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NUCLEAR FAMILY 19769

 

 

 

 

 

 

 

 

THE

MAGICALALPHABET

 

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

 

 

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

 

 

 

 

 

 

 

THIS IS THE SCENE OF THE SCENE UNSEEN

THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE

 

 

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

 

 

THE DIVINE COMEDY

OF

DANTE ALIGHIERI (1265-1321)

THE FLORENTINE

CANTICA I

HELL

(L'INFERNO)

INTRODUCTION

Page 9

"Midway this way of life we're bound upon

I woke to find myself in a dark wood,

Where the right road was wholly lost and gone."

 

 

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

 

 

THE DIVINE COMEDY

OF

DANTE ALIGHIERI (1265-1321)

THE FLORENTINE

CANTICA I

HELL

(L'INFERNO)

INTRODUCTION

Page 9

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

Even as a wheel moves equal, free from jars,

Already my heart and will were wheeled by love,

The Love that moves the sun and other stars."

 

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

 

 

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

 

 

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

 

 

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

 

 

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

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

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

AT THE THROW OF THE NINTH NUMBER WHEN IN CONJUNCTION SET

THE

FAR YONDER SCRIBE

MADE RECORD OF THEIR FALL

 

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

 

 

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

 

 

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

 

ADVENT 2208 ADVENT

 

RA OSIRIS 2 OSIRIS RA

 

-
RA OSIRIS
-
-
-
1
R
18
9
9
3
AOS
35
17
8
1
I
9
9
9
1
R
18
9
9
1
I
9
9
9
1
S
19
10
1
8
RA OSIRIS
108
45
36
-
1+0+8
4+6
3+6
8
RA OSIRIS
9
9
9

 

 

4
LORD
49
22
4
2
OF
21
12
3
5
DEATH
38
20
2
11
F
108
54
9
1+1
-
1+0+8
5+4
-
2
-
9
9
9

 

 

13
SETI-MER-EN-PTAH
-
-
-
-
SETI
53
26
8
-
MER
36
18
9
-
EN
19
10
1
-
PTAH
45
18
9
13
SETI-MER-EN-PTAH
153
72
27
1+3
-
1+5+3
7+2
2+7
4
SETI-MER-EN-PTAH
9
9
9

 

 

THE EGYPTIAN HEAVEN AND HELL

BOOK OF THE DEAD

E. A.Wallis Budge 1857 - 1934

Page 59/60

CHAPTER OF COMING FORTH BY DAY AND OF MAKING A WAY THROUGH AMMEHET.1

"Saith Osiris, the king, the lord of the two lands, MEN-MAAT-RA, whose word is maat, the son of the Sun, [procceeding] from his body loving him, the lord of crowns, SETI MER-EN-PTAH, whose word is maat, "Homage to "you, O ye lords of maat, who are free from iniquity, "who exist and live for ever and to the double henti "period of everlastingness, MEN-MAAT-, whose word is "maat, the son of the Sun, [proceeding] from his body, " loving him, the lord of diadems, SETI MER-EN-PTAH, / "whose word is maat, before you hath become a "khu (i.e., a spirit) in his attributes, he hath gained " the mastery through his words of power, and he "is laden with his splendours. O deliver ye the "Osiris, the king, the lord of the two lands, MEN-MAAT-RA, whose word is maat, the son of the Sun, lord of diadems, SETI MER-EN-PTAH, whose word is maat, from the crocodile of this Pool of "Matti. He hath his mouth, let him speak there-"with. Let there be granted unto him broad-handed- "ness in your presence, because I know you, and I "know your names. . ."

 

 

THE PERIODIC TABLE

 

Periodic table - Wikipedia

https://en.wikipedia.org › wiki › Periodic_table

The periodic table, also known as the periodic table of elements, is a tabular display of the chemical elements, which are arranged by atomic number, electron? ...

