Previous Page
   Next Page
 
Evokation
  
 
Index
  

 

A

MAZE

IN

ZAZAZA ENTER AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

WORK DAYS OF GOD

Herbert W Morris D.D.circa 1883

Page 22

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

 

 

LIGHT AND LIFE

Lars Olof Bjorn 1976

Page 197

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

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

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

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

 

 

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

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

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

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

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

 

THIS IS THE SCENE OF THE SCENE UNSEEN

THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE

 

 

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

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

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

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

Searching out patterns is a pure delight.

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

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

 

 

YOU ARE GOING ON A JOURNEY A VERY SPECIAL JOURNEY DO HAVE A PLEASANT JOURNEY DO

 

 

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

 

 

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

 

 

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

 

 

 

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

THE

BOOK OF CREATION

Page 250

"THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

THE ACCOUNT IS UNASHAMEDLY SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS

OF LANGUAGE AS THOUGH HE WERE WRITING A BOOK BUT LANGUAGE HAS BEEN ENTIRELY

TRANSFORMED AND THE MESSAGE OF CREATION IS NO LONGER CLEAR EACH LETTER OF THE

HEBREW ALPHABET IS GIVEN A NUMERICAL VALUE BY COMBINING THE LETTERS WITH THE

SACRED NUMBERS REARRANGING THEM IN ENDLESS CONFIGURATIONS THE MYSTIC WEANED

THE MIND AWAY ROM THE NORMAL CONNOTATIONS OF WORDS"

 

 

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

 

 

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

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

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

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

 

 

 

 

 

 

 

 

 

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

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN SOME AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

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

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

Searching out patterns is a pure delight.

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

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

 

 

CITY OF REVELATION

John Michell 1972

Page 77

CHAPTER SEVEN

3168, The Perimeter of the Temple

"If the numbers of the sacred principles, mentioned by St John in connection with the New Jerusalem, are obtained from the Greek text by the cabalistic method of gematria, it is found that they correspond to the dimensions of the city, set out in Fig 16. (Figure omitted) For example, the perimeter of a hexagon contained within the circle representing the earth, 7920 feet in diameter, measures 2376 feet, and 2376 is the number of (Greek text omitted), the twelve apostles of the Lamb (Revelation 21.14). 2376 x 2 feet is equal to 1746 MY, and 1745 = (Greek text omitted), the twelve apostles. The names of the apostles are said to be in the twelve foundations of the wall of the city. The wall is the circle of diameter 7920 feet and 14,400 cubits in circumference, and the foundations are the twelve corners of the double hexagon inscribed within it, fonowing the customary pattern of an astrological chart. The position of the twelve apostles in the scheme is thus clearly defined.
Of all the canonical numbers the most notable is 3168. The New Jerusalem measures 48,000 furlongs or 31,680,000 feet round the perimeter of its four sides; the mean perimeter of the Stonehenge sarsen circle is 316.8 feet; the perimeter of the square 12 hides of Glastonbury is 31,680 feet; the significance of 31,680 in the canon of cosmology is illustrated in Fig.11, and we shall also find this number set round the border of Plato's mystical city, described in Laws.
Obviously the number 3168 had an important symbolic meaning, the Christian interpretation of which is provided in New Testament
gematria. The most sacred name of Christianity is (Greek text omitted);
(Greek text omitted), Lord Jesus Christ, and the number of these three words together is 3168. (Greek text omitted) is an astrological term meaning the ruler or dominant influence.
Another sacred phrase from the New Testament, (Greek text omitted) the Power of Christ (2 Corinthians 12.9) has the value 3168 if the alternative spelling of Christos, (Greek text omitted) is adopted.

Page 78

The perimeter of the temple is 3168, Lord Jesus Christ, when the temple is measured by the foot, the most sacred unit of ancient metrology. In terms of the megalithic yard (2.72 feet), however, the perimeter measures 1164, because 3168 feet = 1164 MY. Yet this makes no difference to the symbolic interpretation by gematria, for 1164 is the number of another name of Christ, (Greek text omitted) Son of God.

As a geodetic or earth-measuring number, 3168 also demonstrates the antiquity and sacred origin of British metrology, for
31,680 inches = half a mile

31,680 ft. = 6 miles.

31,680 furlongs = 3960 miles = radius of the earth.

31,680 miles = perimeter of square containing the terrestrial sphere.

31,680 miles = circumference of circle drawn on the combined diameters of the earth and moon (10,080 miles)

Other cosmological correspondences of 3168 are given on page 109.

The Stonehenge sarsen circle with circumference of 316.8 feet
contains an area of 888 square yards, 888 being the number of Jesus, which is equal to 1080 square MY. The circle contained within a square of perimeter 316.8 feet, corresponding to the bluestone circle at Stonehenge, has an area of 666 square MY. Thus the two stone circles at Stonehenge have areas of 1080 and 666 square MY, these two numbers representing the opposite poles of lunar and solar or negative and positive energy.
The number 144 or 122 is characteristic of the New Jerusalem scheme, and 3168 demonstrates the value of (pi symbol 22/7 omitted) in terms of this number, for 144 x 7 = 1008 and 144 x 22 = 3168.

3168 in Plato's city
A remarkable use of the number 3168 occurs in Plato's account in Book V of.Laws of the mystical dimensions of the perfect city. Throughout his work Plato makes guarded reference to a secret canon of numbers that applies universally to every aspect of human life and activity, including government, astronomy, acoustics, kinetics, plane and solid geometry and divination. Linear measurements, areas and volumes are obviously incommensurable, but Plato declares that there are certain numbers that link these with each other and with all phenomena capable of being measured. As an example of these numbers, the study of which Plato recommends as the most sanctifying of all pursuits, he gives 5040. This is the ideal number of citizens in the state and serves other purposes in con­/ Page 79 / nection with the framing of laws and standards. The reason why it is most suitable for all matters of division is that for its size it has the greatest number of divisors, 60 in all, including the entire decad, the numbers 1 - 10. Another property of the number 5040 is that it is the radius of a circle with circumference 31,680. Further examina­tion of the numerical foundations of Plato's state shows that the scheme to which he refers is the ancient plan of the cosmic temple.
The lawgivers in Plato's state are reminded that the perfect human society would be one in which all possessions, wives, children, land and chattels were held in common, where all the citizens were of one mind and acted together so harmoniously that it were as if eyes, ears and hands were also common property. To keep this ideal alive is the function of the prophet. Human nature and conditioning, however, demand a more practical alternative, 'very near to the first in immortality and second to it in merit'. This is provided in Laws V.
Plato's state is arranged in a manner that can scarcely be under­stood literally, and is obviously intended, like the New Jerusalem, as a geometer's allegory. The land is all divided into twelve parts, each dedicated to one of the twelve gods and populated by one of the twelve tribes of the 5040 households. The city is similarly divided, forming a microcosm of the state as a whole. In the centre of the city is the acropolis and 'from this centre he must divide up the city itself and the whole country into twelve parts. The twelve parts must be equalised by making those of good land small and those of inferior land greater. He must mark off 5040 allotments, and each of these he must cut in two and join two pieces to form the allotments, so that each contains a near piece and a distant piece - joining the piece next to the city with the piece furthest off, the second nearest with the second furthest, and so on with the rest.'
The only way in which this division can be represented is by a circle of radius 5040, a hundred times larger than that of Stonehenge measured in feet; the perimeter of this circle is 31 ,680. In Fig. 24 (Figure 24 omitted) the radius of the circle should be divided equally into 5040 parts to produce 5040 concentric circles. These are bisected into 10,080 semi­circles by the diameter and positioned out in Plato's manner into 5040 double allotments, each of equal area.
In this scheme 31,680 is not only the circumferenceof the circular state, but also the area of each of its 2520 pairs of rings, proving Plato's assertion that linear and area pleasurements can be made / Page 80 / (Figure 24 omitted) commensurable by number. The entire circle is divided into two halves, each containing 39,916,800 square units of land. These numbers, which are inherent in the New Jerusalem scheme, have the following significance:
31,680 is divisible by all the numbers1-12 with the exception of 7

5040 = 1 x 2 x 3 x 4 x 5 x 6 x 7

39,916,800 = 1 x 2 x 3 x 4 x 5x 6 x 7 x 8 x 9 x.10 x 11

5040, the radius of the circular city, is the product of the numbers1 - 7; 7920, the side of the square city, is the product of numbers 8 - 11. In each case the perimeter of the city is 31,680. In Plato's Republic is the famous, cryptic reference to the 'marriage number', which should be consulted by the guardians of the state in all matters relating to the seasonal union of male and female. There appear to be two numbers involved, adding up to a third, but the riddle is so obscure that no firm solution has been reached despite the vast literature on the subject. For various reasons the number 12,960,000 or 36002 is most commonly proposed, and this would seem appropriate, for 12,960 = 5040 + 7920. 12,960 therefore represents the union of square and circle, symbol of the sacred marriage, and the gematria is also appropriate, for 1296 = (Greek text omitted) Mary mother of Jesus.

FIGURE 24 (Figure omitted) Plato's city divided into 5040 rings, Perimeter = 31,680, Areas: A + a = B + b = C + c = 31,680.

 

 

INCLUDE EUCLID INCLUDE

 

 

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

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

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

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

 

....

 

THE DEATH OF GODS IN ANCIENT EGYPT

Jane B. Sellars 1992

Page 204

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

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

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

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

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

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

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

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

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

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

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

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

AN IMPORTANT POSTSCRIPT

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

Page 206

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

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

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

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

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

Page 207

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

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

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

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

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

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

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

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

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

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

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

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

 

 

 

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

 

 

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
-
-
-
34
-
36
Add
458
197
44
4+5
-
-
-
3+4
-
3+6
Reduce
4+5+8
1+9+7
4+4
9
-
-
-
7
4
9
Deduce
17
17
8
-
-
-
-
-
-
-
Produce
1+7
1+7
-
9
-
-
-
7
-
9
Essence
8
8
8

 

 

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

 

 

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

 

RAINBOW

RA

IN

BOW

 

7
WEATHER
-
-
-
-
W
23
5
5
-
EARTH
52
25
7
-
E
9
9
9
7
WEATHER
84
39
21
-
-
8+4
3+9
1+2
7
WEATHER
12
12
3
-
-
1+2
1+2
-
7
WEATHER
3
3
3

 

 

3
3
-
A+B+C
6
6
6
2
-
2
D+E
9
9
9
3
3
-
F+G+H
21
21
3
1
-
1
I
9
9
9
3
3
-
J+K+L
33
6
6
2
-
2
M+N
27
9
9
2
2
O+P
31
13
4
3
-
3
QRS
54
18
9
3
-
3
TUV
63
9
9
3
-
3
WXY
72
18
9
1
1
-
Z
26
8
8
26
12
14
First Total
351
126
81
2+6
1+2
1+4
Add to Reduce
3+5+1
1+2+6
8+1
8
3
5
Reduce to Deduce
9
9
9

 

 

3
A+B+C
6
6
6
-
D+E
-
-
-
3
F+G+H
21
21
3
-
I
-
-
-
3
J+K+L
33
6
6
-
M+N
-
-
-
2
O+P
31
13
4
-
QRS
-
-
-
-
TUV
-
-
-
-
WXY
-
-
-
1
Z
26
8
8
12
First Total
117
54
27
2+6
Add to Reduce
1+1+7
5+4
2+7
8
Reduce to Deduce
9
9
9

 

 

-
A+B+C
-
-
-
2
D+E
9
9
9
-
F+G+H
-
-
-
1
I
9
9
9
-
J+K+L
-
-
-
2
M+N
27
9
9
-
O+P
-
-
-
3
QRS
54
18
9
3
TUV
63
9
9
3
WXY
72
18
9
-
Z
-
-
-
14
First Total
234
72
54
1+4
Add to Reduce
2+3+4
7+2
5+4
5
Reduce to Deduce
9
9
9

 

 

MAGNETIC, THOUGHT, GODS, DIVINE, REALITY
GEOMETRY
GEOMETRICAL
GEO
GEOCENTRIC
GEO, GEOMETRY, GEOMETRICAL, GEOCENTRIC
OMPHALOS
ORACLE
ACE
DICE
IMAN
CABBALA
KABBALA
GREAT GREEN, GREAT, GREEN
TURN, TURNS, TURNED, TURNING
ASEKHA, ASEK, HA
RACE
JEWISH, NEW, YEAR
NEW, OLD
I, AM, NEUTRAL
VRIL, VIRILITY
VRIL, VIRILE
POLYHEDRA
PYTHAGORUS
OURABORUS
THE MAGIC OF PERSPECTIVE

 

SACRED GEOMETRY

 

acred geometry - Wikipediahttps://en.wikipedia.org › wiki › Sacred_geometry
sacred geometry from en.wikipedia.org
Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions. ... It is associated with the belief that a ...
?Shield of the Trinity · ?Temenos · ?108 (number) · ?Sacral architecture

Sacred geometry ascribes symbolic and sacred meanings to certain geometric shapes and certain geometric proportions.[1] It is associated with the belief that a god is the creator of the universal geometer. The geometry used in the design and construction of religious structures such as churches, temples, mosques, religious monuments, altars, and tabernacles has sometimes been considered sacred. The concept applies also to sacred spaces such as temenoi, sacred groves, village greens, pagodas and holy wells, Mandala Gardens and the creation of religious and spiritual art.
As worldview and cosmology

Further information: Mathematics and art
The belief that a god created the universe according to a geometric plan has ancient origins. Plutarch attributed the belief to Plato, writing that "Plato said god geometrizes continually" (Convivialium disputationum, liber 8,2). In modern times, the mathematician Carl Friedrich Gauss adapted this quote, saying "God arithmetizes".[2]

Johannes Kepler (1571–1630) believed in the geometric underpinnings of the cosmos.[3] Harvard mathematician Shing-Tung Yau expressed a belief in the centrality of geometry in 2010: "Lest one conclude that geometry is little more than a well-calibrated ruler – and this is no knock against the ruler, which happens to be a technology I admire – geometry is one of the main avenues available to us for probing the universe. Physics and cosmology have been, almost by definition, absolutely crucial for making sense of the universe. Geometry's role in this may be less obvious, but is equally vital. I would go so far as to say that geometry not only deserves a place at the table alongside physics and cosmology, but in many ways it is the table."[4]

Natural forms
Further information: Patterns in nature

Nautilus shell's logarithmic growth spiral
According to Stephen Skinner, the study of sacred geometry has its roots in the study of nature, and the mathematical principles at work therein.[5] Many forms observed in nature can be related to geometry; for example, the chambered nautilus grows at a constant rate and so its shell forms a logarithmic spiral to accommodate that growth without changing shape. Also, honeybees construct hexagonal cells to hold their honey. These and other correspondences are sometimes interpreted in terms of sacred geometry and considered to be further proof of the natural significance of geometric forms.

Art and architecture
Further information: Mathematics and architecture, Mathematics and art, and Islamic geometric patterns
Geometric ratios, and geometric figures were often employed in the designs of ancient Egyptian, ancient Indian, Greek and Roman architecture. Medieval European cathedrals also incorporated symbolic geometry. Indian and Himalayan spiritual communities often constructed temples and fortifications on design plans of mandala and yantra. Mandala Vaatikas or Sacred Gardens were designed using the same principles.

Many of the sacred geometry principles of the human body and of ancient architecture were compiled into the Vitruvian Man drawing by Leonardo da Vinci. The latter drawing was itself based on the much older writings of the Roman architect Vitruvius.

In Islam
The geometric designs in Islamic art are often built on combinations of repeated squares and circles, which may be overlapped and interlaced, as can arabesques (with which they are often combined), to form intricate and complex patterns, including a wide variety of tessellations. These may constitute the entire decoration, may form a framework for floral or calligraphic embellishments, or may retreat into the background around other motifs. The complexity and variety of patterns used evolved from simple stars and lozenges in the ninth century, through a variety of 6- to 13-point patterns by the 13th century, and finally to include also 14- and 16-point stars in the sixteenth century.

Geometric patterns occur in a variety of forms in Islamic art and architecture including kilim carpets, Persian girih and Moroccan/Algerian zellige tilework, muqarnas decorative vaulting, jali pierced stone screens, ceramics, leather, stained glass, woodwork, and metalwork.

Islamic geometric patterns are used in the Quran, Mosques and even in the calligraphies.

In Hinduism

A Hindu Ma??ala
The Agamas are a collection of Sanskrit,[6] Tamil, and Grantha[7] scriptures chiefly constituting the methods of temple construction and creation of idols, worship means of deities, philosophical doctrines, meditative practices, attainment of sixfold desires, and four kinds of yoga.[6]

Elaborate rules are laid out in the Agamas for Shilpa (the art of sculpture) describing the quality requirements of such matters as the places where temples are to be built, the kinds of image to be installed, the materials from which they are to be made, their dimensions, proportions, air circulation, and lighting in the temple complex. The Manasara and Silpasara are works that deal with these rules. The rituals of daily worship at the temple also follow rules laid out in the Agamas.

Hindu temples, the symbolic representation of cosmic model is then projected onto Hindu temples using the Vastu Shastra principle of Sukha Darshan, which states that smaller parts of the temple should be self-similar and a replica of the whole. The repetition of these replication parts symbolizes the natural phenomena of fractal patterns found in nature. These patterns make up the exterior of Hindu temples. Each element and detail are proportional to each other, this occurrence is also known as the sacred geometry.[8]

In Christianity
The construction of Medieval European cathedrals was often based on geometries intended to make the viewer see the world through mathematics, and through this understanding, gain a better understanding of the divine.[9] These churches frequently featured a Latin Cross floor-plan.[10]

At the beginning of the Renaissance in Europe, views shifted to favor simple and regular geometries. The circle in particular became a central and symbolic shape for the base of buildings, as it represented the perfection of nature and the centrality of man's place in the universe.[10] The use of the circle and other simple and symmetrical geometric shapes was solidified as a staple of Renaissance sacred architecture in Leon Battista Alberti's architectural treatise, which described the ideal church in terms of spiritual geometry.[11]

Unanchored geometry
Stephen Skinner discusses the tendency of some writers to place a geometric diagram over virtually any image of a natural object or human created structure, find some lines intersecting the image and declare it based on sacred geometry. If the geometric diagram does not intersect major physical points in the image, the result is what Skinner calls "unanchored geometry".[12]

See also
Circle dance
Harmony of the spheres
Lu Ban and Feng shui
Magic circle
Shield of the Trinity
108 (number)
References
"dartmouth.edu: Paul Calter, Polygons, Tilings, & Sacred Geometry".
Cathérine Goldstein, Norbert Schappacher, Joachim Schwermer, The shaping of arithmetic, p. 235.
Calter, Paul (1998). "Celestial Themes in Art & Architecture". Dartmouth College. Retrieved 5 September 2015.
Shing-Tung Yau and Steve Nadis, The Shape of Inner Space, (New York: Basic Books, 2010), 18.
Skinner, Stephen (2009). Sacred Geometry: Deciphering the Code. Sterling. ISBN 978-1-4027-6582-7.
Grimes, John A. (1996). A Concise Dictionary of Indian Philosophy: Sanskrit Terms Defined in English. State University of New York Press. ISBN 9780791430682. LCCN 96012383. [1]
Nagalingam, Pathmarajah (2009). The Religion of the Agamas. Siddhanta Publications. [2]
"Sacred Geometry Of Hindu Temples". Indic Today. 2019-10-22. Retrieved 2021-04-14.
Petersen, Toni (2003), "A(rt and) A(rchitecture) T(hesaurus)", Oxford Art Online, Oxford University Press, doi:10.1093/gao/9781884446054.article.t000037
CUMMINGS, L.A. (1986), "A RECURRING GEOMETRICAL PATTERN IN THE EARLY RENAISSANCE IMAGINATION", Symmetry, Elsevier, pp. 981–997, doi:10.1016/b978-0-08-033986-3.50067-7, ISBN 9780080339863
Rudolf., Wittkower (1998). Architectural principles in the age of humanism. Academy Editions. ISBN 978-0471977636. OCLC 981109542.
Skinner, Stephen (2006). Stephen Skinner, Sacred geometry: deciphering the code, p91. ISBN 9781402741296.
Further reading
Bain, George. Celtic Art: The Methods of Construction. Dover, 1973. ISBN 0-486-22923-8.
Bromwell, Henry P. H. (2010). Townley, Kevin (ed.). Restorations of Masonic Geometry and Symbolry: Being a Dissertation on the Lost Knowledges of the Lodge. Lovers of the Craft. ISBN 978-0-9713441-5-0. Archived from the original on 2012-02-03. Retrieved Jan 7, 2012.
Bamford, Christopher, Homage to Pythagoras: Rediscovering Sacred Science, Lindisfarne Press, 1994, ISBN 0-940262-63-0
Critchlow, Keith (1970). Order In Space: A Design Source Book. New York: Viking.
Critchlow, Keith (1976). Islamic Patterns: An Analytical and Cosmological Approach. Schocken Books. ISBN 978-0-8052-3627-9.
Iamblichus; Robin Waterfield; Keith Critchlow; Translated by Robin Waterfield (1988). The Theology of Arithmetic: On the Mystical, Mathematical and Cosmological Symbolism of the First Ten Numbers. Phanes Press. ISBN 978-0-933999-72-5.
Johnson, Anthony: Solving Stonehenge, the New Key to an Ancient Enigma. Thames & Hudson 2008 ISBN 978-0-500-05155-9
Lesser, George (1957–64). Gothic cathedrals and sacred geometry. London: A. Tiranti.
Lawlor, Robert. Sacred Geometry: Philosophy and practice (Art and Imagination). Thames & Hudson, 1989 (1st edition 1979, 1980, or 1982). ISBN 0-500-81030-3.
Lippard, Lucy R. Overlay: Contemporary Art and the Art of Prehistory. Pantheon Books New York 1983 ISBN 0-394-51812-8
Mann, A. T. Sacred Architecture, Element Books, 1993, ISBN 1-84333-355-4.
Michell, John. City of Revelation. Abacus, 1972. ISBN 0-349-12320-9.
Schneider, Michael S. A Beginner's Guide to Constructing the Universe: Mathematical Archetypes of Nature, Art, and Science. Harper, 1995. ISBN 0-06-092671-6
Steiner, Rudolf; Creeger, Catherine (2001). The Fourth Dimension : Sacred Geometry, Alchemy, and Mathematics. Anthroposophic Press. ISBN 978-0-88010-472-2.
The Golden Mean, Parabola magazine, v.16, n.4 (1991)
West, John Anthony, Inaugural Lines: Sacred geometry at St. John the Divine, Parabola magazine, v.8, n.1, Spring 1983.

