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p. 69
Pythagorean
Mathematics
CONCERNING the secret
significance of numbers there has been much speculation. Though many
interesting discoveries have been made, it may be safely said that with the
death of Pythagoras the great key to this science was lost. For nearly 2500
years philosophers of all nations have attempted to unravel the Pythagorean
skein, but apparently none has been successful. Notwithstanding attempts made
to obliterate all records of the teachings of Pythagoras, fragments have
survived which give clues to some of the simpler parts of his philosophy. The
major secrets were never committed to writing, but were communicated orally to
a few chosen disciples. These apparently dated not divulge their secrets to
the profane, the result being that when death sealed their lips the arcana
died with diem.
Certain of the secret schools
in the world today are perpetuations of the ancient Mysteries, and although it
is quite possible that they may possess some of the original numerical formulæ,
there is no evidence of it in the voluminous writings which have issued from
these groups during the last five hundred years. These writings, while
frequently discussing Pythagoras, show no indication of a more complete
knowledge of his intricate doctrines than the postPythagorean Greek
speculators had, who talked much, wrote little, knew less, and concealed their
ignorance under a series of mysterious hints and promises. Here and there
among the literary products of early writers are found enigmatic statements
which they made no effort: to interpret. The following example is quoted from
Plutarch:
"The Pythagoreans indeed go
farther than this, and honour even numbers and geometrical diagrams with the
names and titles of the gods. Thus they call the equilateral triangle
headborn Minerva and Tritogenia, because it may be equally divided by three
perpendiculars drawn from each of the angles. So the unit they term Apollo, as
to the number two they have affixed the name of strife and audaciousness, and
to that of three, justice. For, as doing an injury is an extreme on the one
side, and suffering one is an extreme on the on the one side, and suffering in
the middle between them. In like manner the number thirtysix, their Tetractys,
or sacred Quaternion, being composed of the first four odd numbers added to
the first four even ones, as is commonly reported, is looked upon by them as
the most solemn oath they can take, and called Kosmos." (Isis and Osiris.)
Earlier in the same work,
Plutarch also notes: "For as the power of the triangle is expressive of the
nature of Pluto, Bacchus, and Mars; and the properties of the square of Rhea,
Venus, Ceres, Vesta, and Juno; of the Dodecahedron of Jupiter; so, as we are
informed by Eudoxus, is the figure of fiftysix angles expressive of the
nature of Typhon." Plutarch did not pretend to explain the inner significance
of the symbols, but believed that the relationship which Pythagoras
established between the geometrical solids and the gods was the result of
images the great sage had seen in the Egyptian temples.
Albert Pike, the great Masonic
symbolist, admitted that there were many points concerning which he could
secure no reliable information. In his Symbolism, for the 32° and 33°,
he wrote: "I do not understand why the 7 should be called Minerva, or the
cube, Neptune." Further on he added: "Undoubtedly the names given by the
Pythagoreans to the different numbers were themselves enigmatical and
symbolicand there is little doubt that in the time of Plutarch the meanings
these names concealed were lost. Pythagoras had succeeded too well in
concealing his symbols with a veil that was from the first impenetrable,
without his oral explanation * * *."
This uncertainty shared by all
true students of the subject proves conclusively that it is unwise to make
definite statements founded on the indefinite and fragmentary information
available concerning the Pythagorean system of mathematical philosophy. The
material which follows represents an effort to collect a few salient points
from the scattered records preserved by disciples of Pythagoras and others who
have since contacted his philosophy.
METHOD OF SECURING THE
NUMERICAL POWER OF WORDS
The first step in obtaining the
numerical value of a word is to resolve it back into its original tongue. Only
words of Greek or Hebrew derivation can be successfully analyzed by this
method, and all words must be spelled in their most ancient and complete
forms. Old Testament words and names, therefore, must be translated back
into the early Hebrew characters and New Testament words into the Greek. Two
examples will help to clarify this principle.
