
19
-
3
SIX
52
16
7
-
6
-
-
57
-
17
Add
208
91
19
-
30
-
-
5+7
-
1+7
Reduce
2+0+8
9+1
1+9
-
3+0
-
-
12
-
8
Deduce
10
10
10
-
3
-
-
1+2
-
-
Produce
1+0
1+0
1+0
-
-
-
-
3
-
8
Essence
1
1
1
-
3

 

 

4
FIVE
42
24
6

 

1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1

 

15
ONE TWO THREE FOUR
208
82
1
4
FIVE
42
24
6
17
NINE EIGHT SEVEN SIX
208
91
1

 

 

3
ONE
34
16
7
-
3
SIX
52
16
7
3
TWO
58
13
4
-
5
SEVEN
65
20
2
5
THREE
56
29
2
-
5
EIGHT
49
31
4
4
FOUR
60
24
6
-
4
NINE
42
24
6
15
Add
208
82
19
-
17
Add
208
91
19
1+5
Reduce
2+0+8
8+2
1+9
-
1+7
Reduce
2+0+8
9+1
1+9
6
Deduce
10
10
10
-
8
Deduce
10
10
10
-
Produce
1+0
1+0
1+0
-
-
Produce
1+0
1+0
1+0
6
Essence
1
1
1
-
8
Essence
1
1
1

 

ALWAYS BALANCING IS THAT FIVE THAT FIVE IS BALANCING ALWAYS

 

 

F
=
6
-
3
FOR
39
21
3
E
=
5
-
5
EVERY
75
30
3
A
=
1
-
6
ACTION
62
26
8
T
=
2
-
5
THERE
56
29
2
I
=
9
-
2
IS
28
19
1
A
=
1
-
2
AN
15
6
6
E
=
5
-
5
EQUAL
56
20
2
A
=
1
-
3
AND
19
10
1
O
=
6
-
8
OPPOSITE
115
43
7
R
=
9
-
8
REACTION
85
40
4
-
-
45
4
46
First Total
550
244
37
-
-
4+5
-
4+6
Add to Reduce
5+5+0
2+4+4
3+7
Q
-
9
-
10
Second Total
10
10
10
-
-
-
4
1+0
Reduce to Deduce
1+0
1+0
1+0
-
-
9
5
1
Essence of Number
1
1
1

 

 

 

-
-
-
-
-
-
-
-
-
-
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
-
9
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
57
First Total
995
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+7
Add to Reduce
9+9+5
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
12
Second Total
23
14
14
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+2
Reduce to Deduce
2+3
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
3
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
8
9

 

 

O
=
6
-
6
OSIRIS
89
35
8
I
=
9
-
4
ISIS
56
20
2
S
=
1
4
6
SIRIUS
95
32
5
-
-
16
-
16
First Total
240
87
15
-
-
1+6
-
1+6
Add to Reduce
2+4+0
8+7
1+5
4
4
7
4
7
Second Total
6
15
6
-
-
-
-
-
Reduce to Deduce
-
1+5
-
-
-
7
-
7
Essence of Number
6
6
6

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
O
=
6
2
6
OSIRIS
89
53
8
-
-
-
-
-
-
-
-
-
-
I
=
9
3
4
ISIS
56
38
2
-
-
-
-
-
-
-
-
-
-
S
=
1
4
6
SIRIUS
95
50
5
-
-
-
-
-
-
-
-
-
-
-
-
16
-
16
-
240
141
15
-
1
2
3
4
5
6
7
8
9
-
-
1+6
-
1+6
-
4+0+5
1+4+1
1+5
-
-
-
-
-
-
-
-
-
-
-
-
7
-
7
-
6
6
6
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
O
=
6
1
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
S
=
1
2
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
7
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
10
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
I
=
9
12
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
13
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
U
=
3
15
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
S
=
1
16
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
87
-
16
-
240
141
87
-
6
2
3
4
5
6
7
8
72
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7+2
-
-
-
-
-
-
-
-
-
-
6
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
O
=
6
2
6
OSIRIS
89
53
8
-
-
-
-
-
-
-
-
-
-
I
=
9
3
4
ISIS
56
38
2
-
-
-
-
-
-
-
-
-
-
S
=
1
4
6
SIRIUS
95
50
5
-
-
-
-
-
-
-
-
-
-
-
-
16
-
16
-
240
141
15
-
1
2
3
4
5
6
7
8
9
-
-
1+6
-
1+6
-
2+4+0
1+4+1
1+5
-
-
-
-
-
-
-
-
-
-
-
-
7
-
7
-
6
6
6
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
O
=
6
-
6
OSIRIS
89
53
8
-
-
-
-
-
-
-
-
-
-
I
=
9
-
4
ISIS
56
38
2
-
-
-
-
-
-
-
-
-
-
S
=
1
-
6
SIRIUS
95
50
5
-
-
-
-
-
-
-
-
-
-
-
-
16
-
16
-
240
141
15
-
1
2
3
4
5
6
7
8
9
-
-
1+6
-
1+6
-
4+0+5
1+4+1
1+5
-
-
-
-
-
-
-
-
-
-
-
-
7
-
7
-
6
6
6
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
2
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
10
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
S
=
1
16
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
U
=
3
15
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
O
=
6
1
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
I
=
9
3
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
7
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
12
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
13
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
87
-
16
-
240
141
87
-
6
2
3
4
5
6
7
8
72
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7+2
-
-
-
-
-
-
-
-
-
-
6
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
O
=
6
2
6
OSIRIS
89
53
8
-
-
-
-
-
-
-
-
-
-
I
=
9
3
4
ISIS
56
38
2
-
-
-
-
-
-
-
-
-
-
S
=
1
4
6
SIRIUS
95
50
5
-
-
-
-
-
-
-
-
-
-
-
-
16
-
16
-
240
141
15
-
1
2
3
4
5
6
7
8
9
-
-
1+6
-
1+6
-
2+4+0
1+4+1
1+5
-
-
-
-
-
-
-
-
-
-
-
-
7
-
7
-
6
6
6
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
3
6
9
O
=
6
-
6
OSIRIS
89
53
8
-
-
-
-
-
I
=
9
-
4
ISIS
56
38
2
-
-
-
-
-
S
=
1
-
6
SIRIUS
95
50
5
-
-
-
-
-
-
-
16
-
16
-
240
141
15
-
1
3
6
9
-
-
1+6
-
1+6
-
4+0+5
1+4+1
1+5
-
-
-
-
-
-
-
7
-
7
-
6
6
6
-
1
3
6
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
3
6
9
S
=
1
2
1
S
19
10
1
-
1
-
-
-
S
=
1
6
1
S
19
10
1
-
1
-
-
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
S
=
1
10
1
S
19
10
1
-
1
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
-
-
S
=
1
16
1
S
19
10
1
-
1
-
-
-
U
=
3
15
1
U
21
3
3
-
-
3
-
-
O
=
6
1
1
O
15
6
6
-
-
-
6
-
I
=
9
3
1
I
9
9
9
-
-
-
-
9
R
=
9
4
1
R
18
9
9
-
-
-
-
9
I
=
9
5
1
I
9
9
9
-
-
-
-
9
I
=
9
7
1
I
9
9
9
-
-
-
-
9
I
=
9
9
1
I
9
9
9
-
-
-
-
9
I
=
9
12
1
I
9
9
9
-
-
-
-
9
R
=
9
13
1
R
18
9
9
-
-
-
-
9
I
=
9
14
1
I
9
9
9
-
-
-
-
9
-
-
87
-
16
-
240
141
87
-
6
3
6
72
-
-
-
-
-
-
-
-
-
-
-
-
-
7+2
-
-
-
-
-
-
-
-
-
-
6
3
6
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
3
6
9
O
=
6
2
6
OSIRIS
89
53
8
-
-
-
-
-
I
=
9
3
4
ISIS
56
38
2
-
-
-
-
-
S
=
1
4
6
SIRIUS
95
50
5
-
-
-
-
-
-
-
16
-
16
-
240
141
15
-
1
3
6
9
-
-
1+6
-
1+6
-
2+4+0
1+4+1
1+5
-
-
-
-
-
-
-
7
-
7
-
6
6
6
-
1
3
6
9

 

 

I
=
9
-
1
I
9
9
9
T
=
2
-
4
THAT
49
13
4
A
=
1
-
2
AM
14
5
5
T
=
2
-
4
THAT
49
13
4
T
=
2
-
4
TIME
47
20
2
E
=
7
-
5
EMITS
66
21
3
-
-
23
4
20
Add to Reduce
234
81
27
-
-
2+3
-
2+0
Reduce to Deduce
2+3+4
8+1
2+7
-
-
5
-
2
Essence of Number
9
9
9

 

 

S
=
1
-
-
SOPHIA
-
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
1
O
15
6
6
-
-
-
-
1
P
16
7
7
-
-
-
-
1
H
8
8
8
-
-
-
-
1
I
9
9
9
-
-
-
-
1
A
1
1
1
S
=
1
-
6
SOPHIA
68
41
32
-
-
-
-
-
-
6+8
6+3
3+2
S
=
1
-
6
SOPHIA
14
5
5
-
-
-
-
-
-
1+4
-
-
S
=
1
-
6
SOPHIA
5
5
5

 

 

-
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
1
6
-
8
9
-
+
=
24
2+4
=
6
=
6
-
-
19
15
-
8
9
-
+
=
51
5+1
=
6
=
6
-
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
1
+
=
8
-
=
8
=
8
-
-
-
-
16
-
-
1
+
=
17
1+7
=
17
1+7
8
-
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
19
15
16
8
9
1
+
=
68
6+8
=
14
1+4
5
-
-
1
6
7
8
9
1
+
=
32
3+2
=
5
-
5
-
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
2
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
--
-
-
-
-
-
-
3
THREE
3
-
-
-
4
-
-
-
-
-
-
-
-
-
4
FOUR
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
occurs
x
1
=
9
14
20
S
O
P
H
I
A
-
-
31
-
-
6
-
32
1+4
2+0
-
-
-
-
9
-
-
-
3+1
-
-
-
-
3+2
5
2
S
O
P
H
I
A
-
-
3
-
-
6
-
5
-
-
1
6
7
8
9
1
-
-
-
-
-
-
-
-
5
2
S
O
P
H
I
A
-
-
3
-
-
6
-
5

 

 

20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
1
6
-
8
9
-
+
=
24
2+4
=
6
=
6
-
19
15
-
8
9
-
+
=
51
5+1
=
6
=
6
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
-
-
7
-
-
1
+
=
8
-
=
8
=
8
-
-
-
16
-
-
1
+
=
17
1+7
=
17
1+7
8
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
19
15
16
8
9
1
+
=
68
6+8
=
14
1+4
5
-
1
6
7
8
9
1
+
=
32
3+2
=
5
-
5
20
S
O
P
H
I
A
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
6
-
-
-
-
-
-
6
occurs
x
1
=
6
-
-
-
7
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
8
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
20
S
O
P
H
I
A
-
-
31
-
-
6
-
32
2+0
-
-
-
-
9
-
-
-
3+1
-
-
-
-
3+2
2
S
O
P
H
I
A
-
-
3
-
-
6
-
5
-
1
6
7
8
9
1
-
-
-
-
-
-
-
-
2
S
O
P
H
I
A
-
-
3
-
-
6
-
5

 

 

 

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

 

 

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

 

AUTUMN ATUM AUTUMN

 

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

 

 

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

 

 

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

 

AUTUMN ATUM AUTUMN

QUANTUM ATUM QUANTUM

 

AUTUMN ATUM AUTUMN

QUANTUM ATUM QUANTUM

 

 

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

 

 

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

 

 

RA ATUM ATUM RA

 

 

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

 

 

8
CALCULUS
-
-
-
-
-
C
3
3
-
3
-
A
1
1
1
-
-
L
12
3
-
3
-
C
3
3
-
3
-
U
21
3
-
3
-
L
12
3
-
3
-
U
21
3
-
3
-
S
19
10
1
-
8
CALCULUS
92
29
2
18
-
-
9+2
2+9
-
1+8
8
CALCULUS
11
11
2
9
-
-
-
-
-
-
8
CALCULUS
2
2
2
9

 

 

-
-
-
-
-
PROBLEMS
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
S
=
1
8
1
S
1
10
1
-
1
-
-
-
-
-
-
8
-
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
PROBLEMS
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
8
1
S
1
10
1
-
1
-
-
-
-
-
-
8
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 

D
=
4
-
-
DICTIONARY
-
-
-
-
-
-
-
1

D

4
4
4
-
-
-
-
1

I

9
9
9
-
-
-
-
1

C

3
3
3
-
-
-
-
1

T

20
2
2
-
-
-
-
1

I

9
9
9
-
-
-
-
1

O

15
6
6
-
-
-
-
1

N

14
5
5
-
-
-
-
1

A

1
1
1
-
-
-
-
1

R

18
9
9
-
-
-
-
1

Y

25
7
7
D
=
4
Q
10
DICTIONARY
118
55
55
-
-
-
-
1+0
-
1+1+8
5+5
5+5
D
=
4
Q
1
DICTIONARY
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
D
=
4
Q
1
DICTIONARY
1
1
1

 

 

-
-
-
-
-
DICTIONARY
-
-
-
-
1
2
3
4
5
6
7
8
9
D
=
4
1
1

D

4
4
4
-
-
-
-
4
-
-
-
8
-
I
=
9
2
1

I

9
9
9
-
-
-
-
-
-
-
-
8
9
C
=
3
3
1

C

3
3
3
-
-
-
3
-
-
-
-
8
-
T
=
2
4
1

T

20
2
2
-
-
2
-
-
-
-
-
8
-
I
=
9
5
1

I

9
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
6
1

O

15
6
6
-
-
-
-
-
-
6
-
8
-
N
=
5
7
1

N

14
5
5
-
-
-
-
-
5
-
-
8
-
A
=
1
8
1

A

1
1
1
-
1
-
-
-
-
-
-
8
-
R
=
9
9
1

R

18
9
9
-
-
-
-
-
-
-
-
8
9
Y
=
7
10
1

Y

25
7
7
-
-
-
-
-
-
-
7
8
-
-
-
55
-
10
DICTIONARY
118
55
55
-
1
2
3
4
5
6
7
8
27
-
-
5+5
-
1+0
-
1+1+8
5+5
5+5
-
-
-
-
-
-
-
-
-
2+7
-
-
10
-
1
DICTIONARY
10
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
DICTIONARY
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
DICTIONARY
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
C
=
3
3
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
D
=
4
4
1
D
4
4
4
-
-
-
-
4
-
-
-
8
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
6
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
Y
=
7
7
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
8
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
55
-
10
DICTIONARY
118
55
55
-
1
2
3
4
5
6
7
8
27
-
-
5+5
-
1+0
-
1+1+8
5+5
5+5
-
-
-
-
-
-
-
-
-
2+7
-
-
10
-
1
DICTIONARY
10
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
1
DICTIONARY
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 

D
=
4
-
-
DICTIONARY
-
-
-
-
-
-
-
1
D
4
4
4
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C+T
23
5
5
-
-
-
-
1
I
9
9
9
-
-
-
-
1
O+N+A
30
12
3
-
-
-
-
1
R
18
9
9
-
-
-
-
1
Y
25
7
7
D
=
4
Q
10
DICTIONARY
118
55
46
-
-
-
-
1+0
-
1+1+8
5+5
4+6
D
=
4
Q
1
DICTIONARY
10
10
10
-
-
-
-
-
-
1+0
1+0
-
D
=
4
Q
1
DICTIONARY
1
1
1

 

 

T
=
2
Q
3
THE
33
15
6
E
=
5
Q
7
ENGLISH
74
38
2
D
=
4
Q
10
DICTIONARY
118
55
1
-
-
11
Q
20
Add to Reduce
225
108
9
Q
Q
1+1
-
2+0
Reduce to Deduce
1+1+8
1+0+8
-
-
-
2
Q
2
Essence of Number
9
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 = 351 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

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 = 126 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

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 = Z Y X W V U T S R Q P O N M L K J I H G F E D C B A

 

 

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 351 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 126 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

ABCDEFGH I JKLMNOPQ R STUVWXYZ = 9 = ZYXWVUTS R QPONMLKJ I HGFEDCBA

 

 

L
=
3
6
LETTER
73
28
1
N
=
6
6
NUMBER
80
26
8
T
=
2
13
TRANSPOSITION
189
63
9
-
-
11
25
Add to Reduce
342
117
18
-
-
1+1
2+5
Reduce to Deduce
3+4+2
1+1+7
1+8
=-
-
2
7
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
CRYPTOGRAPHY
-
-
-
-
-
-
-
4
5
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
-
3
4
5
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
4
5
-
-
-
9
Y
=
7
-
7
Y
25
7
7
-
-
-
-
4
5
-
7
-
-
P
=
7
-
1
P
16
7
7
-
-
-
-
4
5
-
7
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
4
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
4
5
6
-
-
-
G
=
7
-
7
G
7
7
7
-
-
-
-
4
5
-
7
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
4
5
-
-
-
9
A
=
1
-
1
A
1
1
1
-
1
-
-
4
5
-
-
-
-
P
=
7
-
1
P
16
7
7
-
-
-
-
4
5
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
4
5
-
-
8
-
Y
=
7
-
7
Y
25
7
7
-
-
-
-
4
5
-
7
-
-
-
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12
CRYPTOGRAPHY
172
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7+3
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3
CRYPTOGRAPHY
10
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1
2
3
4
5
6
8
8
9
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1
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3
CRYPTOGRAPHY
1
1
1
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1
2
3
4
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6
8
8
9

 

 

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CRYPTOGRAPHY
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CRYPTOGRAPHY
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CRYPTOGRAPHY
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CRYPTOGRAPHY
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CRYPTOGRAPHY
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3
CRYPTOGRAPHY
10
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8
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9
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CRYPTOGRAPHY
1
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1
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6
8
8
9

 

 

www.merriam-webster.com/dictionary/algorithm

a procedure for solving a mathematical problem (as of finding the greatest common divisor) in a finite number of steps that frequently involves repetition of an ...

algorithm [ˈælgəˌrɪðəm]
n
1. (Mathematics) a logical arithmetical or computational procedure that if correctly applied ensures the solution of a problem Compare heuristic
2. (Mathematics) Logic Maths a recursive procedure whereby an infinite sequence of terms can be generated Also called algorism
[changed from algorism, through influence of Greek arithmos number]
algorithmic adj
aal·go·rithm (lg-rm)
n.
A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.
algorithmically adv

algorithm (lg-rthm)
A finite set of unambiguous instructions performed in a prescribed sequence to achieve a goal, especially a mathematical rule or procedure used to compute a desired result. Algorithms are the basis for most computer programming.

Noun 1. algorithm - a precise rule (or set of rules) specifying how to solve some problem
algorithmic program, algorithmic rule
formula, rule - (mathematics) a standard procedure for solving a class of mathematical problems; "he determined the upper bound with Descartes' rule of signs"; "he gave us a general formula for attacking polynomials"
sorting algorithm - an algorithm for sorting a list
stemming algorithm, stemmer - an algorithm for removing inflectional and derivational endings in order to reduce word forms to a common stem algorithm
any methodology for solving a certain kind of problem.
See also: Mathematics

 

 

Algorithm
From Wikipedia, the free encyclopedia

Flow chart of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers a and b in locations named A and B. The algorithm proceeds by successive subtractions in two loops: IF the test B ≥ A yields "yes" (or true) (more accurately the number b in location B is greater than or equal to the number a in location A) THEN, the algorithm specifies B ← B − A (meaning the number b − a replaces the old b). Similarly, IF A > B, THEN A ← A − B. The process terminates when (the contents of) B is 0, yielding the g.c.d. in A. (Algorithm derived from Scott 2009:13; symbols and drawing style from Tausworthe 1977).
In mathematics and computer science, an algorithm (i/ˈælɡərɪðəm/) is a step-by-step procedure for calculations. Algorithms are used for calculation, data processing, and automated reasoning.

More precisely, an algorithm is an effective method expressed as a finite list[1] of well-defined instructions[2] for calculating a function.[3] Starting from an initial state and initial input (perhaps empty),[4] the instructions describe a computation that, when executed, will proceed through a finite [5] number of well-defined successive states, eventually producing "output"[6] and terminating at a final ending state. The transition from one state to the next is not necessarily deterministic; some algorithms, known as randomized algorithms, incorporate random input.[7]

Though al-Khwārizmī's algorism referred to the rules of performing arithmetic using Hindu-Arabic numerals and the systematic solution of linear and quadratic equations, a partial formalization of what would become the modern algorithm began with attempts to solve the Entscheidungsproblem (the "decision problem") posed by David Hilbert in 1928. Subsequent formalizations were framed as attempts to define "effective calculability"[8] or "effective method";[9] those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's "Formulation 1" of 1936, and Alan Turing's Turing machines of 1936–7 and 1939. Giving a formal definition of algorithms, corresponding to the intuitive notion, remains a challenging problem.[10]

Informal definition
For a detailed presentation of the various points of view around the definition of "algorithm" see Algorithm characterizations. For examples of simple addition algorithms specified in the detailed manner described in Algorithm characterizations, see Algorithm examples.
While there is no generally accepted formal definition of "algorithm," an informal definition could be "a set of rules that precisely defines a sequence of operations."[11] For some people, a program is only an algorithm if it stops eventually; for others, a program is only an algorithm if it stops before a given number of calculation steps.[12]

A prototypical example of an algorithm is Euclid's algorithm to determine the maximum common divisor of two integers; an example (there are others) is described by the flow chart above and as an example in a later section.

Boolos & Jeffrey (1974, 1999) offer an informal meaning of the word in the following quotation:

No human being can write fast enough, or long enough, or small enough† ( †"smaller and smaller without limit ...you'd be trying to write on molecules, on atoms, on electrons") to list all members of an enumerably infinite set by writing out their names, one after another, in some notation. But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.[13]

The term "enumerably infinite" means "countable using integers perhaps extending to infinity." Thus, Boolos and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be chosen from 0 to infinity. Thus an algorithm can be an algebraic equation such as y = m + n—two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of (for the addition example):
Precise instructions (in language understood by "the computer")[14] for a fast, efficient, "good"[15] process that specifies the "moves" of "the computer" (machine or human, equipped with the necessary internally contained information and capabilities)[16] to find, decode, and then process arbitrary input integers/symbols m and n, symbols + and = ... and "effectively"[17] produce, in a "reasonable" time,[18] output-integer y at a specified place and in a specified format.
The concept of algorithm is also used to define the notion of decidability. That notion is central for explaining how formal systems come into being starting from a small set of axioms and rules. In logic, the time that an algorithm requires to complete cannot be measured, as it is not apparently related with our customary physical dimension. From such uncertainties, that characterize ongoing work, stems the unavailability of a definition of algorithm that suits both concrete (in some sense) and abstract usage of the term.

[edit] Formalization

Algorithms are essential to the way computers process data. Many computer programs contain algorithms that detail the specific instructions a computer should perform (in a specific order) to carry out a specified task, such as calculating employees' paychecks or printing students' report cards. Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system. Authors who assert this thesis include Minsky (1967), Savage (1987) and Gurevich (2000):

Minsky: "But we will also maintain, with Turing . . . that any procedure which could "naturally" be called effective, can in fact be realized by a (simple) machine. Although this may seem extreme, the arguments . . . in its favor are hard to refute".[19]

Gurevich: "...Turing's informal argument in favor of his thesis justifies a stronger thesis: every algorithm can be simulated by a Turing machine ... according to Savage [1987], an algorithm is a computational process defined by a Turing machine".[20]

Typically, when an algorithm is associated with processing information, data is read from an input source, written to an output device, and/or stored for further processing. Stored data is regarded as part of the internal state of the entity performing the algorithm. In practice, the state is stored in one or more data structures.

