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Evokation
 
 
Index
 

 

 

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

 

 

THE LIGHT IS RISING NOW RISING IS THE LIGHT

 

 

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

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0
-
Z
=
8
-
4
ZERO
64
28
1
-
1
-
-
-
-
-
-
-
-
1
-
O
=
6
-
3
ONE
34
16
7
-
-
-
-
-
-
-
7
-
-
2
-
T
=
2
-
3
TWO
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
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
58
13
4
3
-
T
=
2
-
5
THREE
56
29
2
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
42
24
6
6
-
S
=
1
-
3
SIX
52
16
7
7
-
S
=
1
-
5
SEVEN
65
20
2
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

 

 

-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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
-
-
-
34
-
36
Add
458
197
44
-
1
4
3
8
5
18
14
8
9
4+5
-
-
-
3+4
-
3+6
Reduce
4+5+8
1+9+7
4+4
-
-
-
-
-
-
1+8
1+4
-
-
9
-
-
-
7
4
9
Deduce
17
17
8
-
1
4
3
8
5
18
14
8
9
-
-
-
-
-
-
-
Produce
1+7
1+7
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
7
-
9
Essence
8
8
8
-
1
4
3
8
5
9
5
8
9

 

 

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

 

 

-
THE SOLAR SYSTEM
-
-
-
3
THE
33
15
6
5
SOLAR
65
29
2
6
SYSTEM
101
38
2
14
THE SOLAR SYSTEM
199
82
10
1+4
-
1+9+9
8+2
1+0
5
THE SOLAR SYSTEM
19
10
1
-
-
1+9
1+0
-
5
THE SOLAR SYSTEM
10
1
1
-
-
1+0
-
-
5
THE SOLAR SYSTEM
1
1
1

 

 

3
SUN
54
9
9
7
MERCURY
103
40
4
5
VENUS
81
18
9
5
EARTH
52
25
7
4
MOON
57
21
3
4
MARS
51
15
6
7
JUPITER
99
36
9
6
SATURN
93
21
3
6
URANUS
94
22
4
7
NEPTUNE
95
32
5
5
PLUTO
84
21
3
59
First Total
863
260
62
5+9
Add to Reduce
8+6+3
2+6
6+2
14
Second Total
17
8
8
1+4
Reduce to Deduce
1+7
-
-
5
Essence of Number
8
8
8

 

 

3
SUN
54
9
9
5
EARTH
52
25
7
4
MOON
57
21
3
12
First Total
163
55
19
1+2
Add to Reduce
1+6+3
5+5
1+9
3
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
3
Essence of Number
1
1
1

 

 

4
GODS
45
18
9
6
SPIRIT
91
37
1
4
ISIS
89
35
8
6
OSIRIS
89
35
8
6
VISHNU
93
30
3
5
SHIVA
59
59
4
7
KRISHNA
80
35
3
7
SHRISTI
102
39
3
5
RISHI
63
36
9
4
ISHI
45
27
9
6
CHRIST
77
32
5

 

 

GODS SPIRIT GODS

ISIS OSIRIS VISHNU SHIVA SHRI KRISHNA SHRISTI RISHI ISHI CHRIST

SING A SONG OF NINES OF NINES A SONG SING

 

 

ADVENT 932 ADVENT

 

 

S
=
1
-
12
SCHRODINGERS
139
67
4
C
=
3
-
3
CAT
24
6
6
D
=
4
-
3
DID
17
17
8
C
=
3
-
10
CURIOUSITY
160
52
7
K
=
2
-
4
KILL
44
17
6
T
=
2
-
3
THE
33
15
6
C
=
3
-
3
CAT
24
6
6
-
-
18
4
38
First Total
441
180
19
-
-
1+8
-
3+8
Add to Reduce
4+4+1
1+8+0
6+2
-
-
9
-
11
Second Total
9
10
10
4
4
-
4
1+1
Reduce to Deduce
-
1+0
1+0
Q
-
9
5
2
Essence of Number
9
7
7

 

 

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

 

 

WISDOM OF THE EAST

by Hari Prasad Shastri 1948

Page 8

"There is no such word in Sanscrita as 'Creation' applied to the universe. The Sanscrita word for Creation is Shristi, which means 'projection' Creation means to bring something into being out /Page 9/ of nothing, to create, as a novelist creates a character. There was no Miranda, for example, until Shakespeare created her. Similarly the ancient Indians (this term is innacurately used as there was no India at that time). who were our ancestors long, long ago. used a word for creation that means 'projection'.