?Periodic trends · ?Extended periodic table · ?Alternative periodic tables · ?Block
chemical properties. The structure of the table shows periodic trends. The seven rows of the table, called periods, generally have metals on the left and nonmetals on the right. The columns, called groups, contain elements with similar chemical behaviours. Six groups have accepted names as well as assigned numbers: for example, group 17 elements are the halogens; and group 18 are the noble gases. Also displayed are four simple rectangular areas or blocks associated with the filling of different atomic orbitals.

The elements from atomic numbers 1 (hydrogen) to 118 (oganesson) have all been discovered or synthesized, completing seven full rows of the periodic table.[1][2] The first 94 elements, hydrogen to plutonium, all occur naturally, though some are found only in trace amounts and a few were discovered in nature only after having first been synthesized.[n 1] Elements 95 to 118 have only been synthesized in laboratories, nuclear reactors, or nuclear explosions.[3] The synthesis of elements having higher atomic numbers is currently being pursued: these elements would begin an eighth row, and theoretical work has been done to suggest possible candidates for this extension. Numerous synthetic radioisotopes of naturally occurring elements have also been produced in laboratories.

The organization of the periodic table can be used to derive relationships between the various element properties, and also to predict chemical properties and behaviours of undiscovered or newly synthesized elements. Russian chemist Dmitri Mendeleev published the first recognizable periodic table in 1869, developed mainly to illustrate periodic trends of the then-known elements. He also predicted some properties of unidentified elements that were expected to fill gaps within the table. Most of his forecasts soon proved to be correct, culminating with the discovery of gallium and germanium in 1875 and 1886 respectively, which corroborated his predictions.[4] Mendeleev's idea has been slowly expanded and refined with the discovery or synthesis of further new elements and the development of new theoretical models to explain chemical behaviour. The modern periodic table now provides a useful framework for analyzing chemical reactions, and continues to be widely used in chemistry, nuclear physics and other sciences. Some discussion remains ongoing regarding the placement and categorisation of specific elements, the future extension and limits of the table, and whether there is an optimal form of the table.

Each chemical element has a unique atomic number (Z) representing the number of protons in its nucleus.[n 2] Most elements have differing numbers of neutrons among different atoms, with these variants being referred to as isotopes. For example, carbon has three naturally occurring isotopes: all of its atoms have six protons and most have six neutrons as well, but about one per cent have seven neutrons, and a very small fraction have eight neutrons. Isotopes are never separated in the periodic table; they are always grouped together under a single element. Elements with no stable isotopes have the atomic masses of their most stable isotopes, where such masses are shown, listed in parentheses.[8]

In the standard periodic table, the elements are listed in order of increasing atomic number Z. A new row (period) is started when a new electron shell has its first electron. Columns (groups) are determined by the electron configuration of the atom; elements with the same number of electrons in a particular subshell fall into the same columns (e.g. oxygen and selenium are in the same column because they both have four electrons in the outermost p-subshell). Elements with similar chemical properties generally fall into the same group in the periodic table, although in the f-block, and to some respect in the d-block, the elements in the same period tend to have similar properties, as well. Thus, it is relatively easy to predict the chemical properties of an element if one knows the properties of the elements around it.[9]

Since 2016, the periodic table has 118 confirmed elements, from element 1 (hydrogen) to 118 (oganesson). Elements 113, 115, 117 and 118, the most recent discoveries, were officially confirmed by the International Union of Pure and Applied Chemistry (IUPAC) in December 2015. Their proposed names, nihonium (Nh), moscovium (Mc), tennessine (Ts) and oganesson (Og) respectively, were made official in November 2016 by IUPAC.[10][11][12][13]

The first 94 elements occur naturally; the remaining 24, americium to oganesson (95–118), occur only when synthesized in laboratories. Of the 94 naturally occurring elements, 83 are primordial and 11 occur only in decay chains of primordial elements.[3] No element heavier than einsteinium (element 99) has ever been observed in macroscopic quantities in its pure form, nor has astatine (element 85); francium (element 87) has been only photographed in the form of light emitted from microscopic quantities (300,000 atoms).[14]