 

 

-
-
-
-
-
SACRED GEOMETRY
-
-
-
S
=
1
-
6
SACRED
50
32
5
G
=
7
-
8
GEOMETRY
108
45
9
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
-
-
-
1+4
-
1+8+5
7+7
1+4
-
-
8
-
5
SACRED GEOMETRY
14
14
5
-
-
-
-
-
-
1+4
1+4
-
-
-
8
-
5
SACRED GEOMETRY
5
5
5

 

 

-
-
-
-
-
SACRED GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
6
SACRED
50
32
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
8
GEOMETRY
108
45
9
-
-
-
-
-
-
-
-
-
-
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SACRED
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
C
=
3
3
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
23
-
6
SACRED
50
32
23
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
GEOMETRY
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
13
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
Y
=
7
14
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
45
-
5
GEOMETRY
108
45
45
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
3
8
15
6
14
8
18
-
-
-
-
-
SACRED GEOMETRY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
S
=
1
-
6
SACRED
50
32
5
-
2
2
3
8
6
6
5
8
9
G
=
7
-
8
GEOMETRY
108
45
9
-
-
-
-
-
-
-
-
-
-
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
2
2
3
8
6
6
5
8
9
-
-
-
-
1+4
-
1+8+5
7+7
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
SACRED GEOMETRY
14
14
5
-
2
2
3
8
6
6
5
8
9
-
-
-
-
-
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
SACRED GEOMETRY
5
5
5
-
2
2
3
8
6
6
5
8
9

 

 

-
-
-
-
-
SACRED GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
6
SACRED
50
32
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
8
GEOMETRY
108
45
9
-
-
-
-
-
-
-
-
-
-
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SACRED
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
C
=
3
3
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
8
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
R
=
9
13
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
Y
=
7
14
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
-
-
-
-
-
-
GEOMETRY
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
3
8
15
6
14
8
18
-
-
-
-
-
SACRED GEOMETRY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
S
=
1
-
6
SACRED
50
32
5
-
2
2
3
8
6
6
5
8
9
G
=
7
-
8
GEOMETRY
108
45
9
-
-
-
-
-
-
-
-
-
-
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
2
2
3
8
6
6
5
8
9
-
-
-
-
1+4
-
1+8+5
7+7
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
SACRED GEOMETRY
14
14
5
-
2
2
3
8
6
6
5
8
9
-
-
-
-
-
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
SACRED GEOMETRY
5
5
5
-
2
2
3
8
6
6
5
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 3 = 15

"The most common letter in the English alphabet is E."

-
-
-
-
-
SACRED GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
6
SACRED
50
32
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
8
GEOMETRY
108
45
9
-
-
-
-
-
-
-
-
-
-
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
SACRED
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
8
-
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
8
-
C
=
3
3
1
C
3
3
3
-
-
-
3
-
-
-
-
8
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
8
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
Y
=
7
14
1
Y
25
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
4
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
13
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
GEOMETRY
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
3
8
15
6
14
8
18
-
-
-
-
-
SACRED GEOMETRY
-
-
-
-
-
-
-
-
1+5
-
1+4
-
1+8
S
=
1
-
6
SACRED
50
32
5
-
2
2
3
8
6
6
5
8
9
G
=
7
-
8
GEOMETRY
108
45
9
-
-
-
-
-
-
-
-
-
-
-
-
8
-
14
SACRED GEOMETRY
185
77
14
-
2
2
3
8
6
6
5
8
9
-
-
-
-
1+4
-
1+8+5
7+7
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
SACRED GEOMETRY
14
14
5
-
2
2
3
8
6
6
5
8
9
-
-
-
-
-
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
SACRED GEOMETRY
5
5
5
-
2
2
3
8
6
6
5
8
9

 

 

Golden ratio - Wikipedia

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

The Parthenon - In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[1][2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number (not to be confused with the related Fibonacci numbers).[6][7][8]

Mathematicians since Euclid—and perhaps earlier—have studied the properties of the golden ratio, including its self-similarity and appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[9]

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

 

What is the Golden Ratio? - Live Science
https://www.livescience.com › Tech

24 Jun 2013 - The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet.

 

The designer's guide to the Golden Ratio | Creative Bloq
https://www.creativebloq.com/design/designers-guide-golden-ratio-12121546

24 Oct 2018 - We call it the Golden Ratio, although it's also known as the Golden Mean, The Golden Section, or the Greek letter phi. Whether you're an ...

 

Golden Ratio - Math is Fun
https://www.mathsisfun.com/numbers/golden-ratio.html

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618. It appears many times in geometry,
art, ...
phi
The golden ratio (symbol is the Greek letter "phi" shown at left)
is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

 

The Golden Ratio
https://www.goldennumber.net/

Golden Ratio, Phi, 1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the Face. One source with over 100 articles and latest findings.
ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it? It's a number that goes by many names. This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section. It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in "De Divina Proportione" in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know Phi was brought even more into the mainstream of pop culture. The allure of “The Da Vinci Code” was that it creatively integrated fiction with both fact and myth from art, history, theology and mathematics, leaving the reader never really knowing what was truth and …

 

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
T
=
2
-
3
THE
33
15
6
G
=
7
-
6
GOLDEN
57
30
3
R
=
9
-
5
RATIO
63
27
9
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
9
-
5
THE GOLDEN RATIO
9
9
9

 

 

THE GOLDEN RATIO

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
1
2
3
4
5
6
7
8
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
-
-
-
-
-
-
15
-
3
THE
33
15
15
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
GOLDEN
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
16
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
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
30
-
6
GOLDEN
57
30
30
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
RATIO
-
-
-
-
-
-
-
-
-
-
-
-
-
R
=
9
10
1
R
9
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
-
-
-
-
-
27
-
5
RATIO
63
27
27
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
1
4
3
4
15
12
7
8
18
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
1+2
-
-
1+8
G
=
7
-
6
GOLDEN
57
30
3
-
1
4
3
4
6
3
7
8
9
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
1
4
3
4
6
3
7
8
9
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
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
-
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
16
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
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
10
1
R
9
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
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
1
4
3
4
15
12
7
8
18
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
1+2
-
-
1+8
G
=
7
-
6
GOLDEN
57
30
3
-
1
4
3
4
6
3
7
8
9
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
1
4
3
4
6
3
7
8
9
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 3 = 15

"The most common letter in the English alphabet is E."

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
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
-
-
-
-
G
=
7
4
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
5
1
O
16
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
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
10
1
R
9
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
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
1
4
3
4
15
12
7
8
18
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
1+2
-
-
1+8
G
=
7
-
6
GOLDEN
57
30
3
-
1
4
3
4
6
3
7
8
9
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
1
4
3
4
6
3
7
8
9
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
T
=
2
12
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
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
5
1
O
16
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
14
1
O
15
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
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN RATIO
-
-
-
-
1
4
3
4
15
12
7
8
18
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
1+5
1+2
-
-
1+8
G
=
7
-
6
GOLDEN
57
30
3
-
1
4
3
4
6
3
7
8
9
R
=
9
-
5
RATIO
63
27
9
-
-
-
-
-
-
-
-
-
-
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
1
4
3
4
6
3
7
8
9
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

THE GOLDEN MEAN

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
T
=
2
-
3
THE
33
15
6
G
=
7
-
6
GOLDEN
57
30
3
M
=
4
-
4
MEAN
33
15
6
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
4
-
4
THE GOLDEN MEAN
6
6
6

 

 

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
23
-
6
THE
50
32
23
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
GOLDEN
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
45
-
5
GOLDEN
108
45
45
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
MEAN
-
-
-
-
-
-
-
-
-
-
-
-
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
23
-
6
MEAN
50
32
23
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
8
25
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
2+5
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
1
2
3
8
7
6
7
8
9
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
8
7
6
7
8
9
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE GOLDEN MEAN
6
6
6
-
1
2
3
8
7
6
7
8
9

 

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
9
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
9
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
9
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
9
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
9
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
8
25
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
2+5
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
1
2
3
8
7
6
7
8
9
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
8
7
6
7
8
9
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE GOLDEN MEAN
6
6
6
-
1
2
3
8
7
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 5 = 25

"The most common letter in the English alphabet is E."

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
9
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
9
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
9
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
9
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
9
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
9
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
8
25
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
2+5
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
1
2
3
8
7
6
7
8
9
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
8
7
6
7
8
9
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE GOLDEN MEAN
6
6
6
-
1
2
3
8
7
6
7
8
9

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
0
-
Z
=
8
1
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
1
-
F
=
6
2
5
FIRST
72
27
9
-
-
-
-
-
-
-
-
-
9
2
-
S
=
1
3
6
SECOND
60
24
6
-
-
-
-
-
-
6
-
-
-
3
-
T
=
2
4
5
THIRD
59
32
5
-
-
-
-
-
5
-
-
-
-
4
-
F
=
6
5
6
FOURTH
88
34
7
-
-
-
-
-
-
-
7
-
-
5
-
F
=
6
6
5
FIFTH
49
31
4
-
-
-
-
4
-
-
-
-
-
6
-
S
=
1
7
5
SIXTH
80
26
8
-
-
-
-
-
-
-
-
8
-
7
-
S
=
1
8
7
SEVENTH
93
30
3
-
-
-
3
-
-
-
-
-
-
8
-
E
=
5
9
6
EIGHTH
57
39
3
-
-
-
3
-
-
-
-
-
-
9
-
N
=
5
10
5
NINTH
65
29
2
-
-
2
-
-
-
-
-
-
-
45
-
-
-
41
-
54
Add
687
300
48
-
1
2
6
4
5
6
7
8
9
4+5
-
-
-
4+1
-
5+4
Reduce
6+8+7
3+0+0
4+8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
5
-
9
Deduce
21
3
12
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
-
Reduce
2+1
-
1+2
-
-
-
-
-
-
-
-
-
-
9
-
-
-
5
-
9
Essence
3
3
3
-
1
2
6
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
0
-
Z
=
8
1
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
9
-
N
=
5
10
5
NINTH
65
29
2
-
-
2
-
-
-
-
-
-
-
7
-
S
=
1
8
7
SEVENTH
93
30
3
-
-
-
3
-
-
-
-
-
-
8
-
E
=
5
9
6
EIGHTH
57
39
3
-
-
-
3
-
-
-
-
-
-
5
-
F
=
6
6
5
FIFTH
49
31
4
-
-
-
-
4
-
-
-
-
-
3
-
T
=
2
4
5
THIRD
59
32
5
-
-
-
-
-
5
-
-
-
-
2
-
S
=
1
3
6
SECOND
60
24
6
-
-
-
-
-
-
6
-
-
-
4
-
F
=
6
5
6
FOURTH
88
34
7
-
-
-
-
-
-
-
7
-
-
6
-
S
=
1
7
5
SIXTH
80
26
8
-
-
-
-
-
-
-
-
8
-
1
-
F
=
6
2
5
FIRST
72
27
9
-
-
-
-
-
-
-
-
-
9
45
-
-
-
41
-
54
Add
687
300
48
-
1
2
6
4
5
6
7
8
9
4+5
-
-
-
4+1
-
5+4
Reduce
6+8+7
3+0+0
4+8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
5
-
9
Deduce
21
3
12
-
1
2
6
4
5
6
7
8
9
-
-
-
-
-
-
-
Reduce
2+1
-
1+2
-
-
-
-
-
-
-
-
-
-
9
-
-
-
5
-
9
Essence
3
3
3
-
1
2
6
4
5
6
7
8
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
58
13
4
-
-
-
-
4
-
-
-
-
-
3
-
T
=
2
-
5
THREE
56
29
2
-
-
2
-
-
-
-
-
-
-
4
-
F
=
6
-
4
FOUR
60
24
6
-
-
-
-
-
-
6
-
-
-
5
-
F
=
6
-
4
FIVE
42
24
6
-
-
-
-
-
-
6
-
-
-
6
-
S
=
1
-
3
SIX
52
16
7
-
-
-
-
-
-
-
7
-
-
7
-
S
=
1
-
5
SEVEN
65
20
2
-
-
2
-
-
-
-
-
-
-
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
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
1
4
ZERO
64
28
1
-
1
-
3
-
5
-
-
8
9
1
-
O
=
6
2
3
ONE
34
16
7
-
-
-
3
-
5
-
7
8
9
2
-
T
=
2
3
3
TWO
58
13
4
-
-
-
3
4
5
-
-
8
9
3
-
T
=
2
4
5
THREE
56
29
2
-
-
2
3
-
5
-
-
8
9
4
-
F
=
6
5
4
FOUR
60
24
6
-
-
-
3
-
5
6
-
8
9
5
-
F
=
6
6
4
FIVE
42
24
6
-
-
-
3
-
5
6
-
8
9
6
-
S
=
1
7
3
SIX
52
16
7
-
-
-
3
-
5
-
7
8
9
7
-
S
=
1
8
5
SEVEN
65
20
2
-
-
2
3
-
5
-
-
8
9
8
-
E
=
5
9
5
EIGHT
49
31
4
-
-
-
3
4
5
-
-
8
9
9
-
N
=
5
10
4
NINE
42
24
6
-
-
-
3
-
5
6
-
8
9
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
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
1
4
ZERO
64
28
1
-
1
-
3
-
5
-
-
8
9
3
-
T
=
2
4
5
THREE
56
29
2
-
-
2
3
-
5
-
-
8
9
7
-
S
=
1
8
5
SEVEN
65
20
2
-
-
2
3
-
5
-
-
8
9
2
-
T
=
2
3
3
TWO
58
13
4
-
-
-
3
4
5
-
-
8
9
8
-
E
=
5
9
5
EIGHT
49
31
4
-
-
-
3
4
5
-
-
8
9
4
-
F
=
6
5
4
FOUR
60
24
6
-
-
-
3
-
5
6
-
8
9
5
-
F
=
6
6
4
FIVE
42
24
6
-
-
-
3
-
5
6
-
8
9
9
-
N
=
5
10
4
NINE
42
24
6
-
-
-
3
-
5
6
-
8
9
1
-
O
=
6
2
3
ONE
34
16
7
-
-
-
3
-
5
-
7
8
9
6
-
S
=
1
7
3
SIX
52
16
7
-
-
-
3
-
5
-
7
8
9
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
1
4
ZERO
64
28
1
-
1
-
-
-
-
3
-
T
=
2
4
5
THREE
56
29
2
-
-
2
-
-
-
7
-
S
=
1
8
5
SEVEN
65
20
2
-
-
2
-
-
-
2
-
T
=
2
3
3
TWO
58
13
4
-
-
-
4
-
-
8
-
E
=
5
9
5
EIGHT
49
31
4
-
-
-
4
-
-
4
-
F
=
6
5
4
FOUR
60
24
6
-
-
-
-
6
-
5
-
F
=
6
6
4
FIVE
42
24
6
-
-
-
-
6
-
9
-
N
=
5
10
4
NINE
42
24
6
-
-
-
-
6
-
1
-
O
=
6
2
3
ONE
34
16
7
-
-
-
-
-
7
6
-
S
=
1
7
3
SIX
52
16
7
-
-
-
-
-
7
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

 

 

M
=
4
8
MAGNETIC
72
36
9
T
=
2
7
THOUGHT
99
36
9
G
=
7
4
GODS
45
18
9
D
=
4
6
DIVINE
63
36
9
R
=
9
7
REALITY
90
36
9

 

 

E
=
5
6
EUCLID
54
27
9
T
=
2
3
THE
33
15
6
E
=
5
8
ELEMENTS
93
30
3
-
-
12
17
Add to Reduce
180
72
18
-
-
1+2
1+7
Add to Reduce
1+8+0
7+2
1+8
Q
-
3
8
Essence of Number
9
9
9

 

 

E
=
5
7
EUCLID'S
73
28
1
T
=
2
3
THE
33
15
6
E
=
5
8
ELEMENTS
93
30
3
-
-
12
18
First Total
199
73
10
-
-
1+2
1+8
Add to Reduce
1+9+9
7+3
1+0
Q
-
3
9
Second Total
19
10
1
-
-
-
-
Reduce to Deduce
1+9
1+0
-
Q
-
3
9
Third Total
10
1
1
-
-
-
-
Add to Reduce
1+0
1+0
-
Q
-
3
9
Essence of Number
1
1
1

 

 

INCLUDE EUCLID INCLUDE

 

Euclid's Elements - Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Euclid's_Elements

 

Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria c. 300 BC.

Euclid's Elements (Ancient Greek: Στοιχεῖα Stoicheia) is a mathematical and geometric treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt c. 300 BC. It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. The work also includes an algebraic system that has become known as geometric algebra, which is powerful enough to solve many algebraic problems,[1] including the problem of finding the square root of a number.[2] The Elements is the second oldest extant Greek mathematical treatise after Autolycus' On the Moving Sphere,[3] and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science. According to Proclus the term "element" was used to describe a theorem that is all-pervading and helps furnishing proofs of many other theorems. The word 'element' is in the Greek language the same as 'letter'. This suggests that theorems in the Elements should be seen as standing in the same relation to geometry as letters to language. Later commentators give a slightly different meaning to the term 'element', emphasizing how the propositions have progressed in small steps, and continued to build on previous propositions in a well-defined order.[4]

Euclid's Elements has been referred to as the most successful[5][6] and influential[7] textbook ever written. Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and was estimated by Carl Benjamin Boyer to be second only to the Bible in the number of editions published,[7] with the number reaching well over one thousand.[8] For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.[

Basis in earlier work
Scholars believe that the Elements is largely a collection of theorems proven by other mathematicians, supplemented by some original work.

Proclus (412 – 485 AD), a Greek mathematician who lived around seven centuries after Euclid, wrote in his commentary on the Elements: "Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors".