 

 

 

THOUGHT NEGATIVE BALANCING POSITIVE THOUGHT

THOUGHT POSITIVE BALANCING NEGATIVE THOUGHT

FIRST LAW GODS LAW FIRST

 

 

 

I

ME

MIN

MINE OF MIND GODS MIND OF MINE

FROM FORM TRANSFORM TRANSFORM FORM FROM

REAL REALITY REVEALED DIVINE THOUGHT DIVINE REVEALED REAL REALITY

 

 

I

I ME I

SAY

SEE HEAR HEAR SEE

SEE THAT SEA HEAR THAT SEA

SEA THE SEE HERE HERE SEE THE SEA

THAT HERE THAT HEAR THAT HERE THAT

THAT I THAT SEES THAT I THAT SEES THAT I THAT

THE EYES THAT SEE THE SEA THE EARS THAT HEAR THE SEA

MIND CREATORS THAT MIND GODS MIND THAT CREATORS MIND

ACTIONS IN ACTIONS IN ACTIONS GODS ACTIONS IN ACTIONS IN ACTIONS

 

 

Fluorescence spectroscopy or fluorometry or spectrofluorimetry is a type of electromagnetic spectroscopy which analyzes fluorescence from a sample. ...en wikipedia.org/wiki/Fluorescence_spectroscopy

 

Fluorescence is a process distinct from incandescence (ie emission of light due to high temperatures). Heating is generally detrimental to the process of ... www.berthold.com/ww/en/pub/bioanalytik/biomethods/fluor.cfm

 

Fluorescence is the most rapidly expanding microscopy technique in both the medical and biological sciences, a fact which has spurred the development of ... micro.magnet.fsu.edu/primer/techniques/fluorescence/fluorhome.

 

In fluorescence microscopy, the sample you want to study is itself the light source. The technique is used to study specimens, which can be made to ...nobelprize.org/educational_games/

physics/microscopes/fluorescence/index.html

 

 

Daily Mail

Monday February 23, 2009

ANSWERS TO CORRESPONDENTS

Compiled by Charles Legge

Page 57

QUESTION Are the three stars that make up Orion's belt a similar distance from the Earth or do they just appear that way?

ORION, the giant huntsman of Greek mythology whom Zeus placed among the stars as the constellation, has three stars of apparently similar brightness and colour (bluish-white) in his belt, given the Arabic names (from left to right) Alnitak, Alnilam and Mintaka.

In fact, Alnitak is 800 light years away from us, Alnilam 1,300 light years and Mintaka 900 light years. They appear in a straight line only in our line of sight.
Despite this, the three stars appear to be closely associated. Each is a luminous, hot supergiant having luminosities at least 100,000 times that of the Sun, and surface temperatures exceeding 25,000c (our Sun is 5,500c).

It's believed the three stars, and several other equally hot and luminous stars in the constellation Orion, were formed together as a close cluster. The passage of time has seen them drift apart.

Such luminous stars use up hydrogen at a prodigious rate so they're only a few million years old and have no more than a few million years to live before blowing themselves up in a supernova explosion.

These timescales are short in astronomical terms. Our Sun, with its far lower luminosity and lower fuel consumption, has shone for five billion years and is expected to shine steadily for the same amount of time before it, too, dies, in a much more sedate fashion than a supernova explosion.

The five billion years that our Sun has been around has meant that life has had time to develop on one of its planets — Earth.
It seems most unlikely that any hypopthetical planet around a star in Orion's belt could have developed sort of life when the history of the star, from birth to death, is only a few million years.

Norman Wallace, Sutton Coldfield, W Mids

 

7
ALNITAK
68
23
5
7
ALNILAM
62
26
8
7
MINTAKA
69
24
6
21
First Total
199
73
19
2+1
Add to Reduce
1+9+9
7+3
1+9
3
Second Total
19
10
10
-
Reduce to Deduce
1+9
1+0
1+0
3
Third Totalr
10
1
1
-
Reduce to Produce
1+0
-
-
3
Essence of Number
1
1
1

 

 

T
=
2
-
3
TAK
32
5
5
L
=
3
-
3
LAM
26
8
8
T
=
2
-
4
TAKA
33
6
6
-
-
7
4
10
First Total
91
19
19
-
-
-
-
1+0
Add to Reduce
9+1
1+9
1+9
Q
-
7
-
1
Second Total
10
10
10
-
-
-
-
-
Reduce to Deduce
1+0
1+0
1+0
Q
-
7
-
1
Essence of Number
10
10
10

 

 

7
ALNITAK
68
23
5
7
ALNILAM
62
26
8
7
MINTAKA
69
24
6
21
First Total
199
73
19
2+1
Add to Reduce
1+9+9
7+3
1+9
3
Second Total
19
10
10
-
Reduce to Deduce
1+9
1+0
1+0
3
Third Totalr
10
1
1
-
Reduce to Produce
1+0
-
-
3
Essence of Number
1
1
1

 

 

ALNITAK ALNILAM MINTAKA

ALNI TAK ALNI LAM MIN TAKA

1+3+5+9 2+1+2 1+3+5+9 3+1+4 4+9+5 2+1+2+1

ALNI TAK ALNI LAM MIN TAKA

ALNITAK ALNILAM MINTAKA

 

 