Sets of elements

In chronological order, this section discusses metals and nonmetals (and metalloids); categories of elements; groups and periods; and periodic table blocks. While the recognition of metals as solid, fusible and generally malleable substances dates from antiquity,[15][16] Antoine Lavoisier may have the first to formally distinguish between metals and nonmetals ('non-métalliques') in 1789 with the publication of his 'revolutionary'[17] Elementary Treatise on Chemistry.[18] In 1811, Berzelius referred to nonmetallic elements as metalloids,[19][20] in reference to their ability to form oxyanions.[21][22] In 1825, in a revised German edition of his Textbook of Chemistry,[23][24] he subdivided the metalloids into three classes. These were: constantly gaseous 'gazolyta' (hydrogen, nitrogen, oxygen); real metalloids (sulfur, phosphorus, carbon, boron, silicon); and salt-forming 'halogenia' (fluorine, chlorine, bromine, iodine).[25] Only recently, since the mid-20th century, has the term metalloid been widely used to refer to elements with intermediate or borderline properties between metals and nonmetals. Mendeleev published his periodic table in 1869, along with references to groups of families of elements, and rows or periods of his periodic table. At the same time, Hinrichs wrote that simple lines could be drawn on a periodic table in order to delimit properties of interest, such as elements having metallic lustre (in contrast to those not having such lustre).[26] Charles Janet, in 1928, appears to have been the first to refer to the periodic table's blocks.[27]

Metals, metalloids and nonmetals


Metals, metalloids, nonmetals, and elements with unknown chemical properties.
Classification may vary depending on author's focus.
According to their shared physical and chemical properties, the elements can be classified into the major categories of metals, metalloids and nonmetals. Metals are generally shiny, highly conducting solids that form alloys with one another and salt-like ionic compounds with nonmetals (other than noble gases). A majority of nonmetals are colored or colorless insulating gases; nonmetals that form compounds with other nonmetals feature covalent bonding. In between metals and nonmetals are metalloids, which have intermediate or mixed properties.[28]

Metal and nonmetals can be further classified into subcategories that show a gradation from metallic to non-metallic properties, when going left to right in the rows. The metals may be subdivided into the highly reactive alkali metals, through the less reactive alkaline earth metals, lanthanides and actinides, via the archetypal transition metals, and ending in the physically and chemically weak post-transition metals. Nonmetals may be simply subdivided into the polyatomic nonmetals, being nearer to the metalloids and show some incipient metallic character; the essentially nonmetallic diatomic nonmetals, nonmetallic and the almost completely inert, monatomic noble gases. Specialized groupings such as refractory metals and noble metals, are examples of subsets of transition metals, also known[29] and occasionally denoted.[30]

Placing elements into categories and subcategories based just on shared properties is imperfect. There is a large disparity of properties within each category with notable overlaps at the boundaries, as is the case with most classification schemes.[31] Beryllium, for example, is classified as an alkaline earth metal although its amphoteric chemistry and tendency to mostly form covalent compounds are both attributes of a chemically weak or post-transition metal. Radon is classified as a nonmetallic noble gas yet has some cationic chemistry that is characteristic of metals. Other classification schemes are possible such as the division of the elements into mineralogical occurrence categories, or crystalline structures. Categorizing the elements in this fashion dates back to at least 1869 when Hinrichs[32] wrote that simple boundary lines could be placed on the periodic table to show elements having shared properties, such as metals, nonmetals, or gaseous elements.

Categories
Categorisation as described here can vary among authors
The elements of the periodic table shown here are divided into nine categories; six for the metals, and two for nonmetals, and a metalloid category. The nine categories (or sets) correspond to those found in the literature for the applicable part of the periodic table. Different authors may use different categorisation schema depending on the properties of interest.