For some such computational process, the algorithm must be rigorously defined: specified in the way it applies in all possible circumstances that could arise. That is, any conditional steps must be systematically dealt with, case-by-case; the criteria for each case must be clear (and computable).

Because an algorithm is a precise list of precise steps, the order of computation will always be critical to the functioning of the algorithm. Instructions are usually assumed to be listed explicitly, and are described as starting "from the top" and going "down to the bottom", an idea that is described more formally by flow of control.

So far, this discussion of the formalization of an algorithm has assumed the premises of imperative programming. This is the most common conception, and it attempts to describe a task in discrete, "mechanical" means. Unique to this conception of formalized algorithms is the assignment operation, setting the value of a variable. It derives from the intuition of "memory" as a scratchpad. There is an example below of such an assignment.

For some alternate conceptions of what constitutes an algorithm see functional programming and logic programming.

[edit] Expressing algorithms

Algorithms can be expressed in many kinds of notation, including natural languages, pseudocode, flowcharts, programming languages or control tables (processed by interpreters). Natural language expressions of algorithms tend to be verbose and ambiguous, and are rarely used for complex or technical algorithms. Pseudocode, flowcharts and control tables are structured ways to express algorithms that avoid many of the ambiguities common in natural language statements. Programming languages are primarily intended for expressing algorithms in a form that can be executed by a computer, but are often used as a way to define or document algorithms.

There is a wide variety of representations possible and one can express a given Turing machine program as a sequence of machine tables (see more at finite state machine, state transition table and control table), as flowcharts (see more at state diagram), or as a form of rudimentary machine code or assembly code called "sets of quadruples" (see more at Turing machine).

Representations of algorithms can be classed into three accepted levels of Turing machine description:[21]
1 High-level description:
"...prose to describe an algorithm, ignoring the implementation details. At this level we do not need to mention how the machine manages its tape or head." 2 Implementation description:
"...prose used to define the way the Turing machine uses its head and the way that it stores data on its tape. At this level we do not give details of states or transition function." 3 Formal description:
Most detailed, "lowest level", gives the Turing machine's "state table". For an example of the simple algorithm "Add m+n" described in all three levels see Algorithm examples.
[edit] Implementation

Most algorithms are intended to be implemented as computer programs. However, algorithms are also implemented by other means, such as in a biological neural network (for example, the human brain implementing arithmetic or an insect looking for food), in an electrical circuit, or in a mechanical device.

[edit] Computer algorithms

Flowchart examples of the canonical Böhm-Jacopini structures: the SEQUENCE (rectangles descending the page), the WHILE-DO and the IF-THEN-ELSE. The three structures are made of the primitive conditional GOTO (IF test=true THEN GOTO step xxx) (a diamond), the unconditional GOTO (rectangle), various assignment operators (rectangle), and HALT (rectangle). Nesting of these structures inside assignment-blocks result in complex diagrams (cf Tausworthe 1977:100,114).
In computer systems, an algorithm is basically an instance of logic written in software by software developers to be effective for the intended "target" computer(s), in order for the target machines to produce output from given input (perhaps null).

"Elegant" (compact) programs, "good" (fast) programs : The notion of "simplicity and elegance" appears informally in Knuth and precisely in Chaitin:
Knuth: ". . .we want good algorithms in some loosely defined aesthetic sense. One criterion . . . is the length of time taken to perform the algorithm . . .. Other criteria are adaptability of the algorithm to computers, its simplicity and elegance, etc"[22] Chaitin: " . . . a program is 'elegant,' by which I mean that it's the smallest possible program for producing the output that it does"[23]
Chaitin prefaces his definition with: "I'll show you can't prove that a program is 'elegant'"—such a proof would solve the Halting problem (ibid).

Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is . . . important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms".[24]

Unfortunately there may be a tradeoff between goodness (speed) and elegance (compactness)—an elegant program may take more steps to complete a computation than one less elegant. An example of using Euclid's algorithm will be shown below.

Computers (and computors), models of computation: A computer (or human "computor"[25]) is a restricted type of machine, a "discrete deterministic mechanical device"[26] that blindly follows its instructions.[27] Melzak's and Lambek's primitive models[28] reduced this notion to four elements: (i) discrete, distinguishable locations, (ii) discrete, indistinguishable counters[29] (iii) an agent, and (iv) a list of instructions that are effective relative to the capability of the agent.[30]

Minsky describes a more congenial variation of Lambek's "abacus" model in his "Very Simple Bases for Computability".[31] Minsky's machine proceeds sequentially through its five (or six depending on how one counts) instructions unless either a conditional IF–THEN GOTO or an unconditional GOTO changes program flow out of sequence. Besides HALT, Minsky's machine includes three assignment (replacement, substitution)[32] operations: ZERO (e.g. the contents of location replaced by 0: L ← 0), SUCCESSOR (e.g. L ← L+1), and DECREMENT (e.g. L ← L − 1).[33] Rarely will a programmer have to write "code" with such a limited instruction set. But Minsky shows (as do Melzak and Lambek) that his machine is Turing complete with only four general types of instructions: conditional GOTO, unconditional GOTO, assignment/replacement/substitution, and HALT.[34]

Simulation of an algorithm: computer (computor) language: Knuth advises the reader that "the best way to learn an algorithm is to try it . . . immediately take pen and paper and work through an example".[35] But what about a simulation or execution of the real thing? The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't then for the algorithm to be effective it must provide a set of rules for extracting a square root.[36]

This means that the programmer must know a "language" that is effective relative to the target computing agent (computer/computor).

But what model should be used for the simulation? Van Emde Boas observes "even if we base complexity theory on abstract instead of concrete machines, arbitrariness of the choice of a model remains. It is at this point that the notion of simulation enters".[37] When speed is being measured, the instruction set matters. For example, the subprogram in Euclid's algorithm to compute the remainder would execute much faster if the programmer had a "modulus" (division) instruction available rather than just subtraction (or worse: just Minsky's "decrement").

Structured programming, canonical structures: Per the Church-Turing thesis any algorithm can be computed by a model known to be Turing complete, and per Minsky's demonstrations Turing completeness requires only four instruction types—conditional GOTO, unconditional GOTO, assignment, HALT. Kemeny and Kurtz observe that while "undisciplined" use of unconditional GOTOs and conditional IF-THEN GOTOs can result in "spaghetti code" a programmer can write structured programs using these instructions; on the other hand "it is also possible, and not too hard, to write badly structured programs in a structured language".[38] Tausworthe augments the three Böhm-Jacopini canonical structures:[39] SEQUENCE, IF-THEN-ELSE, and WHILE-DO, with two more: DO-WHILE and CASE.[40] An additional benefit of a structured program will be one that lends itself to proofs of correctness using mathematical induction.[41]

Canonical flowchart symbols[42]: The graphical aide called a flowchart offers a way to describe and document an algorithm (and a computer program of one). Like program flow of a Minsky machine, a flowchart always starts at the top of a page and proceeds down. Its primary symbols are only 4: the directed arrow showing program flow, the rectangle (SEQUENCE, GOTO), the diamond (IF-THEN-ELSE), and the dot (OR-tie). The Böhm-Jacopini canonical structures are made of these primitive shapes. Sub-structures can "nest" in rectangles but only if a single exit occurs from the superstructure. The symbols and their use to build the canonical structures are shown in the diagram.

 

 

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 ...

 

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ALGORITHM
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ALGORITHMS
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-
-
-
-
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
-
-
27
-
-
10
-
41
-
1+0
-
-
-
-
9
9
-
-
-
-
-
-
2+7
-
-
1+0
-
4+1
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
1
-
5
-
-
1
3
7
6
9
9
2
8
4
1
-
-
-
-
-
-
-
-
5
1
A
L
G
O
R
I
T
H
M
S
-
-
9
-
-
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
-
-
-
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
-
-
-
-
S
=
1
10
1
S
19
10
1
-
1
-
-
-
5
-
-
-
-
-
-
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
19
10
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

 

 

-
-
-
-
-
-
-
-
-
-
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
19
10
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
19
10
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

 

 

logarithm
/ˈlɒɡərɪð(ə)m,-rɪθ-/
noun
noun: logarithm; plural noun: logarithms; noun: log

a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.
"proportional to the logarithm to the base 10 of the concentration"

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
LOGARITHM
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
A
=
1
4
1
A
1
1
1
-
1
-
3
-
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
-
1
2
3
4
5
6
7
8
18
-
-
4+9
-
-
-
1+0+2
4+9
4+9
-
-
-
-
-
-
-
-
-
1+8
-
-
13
-
9
LOGARITHM
4
13
13
-
1
2
3
4
5
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
9
LOGARITHM
4
4
4
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
LOGARITHM
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
-
-
-
-
-
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
-
-
-
-
-
-
49
-
9
LOGARITHM
103
49
49
-
1
2
3
4
6
7
8
18
-
-
4+9
-
-
-
1+0+2
4+9
4+9
-
-
-
-
-
-
-
-
1+8
-
-
13
-
9
LOGARITHM
4
13
13
-
1
2
3
4
6
7
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
9
LOGARITHM
4
4
4
-
1
2
3
4
6
7
8
9

 

 

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

 

 

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

 

 

LOGARITHMS

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
LOGARITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
5
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
5
-
7
-
-
A
=
1
4
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
-
-
-
-
S
=
1
10
1
S
19
10
1
-
1
-
-
-
5
-
-
-
-
-
-
41
-
10
LOGARITHMS
122
50
41
-
1
2
3
4
5
6
7
8
18
-
-
4+1
-
1+0
-
1+2+2
5+0
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
9
LOGARITHMS
5
5
5
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
LOGARITHMS
-
-
-
-
-
-
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
-
-
-
-
-
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
19
10
1
-
1
-
-
-
-
-
-
-
-
-
41
-
10
LOGARITHMS
122
50
41
-
1
2
3
4
6
7
8
18
-
-
4+1
-
1+0
-
1+2+2
5+0
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
9
LOGARITHMS
5
5
5
-
1
2
3
4
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
LOGARITHMS
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
-
5
-
-
-
-
S
=
1
10
1
S
19
10
1
-
1
-
-
-
5
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
5
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
5
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
5
-
-
-
-
O
=
6
2
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
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
5
-
-
-
9
R
=
9
5
1
R
18
9
9
-
-
-
-
-
5
-
-
-
9
-
-
41
-
10
LOGARITHMS
122
50
41
-
1
2
3
4
5
6
7
8
18
-
-
4+1
-
1+0
-
1+2+2
5+0
5+0
-
-
-
-
-
-
-
-
-
1+8
-
-
5
-
9
LOGARITHMS
5
5
5
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
6
7
8
9
-
-
-
-
-
LOGARITHMS
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
-
-
-
-
-
S
=
1
10
1
S
19
10
1
-
1
-
-
-
-
-
-
-
T
=
2
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
L
=
3
1
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
9
1
M
13
4
4
-
-
-
-
4
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
6
-
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
-
7
-
-
H
=
8
8
1
H
8
8
8
-
-
-
-
-
-
-
8
-
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
9
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
41
-
10
LOGARITHMS
122
50
41
-
1
2
3
4
6
7
8
18
-
-
4+1
-
1+0
-
1+2+2
5+0
5+0
-
-
-
-
-
-
-
-
1+8
-
-
5
-
9
LOGARITHMS
5
5
5
-
1
2
3
4
6
7
8
9

 

 

Math Forum: Ask Dr. Math FAQ: About e

mathforum.org/dr.math/faq/faq.e.html

e = 2.71828..., the Base of Natural Logarithms. e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those ...

_____________________________________________

What is e? Who first used e? How do you find it? How many digits does it have?

e = 2.71828..., the Base of Natural Logarithms

e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between e and complex numbers, via Euler's Equation.

e is usually defined by the following equation:

e = limn->infinity (1 + 1/n)n.

Its value is approximately 2.718281828459045... and has been calculated to 869,894,101 decimal places by Sebastian Wedeniwski (you'll find the first 50 digits in a Table of constants with 50 decimal places, from the Numbers, constants and computation site, by Xavier Gourdon and Pascal Sebah).

The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).

An effective way to calculate the value of e is not to use the defining equation above, but to use the following infinite sum:

e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

If you need K decimal places, compute each term to K+3 decimal places and add them up. You can stop adding after the term 1/n! where n! > 10K+3, because, to K+3 decimal places, the rest of the terms are all zero. Even though there are infinitely many of them, they will not change the decimal places you have already calculated. Now the last decimal place or two of the resulting sum may be off due to truncation or rounding of each term, but the first K places should be correct. That is why the computation uses extra decimal places.

As an example, here is the computation of e to 22 decimal places:
1/0! = 1/1 = 1.0000000000000000000000000
1/1! = 1/1 = 1.0000000000000000000000000
1/2! = 1/2 = 0.5000000000000000000000000
1/3! = 1/6 = 0.1666666666666666666666667
1/4! = 1/24 = 0.0416666666666666666666667
1/5! = 1/120 = 0.0083333333333333333333333
1/6! = 1/720 = 0.0013888888888888888888889
1/7! = 1/5040 = 0.0001984126984126984126984
1/8! = 1/40320 = 0.0000248015873015873015873
1/9! = 1/362880 = 0.0000027557319223985890653
1/10! = 1/3628800 = 0.0000002755731922398589065
1/11! = 0.0000000250521083854417188
1/12! = 0.0000000020876756987868099
1/13! = 0.0000000001605904383682161
1/14! = 0.0000000000114707455977297
1/15! = 0.0000000000007647163731820
1/16! = 0.0000000000000477947733239
1/17! = 0.0000000000000028114572543
1/18! = 0.0000000000000001561920697
1/19! = 0.0000000000000000082206352
1/20! = 0.0000000000000000004110318
1/21! = 0.0000000000000000000195729
1/22! = 0.0000000000000000000008897
1/23! = 0.0000000000000000000000387
1/24! = 0.0000000000000000000000016
1/25! = 0.0000000000000000000000001
-----------------------------
2.7182818284590452353602875

Then to 22 decimal places, e = 2.7182818284590452353603, which is correct. (Actually,it's correct to 25 places, but that was luck!).
There have been recent discoveries of even more efficient ways of computing e, one of which was used for setting the record mentioned above.

It is a fact (proved by Euler) that e is an irrational number, so its decimal expansion never terminates, nor is it eventually periodic. Thus no matter how many digits in the expansion of e you know, the only way to predict the next one is to compute e using the method above using more accuracy.

It is also true that e is a transcendental number (a fact first proved in 1873 by the French mathematician Charles Hermite), which means that e is not the root of any polynomial with rational number coefficients. These are properties that e shares with pi. The Dr. Math archives contain one proof of The Irrationality of e, and on the Web is another by Kevin Brown.

e is also the base of natural logarithms. The natural logarithm function ln(x) is defined that way: ln(x) = loge(x). This is "natural" for several reasons. One is the following limit:

ln(x) = limk->0 (xk-1)/k.

Another example from calculus is that if y = ln(x) + c, for c constant, then dy/dx = 1/x, and these are the only functions for which this is true. Another is that the curve y = ln(x) has a tangent at (1,0) with slope 1, and among all logarithmic functions, it is the only one that does.

Note: The term Euler's Constant is usually reserved for another number also first studied by Euler, 0.5772156649... = limn->infinity [1/1 + 1/2 + 1/3 + ... + 1/n - ln(n)]. [Return to text.]

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGEBRA
-
-
-
-
-
-
-
-
-
-
-
-
-
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
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
B
=
2
5
1
B
20
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
6
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
4
5
6
7
8
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGEBRA
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
L
=
3
2
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
G
=
7
3
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
E
=
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
B
=
2
5
1
B
20
2
2
-
-
2
-
4
-
6
-
8
-
R
=
9
6
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
A
=
1
7
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
4
5
6
7
8
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
ALGEBRA
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
A
=
1
7
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
B
=
2
5
1
B
20
2
2
-
-
2
-
4
-
6
-
8
-
L
=
3
2
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
E
=
5
4
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
G
=
7
3
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
R
=
9
6
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
4
5
6
7
8
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
9
-
-
-
-
-
ALGEBRA
-
-
-
-
-
-
-
-
-
-
A
=
1
1
1
A
1
1
1
-
1
-
-
-
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
B
=
2
5
1
B
20
2
2
-
-
2
-
-
-
-
L
=
3
2
1
L
12
3
3
-
-
-
3
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
5
-
-
G
=
7
3
1
G
7
7
7
-
-
-
-
-
7
-
R
=
9
6
1
R
18
9
9
-
-
-
-
-
-
9
-
-
28
-
7
ALGEBRA
46
28
28
-
2
2
3
5
7
9
-
-
2+8
-
-
-
4+6
2+8
2+8
-
-
-
-
-
-
-
-
-
10
-
7
ALGEBRA
10
10
10
-
2
2
3
5
7
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
1
-
7
ALGEBRA
1
1
1
-
2
2
3
5
7
9

 

 

-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
8
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
18
7
A
L
G
E
B
R
A
-
-
27
-
-
7
-
28
1+8
-
-
-
-
-
-
9
-
-
-
2+7
-
-
-
-
2+8
9
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
10
-
-
1
3
7
5
2
9
1
-
-
-
-
-
-
-
1+0
9
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
1

 

 

7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
12
7
5
2
18
1
+
=
46
4+6
=
10
1+0
1
-
1
3
7
5
2
9
1
+
=
28
2+8
=
10
1+0
1
7
A
L
G
E
B
R
A
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
7
A
L
G
E
B
R
A
-
-
27
-
-
7
-
28
-
-
-
-
-
-
9
-
-
-
2+7
-
-
-
-
2+8
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
10
-
1
3
7
5
2
9
1
-
-
-
-
-
-
-
1+0
7
A
L
G
E
B
R
A
-
-
9
-
-
7
-
1

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
6
-
-
-
-
15
-
3
THE
33
15
15
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
GOLDEN
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
L
=
3
6
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
6
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
6
-
-
-
-
14
-
5
GOLDEN
57
30
30
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
RATIO
-
-
-
-
-
-
-
-
-
-
-
-
-
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
14
1
O
15
6
6
-
-
-
-
-
-
6
6
-
-
-
-
11
-
4
RATIO
63
27
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
15
12
7
8
18
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
1+2
-
-
1+8
-
-
-
-
-
-
-
-
-
-
1
4
3
4
6
3
7
8
9
T
=
2
-
3
THE
33
15
6
4
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
9
4
-
-
-
-
-
-
-
-
-
-
-
18
-
14
Add to Reduce
153
72
18
4
1
2
3
4
5
6
7
8
9
-
-
1+8
-
1+4
Reduce to Deduce
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
5
Essence of Number
9
9
9
4
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
6
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
L
=
3
6
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
6
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
6
-
-
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
14
1
O
15
6
6
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
15
12
7
8
18
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
1+2
-
-
1+8
-
-
-
-
-
-
-
-
-
-
1
4
3
4
6
3
7
8
9
T
=
2
-
3
THE
33
15
6
4
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
9
4
-
-
-
-
-
-
-
-
-
-
-
18
-
14
Add to Reduce
153
72
18
4
1
2
3
4
5
6
7
8
9
-
-
1+8
-
1+4
Reduce to Deduce
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
5
Essence of Number
9
9
9
4
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
6
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
6
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
6
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
6
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
14
1
O
15
6
6
-
-
-
-
-
-
6
6
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
10
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
1
4
3
4
15
12
7
8
18
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+5
1+2
-
-
1+8
-
-
-
-
-
-
-
-
-
-
1
4
3
4
6
3
7
8
9
T
=
2
-
3
THE
33
15
6
4
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
9
4
-
-
-
-
-
-
-
-
-
-
-
18
-
14
Add to Reduce
153
72
18
4
1
2
3
4
5
6
7
8
9
-
-
1+8
-
1+4
Reduce to Deduce
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
Q
-
9
-
5
Essence of Number
9
9
9
4
1
2
3
4
5
6
7
8
9

 

 

-
KINETICS
--
-
-
-
-
--
-
-
-
-
-
-
-
KINETICS
--
-
-
-
-
1
K
11
2
2
-
2
1
I
9
9
9
9
-
3
N+E+T
39
12
3
-
3
1
I
9
9
9
9
-
2
C+S
22
4
4
-
4
8
KINETICS
-
-
-
-
-

 

 

K
=
2
-
-
KINETICS
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
-
-
-
-
1
S
19
10
1
K
=
2
Q
8
KINETICS
90
45
36
-
-
-
-
-
-
9+0
4+5
3+6
K
=
2
Q
8
KINETICS
9
9
9

 

 

K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
7+1
3+5
3+5
K
=
2
Q
7
KINETIC
8
8
8

 

 

E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
8+2
5+5
4+6
E
=
5
Q
8
ENERGIES
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
E
=
5
Q
8
ENERGIES
1
1
1

 

 

K
=
2
Q
7
KINETIC
71
35
8
E
=
5
Q
8
ENERGIES
82
55
1
Q
Q
7
Q
15
Add to Reduce
153
90
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
-
Q
Q
7
Q
6
Essence of Number
9
9
9

 

 

K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
1
K
11
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
T
20
2
2
-
-
-
-
1
I
9
9
9
-
-
-
-
1
C
3
3
3
K
=
2
-
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
1
R
18
9
9
-
-
-
-
1
G
7
7
7
-
-
-
-
1
I
9
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
S
19
10
1
E
=
5
-
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
K
=
2
Q
7
KINETIC
71
35
35
E
=
5
Q
8
ENERGIES
82
55
46
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
Q
Q
7
Q
6
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
-
-
KINETIC
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
K
11
2
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
K
=
2
-
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
-
ENERGIES
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
30
6
7
8
36
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
-
-
3+6
-
-
-
-
-
-
-
-
-
-
1
4
3
4
3
6
7
8
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
K
11
2
2
-
-
2
-
4
-
6
-
8
-
-
-
-
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
-
-
-
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
C
3
3
3
-
-
-
3
4
-
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
-
-
-
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
S
19
10
1
-
1
-
-
4
-
6
-
8
-
E
=
5
-
8
ENERGIES
82
55
46
-
-
-
-
4
-
6
-
8
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
30
6
7
8
36
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
-
-
3+6
-
-
-
-
-
-
-
-
-
-
1
4
3
4
3
6
7
8
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
9
-
-
-
-
1
K
11
2
2
-
-
2
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
1
N
14
5
5
-
-
-
-
5
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
N
14
5
5
-
-
-
-
5
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
-
-
1
G
7
7
7
-
-
-
-
-
7
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
4
3
30
7
36
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
3+6
-
-
-
-
-
-
-
-
-
-
1
4
3
3
7
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
2
3
5
7
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
2
3
5
7
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1
S
19
10
1
-
1
-
-
4
-
6
-
8
-
-
-
-
-
1
K
11
2
2
-
-
2
-
4
-
6
-
8
-
-
-
-
-
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
-
-
-
-
1
C
3
3
3
-
-
-
3
4
-
6
-
8
-
-
-
-
-
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
-
-
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
-
-
-
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
-
-
-
-
-
-
-
1
4
3
4
30
6
7
8
36
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
-
-
3+6
-
-
-
-
-
-
-
-
-
-
1
4
3
4
3
6
7
8
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
9
-
-
-
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
1
K
11
2
2
-
-
2
-
-
-
-
-
-
-
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
-
1
N
14
5
5
-
-
-
-
5
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
N
14
5
5
-
-
-
-
5
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
5
-
-
-
-
-
-
1
G
7
7
7
-
-
-
-
-
7
-
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
-
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
1
4
3
30
7
36
-
-
-
-
-
-
-
-
-
-
-
-
-
3+0
-
3+6
-
-
-
-
-
-
-
-
-
-
1
4
3
3
7
9
K
=
2
Q
7
KINETIC
71
35
35
-
-
-
-
-
-
-
E
=
5
Q
8
ENERGIES
82
55
46
-
-
-
-
-
-
-
Q
Q
7
Q
15
Add to Reduce
153
90
81
-
1
2
3
5
7
9
-
-
-
-
1+5
Reduce to Deduce
1+5+3
9+0
8+1
-
-
-
-
-
-
-
Q
Q
7
Q
6
Essence of Number
9
9
9
-
1
2
3
5
7
9