I SAY HALLOWED IS APOLLO APOLLO IS HALLOWED SAY I

SAY I HALLOWED IS DIONYSUS IS HALLOWED I SAY

 

 

10
PRINCIPIUM
128
65
2
15
INDIVIDUATIONIS
179
80
8
25
First Total
307
145
10
2+5
Add to Reduce
3+0+7
1+4+5
1+0
7
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
7
Essence of Number
1
1
1

 

 

PRINCIPIUM INDIVIDUATIONIS

-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
1
R
18
9
9
--
--
--
--
--
--
--
-
--
9
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
2
N+C
17
8
8
--
--
--
--
--
--
--
-
8
-
1
I
9
9
9
--
--
--
--
--
--
--
-
9
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
1
I
9
9
9
--
--
--
--
--
--
--
-
7
9
2
U+M
34
7
7
-
-
-
-
-
-
-
7
-
-
1
I
9
9
9
--
--
--
--
--
--
--
-
-
9
2
N+D
18
9
9
--
--
--
--
--
--
--
-
-
9
1
I
9
9
9
-
-
-
-
-
-
-
-
--
9
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
1
I
9
9
9
--
--
--
--
--
--
--
-
--
9
1
D
4
4
4
-
-
-
-
4
-
-
-
--
-
3
U+A+T
42
6
6
--
--
--
-
-
-
6
-
--
-
1
I
9
9
9
--
--
--
-
-
-
-
-
--
9
2
O+N
29
11
2
--
--
2
-
-
-
-
-
--
-
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
25
First Total
307
145
10
-
1
2
3
8
5
6
21
8
90
2+5
Add to Reduce
3+0+7
1+4+5
1+0
-
-
-
-
-
-
-
2+1
-
9+0
7
Second Total
10
10
1
-
1
2
3
8
5
6
3
8
9
-
Reduce to Deduce
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
7
Essence of Number
1
1
1
-
1
2
3
8
5
6
3
8
9

 

PRINCIPIUM INDIVIDUATIONIS

 

-
-
-
-
-
-
4
7
8
9
1
P
16
7
7
-
-
-
-
7
-
-
1
R
18
9
9
--
--
--
--
-
--
9
1
I
9
9
9
-
-
-
-
-
-
9
2
N+C
17
8
8
--
--
--
--
-
8
-
1
I
9
9
9
--
--
--
--
-
9
1
P
16
7
7
-
-
-
-
7
-
-
1
I
9
9
9
--
--
--
--
-
7
9
2
U+M
34
7
7
-
-
-
-
7
-
-
1
I
9
9
9
--
--
--
--
-
-
9
2
N+D
18
9
9
--
--
--
--
-
-
9
1
I
9
9
9
-
-
-
-
-
--
9
1
V
22
4
4
-
-
4
-
-
-
-
1
I
9
9
9
--
--
--
--
-
--
9
1
D
4
4
4
-
-
4
-
-
--
-
3
U+A+T
42
6
6
--
--
-
6
-
--
-
1
I
9
9
9
--
--
-
-
-
--
9
2
O+N
29
11
2
--
2
-
-
-
--
-
1
I
9
9
9
-
-
-
-
-
-
9
25
First Total
307
145
10
-
2
8
6
21
8
90
2+5
Add to Reduce
3+0+7
1+4+5
1+0
-
-
-
-
2+1
-
9+0
7
Second Total
10
10
1
-
2
8
6
3
8
9
-
Reduce to Deduce
1+0
1+0
-
-
-
-
-
-
-
-
7
Essence of Number
1
1
1
-
2
8
6
3
8
9

PRINCIPIUM INDIVIDUATIONIS

 

 

THE MILKY WASTE

 

N
=
5
-
5
NAKED
35
17
8
I
=
9
-
I
I
9
9
9
C
=
3
-
4
CAME
22
13
4
O
=
6
-
3
OUT
56
11
2
O
=
6
-
2
OF
21
12
3
M
=
4
-
2
MY
38
11
2
M
=
4
-
7
MOTHERS
98
35
8
W
=
5
-
4
WOMB
53
17
8
A
=
1
-
3
AND
19
10
1
N
=
5
-
5
NAKED
35
17
8
S
=
1
-
5
SHALL
52
16
7
I
=
9
-
1
I
9
9
9
R
=
9
-
6
RETURN
96
33
6
T
=
2
-
2
TO
35
8
8
E
=
5
-
5
EARTH
52
25
7
-
-
74
-
55
First Total
630
243
90
-
-
7+4
-
5+5
Add to Reduce
6+3+0
2+4+3
9+0
-
-
11
-
10
Second Total
9
9
9
-
-
1+1
-
1+0
Reduce to Deduce
-
-
-
-
-
2
-
1
Essence of Number
9
9
9

 

 