Pythagoras (c. 570 – c. 495 BCE) was probably the source for most of books I and II, Hippocrates of Chios (c. 470 – c. 410 BCE, not the better known Hippocrates of Kos) for book III, and Eudoxus of Cnidus (c. 408 – c. 355 BC) for book V, while books IV, VI, XI, and XII probably came from other Pythagorean or Athenian mathematicians.[11] The Elements may have been based on an earlier textbook by Hippocrates of Chios, who also may have originated the use of letters to refer to figures.[12]

 

 

-
-
-
-
-
EUCLID
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
5
5
5
-
E
-
-
5
-
E
5
E
-
-
-
-
-
U
21
3
3
-
U
3
-
-
-
U
3
U
-
-
-
-
-
C
3
3
3
-
C
3
-
-
-
C
3
C
-
-
-
-
-
L
12
3
3
-
L
3
-
-
-
L
3
L
-
-
-
-
-
I
9
9
9
-
I
-
-
-
9
I
9
I
-
-
-
-
-
D
4
4
4
-
D
-
4
-
-
D
4
D
E
=
5
-
6
EUCLID
54
27
27
-
-
9
4
5
9
-
27
-
-
-
-
-
-
-
5+4
2+7
2+7
-
-
-
-
-
-
-
2+7
-
-
-
5
-
6
EUCLID
9
9
9
-
-
9
4
5
9
-
9
-

 

 

-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
3
3
3
-
4
+
=
18
1+8
=
9
-
9
-
-
5
21
3
12
-
4
+
=
45
4+5
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
21
3
12
9
4
+
=
54
5+4
=
9
-
9
-
-
5
3
3
3
9
4
+
=
27
2+7
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
-
-
-
1
-
-
-
-
-
--
-
-
--
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
3
3
3
-
-
-
-
3
occurs
x
3
=
9
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
5
-
-
-
-
-
-
--
5
occurs
x
1
=
5
6
-
-
--
-
--
--
-
-
-
6
-
-
-
-
--
7
-
-
--
-
--
--
-
-
--
7
-
-
-
-
--
8
-
-
--
-
--
--
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
24
6
E
U
C
L
I
D
-
-
21
-
-
6
-
27
2+4
-
-
3
3
3
-
-
-
-
2+1
-
-
-
-
2+7
6
6
E
U
C
L
I
D
-
-
3
-
-
6
-
9
-
-
5
3
3
3
9
4
-
-
-
-
-
-
-
-
6
6
E
U
C
L
I
D
-
-
3
-
-
6
-
9

 

 

6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
5
3
3
3
-
4
+
=
18
1+8
=
9
-
9
-
5
21
3
12
-
4
+
=
45
4+5
=
9
-
9
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
5
21
3
12
9
4
+
=
54
5+4
=
9
-
9
-
5
3
3
3
9
4
+
=
27
2+7
=
9
-
9
6
E
U
C
L
I
D
-
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
-
3
occurs
x
3
=
9
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
6
E
U
C
L
I
D
-
-
21
-
-
6
-
27
-
-
3
3
3
-
-
-
-
2+1
-
-
-
-
2+7
6
E
U
C
L
I
D
-
-
3
-
-
6
-
9
-
5
3
3
3
9
4
-
-
-
-
-
-
-
-
6
E
U
C
L
I
D
-
-
3
-
-
6
-
9

 

 

E
=
5
6
EUCLID
54
27
9
T
=
2
3
THE
33
15
6
E
=
5
8
ELEMENTS
93
30
3
-
-
12
17
Add to Reduce
180
72
18
-
-
1+2
1+7
Add to Reduce
1+8+0
7+2
1+8
Q
-
3
8
Essence of Number
9
9
9

 

 

T
=
2
3
THE
33
15
6
E
=
5
8
ELEMENTS
93
30
3
-
-
7
11
Add to Reduce
126
45
9
-
-
-
1+1
Reduce to Deduce
1+2+6
4+5
-
Q
-
7
2
Essence of Number
9
9
9

 

 

-
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
5
-
1
+
=
14
1+4
=
5
-
5
-
5
-
-
-
8
-
-
-
-
-
-
-
14
-
19
+
=
41
4+1
=
5
-
5
-
5
-
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
5
3
5
4
5
-
2
-
+
=
31
3+1
=
4
-
4
-
4
-
-
20
-
5
-
5
12
5
13
5
-
20
-
+
=
85
8+5
=
13
1+3
4
-
4
-
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
5
12
5
13
5
14
20
19
+
=
126
1+2+6
=
9
-
9
-
9
-
-
2
8
5
-
5
3
5
4
5
5
2
1
+
=
45
4+5
=
9
-
9
-
9
-
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
--
3
--
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
5
-
5
-
5
-
5
5
-
-
-
-
5
occurs
x
5
=
25
2+5
7
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
22
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
23
-
-
11
-
45
-
27
2+2
1+1
-
-
5
-
5
-
5
-
5
5
-
-
-
-
2+3
-
-
1+1
-
4+5
-
2+7
4
2
T
H
E
-
E
L
E
M
E
N
T
S
-
-
5
-
-
2
-
9
-
9
-
-
2
8
5
-
5
3
5
4
5
5
2
1
-
-
-
-
-
-
-
-
-
-
4
2
T
H
E
-
E
L
E
M
E
N
T
S
-
-
5
-
-
2
-
1
-
1

 

 

11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
-
-
-
5
-
1
+
=
14
1+4
=
5
-
5
-
5
-
-
8
-
-
-
-
-
-
-
14
-
19
+
=
41
4+1
=
5
-
5
-
5
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
5
3
5
4
5
-
2
-
+
=
31
3+1
=
4
-
4
-
4
-
20
-
5
-
5
12
5
13
5
-
20
-
+
=
85
8+5
=
13
1+3
4
-
4
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
5
12
5
13
5
14
20
19
+
=
126
1+2+6
=
9
-
9
-
9
-
2
8
5
-
5
3
5
4
5
5
2
1
+
=
45
4+5
=
9
-
9
-
9
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
--
3
--
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
5
-
5
-
5
-
5
5
-
-
-
-
5
occurs
x
5
=
25
2+5
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
11
T
H
E
-
E
L
E
M
E
N
T
S
-
-
23
-
-
11
-
45
-
27
1+1
-
-
5
-
5
-
5
-
5
5
-
-
-
-
2+3
-
-
1+1
-
4+5
-
2+7
2
T
H
E
-
E
L
E
M
E
N
T
S
-
-
5
-
-
2
-
9
-
9
-
2
8
5
-
5
3
5
4
5
5
2
1
-
-
-
-
-
-
-
-
-
-
2
T
H
E
-
E
L
E
M
E
N
T
S
-
-
5
-
-
2
-
1
-
1

 

 

E
=
5
7
EUCLID'S
73
28
1
T
=
2
3
THE
33
15
6
E
=
5
8
ELEMENTS
93
30
3
-
-
12
18
First Total
199
73
10
-
-
1+2
1+8
Add to Reduce
1+9+9
7+3
1+0
Q
-
3
9
Second Total
19
10
1
-
-
-
-
Reduce to Deduce
1+9
1+0
-
Q
-
3
9
Third Total
10
1
1
-
-
-
-
Add to Reduce
1+0
1+0
-
Q
-
3
9
Essence of Number
1
1
1

 

 

E
=
5
7
EUCLID'S
73
28
1
=
5
8
ELEMENTS
93
30
3
-
-
10
15
First Total
166
58
4
-
-
1+0
1+5
Add to Reduce
1+6+6
5+8
1+0
Q
-
1
6
Second Total
13
13
4
-
-
-
-
Reduce to Deduce
1+3
1+3
-
Q
-
1
6
Essence of Number
4
4
4

 

 

-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
4
4
7
EUCLID'S
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
76
4
4
1
2
3
4
5
6
7
8
9
-
-
1+0
4
4
1+5
-
1+6+6
7+6
1+0
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
-
2
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
C
=
3
-
3
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
L
=
3
-
4
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
-
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
D
=
7
-
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
-
7
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
28
4
4
7
-
73
37
28
-
-
-
-
-
-
-
-
-
-
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
L
=
3
-
9
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
-
11
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
12
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
13
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
14
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
S
=
1
-
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
28
4
4
8
-
93
39
30
-
2
2
12
8
25
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
2
2
3
8
7
6
7
8
9
E
=
5
4
4
7
EUCLID'S
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
76
4
4
-
-
-
-
-
-
-
-
-
-
-
1+0
4
4
1+5
-
1+6+6
7+6
4
-
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
4
4
7
EUCLID'S
73
28
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
1
2
3
4
5
6
7
8
9
-
-
1+0
4
4
1+5
-
1+6+6
5+8
1+0
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
-
2
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
C
=
3
-
3
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
L
=
3
-
4
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
-
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
D
=
7
-
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
-
7
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
L
=
3
-
9
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
-
11
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
12
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
13
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
14
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
S
=
1
-
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
58
4
4
15
-
166
76
58
-
2
2
12
8
25
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
2
2
3
8
7
6
7
8
9
E
=
5
4
4
7
EUCLID'S
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
76
4
4
-
-
-
-
-
-
-
-
-
-
-
1+0
4
4
1+5
-
1+6+6
7+6
4
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
1
2
3
4
5
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 5 = 25

"The most common letter in the English alphabet is E."

 

-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
1
2
3
4
5
6
7
8
9
E
=
5
4
4
7
EUCLID'S
73
28
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
1
2
3
4
5
6
7
8
9
-
-
1+0
4
4
1+5
-
1+6+6
5+8
1+0
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
7
1
S
19
10
1
-
1
-
-
-
-
6
7
8
-
S
=
1
-
15
1
S
19
10
1
-
1
-
-
-
-
6
7
8
-
T
=
2
-
14
1
T
20
2
2
-
-
2
-
-
-
6
7
8
-
U
=
3
-
2
1
U
21
3
3
-
-
-
3
-
-
6
7
8
-
C
=
3
-
3
1
C
3
3
3
-
-
-
3
-
-
6
7
8
-
L
=
3
-
4
1
L
12
3
3
-
-
-
3
-
-
6
7
8
-
L
=
3
-
9
1
L
12
3
3
-
-
-
3
-
-
6
7
8
-
D
=
7
-
6
1
D
4
4
4
-
-
-
-
4
-
6
7
8
-
M
=
4
-
11
1
M
13
4
4
-
-
-
-
4
-
6
7
8
-
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
E
=
5
-
10
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
E
=
5
-
12
1
E
5
5
5
-
-
-
-
-
5
6
7
8
-
N
=
5
-
13
1
N
14
5
5
-
-
-
-
-
5
6
7
8
-
I
=
9
-
5
1
I
9
9
9
-
-
-
-
-
-
6
7
8
9
-
-
58
4
4
15
-
166
76
58
-
2
2
12
8
25
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
2
2
3
8
7
6
7
8
9
E
=
5
4
4
7
EUCLID'S
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
76
4
4
-
-
-
-
-
-
-
-
-
-
-
1+0
4
4
1+5
-
1+6+6
7+6
4
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
1
2
3
4
5
9
E
=
5
4
4
7
EUCLID'S
73
28
1
4
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
1
2
3
4
5
9
-
-
1+0
4
4
1+5
-
1+6+6
5+8
1+0
4
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
9
-
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
4
1
2
3
4
5
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
9
S
=
1
-
7
1
S
19
10
1
-
1
-
-
-
-
-
S
=
1
-
15
1
S
19
10
1
-
1
-
-
-
-
-
T
=
2
-
14
1
T
20
2
2
-
-
2
-
-
-
-
U
=
3
-
2
1
U
21
3
3
-
-
-
3
-
-
-
C
=
3
-
3
1
C
3
3
3
-
-
-
3
-
-
-
L
=
3
-
4
1
L
12
3
3
-
-
-
3
-
-
-
L
=
3
-
9
1
L
12
3
3
-
-
-
3
-
-
-
D
=
7
-
6
1
D
4
4
4
-
-
-
-
4
-
-
M
=
4
-
11
1
M
13
4
4
-
-
-
-
4
-
-
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
-
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
-
E
=
5
-
10
1
E
5
5
5
-
-
-
-
-
5
-
E
=
5
-
12
1
E
5
5
5
-
-
-
-
-
5
-
N
=
5
-
13
1
N
14
5
5
-
-
-
-
-
5
-
I
=
9
-
5
1
I
9
9
9
-
-
-
-
-
-
9
-
-
58
4
4
15
-
166
76
58
-
2
2
12
8
25
9
-
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
2
2
3
8
7
9
E
=
5
4
4
7
EUCLID'S
73
37
1
4
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
3
4
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
76
4
4
-
-
-
-
-
-
-
-
1+0
4
4
1+5
-
1+6+6
7+6
4
4
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
13
13
4
4
1
2
3
4
5
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
1
2
3
4
5
9

 

 

Daily Mail, Thursday, May 12, 2016
ANSWERS TO CORRESPONDENTS
Compiled by Charles Legge

Page 58

QUESTION Did ancient scholars such as Aristotle, Socrates and Euclid make a good living through their learned works? What sort of income did they make compared with the general population?

SOCRATES, like his father, was a stonemason for many years before devoting his life to philosophy.
Contemporaries differ in their account of how Socrates supported himself as a philosopher. Both Xenophon, a student of his, and Aristophanes say Socrates received payment for teaching, while Plato, who had also been one of his students, writes that Socrates explicitly denied accepting payment, citing his poverty as proof.

Socrates's wife Xanthippe, a younger woman who bore him three sons, was described by Xenophon as 'undesirable'. She often complained that Socrates wasn't supporting his family.

Socrates's marriage was not a happy one, hence his lament: 'By all means, marry. If you get a good wife, you'll become happy; if you get a bad one, you'll become a philosopher.'

Plato believed that virtue could not be taught and it was immoral to charge pupils tuition fees. Lord Byron once held a similar -position, declaring that he would not `prostitute his genius for hire'.

Schools of philosophy such as Plato's Academy, Aristotle's Lyceum, the Epicureans' Garden, or the Stoics' Porch all offered free room and board plus free tuition. These `schools' had wealthy patrons who supported them, much as ballet and opera companies do today, and the teachers received food and board but no pay.

By the time Aristotle had established his Lyceum, he probably didn't need payment. He had previously been hired by Philip of Macedon to teach his precocious son Alexander (later the Great').

Philip built a whole institution for Aristotle to teach Alexander and the other ( nobles at Mieza. The ruins are an hour's drive from Thessaloniki today. Philip also rebuilt Aristotle's home-town Stageira and bought back and freed all of Aristotle's compatriots whom he had previously sold into slavery.

The Cynics lived without personal wealth or property and begged for a living. The most famous cynic was Diogenes of Sinope, who supposedly lived in a pithos (a big wine vat). When Alexander asked him: 'What can I do-for you?' he answered: `Move. You're blocking the sun.'

At the other end of the spectrum were the Sophists, itinerant professional teachers and intellectuals who frequented Greek cities in the second half of the fifth century BC. In return for a fee, the sophists offered young wealthy Greek men an education in aret (virtue or excellence), thereby attaining wealth and fame.

This payment for knowledge aroused significant antipathy from Plato and Aristotle — the term sophistry has since come to signify the deliberate use of fallacious reasoning, intellectual charlatanism and moral unscrupulousness.

Euclid was a great Greek mathematician, 'father of geometry' and author of Elements. Practically nothing is known of Euclid's life but it is presumed that he was born around 330 BC in Tyre.

According to various Arabic authors, he came from a wealthy background, but the accounts are thought to be fictitious. As a mathematician he would not have had philosophical difficulties with money, but his financial status is unknown.
Dr P L. Cray, Nottingham.

 

-
-
-
-
-
EUCLID
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
5
5
5
-
E
-
-
5
-
E
5
E
-
-
-
-
-
U
21
3
3
-
U
3
-
-
-
U
3
U
-
-
-
-
-
C
3
3
3
-
C
3
-
-
-
C
3
C
-
-
-
-
-
L
12
3
3
-
L
3
-
-
-
L
3
L
-
-
-
-
-
I
9
9
9
-
I
-
-
-
9
I
9
I
-
-
-
-
-
D
4
4
4
-
D
-
4
-
-
D
4
D
E
=
5
-
6
EUCLID
54
27
27
-
-
9
4
5
9
-
27
-
-
-
-
-
-
-
5+4
2+7
2+7
-
-
-
-
-
-
-
2+7
-
E
=
5
-
6
EUCLID
9
9
9
-
-
9
4
5
9
-
9
-

 

3 4 5 - 9

 

-
-
-
-
-
EUCLID
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
E
5
5
5
-
E
-
1
2
-
-
5
6
7
8
-
E
-
E
5
E
-
-
-
-
-
U
21
3
3
-
U
-
1
2
3
-
-
6
7
8
-
U
-
U
3
U
-
-
-
-
-
C
3
3
3
-
C
-
1
2
3
-
-
6
7
8
-
C
-
C
3
C
-
-
-
-
-
L
12
3
3
-
L
-
1
2
3
-
-
6
7
8
-
L
-
L
3
L
-
-
-
-
-
I
9
9
9
-
I
-
1
2
-
-
-
6
7
8
9
I
9
I
9
I
-
-
-
-
-
D
4
4
4
-
D
-
1
2
-
4
-
6
7
8
-
D
-
D
4
D
E
=
5
-
6
EUCLID
54
27
27
-
-
-
1
2
3
4
5
6
7
8
9
-
9
-
27
-
-
-
-
-
-
-
5+4
2+7
2+7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2+7
-
E
=
5
-
6
EUCLID
9
9
9
-
-
-
1
2
3
4
5
6
7
8
9
-
9
-
9
-

 

3 4 5 - 9

 

-
-
-
-
-
EUCLID
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
E
5
5
5
-
E
-
1
2
-
-
5
6
7
8
-
-
-
-
-
-
U
21
3
3
-
U
-
1
2
3
-
-
6
7
8
-
-
-
-
-
-
C
3
3
3
-
C
-
1
2
3
-
-
6
7
8
-
-
-
-
-
-
L
12
3
3
-
L
-
1
2
3
-
-
6
7
8
-
-
-
-
-
-
I
9
9
9
-
I
-
1
2
-
-
-
6
7
8
9
-
-
-
-
-
D
4
4
4
-
D
-
1
2
-
4
-
6
7
8
-
E
=
5
-
6
EUCLID
54
27
27
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
5+4
2+7
2+7
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
6
EUCLID
9
9
9
-
-
-
1
2
3
4
5
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
EUCLID
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
U
21
3
3
-
U
-
1
2
3
-
-
6
7
8
-
-
-
-
-
-
C
3
3
3
-
C
-
1
2
3
-
-
6
7
8
-
-
-
-
-
-
L
12
3
3
-
L
-
1
2
3
-
-
6
7
8
-
-
-
-
-
-
E
5
5
5
-
E
-
1
2
-
-
5
6
7
8
-
-
-
-
-
-
D
4
4
4
-
D
-
1
2
-
4
-
6
7
8
-
-
-
-
-
-
I
9
9
9
-
I
-
1
2
-
-
-
6
7
8
9
E
=
5
-
6
EUCLID
54
27
27
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
5+4
2+7
2+7
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
6
EUCLID
9
9
9
-
-
-
1
2
3
4
5
6
7
8
9

 

3 4 5 - 9

 

-
-
-
-
-
EUCLID
-
-
-
-
-
-
3
4
5
9
-
-
-
-
-
U
21
3
3
-
U
-
3
-
-
-
-
-
-
-
-
C
3
3
3
-
C
-
3
-
-
-
-
-
-
-
-
L
12
3
3
-
L
-
3
-
-
-
-
-
-
-
-
E
5
5
5
-
E
-
-
-
5
-
-
-
-
-
-
D
4
4
4
-
D
-
-
4
-
-
-
-
-
-
-
I
9
9
9
-
I
-
-
-
-
9
E
=
5
-
6
EUCLID
54
27
27
-
-
-
3
4
5
9
-
-
-
-
-
-
5+4
2+7
2+7
-
-
-
-
-
-
-
E
=
5
-
6
EUCLID
9
9
9
-
-
-
3
4
5
9

 

3 4 5 - 9

 

 

E
=
5
4
4
8
ELEMENTS
93
30
3
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
L
=
3
-
2
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
-
4
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
7
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
S
=
1
-
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
30
4
1
2
3
4
20
6
7
8
9
-
-
-
4
4
4
-
9+3
3+9
3+0
-
-
-
-
-
2+0
-
-
-
-
S
-
7
4
4
8
ELEMENTS
12
3
3
4
1
2
3
4
2
6
7
8
9
-
-
-
4
4
4
-
1+2
-
-
-
-
-
-
-
-
-
-
-
-
S
-
7
4
4
8
ELEMENTS
3
3
3
4
1
2
3
4
2
6
7
8
9

 

 

E
=
5
4
4
8
ELEMENTS
93
30
3
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
6
7
8
9
L
=
3
-
2
1
L
12
3
3
-
-
-
3
-
-
6
7
8
9
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
6
7
8
9
M
=
4
-
4
1
M
13
4
4
-
-
-
-
4
-
6
7
8
9
E
=
5
-
5
1
E
5
5
5
-
-
-
-
-
5
6
7
8
9
N
=
5
-
6
1
N
14
5
5
-
-
-
-
-
5
6
7
8
9
T
=
2
-
7
1
T
20
2
2
-
-
2
-
-
-
6
7
8
9
S
=
1
-
8
1
S
19
10
1
-
1
-
-
-
-
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
30
4
1
2
3
4
20
6
7
8
9
-
-
-
4
4
4
-
9+3
3+9
3+0
-
-
-
-
-
2+0
-
-
-
-
S
-
7
4
4
8
ELEMENTS
12
3
3
4
1
2
3
4
2
6
7
8
9
-
-
-
4
4
4
-
1+2
-
-
-
-
-
-
-
-
-
-
-
-
S
-
7
4
4
8
ELEMENTS
3
3
3
4
1
2
3
4
2
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

E
=
5
4
4
8
ELEMENTS
93
30
3
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
8
1
S
19
10
1
-
1
-
-
-
-
6
7
8
9
T
=
2
-
7
1
T
20
2
2
-
-
2
-
-
-
6
7
8
9
L
=
3
-
2
1
L
12
3
3
-
-
-
3
-
-
6
7
8
9
M
=
4
-
4
1
M
13
4
4
-
-
-
-
4
-
6
7
8
9
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
6
7
8
9
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
6
7
8
9
E
=
5
-
5
1
E
5
5
5
-
-
-
-
-
5
6
7
8
9
N
=
5
-
6
1
N
14
5
5
-
-
-
-
-
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
30
4
1
2
3
4
20
6
7
8
9
-
-
-
4
4
4
-
9+3
3+9
3+0
-
-
-
-
-
2+0
-
-
-
-
S
-
7
4
4
8
ELEMENTS
12
3
3
4
1
2
3
4
2
6
7
8
9
-
-
-
4
4
4
-
1+2
-
-
-
-
-
-
-
-
-
-
-
-
S
-
7
4
4
8
ELEMENTS
3
3
3
4
1
2
3
4
2
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 4 = 25

"The most common letter in the English alphabet is E."