The Demiurgus of the
Jews is called in English Jehovah, but when seeking the numerical value
of the name Jehovah it is necessary to resolve the name into its Hebrew
letters. It becomes יהוה, and is read from right to left. The Hebrew letters
are: ה, He; ו, Vau; ה, He; י, Yod; and when reversed into the English order
from left to right read: YodHeVauHe. By consulting the foregoing
table of letter values, it is found that the four characters of this sacred
name have the following numerical significance: Yod equals 10. He
equals 5, Vau equals 6, and the second He equals 5. Therefore,
10+5+6+5=26, a synonym of Jehovah. If the English letters were used,
the answer obviously would not be correct.
The second example is the
mysterious Gnostic pantheos Abraxas. For this name the Greek table is
used. Abraxas in Greek is Ἀβραξας. Α = 1, β = 2, ρ = 100, α = 1, ξ =60, α = 1,
ς = 200, the sum being 365, the number of days in the year. This number
furnishes the key to the mystery of Abraxas, who is symbolic of the 365 Æons,
or Spirits of the Days, gathered together in one composite personality.
Abraxas is symbolic of five creatures, and as the circle of the year
actually consists of 360 degrees, each of the emanating deities is onefifth
of this power, or 72, one of the most sacred numbers in the Old Testament of
the Jews and in their Qabbalistic system. This same method is used in finding
the numerical value of the names of the gods and goddesses of the Greeks and
Jews.
All higher numbers can be
reduced to one of the original ten numerals, and the 10 itself to 1.
Therefore, all groups of numbers resulting from the translation of names of
deities into their numerical equivalents have a basis in one of the first ten
numbers. By this system, in which the digits are added together, 666 becomes
6+6+6 or 18, and this, in turn, becomes 1+8 or 9. According to Revelation,
144,000 are to be saved. This number becomes 1+4+4+0+0+0, which equals 9, thus
proving that both the Beast of Babylon and the number of the saved refer to
man himself, whose symbol is the number 9. This system can be used
successfully with both Greek and Hebrew letter values.
The original Pythagorean system
of numerical philosophy contains nothing to justify the practice now in vogue
of changing the given name or surname in the hope of improving the temperament
or financial condition by altering the name vibrations.
There is also a system of
calculation in vogue for the English language, but its accuracy is a matter of
legitimate dispute. It is comparatively modern and has no relationship either
to the Hebrew Qabbalistic system or to the Greek procedure. The claim made by
some that it is Pythagorean is not supported by any tangible evidence, and
there are many reasons why such a contention is untenable. The fact that
Pythagoras used 10 as the basis of calculation, while this system uses 9an
imperfect numberis in itself almost conclusive. Furthermore, the arrangement
of the Greek and Hebrew letters does not agree closely enough with the English
to permit the application of the number sequences of one language to the
number sequences of the others. Further experimentation with
THE NUMERICAL VALUES OF THE HEBREW, GREEK, AND SAMARITAN ALPHABETS.
From Higgins' Celtic Druids.
Column 

1 
Names of the Hebrew
letters. 
2 
Samaritan Letters. 
3 
Hebrew and Chaldean
letters. 
4 
Numerical equivalents of
the letters. 
5 
Capital and small Greek
letters. 
6 
The letters marked with
asterisks are those brought to Greece from Phnicia by Cadmus. 
7 
Name of the Greek letters. 
8 
Nearest English equivalents
to the Hebrew, Greek, and Samaritan Letters. 
NOTE. When used at the end of a
word, the Hebrew Tau has the numerical value 440, Caph 500,
Mem 600, Nun 700, Pe 800, Tzadi 900. A dotted
Alpha and a dashed Aleph have the value of 1,000.
p. 70
the system may prove
profitable, but it is without basis in antiquity. The arrangement of the
letters and numbers is as follows:
1 
2 
3 
4 
5 
6 
7 
8 
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 

The letters under each of the
numbers have the value of the figure at: the top of the column. Thus, in the
word man, M = 4, A = 1, N = 5: a total of 10. The
values of the numbers are practically the same as those given by the
Pythagorean system.