-
-
-
-
-
GODS I BE REAL
-
-
-
G
=
7
-
4
G+O+D+S
45
18
9
I
=
9
-
1
I
9
9
9
B
=
2
-
2
B+E
7
7
7
R
=
9
-
4
R+E+A+L
36
18
9
-
-
27
4
11
GODS I BE REAL
97
52
34
-
-
2+7
-
1+1
-
9+7
5+2
3+4
Q
-
9
-
2
GODS I BE REAL
16
7
7
-
-
-
-
-
-
1+6
-
-
Q
-
9
-
2
GODS I BE REAL
7
7
7

 

 

C
=
3
-
7
CLARIOR
76
40
4
E
=
5
--
2
EX
29
11
2
I
=
9
-
7
IGNIBUS
81
36
9
B
=
2
-
8
BRIGHTER
87
51
6
F
=
6
--
4
FROM
52
25
7
T
=
2
-
3
THE
33
15
6
F
=
6
-
6
FLAMES
56
20
2
-
-
33
--
37
First Total
414
198
36
-
-
3+3
-
3+7
Add to Reduce
4+1+4
1+9+8
3+6
-
-
6
-
10
Second Total
9
18
9
-
-
-
--
1+0
Reduce to Deduce
-
1+8
-
-
-
6
-
1
Essence of Number
9
9
9

 

 

3
THE
33
15
6
7
PHOENIX
91
46
1
2
IS
28
10
1
6
RISING
76
31
4

 

 

7
PHOENIX
91
46
1
6
SPIRIT
91
37
1

 

 

3
THE
33
15
6
7
PHOENIX
-
-
-
-
P+H+O+E+N
58
31
4
5
I
9
9
9
5
X
24
6
6
7
PHOENIX
91
46
19
-
-
9+1
4+6
1+9
7
PHOENIX
10
10
10
-
-
1+0
1+0
1+0
7
PHOENIX
1
1
1

 

 

5
RISES
70
25
7
6
RISING
76
31
4

 

 

6
RISING
76
31
4

 

 

10
BETELGEUSE
101
38
2
6
BEETLE
49
22
4
5
JUICE
48
21
3
21
First Total
198
81
9
2+1
Add to Reduce
1+9+8
8+1
-
3
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
3
Essence of Number
9
9
9

 

 

I
=
9
M
2
IF
15
15
6
T
=
2
O
3
THE
33
15
6
D
=
4
T
5
DOORS
71
26
8
O
=
5
M
2
OF
21
12
3
P
=
7
-
4
PERCEPTION
51
24
6
W
=
5
--
4
WERE
51
24
6
C
=
3
-
8
CLEANSED
63
27
9
E
=
5
M
10
EVERYTHING
133
61
7
W
=
5
O
5
WOULD
75
21
3
A
=
1
M
6
APPEAR
57
30
3
I
=
9
O
8
INFINITE
86
50
5
-
-
56
T
63
First Total
726
339
60
-
-
5+6
M
6+3
Add to Reduce
7+2+6
3+3+9
6+0
-
-
11
-
9
Second Total
15
15
6
-
-
1+1
--
-
Reduce to Deduce
1+5
1+5
-
-
-
2
-
9
Essence of Number
6
6
6

 

 

7
WILLIAM
79
34
7
5
BLAKE
31
13
4
12
First Total
110
47
11
1+2
Add to Reduce
1+1+0
4+7
1+1
3
Second Total
2
11
2
-
Reduce to Deduce
-
1+1
-
3
Essence of Number
2
2
2

 

 

6
CREATE
52
25
7
7
CREATES
71
26
8
7
CREATED
56
29
2
7
CREATOR
80
35
8
7
CREATORS
99
36
9
8
CREATING
77
41
5
10
CREATIVITY
132
51
6

 

 

3
GOD
45
178
9
2
IS
28
10
1
3
THE
33
15
6
6
SPIRIT
91
37
1
2
OF
21
12
3
3
THE
33
15
6
7
PHOENIX
91
37
1
26
Add to Reduce
342
153
27
2+6
Reduce to Deduce
3+4+2
1+5+3
2+7
7
Essence of Number
9
9
9

 

 

T
=
2
-
3
THE
33
15
6
G
=
7
--
3
GOD
26
17
8
C
=
3
--
9
CONSCIOUS
118
37
1
H
=
8
-
5
HUMAN
57
21
3
-
-
20
--
20
Add to Reduce
234
90
18
-
--
2+0
-
2+0
Reduce to Deduce
2+3+4
9+0
1+8
-
-
2
--
2
Essence of Number
9
9
9

 

 

7
MYSTIC
-
-
-
-
MY
38
11
2
-
STICK
62
17
8
7
MY STICK
100
28
10
-
-
1+0+0
2+8
1+0
7
MY STICK
1
10
1
-
-
-
1+0
-
7
MY STICK
1
1
1

 

MY WAND WITH HAND

 

5
MAGUS
61
16
3
6
MYSTIC
89
26
8
6
MYSTICK
100
28
1
5
MAGIC
33
24
6
2
MY
38
11
2
5
STICK
62
17
8
7
MYSTICS
108
27
9
4
MAGI
30
21
3
5
IMAGE
35
26
8
6
IMAGES
54
27
9
7
IMAGERS
72
27
9
8
CREATORS
99
36
9