An individual category is not necessarily exclusive according to its name, boundary, or shared properties. For example, while beryllium in Group 2 is colored as an alkaline earth metal, it is amphoteric rather than alkaline in nature

 

118 Elements and Their Symbols and Atomic Numbers

The periodic table of elements is widely used in the field of Chemistry to look up chemical elements as they are arranged in a manner that displays periodic trends in the chemical properties of the elements. However, the Periodic table generally displays only the symbol of the element and not its entire name.

Most of the symbols are similar to the name of the element but some symbols of elements have Latin roots. An example for this is silver which denoted by Ag from its Latin name “Argentum”. Another such example would be the symbol ‘Fe’ which is used to denote Iron and can be traced to the Latin word for iron, “Ferrum”. It could prove difficult for a beginner in chemistry to learn the names of all the elements in the periodic table because these symbols do not always correspond to the English names of the elements. Therefore, a list of 118 elements and their symbols and atomic numbers is provided below:

118 Elements and Their Symbols and Atomic Numbers

Name of the Element Symbol of the Element Atomic Number


Hydrogen H 1
Helium He 2
Lithium Li 3
Beryllium Be 4
Boron B 5
Carbon C 6
Nitrogen N 7
Oxygen O 8
Fluorine F 9
Neon Ne 10
Sodium Na 11
Magnesium Mg 12
Aluminium Al 13
Silicon Si 14
Phosphorus P 15
Sulfur S 16
Chlorine Cl 17
Argon Ar 18
Potassium K 19
Calcium Ca 20
Scandium Sc 21
Titanium Ti 22
Vanadium V 23
Chromium Cr 24
Manganese Mn 25
Iron Fe 26
Cobalt Co 27
Nickel Ni 28
Copper Cu 29
Zinc Zn 30
Gallium Ga 31
Germanium Ge 32
Arsenic As 33
Selenium Se 34
Bromine Br 35
Krypton Kr 36
Rubidium Rb 37
Strontium Sr 38
Yttrium Y 39
Zirconium Zr 40
Niobium Nb 41
Molybdenum Mo 42
Technetium Tc 43
Ruthenium Ru 44
Rhodium Rh 45
Palladium Pd 46
Silver Ag 47
Cadmium Cd 48
Indium In 49
Tin Sn 50
Antimony Sb 51
Tellurium Te 52
Iodine I 53
Xenon Xe 54
Cesium Cs 55
Barium Ba 56
Lanthanum La 57
Cerium Ce 58
Praseodymium Pr 59
Neodymium Nd 60
Promethium Pm 61
Samarium Sm 62
Europium Eu 63
Gadolinium Gd 64
Terbium Tb 65
Dysprosium Dy 66
Holmium Ho 67
Erbium Er 68
Thulium Tm 69
Ytterbium Yb 70
Lutetium Lu 71
Hafnium Hf 72
Tantalum Ta 73
Tungsten W 74
Rhenium Re 75
Osmium Os 76
Iridium Ir 77
Platinum Pt 78
Gold Au 79
Mercury Hg 80
Thallium Tl 81
Lead Pb 82
Bismuth Bi 83
Polonium Po 84
Astatine At 85
Radon Rn 86
Francium Fr 87
Radium Ra 88
Actinium Ac 89
Thorium Th 90
Protactinium Pa 91
Uranium U 92
Neptunium Np 93
Plutonium Pu 94
Americium Am 95
Curium Cm 96
Berkelium Bk 97
Californium Cf 98
Einsteinium Es 99
Fermium Fm 100
Mendelevium Md 101
Nobelium No 102
Lawrencium Lr 103
Rutherfordium Rf 104
Dubnium Db 105
Seaborgium Sg 106
Bohrium Bh 107
Hassium Hs 108
Meitnerium Mt 109
Darmstadtium Ds 110
Roentgenium Rg 111
Copernicium Cn 112
Nihonium Nh 113
Flerovium Fl 114
Moscovium Mc 115
Livermorium Lv 116
Tennessine Ts 117
Oganesson Og 118

 

 