 

E
=
5
4
4
8
ELEMENTS
93
30
3
4
1
2
3
4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
S
=
1
-
8
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
-
4
1
M
13
4
4
-
-
-
-
4
-
E
=
5
-
1
1
E
5
5
5
-
-
-
-
-
5
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
E
=
5
-
5
1
E
5
5
5
-
-
-
-
-
5
N
=
5
-
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
39
30
4
1
2
3
4
20
-
-
-
4
4
4
-
9+3
3+9
3+0
-
-
-
-
-
2+0
S
-
7
4
4
8
ELEMENTS
12
3
3
4
1
2
3
4
2
-
-
-
4
4
4
-
1+2
-
-
-
-
-
-
-
-
S
-
7
4
4
8
ELEMENTS
3
3
3
4
1
2
3
4
2

 

 

T
=
2
3
THE
33
15
6
E
=
5
8
ELEMENTS
93
30
3
-
-
7
11
Add to Reduce
126
45
9
-
-
-
1+1
Reduce to Deduce
1+2+6
4+5
-
Q
-
7
2
Essence of Number
9
9
9

 

 

-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
4
4
3
THE
33
15
6
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
11
THE ELEMENTS
126
45
9
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
3
-
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
15
4
4
3
-
33
15
15
-
-
-
-
-
-
-
-
-
-
E
=
5
-
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
L
=
3
-
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
-
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
-
9
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
10
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
S
=
1
-
11
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
30
4
4
8
-
93
39
30
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
25
6
7
8
9
-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
-
-
-
-
2+5
-
-
-
-
T
=
2
4
4
3
THE
33
15
6
4
1
4
3
4
7
6
7
8
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
7
4
4
11
THE ELEMENTS
126
45
9
4
1
4
3
4
7
6
7
8
9
-
-
-
4
4
1+1
-
1+2+6
4+5
-
-
-
-
-
-
-
-
-
-
-
S
-
7
4
4
2
THE ELEMENTS
9
9
9
4
1
4
3
4
7
6
7
8
9

 

 

-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
4
4
3
THE
33
15
6
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
11
THE ELEMENTS
126
45
9
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
1
1
T
20
2
2
-
-
2
-
-
-
6
7
-
9
H
=
3
-
2
1
H
8
8
8
-
-
-
-
-
-
6
7
8
9
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
E
=
5
-
4
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
L
=
3
-
5
1
L
12
3
3
-
-
-
3
-
-
6
7
-
9
E
=
5
-
6
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
M
=
4
-
7
1
M
13
4
4
-
-
-
-
4
-
6
7
-
9
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
N
=
5
-
9
1
N
14
5
5
-
-
-
-
-
5
6
7
-
9
T
=
2
-
10
1
T
20
2
2
-
-
2
-
-
-
6
7
-
9
S
=
1
-
11
1
S
19
10
1
-
1
-
-
-
-
6
7
-
9
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
25
6
7
8
9
-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
-
-
-
-
2+5
-
-
-
-
T
=
2
4
4
3
THE
33
15
6
4
1
4
3
4
7
6
7
8
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
7
4
4
11
THE ELEMENTS
126
45
9
4
1
4
3
4
7
6
7
8
9
-
-
-
4
4
1+1
-
1+2+6
4+5
-
-
-
-
-
-
-
-
-
-
-
S
-
7
4
4
2
THE ELEMENTS
9
9
9
4
1
4
3
4
7
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 5 = 25

"The most common letter in the English alphabet is E."

 

-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
4
4
3
THE
33
15
6
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
11
THE ELEMENTS
126
45
9
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
11
1
S
19
10
1
-
1
-
-
-
-
6
7
-
9
T
=
2
-
1
1
T
20
2
2
-
-
2
-
-
-
6
7
-
9
T
=
2
-
10
1
T
20
2
2
-
-
2
-
-
-
6
7
-
9
L
=
3
-
5
1
L
12
3
3
-
-
-
3
-
-
6
7
-
9
M
=
4
-
7
1
M
13
4
4
-
-
-
-
4
-
6
7
-
9
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
E
=
5
-
4
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
E
=
5
-
6
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
6
7
-
9
N
=
5
-
9
1
N
14
5
5
-
-
-
-
-
5
6
7
-
9
H
=
3
-
2
1
H
8
8
8
-
-
-
-
-
-
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
25
6
7
8
9
-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
-
-
-
-
2+5
-
-
-
-
T
=
2
4
4
3
THE
33
15
6
4
1
4
3
4
7
6
7
8
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
7
4
4
11
THE ELEMENTS
126
45
9
4
1
4
3
4
7
6
7
8
9
-
-
-
4
4
1+1
-
1+2+6
4+5
-
-
-
-
-
-
-
-
-
-
-
S
-
7
4
4
2
THE ELEMENTS
9
9
9
4
1
4
3
4
7
6
7
8
9

 

 

-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
1
2
3
4
5
8
T
=
2
4
4
3
THE
33
15
6
4
-
-
-
-
-
-
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
S
-
10
4
4
11
THE ELEMENTS
126
45
9
4
1
2
3
4
5
8
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
8
S
=
1
-
11
1
S
19
10
1
-
1
-
-
-
-
-
T
=
2
-
1
1
T
20
2
2
-
-
2
-
-
-
-
T
=
2
-
10
1
T
20
2
2
-
-
2
-
-
-
-
L
=
3
-
5
1
L
12
3
3
-
-
-
3
-
-
-
M
=
4
-
7
1
M
13
4
4
-
-
-
-
4
-
-
E
=
5
-
3
1
E
5
5
5
-
-
-
-
-
5
-
E
=
5
-
4
1
E
5
5
5
-
-
-
-
-
5
-
E
=
5
-
6
1
E
5
5
5
-
-
-
-
-
5
-
E
=
5
-
8
1
E
5
5
5
-
-
-
-
-
5
-
N
=
5
-
9
1
N
14
5
5
-
-
-
-
-
5
-
H
=
3
-
2
1
H
8
8
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
1
4
3
4
25
8
-
-
-
-
-
-
THE ELEMENTS
-
-
-
-
-
-
-
-
2+5
-
T
=
2
4
4
3
THE
33
15
6
4
1
4
3
4
7
8
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
S
-
7
4
4
11
THE ELEMENTS
126
45
9
4
1
4
3
4
7
8
-
-
-
4
4
1+1
-
1+2+6
4+5
-
-
-
-
-
-
-
-
S
-
7
4
4
2
THE ELEMENTS
9
9
9
4
1
4
3
4
7
8

 

Sacred Geometry is the theory of dimensional evolution which assumes the ... door to his academy stating,

"Let no one unacquainted with geometry enter here.". ...
ezinearticles.com/?An-Overview-Of-Sacred-Geometry

 

AN OVERVIEW OF SACRED GEOMETRY

Gregg Hall

Sacred Geometry is the theory of dimensional evolution which assumes the universe is a living system kept together by the existence of a sacred geometry that encompasses the entire cosmos and makes for the blueprint for the mani-festation of what we know as our material universe and in addition organizes the context through which all love evolves.

Our universe was designed to be highly efficient and is capable of performing a wide range of multiple functions at the same time. The very same geometry which provides structure to physical reality also allows for the perceptual environments that people and civilizations must move through as part of a systematic learning process on the path towardsevolution.

Each dimension of this sacred geometry holds a unique place of perceptual space and a context of learning both for personal and social evolution. As each new dimension appears a new set of perceptions and potentials is awakened which we are free to accept and actualize or ignore. It is in understanding the dimensional structure which exists all around us that allows us to be able to understand the path and direction of personal and social evolution.

Even though our modern science generally believes there is nothing of deeper meaning to the dimensional geometry of the universe other than the actual physical aspects, there is a view that is almost diametrically opposed to this that began with the Greek philosopher Pythagoras in 500 B.C. Pythagoras believed and taught the theory or belief that all of the mathematical patterns in the universe were actually expressions of divine intelligence and signified a divine intention.

According to Pythagoras, we are surrounded by organizational intelligence that is shown in its purest from through mathematical formulas and musical harmonies and allowing ourselves to be at the center of our experience; we can know and share the organizing patterns and principles that pervade the universe. This is a thought that was even held by Albert Einstein, who stated that he received his greatest breakthroughs after praying and sleeping. The answers to the questions he was seeking came to him from the Universe while he slept! This is also the way that it can be explained for someone who is blind to be able to sculpt and for a deaf person, such as Beethoven to be able to compose intricate musical scores.

Plato, who taught over a hundred years after Pythagoras, continued in the teachings of Pythagorean thought in espousing that the universe or cosmos as Pythagoras termed it was a place of "harmonious and beautiful order" and placed such a high regard on geometry that he placed a sign above the door to his academy stating, "Let no one unacquainted with geometry enter here."

 

 

L
=
3
3
LET
37
10
1
N
=
5
2
NO
29
11
2
O
=
6
3
ONE
34
16
7
E
=
5
5
ENTER
62
26
8
H
=
8
4
HERE
36
27
9
W
=
5
3
WHO
46
19
1
I
=
9
2
IS
28
19
1
U
=
3
11
UNAQUAINTED
127
46
1
W
=
5
4
WITH
60
24
6
G
=
7
8
GEOMETRY
108
45
9
-
-
56
45
First Total
567
243
45
-
-
5+6
4+5
Add to Reduce
5+6+7
2+4+3
4+5
-
-
11
9
Second Total
18
9
9
-
-
1+1
-
Reduce to Deduce
1+8
-
-
-
-
2
9
Essence of Number
9
9
9

 

 

9
GEOGRAPHY
-
-
-
-
G
7
7
7
-
E
5
5
5
-
O
15
6
6
-
G
7
7
7
-
R
18
9
9
-
A
1
1
1
-
P
16
7
7
-
H
8
8
8
-
Y
25
7
7
9
GEOGRAPHY
102
57
57
-
-
1+0+2
5+7
5+7
9
GEOGRAPHY
12
12
12
-
-
1+2
1+2
1+2
9
GEOGRAPHY
3
3
3

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
M
=
4
-
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
Y
=
7
-
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
45
-
8
GEOMETRY
108
45
45
-
1
2
3
4
10
6
14
8
9
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
-
-
1+0
-
1+4
-
-
-
-
9
-
8
GEOMETRY
9
9
9
-
1
2
3
4
1
6
5
8
9

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
1
T
20
2
2
-
-
2
3
-
-
-
-
8
-
M
=
4
-
1
M
13
4
4
-
-
-
3
4
-
-
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
3
-
5
-
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
3
-
5
-
-
8
-
O
=
6
-
1
O
15
6
6
-
-
-
3
-
-
6
-
8
-
G
=
7
-
1
G
7
7
7
-
-
-
3
-
-
-
7
8
-
Y
=
7
-
1
Y
25
7
7
-
-
-
3
-
-
-
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
3
-
-
-
-
8
9
-
-
45
-
8
GEOMETRY
108
45
45
-
1
2
3
4
10
6
14
8
9
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
-
-
1+0
-
1+4
-
-
-
-
9
-
8
GEOMETRY
9
9
9
-
1
2
3
4
1
6
5
8
9

 

 

-
-
-
-
-
GEOMETRY
-
-
-
-
2
4
5
6
7
9
T
=
2
-
1
T
20
2
2
-
2
-
-
-
-
-
M
=
4
-
1
M
13
4
4
-
-
4
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
5
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
5
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
6
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
7
-
Y
=
7
-
1
Y
25
7
7
-
-
-
-
-
7
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
45
-
8
GEOMETRY
108
45
45
-
2
4
10
6
14
9
-
-
4+5
-
-
-
1+0+8
4+5
4+5
-
-
-
1+0
-
1+4
-
-
-
9
-
8
GEOMETRY
9
9
9
-
2
4
1
6
5
9

 

 

-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
-
15
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
7
5
-
4
5
2
9
25
+
=
39
3+9
=
12
1+2
3
=
3
-
-
7
5
-
13
5
20
18
7
+
=
93
9+3
=
12
1+2
3
=
3
-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
7
5
15
13
5
20
18
25
+
=
108
1+0+8
=
9
=
9
=
9
-
-
7
5
6
4
5
2
9
7
+
=
45
4+5
=
9
=
9
=
9
-
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
5
-
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
12
8
G
E
O
M
E
T
R
Y
-
-
33
-
-
8
-
45
-
27
1+2
-
7
5
6
4
5
2
9
7
-
-
3+3
-
-
-
-
4+5
-
2+7
3
8
G
E
O
M
E
T
R
Y
-
-
6
-
-
8
-
9
-
9

 

 

8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
+
=
6
-
=
6
=
6
=
6
-
-
-
15
-
-
-
-
-
+
=
15
1+5
=
6
=
6
=
6
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
7
5
-
4
5
2
9
25
+
=
39
3+9
=
12
1+2
3
=
3
-
7
5
-
13
5
20
18
7
+
=
93
9+3
=
12
1+2
3
=
3
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
7
5
15
13
5
20
18
25
+
=
108
1+0+8
=
9
=
9
=
9
-
7
5
6
4
5
2
9
7
+
=
45
4+5
=
9
=
9
=
9
8
G
E
O
M
E
T
R
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
5
-
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
7
-
-
-
-
-
-
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
8
G
E
O
M
E
T
R
Y
-
-
33
-
-
8
-
45
-
27
-
7
5
6
4
5
2
9
7
-
-
3+3
-
-
-
-
4+5
-
2+7
8
G
E
O
M
E
T
R
Y
-
-
6
-
-
8
-
9
-
9

 

 

-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
-
-
9
-
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
15
-
-
-
-
9
-
-
-
+
=
24
2+4
=
6
=
6
=
6
-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
7
5
-
4
5
2
9
-
3
1
3
+
=
39
3+9
=
12
1+2
3
=
3
-
-
7
5
-
13
5
20
18
-
3
1
12
+
=
84
8+4
=
12
1+2
3
=
3
-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
7
5
15
13
5
20
18
9
3
1
12
+
=
108
1+0+8
=
9
=
9
=
9
-
-
7
5
6
4
5
2
9
9
3
1
3
+
=
54
5+4
=
9
=
9
=
9
-
11
G
E
O
M
E
T
R
I
C
A
L
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
3
-
3
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
5
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
11
G
E
O
M
E
T
R
I
C
A
L
-
-
37
-
-
11
-
54
-
36
-
1+1
-
-
-
-
-
-
9
9
-
-
-
-
-
1+0
-
-
1+1
-
5+4
-
3+6
8
2
G
E
O
M
E
T
R
I
C
A
L
-
-
1
-
-
2
-
9
-
9

 

 

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

 

 

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

 

 

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

 

 

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

 

 

G
=
7
3
GEO
27
18
9
G
=
7
8
GEOMETRY
108
45
9
G
=
7
11
GEOMETRICAL
108
54
9
G
=
7
10
GEOCENTRIC
99
54
9

 

 

-
-
-
-
-
GEOGRAPHY
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
P
=
7
-
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
Y
=
7
-
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
-
-
57
-
9
GEOGRAPHY
102
57
57
-
1
2
3
4
5
6
28
8
9
-
-
5+7
-
-
-
1+0+2
5+7
5+7
-
-
-
-
-
-
-
2+8
-
-
-
-
12
-
9
GEOGRAPHY
3
12
12
-
1
2
3
4
5
6
10
8
9
-
-
1+2
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
1+0
-
-
-
-
3
-
9
GEOGRAPHYY
3
3
3
-
1
2
3
4
5
6
1
8
9

 

 

-
-
-
-
-
GEOGRAPHY
-
-
-
-
1
2
3
4
5
6
7
8
9
G
=
7
-
1
G
7
7
7
-
-
2
3
4
-
-
7
-
-
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
2
3
4
-
6
-
-
-
G
=
7
-
1
G
7
7
7
-
-
2
3
4
-
-
7
-
-
R
=
9
-
1
R
18
9
9
-
-
2
3
4
-
-
-
-
9
A
=
1
-
1
A
1
1
1
-
1
2
3
4
-
-
-
-
-
P
=
7
-
1
P
16
7
7
-
-
2
3
4
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
2
3
4
-
-
-
8
-
Y
=
7
-
1
Y
25
7
7
-
-
2
3
4
-
-
7
-
-
-
-
57
-
9
GEOGRAPHY
102
57
57
-
1
2
3
4
5
6
28
8
9
-
-
5+7
-
-
-
1+0+2
5+7
5+7
-
-
-
-
-
-
-
2+8
-
-
-
-
12
-
9
GEOGRAPHY
3
12
12
-
1
2
3
4
5
6
10
8
9
-
-
1+2
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
1+0
-
-
-
-
3
-
9
GEOGRAPHY
3
3
3
-
1
2
3
4
5
6
1
8
9

 

 

-
-
-
-
-
GEOGRAPHY
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
2
3
4
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
2
3
4
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
2
3
4
-
6
-
-
-
G
=
7
-
1
G
7
7
7
-
-
2
3
4
-
-
7
-
-
G
=
7
-
1
G
7
7
7
-
-
2
3
4
-
-
7
-
-
P
=
7
-
1
P
16
7
7
-
-
2
3
4
-
-
7
-
-
Y
=
7
-
1
Y
25
7
7
-
-
2
3
4
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
2
3
4
-
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
2
3
4
-
-
-
-
9
-
-
57
-
9
GEOGRAPHY
102
57
57
-
1
2
3
4
5
6
28
8
9
-
-
5+7
-
-
-
1+0+2
5+7
5+7
-
-
-
-
-
-
-
2+8
-
-
-
-
12
-
9
GEOGRAPHY
3
12
12
-
1
2
3
4
5
6
10
8
9
-
-
1+2
-
-
-
-
1+2
1+2
-
-
-
-
-
-
-
1+0
-
-
-
-
3
-
9
GEOGRAPHY
3
3
3
-
1
2
3
4
5
6
1
8
9

 

 

-
-
-
-
-
GEOGRAPHY
-
-
-
-
1
5
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
6
-
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
7
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
7
-
-
P
=
7
-
1
P
16
7
7
-
-
-
-
7
-
-
Y
=
7
-
1
Y
25
7
7
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
57
-
9
GEOGRAPHY
102
57
57
-
1
5
6
28
8
9
-
-
5+7
-
-
-
1+0+2
5+7
5+7
-
-
-
-
2+8
-
-
-
-
12
-
9
GEOGRAPHY
3
12
12
-
1
5
6
10
8
9
-
-
1+2
-
-
-
-
1+2
1+2
-
-
-
-
1+0
-
-
-
-
3
-
9
GEOGRAPHY
3
3
3
-
1
5
6
1
8
9

 

 

G
=
7
3
GEO
27
18
9
G
=
7
8
GEOMETRY
108
45
9
G
=
7
11
GEOMETRICAL
108
54
9
G
=
7
10
GEOCENTRIC
99
54
9

 

 

-
SQUARE
-
-
-
2
S+Q
36
18
9
2
U+A
22
4
4
1
R
18
9
9
1
E
5
5
5
6
SQUARE
81
36
27
-
-
8+1
3+6
2+7
6
SQUARE
9
9
9

 

 

-
6
S
Q
U
A
R
E
-
-
--
-
-
-
-
-
-
-
19
17
21
1
18
5
+
=
81
8+1
=
9
=
9
-
-
1
8
3
1
9
5
+
=
27
2+7
=
9
=
9
-
6
S
Q
U
A
R
E
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
occurs
x
2
=
2
2
-
--
--
--
--
--
--
--
--
2
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
4
-
--
--
--
--
--
--
--
--
4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
--
--
--
--
--
--
-
-
6
-
-
-
-
-
7
-
--
--
--
--
--
--
--
--
7
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
-
--
--
9
occurs
x
1
=
9
19
6
S
Q
U
A
R
E
-
-
26
-
-
6
-
27
10
-
-
-
-
-
9
-
-
-
2+6
-
-
-
-
2+7
1
6
S
Q
U
A
R
E
-
-
8
-
-
6
-
9

 

 