 

 

M
=
4
-
5
MAGIC
33
24
6
O
=
6
--
2
OF
21
12
3
T
=
2
--
3
THE
33
15
6
M
=
4
-
4
MAGI
30
21
3
-
-
16
--
14
Add to Reduce
117
72
18
--
--
1+6
-
1+4
Reduce to Deduce
1+1+7
7+2
1+8
-
-
7
--
5
Essence of Number
9
9
9

 

 

3
THE
33
15
6
4
MAGI
30
21
3
7
IMAGERS
72
36
9
14
Add to Reduce
135
72
18
1+4
Reduce to Deduce
1+3+5
7+2
1+8
5
Essence of Number
9
9
9

 

 

11
MAGNIFICENT
101
56
2
2
IS
28
10
1
3
THE
33
15
6
5
MAGIC
33
24
6
4
MAGI
30
21
3
25
Add to Reduce
225
126
18
2+5
Reduce to Deduce
2+2+5
1+2+6
1+8
7
Essence of Number
9
9
9

 

 

I
=
9
-
5
IMAGE
35
26
8
I
=
9
--
6
IMAGES
54
27
9
I
=
9
--
7
IMAGERS
72
36
9

 

 

2
MY
38
11
2
4
WAND
42
15
6

 

 

3
THE
33
15
6
4
WAND
42
15
6
4
WAVE
51
15
6
4
HAND
27
18
9
15
Add to Reduce
153
63
27
1+5
Reduce to Deduce
1+5+3
6+3
2+7
6
Essence of Number
9
9
9

 

 

4
WAVE
51
15
6
4
WITH
60
24
6
4
HAND
27
18
9
4
WAND
42
15
6
16
Add to Reduce
180
72
27
1+6
Reduce to Deduce
1+8+0
7+2
2+7
7
Essence of Number
9
9
9

 

 

M
=
4
-
4
MAGI
30
21
3
T
=
2
-
3
THE
33
15
6
M
=
4
-
5
MAGIC
33
24
6
C
=
3
-
1
C
3
3
3
-
-
13
-
13
First Total
99
63
18
-
-
1+3
-
1+3
Add to Reduce
9+9
6+3
1+8
-
-
4
-
4
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
-
4
Essence of Number
9
9
9

 

 

T
=
2
--
3
THE
33
15
6
H
=
8
-
4
HOLY
60
24
6
M
=
4
--
5
MAGUS
61
16
7
-
--
14
-
12
First Total
154
55
19
--
-
1+4
--
1+2
Add to Reduce
1+5+4
5+5
1+9
-
--
5
--
3
Second Total
10
10
10
--
-
---
-
-
Reduce to Deduce
1+0
1+0
1+0
-
--
5
--
3
Essence of Number
1
1
1

 

 

T
=
2
--
3
THE
33
15
6
M
=
4
-
4
MAGI
30
21
3
-
--
6
--
7
Add to Reduce
117
72
18
--
-
-
-
-
Reduce to Deduce
1+1+7
7+2
1+8
-
--
6
--
7
Essence of Number
9
9
9

 

 

-
OPTIONS
-
-
-
1
O
15
6
6
2
P+T
36
9
9
1
I
9
9
9
3
O+N+S
48
12
3
7
OPTIONS
108
36
27
-
-
1+0+8
3+6
2+7
7
OPTIONS
9
9
9

 

 

4
SOFT
60
15
6
6
OPTION
89
35
8
10
First Total
149
50
14
1+0
Add to Reduce
1+4+9
5+0
1+4
1
Second Total
14
5
5
-
Reduce to Deduce
1+4
-
-
1
Essence of Number
5
5
5

 

 

4
SELF
42
15
6
10
INDULGENCE
94
49
4
14
First Total
136
64
10
1+4
Add to Reduce
1+3+6
6+4
1+0
5
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
5
Essence of Number
1
1
1

 

 

9
SPIRITUAL
125
44
8
2
IS
28
10
1
11
Add to Reduce
153
54
9
1+1
Reduce to Deduce
1+5+3
5+4
-
2
Essence of Number
8
8
8

 

 

A
=
1
-
3
ALL
25
7
7
U
=
7
--
3
YOU
61
16
7
S
=
1
-
3
SEE
29
11
2
I
=
9
--
2
IS
28
10
1
S
=
1
--
6
SPIRIT
91
37
1
-
-
19
-
17
First Total
234
81
18
-
-
1+9
-
1+7
Add to Reduce
2+3+4
8+1
1+8
-
-
10
-
8
Second Total
9
9
9
-
-
1+0
-
-
Reduce to Deduce
-
-
-
-
-
1
-
8
Essence of Number
9
9
9

 

 