AN INTRODUCTION TO THE
PYTHAGOREAN THEORY OF NUMBERS
(The following outline of
Pythagorean mathematics is a paraphrase of the opening chapters of Thomas
Taylor's Theoretic Arithmetic, the rarest and most important
compilation of Pythagorean mathematical fragments extant.)
The Pythagoreans declared
arithmetic to be the mother of the mathematical sciences. This is proved by
the fact that geometry, music, and astronomy are dependent upon it but it is
not dependent upon them. Thus, geometry may be removed but arithmetic will
remain; but if arithmetic be removed, geometry is eliminated. In the same
manner music depends upon arithmetic, but the elimination of music affects
arithmetic only by limiting one of its expressions. The Pythagoreans also
demonstrated arithmetic to be prior to astronomy, for the latter is dependent
upon both geometry and music. The size, form, and motion of the celestial
bodies is determined by the use of geometry; their harmony and rhythm by the
use of music. If astronomy be removed, neither geometry nor music is injured;
but if geometry and music be eliminated, astronomy is destroyed. The priority
of both geometry and music to astronomy is therefore established. Arithmetic,
however, is prior to all; it is primary and fundamental.
Pythagoras instructed his
disciples that the science of mathematics is divided into two major parts. The
first is concerned with the multitude, or the constituent parts of a
thing, and the second with the magnitude, or the relative size or
density of a thing.
Magnitude is divided
into two partsmagnitude which is stationary and magnitude which is movable,
the stationary pare having priority. Multitude is also divided into two
parts, for it is related both to itself and to other things, the first
relationship having priority. Pythagoras assigned the science of arithmetic to
multitude related to itself, and the art of music to multitude related to
other things. Geometry likewise was assigned to stationary magnitude, and
spherics (used partly in the sense of astronomy) to movable magnitude. Both
multitude and magnitude were circumscribed by the circumference of mind. The
atomic theory has proved size to be the result of number, for a mass is made
up of minute units though mistaken by the uninformed for a single simple
substance.
Owing to the fragmentary
condition of existing Pythagorean records, it is difficult to arrive at exact
definitions of terms. Before it is possible, however, to unfold the subject
further some light must he cast upon the meanings of the words number, monad,
and one.
The monad signifies (a)
the allincluding ONE. The Pythagoreans called the monad the "noble number,
Sire of Gods and men." The monad also signifies (b) the sum of any combination
of numbers considered as a whole. Thus, the universe is considered as a monad,
but the individual parts of the universe (such as the planets and elements)
are monads in relation to the parts of which they themselves are composed,
though they, in turn, are parts of the greater monad formed of their sum. The
monad may also be likened (c) to the seed of a tree which, when it has grown,
has many branches (the numbers). In other words, the numbers are to the monad
what the branches of the tree are to the seed of the tree. From the study of
the mysterious Pythagorean monad, Leibnitz evolved his magnificent theory of
the world atomsa theory in perfect accord with the ancient teachings of the
Mysteries, for Leibnitz himself was an initiate of a secret school. By some
Pythagoreans the monad is also considered (d) synonymous with the one.
Number is the term
applied to all numerals and their combinations. (A strict interpretation of
the term number by certain of the Pythagoreans excludes 1 and 2.) Pythagoras
defines number to be the extension and energy of the spermatic reasons
contained in the monad. The followers of Hippasus declared number to be the
first pattern used by the Demiurgus in the formation of the universe.
The one was defined by
the Platonists as "the summit of the many." The one differs from the
monad in that the term monad is used to designate the sum of the parts
considered as a unit, whereas the one is the term applied to each of
its integral parts.
There are two orders of number:
odd and even. Because unity, or 1, always remains indivisible,
the odd number cannot be divided equally. Thus, 9 is 4+1+4, the unity in the
center being indivisible. Furthermore, if any odd number be divided into two
parts, one part will always be odd and the other even. Thus, 9 may be 5+4,
3+6, 7+2, or 8+1. The Pythagoreans considered the odd numberof which the
monad was the prototypeto be definite and masculine. They were not all
agreed, however, as to the nature of unity, or 1. Some declared it to be
positive, because if added to an even (negative) number, it produces an odd
(positive) number. Others demonstrated that if unity be added to an odd
number, the latter becomes even, thereby making the masculine to be feminine.