6
S
Q
U
A
R
E
-
-
--
-
-
-
-
-
-
19
17
21
1
18
5
+
=
81
8+1
=
9
=
9
-
1
8
3
1
9
5
+
=
27
2+7
=
9
=
9
6
S
Q
U
A
R
E
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
occurs
x
2
=
2
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
6
S
Q
U
A
R
E
-
-
26
-
-
6
-
27
-
-
-
-
-
9
-
-
-
2+6
-
-
-
-
2+7
6
S
Q
U
A
R
E
-
-
8
-
-
6
-
9

 

 

-
-
-
-
-
TRIANGLE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
L
=
6
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
41
-
8
TRIANGLE
86
41
41
-
1
2
3
4
10
6
7
8
18
-
-
4+1
-
-
-
8+6
4+1
4+1
-
-
-
-
-
1+0
-
-
-
1+8
-
-
5
-
8
TRIANGLE
14
5
5
-
1
2
3
4
1
6
7
8
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
8
TRIANGLE
5
5
5
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
TRIANGLE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
I
=
9
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
A
=
1
-
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
N
=
5
-
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
G
=
7
-
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
L
=
6
-
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
-
-
41
-
8
TRIANGLE
86
41
41
-
1
2
3
4
10
6
7
8
18
-
-
4+1
-
-
-
8+6
4+1
4+1
-
-
-
-
-
1+0
-
-
-
1+8
-
-
5
-
8
TRIANGLE
14
5
5
-
1
2
3
4
1
6
7
8
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
8
TRIANGLE
5
5
5
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
TRIANGLE
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
-
-
4
-
6
-
8
-
T
=
2
-
1
T
20
2
2
-
-
2
-
4
-
6
-
8
-
L
=
6
-
1
L
12
3
3
-
-
-
3
4
-
6
-
8
-
N
=
5
-
1
N
14
5
5
-
-
-
-
4
5
6
-
8
-
E
=
5
-
1
E
5
5
5
-
-
-
-
4
5
6
-
8
-
G
=
7
-
1
G
7
7
7
-
-
-
-
4
-
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
4
-
6
-
8
9
I
=
9
-
1
I
9
9
9
-
-
-
-
4
-
6
-
8
9
-
-
41
-
8
TRIANGLE
86
41
41
-
1
2
3
4
10
6
7
8
18
-
-
4+1
-
-
-
8+6
4+1
4+1
-
-
-
-
-
1+0
-
-
-
1+8
-
-
5
-
8
TRIANGLE
14
5
5
-
1
2
3
4
1
6
7
8
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
8
TRIANGLE
5
5
5
-
1
2
3
4
1
6
7
8
9

 

 

-
-
-
-
-
TRIANGLE
-
-
-
-
1
2
3
5
7
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
L
=
6
-
1
L
12
3
3
-
-
-
3
-
-
-
N
=
5
-
1
N
14
5
5
-
-
-
-
5
-
-
E
=
5
-
1
E
5
5
5
-
-
-
-
5
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
-
7
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
I
=
9
-
1
I
9
9
9
-
-
-
-
-
-
9
-
-
41
-
8
TRIANGLE
86
41
41
-
1
2
3
10
7
18
-
-
4+1
-
-
-
8+6
4+1
4+1
-
-
-
-
1+0
-
1+8
-
-
5
-
8
TRIANGLE
14
5
5
-
1
2
3
1
7
9
-
-
-
-
-
-
1+4
-
-
-
-
-
-
-
-
-
-
-
5
-
8
TRIANGLE
5
5
5
-
1
2
3
1
7
9

 

 

WHY SMASH ATOMS

A. K. Solomon 1940

VAN DE GRAAFF GENERATOR

Page 77

"Once the fairy tale hero has penetrated the ring of fire round The Magic Mountain he is free to woo the heroine in her castle on the mountain top."

 

 

SACRED GEOMETRY

Miranda Lundy 2006

CIRCLING THE SQUARE

the marriage of heaven and earth

 

T
=
2
-
3
THE
33
15
6
M
=
4
-
8
MARRIAGE
72
45
9
O
=
6
-
2
OF
21
12
3
H
=
8
-
6
HEAVEN
55
28
1
A
=
1
-
3
AND
19
10
1
E
=
5
-
5
EARTH
52
25
7
-
-
26
-
27
Add to Reduce
252
135
27
-
-
2+6
-
2+7
Reduce to Deduce
2+5+2
1+3+5
2+7
-
-
8
-
9
Essence of Number
9
9
9

 

The circle is the shape traditionally assigned to the Heavens, and the square to the Earth. When these two shapes are unified by being made equal in area or perimeter we speak of 'squaring the circle', meaning that Heaven and Earth, or Spirit and Matter, are symbolically combined, or married. Fivefold Man exists between sixfold Heaven and fourfold Earth and Leonardo da Vinci's image opposite also shows how a man's span equals his height, that this measure equals seven of his feet and other important ratios.
Amazingly, the Earth and the Moon square the circle, for if the Moon (diameter 3) is drawn down to the Earth (diameter II) then a heavenly circle through the Moon (dotted below centre) has radius 7, and so circumference 44, the same as the perimeter of the square around the Earth. This works because irjelating the circumference of a circle to its diameter is practically 22/7. In Leonardo's image the Moon would fit above the man's head.
Also shown (below left and right) is a construction for a square using ruler and compasses. Octagons soon follow.

 

C
=
3
-
8
CIRCLING
75
48
3
T
=
2
-
3
THE
33
15
6
S
=
1
-
6
SQUARE
81
36
9
-
-
6
-
45
First Total
189
99
18
-
-
-
-
4+5
Add to Reduce
1+8+9
9+9
1+8
-
-
6
-
9
Second Total
18
18
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
6
-
9
Essence of Number
9
9
9

 

 

S
=
1
-
8
SQUARING
106
52
7
T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CIRCLE
50
32
5
-
-
6
-
17
First Total
189
99
18
-
-
-
-
1+7
Add to Reduce
1+8+9
9+9
1+8
-
-
6
-
8
Second Total
18
18
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
6
-
8
Essence of Number
9
9
9

 

 

-
17
S
Q
U
A
R
I
N
G
-
T
H
E
C
I
R
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
-
9
5
-
-
-
8
-
-
9
-
-
-
-
+
=
32
3+2
=
5
=
5
=
5
-
-
19
-
-
-
-
9
14
-
-
-
8
-
-
9
-
-
-
-
+
=
59
5+9
=
14
1+4
5
=
5
-
17
S
Q
U
A
R
I
N
G
-
T
H
E
C
I
R
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
=
8
3
1
9
-
-
7
-
2
-
5
3
-
9
3
3
5
+
=
58
5+8
=
13
1+3
4
=
4
-
-
-
17
21
1
18
-
-
7
-
20
-
5
3
-
18
3
12
5
+
=
130
1+3+0
=
4
=
4
=
4
-
17
S
Q
U
A
R
I
N
G
-
T
H
E
C
I
R
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
19
17
21
1
18
9
14
7
-
20
8
5
3
9
18
3
12
5
+
=
189
1+8+9
=
18
1+8
9
=
9
-
-
1
8
3
1
9
9
5
7
-
2
8
5
3
9
9
3
3
5
+
=
90
9+0
=
9
=
9
=
9
-
17
S
Q
U
A
R
I
N
G
-
T
H
E
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
-
--
-
-
--
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
4
=
36
3+6
9
10
17
S
Q
U
A
R
I
N
G
-
T
H
E
C
I
R
C
L
E
-
-
35
-
-
17
-
90
-
36
1+0
1+7
--
-
-
--
9
9
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
3+5
-
-
1+7
-
9+0
-
3+6
1
8
S
Q
U
A
R
I
N
G
-
T
H
E
C
I
R
C
L
E
-
-
8
-
-
8
-
9
-
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
8
SQUARING
106
52
7
-
-
-
-
-
-
-
7
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
5
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
1+7
Add to Reduce
1+8+9
9+9
1+8
-
-
-
-
-
-
-
-
-
-
-
-
6
-
8
Second Total
18
18
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
8
Essence of Number
9
9
9
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
8
SQUARING
106
52
7
-
-
-
-
-
-
-
7
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
5
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
4
-
6
-
-
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
4
-
6
-
8
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
6
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
4
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
4
-
6
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
N
=
5
7
1
N
14
5
5
-
-
-
-
4
5
6
-
-
-
G
=
7
8
1
G
7
7
7
-
-
-
-
4
-
6
7
-
-
43
-
8
-
106
52
43
-
-
-
-
-
-
-
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
4
-
6
-
-
-
H
=
8
10
1
H
8
8
8
-
-
-
-
4
-
6
-
8
-
E
=
5
11
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
15
-
3
-
33
15
15
-
-
-
-
-
-
-
-
-
C
=
3
12
1
C
3
3
3
-
-
-
3
4
-
6
-
-
-
I
=
9
13
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
R
=
9
14
1
R
18
9
9
-
-
-
-
4
-
6
-
-
9
C
=
3
15
1
C
3
3
3
-
-
-
3
4
-
6
-
-
-
L
=
3
16
1
L
12
3
3
-
-
-
3
4
-
6
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
32
-
6
-
50
32
32
-
2
2
12
4
15
6
7
16
36
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
1+5
-
-
1+6
3+6
S
=
1
-
8
SQUARING
106
52
7
-
2
2
3
4
6
6
7
7
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
2
2
3
4
6
6
7
7
9
-
-
-
-
2+6
Add to Reduce
1+8+9
9+9
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Second Total
18
18
9
-
2
2
3
4
6
6
7
7
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Essence of Number
9
9
9
-
2
2
3
4
6
6
7
7
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
8
SQUARING
106
52
7
-
-
-
-
-
-
-
7
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
5
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
4
-
6
-
-
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
4
-
6
-
8
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
6
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
4
-
6
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
4
-
6
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
N
=
5
7
1
N
14
5
5
-
-
-
-
4
5
6
-
-
-
G
=
7
8
1
G
7
7
7
-
-
-
-
4
-
6
7
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
4
-
6
-
-
-
H
=
8
10
1
H
8
8
8
-
-
-
-
4
-
6
-
8
-
E
=
5
11
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
C
=
3
12
1
C
3
3
3
-
-
-
3
4
-
6
-
-
-
I
=
9
13
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
R
=
9
14
1
R
18
9
9
-
-
-
-
4
-
6
-
-
9
C
=
3
15
1
C
3
3
3
-
-
-
3
4
-
6
-
-
-
L
=
3
16
1
L
12
3
3
-
-
-
3
4
-
6
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
-
-
-
-
-
-
-
-
2
2
12
4
15
6
7
16
36
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
1+5
-
-
1+6
3+6
S
=
1
-
8
SQUARING
106
52
7
-
2
2
3
4
6
6
7
7
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
2
2
3
4
6
6
7
7
9
-
-
-
-
2+6
Add to Reduce
1+8+9
9+9
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Second Total
18
18
9
-
2
2
3
4
6
6
7
7
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Essence of Number
9
9
9
-
2
2
3
4
6
6
7
7
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
-
8
SQUARING
106
52
7
-
-
-
-
-
-
-
7
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
6-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
5
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
4
-
6
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
4
-
6
-
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
4
-
6
-
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
4
-
6
-
-
-
C
=
3
12
1
C
3
3
3
-
-
-
3
4
-
6
-
-
-
C
=
3
15
1
C
3
3
3
-
-
-
3
4
-
6
-
-
-
L
=
3
16
1
L
12
3
3
-
-
-
3
4
-
6
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
4
5
6
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
4
5
6
-
-
-
G
=
7
8
1
G
7
7
7
-
-
-
-
4
-
6
7
-
-
H
=
8
10
1
H
8
8
8
-
-
-
-
4
-
6
-
8
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
4
-
6
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
4
-
6
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
-
4
-
6
-
-
9
R
=
9
14
1
R
18
9
9
-
-
-
-
4
-
6
-
-
9
-
-
-
-
-
-
-
-
2
2
12
4
15
6
7
16
36
-
-
-
-
-
-
-
-
-
-
-
-
1+2
-
1+5
-
-
1+6
3+6
S
=
1
-
8
SQUARING
106
52
7
-
2
2
3
4
6
6
7
7
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
2
2
3
4
6
6
7
7
9
-
-
-
-
2+6
Add to Reduce
1+8+9
9+9
1+8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Second Total
18
18
9
-
2
2
3
4
6
6
7
7
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Essence of Number
9
9
9
-
2
2
3
4
6
6
7
7
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
8
9
S
=
1
-
8
SQUARING
106
52
7
-
-
-
-
-
7
-
-
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
5
-
-
-
-
-
6
-
17
First Total
189
99
18
-
1
2
3
5
7
8
9
-
-
-
-
-
-
-
-
-
-
1
2
3
5
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
A
=
1
4
1
A
1
1
1
-
1
-
-
-
-
-
-
T
=
2
9
1
T
20
2
2
-
-
2
-
-
-
-
-
U
=
3
3
1
U
21
3
3
-
-
-
3
-
-
-
-
C
=
3
12
1
C
3
3
3
-
-
-
3
-
-
-
-
C
=
3
15
1
C
3
3
3
-
-
-
3
-
-
-
-
L
=
3
16
1
L
12
3
3
-
-
-
3
-
-
-
-
N
=
5
7
1
N
14
5
5
-
-
-
-
5
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
5
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
5
-
-
-
G
=
7
8
1
G
7
7
7
-
-
-
-
-
7
-
-
H
=
8
10
1
H
8
8
8
-
-
-
-
-
-
8
-
Q
=
8
2
1
Q
17
8
8
-
-
-
-
-
-
8
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
9
R
=
9
14
1
R
18
9
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
2
2
12
15
7
16
36
-
-
-
-
-
-
-
-
-
-
-
-
1+2
1+5
-
1+6
3+6
S
=
1
-
8
SQUARING
106
52
7
-
2
2
3
6
7
7
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
C
=
3
-
6
CIRCLE
50
32
5
-
-
-
-
-
-
-
-
-
-
6
-
17
First Total
189
99
18
-
2
2
3
6
7
7
9
-
-
-
-
2+6
Add to Reduce
1+8+9
9+9
1+8
-
-
-
-
-
-
-
-
-
-
9
-
8
Second Total
18
18
9
-
2
2
3
6
7
7
9
-
-
-
-
-
Reduce to Deduce
1+8
1+8
-
-
-
-
-
-
-
-
-
-
-
9
-
8
Essence of Number
9
9
9
-
2
2
3
6
7
7
9

 

 

-
6
S
Q
U
A
R
E
-
-
--
-
-
-
-
-
-
-
1
-
-
--
-
-
+
=
1
--
=
1
=
1
-
-
19
-
-
--
-
-
+
=
19
1+9
=
10
1+0
1
-
6
S
Q
U
A
R
E
-
-
-
-
-
-
-
-
-
-
-
8
3
1
9
5
+
=
26
2+6
=
8
=
8
-
-
-
17
21
1
18
5
+
=
62
6+2
=
8
=
8
-
6
S
Q
U
A
R
E
-
-
--
-
-
-
-
-
-
-
19
17
21
1
18
5
+
=
81
8+1
=
9
=
9
-
-
1
8
3
1
9
5
+
=
27
2+7
=
9
=
9
-
6
S
Q
U
A
R
E
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
occurs
x
2
=
2
2
-
--
--
--
--
--
--
--
--
2
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
4
-
--
--
--
--
--
--
--
--
4
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
--
--
--
--
--
--
--
--
6
-
6
-
-
-
7
-
--
--
--
--
--
--
--
--
7
-
7
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
19
6
S
Q
U
A
R
E
-
-
26
-
-
6
-
27
1+9
-
-
-
-
-
9
-
-
-
2+6
-
-
-
-
2+7
10
6
S
Q
U
A
R
E
-
-
7
-
-
6
-
9
1+0
-
1
8
3
1
9
5
-
-
-
-
-
-
-
-
1
6
S
Q
U
A
R
E
-
-
7
-
-
6
-
9

 

 

6
S
Q
U
A
R
E
-
-
--
-
-
-
-
-
-
1
-
-
--
-
-
+
=
1
--
=
1
=
1
-
19
-
-
--
-
-
+
=
19
1+9
=
10
1+0
1
6
S
Q
U
A
R
E
-
-
-
-
-
-
-
-
-
-
8
3
1
9
5
+
=
26
2+6
=
8
=
8
-
-
17
21
1
18
5
+
=
62
6+2
=
8
=
8
6
S
Q
U
A
R
E
-
-
--
-
-
-
-
-
-
19
17
21
1
18
5
+
=
81
8+1
=
9
=
9
-
1
8
3
1
9
5
+
=
27
2+7
=
9
=
9
6
S
Q
U
A
R
E
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
1
occurs
x
2
=
2
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
6
S
Q
U
A
R
E
-
-
26
-
-
6
-
27
-
-
-
-
-
9
-
-
-
2+6
-
-
-
-
2+7
6
S
Q
U
A
R
E
-
-
7
-
-
6
-
9
-
1
8
3
1
9
5
-
-
-
-
-
-
-
-
6
S
Q
U
A
R
E
-
-
7
-
-
6
-
9

 

 

-
CYLINDER
-
-
-
3
C+Y+L
40
13
4
1
I
9
9
9
1
N
14
5
5
2
D+E
9
9
9
1
R
18
9
9
8
CYLINDER
81
36
27
-
-
8+1
3+6
2+7
8
CYLINDER
9
9
9

 

 

-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
9
5
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
-
-
9
14
-
-
-
+
=
23
2+3
=
5
=
5
=
5
-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
3
7
3
-
-
4
5
9
+
=
31
3+1
=
4
=
4
=
4
-
-
3
25
12
-
-
4
5
18
+
=
67
6+7
=
13
1+3
4
=
4
-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
3
25
12
9
14
4
5
18
+
=
90
9+0
=
9
=
9
=
9
-
-
3
7
3
9
5
4
5
9
+
=
45
4+5
=
9
=
9
=
9
-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
17
8
C
Y
L
I
N
D
E
R
-
-
28
-
-
8
-
45
-
27
1+7
-
-
 -
-
9
-
-
-
9
-
-
2+8
-
-
-
-
4+5
-
2+7
8
8
C
Y
L
I
N
D
E
R
-
-
10
-
-
8
-
9
-
9

 

 

-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
9
5
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
-
-
9
14
-
-
-
+
=
23
2+3
=
5
=
5
=
5
-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
3
7
3
-
-
4
5
9
+
=
31
3+1
=
4
=
4
=
4
-
-
3
25
12
-
-
4
5
18
+
=
67
6+7
=
13
1+3
4
=
4
-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
3
25
12
9
14
4
5
18
+
=
90
9+0
=
9
=
9
=
9
-
-
3
7
3
9
5
4
5
9
+
=
45
4+5
=
9
=
9
=
9
-
8
C
Y
L
I
N
D
E
R
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
9
occurs
x
2
=
18
1+8
9
17
8
C
Y
L
I
N
D
E
R
-
-
28
-
-
8
-
45
-
27
1+7
-
-
 -
-
9
-
-
-
9
-
-
2+8
-
-
-
-
4+5
-
2+7
8
8
C
Y
L
I
N
D
E
R
-
-
10
-
-
8
-
9
-
9

 

 

-
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
22
1
3
21
21
13
+
=
81
8+1
=
9
=
9
-
-
4
1
3
3
3
4
+
=
18
1+8
=
9
=
9
-
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
3
occurs
x
3
=
9
-
-
4
-
-
-
-
4
-
-
4
occurs
x
2
=
8
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
37
6
V
A
C
U
U
M
-
-
8
-
-
6
-
18
10
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+8
1
6
V
A
C
U
U
M
-
-
8
-
-
6
-
9

 

 

6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
22
1
3
21
21
13
+
=
81
8+1
=
9
=
9
-
4
1
3
3
3
4
+
=
18
1+8
=
9
=
9
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
3
3
3
-
-
-
3
occurs
x
3
=
9
-
4
-
-
-
-
4
-
-
4
occurs
x
2
=
8
6
V
A
C
U
U
M
-
-
8
-
-
6
-
18
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+8
6
V
A
C
U
U
M
-
-
8
-
-
6
-
9

 

 

-`
4
C
U
B
E
-
-
-
-
-
--
-
-
-
-
3
21
2
5
+
=
31
3+1
=
4
=
4
-
-
3
3
2
5
+
=
13
1+3
=
4
=
4
-
4
C
U
B
E
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
--
-
-
-
2
-
-
-
2
occurs
x
1
=
2
-
-
3
3
-
-
-
-
3
occurs
x
2
=
6
4
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
35
4
C
U
B
E
-
-
10
-``
-
4
-
13
3+5
-
-
-
-
-
-
-
1+0
-
-
-
-
1+3
8
4
C
U
B
E
-
-
1
-
-
4
-
4

 

 