-
SPIRIT
-
-
-
1
S
19
10
1
1
P
16
7
7
1
I
9
9
9
1
R
18
9
9
1
I
9
9
9
1
T
20
2
2
6
SPIRIT
-
-
-

 

 

-
SPIRIT
-
-
-
2
S+P
35
17
8
1
I
9
9
9
1
R
18
9
9
1
I
9
9
9
1
T
20
2
2
6
SPIRIT
91
46
37
-
-
9+1
4+6
3+7
6
SPIRIT
10
10
10
-
-
1+0
1+0
1+0
6
SPIRIT
1
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

 

 

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
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
-
-
-
-
-
-
-
-
-
-
28
4
4
7
-
73
28
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
-
-
-
-
-
-
-
-
-
-
33
4
4
8
-
93
33
33
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
12
8
25
6
7
8
9
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
E
=
5
4
4
7
EUCLID'S
73
28
1
4
2
2
3
8
7
6
7
8
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
2
2
3
8
7
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
2
2
3
8
7
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
2
2
3
8
7
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
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
2
12
8
25
6
7
8
9
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
E
=
5
4
4
7
EUCLID'S
73
28
1
4
2
2
3
8
7
6
7
8
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
2
2
3
8
7
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
2
2
3
8
7
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
2
2
3
8
7
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
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
-
-
-
-
-
-
-
-
-
-
-
2
2
12
8
25
6
7
8
9
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
-
-
1+2
-
2+5
-
-
-
-
E
=
5
4
4
7
EUCLID'S
73
28
1
4
2
2
3
8
7
6
7
8
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
2
2
3
8
7
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
2
2
3
8
7
6
7
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
2
2
3
8
7
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
-
-
-
-
-
-
-
-
-
-
-
2
2
12
8
25
9
-
-
-
-
-
-
EUCLID'S ELEMENTS
-
-
-
-
-
-
1+2
-
2+5
-
E
=
5
4
4
7
EUCLID'S
73
28
1
4
2
2
3
8
7
9
E
=
5
4
4
8
ELEMENTS
93
30
3
4
-
-
-
-
-
-
S
-
10
4
4
15
EUCLID'S ELEMENTS
166
58
4
4
2
2
3
8
7
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
2
2
3
8
7
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
S
-
1
4
4
6
EUCLID'S ELEMENTS
4
4
4
-
2
2
3
8
7
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.

 

 

-
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
9
-
9
-
-
9
-
-
+
=
27
2+7
=
9
=
9
=
9
-
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
8
3
-
3
-
2
9
-
3
4
+
=
37
3+7
=
10
1+0
1
=
1
-
-
5
17
21
-
12
-
2
18
-
21
13
+
=
109
1+0+9
=
10
1+0
1
=
1
-
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
5
17
21
9
12
9
2
18
9
21
13
+
=
136
1+3+6
=
10
1+0
1
=
1
-
-
5
8
3
9
3
9
2
9
9
3
4
+
=
64
6+4
=
10
1+0
1
=
1
-
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
-
-
-
-
-
1
-
-
--
--
--
--
--
--
--
--
--
--
--
--
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
--
-
3
-
3
-
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
6
-
-
--
--
--
--
--
--
--
--
--
--
--
--
6
-
-
-
-
-
-
-
7
-
-
--
--
--
--
--
--
--
--
--
--
--
--
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
9
-
9
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
14
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
31
-
-
11
-
64
-
37
1+4
1+1
-
-
-
9
-
9
-
9
9
-
-
-
-
3+1
-
-
1+1
-
6+4
-
3+7
5
2
E
Q
U
I
L
I
B
R
I
U
M
-
-
4
-
-
2
-
10
-
10
-
-
5
8
3
9
3
9
2
9
9
3
4
-
-
-
-
-
-
-
1+0
-
1+0
5
2
E
Q
U
I
L
I
B
R
I
U
M
-
-
4
-
-
2
-
1
-
1

 

 

11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
--
-
-
-
-
-
-
-
-
9
-
9
-
-
9
-
-
+
=
27
2+7
=
9
=
9
=
9
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
--
-
-
-
-
-
5
8
3
-
3
-
2
9
-
3
4
+
=
37
3+7
=
10
1+0
1
=
1
-
5
17
21
-
12
-
2
18
-
21
13
+
=
109
1+0+9
=
10
1+0
1
=
1
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
-
-
-
--
-
-
-
-
-
5
17
21
9
12
9
2
18
9
21
13
+
=
136
1+3+6
=
10
1+0
1
=
1
-
5
8
3
9
3
9
2
9
9
3
4
+
=
64
6+4
=
10
1+0
1
=
1
11
E
Q
U
I
L
I
B
R
I
U
M
--
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
--
-
3
-
3
-
-
-
-
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
-
5
-
-
-
-
-
-
-
-
-
-
--
--
5
occurs
x
1
=
5
=
5
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
9
-
9
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
11
E
Q
U
I
L
I
B
R
I
U
M
-
-
31
-
-
11
-
64
-
37
1+1
-
-
-
9
-
9
-
9
9
-
-
-
-
3+1
-
-
1+1
-
6+4
-
3+7
2
E
Q
U
I
L
I
B
R
I
U
M
--
--
4
-
-
2
-
10
-
10
-
5
8
3
9
3
9
2
9
9
3
4
--
--
-
-
-
-
-
1+0
-
1+0
2
E
Q
U
I
L
I
B
R
I
U
M
-
-
4
-
-
2
-
1
-
1