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=
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
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7+2
1+8
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-
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8
Essence of Number
9
9
9

 

 

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=
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
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9
Second Total
19
10
1
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Third Total
10
1
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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]

 

 

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

 

 

T
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THE
33
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6
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ELEMENTS
93
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Add to Reduce
126
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Essence of Number
9
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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

 

-
-
-
-
-
-
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
-
-
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
-

 


Pythagoras Euclid history
In particular, most of the story of Pythagoras (500 BC) and Euclid (300 BC) is based on legend. Pythagoras founded a secret society whose mathematics was just one part of their activity. ... The first of the 13 books that make up Euclid's ELEMENTS is devoted to a proof of theorem 47, which is the theorem of Pythagoras.

Euclid and Pythagoras - Math Central

 

PYTHAGORAS AND EUCLID

Question 1.
In about 300 BC Euclid recorded a proof of Pythagoras rule. Disscuss Euclid's contribution to developing the theroem.

Question 2.
Why was it named after Pyhagoras if he did not orginally discover it?

If you could possibly forward me some websites which could give an indicator to these answers, I would be most grateful or if you do have the answer yourself, that would be even better.

Yours faithfully,

Scott
Melbourne, Australia.
Hi Scott,
First, some general comments about the history of Greek mathematics. Almost the documents have been lost. There are almost no historical FACTS -- all that is "known" comes from conjectures made by clever and wise scholars who have studied these things all their lives. The stories are nice and carefully conceived, but how close they come to the truth is anybody's guess.

In particular, most of the story of Pythagoras (500 BC) and Euclid (300 BC) is based on legend. Pythagoras founded a secret society whose mathematics was just one part of their activity. Although it is possible that Pythagoras was a brilliant mathematician, it is also possible that all the mathematical discoveries were made by his followers and attributed to the leader. It is certain that the theorem and some sort of proof came from the Pythagoreans at the time of Pythagoras. (The theorem itself has been discovered independently both before and after by various other cultures; of course, those cultures had no real notion of a mathematical proof -- the concept of "proof" originated with the Greeks shortly before the time of Pythagoras.)

Euclid is best known for having compiled a geometry text that gathered together all that was known at the time about geometry and proofs. The book was so brilliantly written that it replaced all the mathematics that preceded it, and all the earlier work was subsequently lost. As a consequence, there is no way to be certain about what Euclid himself contributed. The first of the 13 books that make up Euclid's ELEMENTS is devoted to a proof of theorem 47, which is the theorem of Pythagoras. That proof is generally thought to have been devised by Euclid himself for his book. It depends on most of the 46 theorems that precede it. Of course, there are hunreds of different ways to prove the Pythagorean theorem. There is no way of knowing how or if Pythagoras proved it.

Chris

 

-
-
-
-
-
GEO
-
-
-
G
=
7
-
1
G
7
7
7
E
=
5
-
1
E
5
5
5
O
=
6
-
1
O
15
6
6
-
-
18
-
8
GEO
27
18
18
-
-
1+8
-
-
-
2+7
1+8
1+8
-
-
9
-
8
GEO
9
9
9

 

 

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

 

 

G
=
7
-
-
GEOMETRY
-
-
-
-
-
-
-
3
G+E+O
27
18
9
-
-
-
-
2
M+E
18
9
9
-
-
-
-
2
T+Y
45
9
9
-
-
-
-
1
R
18
9
9
G
=
7
-
8
GEOMETRY
108
45
45
-
-
-
-
-
-
1+0+8
4+5
4+5
G
=
7
-
8
GEOMETRY
9
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

 

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

THE

NAME

EUCLID 533394 EUCLID

THE

EVER

PRESENT

NUMBERS

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

 

 

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

 

-
-
-
-
-
-
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

 

 

-
-
-
-
-
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

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
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

 

 

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

 

 

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."