 

 

-
-
-
-
-
SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
5
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
19
-
5
SOUND
73
28
19
-
1
2
3
4
5
6
7
8
9
-
-
1+9
-
-
-
7+3
2+8
1+9
-
-
-
-
-
-
-
-
-
-
-
-
10
-
5
SOUND
10
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
5
SOUND
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
SOUND
-
-
-
-
1
3
4
5
6
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
6
U
=
3
3
1
U
21
3
3
-
-
3
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
5
-
D
=
4
5
1
D
4
4
4
-
-
-
4
-
-
-
-
19
-
5
SOUND
73
28
19
-
1
3
4
5
6
-
-
1+9
-
-
-
7+3
2+8
1+9
-
-
-
-
-
-
-
-
10
-
5
SOUND
10
10
10
-
1
3
4
5
6
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
1
-
5
SOUND
1
1
1
-
1
3
4
5
6

 

 

-
-
-
-
-
SOUND
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
2
-
-
-
-
7
8
9
U
=
3
3
1
U
21
3
3
-
-
2
3
-
-
-
7
8
9
D
=
4
5
1
D
4
4
4
-
-
2
-
4
-
-
7
8
9
N
=
5
4
1
N
14
5
5
-
-
2
-
-
5
-
7
8
9
O
=
6
2
1
O
15
6
6
-
-
2
-
-
-
6
7
8
9
-
-
19
-
5
SOUND
73
28
19
-
1
2
3
4
5
6
7
8
9
-
-
1+9
-
-
-
7+3
2+8
1+9
-
-
-
-
-
-
-
-
-
-
-
-
10
-
5
SOUND
10
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
5
SOUND
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
SOUND
-
-
-
-
1
3
4
5
6
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
U
=
3
3
1
U
21
3
3
-
-
3
-
-
-
D
=
4
5
1
D
4
4
4
-
-
-
4
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
5
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
6
-
-
19
-
5
SOUND
73
28
19
-
1
3
4
5
6
-
-
1+9
-
-
-
7+3
2+8
1+9
-
-
-
-
-
-
-
-
10
-
5
SOUND
10
10
10
-
1
3
4
5
6
-
-
1+0
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
1
-
5
SOUND
1
1
1
-
1
3
4
5
6

 

 

-
-
-
-
-
HARMONIC
-
-
-
-
1
2
3
4
5
6
7
8
9
H
=
8
1
1
H
8
8
8
-
-
2
-
-
-
-
7
8
-
A
=
1
2
1
A
1
1
1
-
1
2
-
-
-
-
7
-
-
R
=
9
3
1
R
18
9
9
-
-
2
-
-
-
-
7
-
9
M
=
4
4
1
M
13
4
4
-
-
2
-
4
-
-
7
-
-
O
=
6
5
1
O
15
6
6
-
-
2
-
-
-
6
7
-
-
N
=
5
6
1
N
14
5
5
-
-
2
-
-
5
-
7
-
-
I
=
9
7
1
I
9
9
9
-
-
2
-
-
-
-
7
-
9
C
=
3
8
1
C
C
3
3
-
-
2
3
-
-
-
7
-
-
-
-
45
-
8
HARMONIC
81
45
45
-
1
2
3
4
5
6
7
8
9
-
-
4+5
-
-
-
8+1
4+5
4+5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
HARMONIC
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
HARMONIC
-
-
-
-
1
3
4
5
6
8
9
H
=
8
1
1
H
8
8
8
-
-
-
-
-
-
8
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
R
=
9
3
1
R
18
9
9
-
-
-
-
-
-
-
9
M
=
4
4
1
M
13
4
4
-
-
-
4
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
6
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
5
-
-
-
I
=
9
7
1
I
9
9
9
-
-
-
-
-
-
-
9
C
=
3
8
1
C
C
3
3
-
-
3
-
-
-
-
-
-
-
45
-
8
HARMONIC
81
45
45
-
1
3
4
5
6
8
9
-
-
4+5
-
-
-
8+1
4+5
4+5
-
-
-
-
-
-
-
-
-
-
9
-
8
HARMONIC
9
9
9
-
1
3
4
5
6
8
9

 

 

-
-
-
-
-
HARMONIC
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
2
1
A
1
1
1
-
1
2
-
-
-
-
7
-
-
C
=
3
8
1
C
C
3
3
-
-
2
3
-
-
-
7
-
-
M
=
4
4
1
M
13
4
4
-
-
2
-
4
-
-
7
-
-
N
=
5
6
1
N
14
5
5
-
-
2
-
-
5
-
7
-
-
O
=
6
5
1
O
15
6
6
-
-
2
-
-
-
6
7
-
-
H
=
8
1
1
H
8
8
8
-
-
2
-
-
-
-
7
8
-
R
=
9
3
1
R
18
9
9
-
-
2
-
-
-
-
7
-
9
I
=
9
7
1
I
9
9
9
-
-
2
-
-
-
-
7
-
9
-
-
45
-
8
HARMONIC
81
45
45
-
1
2
3
4
5
6
7
8
9
-
-
4+5
-
-
-
8+1
4+5
4+5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
HARMONIC
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
HARMONIC
-
-
-
-
1
3
4
5
6
8
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
C
=
3
8
1
C
C
3
3
-
-
3
-
-
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
4
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
5
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
6
-
-
H
=
8
1
1
H
8
8
8
-
-
-
-
-
-
8
-
R
=
9
3
1
R
18
9
9
-
-
-
-
-
-
-
9
I
=
9
7
1
I
9
9
9
-
-
-
-
-
-
-
9
-
-
45
-
8
HARMONIC
81
45
45
-
1
3
4
5
6
8
9
-
-
4+5
-
-
-
8+1
4+5
4+5
-
-
-
-
-
-
-
-
-
-
9
-
8
HARMONIC
9
9
9
-
1
3
4
5
6
8
9

 

 

WORLD THE L WORD L THE WORLD

 

W
=
5
-
-
WORLD
-
-
-
-
-
-
-
1
L
12
3
3
-
-
-
-
-
-
-
-
-
-
-
-
-
1
W
23
5
5
-
-
-
-
1
O
15
6
6
-
-
-
-
1
R
18
9
9
-
-
-
-
1
D
4
4
4
W
=
5
Q
5
WORLD
72
27
27
-
-
-
-
-
-
7+2
2+7
2+7
W
=
5
Q
5
WORLD
9
9
9

 

 


Cogito ergo sum - Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Cogito_ergo_sum

Cogito ergo sum[a] is a Latin philosophical proposition by René Descartes usually translated into English as "I think, therefore I am". The phrase originally appeared in French as je pense, donc je suis in his Discourse on the Method, so as to reach a wider audience than Latin would have allowed.[1] It appeared in Latin in his later Principles of Philosophy. As Descartes explained, "[W]e cannot doubt of our existence while we doubt … ." A fuller form, dubito, ergo cogito, ergo sum ("I doubt, therefore I think, therefore I am"),[b] aptly captures Descartes' intent.

This proposition became a fundamental element of Western philosophy, as it purported to form a secure foundation for knowledge in the face of radical doubt. While other knowledge could be a figment of imagination, deception, or mistake, Descartes asserted that the very act of doubting one's own existence served—at minimum—as proof of the reality of one's own mind; there must be a thinking entity—in this case the self—for there to be a thought.

 

Rene Descartes: 'I think therefore I am' - Public Bookshelf

www.publicbookshelf.com/public_html/...of...I/ithinkth_bga.html

Taken from Rene Descartes' philosophical classic, Discourse on Method, the chapter that contains his famous supposition, 'I think therefore I am.'

Cogito ergo sum
I think, therefore I am

 

 

I

THINK

THEREFORE

I

AM

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
5
THINK
62
26
8
-
-
-
-
-
-
-
-
8
-
T
=
2
-
9
THEREFORE
100
55
1
-
1
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
-
2
AM
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
23
-
18
First Total
194
104
32
-
1
2
3
4
5
6
7
8
18
-
-
5+9
-
1+8
Add to Reduce
1+9+4
1+0+4
5+9
-
-
-
-
-
-
-
-
-
1+8
-
-
14
-
9
Second Total
5
14
14
-
1
2
3
4
5
6
7
8
9
-
-
1+4
-
-
Reduce to Deduce
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
Essence of Number
5
5
5
-
1
2
3
4
5
6
7
8
9

 

 

I
=
9
-
18
I THINK THEREFORE I AM
194
104
5
I
=
9
-
21
THINK THEREFORE I AM NOT
243
117
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
5
THINK
62
26
8
-
-
-
-
-
-
-
-
8
-
T
=
2
-
9
THEREFORE
100
55
1
-
1
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
-
2
AM
14
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
3
NOT
49
13
4
-
-
-
-
4
-
-
-
-
-
-
-
28
-
21
First Total
243
117
36
-
1
2
3
4
5
6
7
8
18
-
-
2+8
-
2+1
Add to Reduce
2+4+3
1+1+7
3+6
-
-
-
-
-
-
-
-
-
1+8
-
-
10
-
3
Second Total
9
9
9
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
Reduce to Deduce
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
3
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
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

 

 

BURIED ANGELS

Camilla Lackburg 2011

Translated from the Swedish by Tiina Nunnally 2014

Page 189

"...A mobile began ringing in the living room, and he jumped up. 'Sorry, but do you mind if I answer that?'

Without waiting for her reply, he left the kitchen, and Erica heard him speaking in a low voice. No one else seemed to be at home. She gazed around the room while she waited. A pile of documents stacked up on a kitchen chair caught her interest. Casting a quick glance over her shoulder, she began leafing through the pages. They seemed for the most part to be records of parliamentary proceedings and meetings, but then she gave a start. Between two printouts she found a piece of paper covered with scribbles that she couldn't decipher. From the living room she heard Holm saying goodbye, so she quickly pulled the page out of the pile of documents and slipped it into her handbag. When he returned to the kitchen, she gave him an innocent smile.

'Everything okay?'

He nodded and sat down again."

Page 209

Several months later, John Holm's name appeared for the first time in connection with the National Socialists.

'And his hatred hasn't diminished,' said Erica, reaching for her handbag. She took out the note and handed it to Kjell. 'I found this in Holm's house yesterday. [can't read what it says, but maybe it's important.'

He laughed. 'Define what you mean by found.'

'You sound exactly like Patrik,' said Erica, smiling. 'It was just lying there. I'm sure it's only a scribbled note that nobody will ever miss.'

'Let me see.' Kjell put on his reading glasses, which he'd pushed up on his forehead. 'Gimle,' he read aloud, frowning.

'Yes. What does that mean? I've never come across the word before. Is it an abbreviation of some sort?'

Kjell shook his head. 'Gimle is what comes after Ragnarok, the end of the world in Nordic mythology. A sort of heaven or paradise. It's a well-known concept and frequently used in neo-Nazi circles. It's also the name of a cultural association. They claim not to be affiliated with any political party, but I have my doubts on that score. They're certainly popular with both the Friends of Sweden and the Danish People's Party.'

And what do they do?'

'According to their literature, their aim is to revive nationalistic feelings and a shared identity through reviving old Swedish traditions, folk dancing, ancient Swedish poetry, relics of antiquity, and so on. All of which fits in with the purported goal of the Friends of Sweden to promote Swedish traditions.'

'So Gimle might also be a reference to that association?' She pointed at the paper.
'It's impossible to tell. It could mean anything. And /Page 210/ it's hard to know what these numbers signify: 1920211851612114. And then it says: 5 08 1400.'

Erica shrugged. 'I haven't a clue. I thought they might have been scribbled down in a hurry, the way you do when you're on the phone.'

'Could be,' said Kjell. He waved the paper in the air. 'Can I keep this?'

'Sure, go ahead. I'll just use my mobile to take a picture, in case I suddenly get a flash of inspiration and crack the code.'

'Good idea.' He pushed the paper across to her, and she took a picture. Then she knelt down on the rug and began tidying up after the children.

'Do you have any idea what you're going to do with that?'

'No, not really. I might start by exploring a few archives, see if I can come up with more information.'
'So you think it's more than a phone doodle?' she said.

'It could be. In any case, it's worth checking out.'

'Keep me posted, and I'll let you know if I find anything new.' She began ushering the kids towards the hall.

'Of course. We'll keep in touch,' Kjell said, reaching for the phone.

It was so typical. If Gosta arrived late, there was hell to pay, while Patrik could be gone half the morning and nobody raised an eyebrow. Erica had phoned last night and told Gosta about her visits to Ove Linder and John Holm. Now he was impatiently awaiting Patrik's arrival so they could go see Leon. Sighing at the unfairness of life, he returned to studying the list on his desk.

Page390

"...This was more than a scoop for him as a journalist — this story was going to have reverberations in Swedish politics and shock the entire country. 'Thanks for including me,' he muttered, feeling suddenly embarrassed.

Sven shrugged. 'We wouldn't have been able to finalize things if you hadn't provided the information that you did.'

'So you were able to decode the numbers?' Kjell was practically bursting with curiosity. Sven hadn't told him all the details on the phone.

'It was a ridiculously simple code.' Sven -laughed. 'My kids could have cracked it in fifteen minutes.'

'What do you mean?'

'One was "A", two was "B". And so on.'

'You're joking.' Kjell glanced over at Sven and almost drove off the road.

'No, I wish I was. It just shows how stupid they think we are.'

So what did you find out?' In his mind Kjell tried to picture the numbers, but he'd never been good at maths in school. Nowadays he could barely remember his own phone number.

'Stureplan. It said Stureplan. Followed by a date and a time.'

'Jesus Christ,' Kjell said, turning right on to the roundabout near Torp. 'That could have been disastrous.'

'Yes, but the police went in early this morning to pick up the people who were going to carry out the attack. Now they can't communicate with anyone to reveal that the police know all about the plan. That's why this is so urgent. It won't be long before the responsible individuals in the party notice that they haven't heard from the attackers, nor can they contact them. And then they'll be on guard, and we won't have another chance.'

 

9 STUREPLAN 126 STUREPLAN 9

9 STUREPLAN 45 STUREPLAN 9

9 STUREPLAN 36 STUREPLAN 9

9 STUREPLAN 9 STUREPLAN 9

 

-
-
-
-
-
STUREPLAN
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
4
-
6
-
1
-
T
=
2
2
1
T
20
2
2
-
-
2
-
4
-
6
-
2
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
6
-
8
-
R
=
9
4
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
E
=
5
5
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
P
=
7
6
1
P
16
7
7
-
-
-
-
4
-
6
7
8
-
L
=
3
7
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
A
=
1
8
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
N
=
5
9
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
-
-
36
-
9
STUREPLAN
126
45
36
-
2
2
6
4
10
6
7
8
9
-
-
3+6
-
-
-
1+2+6
4+5
3+6
-
-
-
-
-
1+0
-
-
-
-
-
-
9
-
9
STUREPLAN
9
9
9
-
2
2
6
4
1
6
7
8
9

 

 

-
-
-
-
-
STUREPLAN
-
-
-
-
1
2
3
5
7
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
-
-
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
5
-
-
P
=
7
6
1
P
16
7
7
-
-
-
-
-
7
-
L
=
3
7
1
L
12
3
3
-
-
-
3
-
-
-
A
=
1
8
1
A
1
1
1
-
1
-
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
5
-
-
-
-
36
-
9
STUREPLAN
126
45
36
-
2
2
6
10
7
9
-
-
3+6
-
-
-
1+2+6
4+5
3+6
-
-
-
-
1+0
-
-
-
-
9
-
9
STUREPLAN
9
9
9
-
2
2
6
1
7
9

 

 

-
-
-
-
-
STUREPLAN
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
4
-
6
-
1
-
A
=
1
8
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
T
=
2
2
1
T
20
2
2
-
-
2
-
4
-
6
-
2
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
6
-
8
-
L
=
3
7
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
N
=
5
9
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
P
=
7
6
1
P
16
7
7
-
-
-
-
4
-
6
7
8
-
R
=
9
4
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
-
-
36
-
9
STUREPLAN
126
45
36
-
2
2
6
4
10
6
7
8
9
-
-
3+6
-
-
-
1+2+6
4+5
3+6
-
-
-
-
-
1+0
-
-
-
-
-
-
9
-
9
STUREPLAN
9
9
9
-
2
2
6
4
1
6
7
8
9

 

 

-
-
-
-
-
STUREPLAN
-
-
-
-
1
2
3
5
7
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
A
=
1
8
1
A
1
1
1
-
1
-
-
-
-
-
T
=
2
2
1
T
20
2
2
-
-
2
-
-
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
-
-
-
L
=
3
7
1
L
12
3
3
-
-
-
3
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
5
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
5
-
-
P
=
7
6
1
P
16
7
7
-
-
-
-
-
7
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
9
-
-
36
-
9
STUREPLAN
126
45
36
-
2
2
6
10
7
9
-
-
3+6
-
-
-
1+2+6
4+5
3+6
-
-
-
-
1+0
-
-
-
-
9
-
9
STUREPLAN
9
9
9
-
2
2
6
1
7
9

 

 

PROPER PEOPLE

EARLY ASYLUM LIFE IN THE WORDS

OF

THOSE WHO WERE THERE

David Scrimgeour 2015

Proper People takes the reader back in time to the early nineteenth century to meet some of the mentally ill patients who passed through Yorkshire's West Riding Pauper Lunatic Asylum in Wakefield during the first 50 years of its existence. Using transcriptions from the celebrated collection of original asylum records held by the West Yorkshire Archive Service - casebooks, reception orders, annual reports, journals together with contemporary newspaper articles, the patients' stories are told using the actual words of those who were there at that time, their physicians, poor law officers, reporters and sometimes even the patients themselves...."

Page 1

Introduction

"Many excellent books have already been published about the county asylums of the nineteenth century, but little has been revealed of the "pauper lunatics" themselves - the tens of thousands of people receiving poor relief and suffering from mental illness who passed through the asylum doors. While there is a lot of general information available, social historians are, relatively speaking, starved of material providing a deeper insight into the lives of the patients, material that might allow them to answer key questions, such as: Why were they in an asylum? What was life really like for them while they were there? What happened to them in the long term? What was the impact on their families? Proper People is my contribution towards gaining a better understanding by looking at the lives of just a few of those patients, all admissions to the West Riding Pauper Lunatic Asylum in Yorkshire, England, in the early part of the 1800s.

Why this title, Proper People? In 1819, a magistrate committing someone to an asylum would sign a warrant declaring that the person was a "proper object" to be admitted. Fifty years later, the expression used had become "proper person". Proper People has been written about some of those early asylum patients, and through their stories the reader is very much reminded that they Were, as all patients are today, real people, proper people, not objects, suffering from a period of mental illness.

Page 2

"I chose Stanley Royd Hospital in Wakefield, originally known as the West Riding Pauper Lunatic Asylum...."

Page 17

Patients Admitted 1818-1829

  • Thirty-five year old butcher John FIELD from Rothwell had been committed to the House of Correction, Wakefield, for assaulting a man before being removed to the Asylum on 18th January, 1819, by George Wadsworth, Overseer of the Poor.' Described as being tall and thin with an irritable temper, John was said to have been insane for many years. His early case notes suggest a man who was always troubled by something.

2nd June, 1819. Pulse near 100, complains of debility.

4th December, 1819. Complains of pain at his stomach which he thinks will be cured by tincture of rhubarb.

22nd March, 1820. Appetite impaired - bowels regular.

14th April, 1820. Pulse 90. Urine high coloured - complains of pain in his back, heat natural.

17th April, 1820. Pulse 80 - looks better and has less pain in his back.

Later that year John FIELD, presumably by then free from any problems with his stomach and back, decided to take a short break from the Asylum.

10th November, 1820. Escaped over the airing court wall last night about six o'clock. He returned of his own accord about 5 o'clock this morning.

11th November, 1820. Has not been well since his return - complains of sickness - tongue white.

It hadn't taken John long to find something else to complain about. John's case notes from that time onwards record nothing particularly unusual until 25 years after his admission when his doctor added the following extraordinary description of how John used to spend some of his time in the Asylum.

June, 1844. For years his frequent occupation has been to sit down on the flags, take a piece of tobacco pipe and commence figuring. He /page 18/ uses the Arabic numerals in order, for the letters of the Alphabet; A is represented by figure 1, B by 2, C by 3, D by 4, etc; thus his rows of figures are in the place of letters, and his occupation is writing by cyphers. For example, the name John Field stands thus, 10 15 8 14 6 9 5 12 4. Having obtained these numbers, or others furnished by any other words, he multiplies, adds, divides or subtracts them in numerous ways. Hence it will be easily perceived that the combination being unlimited, an infinite variety of results may be obtained. These resulting figures are, in their turn, considered as letters, which are formed into words, sometimes rational, but generally most anomalous and unheard of sounds. They have usually some personal signification - the name of an estate to which he is entitled; the person who left it to him, the act of Parliament according to which he ought to be set at liberty, the time of his liberation, the money owing to him, the name of his secret enemy, etc, etc, etc. It is unknown how he got this system of cyphers; the probability is that he has learnt it from someone; he appears to regard it as a means of arriving at secret knowledge, and practices it as an astrologer in former times would consult the stars.

The following is an instance - he said "N o n e spells none, or no ne, and it is in the last clause of the act, and signifies that they have no right to keep me here". He then writes it down, 14 15 14 5; these figures he divides, with great facility of manner, by 12 (the number of pence in a shilling) and by 20 (the number of shillings in a pound)

12\ 14 15 14 5
20\ 117928 — 9
5896 -8-9

This he reads "Five thousand eight hundred and ninety-six stops. This is what they have to pay me an hour. These marks are against them". But the irrational, incoherent explanations are too intricate, too extensive, and moreover too mystical to be followed any length. All the concerns of life undergo this perplexing process, and to a listener he will discourse in this way for a very long time, yet in many matters of everyday life he is shrewd and intelligent.
Page 19
His moral feelings are undoubtedly benevolent and highly conscientious. Indeed instances of genuine compassion, and of a rigid equity, occur frequently. The disease is mental. He is said to be a Deist, but his religious opinions have never fairly manifested themselves; he seems to think seldom of such things - he sometimes speaks of the Justice and of the Benevolence of God. He says "God is said to be merciful; I hope he is, for if he punishes those who keep me here with strict justice, the punishment will be terrible. I hope he is merciful!"

John's case notes show that there was to be little change in his health or behaviour during the next eleven years and that is also reflected in the Chaplain's Journal. The Asylum chaplain, Reverend Thomas Bayley Clarkson, had been appointed by the Committee of Visitors in 1843, when aged 48. He kept a journal during the 26 years of his tenure and over time it had spread to multiple volumes but only one leather bound volume has survived, covering 1853-1854. It is written very neatly and every patient mentioned appears in an index at the front. The Reverend Clarkson's journal was intended to be inspected by the visiting magistrates so in it he recorded all the services which had been conducted, statistics of attendees, the names of patients and staff who had taken communion and, of great value in better understanding the "proper people", his insightful reflections on the patients with whom he had come into contact.

John FIELD features in the Chaplain's Journal in an entry made on 17th March, 1854.3

It is as long ago as the month of January, in the last year V. VII P. 455 since allusion has been made to John Field now in the 36th year of his captivity! So mutch has in earlier volumes been said about him and so little change has taken place, physically or mentally, during the eleven summers and winters which the Writer has passed in close neighbourhood with him that a few sentences would comprise all that needs be added in order to bring down his uneventful history to the present date.