Unity, or 1, therefore, was considered an androgynous number, partaking of
both the masculine and the feminine attributes; consequently both odd and
even. For this reason the Pythagoreans called it evenlyodd. It was
customary for the Pythagoreans to offer sacrifices of an uneven number of
objects to the superior gods, while to the goddesses and subterranean spirits
an even number was offered.
Any even number may be divided
into two equal parts, which are always either both odd or both even. Thus, 10
by equal division gives 5+5, both odd numbers. The same principle holds true
if the 10 be unequally divided. For example, in 6+4, both parts are even; in
7+3, both parts are odd; in 8+2, both parts are again even; and in 9+1, both
parts are again odd. Thus, in the even number, however it may be divided, the
parts will always be both odd or both even. The Pythagoreans considered the
even numberof which the duad was the prototypeto be indefinite and
feminine.
The odd numbers are divided by
a mathematical contrivancecalled "the Sieve of Eratosthenes"into three
general classes: incomposite, composite, and incompositecomposite.
The incomposite numbers
are those which have no divisor other than themselves and unity, such as 3, 5,
7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, and so forth. For example, 7 is
divisible only by 7, which goes into itself once, and unity, which goes into 7
seven times.
The composite numbers
are those which are divisible not only by themselves and unity but also by
some other number, such as 9, 15, 21, 25, 27, 33, 39, 45, 51, 57, and so
forth. For example, 21 is divisible not only by itself and by unity, but also
by 3 and by 7.
The incompositecomposite
numbers are those which have no common divisor, although each of itself is
capable of division, such as 9 and 25. For example, 9 is divisible by 3 and 25
by 5, but neither is divisible by the divisor of the other; thus they have no
common divisor. Because they have individual divisors, they are called
composite; and because they have no common divisor, they are called in,
composite. Accordingly, the term incompositecomposite was created to
describe their properties.
Even numbers are divided
into three classes: evenlyeven, evenlyodd, and oddlyodd.
The evenlyeven numbers
are all in duple ratio from unity; thus: 1, 2, 4, 8, 16, 32, 64, 128, 256,
512, and 1,024. The proof of the perfect evenlyeven number is that it
can be halved and the halves again halved back to unity, as 1/2 of 64 = 32;
1/2 of 32 = 16; 1/2 of 16 = 8; 1/2 of 8 = 4; 1/2 of 4 = 2; 1/2 of 2 = 1;
beyond unity it is impossible to go.
The evenlyeven numbers
possess certain unique properties. The sum of any number of terms but the last
term is always equal to the last term minus one. For example: the sum of the
first and second terms (1+2) equals the third term (4) minus one; or, the sum
of the first, second, third, and fourth terms (1+2+4+8) equals the fifth term
(16) minus one.
In a series of evenlyeven
numbers, the first multiplied by the last equals the last, the second
multiplied by the second from the last equals the last, and so on until in an
odd series one number remains, which multiplied by itself equals the last
number of the series; or, in an even series two numbers remain, which
multiplied by each other give the last number of the series. For example: 1,
2, 4, 8, 16 is an odd series. The first number (1) multiplied by the last
number (16) equals the last number (16). The second number (2) multiplied by
the second from the last number (8) equals the last number (16). Being an odd
series, the 4 is left in the center, and this multiplied by itself also equals
the last number (16).
The evenlyodd numbers
are those which, when halved, are incapable of further division by halving.
They are formed by taking the odd numbers in sequential order and multiplying
them by 2. By this process the odd numbers 1, 3, 5, 7, 9, 11 produce the
evenlyodd numbers, 2, 6, 10, 14, 18, 22. Thus, every fourth number is
evenlyodd. Each of the evenodd numbers may be divided once, as 2, which
becomes two 1's and cannot be divided further; or 6, which becomes two 3's and
cannot be divided further.