 

THE

NAME

PYTHAGORAS 7728176911 PYTHAGORAS

THE

MISSING

NUMBERS

3 4 5

THE

NAME

EUCLID 533394 EUCLID

THE

EVER

PRESENT

NUMBERS

3 4 5 - 9

 

 

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.

 

THE

NAME

PYTHAGORAS 7728176911 PYTHAGORAS

THE

MISSING

NUMBERS

3 4 5

THE

NAME

EUCLID 533394 EUCLID

THE

EVER

PRESENT

NUMBERS

3 4 5 - 9

 

 

-
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

 

 

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."

 

THE

NAME

PYTHAGORAS 7728176911 PYTHAGORAS

THE

MISSING

NUMBERS

3 4 5

THE

NAME

EUCLID 533394 EUCLID

THE

EVER

PRESENT

NUMBERS

3 4 5 - 9

 

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 7728176911 PYTHAGORAS

THE

MISSING

NUMBERS

3 4 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

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
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

 

 

-
-
-
-
-
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 EUCLID
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
4
4
10
PYTHAGORAS
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
6
EUCLID
54
27
9
4
-
-
-
-
-
-
-
-
-
S
-
12
4
4
16
PYTHAGORAS EUCLID
127
64
10
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
-
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
Y
=
7
-
2
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
T
=
2
-
3
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
-
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
5
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
G
=
7
-
6
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
-
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
R
=
9
-
8
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
A
=
1
-
9
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
S
=
1
4
10
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
49
-
4
8
-
130
58
49
-
-
-
-
-
-
-
-
-
-
E
=
5
-
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
U
=
3
-
12
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
C
=
3
-
13
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
L
=
3
-
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
-
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
D
=
7
-
15
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
-
-
27
4
4
8
-
54
27
27
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
2
9
4
5
6
21
8
18
-
-
-
-
-
-
PYTHAGORAS EUCLID
-
-
-
-
-
-
-
-
-
-
2+1
-
1+8
P
=
7
4
4
10
PYTHAGORAS
130
58
4
4
3
2
9
4
5
6
3
8
9
E
=
5
4
4
6
EUCLID
54
27
9
4
-
-
-
-
-
-
-
-
-
S
-
12
4
4
16
PYTHAGORAS EUCLID
184
85
13
4
3
2
9
4
5
6
3
8
9
-
-
1+2
4
4
1+6
-
1+8+4
8+5
1+3
-
-
-
-
-
-
-
-
-
-
S
-
3
4
4
7
PYTHAGORAS EUCLID
13
13
4
4
3
2
3
4
5
6
3
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
34
4
7
PYTHAGORAS EUCLID
4
4
4
-
3
2
9
4
5
6
3
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
-
PYTHAGORAS EUCLID
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
4
4
10
PYTHAGORAS
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
6
EUCLID
54
27
9
4
-
-
-
-
-
-
-
-
-
S
-
12
4
4
16
PYTHAGORAS EUCLID
127
64
10
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
PYTHAGORAS
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
A
=
1
-
9
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
S
=
1
4
10
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
-
3
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
O
=
6
-
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
P
=
7
-
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
Y
=
7
-
2
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
G
=
7
-
6
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
-
4
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
R
=
9
-
8
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
49
-
-
10
PYTHAGORAS
130
58
49
-
3
2
3
4
5
6
21
8
9
-
-
-
-
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
EUCLID
-
-
-
-
1
2
3
4
5
6
7
8
9
U
=
3
-
12
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
C
=
3
-
13
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
L
=
3
-
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
D
=
7
-
15
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
-
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
-
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
-
-
27
4
4
8
EUCLID-
54
27
27
4
1
2
9
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
PYTHAGORAS EUCLID
-
-
-
-
-
-
-
-
-
-
-
-
-
P
=
7
4
4
10
PYTHAGORAS
130
58
4
4
3
2
9
4
5
6
3
8
9
E
=
5
4
4
6
EUCLID
54
27
9
4
-
-
-
-
-
-
-
-
-
S
-
12
4
4
16
PYTHAGORAS EUCLID
184
85
13
4
3
2
9
4
5
6
3
8
9
-
-
1+2
4
4
1+6
-
1+8+4
8+5
1+3
-
-
-
-
-
-
-
-
-
-
S
-
3
4
4
7
PYTHAGORAS EUCLID
13
13
4
4
3
2
3
4
5
6
3
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
34
4
7
PYTHAGORAS EUCLID
4
4
4
-
3
2
9
4
5
6
3
8
9