Page 20 from the Chaplain's Journal 1854 (omitted?)

The little change which has occurred in John's outward man is rather for the better; for, contrary to the usual course of nature, his wrinkled face is looking fresher, and, his person stouter, than was the case years /page 21/ ago; and this notwithstanding considerable peevish irregularity in the eating of his meals. His age is about 75, and he seems to promise fair for an extreme old age. He has a good deal of personal strength remaining and the tones of his voice are in good force, though not always well employed.

As for his delusions they are just as extravagant and multitudinous as ever, embracing every subject in creation, and consequently admitting of no increases. To argue with him on any supposable topic would be as hopeless as the attempt to cut down an aged oak with a pen knife.

The most courteous contradiction of any of his absurdities is what he does not at all like but perseverance in argument is sure to put him into a pet, and perhaps to elicit an oath. "Vocas in certamina divos." The latter part of the charge against him the Writer, however, knows only from report: he has always succeeded in living on good terms with the irritable old man, and has rarely heard him swear. His most troublesome hallucination - as is well known - is to the effect that he is the most injured individual on the face of the earth; inasmuch as, being a man of immense wealth and lawful master of the Institution and of every thing contained in it, his servants (the Chaplain included) are wickedly and for interested purposes forcibly debarring him from the enjoyment of his own by means of false accusations; - he himself maintaining, and firmly believing that his own sense amounts to more than the "collective wisdom" of the whole community. "Give me a key and let me go down to the Bank and see if my money is safe", such is the burden of his cry, often accompanied with violent threats in the event of non-compliance. Occasionally, he puts this demand in a form which it is difficult to evade. "My excellent John", the Chaplain said to him the other day, "I assure you I have no more power to let you out, than I have to jump over the moon." "Oh! You haven't" was his reply; "well then, say that is not a key that you have in your hand, and I'll believe you" - "I can't spare my key, John" - "Why, bless you! (1 give it you again as soon as ever I came back from the Bank", was his rejoinder. This appeal being insufficient, he turned abruptly away in dudgeon and mal content.

This is a point to which it is as well to bring the poor man at once, for let him lay hold of a button, or thrust his person within the half-/page 21/opened door, and there is no end of the discussion which follows, and which cannot be terminated without an appearance of discourtesy which it is always desirable to avoid with reference to insane patients.

It will be concluded that this venerable inmate is not at all a subject for the Chaplain's ministry; and this admission is made on his part in the widest sense. The attempt to teach him a religious idea would be as insensate a thing as to pour water into a sieve in the expectation that it would prove a retentive receptacle of the fluid. John, in fact, knows this, as well as every other thing, better than any one else, and disdains to be a scholar in any respect. Assert in his hearing that black is not white, and he will forthwith engage to prove to you by arithmetic, his usual recourse, that you are entirely in the wrong. Sometimes, when, in the earlier stage of his acquaintance with Field the Chaplain has addressed him seriously on the subject of eternity, and perhaps, proposed to read him a chapter of the Bible, - putting on an expression of mingled surprise, pity, and profound sagacity, he would exclaim, "The Bible! Why I could teach you things out of the Bible that you never knew, and that nobody knows but me." And he would sometimes proceed to demonstrations, which, in the mouth of a sane person, would amount to blasphemy. He is, therefore, better let alone, and left to Almighty God, from whom he will receive that measure of mercy through Christ, of which all stand in need, when his failings shall be weighed against his opportunities in an answering balance.

It will perhaps be recollected, that the Chaplain when speaking of this well known aboriginal inmate of this Asylum, has always given him credit for the possession of several good qualities; more especially, for an absence of selfishness, which is not always, unusually, a characteristic of the Insane; and for open-handed munificence. This latter endowment sounds oddly as an attribute of a pauper-lunatic, not possessed of a groat : but if destitute of tangible, he is richly endowed with ideal wealth, and has the heart, if not the power to dispense it liberally. Often has the Writer felt pain, and some degree of shame, when compelled to turn the remorseless key upon one whom the moment before has, in his own belief conferred upon him a noble gift. "Take this piece of paper", he would say "and go with it to the Bank; its for a thousand pounds; it will do you good, and when you've /page 23/spent it, come to me, and you shall have some more." Sometimes his benevolence puts on a humbler shape: "You look poorly this morning, as if you hadn't had a good breakfast: go down to the house-keeper's room, and get some ham and eggs, and say I sent you".

Such redeeming traits go far to ensure a certain measure of respect for John, notwithstanding not a few faults in his disposition, and more tolerance than he would otherwise gain as an inordinate abuser of the time and patience of others.

Despite the Chaplain's prediction for John looking as if he would reach "an extreme old age" that was not to be as, in January the following year, John's health began to decline. He started to refuse medicine, his language became incoherent and he expressed ever more deluded notions. John FIELD, cypher writer, passed away on 9th May, 1855, at the age of 71 years having spent over half of his life in the Asylum. He was buried at Holy Trinity Church in nearby Rothwell, four days later.'

 

 

PROPER PEOPLE EARLY ASYLUM LIFE IN THE WORDS

OF

THOSE WHO WERE THERE

David Scrimgeour 2015

Page 20

June, 1844. For years his frequent occupation has been to sit down on the flags, take a piece of tobacco pipe and commence figuring. He /page 18/ uses the Arabic numerals in order, for the letters of the Alphabet; A is represented by figure 1, B by 2, C by 3, D by 4, etc; thus his rows of figures are in the place of letters, and his occupation is writing by cyphers. For example, the name John Field stands thus, 10 15 8 14 6 9 5 12 4. Having obtained these numbers, or others furnished by any other words, he multiplies, adds, divides or subtracts them in numerous ways. Hence it will be easily perceived that the combination being unlimited, an infinite variety of results may be obtained. These resulting figures are, in their turn, considered as letters, which are formed into words, sometimes rational, but generally most anomalous and unheard of sounds. They have usually some personal signification - the name of an estate to which he is entitled; the person who left it to him, the act of Parliament according to which he ought to be set at liberty, the time of his liberation, the money owing to him, the name of his secret enemy, etc, etc, etc. It is unknown how he got this system of cyphers; the probability is that he has learnt it from someone; he appears to regard it as a means of arriving at secret knowledge, and practices it as an astrologer in former times would consult the stars.

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

....

 

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

 

 

 

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

 

 

 

DARKNESS AT NOON

Arthur Koestler

THE FIRST HEARING

Page 19
... Rubashov noticed suddenly that he himself had been walking up and down for the last five minutes; he had risen from his bed without realizing it. He was caught again by his old ritual of never walking on the edges of the paving stones, and he already knew the pattern by heart. But his thoughts had not left No. I for a second, No. 1, who, sitting at his desk and dictating immovably, had gradually turned into his own portrait, into that well-known colour-print, which hung over every bed or sideboard in the country and stared at people with its frozen eyes.

Rubashov walked up and down in the cell, from the door to the window and back, between bunk, wash-basin and bucket, six and a half steps there, six and a half steps back. At the door he turned to the right, at the window to the left: it was an old prison habit; if one did not change the direction of the turn one rapidly became dizzy. What went on in No. l's brain? He pictured to himself a cross-section through that brain, painted neatly with grey water-colour on a sheet of paper stretched on a drawing-board with drawing-pins. The whorls of grey matter swelled to entrails, they curled round one /page 20/ another like muscular snakes, became vague and misty like the spiral nebulas on astronomical charts. . . . What went on in the inflated grey whorls? One knew everything about the far-away spiral nebulas, but about them nothing. That was probably the reason that history was more of an oracle than a science. Perhaps later, much later, it would be taught by means of tables of statistics, supplemented by such anatomical sections. The teacher would draw on the blackboard an algebraic formula representing the conditions of life of the masses of a particular nation at a particular period: 'Here, citizens, you see the objective factors which conditioned this historical process.' And, pointing with his ruler to a grey foggy landscape between the second and third lobe of No. l's brain: 'Now here you see the subjective reflection of these factors. It was this which in the second quarter of the twentieth century led to the triumph of the totalitarian principle in the East of Europe.' Until this stage was reached, politics would remain bloody dilettantism, mere superstition and black magic. . . .

Rubashov heard the sound of several people marching down the corridor in step. His first thought was: now the beating-up will start. He stopped in the middle of the cell, listening, his chin pushed forward. The marching steps came to a halt before one of the neighbouring cells, a low command was heard, the keys jangled. Then there was silence.

Rubashov stood stiffly between the bed and the bucket, held his breath, and waited for the first scream. He remembered that the first scream, in which terror still predominated over physical pain, was usually the worst; what followed was already more bearable, one got used to it and after a time one could even draw conclusions on the method of torture from the tone and rhythm of the screams. Towards the end, most people behaved in the same way, however different they were in temperament and voice: the screams became weaker, changed over into whining and choking. Usually the door would slam soon after. The keys would jangle again; and the first scream of the next victim often came even before they had touched him. at the mere sight of the men in the doorway.

Rubashov stood in the middle of his cell and waited for the first scream. He rubbed his glasses on his sleeve and said to /page 21/ himself that he would not scream this time either, whatever happened to him. He repeated this sentence as if praying with a rosary. He stood and waited: the scream still did not come. Then he heard a faint clanging, a voice murmured something, the cell-door slammed. The footsteps moved to the next cell.

Rubashov went to the spy-hole and looked into the corridor. The men stopped nearly opposite his cell, at No. 407. There was the old warder with two orderlies dragging a tub of tea, a third carrying a basket with slices of black bread, and two uniformed officials with pistols. There was no beating-up; they were bringing breakfast....

No. 407 was just being given bread. Rubashov could not see him. No. 407 was presumably standing in the regulation position, a step behind the door; Rubashov could only see his forearms and hands. The arms were bare and very thin; like two parallel sticks, they stuck out of the doorway into the corridor. The palms of the invisible No. 407 were turned upwards, curved in the shape of a bowl. When he had taken the bread, he clasped his hands and withdrew into the darkness of his cell. The door slammed.

Rubashov abandoned the spy-hole and resumed his marching up and down. He ceased rubbing his spectacles on his sleeve, put them in place, breathed deeply and with relief. He whistled a tune and waited for his breakfast. He remembered with a slight feeling of uneasiness those thin arms and the curved hands; they reminded him vaguely of something he could not define. The outlines of those stretched-out hands and even the shadows on them were familiar to him — familiar and yet gone from his memory like an old tune or the smell of a narrow street in a harbour.

7

The procession had unlocked and slammed a row of doors, but not yet his. Rubashov went back to the judas, to see whether they were coming at last; he was looking forward to the hot tea. The tub had been steaming, and thin slices of lemon had floated on its surface. He took off his pince-nez and pressed his eye to the spy-hole. His range of sight held four of the /page 22/ cells opposite : Nos. 401 to 407. Above the cells ran a narrow iron gallery; behind it were more cells, those of the second
floor. The procession was just coming back along the corridor from the right; evidently they first did the odd numbers, then the even. Now they stood at No. 408; Rubashov only saw the backs of the two uniformed men with the revolver belts; the rest of the procession stood outside his view-range. The door slammed; now they all came to No. 406. Rubashov saw again the steaming tub and the orderly with the bread basket in which only a few slices were left. The door of No. 406 slam-
med instantly; the cell was uninhabited. The procession approached, passed his door and stopped at No. 402.

Rubashov began to drum on the door with his fists. He saw that the two orderlies with the tub looked at each other and glanced at his door. The warder busied himself with the lock on the door of No. 402 and pretended not to hear. The two men in uniform stood with their backs to Rubashov's spy-hole. Now the bread was being passed in through the door of No. 402; the procession started to move on. Rubashov drummed more loudly. He toqk a shoe off and banged on the door with it.

The bigger of the two men in uniform turned round, stared expressionlessly at Rubashov's door and turned away again.
The warder slammed the door of No. 402. The orderlies with the tub of tea stood about hesitantly. The man in uniform who had turned round said something to the older warder, who shrugged his shoulders and with jangling keys shuffled to Rubashov's door. The orderlies with the tub followed him;
the orderly with the bread said something through the spy-hole to No. 402.

Rubashov drew back a step from his door and waited for it to open. The tension inside him gave way suddenly; he did not care any more whether he was given tea or not. The tea in the tub had no longer steamed on the way back and the slices of lemon on the rest of the pale yellow liquid had looked limp and shrunken.

The key was turned in his door, then a staring pupil appeared in the spy-hole and disappeared again. The door flew open. Rubashov had seated himself on the bed and was put-/page 23/ ing his shoe on again. The warder held the door open for the big man in uniform who entered the cell. He had a round, clean-shaven skull and expressionless eyes. His stiff uniform creaked; so did his boots; Rubashov thought he could smell the leather of his revolver belt. He stopped next to the bucket and looked round the cell, which seemed to have become smaller through his presence.

'You have not cleaned up your cell,' he said to Rubashov.
'You know the regulations, surely.'

'Why was I omitted at breakfast?' said Rubashov, examining the officer through his pince-nez.

'If you want to argue with me, you will have to stand up,' said the officer.

'I haven't got the slightest desire to argue or even to speak to you,' said Rubashov, and laced up his shoe.

'Then don't bang on the door next time, else the usual disciplinary measures will have to be applied to you,' said the officer. He looked round the cell again. 'The prisoner has no mop to clean the floor,' he said to the warder.

The warder said something to the bread-orderly, who vanished down the corridor at a trot. The two other orderlies stood in the open doorway and gazed into the cell with curiosity. The second officer had his back turned; he stood in the corridor with his legs straddled and his hands behind his back.

'The prisoner has no eating bowl either,' said Rubashov, still busied with the lacing of his shoe. 'I suppose you want to save me the trouble of a hunger-strike. I admire your new methods.'

'You are mistaken,' said the officer, looking at him expressionlessly. He had a broad scar on his shaven skull and wore the ribbon of the Revolutionary Order in his button-hole. So he was in the Civil War, after all, thought Rubashov. But that is long ago and makes no difference now. . . .

'You are mistaken. You were left out at breakfast because you had reported yourself sick.'

'Toothache,' said the old warder, who stood leaning against the door. He still wore slippers, his uniform was crumpled and spotted with grease.

'As you like,' said Rubashov. It was on the tip of his tongue /page 24/ to ask whether it was the latest achievement of the regime to treat invalids by compulsory fasting, but he controlled himself. He was sick of the whole scene.

The bread-orderly came running, panting, and flapping a dirty rag. The warder took the rag out of his hand and threw it in a corner next to the bucket.
'Have you any more requests?' asked the officer without irony.

'Leave me alone and stop this comedy,' said Rubashov. The officer turned to go, the warder jangled his bunch of keys. Rubashov went to the window, turning his back on them. When the door had slammed he remembered that he had forgotten the chief thing and with a bound he was back at the door.
Paper and pencil,' be shouted through the spy-bole. He took off his pince-nez and stuck his eye to the bole to see whether they turned round. He had shouted very loudly, but the procession moved down the corridor as if it had heard nothing. The last he saw of it was the back of the officer with the shaven skull and the broad leather belt with the revolver-case attached to it.

8

Rubashov resumed walking up and down his cell, six and a half steps to the window, six and a half steps back. The scene had stirred him; he recapitulated it in minute detail while rubbing his pince-nez on his sleeve. He tried to hold on to the hatred he had for a few minutes felt for the officer with the scar; he thought it might stiffen him for the coming struggle. Instead, he fell once more under the familiar and fatal constraint to put himself in the position of his opponent, and to see the scene through the other's eyes. There he had sat, this man Rubashov, on the bunk — small, bearded and arrogant, and in an obviously provocative manner, and put his shoe on over the sweaty sock. Of course, this man Rubashov had his merits and a great past, but it was a different thing to see him on the platform at a congress or on a palliasse in a cell. So that is the legendary Rubasbov, thought Rubashov in the name of/page 25/ the officer with the expressionless eyes. Screams for his breakfast like a schoolboy and isn't even ashamed. Cell not cleaned up. Holes in his socks. Querulous intellectual.Conspired against law and order : whether for money or on principle makes no
difference. We did not make the revolution for cranks. True, he helped to make it; at that time he was a man; but now he is
old and self-righteous, ripe for liquidation. Perhaps he was so even at that time; there were many soap bubbles in the revolution which burst afterwards. If he still had a vestige of self-respect, he would clean his cell.

For a few seconds Rubashov wondered whether he should really scrub the tiles. He stood hesitantly in the middle of the cell, then put his pince-nez on again and propped himself at the window.

The yard was now in daylight, a greyish light tinged with yellow, not unfriendly, promising more snow. It was about eight — only three hours had passed since he first entered the cell. The walls surrounding the yard looked like those of barracks; iron gates were in front of all the windows, the cells behind them were too dark for one to see into them. It was impossible even to see whether anyone stood directly behind his window, looking down, like him, at the snow in the yard.
It was nice snow, slightly frozen; it would crackle if one walked on it. On both sides of the path which ran round the yard at a distance of ten paces from the walls, a hilly parapet of snow had been shovelled up. On the rampart opposite the sentinel was pacing up and down. Once, when turning, he spat in a wide arc into the snow; then leant over the ramp to see where it had fallen and frozen.

The old disease, thought Rubashov. Revolutionaries should not think through other people's minds.

Or, perhaps they should? Or even ought to?

How can one change the world if one identifies oneself with everybody?

How else can one change it?

He who understands and forgives — where would he find a motive to act?

Where would he not?

They will shoot me, thought Rubashov. My motives will /page 26/ be of no interest to them. He leaned his forehead on the window pane. The yard lay white and still.

So he stood for a while, without thinking, feeling the cool glass on his forehead. Gradually, he became conscious of a small but persistent ticking sound in his cell.

He turned round listening. The knocking was so quiet that at first he could not distinguish from which wall it came. While be was listening, it stopped. He started tapping himself, first on the wall over the bucket, in the direction of No. 406, but got no answer. He tried the other wall, which separated him from No. 402, next to his bed. Here he got an answer. Rubashov sat down comfortably on the bunk, from where he could keep an eye on the spy-hole, his heart beating. The first contact was always very exciting.

No. 402 was now tapping regularly; three times with short intervals, then a pause, then again three times, then again a pause, then again three times. Rubashov repeated the same series to indicate that he heard. He was anxious to find out whether the other knew the 'quadratic alphabet' — otherwise there would be a lot of fumbling until he had taught it to him. The wall was thick, with poor resonance; he had to put his head close to it to hear clearly and at the same time he had to watch the spy-hole. No. 402 had obviously had a lot of practice; he tapped distinctly and unhurriedly, probably with some hard object such as a pencil. While Rubashov was memorizing the numbers, he tried, being out of practice, to visualize the square of letters with the twenty-five compartments — five horizontal rows with five letters in each. No. 402 first tapped five times — accordingly the fifth row: v to z; then twice; so it was the second letter of the row : w. Then a pause; then two taps — the second row, F - j; then three taps — the third letter of the row : H. Then three times and then five times; so fifth letter of the third row: o. He stopped.

WHO?

A practical person, thought Rubashov; he wants to know at once whom he has to deal with. According to the revolutionary etiquette, he should have started with a political tag; then given the news; then talked of food and tobacco; much later only, days later, if at all, did one introduce oneself. However, /Page 27/ Rubashov's experience had been so far confined to countries in which the Party was persecuted, not persecutor, and the members of the Party, for conspiratorial reasons, knew each other only by their Christian names — and changed even these so often that a name lost all meaning. Here, evidently, it was different, Rubashov hesitated as to whether he should give his name. No. 402 became impatient; he knocked again :

WHO?

Well, why not? thought Rubashov. He tapped out his full name: NICOLAS SALMANOVICH RUBASHOV, and waited for the result.

For a long time there was no answer. Rubashov smiled; he could appreciate the shock it had given his neighbour. He waited a full minute and then another; finally, he shrugged his shoulders and stood up from the bunk. He resumed his walk through the cell, but at every turn he stopped listening to the wall. The wall remained mute. He rubbed his pince-nez on his sleeve, went slowly, with tired steps, to the door and looked through the spy-hole into the corridor.

The corridor was empty; the electric lamps spread their stale, faded light; one did not hear the slightest sound. Why had No. 402 become dumb?

Probably from fear; he was afraid of compromising himself through Rubashov. Perhaps No. 402 was an unpolitical doctor or engineer who trembled at the thought of his dangerous neighbour. Certainly without political experience, else he would not have asked for the name as a start. Presumably mixed up in some affair of sabotage. Has obviously been in prison quite a time already, has perfected his tapping and is devoured by the wish to prove his innocence. Still in the simple belief that his subjective guilt or innocence makes a difference, and with no idea of the higher interests which are really at stake. In all probability he was now sitting on his bunk, writing his hundredth protest to the authorities, who will never read it, or the hundredth letter to his wife, who will never receive it; has in despair grown a beard — a black Pushkin beard — has given up washing and fallen into the habit of biting his nails and of erotic day-dreams. Nothing is worse in prison than the consciousness of one's innocence; it prevents acclimatization /page 28/ and undermine's one's morale. . . . Suddenly the ticking started again.

Rubashov sat down quickly on the bunk; but he had already missed the first two letters. No. 402 was now tapping quickly and less clearly, he was obviously very excited :

. . . . RVES YOU RIGHT.

`Serves you right.'

That was unexpected. No. 402 was a conformist. He hated the oppositional heretics, as one should, believed that history ran on rails according to an infallible plan and an infallible pointsman, No. 1. He believed that his own arrest was merely the result of a misunderstanding, and that all the catastrophes of the last years — from China to Spain, from the famine to the extermination of the old guard — were either regrettable accidents or caused by the devilish tricks of Rubashov and his oppositional friends. No. 402's Pushkin beard vanished; he now had a clean-shaven, fanatical face; he kept his cell painfully tidy and conformed strictly to the regulations. There was no sense in arguing with him; this kind was unteachable. But neither was there any sense in cutting off the only and perhaps the last contact with the world.

WHO? knocked Rubashov very clearly and slowly.

The answer came in agitated fits and starts :

NONE OF YOUR BUSINESS.

AS YOU LIKE, tapped Rubashov, and stood up to resume his wandering through the cell, taking the conversation to be ended. But the tapping started again, this time very loudly and ringingly — No. 402 had obviously taken off a shoe in order to give more weight to his words :

LONG LIVE H.M. THE EMPEROR!

So that's it, thought Rubashov. There still exist genuine and authentic counter-revolutionaries — and we thought that nowadays they only occurred in the speeches of No. 1, as scapegoats for his failures. But there sits a real one, an alibi for No. 1 in flesh and blood, roaring; just as he should : long live the Monarch. . . .

AMEN, tapped out Rubashov, grinning. The answer came immediately, still louder if possible :

SWINE!

Page 29

Rubashov was amusing himself. He took off his pince-nez and tapped with the metal edge, in order to change the tone, with a drawling and distinguished intonation :

DIDN'T QUITE UNDERSTAND.

No. 402 seemed to go into a frenzy. He hammered out NOUN' -, but the D did not come. Instead, his fury suddenly flown, he tapped :

WHY HAVE YOU BEEN LOCKED UP?

What touching simplicity„ .. The face of No. 402 underwent a new transformation. It became that of a young Guards officer, handsome and stupid. Perhaps he even wore a monocle. Rubashov tapped with his pince-nez :

POLITICAL DIVERGENCES.

A short pause. No. 402 was obviously searching his brain for a sarcastic answer. It came at last :

BRAVO! THE WOLVES DEVOUR EACH OTHER.

Rubashov gave no answer. He had enough of this sort of entertainment and started on his wanderings again. But the officer in 402 had become conversational. He tapped :

RUBASHOV

Well, this was just about verging on familiarity.

YES? answered Rubashov.