4
C
U
B
E
-
-
-
-
-
--
-
-
-
3
21
2
5
+
=
31
3+1
=
4
=
4
-
3
3
2
5
+
=
13
1+3
=
4
=
4
4
C
U
B
E
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
2
occurs
x
1
=
2
-
3
3
-
-
-
-
3
occurs
x
2
=
6
-
-
-
-
5
-
-
5
occurs
x
1
=
5
4
C
U
B
E
-
-
10
-``
-
4
-
13
-
-
-
-
-
-
-
1+0
-
-
-
-
1+3
4
C
U
B
E
-
-
1
-``
-
4
-
4

 

 

-
6
O
B
L
O
N
G
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
6
5
-
+
=
17
1+7
=
8
=
8
-
8
-
-
15
-
-
15
14
-
+
=
44
4+4
=
8
=
8
-
8
-
6
O
B
L
O
N
G
-
-
-
-
-
--
-
-
-
-
-
-
-
2
3
-
-
7
+
=
12
1+2
=
3
=
3
-
3
-
-
-
2
12
-
-
7
+
=
21
2+1
=
3
=
3
-
3
-
6
O
B
L
O
N
G
-
-
-
-
-
--
-
-
-
-
-
-
15
2
12
15
14
7
+
=
65
6+5
=
11
1+1
2
-
2
-
-
6
2
3
6
5
7
+
=
29
2+9
=
11
1+1
2
-
2
-
6
O
B
L
O
N
G
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
--
-
3
--
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
=
5
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
22
6
O
B
L
O
N
G
-
-
23
-
-
6
-
29
-
20
2+2
-
-
-
-
-
-
-
-
-
2+3
-
-
-
-
2+9
-
2+0
4
6
O
B
L
O
N
G
-
-
5
-
-
6
-
11
-
2
-`
-
6
2
3
6
5
7
-
-
-
-
-
-
-
1+1
-
-
3
6
O
B
L
O
N
G
-
-
5
-
-
6
-
2
-
2

 

 

-
6
O
B
L
O
N
G
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
6
5
-
+
=
17
1+7
=
8
=
8
-
8
-
-
15
-
-
15
14
-
+
=
44
4+4
=
8
=
8
-
8
-
6
O
B
L
O
N
G
-
-
-
-
-
--
-
-
-
-
-
-
-
2
3
-
-
7
+
=
12
1+2
=
3
=
3
-
3
-
-
-
2
12
-
-
7
+
=
21
2+1
=
3
=
3
-
3
-
6
O
B
L
O
N
G
-
-
-
-
-
--
-
-
-
-
-
-
15
2
12
15
14
7
+
=
65
6+5
=
11
1+1
2
-
2
-
-
6
2
3
6
5
7
+
=
29
2+9
=
11
1+1
2
-
2
-
6
O
B
L
O
N
G
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
--
-
3
--
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
=
5
-
-
6
-
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
-
7
-
-
7
occurs
x
1
=
7
=
7
22
6
O
B
L
O
N
G
-
-
23
-
-
6
-
29
-
20
2+2
-
-
-
-
-
-
-
-
-
2+3
-
-
-
-
2+9
-
2+0
4
6
O
B
L
O
N
G
-
-
5
-
-
6
-
11
-
2
-`
-
6
2
3
6
5
7
-
-
-
-
-
-
-
1+1
-
-
3
6
O
B
L
O
N
G
-
-
5
-
-
6
-
2
-
2

 

 

-
-
-
-
-
FRACTALS
-
-
-
-
1
2
3
4
5
6
7
8
9
F
=
6
-
1
F
6
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
C
=
3
-
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
L
=
3
-
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
26
-
8
FRACTALS
80
35
26
-
3
2
6
4
5
6
7
8
9
-
-
2+6
-
-
-
8+0
3+5
2+6
-
-
-
-
-
-
-
-
-
-
-
-
8
-
8
FRACTALS
8
8
8
-
3
2
6
4
5
6
7
8
9

 

 

-
-
-
-
-
FRACTALS
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
1
A
1
1
1
-
1
-
-
4
5
-
7
8
-
A
=
1
-
1
A
1
1
1
-
1
-
-
4
5
-
7
8
-
S
=
1
-
1
S
19
10
1
-
1
-
-
4
5
-
7
8
-
T
=
2
-
1
T
20
2
2
-
-
2
-
4
5
-
7
8
-
C
=
3
-
1
C
3
3
3
-
-
-
3
4
5
-
7
8
-
L
=
3
-
1
L
12
3
3
-
-
-
3
4
5
-
7
8
-
F
=
6
-
1
F
6
6
6
-
-
-
-
4
5
6
7
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
4
5
-
7
8
9
-
-
26
-
8
FRACTALS
80
35
26
-
3
2
6
4
5
6
7
8
9
-
-
2+6
-
-
-
8+0
3+5
2+6
-
-
-
-
-
-
-
-
-
-
-
-
8
-
8
FRACTALS
8
8
8
-
3
2
6
4
5
6
7
8
9

 

 

-
-
-
-
-
FRACTALS
-
-
-
-
1
2
3
6
9
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
C
=
3
-
1
C
3
3
3
-
-
-
3
-
-
L
=
3
-
1
L
12
3
3
-
-
-
3
-
-
F
=
6
-
1
F
6
6
6
-
-
-
-
6
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
9
-
-
26
-
8
FRACTALS
80
35
26
-
3
2
6
6
9
-
-
2+6
-
-
-
8+0
3+5
2+6
-
-
-
-
-
-
-
-
8
-
8
FRACTALS
8
8
8
-
3
2
6
6
9

 

 

Golden ratio - Wikipedia

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

The Parthenon - In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. The figure on the right illustrates the geometric relationship. The golden ratio is also called the golden mean or golden section (Latin: sectio aurea).[1][2][3] Other names include extreme and mean ratio,[4] medial section, divine proportion, divine section (Latin: sectio divina), golden proportion, golden cut,[5] and golden number (not to be confused with the related Fibonacci numbers).[6][7][8]

Mathematicians since Euclid—and perhaps earlier—have studied the properties of the golden ratio, including its self-similarity and appearance in the dimensions of a regular pentagon and in a golden rectangle, which may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio appears in some patterns in nature, including the spiral arrangement of leaves and other plant parts. The golden ratio has also been used to analyze the proportions of natural objects as well as man-made systems such as financial markets, in some cases based on dubious fits to data.[9]

Some twentieth-century artists and architects, including Le Corbusier and Salvador Dalí, have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing.

 

What is the Golden Ratio? - Live Science
https://www.livescience.com › Tech

24 Jun 2013 - The Golden ratio is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part. It is often symbolized using phi, after the 21st letter of the Greek alphabet.

 

The designer's guide to the Golden Ratio | Creative Bloq
https://www.creativebloq.com/design/designers-guide-golden-ratio-12121546

24 Oct 2018 - We call it the Golden Ratio, although it's also known as the Golden Mean, The Golden Section, or the Greek letter phi. Whether you're an ...

 

Golden Ratio - Math is Fun
https://www.mathsisfun.com/numbers/golden-ratio.html

The golden ratio (symbol is the Greek letter "phi" shown at left) is a special number approximately equal to 1.618. It appears many times in geometry,
art, ...
phi
The golden ratio (symbol is the Greek letter "phi" shown at left)
is a special number approximately equal to 1.618

It appears many times in geometry, art, architecture and other areas.

 

The Golden Ratio
https://www.goldennumber.net/

Golden Ratio, Phi, 1.618, and Fibonacci in Math, Nature, Art, Design, Beauty and the Face. One source with over 100 articles and latest findings.
ancient Greeks, Renaissance artists, a 17th century astronomer and a 21st century novelist all would write about it? It's a number that goes by many names. This “golden” number, 1.61803399, represented by the Greek letter Phi, is known as the Golden Ratio, Golden Number, Golden Proportion, Golden Mean, Golden Section, Divine Proportion and Divine Section. It was written about by Euclid in “Elements” around 300 B.C., by Luca Pacioli, a contemporary of Leonardo Da Vinci, in "De Divina Proportione" in 1509, by Johannes Kepler around 1600 and by Dan Brown in 2003 in his best selling novel, “The Da Vinci Code.” With the movie release of the “The Da Vinci Code”, the quest to know Phi was brought even more into the mainstream of pop culture. The allure of “The Da Vinci Code” was that it creatively integrated fiction with both fact and myth from art, history, theology and mathematics, leaving the reader never really knowing what was truth and …

 

-
-
-
-
-
THE GOLDEN RATIO
-
-
-
T
=
2
-
3
THE
33
15
6
G
=
7
-
6
GOLDEN
57
30
3
R
=
9
-
5
RATIO
63
27
9
-
-
18
-
14
THE GOLDEN RATIO
153
72
18
-
-
1+8
-
1+4
-
1+5+3
7+2
1+8
-
-
9
-
5
THE GOLDEN RATIO
9
9
9

 

 

-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
6
-
-
-
5
-
-
-
-
9
6
+
=
34
3+4
=
7
-
7
=
7
-
-
-
8
-
-
-
15
-
-
-
14
-
-
-
-
9
15
+
=
61
6+1
=
7
-
7
=
7
-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
7
-
3
4
5
-
-
9
1
2
-
-
+
=
38
3+8
=
11
1+1
2
=
2
-
-
20
-
5
-
7
-
12
4
5
-
-
18
1
20
-
-
+
=
92
9+2
=
11
1+1
2
=
2
-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
-
-
20
8
5
-
7
15
12
4
5
14
-
18
1
20
9
15
+
=
153
1+5+3
=
9
=
9
=
9
-
-
2
8
5
-
7
6
3
4
5
5
-
9
1
2
9
6
+
=
72
7+2
=
9
=
9
=
9
-
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
2
-
-
-
-
-
-
--
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
2
3
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
-
-
-
-
--
5
5
-
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
--
-
-
-
9
-
-
9
-
-
-
9
occurs
x
2
=
18
1+8
9
45
14
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
45
-
-
14
-
72
-
36
4+5
1+4
-
-
5
-
-
-
-
--
5
5
-
-
-
-
-
-
-
-
4+5
-
-
1+4
-
7+2
-
3+6
9
5
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
9
-
-
5
-
9
-
9
-
-
2
8
5
-
7
6
3
4
5
5
-
9
1
2
9
6
-
-
-
-
-
-
-
-
-
-
9
5
T
H
E
-
G
O
L
D
E
N
-
R
A
T
I
O
-
-
9
-
-
5
-
9
-
9

 

 

THE GOLDEN RATIO

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
95
41
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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
-
-
-
-
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
-
-
-
-
N
=
5
9
1
N
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
10
1
R
9
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
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
72
4
14
THE GOLDEN RATIO
153
72
72
-
1
4
3
4
15
12
7
8
18
-
-
7+2
-
1+4
-
1+5+3
7+2
7+2
-
-
-
-
-
1+5
1+2
-
-
1+8
Q
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
95
41
5
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
R
=
9
-
5
RATIO
63
27
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
11
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
T
=
2
12
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
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
9
1
N
5
5
5
-
-
-
-
-
5
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
O
=
6
14
1
O
15
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
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
13
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
72
4
14
THE GOLDEN RATIO
153
72
72
-
1
4
3
4
15
12
7
8
18
-
-
7+2
-
1+4
-
1+5+3
7+2
7+2
-
-
-
-
-
1+5
1+2
-
-
1+8
Q
-
9
-
5
THE GOLDEN RATIO
9
9
9
-
1
4
3
4
6
3
7
8
9

 

 

THE GOLDEN MEAN

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
T
=
2
-
3
THE
33
15
6
G
=
7
-
6
GOLDEN
57
30
3
M
=
4
-
4
MEAN
33
15
6
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
4
-
4
THE GOLDEN MEAN
6
6
6

 

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
23
-
6
THE
50
32
23
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
GOLDEN
-
-
-
-
-
-
-
-
-
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
45
-
5
GOLDEN
108
45
45
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
MEAN
-
-
-
-
-
-
-
-
-
-
-
-
-
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
-
-
23
-
6
MEAN
50
32
23
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
8
25
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
2+5
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
1
2
3
8
7
6
7
8
9
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
8
7
6
7
8
9
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE GOLDEN MEAN
6
6
6
-
1
2
3
8
7
6
7
8
9

 

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
9
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
9
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
9
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
9
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
9
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
8
25
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
2+5
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
1
2
3
8
7
6
7
8
9
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
8
7
6
7
8
9
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE GOLDEN MEAN
6
6
6
-
1
2
3
8
7
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S LOOK AT THE 5FIVE5S THE 5FIVE5S THE 5FIVE5S

5 x 5 = 25

"The most common letter in the English alphabet is E."

 

-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
-
-
-
-
-
-
-
-
-
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
2
1
A
1
1
1
-
1
-
-
-
-
-
-
-
9
T
=
2
12
1
T
20
2
2
-
-
2
-
-
-
-
-
-
9
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
9
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
9
M
=
4
10
1
M
13
4
4
-
-
-
-
4
-
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
E
=
5
5
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
9
N
=
5
5
1
N
14
5
5
-
-
-
-
-
5
-
-
-
9
O
=
6
9
1
O
16
6
6
-
-
-
-
-
-
6
-
-
9
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
9
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
THE GOLDEN MEAN
-
-
-
-
1
2
3
8
25
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
2+5
-
-
-
-
G
=
7
-
6
GOLDEN
57
30
3
-
1
2
3
8
7
6
7
8
9
M
=
4
-
4
MEAN
33
15
6
-
-
-
-
-
-
-
-
-
-
-
-
13
-
13
THE GOLDEN MEAN
123
60
15
-
1
2
3
8
7
6
7
8
9
-
-
1+3
-
1+3
-
1+2+3
6+0
1+5
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE GOLDEN MEAN
6
6
6
-
1
2
3
8
7
6
7
8
9

 

 

-
8
O
M
P
H
A
L
O
S
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
8
-
-
6
1
+
=
21
2+1
=
3
=
3
=
3
-
-
15
-
-
8
-
-
15
19
+
=
57
5+7
=
12
1+2
3
=
3
-
8
O
M
P
H
A
L
O
S
-
-
-
-
-
--
-
-
-``
-
-
-
-
4
7
-
1
3
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
13
16
-
1
12
-
-
+
=
42
4+2
=
6
=
6
=
6
-
8
O
M
P
H
A
L
O
S
-
-
-
-
-
--
-
-
-
-
-
-
15
13
16
8
1
12
15
19
+
=
99
9+9
=
18
1+8
9
=
9
-
-
6
4
7
8
1
3
6
1
+
=
36
3+6
=
9
=
9
=
9
-
8
O
M
P
H
A
L
O
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
2
-
2
-
-
-
-
-
-
--
-
-
-
-
-
3
-
-
-
-
3
occurs
x
1
=
3
=
3
-
--
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
-
-
-
5
-
5
-
-
-
-
-
-
-
6
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
7
--
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
16
8
O
M
P
H
A
L
O
S
-
-
29
-
-
8
-
36
-
27
1+6
-
-
-
-
-
-
-
-
-
-
-
1+1
-
-
-
-
3+6
-
2+7
7
8
O
M
P
H
A
L
O
S
-
-
2
-
-
8
-
9
-
9

 

 

8
O
M
P
H
A
L
O
S
-
-
-
-
-
--
-
-
-
-
-
6
-
-
8
-
-
6
1
+
=
21
2+1
=
3
=
3
=
3
-
15
-
-
8
-
-
15
19
+
=
57
5+7
=
12
1+2
3
=
3
8
O
M
P
H
A
L
O
S
-
-
-
-
-
--
-
-
-``
-
-
-
4
7
-
1
3
-
-
+
=
15
1+5
=
6
=
6
=
6
-
-
13
16
-
1
12
-
-
+
=
42
4+2
=
6
=
6
=
6
8
O
M
P
H
A
L
O
S
-
-
-
-
-
--
-
-
-
-
-
15
13
16
8
1
12
15
19
+
=
99
9+9
=
18
1+8
9
=
9
-
6
4
7
8
1
3
6
1
+
=
36
3+6
=
9
=
9
=
9
8
O
M
P
H
A
L
O
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
--
-
-
-
-
-
3
-
-
-
-
3
occurs
x
1
=
3
=
3
--
-
4
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
6
-
-
-
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
7
--
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
8
O
M
P
H
A
L
O
S
-
-
29
-
-
8
-
36
-
27
-
-
-
-
-
-
-
-
-
-
-
1+1
-
-
-
-
3+6
-
2+7
8
O
M
P
H
A
L
O
S
-
-
2
-
-
8
-
9
-
9

 

 

-
6
O
R
A
C
L
E
-
-
-
-
-
--
-
-
-
-
6
-
-
-
-
-
+
=
6
-
=
6
=
6
-
-
15
-
-
-
-
-
+
=
15
1+5
=
6
=
6
-
6
O
R
A
C
L
E
-
-
-
-
-
--
-
-
-
-
-
9
1
3
3
5
+
=
21
2+1
=
3
=
3
-
-
-
18
1
3
12
5
+
=
39
3+9
=
12
1+2
3
-
6
O
R
A
C
L
E
-
-
-
-
-
--
=
-
-
-
15
18
1
3
12
5
+
=
54
5+4
=
9
=
9
-
-
6
9
1
3
3
5
+
=
27
2+7
=
9
-
9
-
6
O
R
A
C
L
E
-
-
-
-
-
--
=
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
-
2
-
2
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
4
-
-
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
-
-
7
-
7
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
8
-
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
21
6
O
R
A
C
L
E
-
-
24
-
-
6
-
27
2+1
-
-
9
-
-
-
-
-
-
2+4
-
-
-
-
2+7
3
6
O
R
A
C
L
E
-
-
6
-
-
6
-
9

 

 

6
O
R
A
C
L
E
-
-
-
-
-
--
-
-
-
6
-
-
-
-
-
+
=
6
-
=
6
=
6
-
15
-
-
-
-
-
+
=
15
1+5
=
6
=
6
6
O
R
A
C
L
E
-
-
-
-
-
--
-
-
-
-
9
1
3
3
5
+
=
21
2+1
=
3
=
3
-
-
18
1
3
12
5
+
=
39
3+9
=
12
1+2
3
6
O
R
A
C
L
E
-
-
-
-
-
--
=
-
-
15
18
1
3
12
5
+
=
54
5+4
=
9
=
9
-
6
9
1
3
3
5
+
=
27
2+7
=
9
-
9
6
O
R
A
C
L
E
-
-
-
-
-
--
=
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
6
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
6
O
R
A
C
L
E
-
-
24
-
-
6
-
27
-
-
9
-
-
-
-
-
-
2+4
-
-
-
-
2+7
6
O
R
A
C
L
E
-
-
6
-
-
6
-
9

 

 

-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
3
3
3
-
4
+
=
18
1+8
=
9
-
9
-
-
5
21
3
12
-
4
+
=
45
4+5
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
21
3
12
9
4
+
=
54
5+4
=
9
-
9
-
-
5
3
3
3
9
4
+
=
27
2+7
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
-
-
-
1
-
-
--
-
--
--
-
--
-
1
--
1
--
--
-
2
-
-
--
-
--
--
-
--
-
2
--
2
--
--
-
-
-
-
3
3
3
-
-
-
-
3
occurs
x
3
=
9
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
--
-
--
--
-
-
-
6
--
6
--
--
-
7
-
-
--
-
--
--
-
-
-
7
--
7
--
--
-
8
-
-
--
-
--
--
-
-
-
8
--
8
--
--
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
24
6
E
U
C
L
I
D
-
-
21
-
-
6
-
27
2+4
-
-
-
-
-
9
-
-
-
2+1
-
-
-
-
2+7
6
6
E
U
C
L
I
D
-
-
2
-
-
6
-
9
-
-
5
3
3
3
9
4
-
-
-
-
-
-
-
-
6
6
E
U
C
L
I
D
-
-
2
-
-
6
-
9

 

 

-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
-
-
-
-
9
-
+
=
9
-
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
3
3
3
-
4
+
=
18
1+8
=
9
-
9
-
-
5
21
3
12
-
4
+
=
45
4+5
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
--
-
-
-
-
5
21
3
12
9
4
+
=
54
5+4
=
9
-
9
-
-
5
3
3
3
9
4
+
=
27
2+7
=
9
-
9
-
6
E
U
C
L
I
D
-
-
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
-
3
occurs
x
3
=
9
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
24
6
E
U
C
L
I
D
-
-
21
-
-
6
-
27
2+4
-
-
-
-
-
9
-
-
-
2+1
-
-
-
-
2+7
6
6
E
U
C
L
I
D
-
-
2
-
-
6
-
9
-
-
5
3
3
3
9
4
-
-
-
-
-
-
-
-
6
6
E
U
C
L
I
D
-
-
2
-
-
6
-
9

 

 