 

 

-
-
-
-
-
-
PYTHAGORAS EUCLID
-
-
-
-
1
2
3
4
5
6
7
8
9
P
=
7
4
4
10
PYTHAGORAS
73
37
1
4
-
-
-
-
-
-
-
-
-
E
=
5
4
4
6
EUCLID
54
27
9
4
-
-
-
-
-
-
-
-
-
S
-
12
4
4
16
PYTHAGORAS EUCLID
127
64
10
4
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
PYTHAGORAS
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
1
A
1
1
1
-
1
-
3
4
5
-
-
-
-
A
=
1
-
9
1
A
1
1
1
-
1
-
3
4
5
-
-
-
-
S
=
1
4
10
1
S
19
10
1
-
1
-
3
4
5
-
-
-
-
T
=
2
-
3
1
T
20
2
2
-
-
2
3
4
5
-
-
-
-
O
=
6
-
7
1
O
15
6
6
-
-
-
3
4
5
6
-
-
-
P
=
7
-
1
1
P
16
7
7
-
-
-
3
4
5
-
7
-
-
Y
=
7
-
2
1
Y
25
7
7
-
-
-
3
4
5
-
7
-
-
G
=
7
-
6
1
G
7
7
7
-
-
-
3
4
5
-
7
-
-
H
=
8
-
4
1
H
8
8
8
-
-
-
3
4
5
-
-
8
-
R
=
9
-
8
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
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
EUCLID
-
-
-
-
1
2
3
4
5
6
7
8
9
U
=
3
-
12
1
U
21
3
3
-
1
2
3
-
-
6
7
8
-
C
=
3
-
13
1
C
3
3
3
-
1
2
3
-
-
6
7
8
-
L
=
3
-
12
1
L
12
3
3
-
1
2
3
-
-
6
7
8
-
D
=
7
-
15
1
D
4
4
4
-
1
1
-
4
-
6
7
8
-
E
=
5
-
11
1
E
5
5
5
-
1
2
-
-
5
6
7
8
-
I
=
9
-
14
1
I
9
9
9
-
1
2
-
-
-
6
7
8
9
-
-
27
4
4
8
EUCLID-
54
27
27
4
1
2
9
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
PYTHAGORAS EUCLID
-
-
-
-
-
-
-
-
-
-
-
-
-
P
=
7
4
4
10
PYTHAGORAS
130
58
4
4
3
2
9
4
5
6
3
8
9
E
=
5
4
4
6
EUCLID
54
27
9
4
-
-
-
-
-
-
-
-
-
S
-
12
4
4
16
PYTHAGORAS EUCLID
184
85
13
4
3
2
9
4
5
6
3
8
9
-
-
1+2
4
4
1+6
-
1+8+4
8+5
1+3
-
-
-
-
-
-
-
-
-
-
S
-
3
4
4
7
PYTHAGORAS EUCLID
13
13
4
4
3
2
3
4
5
6
3
8
9
-
-
-
-
-
-
-
1+3
1+3
-
4
-
-
-
-
-
-
-
-
-
S
-
1
34
4
7
PYTHAGORAS EUCLID
4
4
4
-
3
2
9
4
5
6
3
8
9

 

 

THE

NAME

PYTHAGORAS 7728176911 PYTHAGORAS

THE

MISSING

NUMBERS

3 4 5

THE

NAME

EUCLID 533394 EUCLID

THE

EVER

PRESENT

NUMBERS

3 4 5 - 9

 

 
 
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