No. 402 seemed to hesitate; then came quite a long sen tence :

WHEN DID YOU LAST SLEEP WITH A WOMAN?

Certainly No. 402 wore an eye-glass; probably he was tapping with it and the bared eye was twitching nervously. Rubashov did not feel repelled. The man at least showed himself as he was; which was pleasanter than if he had tapped out monarchist manifestos. Rubashov thought it over for a bit, and then tapped :

THREE WEEKS AGO.

The answer came at once :

TELL ME ALL ABOUT IT.

Well, really, that was going a bit far. Rubasbov's first impulse was to break off the conversation; but he remembered the man might later become very useful as a connecting link to No. 400 and the cells beyond. The cell to the left was obviously uninhabited; there the chain broke off. Rubashov /page 30/ racked his brain. An old pre-war song came to his memory, which he had heard as a student, in some cabaret where black-stockinged ladies danced the French cancan. He sighed resignedly and tapped with his pince-nez :

SNOWY BREASTS FITTING INTO CHAMPAGNE GLASSES. . . .

He hoped that was the right tone. It was apparently, for No. 402 urged :

GO ON. DETAILS.

By this time he was doubtless plucking nervously at his moustache. He certainly had a little moustache with twirled-up ends. The devil take the man; he was the only connecting link; one had to keep up with him. What did officers talk about in the mess? Women and horses. Rubashov rubbed his pince-nez on his sleeve and tapped conscientiously :

THIGHS LIKE A WILD MARE.

He stopped, exhausted. With the best will in the world he could not do more. But No. 402 was highly satisfied.

GOOD CHAP! he tapped enthusiastically. He was doubtless laughing boisterously, but one heard nothing; he slapped his thighs and twirled his moustache, but one saw nothing. The abstract obscenity of the dumb wall was embarrassing to Rubashov.

GO ON, urged No. 402.

He couldn't. THAT'S ALL — tapped Rubashov and regretted it immediately. No. 402 must not be offended. But fortunately No. 402 did not let himself be offended. He tapped on obstinately with his monocle :

GO ON - PLEASE, PLEASE. . . .

Rubashov was now again practised to the extent of no longer having to count the signs; he transformed them automatically into acoustic perception. It seemed to him that he actually heard the tone of voice in which No. 402 begged for more erotic material. The begging was repeated:
PLEASE - PLEASE....
No. 402 was obviously still young — probably grown up in exile, sprung from an old Army family, sent back into his country with a false passport — and he was obviously tormenting himself badly. He was doubtless plucking at his little mous- /page 31/tache, had stuck his monocle to his eye again and was staring hopelessly at the whitewashed wall.

MORE—PLEASE, PLEASE.

. . . .Hopelessly staring at the dumb, whitewashed wall, staring at the stains caused by the damp, which gradually began to assume the outlines of the woman with champagne-cup breasts and the thighs of a wild mare.

TELL ME MORE — PLEASE.

Perhaps he was kneeling on the bunk with his hands folded like the prisoner in No. 407 had folded them to receive his piece of bread.
And now at last Rubashov knew of what experience this gesture had reminded him - the imploring gesture of the meagre, stretched-out hands. Pieta. . . .

9

Pieta. . . .

 

https://en.wikipedia.org/wiki/Tap_code

The tap code, sometimes called the knock code, is a way to encode text messages on a ... It was introduced in June 1965 by four POWs held in the Hỏa Lò Prison "Hanoi Hilton" prison: Captain Carlyle "Smitty" Harris, Lieutenant Phillip Butler, ...

Tap code
From Wikipedia, the free encyclopedia

 

Roman alphabet tap code
1
2
3
4
5
1
2
3
4
5
The tap code table

The tap code table

The tap code, sometimes called the knock code, is a way to encode text messages on a letter-by-letter basis in a very simple way. The message is transmitted using a series of tap sounds, hence its name.

The tap code has been commonly used by prisoners to communicate with each other. The method of communicating is usually by tapping either the metal bars, pipes, or the walls inside a cell.

The tap code is based on a Polybius square using a 5×5 grid of letters representing all the letters of the Latin alphabet, except for K, which is represented by C.

The listener only needs to discriminate the timing of the taps to isolate letters.

Each letter is communicated by tapping two numbers
the first designating the row
the second designating the column

For example, to specify the letter "B", one taps once, pauses, and then taps twice.

Or to communicate the word "water", the cipher would be the following (the pause between each number in a pair is smaller than the pause between letters):

 

W A T E R
5, 2
1, 1
4, 4
1, 5
4, 2
····· ··
· ·
···· ····
· ·····
···· ··

 

The letter "X" is used to break up sentences, and "K" for acknowledgements.

Because of the difficulty and length of time required for specifying a single letter, prisoners often devise abbreviations and acronyms for common items or phrases, such as "GN" for Good night, or "GBU" for God bless you.[1]

By comparison, Morse code is harder to send by tapping or banging because it requires the ability to create two differently sounding taps (representing the dits and dahs of Morse code). A Morse code novice would also need to keep a "cheat sheet" until he or she remembers every letter's code, which the captors would likely confiscate. Tap code can be more easily decoded in one's head by mentally using the table. For example, if you hear four knocks, you can think A...F...L...Q; then after the pause, you hear three knocks and think Q...R...S to arrive at the letter S.

History[edit]

The origins of this encoding go back to the Polybius square of Ancient Greece. As the "knock code", a Cyrillic script version is said to have been used by nihilist prisoners of the Russian czars.[2] The knock code is featured in Arthur Koestler's classic 1941 work Darkness at Noon.[3]

United States prisoners of war during the Vietnam War are most known for having used the tap code. It was introduced in June 1965 by four POWs held in the Hỏa Lò Prison "Hanoi Hilton" prison: Captain Carlyle "Smitty" Harris, Lieutenant Phillip Butler, Lieutenant Robert Peel, and Lieutenant Commander Robert Shumaker.[1] Harris had heard of the tap code being used by prisoners in World War II[4] and remembered a United States Air Force instructor who had discussed it as well.[1]

In Vietnam, the tap code became a very successful way for otherwise isolated prisoners to communicate.[4] POWs would use the tap code in order to communicate to each other between cells in a way which the guards would be unable to pick up on. They used it to communicate everything from what questions interrogators were asking (in order for everyone to stay consistent with a deceptive story), to who was hurt and needed others to donate meager food rations. It was easy to teach and newly arrived prisoners became fluent in it within a few days.[5][6] It was even used when prisoners were sitting next to each other but not allowed to talk, by tapping on anothers' thigh.[6] By overcoming isolation with the tap code, prisoners were able to maintain a chain of command and keep up morale.[4]

References[edit]

1.^ Jump up to: a b c "Return with Honor: The Tap Code". American Experience. PBS. 1999. Retrieved 2008-04-08.
2.Jump up ^ David Kahn, The Codebreakers - The Story of Secret Writing. 1967. ISBN 978-0-684-83130-5.
3.Jump up ^ Koestler, Arthur, Darkness at Noon (1941). Translated by Daphne Hardy. See page 19 of the Bantam Publishing paperback, 1981 printing for more info.
4.^ Jump up to: a b c Staff Sgt. Jason Tudor (1998-03-18). "Vets, Flyers discuss ideology, time in POW camps". Air Force News Service. Retrieved 2008-04-08.
5.Jump up ^ McCain, John; Mark Salter (1999). Faith of My Fathers. Random House. ISBN 0-375-50191-6. pp. 211–212.
6.^ Jump up to: a b Brace, Ernest C. (1988). A Code to Keep: The true story of America's longest held civilian prisoner of war in Vietnam. St. Martin's Press. ISBN 0-7090-3560-8. pp. 171–172, 187–188

 

 

The Oracle Forum at 973-Eht-Namuh-973.com

KNOCK THREE TIMES
Postby hope » Sun Jun 19, 2016 3:32 pm

One of the most important parts of a POWs life was communicating with his Fellow captives. The first communication between isolated prisoners of war may have been name scrawled on a piece of toilet paper with the burnt end of a matchstick. Notes and whispers were attempted but both were often detected and severely punished.

In June 1965, four POWs --Captain Carlyle Harris known as Smitty, Lieutenant Robert Peel and Lueitenant Commander Robert Harper Shumaker, who were imprisoned, in the same cell in Hoa Lo, devised a simple,secretive code. The four men , expecting to be split up again, vowed to continue their resistance, to do this they communicating closely would be essential.

Harris remembered an Air Force instructer who had shown him a secret code based on a five-by-five alphabet matrix, each letter was communicated by tapping two
Numbers, the first designated the horizontal row and the second the vertical row, the letter W, for example, would be 5-2, the letter H would be 2-3, the letter X was
used to break up sentences and the letter C replaced the letter K. by August 1965 most of the prisoners had been initiated, and were passing messages
By tapping. It was said that the building sounded like 'a den of runaway woodpeckers'

 

 

Knock Three Times
Tony Orlando & Dawn
Lyrics: Irwin Levine / Larry Russell Brown

Hey girl, what ya doin' down there?
Dancin' alone every night
While I live right above you

I can hear your music playin'
I can feel your body swayin'
One floor below me
You don't even know me, I love you

Oh, my darling, knock three times
On the ceiling if you want me
Hmm, twice on the pipe
If the answer is no

Oh, my sweetness
Means you'll meet me in the hallway
Oh, twice on the pipe
Means you ain't gonna show

If you look out your window tonight
Pull in the string with the note
That's attached to my heart

Read how many times I saw you
How in my silence I adored you
And only in my dreams did
That wall between us come apart

Oh, my darling, knock three times
On the ceiling if you want me
Mmm, twice on the pipe
If the answer is no

Oh, my sweetness
Means you'll meet me in the hallway
Whoa, twice on the pipe
Means you ain't gonna show

Oh, I can hear the music playin'
I can feel your body swayin'
One floor below me
You don't even know me, I love you

Oh, my darling, knock three times
On the ceiling if you want me
Oh, twice on the pipe
If the answer is no
(I love you, I love you, I love you)

Oh, my sweetness
Means you'll meet me in the hallway
Mmm, twice on the pipe
Means you ain't gonna show

 

Knock Three Times .. on the ceiling if you want me
Twice on the floor if the answer is no

Knock 11-14-15-3-11 Knock

Knock 2-5-6-3-2 Knock

Knock=9 Knock=9 Knock=9

999

 

K
=
2
-
5
KNOCK
54
18
9
K
=
2
-
5
KNOCK
54
18
9
K
=
2
-
5
KNOCK
54
18
9
-
-
6
-
15
-
162
54
18
-
-
-
-
1+5
-
1+6+2
5+4
1+8
-
-
6
-
6
-
9
9
9

 

 

Re: KNOCK THREE TIMES

Postby Redbeck » Wed Jun 22, 2016 3:51 pm
Great stuff Hope, well worth the research, examination and explanation of the more practical or perhaps practiced use of numeric coded patterns outside the technology halo to the advantage of us all, thank you.

In fact Tap Codes are spread far and wide in their application; prime example the Code of Safety for Diving, in which as you will see later, 3 x 3 clusters of three taps are an important open and close statement.

divebell-underwater.jpg (omitted)

Diving_dress_2.jpg (omitted)

However, let's get on with the detail, and I quote:

'Chapter 2
Communication
2.2.16 In addition to the communication systems referred to above, a standard bell [diving bell] emergency communication tapping code shoul be adopted, as given below, for use between persons in the bell and rescue divers.

A copy of this tapping code should be displayed both inside and outside the bell, and also in the dive control room.'

'Chapter 3
Evacuation
3.1 An evacuation system should be provided having sufficient capacity to evacuate all divers under pressure, in event of the ship having to be abandoned, and should be in accordance with the provisions of this Code'

As I mentioned the significance of three taps earlier, 3.3.3 in the table below signifies:

Communication opening procedure (inside and out [of the bell]
AND ALSO
Communication closing procedure (inside and out [of the bell]

As there is a fair sized image of the tap codes table available (omitted), I recommend you just click on the picture below for a description of the meaning of each element of a relatively simple yet still esoteric numeric tapping code alternative.

 

 

Meter reading

August 2016

Knock Knock

We popped back to read your meter today but unfortunately you weren't home.

 

Daily Mail, Thursday December 17, 2015

ANSWERS TO CORRESPONDENTS

Compiled by Charles Legge

Page 64

QUESTION When we were kids, we used to knock on people's doors and run and hide. We called this Knock Down Ginger. Why?

AT LEAST 100 terms have been collected for this activity though the variants are sometimes slight; in Coventry they played Rat-tat-tat, in Solihull Rat-a-tat-tat.
Other terms include Ding Dong Ditch, Knock Door Run, Dolly Knock, Chap Door Run or Knock, Knock Ginger Ring and Run, and Nicky Nicky Nine Door.
The most common variant seems to involve the word ginger. A popular suggestion for this is that the doors of council estates were once stained ginger Another suggestion is that it comes from an Old Mother Goose rhyme:
`Ginger, Ginger broke a winder/Hit the winda — crack!/The baker came out to give 'im a clout/And landed on his back.'
Annie Crouch, St Ives, Cambs.

 

K
=
2
-
5
KNOCK
54
18
9
K
=
2
-
5
KNOCK
54
18
9
K
=
2
-
5
KNOCK
54
18
9
-
-
6
-
15
-
162
54
18
-
-
-
-
1+5
-
1+6+2
5+4
1+8
-
-
6
-
6
-
9
9
9

 

 

Meter reading

Wakefield August 2016

Knock Knock

We popped back to read your meter today but unfortunately you weren't home.

 

-
-
-
-
-
KNOCK
-
-
-
K
=
2
-
1
K
11
2
2
N
=
5
-
1
N
14
5
5
O
=
6
-
1
O
15
6
6
C
=
3
-
1
C
3
3
3
K
=
2
-
1
K
11
2
2
-
-
18
Q
5
KNOCK
54
18
18
-
-
1+8
-
-
-
5+4
1+8
1+8
-
-
9
Q
5
KNOCK
9
9
9

 

 

10
CODE DE CODE
67
13
4
6
C+O
18
9
9
4
D+E
9
9
9
4
D+E
9
9
9
6
C+O
18
9
9
4
D+E
9
9
9
10
CODE DE CODE
63
36
36
1+0
-
6+3
3+6
3+6
1
CODE DE CODE
9
9
9

 

 

The Splendour That Was Egypt
Margaret A. Murray

New and Revised Edition 1964

Page 101

"In many countries the Divine King was allowed to reign for a term of years only, usually seven or nine or multiples of those numbers".

 

 

THE TUTANKHAMUN PROPHECIES

Maurice Cotterell

1
999

BEHIND THE WALL OF SILENCE

Page 190

The holy number of sun-worshippers is 9, the highest number that can be reached before becoming one (10) with the creator. This is why Tutankhamun was entombed in nine layers of coffin. This is why the pyramid skirts of the two statues, guarding the entrance to the Burial Chamber, were triangular (base 3), when the all-seeing eye-skirt of Mereruka contained a pyramid skirt with a base of four sides. The message concealed here is that the 3 should be squared, which equals 9"

"The message concealed here is that the 3 should be squared, which equals 9"

 

 

TUTANKHAMEN

Christiane Desroches-Noblecourt 1963

Page177

"The red oxen had been left behind; now the "Nine Friends" and the two viziers-of the North and South- drew the ropes attached to the bier behind which followed a last high dignitary of the royal procession."

Page158 (Chapter 7) 1343

The death of the king and preparations for immortality 

"Analysis of his mummy shows that Tutankhamen was between eighteen and twenty years old when he died. This allows one to set the approximate date of his coronation in his ninth year, since there is no date mentioned in connexion with him after year 9 which appears on wine jars found in his tomb."

 

 

TRANSFORMATION THE BREAKTHROUGH

Whitley Strieber 1988

“nine knocks I was shattered, overwhelmed. I remembered their eerie precision-three groups of three
“nine of them in three groups of three

 

 

THE RISE AND FALL OF ANCIENT EGYPT

The History of a Civilisation from 3000 BC to Cleopatra

Toby Wilkinson 2010

Page 30

"In total, Atum and his immediate descendants numbered nine deities, three times three expressing the the ancient Egyptian concept of completeness.

“nine deities, three times three

 

 

STEPHEN HAWKING

Quest For A Theory Of Everything

Kitty Ferguson 1992

Page 103

"...The square root of 9 is 3. So we know that the third side

This occurs on the 33rd line down of page 103

 

 

TRANSFORMATION THE BREAKTHROUGH

Whitley Strieber 1988

Page 128

"Dr Gliedman had given me his essay "Quantum Entanglements: On Atomic Physics and the Nature of Reality," and I had been reading it..."

"Page 129

"I returned to Dr. Gliedman's essay.

I read the following sentence: "The mind is not the playwright of reality."

At that moment there came a knocking on the side of the house. This was a substantial noise, very regular and sharp. The knocks were so exactly spaced that they sounded like they were being produced by a machine. Both cats were riveted with terror. They stared at the wall. The knocks went on, nine of them in three groups of three, followed by a tenth lighter double-knock that communicated an impression of finality.

These knocks were coming from just below the line of the roof, at a spot approximately eighteen feet above the gravel driveway. Below the point of origin of the knocks were two open windows. Had anybody been out on the driveway with a ladder I would certainly have heard their movements on the gravel.

In addition, to get a ladder to that point they would have activated the movement-sensitive lights. But it was dark beyond the windows.

It would be next to impossible to stand on the sharply angled roof that covers the living room of the cabin. While the angle of the roof above the upstairs bedroom is almost flat, this roof is extremely steep. What's more, I would certainly have heard anybody crawling around on the roof. There would have been creaks and groans from the boards, and there is no question but that I would have noticed the sounds, given the profound silence of the country night.

I am absolutely dead certain about the reality of the knocks. They were not made by the house settling. Nothing but an intentional act could have produced such loud, evenly spaced sounds. They were not a prank being played by neighbors. In the summer of 1986 I had not yet told my neighbors about the visitors. What's more, the prank explanation was hopelessly impractical.

To reach the place from which I heard the knocks..."

Page 131

cannot be put down to disease. Such a thing is not a symptom. My cats would not have reacted to something happening in my mind. I am reporting a true event. It was the first definite, physical indication I had while in a state of commpletely normal consciousness that the visitors were part of this world.

They were responding to my attempts to develop the relationship and accept my fear by making their physical reality more plain.

The stunning event of August 27, 1986, strengthened my wavering resolve to keep the matter where it belongs, which is in question. It is an awfully serious business, and it cannot be removed from question except as we learn more facts. Should we decide to believe something about this that is not true, we will ruin it for ourselves. We will form yet another mythology around the visitors, as I suspect we have been doing throughout our history.

The moment after the nine knocks I thought to go outside. I also thought, You're not ready yet. You just go up to bed.

The next morning I thought that was exactly what I had done. But there was something wrong. While the knocks were taking place I was unquestionably in a normal state of mind. As soon as I began to move from the chair, though, I feel that I may have entered another state.

Unfortunately, I did not remember that something may have happened after the knocks until weeks later. On the morning after, my immediate thought was that I had failed miserably. The visitors had come, had knocked-and I'd just sat there, too scared even to open the door!

I therefore don:t know whether I concocted the subsequent memories to make myself feel better, or if they were hidden by a more prosaic screen memory.

One day I glanced at the clock on our videotape machine and suddenly remembered seeing it when it said 2:18 A.M. An instant later I recalled that I'd seen it reading that time as I went upstairs on the night of the nine knocks.

Page 134 (omitted)

TWELVE

Fire of the Question

"In the days after I heard the nine knocks I was shattered, overwhelmed. I remembered their eerie precision-three groups of three perfectly measured, exactly spaced sounds, each precisely as loud as the one previous. And then there had been a soft double-knock completely different in tone from the others. It had communicated a distinct sense of finality, and seemed by its lightness of tone not to be a part of the group. The nine knocks were a sort of communication. The tenth was punctuation..."

Page 135

""The nine knocks made me struggle even harder to understand. And I did not understand. But I had a few ideas"

"It was as if I had discovered an unknown world that has always been around us, that may be an even greater reality..."
"In the days after I heard the nine knocks I was shattered, overwhelmed. I remembered their eerie precision-three groups of three perfectly measured, exactly spaced sounds,”

 

Page 135

"The nine knocks made me struggle even harder to understand. And I did not understand. But I had a few ideas

It was as if I had discovered an unknown world that has always been around us, that may be an even greater reality..."

 

 

Daily Mail, Thursday December 17, 2015

ANSWERS TO CORRESPONDENTS

Compiled by Charles Legge

Page 64

QUESTION When we were kids, we used to knock on people's doors and run and hide. We called this Knock Down Ginger. Why?

AT LEAST 100 terms have been collected for this activity though the variants are sometimes slight; in Coventry they played Rat-tat-tat, in Solihull Rat-a-tat-tat.
Other terms include Ding Dong Ditch, Knock Door Run, Dolly Knock, Chap Door Run or Knock, Knock Ginger Ring and Run, and Nicky Nicky Nine Door.
The most common variant seems to involve the word ginger. A popular suggestion for this is that the doors of council estates were once stained ginger Another suggestion is that it comes from an Old Mother Goose rhyme:
`Ginger, Ginger broke a winder/Hit the winda — crack!/The baker came out to give 'im a clout/And landed on his back.'
Annie Crouch, St Ives, Cambs.

 

 

https://www.youtube.com/watch?v=amc_7l16LSM

Knock Three Times - Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Knock_Three_Times

"Knock Three Times" is a popular song credited to Tony Orlando and Dawn. ... The song was released as a single in November 1970, paired with Orlando's other hit song, "Candida" (also written by Toni .... Full lyrics of this song at MetroLyrics ...

 

 

Meter reading

Wakefield August 2016

Knock Knock

We popped back to read your meter today but unfortunately you weren't home.

 

-
-
-
-
-
KNOCK
-
-
-
K
=
2
-
1
K
11
2
2
N
=
5
-
1
N
14
5
5
O
=
6
-
1
O
15
6
6
C
=
3
-
1
C
3
3
3
K
=
2
-
1
K
11
2
2
-
-
18
Q
5
KNOCK
54
18
18
-
-
1+8
-
-
-
5+4
1+8
1+8
-
-
9
Q
5
KNOCK
9
9
9

 

 

10
CODE DE CODE
67
13
4
6
C+O
18
9
9
4
D+E
9
9
9
4
D+E
9
9
9
6
C+O
18
9
9
4
D+E
9
9
9
10
CODE DE CODE
63
36
36
1+0
-
6+3
3+6
3+6
1
CODE DE CODE
9
9
9

 

 

THE RISE AND FALL OF ANCIENT EGYPT

The History of a Civilisation from 3000 BC to Cleopatra

Toby Wilkinson 2010

Page 30

"In total, Atum and his immediate descendants numbered nine deities, three times three expressing the ancient Egyptian concept of completeness.

“nine deities, three times three

 

 

STEPHEN HAWKING

Quest For A Theory Of Everything

Kitty Ferguson 1992

Page 103

"...The square root of 9 is 3. So we know that the third side

This occurs on the 33rd line down of page 103

 

 

TRANSFORMATION THE BREAKTHROUGH

Whitley Strieber 1988

Page 134

"In the days after I heard the nine knocks I was shattered, overwhelmed. I remembered their eerie precision-three groups of three perfectly measured, exactly spaced sounds,”

“nine knocks” “three groups of three

 

Whitley Strieber 1988 Page 134 “nine knocks I was shattered, overwhelmed. I remembered their eerie precision-three groups of three
“nine of them in three groups of three

Page 135

"The nine knocks made me struggle even harder to understand. And I did not understand. But I had a few ideas

It was as if I had discovered an unknown world that has always been around us, that may be an even greater reality..."