-
3
A
C
E
-
-
-
-
-
--
-
-
-
-
1
3
5
+
=
9
-
=
9
=
9
-
3
A
C
E
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
2
-
2
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
6
-
6
-
-
-
7
-
-
-
-
-
-
7
-
7
-
-
-
8
-
-
-
-
-
-
8
-
8
-
-
-
9
-
-
-
-
-
-
9
-
9
-
-
-
36
3
A
C
E
-
-
9
-
-
3
-
9
3+6
-
-
-
-
-
-
-
-
-
-
-
-
9
3
A
C
E
-
-
9
-
-
3
-
9

 

 

3
A
C
E
-
-
-
-
-
--
-
-
-
1
3
5
+
=
9
-
=
9
=
9
3
A
C
E
-
-
-
-
-
--
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
5
-
-
5
occurs
x
1
=
5
3
A
C
E
-
-
9
-
-
3
-
9
-
-
-
-
-
-
-
-
-
-
-
-
3
A
C
E
-
-
9
-
-
3
-
9

 

 

-
4
D
I
C
E
-
-
-
-
-
--
-
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
-
4
D
I
C
E
-
-
-
-
-
--
-
-
-
-
4
-
3
5
+
=
12
1+2
=
3
=
3
-
4
D
I
C
E
-
-
-
-
-
--
-
-
-
-
4
9
3
5
+
=
21
2+1
=
3
=
3
-
4
D
I
C
E
-
-
-
-
-
--
-
-
1
-
-
-
-
-
-
-
1
-
1
-
-
-
2
-
-
-
-
-
-
-
2
-
2
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
6
-
6
-
-
-
7
-
-
-
-
-
-
-
7
-
7
-
-
-
8
-
-
-
-
-
-
-
8
-
8
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
24
4
D
I
C
E
-
-
21
-
-
4
-
21
2+4
-
-
9
-
-
-
-
2+1
-
-
-
-
2+1
6
4
D
I
C
E
-
-
3
-
-
4
-
3

 

 

4
D
I
C
E
-
-
-
-
-
--
-
-
-
-
9
-
-
+
=
9
-
=
9
=
9
4
D
I
C
E
-
-
-
-
-
--
-
-
-
4
-
3
5
+
=
12
1+2
=
3
=
3
4
D
I
C
E
-
-
-
-
-
--
-
-
-
4
9
3
5
+
=
21
2+1
=
3
=
3
4
D
I
C
E
-
-
-
-
-
--
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
-
9
-
-
-
-
9
occurs
x
1
=
9
4
D
I
C
E
-
-
21
-
-
4
-
21
-
-
9
-
-
-
-
2+1
-
-
-
-
2+1
4
D
I
C
E
-
-
3
-
-
4
-
3

 

 

-
4
I
M
A
M
-
-
-
-
-
--
-
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
-
4
I
M
A
M
-
-
-
-
-
--
-
-
-
-
-
4
1
4
+
=
9
-
=
9
=
9
-
-
-
13
1
13
+
=
27
2+7
=
9
=
9
-
4
I
M
A
M
-
-
-
-
-
--
-
-
-
-
9
13
1
13
+
=
36
3+6
=
9
=
9
-
-
9
4
1
4
+
=
18
1+8
=
9
=
9
-
4
I
M
A
M
-
-
-
-
-
--
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
2
-
2
-
-
-
3
-
-
-
-
-
-
-
3
-
3
-
-
-
-
-
-
4
-
4
-
-
4
occurs
x
2
=
8
5
-
-
-
-
-
-
-
5
-
5
-
-
-
6
-
-
-
-
-
-
-
6
-
6
-
-
-
7
-
-
-
-
-
-
-
7
-
7
-
-
-
8
-
-
-
-
-
-
-
8
-
8
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
31
4
I
M
A
M
-
-
14
-
-
4
-
18
3+1
-
9
-
-
-
-
-
1+4
-
-
-
-
1+8
4
4
I
M
A
M
-
-
5
-
-
4
-
9

 

 

4
I
M
A
M
-
-
-
-
-
--
-
-
-
9
-
-
-
+
=
9
-
=
9
=
9
4
I
M
A
M
-
-
-
-
-
--
-
-
-
-
4
1
4
+
=
9
-
=
9
=
9
-
-
13
1
13
+
=
27
2+7
=
9
=
9
4
I
M
A
M
-
-
-
-
-
--
-
-
-
9
13
1
13
+
=
36
3+6
=
9
=
9
-
9
4
1
4
+
=
18
1+8
=
9
=
9
4
I
M
A
M
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
-
-
4
-
4
-
-
4
occurs
x
2
=
8
-
9
-
-
-
-
-
9
occurs
x
1
=
9
4
I
M
A
M
-
-
14
-
-
4
-
18
-
9
-
-
-
-
-
1+4
-
-
-
-
1+8
4
I
M
A
M
-
-
5
-
-
4
-
9

 

 

7
C
A
B
B
A
L
A
-
-
-
-
-
--
-
-
-
3
1
2
2
1
12
1
+
=
22
2+2
=
4
=
4
-
3
1
2
2
1
3
1
+
=
13
1+3
=
4
=
4
7
C
A
B
B
A
L
A
-
-
-
-
-
--
-
-
-
-
1
-
-
1
-
1
-
-
1
occurs
x
3
=
3
-
-
-
2
2
-
-
-
-
-
2
occurs
x
2
=
4
-
3
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
7
C
A
B
B
A
L
A
-
-
6
-
-
7
-
13
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+3
7
C
A
B
B
A
L
A
-
-
6
-
-
7
-
4

 

 

7
K
A
B
B
A
L
A
-
-
-
-
-
--
-
-
-
11
1
2
2
1
12
1
+
=
30
3+0
=
3
=
3
-
2
1
2
2
1
3
1
+
=
12
1+2
=
3
=
3
7
K
A
B
B
A
L
A
-
-
-
-
-
--
-
-
-
-
1
-
-
1
-
1
-
-
1
occurs
x
3
=
3
-
2
-
2
2
-
-
-
-
-
2
occurs
x
3
=
6
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
7
K
A
B
B
A
L
A
-
-
6
-
-
7
-
12
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+2
7
K
A
B
B
A
L
A
-
-
6
-
-
7
-
3

 

 

-
-
-
-
GREAT GREEN
-
-
-
G
=
7
5
GREAT
51
24
6
G
=
7
5
GREEN
49
31
4
-
-
14
10
GREAT GREEN
100
55
10
-
-
1+4
1+0
-`
1+0+0
5+5
1+0
-
-
5
1
GREAT GREEN
1
10
1
-
-
-
-
-`
-
1+0
-`
-
-
5
1
GREAT GREEN
1
1
1

 

 

4
TURN
73
19
1
5
TURNS
92
20
2
6
TURNED
82
28
1
7
TURNING
103
40
4

 

 

6
ASEKHA
-
-
-
-
A+S+E+K
36
9
9
R
H+A
9
9
9
6
ASEKHA
45
18
18
-
-
4+5
1+8
1+8
6
ASEKHA
9
9
9

 

 

-
4
R
A
C
E
-
-
-
-
-
--
-
-
-
-
18
1
3
5
+
=
27
2+7
=
9
=
9
-
-
9
1
3
5
+
=
18
1+8
=
9
=
9
-
4
R
A
C
E
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
2
-
2
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
4
-
-
-
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
6
-
6
-
-
-
7
-
-
-
-
-
-
-
7
-
7
-
-
-
8
-
-
-
-
-
-
-
8
-
8
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
27
4
R
A
C
E
-
-
18
-
-
4
-
18
2+7
-
9
-
-
-
-
-
1+8
-
-
-
-
1+8
9
4
R
A
C
E
-
-
9
-
-
4
-
9

 

 

4
R
A
C
E
-
-
-
-
-
--
-
-
-
18
1
3
5
+
=
27
2+7
=
9
=
9
-
9
1
3
5
+
=
18
1+8
=
9
=
9
4
R
A
C
E
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
9
-
-
-
-
-
9
occurs
x
1
=
9
4
R
A
C
E
-
-
18
-
-
4
-
18
-
9
-
-
-
-
-
1+8
-
-
-
-
1+8
4
R
A
C
E
-
-
9
-
-
4
-
9

 

 

6
JEWISH
74
29
2
3
NEW
42
15
6
4
YEAR
49
22
4
13
First Total
165
66
12
1+3
Add to Reduce
1+6+5
6+6
1+2
4
Second Total
12
12
3
-
Reduce to Deduce
1+2
1+2
-
4
Essence of Number
3
3
3

 

 

3
NEW
42
15
6
3
OLD
31
13
4

 

 

1
I
9
9
9
2
AM
14
5
5
7
NEUTRAL
91
28
1

 

 

4
VRIL
61
25
7
8
VIRILITY
124
52
7

 

 

4
VRIL
61
25
7
6
VIRILE
75
39
3

 

 

-
9
P
O
L
Y
H
E
D
R
A
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
8
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
`-
-
15
-
-
8
-
-
-
-
+
=
23
2+3
=
5
=
5
=
5
-
9
P
O
L
Y
H
E
D
R
A
-
-
-
-
-
-
-
-
-
-
-
-
7
-
3
7
-
5
4
9
1
+
=
36
3+6
=
9
=
9
=
9
-
`-
16
-
12
25
-
5
4
18
1
+
=
81
8+1
=
9
=
9
=
9
-
9
P
O
L
Y
H
E
D
R
A
-
-
-
-
-
-
-
-
-
-
-
`-
16
15
12
25
8
5
4
18
1
+
=
104
1+0+4
=
5
=
5
=
5
-
-
7
6
3
7
8
5
4
9
1
+
=
50
5+0
=
5
=
5
=
5
-
9
P
O
L
Y
H
E
D
R
A
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
7
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
=
9
2
9
P
O
L
Y
H
E
D
R
A
-
-
43
-
-
9
-
50
-
41
-
-
-
-
-
-
-
-
-
9
-
-
-
4+3
-
-
-
-
5+0
-
4+1
2
9
P
O
L
Y
H
E
D
R
A
-
-
7
-
-
9
-
5
-
5

 

 

10
PYTHAGORAS
130
49
4
8
OURABORUS
130
31
4
5
SAMOS
67
13
4
7
MILETUS
99
27
9
8
THIRTEEN
99
45
9

 

 

-
PYTHAGORAS
-
-
-
5
PYTHA
70
25
7
5
GORAS
60
24
6
10
PYTHAGORAS
130
49
13
-
-
1+3+0
4+9
3+1
-
-
4
13
4
8
THIRTEEN
99
45
9

 

 

-
PYTHAGORAS
-
-
-
-
PYT
61
16
7
-
HA
9
9
9
-
GO
22
13
4
-
R
18
9
9
-
AS
20
11
2
10
PYTHAGORAS
130
58
31
1+0
  
1+3+0
5+8
3+1
1
PYTHAGORAS
4
13
4
-
-
-
1+3
-
1
PYTHAGORAS
4
4
4

 

 

-
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
`-
16
25
20
8
1
7
15
18
1
19
+
=
130
1+3
=
18
1+8
4
-
4
-
-
7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=
13
1+3
4
-
4
-
8
P
Y
T
H
A
G
O
R
A
S
-T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
1
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
2
-
-
-
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
3
--
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
--
--
5
FIVE
-
-
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
--
7
7
-
-
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
-
9
12
8
P
Y
T
H
A
G
O
R
A
S
-
-
33
-
-
8
-
49
-
31
1+2
-
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
4+9
-
3+1
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
13
-
4
-
-
7
7
2
8
1
7
6
9
1
1
-
-
-
-
-
-
-
1+3
-
-
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
4
-
4

 

 

THE GROWTH OF SCIENCE

A.P. Rossiter 1939

Page 15

"The Egyptians,…" "…made good observations on the stars and were able to say when the sun or moon would become dark in an eclipse (a most surprising event even in our times), and when the land would be covered by the waters of the Nile: they were expert at building and made some discoveries about the relations of lines and angles - among them one very old rule for getting a right-angle by stretching out knotted cords with 5, 4 And 3 units between the knots."

 

"...among them one very old rule for getting a right-angle by stretching out knotted cords with

5, 4 And 3 units between the knots."

 

 

CIVILIZATION, SCIENCE AND RELIGION

A. D. RITCHIE 1945

THE ART OF THINKING

Page 39

"The Egyptians could set out a right-angle on the ground,

for building or for land surveying,

by means of a cord knotted at intervals of

3, 4 and 5 units of length."

 

 

-
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
`-
16
25
20
8
1
7
15
18
1
19
+
=
130
1+3
=
18
1+8
4
-
4
-
-
7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=
13
1+3
4
-
4
-
8
P
Y
T
H
A
G
O
R
A
S
-T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
1
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
2
-
-
-
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
3
--
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
--
--
5
FIVE
-
-
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
--
7
7
-
-
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
-
9
12
8
P
Y
T
H
A
G
O
R
A
S
-
-
33
-
-
8
-
49
-
31
1+2
-
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
4+9
-
3+1
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
13
-
4
-
-
7
7
2
8
1
7
6
9
1
1
-
-
-
-
-
-
-
1+3
-
-
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
4
-
4

 

 

CIVILIZATION, SCIENCE AND RELIGION

A. D. RITCHIE 1945

THE ART OF THINKING

Page 38

"In the sphere of the natural sciences and of mathematics there have been endless disputes as to how much the Greeks borrowed from their neighbours, and the disputes are likely to continue, for the evidence is scanty and unreliable. It is safe to assume that the Greeks (noted then as now for commercial enterprise) took all they could get. Their own writers say as much, for they attribute the origin of very many useful inventions to other peoples. But this one thing, the scientific outlook and method, was not there to take; they had to invent it themselves. It is well to be clear on this point, for European civilization rests on three legs. They are Greek science, Jewish religion and Roman law. / Page 39 / Roman law may well be considered the Roman development of Greek scientific method. I will therefore deal with two examples in some little detail. These are taken from the sphere of mathematics and astronomy, for it was in these two sciences that the Greeks had their most outstanding success, doing about as much as could possibly be done under the conditions of their day and laying the foundations on which all subsequent work has been based.

The Egyptians knew of many useful methods of -geo- metrical calculation, for finding the area of a field, the volume of a barrel and so on. The Babylonians and earlier Mesopotamians had made accurate observations of sun, moon and stars over long periods and developed ingenious methods for calculating their future positions in the sky. In these arts of calculation these people had nothing to learn from the Greeks; it was the other way about. But there is no evidence that they ever dreamt of turning the art of calculation into the science of mathematics. Solving particular problems, however ingeniously, is not necessarily science any more than is playing chess (though all chess problems are geometrical) or keeping accounts (though all money reckoning is arithmetical). Mathematical science in the proper sense of the word attains its end by two means : (1) generalizing as far as is possible all problems and their solutions, so that one solution solves any number of particular cases; (2) finding proofs that solutions are correct as opposed to finding solutions which might be right by chance, not by necessity. The method used is the method of discussion in its specifically mathematical form.

The Egyptians could set out a right-angle on the ground, for building or for land surveying, by means of a cord knotted at intervals of 3, 4 and 5 units of length. They adjusted three pegs to make a triangle with the knots at the pegs when the cord was stretched tight round them. The Greeks, seeing this trick, generalized the problem and looked for a proof of the solution. The final result, after two centuries of effort, is the First Book of Euclid's Elements, leading up to Proposition 47—that the square on the hypotenuse of a right-angled triangle equals the sum of / Page 40 / the squares on the other sides, and that this must be so, granted the assumptions made at the beginning. (The proposition is further generalized in, Euclid VI, 31.) In this way a technical dodge of the land surveyor, depending upon the fact that 32+42= 52, was turned into science.

Page 38 Notes

1 Thucydides IV, 104—V, 26.
2 Hippocrates, Vol. ii, pp. 138 seq. Loeb Classical Library.

 

 

-
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
`-
16
25
20
8
1
7
15
18
1
19
+
=
130
1+3
=
18
1+8
4
-
4
-
-
7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=
13
1+3
4
-
4
-
8
P
Y
T
H
A
G
O
R
A
S
-T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
1
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
2
-
-
-
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
3
--
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
--
--
5
FIVE
-
-
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
--
7
7
-
-
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
-
9
12
8
P
Y
T
H
A
G
O
R
A
S
-
-
33
-
-
8
-
49
-
31
1+2
-
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
4+9
-
3+1
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
13
-
4
-
-
7
7
2
8
1
7
6
9
1
1
-
-
-
-
-
-
-
1+3
-
-
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
4
-
4

 

 

-
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
8
-
-
6
-
-
1
+
=
15
1+5
=
6
-
6
-
6
-
-
-
-
-
8
-
-
15
-
-
19
+
=
42
4+2
=
6
-
6
-
6
-
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
-
7
7
2
-
1
7
-
9
1
-
+
=
34
3+4
=
7
-
7
-
7
-
`-
16
25
20
-
1
7
-
18
1
-
+
=
88
8+8
=
16
1+6
7
-
7
-
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
`-
16
25
20
8
1
7
15
18
1
19
+
=
 130
1+3
=
18
1+8
4
-
4
-
-
7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=
13
1+3
4
-
4
-
8
P
Y
T
H
A
G
O
R
A
S
-T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
1
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
2
-
-
-
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
3
--
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
--
--
5
-
-
-
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
--
7
7
-
-
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
-
9
12
8
P
Y
T
H
A
G
O
R
A
S
-
-
33
-
-
8
-
49
-
31
1+2
-
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
4+9
-
3+1
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
13
-
4
-
-
7
7
2
8
1
7
6
9
1
1
-
-
-
-
-
-
-
1+3
-
-
3
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
4
-
4

 

 

8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
8
-
-
6
-
-
1
+
=
15
1+5
=
6
-
6
-
6
-
-
-
-
8
-
-
15
-
-
19
+
=
42
4+2
=
6
-
6
-
6
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
-
7
7
2
-
1
7
-
9
1
-
+
=
34
3+4
=
7
-
7
-
7
`-
16
25
20
-
1
7
-
18
1
-
+
=
88
8+8
=
16
1+6
7
-
7
8
P
Y
T
H
A
G
O
R
A
S
-
-
-
-
-
--
-
-
-
-
`-
16
25
20
8
1
7
15
18
1
19
+
=
 130
1+3
=
18
1+8
4
-
4
-
7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=
13
1+3
4
-
4
8
P
Y
T
H
A
G
O
R
A
S
-T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
1
-
-
1
occurs
x
3
=
3
-
3
-
-
-
2
-
-
-
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
--
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
-
6
--
7
7
-
-
-
7
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
9
-
9
8
P
Y
T
H
A
G
O
R
A
S
-
-
33
-
-
8
-
49
-
31
-
-
-
-
-
-
-
-
9
-
-
-
-
3+3
-
-
-
-
4+9
-
3+1
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
13
-
4
-
7
7
2
8
1
7
6
9
1
1
-
-
-
-
-
-
-
1+3
-
-
8
P
Y
T
H
A
G
O
R
A
S
-
-
6
-
-
8
-
4
-
4

 

 

THE GROWTH OF SCIENCE

A.P.Rossiter 1939

Page 15

"The Egyptians,…" "…made good observations on the stars and were able to say when the sun or moon would become dark in an eclipse (a most surprising event even in our times), and when the land would be covered by the waters of the Nile: they were expert at building and made some discoveries about the relations of lines and angles - among them one very old rule for getting a right-angle by stretching out knotted cords with 5, 4 And 3 units between the knots."

"...among them one very old rule for getting a right-angle by stretching out knotted cords with

5, 4 And 3 units between the knots."

 

 

CIVILIZATION, SCIENCE AND RELIGION

A. D. RITCHIE 1945

THE ART OF THINKING

Page 39

"The Egyptians could set out a right-angle on the ground,

for building or for land surveying,

by means of a cord knotted at intervals of

3, 4 and 5 units of length."