 

 

-
-
-
-
0
ZERO
-
-
-
-
-
-
2
-
Z+E
31
13
4
-
-
-
1
-
R
18
9
9
-
-
-
1
-
O
15
6
6
Z
=
8
4
4
ZERO
64
28
19
-
-
-
-
-
-
9+1
1+9
1+9
Z
=
8
4
-
ZERO
10
10
10
-
-
-
-
-
-
1+0
1+0
1+0
Z
=
8
4
-
ZERO
10
10
10

 

 

-
-
-
-
1
ONE
-
-
-
-
-
-
1
-
O
15
6
6
-
-
-
1
-
N
14
5
5
-
-
-
1
-
E
5
5
5
O
=
6
3
4
ONE
34
16
16
-
-
-
-
-
-
3+4
1+6
1+6
O
=
6
3
4
ONE
7
7
7

 

 

-
-
-
-
2
TWO
-
-
-
-
-
-
1
-
T
20
2
2
-
-
-
1
-
W
23
5
5
-
-
-
1
-
O
15
6
6
T
=
2
3
4
TWO
58
13
13
-
-
-
-
-
-
5+8
1+3
1+3
T
=
2
3
4
TWO
13
4
4
-
-
-
-
-
-
1+3
-
-
T
=
2
3
4
TWO
4
4
4

 

 

-
-
-
-
3
THREE
-
-
-
-
-
-
1
-
T
20
2
2
-
-
-
1
-
H
8
8
8
-
-
-
1
-
R
18
9
9
-
-
-
2
-
E+E
10
10
1
T
=
2
5
4
THREE
56
29
20
-
-
-
-
-
-
5+6
2+9
2+0
T
=
2
5
4
THREE
11
11
2
-
-
-
-
-
-
1+1
1+1
-
T
=
2
5
4
THREE
2
2
2

 

 

-
-
-
-
4
FOUR
-
-
-
-
-
-
1
-
F
6
6
6
-
-
-
2
-
O+U
36
9
9
-
-
-
1
-
R
18
9
9
F
=
6
4
4
FOUR
60
24
24
-
-
-
-
-
-
6+0
2+4
2+4
F
-
6
4
4
FOUR
6
6
6

 

 

-
-
-
-
5
FIVE
-
-
-
-
-
-
1
-
F
6
6
6
-
-
-
1
-
I
9
9
9
-
-
-
2
-
V+E
27
9
9
F
=
6
4
4
FIVE
60
24
24
-
-
-
-
-
-
6+0
2+4
2+4
F
=
6
4
4
FIVE
6
6
6

 

 

-
-
-
-
6
SIX
-
-
-
-
-
-
1
-
S
19
10
1
-
-
-
1
-
I
9
9
9
-
-
-
1
-
X
24
6
6
S
=
1
3
4
SIX
52
25
16
-
-
-
-
-
-
5+2
2+5
1+6
S
=
1
3
4
SIX
7
7
7

 

 

-
-
-
-
7
SEVEN
-
-
-
-
-
-
1
-
S
19
10
1
-
-
-
2
-
E+V
27
9
9
-
-
-
2
-
E+N
19
10
1
S
=
1
5
4
SEVEN
65
29
11
-
-
-
-
-
-
6+5
2+9
1+1
S
=
2
5
4
SEVEN
11
11
2
-
-
-
-
-
-
1+1
1+1
-
S
=
2
5
4
SEVEN
2
2
2

 

 

-
-
-
-
8
EIGHT
-
-
-
-
-
-
1
-
E
5
5
5
-
-
-
1
-
I
9
9
9
-
-
-
3
-
G+H+T
35
17
8
E
=
5
5
4
EIGHT
49
31
4
-
-
-
-
-
-
4+9
3+1
-
E
=
5
5
4
EIGHT
13
4
4
-
-
-
-
-
-
1+3
-
-
E
=
5
5
4
EIGHT
4
4
4

 

 

-
-
-
-
9
NINE
-
-
-
-
-
-
1
-
N
14
5
5
-
-
-
1
-
I
9
9
9
-
-
-
2
-
N+E
19
10
1
N
=
5
4
4
NINE
42
24
13
-
-
-
-
-
-
4+2
2+4
1+3
N
=
5
3
4
NINE
6
6
6

 

 