 

 

Toby Wilkinson 2010 Page 30 "In total, Atum and his immediate descendants numbered nine deities, three times three expressing the ancient Egyptian concept of completeness.


Kitty Ferguson 1992 Page 103 "...The square root of 9 is 3. So we know that the third side
This occurs on the 33rd line down of page 103

 

 

Daily Mail, Thursday, December 24, 2015

By Ben Wilkinson

Page 124

Life-saving 999 call by girl of 3 after pregnant mother fell down stairs

WHEN pregnant Catherine Bazzard fell downstairs and knocked herself out, she was lucky to have a cool-headed life-saver nearby-her three-year old daughter Emma.

The youngster calmly told the emergency operator that Mrs Bazzard had a baby in her tummy and directed paramedics to her house.

Yesterday, South Western Ambukance Service released the 11 minute transcript of her call as it gave her a bravery award.

Emma found her mother slipping in and out of consciousness at the foot of the stairs after the 27-year old fell while rushing to get her son Harry, five from school.

In the recording of the call on November 27, Emma tells the call handler: 'Mummy fell down a stairs and she has got a baby'

After she was asked whether the baby is 'asleep' and how old it is. Emma reveals: It's in mummy's tummy. It's very very big. It's coming at Christmas.'

Extracts of the call from Emma to 999 call handler Sara Morris:

S:You are so clever to ring up E: Mummy said '999' and I did it.

"Pre-school worker Mrs Bazzard said: 'I remember Emma bringing me the phone. She couldn't get the nine button to work so I pressed it."

"...her mum spent three nights in hospital..."

Mrs Bazzard who is married to Ben, 33, a teacher said that had Emma not responded so quickly her son George would have been born dangerously early ...

"George was born healthy on December 4th four weeks early Ambulance call handler Sarah Morris said: 'I was amazed to discover that she was only three."

 

 

Daily Mail, Friday, June 24, 2016

ANSWERS TO CORRESPONDENTS

Compiled by Charles Legge

Page 68

Tapping in to sabotage

QUESTION When did the term 'phone tapping' appear in print?

THE wire tap or phone tap gained its name because, historically, the monitoring took place via an actual electrical tap on the telephone line. The first direct evidence of wire tapping was recorded in the American Civil War (1861-1865).

The connection of the West Coast of the United States to the East took place in 1861, opening the door to cross-country wire tapping. The advent of war gave both sides the opportunity to intercept messages and disrupt communications.

The earliest written report of wire tapping appeared in September 1862, in the Smoky Hill And Republican Union newspaper published in Junction City, Kansas.

According to this, Confederate Commander John Morgan and his ragtag force had destroyed bridges at Gallatin, Tennessee, cutting the railroad connection north, but 'not the telegraph; which indicated they have tapped the wires and are taking the Union dispatches from them'.

Morgan's was regarded as one of the most effective cavalry units in the war. Among his men was George 'Lightning' Ellsworth, an expert telegrapher and wire-tapper.
In July 1862, Morgan's unit made its first extensive raid into Kentucky. They covered 1,000 miles in 24 days, burning railroad trestles, destroying Union supplies and disrupting communications.

Ellsworth's ability to tap out false commands on the telegraph wires kept the Confederate cavalry a step ahead of the Federal soldiers and allowed Morgan and his men to retire to Tennessee almost unharmed.

With the arrival of the telephone system, concerns about phone tapping and privacy appeared in a Toronto newspaper: 'Another serious defect in the telephone is the ease with which the wires can be tapped.

`Hitherto no one but an experienced telegrapher could steal lightning, to tap a wire that is running under the Morse or Wheatstone code of signals requires knowledge of those signals and the possession of instruments. But the telephone is so easily managed that the understanding of it could be made the duty of every non-
commissioned officer in an army'

Wire tapping was a significant problem during World War I. Both sides laid down lengths of wire in their forward trenches to act as antennae, which could intercept enemy telegraph and telephone messages by induction through the soil.

Some of these security problems were rectified by the introduction of the Fuller-phone, devised by Royal Engineer signals Captain and Institution of Electrical Engineers (IEE) associate member Algernon Clement 'AC' Fuller (1885-1970).

This was a portable Morse telegraph that transmitted a DC signal through a single wire, but incorporated what was, in effect, a method of scrambling messages to prevent enemy interception.

Pole position: A telegraph engineer at work in the U.S. during the mid-1860s (image omitted)

Mr J. A. Saunders, Conwy.

 

1
I
9
9
9
3
SAY
45
9
9
7
DECODER
54
36
9
6
DECODE
36
27
9
4
CODE
27
18
9
-
-
-
-
-
-
-
-
-
-
5
CODED
31
22
4
5
CODES
46
19
1

 

 

-
CODE
--
-
-
2
C+O
18
9
9
2
D+E
9
9
9
-
DECODE
--
-
-
2
D+E
9
9
9
2
C+O
18
9
9
2
D+E
9
9
9
-
DECODER
--
-
-
2
D+E
9
9
9
2
C+O
18
9
9
2
D+E
9
9
9
1
R
18
9
9

 

 

CODE DE CODE

C+O D+E D+E C+O D+E

9+9+9+9+9

C+O D+E D+E C+O D+E

CODE DE CODE

 

 

6
DECODE
-
-
-
-
D+E
9
9
9
1
C+O
18
9
9
-
D+E
9
9
9
6
DECODE
36
27
27
-
-
3+6
2+7
2+7
6
DECODE
9
9
9

 

 

THE ATLANTIS SECRET

A COMPLETE DECODING OF PLATOS LOST CONTINENT

Alan F. Alford 2001

Page v

ATLANTIS RECONSIDERED

"All of these myths, and many others besides, concealed a 'Secret of secrets' that was accepted, unquestioningly, as a true account of the origins of the cosmos and man...."

Page v

"...Until now, this mainstream breakthrough in comparative religion has found limited application, for, if truth be told, the literature of the Near East is as much of a puzzle to scholars as the literature of the Greeks. Now, however, in the light of my recent deciphering of the Egyptian and Mesopotamian myths, the full import of these parallels may be felt For the first time ever, it becomes possible to understand the Greek myths by literally standing under them. For the first time ever, we can get inside Plato's mind and reconsider the story of Atlantis from an ancient, rather than a modern, perspective.
The result is nothing short of a sensation. In this book, I present not only a complete decoding of the lost continent of Atlantis, but also a complete decoding of ancient Greek religion in its entirety. I am able to decode the myths of the Olympian gods and their associated mystery cults; I am able to decode the myth of the golden age and the fall of man; I am able to decode the scientific cosmogoniess of Thales, Arnixim-inder, Heraclitus, Anaxagoras, Empedocies and Philolaos; I am able to decode the 'soul religion' of Orpheus, Pythagoras, Parmenides, Socrates and Plato; and I am able to decode Plato's Theory of Forms, his account nf the creation by the Demiourgos, and his story of Atlantis. Behind all of these ideas there lies a single secret of stunning simplicity - ..."

 

 

THE WORLD IS BUILT UPON THE POWER OF NUMBERS

"Pythagoras, the old master philosopher and mathematician, who lived in the sixth century BC, propounded the theory that nothing in the universe could exist without numbers. He established a Mystery School in Italy when he was 52 years old. He was born in Greece and lived between 582 and 507 BC, much of his life spent in study and travel. His Mystery School taught esoteric knowledge, which included the secret of number and vibration."

“The World is built upon the power of Numbers” ...Pythagoras – 6th century BC.

 

THE WORLD IS BUILT UPON THE POWER OF NUMBERS

 

T
=
2
Q
3
THE
33
15
6
W
=
5
-
5
WORLD
72
27
9
I
=
9
-
2
IS
28
10
1
B
=
2
Q
5
BUILT
64
19
1
U
=
3
-
4
UPON
66
21
3
T
=
2
Q
3
THE
33
15
6
P
=
7
Q
5
POWER
77
32
5
O
=
6
-
2
OF
21
12
3
N
=
5
-
7
NUMBERS
92
29
2
-
-
41
-
36
First Total
486
180
36
-
-
4+1
-
3+6
Add to Reduce
2+7+9
1+8+0
3+6
-
-
5
-
9
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
5
-
9
Essence of Number
9
9
9

 

 

THE

LAWS OF NATURE ARE WRITTEN IN THE LANGUAGE OF MATHEMATICS

Galileo Galilei (1564-1642)

 

T
=
2
Q
3
THE
33
15
6
L
=
3
-
4
LAWS
55
10
1
O
=
6
-
2
OF
21
12
3
N
=
5
-
6
NATURE
79
25
7
A
=
1
Q
3
ARE
24
15
6
W
=
5
-
7
WRITTEN
109
37
1
I
=
9
-
2
IN
23
14
5
T
=
2
Q
3
THE
33
15
6
L
=
3
Q
8
LANGUAGE
68
32
5
O
=
6
-
2
OF
21
12
3
M
=
4
-
11
MATHEMATICS
112
40
4
-
-
46
-
51
First Total
578
227
47
-
-
4+6
-
5+1
Add to Reduce
5+7+8
2+2+7
4+7
-
-
10
-
6
Second Total
20
11
11
-
-
1+0
-
-
Reduce to Deduce
2+0
1+1
1+1
-
-
1
-
6
Essence of Number
2
2
2

 

 

THE

LAWS OF NATURE ARE WRITTEN IN THE LANGUAGE OF MATHEMATICS

Galileo Galilei (1564-1642)

 

-
-
-
-
-
THE
-
-
-
L
=
3
-
4
LAWS
55
10
1
O
=
6
-
2
OF
21
12
3
N
=
5
-
6
NATURE
79
25
7
A
=
1
Q
3
ARE
24
15
6
W
=
5
-
7
WRITTEN
109
37
1
I
=
9
-
2
IN
23
14
5
T
=
2
Q
3
THE
33
15
6
L
=
3
Q
8
LANGUAGE
68
32
5
O
=
6
-
2
OF
21
12
3
M
=
4
-
11
MATHEMATICS
112
40
4
-
-
44
-
48
First Total
545
212
41
-
-
4+4
-
4+8
Add to Reduce
5+4+5
2+1+2
4+1
-
-
8
-
12
Second Total
14
5
5
-
-
-
-
1+2
Reduce to Deduce
1+4
-
-
-
-
8
-
3
Essence of Number
5
5
5

 

 

T
=
2
Q
3
THE
33
15
6
L
=
3
Q
8
LANGUAGE
68
32
5
O
=
6
-
2
OF
21
12
3
M
=
4
-
11
MATHEMATICS
112
40
4
-
-
44
-
48
First Total
234
99
18
-
-
4+4
-
4+8
Add to Reduce
2+3+4
9+9
1+8
-
-
8
-
12
Second Total
9
18
9
-
-
-
-
1+2
Reduce to Deduce
-
1+8
-
-
-
8
-
3
Essence of Number
9
9
9

 

 

NUMEROLOGY
Geddes and Grosset 1999.
Introduction.

Page 5 /

"Nunerology is the name given to an ancient method of studying numbers that has been in use for thousands of years."
"…The most popular form of numerology in use today is based on the work of Pythagoras, the famous Greek mathematician and philosopher who lived during the sixth century bc."
"…It was Pythagora's belief that numbers were the first of all things in nature. It was his belief that numbers were the basis of everything, in the natural, spiritual and scientific world. He believed that everything could be reduced to mathematical terms and that everything had a numerical value. Through studying the world in numerical form, he sought to achieve greater understanding of the world he lived in. Pythagoras, who believed that numbers created order and beauty, founded a school for students to follow his philosophy, and this was known as the Italic or Pythagorean School."

/ Page 6 /

"…Pythagoras formulated the concept called the Music of the Spheres', based on the idea that all the planets in the universe formed a harmonious whole consisting of a mu-sical chorus. He discovered that there was a relationship between sound and numbers, and developed this discov-ery to form his metaphysical concept. He suggested that every planet was a certain distance from a central point in the universe and that if an invisible string connected each planet to the central point, when plucked the string would emit a certain tone or vibration. Each sound or vibration could be associated with a particular number. He also be-lieved that the sound or vibration of the universe dictated by the planets would have a strong influ-ence on the character of an individual born at that particu-lar time.
Numerologists believe that the numbers one to nine have specific characteristics, and these characteristics are the basis for the methods of analysis described in this book. The numbers one to nine are the only numbers that are

/ Page 7 /

believed to be significant to numerology. All numbers greater than nine can be reduced to a single digit by the process of fadic addition, for example:
12 is reduced to 3 by adding 1 and 2;
49 is reduced to 4 by adding 4 and 9 which equals 13 and subsequently adding 1 and 3 to make 4."

 

 

John Michell 1972.

CITY OF REVELATION

Page 36 ..... All numbers greater than nine can be reduced to a single digit by the process of fadic addition, for example: 12 is reduced to 3 by ...

 

 

F
=
6
-
5
FADIC-
23
23
5
A
=
5
-
8
ADDITION
76
40
4
-
-
11
-
13
First Total
99
63
63
-
-
-
-
1+3
Add to Reduce
9+9
6+3
6+3
-
-
2
-
4
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
2
-
4
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
F
=
6
-
5
FADIC
23
23
5
-
-
-
-
-
-
-
-
-
-
A
=
5
-
8
ADDITION
76
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
F
=
6
1
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
D
=
4
3
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
4
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
T
=
2
10
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
11
1
I
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
ADDITION
-
-
-
-
2
2
3
12
5
12
7
8
27
-
-
-
-
-
FADIC-
-
-
-
-
-
-
-
1+2
-
1+2
-
-
2+7
-
-
63
-
13
First Total
99
63
63
-
2
2
3
3
5
3
7
8
9
-
-
6+3
-
1+3
Add to Reduce
9+9
4+5
6+3
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Second Total
18
9
9
-
2
2
3
3
5
3
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
4+5
4+5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
2
3
3
5
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
9
F
=
6
-
5
FADIC
23
23
5
-
-
-
-
-
-
-
-
A
=
5
-
8
ADDITION
76
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
9
F
=
6
1
1
F
6
6
6
-
-
-
-
-
-
6
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
D
=
4
3
1
D
4
4
4
-
-
-
-
4
-
-
-
I
=
9
4
1
I
9
9
9
-
-
-
-
-
-
-
9
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
-
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
9
T
=
2
10
1
T
20
2
2
-
-
2
-
-
-
-
-
I
=
9
11
1
I
18
9
9
-
-
-
-
-
-
-
9
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
-
N
=
5
13
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
ADDITION
-
-
-
-
2
2
3
12
5
12
27
-
-
-
-
-
FADIC-
-
-
-
-
-
-
-
1+2
-
1+2
2+7
-
-
63
-
13
First Total
99
63
63
-
2
2
3
3
5
3
9
-
-
6+3
-
1+3
Add to Reduce
9+9
4+5
6+3
-
-
-
-
-
-
-
-
-
-
9
-
4
Second Total
18
9
9
-
2
2
3
3
5
3
9
-
-
-
-
-
Reduce to Deduce
1+8
4+5
4+5
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
2
3
3
5
3
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
F
=
6
-
5
FADIC
23
23
5
-
-
-
-
-
-
-
-
-
-
A
=
5
-
8
ADDITION
76
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
7
8
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
7
8
-
T
=
2
10
1
T
20
2
2
-
-
2
-
-
-
-
7
8
-
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
-
7
8
-
D
=
4
3
1
D
4
4
4
-
-
-
-
4
-
-
7
8
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
7
8
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
7
8
-
N
=
5
13
1
N
14
5
5
-
-
-
-
-
5
-
7
8
-
F
=
6
1
1
F
6
6
6
-
-
-
-
-
-
6
7
8
-
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
7
8
-
I
=
9
4
1
I
9
9
9
-
-
-
-
-
-
-
7
8
9
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
7
8
9
I
=
9
11
1
I
18
9
9
-
-
-
-
-
-
-
7
8
9
-
-
-
-
-
ADDITION
-
-
-
-
2
2
3
12
5
12
7
8
27
-
-
-
-
-
FADIC-
-
-
-
-
-
-
-
1+2
-
1+2
-
-
2+7
-
-
63
-
13
First Total
99
63
63
-
2
2
3
3
5
3
7
8
9
-
-
6+3
-
1+3
Add to Reduce
9+9
4+5
6+3
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Second Total
18
9
9
-
2
2
3
3
5
3
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
4+5
4+5
-
-
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
2
3
3
5
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
9
F
=
6
-
5
FADIC
23
23
5
-
-
-
-
-
-
-
-
A
=
5
-
8
ADDITION
76
40
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
A
=
1
6
1
A
1
1
1
-
1
-
-
-
-
-
-
T
=
2
10
1
T
20
2
2
-
-
2
-
-
-
-
-
C
=
3
5
1
C
3
3
3
-
-
-
3
-
-
-
-
D
=
4
3
1
D
4
4
4
-
-
-
-
4
-
-
-
D
=
4
7
1
D
4
4
4
-
-
-
-
4
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
-
-
5
-
-
F
=
6
1
1
F
6
6
6
-
-
-
-
-
-
6
-
O
=
6
12
1
O
15
6
6
-
-
-
-
-
-
6
-
I
=
9
4
1
I
9
9
9
-
-
-
-
-
-
-
9
I
=
9
9
1
I
9
9
9
-
-
-
-
-
-
-
9
I
=
9
11
1
I
18
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
ADDITION
-
-
-
-
2
2
3
12
5
12
27
-
-
-
-
-
FADIC-
-
-
-
-
-
-
-
1+2
-
1+2
2+7
-
-
63
-
13
First Total
99
63
63
-
2
2
3
3
5
3
9
-
-
6+3
-
1+3
Add to Reduce
9+9
4+5
6+3
-
-
-
-
-
-
-
-
-
-
9
-
4
Second Total
18
9
9
-
2
2
3
3
5
3
9
-
-
-
-
-
Reduce to Deduce
1+8
4+5
4+5
-
-
-
-
-
-
-
-
-
-
9
-
4
Essence of Number
9
9
9
-
2
2
3
3
5
3
9

 

 

Numerology - Oddx Paranormal Oddities

oddx.com/numerology/‎

Jan 12, 2013 - ... the truth,” while Pythagoras once said that “The world is built on the power of numbers”. Pythagoras also believed that there was nine stages ...

Numerology is the practice of attempting to use numbers derived from people’s names, date of birth, phone number, etc to determine that person’s personality and destiny. It bears many similarities to astrology, and some believe that the two are connected. Numerology is based on the belief that everything in the universe can be expressed by numbers, and many religions have at some point attempted to integrate numerology with their beliefs, claiming that numerology is a message encoded into the universe for them by their deity/deities.

History of Numerology

Numerology originated from ancient Babylonia, but modern Numerology contains elements from many cultures and teachings, including:
Ancient Hebrew Kabbalah
Pythagorean teachings
Hindu Vedas
Early Christian mysticism
Chinese ‘Circle of the Dead’
Egyptian ‘Book of the Masters of the Secret House’

Many ancient philosophers and mathematicians believed that as mathematical concepts were provable, unlike physical ones, numbers could be used to discern links between everything in creation, and predict the future. St. Augustine of Hippo (A.D. 354–430) wrote, “Numbers are the Universal language offered by the deity to humans as confirmation of the truth,” while Pythagoras once said that “The world is built on the power of numbers”. Pythagoras also believed that there was nine stages of the cycle of life, each of which was connected to a number from 1 to 9, and this was the source of all energy in the universe.

During the early stages of Christianity dominance of Europe many attempted to find link biblical concepts to numerology, resulting in such ideas as the ‘Jesus Number’, and the ‘Number of the Beast’. This practice is still found in some Greek Orthodox churches.
Modern numerology started to take shape in the early 1970’s, with the work of numerologists such as Ruth A. Drayer, Dr. Juno Jordan, and Lynn Buess. The work of these numerologists still comprises much of the belief today, and still used as guides for seminars and those new to the belief.

How Numerology Works

Numerology involves turning your name, date of birth, and several other characteristics into a series of numbers between 1 and 9, which are then used to attempt to determine your personality, future, heart’s desire. Supposedly, numerology can even be used to determine what affect things like your house or phone number are having on your life, and how to overcome any problems they may be causing.

First, you have to turn your name into a number. In numerology, every letter corresponds to a number from 1 to 9, as shown below:

A = 1 B = 2 C = 3 D = 4 E = 5 F = 6
G = 7 H = 8 I = 9 J = 1 K = 2 L = 3
M = 4 N = 5 O = 6 P = 7 Q = 8 R = 9
S = 1 T = 2 U = 3 V = 4 W = 5 X = 6
Y = 7 Z = 8

 

 

In Search Of The Miraculous

Fragments of an Unknown Teaching

P.D.Oupensky

Page 96 9 x 6 = 54

" There exist not one, but three universal languages. The first of them can be spoken and written while remaining within the limits of ones' own language. The only difference is that when people speak in their ordinary language they do not understand one another but in this other language they do understand. In the second language, written language is the same for all peoples, like say figures or mathematical formulae; but people still speak their own language yet each of them understands the other even though the other speaks in an unknown language. The third language is the same for all both the written and the spoken. The difference of language disappears altogether on this level."

Page 283

"In western systems of occultism there is a method known by the name of 'theosophical addition', that is, the definition of numbers consisting of two or more digits by the sum of those digits. To people who do not understand the symbolism of numbers this method of synthesizing numbers seems to be absolutely arbitrary and to lead nowhere. But for a man who understands the unity of everything existing and who has the key to this unity the method of theosophical addition has a profound meaning, for it resolves all diversity into the fundamental laws which govern it and which are expressed in the numbers 1 to 10. As was mentioned earlier in symbology,

as represented , numbers are connected with definate geometrical figures and are mutually complimentary one to another. In the Cabala a symbology of letters is also used and in combination with the symbology of letters a symbology of words.A combination of the four methods of symbolism by numbers, geometrical figures, letters and words, give a complicated but more perfect method."

 

 

THE CITY OF REVELATION

John Michell 1972

Gnostic Numbers

Page 118

"Exactly how they came by their science of numbers is not certain, but they appear to have made the discovery that the numerical code of the Hebrew cabala and those of other mystical systems throughout the world were all degenerate versions of the same once universal system of knowledge that returns within the reach of human perception at certain intervals in time. As the revealed books of the Old Testament were written in a code to be interpreted by reference to number, so were the revelations of the gnostic prophets expressed in words and phrases formed on a system of proportion, which gave life and power to the Christian myth, while allowing initiates to gain a further understanding of the balance of forces that produce the world of phenomena."

Page 121 / How it was ever supposed that the Hebrew alphabet of twenty-two letters, together with various geometrical symbols might serve to represent the entire moving pattern of the universe is not now easy to understand; but, since all ancient philosophy, religion, magic, the arts and sciences were based on the concept of a correspondence between numbers and cosmic law, it is impossible to appreciate the history of the past without some actual experience of the fundamental truth behind this approach to cosmology. Plato gives a remarkable account in Cratylos of the origin of language and letters. The philosopher is asked whether there is any particular significance in names, for surely they are simply a matter of convention and one is more or less as good as another. After all, foreigners call things by different names and appear to manage just as well as the Greeks in this respect. The answer given is that despite appearances the matter is by no means so simple. Words are the tools of expression, and the making of these, as of any other tools, is the task of a skilled craftsman, in this case the lawgiver. Language has grown corrupt over the ages, and names have deviated from their original perfect forms, which are those used by the gods. But all names were originally formed on certain principles, through knowledge of which it is possible to discover the archetypal meaning of words in current use. 'So perhaps the man who knows about names considers their value and is not confused if some letter is added, transposed or subtracted, or even if the force of the name is expressed in quite different letters.' This is Plato's clearest reference to the mystical science of the cabala, in which letters, words and whole phrases may be substituted for others of the same numerical value. The force of a name is to be found in its number, and can be expressed through any combination of letters, provided the sum of the letters amounts to the appropriate number by gematria.