 

3
-
5
THREE
56
29
2
4
-
4
FOUR
60
24
6
5
-
4
FIVE
42
24
6
12
-
13
First Total
158
77
14
1+2
-
1+3
Add to Reduce
1+5+8
7+7
1+4
3
-
4
Second Total
14
14
5
-
-
-
Reduce to Deduce
1+4
1+4
--
3
-
4
Essence of Number
5
5
5

 

 

3
-
T
=
2
-
5
THREE
56
29
2
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
42
24
6
12
-
-
-
14
-
13
First Total
158
77
14
1+2
-
-
-
1+4
-
1+3
Add to Reduce
1+5+8
7+7
1+4
12
-
-
-
5
-
4
Second Total
14
14
5
-
-
-
-
-
-
-
Reduce to Deduce
1+4
1+4
--
3
-
-
-
5
-
4
Essence of Number
5
5
5

 

 

-
-
-
-
-
PYTHAGORAS
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
P
=
7
-
1
P
16
7
7
-
-
-
3
4
5
-
7
-
-
Y
=
7
-
1
Y
25
7
7
-
-
-
3
4
5
-
7
-
-
T
=
2
-
1
T
20
2
2
-
-
2
3
4
5
-
-
-
-
H
=
8
-
1
H
8
8
8
-
-
-
3
4
5
-
-
8
-
A
=
1
-
1
A
1
1
1
-
1
-
3
4
5
-
-
-
-
G
=
7
-
1
G
7
7
7
-
-
-
3
4
5
-
7
-
-
O
=
6
-
1
O
15
6
6
-
-
-
3
4
5
6
-
-
-
R
=
9
-
1
R
18
9
9
-
-
-
3
4
5
-
-
-
9
A
=
1
-
1
A
1
1
1
-
1
-
3
4
5
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
3
4
5
-
-
-
-
-
-
49
-
10
PYTHAGORAS
130
58
49
-
3
2
3
4
5
6
21
8
9
-
-
4+9
-
1+0
-
1+3+0
5+8
4+9
-
-
-
-
-
-
-
2+1
-
-
-
-
13
-
1
PYTHAGORAS
4
13
13
-
3
2
3
4
5
6
3
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
PYTHAGORAS
4
4
4
-
3
2
3
4
5
6
3
8
9

 

 

-
-
-
-
-
PYTHAGORAS
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
3
4
5
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
3
4
5
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
3
4
5
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
3
4
5
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
3
4
5
6
-
-
-
P
=
7
-
1
P
16
7
7
-
-
-
3
4
5
-
7
-
-
Y
=
7
-
1
Y
25
7
7
-
-
-
3
4
5
-
7
-
-
G
=
7
-
1
G
7
7
7
-
-
-
3
4
5
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
3
4
5
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
3
4
5
-
-
-
9
-
-
49
-
10
PYTHAGORAS
130
58
49
-
3
2
3
4
5
6
21
8
9
-
-
4+9
-
1+0
-
1+3+0
5+8
4+9
-
-
-
-
-
-
-
2+1
-
-
-
-
13
-
1
PYTHAGORAS
4
13
13
-
3
2
3
4
5
6
3
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
1
PYTHAGORAS
4
4
4
-
3
2
3
4
5
6
3
8
9

 

 

-
-
-
-
-
PYTHAGORAS
-
-
-
-
1
2
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
A
=
1
-
1
A
1
1
1
-
1
-
-
-
-
-
S
=
1
-
1
S
19
10
1
-
1
-
-
-
-
-
T
=
2
-
1
T
20
2
2
-
-
2
-
-
-
-
O
=
6
-
1
O
15
6
6
-
-
-
6
-
-
-
P
=
7
-
1
P
16
7
7
-
-
-
-
7
-
-
Y
=
7
-
1
Y
25
7
7
-
-
-
-
7
-
-
G
=
7
-
1
G
7
7
7
-
-
-
-
7
-
-
H
=
8
-
1
H
8
8
8
-
-
-
-
-
8
-
R
=
9
-
1
R
18
9
9
-
-
-
-
-
-
9
-
-
49
-
10
PYTHAGORAS
130
58
49
-
3
2
6
21
8
9
-
-
4+9
-
1+0
-
1+3+0
5+8
4+9
-
-
-
-
2+1
-
-
-
-
13
-
1
PYTHAGORAS
4
13
13
-
3
2
6
3
8
9
-
-
1+3
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
4
-
1
PYTHAGORAS
4
4
4
-
3
2
6
3
8
9

 

 

A

SERPENT I PRESENT

 

8
OURABORUS
130
31
4
10
PYTHAGORAS
130
49
4
6
PYTHIA
79
34
7
6
PYTHON
98
35
8
7
PYTHONS
117
36
9
5
 EGYPT
73
28
1
8
 EGYPTIAN
97
43
7

 

 

10
PYTHAGORAS
130
49
4
-
-
-
-
-
-
-
-
-
-
10
PYTHAGORAS
130
49
4
5
EGYPT
73
28
1
1
E
5
5
5
4
PYTG
68
23
5
5
EGYPT
73
28
1

 

 

10
PYTHAGORAS
130
49
4
-
SARAH
-
-
-
2
SA
20
11
2
1
R
18
9
9
2
AH
9
9
9
5
SARAH
47
29
20
-
-
4+7
2+9
2+0
5
SARAH
11
11
2
-
-
1+1
1+1
-
5
SARAH
2
2
2
-
-
-
-
-
-
-
-
-
-
10
PYTHAGORAS
130
49
4
5
HAGOR
49
31
4
5
SARAH
47
20
2
7
ABRAHAM
44
26
8
-
BRAHMAN
-
-
-
-
A
1
1
1
7
BRAHMAN
-
-
-
1
B
2
2
2
3
RAH
27
18
9
3
MAN
28
10
1
7
BRAHMAN
57
30
12
-
  
5+7
3+0
-
7
BRAHMAN
12
3
3
-
-
1+2
-
-
7
BRAHMAN
3
3
3

 

 

8
OURABORUS
130
31
4
10
PYTHAGORAS
130
49
4
18
-
260
80
8
1+8
-
2+6+0
8+0
-
9
-
8
8
8

 

 

BE HOURS HOURS BE

HORUS HOURS

 

-
OURABORUS
-
-
-
2
O+U
36
9
9
1
R
18
9
9
3
A+B+O
18
9
9
1
R
18
9
9
2
U+S
40
13
4
9
OURABORUS
130
49
40
-
Add to Reduce
1+3+0
4+9
-
-
Reduce to Deduce
4
13
4
-
  
-
1+3
-
9
OURABORUS
4
4
4

 

 

-
8
O
U
R
A
B
O
R
U
S
-
-
-
-
-
--
-
-
-
-
-
-
6
-
-
-
-
6
-
-
1
+
=
13
1+3
=
4
-
4
-
4
-
-
15
-
-
-
-
15
-
-
19
+
=
49
4+9
=
13
1+3
4
-
4
-
8
O
U
R
A
B
O
R
U
S
-
-
-
-
-
--
-
-
-
-
-
-
-
3
9
1
2
-
9
3
-
+
=
27
2+7
=
9
-
9
-
9
-
`-
-
21
18
1
2
-
18
21
-
+
=
81
8+1
=
9
-
9
-
9
-
8
O
U
R
A
B
O
R
U
S
-
-
-
-
-
--
-
-
-
-
-
`-
15
21
18
1
2
15
18
21
19
+
=
130
1+3
=
4
-
4
-
4
-
-
6
3
9
1
2
6
9
3
1
+
=
40
4+0
=
4
-
4
-
4
-
8
O
U
R
A
B
O
R
U
S
-T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
2
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
-
--
-
3
-
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
6
4
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
--
--
5
-
-
-
-
-
-
-
-
--
6
-
-
-
-
6
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
--
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
24
8
O
U
R
A
B
O
R
U
S
-
-
21
-
-
8
-
49
-
22
2+4
-
-
-
9
-
-
-
9
-
-
-
-
2+1
-
-
-
-
4+9
-
2+2
6
8
O
U
R
A
B
O
R
U
S
-
-
3
-
-
8
-
13
-
4
-
-
6
3
9
1
2
6
9
3
1
-
-
-
-
-
-
-
1+3
-
-
6
8
O
U
R
A
B
O
R
U
S
-
-
3
-
-
8
-
4
-
4

 

 

8
O
U
R
A
B
O
R
U
S
-
-
-
-
-
--
-
-
-
-
-
6
-
-
-
-
6
-
-
1
+
=
13
1+3
=
4
-
4
-
4
-
15
-
-
-
-
15
-
-
19
+
=
49
4+9
=
13
1+3
4
-
4
8
O
U
R
A
B
O
R
U
S
-
-
-
-
-
--
-
-
-
-
-
-
3
9
1
2
-
9
3
-
+
=
27
2+7
=
9
-
9
-
9
`-
-
21
18
1
2
-
18
21
-
+
=
81
8+1
=
9
-
9
-
9
8
O
U
R
A
B
O
R
U
S
-
-
-
-
-
--
-
-
-
-
`-
15
21
18
1
2
15
18
21
19
+
=
130
1+3
=
4
-
4
-
4
-
6
3
9
1
2
6
9
3
1
+
=
40
4+0
=
4
-
4
-
4
8
O
U
R
A
B
O
R
U
S
-T
-T
-
-
-
--
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
2
-
-
-
-
--
--
2
occurs
x
1
=
2
-
2
--
-
3
-
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
-
6
--
6
-
-
-
-
6
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
9
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
O
U
R
A
B
O
R
U
S
-
-
21
-
-
8
-
49
-
22
-
-
-
9
-
-
-
9
-
-
-
-
2+1
-
-
-
-
4+9
-
2+2
8
O
U
R
A
B
O
R
U
S
-
-
3
-
-
8
-
13
-
4
-
6
3
9
1
2
6
9
3
1
-
-
-
-
-
-
-
1+3
-
-
8
O
U
R
A
B
O
R
U
S
-
-
3
-
-
8
-
4
-
4

 

 

10
P
Y
T
H
A
G
O
R
A
S
               

16
25
20
8
1
7
15
18
1
19
+
=
130
1+3+0
=
4

FOUR

4

10
P
Y
T
H
A
G
O
R
A
S
               
                                     

O
U
R
A
B
O
R
U
S
               

15
21
18
1
2
15
18
21
19
+
=
130
1+3+0
=
4

FOUR

4
                                     

O
U
R
A
B
O
R
U
S
               

 

 

10
P
Y
T
H
A
G
O
R
A
S

16
25
20
8
1
7
15
18
1
19
+
=
130

1+3
=
4

4

FOUR

4

1+6

2+5

2+0

1+5

1+8

1+9

7

7

2

 

6

9

10
+
=
                

1+0

 
7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=

13

1+3

4
FOUR

4

10
P
Y
T
H
A
G
O
R
A
S
           
                                         
                                         

10
P
Y
T
H
A
G
O
R
A
S
           

1

1
1
+
=
3

3

3

THREE
3

2

2

2

2

TWO
2

6

6

6

6

SIX

6

7
7

7

+
=
21
2+1
=
3

3

THREE
3

8

8

8

8

EIGHT
8

9

9

9

NINE
9

10
P
Y
T
H
A
G
O
R
A
S
                   
                                         
                                         

10
P
Y
T
H
A
G
O
R
A
S

O
U
R
A
B
O
R
U
S

15
21
18
1
2
15
18
21
19
+
=

130

1+3+0

=

4

4

FOUR

4

1+5
2+1

1+8

1+5
1+8

2+1

1+0

6

3

9

6

9

3

1

+

=

37

3+7

=
10

1+0
1
ONE
1

O
U
R
A
B
O
R
U
S

10
P
Y
T
H
A
G
O
R
A
S
           

16
25
20
8
1
7

+
=
77

7+7

=

14

1+4

5

FIVE

5

7

7

2

1

7

+

=

24

2+4

=
6

6

SIX

6

10
P
Y
T
H
A
G
O
R
A
S
                   
                                         
                                         
3
  
O
U
R
                               
   
15
21
18
           

+

=
54
5+4

=

9

9
NINE
9
   
1+5
2+1

1+8

                                

6

3

9

+

=

18

1+8

=

9

9

NINE

9
3
  
O
U
R
                               

9

O
U
R
A
B
O
R
U
S

O
R
U
S

15
18
21
19
+
=

73

7+3
=
10

1+0
1
ONE
1

1+5

1+8

2+1

1+9
+
=
28

2+8
=
10

1+0
1
ONE
1

6

9

3

1
+
=

19

1+9

=
10

1+0
1
ONE
1

O
R
U
S

10
P
Y
T
H
A
G
O
R
A
S

O
R
A
S

             
15
18
1
19
+
=
53
5+3
=
8
  
8
EIGHT
8
             

1+5

1+8

2+1

1+9

                  
             

6

9

1

1
+
=
17
1+7
=
8
  
8
EIGHT
8
             
O
R
A
S
                 
10
P
Y
T
H
A
G
O
R
A
S
                 

8

8

8

EIGHT

8

6

10
+
=
16
1+6
=
7

7

SEVEN
7

1+5

1+9

8

15

19

+

=

42

4+2

=

6

6

SIX

6

10
P
Y
T
H
A
G
O
R
A
S
                   
                                         
                                         

10
P
Y
T
H
A
G
O
R
A
S

 

16
25
20
8
1
7
15
18
1
19
+
=
130
1+3
=

4

4
FOUR

4

1+6
2+5
2+0

1+5
1+8

1+9

 

7
7
2

6
9

10
+
=
41
4+1

=

5

5

FIVE

5

1+0

1

1

1

1

ONE

1

8
1
7

1

+
=
17
1+7
=
8

8

EIGHT
8

7
7
2
8
1
7
6
9
1
1
+
=
130

1+3
=
4

4

FOUR

4

10
P
Y
T
H
A
G
O
R
A
S

 

 

10
P
Y
T
H
A
G
O
R
A
S

16
25
20
8
1
7
15
18
1
19
+
=
130

1+3
=
4

4

FOUR

4

7
7
2
8
1
7
6
9
1
1
+
=
49
4+9
=

13

1+3

4
FOUR

4

10
P
Y
T
H
A
G
O
R
A
S

O
U
R
A
B
O
R
U
S

15
21
18
1
2
15
18
21
19
+
=

130

1+3+0

=

4

4

FOUR

4
6
3
9
1
2

6

9

3

1
+
=
40
4+0
=
4

4

FOUR

4

O
U
R
A
B
O
R
U
S

O
U
R
A
B
15
21
18
1
2
+
=
57
5+7
=
12
1+2
3
THREE
3
6
3
9
1
2
+
=
21
2+1
=
3
3
THREE
3
O
U
R
A
B
O
R
U
S

10
P
Y
T
H
A
G
O
R
A
S

16
25
20
8
1
7

+
=
77

7+7

=

14

1+4

5

FIVE

5

7

7

2

8

1

7

+

=

32

3+2

=
5

5

FIVE

5
                                         

9

O
U
R
A
B
O
R
U
S

O
R
U
S

15
18
21
19
+
=

73

7+3
=
10

1+0
1
ONE
1

1+5

1+8

2+1

1+9
+
=
28

2+8
=
10

1+0
1
ONE
1

6

9

3

1
+
=

19

1+9

=
10

1+0
1
ONE
1

O
R
U
S

10
P
Y
T
H
A
G
O
R
A
S

O
R
A
S

           
15
18
1
19
+
=
53
5+3
=
8
  
8
EIGHT
8
           

1+5

1+8

2+1

1+9

                  
           

6

9

1

1
+
=
17
1+7
=
8
  
8
EIGHT
8
           
O
R
A
S
                 
10
P
Y
T
H
A
G
O
R
A
S
                   
                                         
                                         
10
P
Y
T
H
A
G
O
R
A
S
                 
         
1
  
6
9
  
1
+
=
17
1+7
=
8
  
8
EIGHT
8

9

O
U
R
A
B
O
R
U
S

                                         
                                         

9

O
U
R
A
B
O
R
U
S
                   
             
O
R
U
                     
             
6
9
3
  
+
=

18

1+8

=

9

9

NINE

9
   
6
3
9
                              
3
  
O
U
R
                             
   
15
21
18
           

+

=
54
5+4

=

9

9
NINE
9
   
1+5
2+1

1+8

                              

6

3

9

+

=

18

1+8

=

9

9

NINE

9
3
  
O
U
R
                               
       
A
B
                            
       
1
2
        

+

=
3
3
3
THREE
3
2
        
A
B
                            
           
O
R
U
S
                   
           
15
18
21
19
+
=

73

7+3
=
10

1+0
1
ONE
1
           

1+5

1+8

2+1

1+9

                  
           

6

9

3

1
+
=

19

1+9

=
10

1+0
1
ONE
1
4
            
O
R
U
S
                   
9
  
O
U
R
A
B
O
R
U
S
                   

 

 

10
P
Y
T
H
A
G
O
R
A
S
                   

16
25
20
8
1
7
15
18
1
19
+
=
130

1+3
=
4

4

FOUR

4

10
P
Y
T
H
A
G
O
R
A
S
                   
                                         

O
U
R
A
B
O
R
U
S
                   
   
15
21
18
1
2
15
18
21
19
+
=

130

1+3+0

=

4

4

FOUR

4

O
U
R
A
B
O
R
U
S
                   

 

 

-
21
T
H
E
-
M
A
G
I
C
-
-
O
F
-
P
E
R
S
P
E
C
T
I
V
E
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
---
-
-
-
9
-
---
---
6
-
---
-
-
-
1
-
-
-
-
9
-
-
+
=
33
3+3
=
6
=
6
=
6
-
-
-
8
-
---
-
-
-
9
-
---
---
15
-
---
-
-
-
19
-
-
-
-
9
-
-
+
=
60
6+0
=
6
=
6
=
6
-
21
T
H
E
--
M
A
G
I
C
--
--
O
F
--
P
E
R
S
P
E
C
T
I
V
E
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
--
4
1
7
-
3
--
--
-
6
--
7
5
9
-
7
5
3
2
-
4
5
+
=
75
7+5
=
12
1+2
3
=
3
-
-
20
-
5
--
13
1
7
-
3
--
--
-
6
--
16
5
18
-
16
5
3
20
-
22
5
+
=
165
1+6+5
=
12
1+2
3
=
3
-
21
T
H
E
--
M
A
G
I
C
--
--
O
F
--
P
E
R
S
P
E
C
T
I
V
E
-
-
-
-
-
-
-
-
-
-
-
-
20
8
5
--
13
1
7
9
3
--
--
15
6
--
16
5
18
19
16
5
3
20
9
22
5
+
=
225
2+2+5
=
9
=
9
=
9
-
-
2
8
5
--
4
1
7
9
3
--
--
6
6
--
7
5
9
1
7
5
3
2
9
4
5
+
=
108
1+0+8
=
9
=
9
=
9
-
21
T
H
E
--
M
A
G
I
C
--
--
O
F
--
P
E
R
S
P
E
C
T
I
V
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
--
-
1
-
-
-
--
--
-
-
--
-
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
-
2
-
-
--
-
-
-
-
-
--
--
-
-
--
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
-
-
-
--
-
-
-
-
3
--
--
-
-
--
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
--
4
-
-
-
-
--
--
-
-
--
-
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
2
=
8
=
8
5
-
-
-
5
--
-
-
-
-
-
--
--
-
-
--
-
5
-
-
-
5
-
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
--
-
-
-
-
-
--
--
6
6
--
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
--
-
-
7
-
-
--
--
-
-
--
7
-
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
8
-
---
-
-
-
-
-
---
---
-
-
---
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
--
-
-
-
9
-
--
--
-
-
--
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
3
=
27
2+7
9
45
21
T
H
E
--
M
A
G
I
C
--
--
O
F
--
P
E
R
S
P
E
C
T
I
V
E
-
-
45
-
-
21
21
108
-
45
4+5
2+1
-
-
5
--
-
-
-
-
-
--
--
-
-
--
-
5
-
-
-
5
-
-
-
-
5
-
-
4+5
-
-
2+1
-
1+0+8
-
4+5
9
3
T
H
E
--
M
A
G
I
C
--
--
O
F
--
P
E
R
S
P
E
C
T
I
V
E
-
-
9
-
-
3
-
9
-
9
-
-
2
8
5
---
4
1
7
9
3
---
---
6
6
---
7
5
9
1
7
5
3
2
9
4
5
-
-
-
-
-
-
-
-
-
-
9
3
T
H
E
--
M
A
G
I
C
--
--
O
F
--
P
E
R
S
P
E
C
T
I
V
E
-
-
9
-
-
3
7
9
7
9

 

 

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

 

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

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

the golden ratio = 1.61803399
More about calculator.

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

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

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

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

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

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

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

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

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

 

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

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

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

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

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

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

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

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

FundamentalsThe letter πMain article: pi (letter)

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

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

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

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

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

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

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

(Diagrams omitted)

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

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

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

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

Estimating πMain article: Numerical approximations of π

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

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

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

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

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

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

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

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

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

PROBLEMS - PROBLEMS

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
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

 

 
8
PROBLEMS
100
46
1
-
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
19
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

 

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
S
=
1
8
1
S
19
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

 

LETTERS RE ARRANGED NUMERICALLY

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
9

 

LETTERS RE ARRANGED NUMERICALLY

 

SOLVE PROBLEMS SOLVE

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
35
-
-
-
98
44
35
-
-
-
-
-
-
-
-
-
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
37
-
-
-
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
8
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

LETTERS RE ARRANGED NUMERICALLY

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
9
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
9

 

 

EVOLVE LOVE EVOLVE

LOVE EVOLVE LOVE

EVOLVE REVOLVE EVOLVE

REVOLVE EVOLVE REVOLVE

SOLVE LOVES SOLVE

 

 

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

 

 

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

 

 

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

 

 

SEVEN EVENS SEVEN

 

 

     

 

 
 
Top
  
 
Evokation
  
Previous Page
Index
 Next Page