-
-
-
10
-
TEN
--
-
-
-
-
-
-
1
T
20
2
2
-
-
-
-
1
E
5
5
5
-
-
-
-
1
N
14
5
5
T
=
2
10
3
TEN
39
12
12
-
-
-
-
-
-
3+9
1+2
1+2
T
=
2
10
3
TEN
12
3
3
-
-
-
-
-
-
1+2
-
-
T
=
2
10
3
TEN
3
3
3

 

 

-
-
-
11
-
ELEVEN
--
-
-
-
-
-
-
3
E+L+E
22
13
4
-
-
-
-
2
V+E
27
9
9
-
-
-
-
1
N
14
5
5
E
=
5
11
6
ELEVEN
63
27
18
-
-
-
-
-
-
6+3
2+7
1+8
E
=
5
11
6
ELEVEN
9
9
9

 

 

-
-
-
12
-
TWELVE
--
-
-
-
-
-
-
4
T+W+E+L
60
15
6
-
-
-
-
2
V+E
27
9
9
T
=
2
12
6
TWELVE
87
24
15
-
-
-
-
-
-
8+7
2+4
1+5
T
=
2
12
6
TWELVE
15
6
6
6
-
-
-
-
-
1+5
-
-
T
=
2
12
6
TWELVE
6
6
6

 

 

-
-
-
13
-
THIRTEEN
--
-
-
-
-
-
-
2
T+H
28
10
1
-
-
-
-
1
I
9
9
9
-
-
-
-
1
R
18
9
9
-
-
-
-
4
T+E+E+N
44
17
8
T
=
2
13
8
THIRTEEN
99
45
27
-
-
-
-
-
-
9+9
4+5
2+7
T
=
2
13
8
THIRTEEN
18
9
9
6
-
-
-
-
-
1+8
-
-
T
=
2
13
8
THIRTEEN
9
9
9

 

 

-
-
-
14
-
FOURTEEN
--
-
-
-
-
-
-
1
F
6
6
6
-
-
-
-
2
O+U
36
9
9
-
-
-
-
1
R
18
9
9
-
-
-
-
4
T+E+E+N
44
17
8
F
=
6
14
8
FOURTEEN
104
41
32
-
-
-
-
-
-
1+0+4
4+1
3+2
F
=
6
14
8
FOURTEEN
5
5
5

 

 

-
-
-
15
-
FIFTEEN
--
-
-
-
-
-
-
1
F
6
6
6
-
-
-
-
1
I
9
9
9
-
-
-
-
5
F+T+E+E+N
50
23
5
F
=
6
15
7
FIFTEEN
65
38
20
-
-
-
-
-
-
6+5
3+8
2+0
F
=
6
15
7
FIFTEEN
11
11
2
6
-
-
-
-
-
1+1
1+1
-
F
=
6
15
7
FIFTEEN
2
2
2

 

 

-
-
-
16
-
SIXTEEN
--
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
1
I
9
9
9
-
-
-
-
4
X+T+E+E
54
18
9
-
-
-
-
1
N
14
5
5
S
=
1
16
7
SIXTEEN
96
42
24
-
-
-
-
-
-
9+6
4+2
2+4
S
=
1
16
7
SIXTEEN
15
6
6
6
-
-
-
-
-
1+5
-
-
S
=
1
16
7
SIXTEEN
6
6
6

 

 

-
-
-
17
-
SEVENTEEN
--
-
-
-
-
-
-
1
S
19
10
1
-
-
-
-
2
E+V
27
9
9
-
-
-
-
6
E+N+T+E+E+N
63
27
9
S
=
1
17
9
SEVENTEEN
109
46
19
-
-
-
-
-
-
1+0+9
4+6
1+9
S
=
1
17
9
SEVENTEEN
10
10
10
6
-
-
-
-
-
1+0
1+0
1+0
S
=
1
17
9
SEVENTEEN
1
1
1

 

 

-
-
-
18
-
EIGHTEEN
--
-
-
-
-
-
-
1
E
5
5
5
-
-
-
-
1
I
9
9
9
-
-
-
-
5
G+H+T+E+E
45
27
9
-
-
-
-
1
N
14
5
5
E
=
5
18
8
EIGHTEEN
73
46
28
-
-
-
-
-
-
7+3
4+6
2+8
E
=
5
18
8
EIGHTEEN
10
10
10
6
-
-
-
-
-
1+0
1+0
1+0
E
=
5
18
8
EIGHTEEN
1
1
1

 

 

-
TEEN
--
-
-
1
T
20
2
2
3
E+E+N
24
15
6
4
TEEN
44
17
8
-
-
4+4
1+7
-
4
TEEN
8
8
8

 

 

-
-
-
-
TEEN
--
-
-
-
-
-
1
T
20
2
2
-
-
-
1
E
5
5
5
-
-
-
1
E
5
5
5
-
-
-
1
N
14
5
5
T
=
2
4
TEEN
44
17
17
-
-
-
-
-
4+4
1+7
1+7
T
=
2
4
TEEN
8
8
8

 

 
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