 

 

REVELATION

John Michell 1972

Introductory Note On Gematria

The Numerical Correspondences

of The Greek Alphabet

Page 7

"...There were formerly two other letters, representing numbers 90 and 900, but they became obsolete in literature, retained only as numerical symbols. Another letter, the digamma of value 6, also fell out of use and was replaced..."

 

 

HOLY BIBLE

Scofield References

Jeremiah B.C. 590

Page 809 8 x 9 + 72 7 + 2 = 9 Chapter 33 Verse 3 x 33 = 99

"Call unto me, and I will answer thee, and shew thee great and mighty things which thou knowest not."

 

 

THE FINGERPRINTS Of THE GODS

Graham Hancock 1998

Page 274 / 275

"The pre-eminent number in the code is 72. To this is frequently added 36, making 108, and it is permissible to multiply 108 by 100 to get 10,800 or to divide it by 2 to get 54, which may then be multiplied by 10 and expressed as 540 (or as 54,000, or as 540,000, or as 5,400,000, and so on). Also highly significant is 2160 ( the number of years required for the equinoctial point to transit one zodiacal constellation), which is sometimes multiplied by 10 and by factors of ten (to give 216,000, 2,160,000, and so on)

" and sometimes by 2 to give 4320, or 43,200, or 432,000, or 4,320,000, ad infinitum."

 

"The pre-eminent number in the code is 72

 

 

Fingerprints of the Gods, Graham Hancock.

Page 274. "Seventy-two = the number of years required for the equinoctial sun to complete a precessional shift of one degree along the ecliptic."

 

 

Fingerprints Of The Gods

Graham Hancock 1995

Page 381 Chapter 41

 "Conscious of being alone, this blessed and immortal being contrived to create two divine offspring, Shu, god of the air and dryness, and Tefnut the goddess of moisture: ' I thrust my phallus into my closed hand. Imade my seed to enter my hand. Ipoured it into my own mouth. I evacuated under the form of Shu, I passed water under the form of Tefnut.' 7

Despite such apparently inauspicious beginnings, Shu and Tefnut (who were always described as 'Twins' and frequently depicted as lions) grew to maturity, copulated and produced offspring of their own: Geb the god of the earth and Nut, the goddess of the sky. These two also mated, creating Osiris and Isis, Set and Nepthys, and so completed the Ennead, the full company of the Nine Gods of Heliopolis. Of the nine, Ra, Shu, Geb and Osiris were said to have ruled in Egypt as Kings, followed by Horus, and lastly - for 3226 years - by the Ibis-headed wisdom god Thoth.8"

" 3226 years "

3 x 2 x 2 x 6 = 72    

In short it seems that secret knowledge is indeed available in the myth of Osiris and in the dimensions of the Great Pyramid. With this secret knowledge, if we wanted to fix a specific date - say 1008 years in the future - and communicate it to other initiates, then we could do so with the 'special number' 14 (72 x 14 = 1008). We would also have to specify the 'zero point' from which they were to make their calculations - i.e the present epoch - and this might be done with  

 

 

Fingerprints of the Gods

Graham Hancock 1995

Page 71

"Osiris, The ancient Egyptian high god of death and resurrection."

"…He was plotted against by seventy-two members of his court, led by his brother- in -law Set..."

"… Set, out hunting in the marshes, discovered the coffer, opened it and in a mad fury cut the royal corpse into fourteen pieces,"

seventy-two x fourteen

72 x 14

1008

Ra and the Eight

 

 

Joseph and His Brothers.

Thomas Mann

Minerva 1997

The Two Fine Gentlemen.

Page 890. 8 x 9 is 72.

"In all there were two and seventy conspirators privy to the plot. It was a proper and a pregnant number, for there had been just seventy-two when red Set lured Usir into the chest. And these seventy-two in their turn had had good cosmic ground to be no more and no less than that number. For it is just that number of groups of five weeks which make up the three hundred and sixty days of the year, not counting the odd days, and there are just seventy-two days in the dry fifth of the year, when the gauge shows that the Nourisher has reached his lowest ebb, and the god sinks into his grave. So where there is conspiracy anywhere in the world it is requisite and customary for the number of conspirators to be seventy-two. And if the plot fail, the failure shows that if this number had not been adhered to it would have failed even worse."

One this page 890, two and seventy occurs once. Seventy-two occurs four times. Five times seventy-two makes three hundred and sixty as in "three hundred and sixty days of the year not counting the odd days,"

Page 891. 8 x 9 x 1 = 72.

Second and third line down.

"Possibly at the last minute one of the seventy-two."#

 

 

https://en.wikipedia.org/wiki/Joseph_and_His_Brothers

Joseph and His Brothers (Joseph und seine Brüder) is a four-part novel by Thomas Mann, written over the course of 16 years. Mann retells the familiar stories of ...

Joseph and His Brothers (Joseph und seine Brüder) is a four-part novel by Thomas Mann, written over the course of 16 years. Mann retells the familiar stories of Genesis, from Jacob to Joseph (chapters 27–50), setting it in the historical context of the Amarna Period. Mann considered it his greatest work.

The tetralogy consists of:
The Tales of Jacob (written December 1926 to October 1930, Genesis 27–36)
The Young Joseph (written January 1931 to June 1932, Genesis 37)
Joseph in Egypt (written July 1932 to 23 August 1936, Genesis 38–40)
Joseph the Provider (written 10 August 1940 to 4. January 1943, Genesis 41–50)

 

 

THE BULL OF MINOS

Leonard Cottrell

"Furthermore, after he (Theseus) was arrived in Creta, he slew there the Minotaur(as the most part of ancient authors do write) by the means and help of Ariadne; who being fallen in fancy with him, did give him a clue of a thread, by the help whereof she taught him, how he might easily wind out the turnings and cranks of the Labyrinth.

Plutarch (North's translation).

Page 207

" In the year 30, on the ninth day of the third month of the inundation, the god entered his horizon"

Page 90

" Out in the dark blue sea there lies a land called Crete,

a rich and lovely land, washed by the waves on every

side,densely peopled and boasting ninety cities . . . .

One of the ninety towns is a great city called Knossos,

and there, for nine years, King Minos ruled and en-

joyed the friendship of almighty Zeus "

 

 

THE

ALPHABET

David Diringer 1947

Page 316

UIGHUR ALPHABET

" The Uighurs, originally Toquz Oghuz, the " Nine Oghuz," were a strong people of Turki speech. They lived in Mongolia and were Shamanists "

 

 

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

 

 

Added to all, minus none, shared by everything, multiplied in abundance.

 

 

THE MAGIC MOUNTAIN

Thomas Mann

1875 - 1955

"I tell them that if they will occupy themselves with the study of

mathematics

they will find in it the best remedy against the lusts of the flesh."

 

 

A TEST OF TIME

The Bible From Myth To History

David M Rohl 1995

Page 123

"the script of the ancient Egyptians was fundamentally a phonetic writing-'form which was based on an alphabet of just twenty-six main signs.!"

 

 

 THE SORCERORS HANDBOOK

Wade Baskin 1974

NUMEROLOGY

"A survival of the ancient magical theory of names. Names are infinite in their diversity but all may be reduced to a finite set of numbers, usually from 1 to 9, occasionally with the addition of 11 and 22. Leonard Bosman, in The Meaning and Philosophy of Numbers (1932), stated:

The power which the student may draw into himself when trying to realise the inner meaning of these great names and posers is sometimes so great as to cause a physical breakdown.

The simplest way to find the number corresponding to a name is to turn each letter into a number. Two systems are used. The Hebrew system, which also relies on knowledge of the Greek alphabet, does not use the figure 9 and writes the letters under the other numbers:"

"The modern system places the letters of the alphabet under the numbers 1-9."

"Using either system, the digital root is obtained by adding the number equivalents for each letter of a person's full name and reducing the sum to one digit."

 

 

THE

SORCERORS HANDBOOK

Wade Baskin 1974

Page 429

Old Religion

Term designating a cult believed by some students of witch- / Page 430 / craft, mainly those inspired by the writings of Margaret Murray and her followers, to incorporate the essential tenets of a prehistoric religion. These students hold that the idea of a Supreme Being antedates the Old Testament and was fairly widespread. Traces of the Old Religion appear in the Vedas of India, in ancient Persian manuscripts, and in the earliest esoteric writings of the Egyptians. Thousands of years before Paul wrote that "In Him we live, and move, and have our being," the ancient adepts or Magi had taught the essential truths of all the great religions of the world. They taught that the physical world and the mental world existed in the continuum of one great mind, the eternal reconciler of all opposites, the source of all things at all levels, the ultimate and absolute repository of wisdom and knowledge. Man with his limited intelligence could never comprehend the incomprehensible. But knowledge of God was accessible to man through his perception of truth and spiritual values; God revealed himself as perfection, love, light, and beyond that, Mystery. The ancient belief was summed up in the formula carved on ruined temples: "I am all that is, all that was, all that will be, and no one shall lift my veil."

"I am all that is, all that was, all that will be, and no one shall lift my veil."

Page 156

"The rite or initiation follows a general pattern to which / Page 157 / each coven may add its own features. A nine-foot circle is outlined with a sacred black-handled knife. An altar placed in the center of the circle contains a knife symbolizing the air, a cauldron symbolizing water and the Great Mother, a wand symbolizing the phallus and fire, and the pentatle symbolizing the earth. Other instruments may include a sword, a burin, a white-handled knife for use in making talismans, and a cord symbolizing the unifying spirit that links all the elements together. The altar has on it lighted candles, incense burners, a vase filled with salt and water, and a whip symbolizing purification. The circle is consecrated, using ritual instruments, salt, and water. An incantation, repeated over and over, asks the an- cient gods of the four cardinal points to appear. During this part of the ceremony the postulant stands outside the circle.. The leader of the coven touches his chest with the point of the blackhandled knife and warns him that it is better to die by the knife than to enter the coven with fear in his heart. The postulant replies with the password "Per-fect love and perfect faith," enters the circle, and has his feet and hands ceremoniously bound with the cord. The leader presents him to the gods of the east, the south, the west, and the north, brings him back to the altar, forces him to kneel, grasps his feet firmly, and asks: "Are you prepared to swear to remain faithful to the Art forever?" When the postulant states that he is ready, the leader tells him that he must first be purified, and applies first three, then seven, then nine, and finally 21 lashes."

"Perfect love and perfect faith"

"A nine-foot circle"

"nine"

9

"first three, then seven, then nine,"

three seven nine

3 7 9

973

 9

 

 

CAT WATCHING

Desmond Morris 1986

Page 9

" The cat-goddess was called Bastet, meaning She-of-Bast. Bast was the city where the main cat temple was situated, and where each spring as many as half a million people converged for the sacred festival. About 100,000 mummified cats were buried at each of these festivals to honour the feline virgin- goddess (who was presumably a forerunner of the Virgin Mary). These Bastet festivals were said to be the most popular and best attended in the whole of ancient Egypt, a success perhaps not unconnected with the fact that they included wild orgiastic celebrations and 'ritual frenzies'. Indeed, the cult of the cat was so popular that it lasted for nearly 2,000 years. It was officially banned in AD 390, but by then it was already in serious decline. In its heyday, however, it reflected the immense esteem in which the cat was held in that ancient civilization, and the many beautiful bronze statues of cats that have survived bear testimony to the Egyptians' appreciation of its graceful form.

Page 105

"Why does a cat have

nine

lives?

The cat's resilience and toughness led to the idea that it had more than one life, but the reason for endowing it with

nine

lives, rather than any other number, has often puzzled people. The answer is simple enough. In ancient times

nine

was considered a particularly lucky number because it was a

'trinity of trinities'

and therefore ideally suited for the 'lucky' cat."

 

 

THE

SORCERORS HANDBOOK

Wade Baskin 1974

Page 429

Old Religion

"first three, then seven, then nine"

three seven nine

 

379 - 973

 

 

HARMONIC 288

THE PULSE OF THE UNIVERSE

Bruce Cathie1977

Page 35

All the work I had done to-date / Page 36 / indicated to me that the mathematicians of old had a knowledge of the universe which we are only once again beginning to understand.

The final solution to this argument could be overcome only by the discovery of a geometric connection between the harmonics of light and the harmonics inherent in the division of a circle. As I had based my light values on minute of arc measure there must be some type of geometric arrangement which would tie them together."

"This was always in the back of my mind during the reading of many research books and finally I came across something which I believe will answer the critics. The friend who came to my rescue was none other than Pythagoras himself, a man of great stature and forceful personality who lived in the sixth century BC. He travelled extensively to enlarge his mathematical knowledge and was said to have gained much information from the priests of Zoroaster, who had in their possession the mathematical lore of the Mesopotamians. He founded a semi-religious, or mystical, cult based on mathematics, round about 540 BC in the town-ship of Crotona, in southern Italy. He taught his disciples to worship numbers, the main idea being that number is the essence of all things, and is the metaphysical principle of rational order in the universe."

 

 

II INVENTIONS 

"To be wise is - one thing - to see the Thought by which all things are guided through all things"

(HERACLITUS, 500 B.C.).

 

 

FINGERPRINTS OF THE GODS

Graham Hancock 1995

Galilei Galileo 1564-1642

Page 286

"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."

 

 

GALILEO

IN

90

MINUTES

John and Mary Gribbin 1997

Page 50

"But in his book The Assayer, published in 1623, Galileo also summed up his understanding of the scientific method. Sarcastically suggesting that his opponents seemed to think that 'phil-osophy is a book of fiction by some author, like The Iliad', he said that the book of the Universe: cannot be understood unless one first learns to comprehend the language and to under-stand the alphabet in which it is composed It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth."

"It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth."

Page 58 5 + 8 = 13

"on 13 th April 1633, when Galileo was in his 70th year, the infamous trial began."

 

 

IN SEARCH OF THE MIRACULOUS

Fragments of an Unknown Teaching

P.D.Oupensky 1947

Page 304

"You must understand ", he said, " that every real religion, that is, one that has been created by learned people for a definite aim, consists of two parts. One part teaches what is to be done. This part becomes common knowledge and in the course of time is distorted and departs from the original. The other part teaches how to do what the first part teaches. This part is preserved in secret in special schools and with its help it is always possible to rectify what has been distorted in the first part or restore what has been forgotten."

"Realizing the weakness and imperfection of ordinary language the people who have possessed objective knowledge have tried to express the idea of unity in "myths" "symbols" and in particular verbal formulas" which, having been transmitted without alteration, have carried on the idea from one school to another, often from one epoch to another."

Page 279

"The aim of "myths" and "symbols" was to reach mans' higher centres, to transmit to him ideas inaccessible to the intellect and to transmit them in such forms as would exclude the possibility of false interpretation."

 

 

LIGHT AND LIFE

Lars Olof Bjorn

Page 197

"By writing the 26 letters of the alphabet in a certain order one may put down almost any message"

 

 

THE DEATH OF FOREVER

A New Future for Human Consciousness

Darryl Reanney (1995 Edition)

Page 33

" The laws of physics have no inbuilt time asymmetery.They work just as well in the future-to-past sense as the past-to-future sense. We see this clearly when we look at the quantum wave .The wave is a ripple of possibility, not a real thing It has neither past nor future;it can be described as travelling forwards in time and backward in time with equal validity. This is true not just of the quantum wave. Subatomic particles exhibit the same disregard for time."

 

 

NUMBER

9

The Search for the Sigma Code

Cecil Balmond 1998

Page 5

"One…two…three….My eye went over the figures. Suddenly I saw something. There were hidden patterns; the old man's story about secret num-bers came back to me and I became curious. I started to look into these simple ideas and the more I searched the more fascinated I became. Something was indeed going on underneath the surface of arithmetic and what appeared as a unique calculation to the outside /

Page 6 / world was something quite different when viewed from below. Looked at another way, six and six was not necessarily twelve but something much more exciting - the number 3, of a secret code…"

Page 5 "…The thing to do is to follow the path until all the clues are in place and let your mind run free. It is only then that you find what the young master saw: the fixed points in the wind."

"…it is in this spirit I dedicate the journey to you. Follow the clues, build up the jigsaw piece by piece and make your own investigations; become part of the search.

Go back in time and let the free spirit in you enter. Talk to it, play ask the strangest questions.

Start to count again in the simplest of ways, one, two, three, four…up to nine.

 

 

FINGERPRINTS OF THE GODS

Graham Hancock

1995

Page 490

Library angels

The missing piece of the puzzle

"The novelist Arthur Koestler, who had a great interest in synchronicity, coined the term 'library angel' to describe the unknown agency responsible for the lucky breaks researchers sometimes get which lead /Page 491/to exactly the right information being placed in their hands at exactly the right moment"

 

 

FINGERPRINTS OF THE GODS

Graham Hancock 1995

Page 189 1 x 8 x 9 = 72 " The Sun and the Moon and the way of the Dead

Pyramid of the Sun, Teotihuacan,

  Having climbed more than 200 feet up a series of flights of stone stairs I reached the summit and looked towards the zenith. It was midday 19 May, and the sun was directly overhead, and the sun was directly overhead, as it would be again on 25 /

Page 190 / July. On these two dates, and not by accident, the west face of the pyramid was oriented precisely to the position of the setting sun. 6

  "A more curious but equally deliberate effect could be observed on the equinoxes. 20 March and 22 September. Then the passage of the sun's rays from south to north resulted at noon in the progressive obliteration of a perfectly straight shadow that ran along one of the lower stages of the western facade. The whole process, from complete shadow to complete illumination, took exactly 66.6 seconds. It had done so without fail, year - in year - out, ever since the pyramid had been built and would continue to do so until the giant edifice crumbled into dust. 7

What this meant of course, was that at least one of the many functions of the pyramid had been to serve as a 'perennial clock', precisely signalling the equinoxes and thus facilating calendar corrections as and when necessary for a people apparently obsessed, like the Maya, with the elapse and measuring of time. Another implication was that the master - builders of Teotihuacan must have possessed an enormouse body of astronomic and geodetic data and refferred to this data to set the Sun Pyramid at the precise orientation necessary to achieve the desired equinoctial effects."

 

 

CITY OF REVELATION

John Michell 1972

Page 36 " St Augustine in The City of God also writes of the perfection of number 6, for 'in this did God make perfect all his works. Wherefore this number is not to be despised, but has the esteem apparently con-firmed by many places of scripture. Nor was it said in vain of God's works: "Thou madest all things in number, weight and measure." ' It is the unique property of number 6, on account of which it was held perfect, that it is both the sum and the product of all its factors excluding itself, for 1 + 2 + 3 = 6 and 1 x 2 x 3 = 6.

6 is the number of the cosmos, and the Greek word " "…sig-nifying the cosmic order, has the value by gematria of 600. The ancient astronomers adopted the mile as the unit which measures the cosmic intervals in terms of the number 6, and procured the following sacred numbers:

Diameter of sun = 864,000 miles ( 12 x 12 x 6000 )

Diameter of moon = 2160 miles ( 6 x 6 x 60 )

Diameter of earth = 7920 miles ( 12 x 660 )

Mean circumference of earth = 24,883.2 miles (12 x 12 x 12 x 12 x 12)

Speed of earth round sun = 66,600 miles per hour

Distance between earth and moon = 6 x 60 x 660 miles or 60 x earth's radius"

Zed Aliz Zed casts an oblique look, a squinting of the other eye bringing a re-focus.

Diameter of sun = 864,000 miles 8 + 6 + 4 = 18 1 + 8 = 9

Diameter of moon = 2160 miles 2 + 1 + 6 = 9

Diameter of earth = 7920 miles 7 + 9 + 2 = 18 1 + 8 = 9

Mean circumference of earth = 24,883.2 miles 2 + 4 + 8 + 8 + 3 + 2 = 27 2 + 7 = 9

Speed of earth round sun = 66,600 miles per hour 6 + 6 + 6 = 18 1 + 8 = 9

Distance between earth and moon = 6 x 60 x 660 miles = 237600" 2 + 3 + 7 + 6 = 18 1 + 8 = 9

 

 

Fingerprints Of The Gods

Graham Hancock.1995

Page 190  "A more curious but equally deliberate effect could be observed on the equinoxes. 20 March and 22 September. Then the passage of the sun's rays from south to north resulted at noon in the progressive obliteration of a perfectly straight shadow that ran along one of the lower stages of the western facade. The whole process, from complete shadow to complete illumination, took exactly 66.6 seconds. It had done so without fail, year - in year - out, ever since the pyramid had been built and would continue to do so until the giant edifice crumbled into dust. 7

 

 

City Of Revelation

John Michell 1972

Page 36

"Speed of earth round sun = 66,600 miles per hour 6 + 6 + 6 = 18 1 + 8 = 9

Distance between earth and moon = 6 x 60 x 660 miles = 237600 " 2 + 3 + 7 + 6 = 18 1 + 8 = 9

 

 

Fingerprints Of The Gods Graham Hancock 1995

Page 190 "...The whole process, from complete shadow to complete illumination, took exactly 66.6 seconds."

 

 

HOLY BIBLE

Scofield References

Chapter 13

18

“Here is wisdom. Let him that hath understanding count the num-ber of the beast: for it is the num-ber of a man; and his number is Six hundred three score and six

 

 

CITY OF REVELATION

John Michell 1972

Page 137 Chapter Thirteen

"666

has been the subject of more comment and speculation than any other cabalistic number, principally on account of the last verse in revelation 13:

Here is wisdom. Let him that hath understanding count the number of the beast: for it is the num-ber of a man; and his number is

six hundred threescore and six.'

In the Greek text the number is spelt in letters,… "

"…or 600, 60, 6, . ."

 

 

HOLY BIBLE

Scofield References

Page 401

Kings Chapter 10 B.C. 992.

14 "Now the weight of gold that came to Solomon in one year was six hundred threescore and six talents"

 

 

The Lure and Romance of Alchemy

C. J. S. Thompson 1990

Page 26 "…There is further evidence given in the Bible of the richness of the country in the precious metal, for it is recorded that the Queen of Sheba brought much gold and precious stones and / Page 27 / gave to King Solomon 120 talents, a sum equivalent to £240,000. The navy of Hiram also brought gold from Ophir, and the weight of gold that came to Solomon in one year was 666 talents,… " "Page 26 Note È 1 Kings x, 10, 14."

 

 

FLYING TO 3000 B.C.

Pierre Jeannerat 1957

Page 124 "…Enters the Queen of Sheba. "And she gave the king an hun-dred and twenty talents of gold, and of spices great abundance, and precious stones. . . .Now the weight of gold that came to Solomon in one year was six hundred and three score and six talents of gold;…"

 

 

HOLY BIBLE

Scofield References

Page 380

Chapter 21 B.C. 1021

20 "And there was yet a battle in Gath, where was a man of great stature, that had on every hand six fingers, and on every foot six toes, four and twenty in number; and he also was born to the giant."

 

 

ALL SCRIPTURE IS INSPIRED Of GOD AND BENEFICIAL

Watch Tower Bible And Tract Society Of Pennsylvania

Page 11

" 24 In what order did the sixty-six Bible books come to us? What part of the endless stream of time do they cover? " 

" 29 In the following pages the sixty-six books of the Sacred Scriptures are examined in turn. " 

 

 

HOLY BIBLE

Scofield References

Page v

"...The Bible is a book of books. Sixty-six books make up the one Book. Considered with reference to the unity of the one book the separate books may be regarded as chapters. But that is but one side of the truth, for each of the sixty-six books is complete in itself, and has its own theme and analysis."

 

 

FINGERPRINTS OF THE GODS

Graham Hancock 1995

Chapter Nineteen

Page 153 1 + 5 + 3 = 9

"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."

 

 

CITY OF REVELATION
John Mitchell 1973

Page 78

"A remarkable use of the number 3168 occurs in Plato's account in Book V of Laws"

 

 

JUST SIX NUMBERS
Martin Rees 1999

Page 24

Chapter 2

" A proton is 1,836 times heavier than an electron, and the number 1,836 would have the same connotations to any 'intelligence'"

 

 

HOLY BIBLE

Scofield References

Page 809 B.C. 590

JEREMIAH

Chapter 33 Verse 3

"Call unto me, and I will answer thee, and shew thee great and mighty things, which thou knowest not"

3 3 + 3 = 36 3 + 6 = 9

3 times 33 equal 99 and 9 add 9 equals 18 and 1 add 8 equal 9

3 add 3 add 3 is 9

3 times 3 times 3 equal 27 and 2 + 7 equal 9

 

 

QUEST FOR A THEORY OF EVERYTHING

STEPHEN HAWKING

Kitty Ferguson 1991

Page 103 33rd line down

"The square root of 9 is 3. So we know that the third side"

 

 

INTO THE SPIRAL

Charles Ashton 1992

Page 120

"I've come to let you in through the door," the creature croaked.
"What door?" Ormand demanded in a shaky voice.

"Why, this door ," the voice grated. And they saw the plain wooden boards of a door where the tunnel wall had been.

"Where does it go?" Lissie whispered.

"Why, it says here on the door, child. Can't you read it?"

"There's nothing there," said Lissie, staring at the door.

"Look," the lantern bearer rasped, with a black, hollow grin. "It's written on the door. A - M - A-" the bony finger moved across the plain wood of the door as the dry mouth spelled out letters - " Z - E - M - E - N - T. What does that say?"

"I don't know," Lissie replied.

"It says Amazement!" the ancient mouth roared, as the door burst open in another shower of earth.

"IT SAYS AMAZEMENT!"

The empty doorway seemed to do a cartwheel towards them. "It says Amazement!" came a third time, muffled now and echoing and mixed with the slamming sound of wood on wood on the "maze" sound. Lissie and Ormand stood alone in a squared corridor of rock, beside a flickering torch fixed into a bracket on the wall."

Page123

CHAPTER
9

"INTO THE SPIRAL"

 

 

EXTENDED SIMILIES

Jenny Joseph 1997

Page 157

The thread
"There was the thread, the thread you see, and she followed it. Curdie, no that was a boy, Curdie and the thread, the good boy, he got her through. Or there was a fall of rock and it was buried, she had to scrabble with her hands and they never got them out those people trapped underneath when the earthquake collapsed the buildings. I can remember the man with his bare hands, they were bare, raw, that's it, skinned - but it must have been a picture of course.

But the thread was there, sometimes - he was losing it, losing his thought.

Yes, that was the way the thread went, it came and went, elusive as thought - now it flashed into focus, now he had it, him sitting reading to his little girl - but he can't have had that book as a child, he hadn't had that sort of childhood.

Thinking about the thread, the idea, myth of the thread was a good way to get you applying yourself, persisting, and he had, hadn't he, he'd gone on searching with his dog in the rubble long after the others had given up.

So that thinking, which he'd thought he'd come to as a solid thing like chipping away shale and muck to get at a bit, of core, a thing like a lump of coal, usable, source of energy, so that it didn't matter what you thought, it was a rope ladder to get you across somewhere, get you through the mess, something you pretended, no, not pretended - made up? - to be doing to give a reason for going on. Made up. Ah perhaps something you made, engineered, he'd like it when they called him Monsieur l'lngenieur, ingenious. Not for a reason - you don't need a reason for going on, you need a road, a way, ah yes a means. A way of going. That was tautology. You could just say 'a way'.

'Tell Alice' (you think I don't know she's dead, he heard his crafty thought within his head and in the same flash behaved as if he didn't), 'keep her fingers on the golden thread.' If it's all a fancy, if there isn't something that's true, then there isn't untrue and you were back where you were. He was getting there, getting down that path and this time he would get there, he could still breathe he could still tell them even though they couldn't move the rock off him.

If there isn't anything that's true, the opposite of true was false. But it couldn't be false because you can't have an opposite to some-thing that doesn't exist. Though what about negative numbers?

Page168

Alice was cleverer than he was he should have asked her. But she could never explain things like he could but after all he'd been a teacher. So if no true, no false and nothing true means everything false. Yes, he'd got it. 'Useful,' he said. They bent low pretending they could hear to encourage him to speak some more. Useful. It was all useful. Alice's knitting had been useful. The thread and the rope ladder and the bridge were useful. Useful was much more useful than true.

If he had realised that it was his son who was holding his hand he might have tried to speak in his type of hearty old reprobate he'd put on for years for young people and said something in character like 'Bugger the truth' because he knew they thought he thought truth was the pearl so he had it both ways. They would have been his next, last words but he kept his secret from them till the end because he had got beyond the division of time that living beings need in order to negotiate it, to a point where command question statement implying continuing into a future from the past were neither true, false or useful."

 

 

IN TUNE WITH THE INFINITE

OR

FULLNESS OF PEACE POWER AND PLENTY

Ralph Waldo Emerson 1906

Preface

"There is a golden thread that runs through every religion in the world. There is a golden thread that runs through the lives and the teachings of all the prophets, seers, sages, and saviours in the world's history, through the lives of all men and women of truly great and lasting power. All that they have ever done or attained to has been done. in full accordance with law. What one has done, all may do.

This same golden thread must enter into the lives of all who today, in this busy work-a-day world of ours, would exchange impotence for power, weakness and suffering for abounding health and strength, pain and unrest for perfect peace, poverty of whatever nature for fullness and plenty.

Each is building his own world. We both build from within and we attract from without. Thought is the force with which we build, for thoughts are forces. Like builds like and like attracts like. In the degree that thought is spiritualized does it become more subtle and powerful in its workings. This spiritualizing is in accordance with law and is within the power of all.

Everything is first worked out in the unseen before it is manifested in the seen, in the ideal before it is realized in the real, in the spiritual before it shows forth in the materiaL The realm of the unseen is the realm of cause. The realm of the. seen is the realm of effect. The nature of effect is always determined and conditioned by the nature of its cause.

To point out the great facts in connection with, and the great laws underlying the workings of the interior, spiritual, thought forces, to point them out so simply and so clearly that even a child can understand, is the author's aim. To point them out so simply and so clearly that all can grasp them, that an can take them and infuse them into every-day life, so as to mould it in all its details in accordance with what they would have it, is his purpose. That life can be thus moulded by them is not a matter of mere speculation or theory with him, but a matter of positive knowledge.

There is a divine sequence running throughout the universe. Within and above and below the human will inces-santly works the Divine will. To come into harmony with it and thereby with all the higher laws and forces, to come then into league and to work in conjunction with them, in order that they can work in league and in conjunction with us. is to come into the chain of this wonderful sequence. This is the secret of all success. This is to come. into the possession of unknown riches, into the realization of undreamed - of powers."

Brahma

If the red slayer think he slays,

Or if the slain think he is slain

They know not well the subtle ways

I keep and pass and turn again.

Ralph Waldo Emerson

 

 

........

 

 

-
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
6
-
6
-
-
+
=
17
1+7
=
8
=
8
=
8
-
-
14
-
-
-
-
15
-
15
-
-
+
=
44
4+4
=
8
=
8
=
8
-
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
3
4
5
9
-
3
-
7
7
+
=
38
3+8
=
11
1+1
2
=
2
-
-
-
21
13
5
18
-
12
-
7
7
+
=
101
1+0+1
=
2
=
2
=
2
-
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
14
21
13
5
18
15
12
15
7
7
+
=
145
1+4+5
=
10
1+0
1
=
1
-
-
5
3
4
5
9
6
3
6
7
7
+
=
55
5+5
=
10
1+0
1
=
1
-
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
11
10
N
U
M
E
R
O
L
O
G
Y
-
-
34
-
-
10
-
55
-
28
1+1
1+0
-
-
-
-
9
-
-
-
-
-
-
-
3+4
-
-
1+0
-
5+5
-
2+8
2
1
N
U
M
E
R
O
L
O
G
Y
-
-
7
-
-
1
-
10
-
10
-
-
5
3
4
5
9
6
3
6
7
7
-
-
-
-
-
-
-
1+0
-
1+0
2
1
N
U
M
E
R
O
L
O
G
Y
-
-
7
-
-
1
-
1
-
1

 

 

10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
6
-
6
-
-
+
=
17
1+7
=
8
=
8
=
8
-
14
-
-
-
-
15
-
15
-
-
+
=
44
4+4
=
8
=
8
=
8
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
3
4
5
9
-
3
-
7
7
+
=
38
3+8
=
11
1+1
2
=
2
-
-
21
13
5
18
-
12
-
7
7
+
=
101
1+0+1
=
2
=
2
=
2
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
14
21
13
5
18
15
12
15
7
7
+
=
145
1+4+5
=
10
1+0
1
=
1
-
5
3
4
5
9
6
3
6
7
7
+
=
55
5+5
=
10
1+0
1
=
1
10
N
U
M
E
R
O
L
O
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
5
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
6
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
10
N
U
M
E
R
O
L
O
G
Y
-
-
34
-
-
10
-
55
-
28
1+0
-
-
-
-
9
-
-
-
-
-
-
-
3+4
-
-
1+0
-
5+5
-
2+8
1
N
U
M
E
R
O
L
O
G
Y
-
-
7
-
-
1
-
10
-
10
-
5
3
4
5
9
6
3
6
7
7
-
-
-
-
-
-
-
1+0
-
1+0
1
N
U
M
E
R
O
L
O
G
Y
-
-
7
-
-
1
-
1
-
1

 

 

-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
NUMEROLOGY
-
-
-
-
-
-
-
-
-
-
-
-
-
N
=
5
-
1
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
U
=
6
-
2
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
M
=
9
-
3
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
1
-
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
2
-
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
-
6
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
L
=
5
-
7
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
O
=
6
-
8
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
5
-
9
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
Y
=
5
-
10
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
55
4
4
10
NUMEROLOGY
145
55
55
4
1
2
6
4
10
12
14
8
9
-
-
5+5
-
-
1+0
-
1+4+5
5+5
5+5
-
-
-
-
-
1+0
1+2
1+4
-
-
Q
-
10
-
-
1
NUMEROLOGY
10
10
10
-
1
2
6
4
1
3
5
8
9
-
-
1+0
-
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
NUMEROLOGY
1
1
1
-
1
2
6
4
1
3
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
-
3
4
5
6
7
9
-
-
-
-
-
-
NUMEROLOGY
-
-
-
-
-
-
-
-
-
-
N
=
5
-
1
1
N
14
5
5
-
-
-
5
-
-
-
U
=
6
-
2
1
U
21
3
3
-
3
-
-
-
-
-
M
=
9
-
3
1
M
13
4
4
-
-
4
-
-
-
-
E
=
1
-
4
1
E
5
5
5
-
-
-
5
-
-
-
R
=
2
-
5
1
R
18
9
9
-
-
-
-
-
-
9
O
=
6
-
6
1
O
15
6
6
-
-
-
-
6
-
-
L
=
5
-
7
1
L
12
3
3
-
3
-
-
-
-
-
O
=
6
-
8
1
O
15
6
6
-
-
-
-
6
-
-
G
=
5
-
9
1
G
7
7
7
-
-
-
-
-
7
-
Y
=
5
-
10
1
Y
25
7
7
-
-
-
-
-
7
-
-
-
55
4
4
10
NUMEROLOGY
145
55
55
4
6
4
10
12
14
9
-
-
5+5
-
-
1+0
-
1+4+5
5+5
5+5
-
-
-
1+0
1+2
1+4
-
Q
-
10
-
-
1
NUMEROLOGY
10
10
10
-
6
4
1
3
5
9
-
-
1+0
-
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
1
-
-
1
NUMEROLOGY
1
1
1
-
6
4
1
3
5
9

 

 

-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
NUMEROLOGY
-
-
-
-
-
-
-
-
-
-
-
-
-
U
=
6
-
2
1
U
21
3
3
-
-
-
3
-
-
-
-
8
-
L
=
5
-
7
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
M
=
9
-
3
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
N
=
5
-
1
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
E
=
1
-
4
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
-
6
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
O
=
6
-
8
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
G
=
5
-
9
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
Y
=
5
-
10
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
-
R
=
2
-
5
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
55
4
4
10
NUMEROLOGY
145
55
55
4
1
2
6
4
10
12
14
8
9
-
-
5+5
-
-
1+0
-
1+4+5
5+5
5+5
-
-
-
-
-
1+0
1+2
1+4
-
-
Q
-
10
-
-
1
NUMEROLOGY
10
10
10
-
1
2
6
4
1
3
5
8
9
-
-
1+0
-
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
NUMEROLOGY
1
1
1
-
1
2
6
4
1
3
5
8
9

 

 

-
-
-
-
-
-
-
-
-
-
-
3
4
5
6
7
9
-
-
-
-
-
-
NUMEROLOGY
-
-
-
-
-
-
-
-
-
-
U
=
6
-
2
1
U
21
3
3
-
3
-
-
-
-
-
L
=
5
-
7
1
L
12
3
3
-
3
-
-
-
-
-
M
=
9
-
3
1
M
13
4
4
-
-
4
-
-
-
-
N
=
5
-
1
1
N
14
5
5
-
-
-
5
-
-
-
E
=
1
-
4
1
E
5
5
5
-
-
-
5
-
-
-
O
=
6
-
6
1
O
15
6
6
-
-
-
-
6
-
-
O
=
6
-
8
1
O
15
6
6
-
-
-
-
6
-
-
G
=
5
-
9
1
G
7
7
7
-
-
-
-
-
7
-
Y
=
5
-
10
1
Y
25
7
7
-
-
-
-
-
7
-
R
=
2
-
5
1
R
18
9
9
-
-
-
-
-
-
9
-
-
55
4
4
10
NUMEROLOGY
145
55
55
4
6
4
10
12
14
9
-
-
5+5
-
-
1+0
-
1+4+5
5+5
5+5
-
-
-
1+0
1+2
1+4
-
Q
-
10
-
-
1
NUMEROLOGY
10
10
10
-
6
4
1
3
5
9
-
-
1+0
-
-
-
-
1+0
1+0
1+0
-
-
-
-
-
-
-
-
-
1
-
-
1
NUMEROLOGY
1
1
1
-
6
4
1
3
5
9

 

 

THE

MAGIKALALPHABET

ROOT

VALUE OF THE WORDS

I = 9 9 = I

ME = 9 9 = ME

EGO = 9 9= EGO

CONSCIENCE = 9 9 = CONSCIENCE

DIVINE = 9 9 = DIVINE

THOUGHT = 9 9 = THOUGHT

OUR = 9 9 = OUR

LOVE = 9 9 = LOVE

REAL = 9 9 = REAL

REALITY = 9 9 = REALITY

SUN = 9 9 = SUN

EARTH = 7 7 = EARTH

MOON = 3 3 = MOON

JUPITER = 9 9 = JUPITER

MAGNETIC = 9 9 = MAGNETIC

FIELD = 9 9 = FIELD

PHYSICS = 9 9 = PHYSICS

ORIONIS = 9 9 = ORIONIS

NAMES OF GOD = 9 9 = NAMES OF GOD

ASCENSION = 99 AND 9 + 9 = 18 AND 1 + 8 = 9 AND 9 = ASCENSION

 

 

HOLY BIBLE

Scofield References

Jeremiah B.C. 590

Page 809 8 x 9 = 72 7 + 2 = 9

Chapter 33 Verse 3 x 33 = 99

CALL UNTO ME AND I WILL ANSWER THEE AND SHEW THEE GREAT AND MIGHTY THINGS WHICH THOU KNOWEST NOT

 

 

I

I = 9 9 = I

ME = 9 9 = ME

BRAIN + BODY = 9 9 = BODY + BRAIN

LIGHT + DARK = 9 9 = DARK + LIGHT

ENERGY + MASS = 9 9 = MASS + ENERGY

MIND + MATTER = 9 9 = MATTER + MIND

MAGNETIC + FIELD = 9 9 = FIELD + MAGNETIC

POSITIVE + NEGATIVE = 9 9 = NEGATIVE + POSITIVE

MYTHS MATHS MATHS MYTHS

 

 

ANUBIS

A

NUMBER

IS

MIN DOTH DREAM WHAT DOTH MIN MEAN

 

 

THE

99 NAMES OF GOD GOD OF NAMES 99

 

 

I

MASS ENERGY

ME

LIGHT + DARK 9 9 DARK + LIGHT

POSITIVE + NEGATIVE 9 9 NEGATIVE + POSITIVE

 

 

1 Wormwood in the Bible; 2 Interpretations of Revelation 8:11 ... A number of Bible scholars consider the term Wormwood to be a purely symbolic ... en.wikipedia.org/wiki/Wormwood_(star)

 

Wormwood (star) From Wikipedia, the free encyclopedia Jump to: navigation, search

Wormwood, αψινθιον (apsinthion) in Greek, is a star, or angel,[1] that appears in the Biblical New Testament Book of Revelation.

 

[edit] Wormwood in the Bible although the word Wormwood appears several times in the Old Testament, translated from the Hebrew term לענה (la'anah), e.g., Deuteronomy 29:18 and Jeremiah 9:15, its only clear reference as a named entity occurs in the New Testament book of Revelation: "And the third angel sounded, and there fell a great star from heaven, burning as it were a lamp, and it fell upon the third part of the rivers, and upon the fountains of waters; And the name of the star is called Wormwood: and the third part of the waters became wormwood; and many men died of the waters, because they were made bitter." (Revelation 8:10, 11 - KJB).

Certain commentators have held that this "great star" represents one of several important figures in political or ecclesiastical history,[2] while other Bible dictionaries and commentaries view the term as a reference to a celestial being.

A Dictionary of The Holy Bible states, "the star called Wormwood seems to denote a mighty prince, or power of the air, the instrument, in its fall, of sore judgments on large numbers of the wicked."[3] Scofield Reference Notes draws a link between the term in Revelation and Isaiah 14:12,[4] which reads, "How you have fallen from heaven,O Lucifer , son of the morning! You have been cast down to the earth, you who once laid low the nations!" (King James Bible) KJB

 

HOW YOU HAVE FALLEN FROM EVEN O LUCIFER BRIGHT SON OF THE MORNING

 

 

-
WORMWOOD
-
-
-
4
WORM
69
24
6
4
WOOD
57
21
3
8
WORMWOOD
126
45
9
-
-
1+2+6
4+5
-
8
WORMWOOD
9
9
9

 

 

-
-
-
-
-
-
-
-
-
W
=
5
-
-
WORMWOOD
-
-
-
-
-
-
-
2
W+M
36
9
9
-
-
-
-
3
O+O+O
45
18
9
-
-
-
-
1
R
18
9
9
-
-
-
-
2
W+D
27
9
9
W
=
5
-
8
WORMWOOD
126
45
27
-
-
-
-
-
-
1+2+6
4+5
2+7
W
=
5
-
8
WORMWOOD
9
9
9

 

 

HOLY BIBLE

Scofield References

REVELATION

C 8 V 11

Page 1338

"AND THE NAME OF THE STAR IS CALLED WORMWOOD. . . "

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
WORMWOOD
-
-
-
-
-
-
-
-
-
-
-
-
-
W
=
5
1
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
3
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
W
=
5
5
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
6
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
45
-
8
WORMWOOD
126
45
45
-
1
2
3
8
10
18
7
8
9
-
-
4+5
-
-
-
1+2+6
4+5
4+5
-
-
-
-
-
1+0
1+8
-
-
-
-
-
9
-
8
WORMWOOD
9
9
9
-
1
2
3
8
1
9
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
WORMWOOD
-
-
-
-
-
-
-
-
-
-
-
-
-
W
=
5
1
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
3
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
W
=
5
5
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
6
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
45
-
8
WORMWOOD
126
45
45
-
1
2
3
8
10
18
7
8
9
-
-
4+5
-
-
-
1+2+6
4+5
4+5
-
-
-
-
-
1+0
1+8
-
-
-
-
-
9
-
8
WORMWOOD
9
9
9
-
1
2
3
8
1
9
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
WORMWOOD
-
-
-
-
-
-
-
-
-
-
-
-
-
M
=
4
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
D
=
4
8
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
W
=
5
1
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
W
=
5
5
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
6
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
3
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
45
-
8
WORMWOOD
126
45
45
-
1
2
3
8
10
18
7
8
9
-
-
4+5
-
-
-
1+2+6
4+5
4+5
-
-
-
-
-
1+0
1+8
-
-
-
-
-
9
-
8
WORMWOOD
9
9
9
-
1
2
3
8
1
9
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
4
5
6
9
-
-
-
-
-
WORMWOOD
-
-
-
-
-
-
-
-
M
=
4
4
1
M
13
4
4
-
4
-
-
-
D
=
4
8
1
D
4
4
4
-
4
-
-
-
W
=
5
1
1
W
23
5
5
-
-
5
-
-
W
=
5
5
1
W
23
5
5
-
-
5
-
-
O
=
6
2
1
O
15
6
6
-
-
-
6
-
O
=
6
6
1
O
15
6
6
-
-
-
6
-
O
=
6
7
1
O
15
6
6
-
-
-
6
-
R
=
9
3
1
R
18
9
9
-
-
-
-
9
-
-
45
-
8
WORMWOOD
126
45
45
-
8
10
18
9
-
-
4+5
-
-
-
1+2+6
4+5
4+5
-
-
1+0
1+8
-
-
-
9
-
8
WORMWOOD
9
9
9
-
8
1
9
9

 

 

"THE WORD FIRST USED FOR MAN IS LULLU"

"THE WORD FIRST USED FOR MAN IS 33333"

"THE WORD FIRST USED FOR MAN IS LULLU"

 

 

ENUMA ELISH - Babylonian Creation Myth - The continued story www.stenudd.com/myth/enumaelish/enumaelish-

The word used for man is lullu, meaning a first, primitive man. The same word is used about the savage Enkidu in the Gilgamesh epic. Since Qingu is found ...

I hereby name it Babylon, home of the great gods.

The word used in the text is written phonetically, ba-ab-i-li, contrary to tradition, maybe to allow for the etymological explanation of the name as the ‘gate of the gods’.
Then he decides to create man, to serve the gods with offerings, so that they can be at leisure. The word used for man is lullu , meaning a first, primitive man. The same word is used about the savage Enkidu in the Gilgamesh epic. Since Qingu is found guilty of the war between the gods, his blood is used to create mankind. Here, it is unclear if Marduk or Ea creates mankind. Later in the text, Ea is specified as the creator of man. Finally, the gods praise Marduk, and give him fifty names that represent different aspects of his powers and sovereignty.
The text ends with instructions on how it should be passed on from generation to generation, and the command to worship Marduk, king of the gods.

 

ENUMA ELISH
The Babylonian Creation Myth

"The word used for man is lullu"

LULLU 33333 LULLU

"The word used for man is lullu"

 

-
-
-
-
-
LULLU
-
-
-
L
3
L
-
1
L
12
3
3
U
3
U
-
1
U
21
3
3
L
3
L
-
1
L
12
3
3
L
3
L
-
1
L
12
3
3
U
3
U
-
1
U
21
3
3
-
15
-
-
6
LULLU
78
15
15
-
1+5
-
-
-
-
7+8
1+5
1+5
-
6
-
-
6
LULLU
15
6
6
-
-
-
-
-
-
1+5
-
-
-
6
-
-
6
LULLU
6
6
6

 

 

 

 

-
ROTATORS
-
-
-
1
R
18
9
9
3
O+T+A
36
9
9
4
T+O+R+S
72
18
9
8
ROTATORS
126
36
27
-
-
1+2+6
3+6
2+7
8
ROTATORS
9
9
9

 

 

 

..

 

 

 
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