Another peculiarity of the
evenlyodd numbers is that if the divisor be odd the quotient is always even,
and if the divisor be even the quotient is always odd. For example: if 18 be
divided by 2 (an even divisor) the quotient is 9 (an odd number); if 18 be
divided by 3 (an odd divisor) the quotient is 6 (an even number).
The evenlyodd numbers
are also remarkable in that each term is onehalf of the sum of the terms on
either side of it. For example: [paragraph
continues]
p. 71
THE SIEVE OF ERATOSTHENES.
Redrawn from Taylor's
Theoretic Arithmetic.
This sieve is a mathematical
device originated by Eratosthenes about 230 B.C. far the purpose of
segregating the composite and incomposite odd numbers. Its use is extremely
simple after the theory has once been mastered. All the odd numbers are first
arranged in their natural order as shown in the second panel from the bottom,
designated Odd Numbers. It will then be seen that every third number
(beginning with 3) is divisible by 3, every fifth number (beginning with 5;)
is divisible by 5, every seventh number (beginning with 7) is divisible by 7,
every ninth number (beginning with 9) is divisible by 9, every eleventh number
(beginning with 11) is divisible by 11, and so on to infinity. This system
finally sifts out what the Pythagoreans called the "incomposite" numbers, or
those having no divisor other than themselves and unity. These will be found
in the lowest panel, designated Primary and Incomposite Numbers. In his
History of Mathematics, David Eugene Smith states that Eratosthenes was
one of the greatest scholars of Alexandria and was called by his admirers "the
second Plato." Eratosthenes was educated at Athens, and is renowned not only
for his sieve but for having computed, by a very ingenious method, the
circumference and diameter of the earth. His estimate of the earth's diameter
was only 50 miles less than the polar diameter accepted by modern scientists.
This and other mathematical achievements of Eratosthenes, are indisputable
evidence that in the third century before Christ the Greeks not only knew the
earth to be spherical in farm but could also approximate, with amazing
accuracy, its actual size and distance from both the sun and the moon.
Aristarchus of Samos, another great Greek astronomer and mathematician, who
lived about 250 B.C., established by philosophical deduction and a few simple
scientific instruments that the earth revolved around the sun. While
Copernicus actually believed himself to be the discoverer of this fact, he but
restated the findings advanced by Aristarchus seventeen hundred years earlier.
__________________________
[paragraph
continues] 10 is onehalf of the sum of 6 and 14; 18 is
onehalf the sum of 14 and 22; and 6 is onehalf the sum of 2 and 10.
The oddlyodd, or
unevenlyeven, numbers are a compromise between the evenlyeven and the
evenlyodd numbers. Unlike the evenlyeven, they cannot be halved back to
unity; and unlike the evenlyodd, they are capable of more than one division
by halving. The oddlyodd numbers are formed by multiplying the evenlyeven
numbers above 2 by the odd numbers above one. The odd numbers above one are 3,
5, 7, 9, 11, and so forth. The evenlyeven numbers above 2 are 4, 8, 16, 32,
64, and soon. The first odd number of the series (3) multiplied by 4 (the
first evenlyeven number of the series) gives 12, the first oddlyodd number.
By multiplying 5, 7, 9, 11, and so forth, by 4, oddlyodd numbers are found.
The other oddlyodd numbers are produced by multiplying 3, 5, 7, 9, 11, and so
forth, in turn, by the other evenlyeven numbers (8, 16, 32, 64, and so
forth). An example of the halving of the oddlyodd number is as follows: 1/2
of 12 = 6; 1/2 of 6 = 3, which cannot be halved further because the
Pythagoreans did not divide unity.
Even numbers are also divided
into three other classes: superperfect, deficient, and
perfect.
Superperfect or
superabundant numbers are such as have the sum of their fractional parts
greater than themselves. For example: 1/2 of 24 = 12; 1/4 = 6; 1/3 = 8; 1/6 =
4; 1/12 = 2; and 1/24 = 1. The sum of these parts (12+6+8+4+2+1) is 33, which
is in excess of 24, the original number.
Deficient numbers are
such as have the sum of their fractional parts less than themselves. For
example: 1/2 of 14 = 7; 1/7 = 2; and 1/14 = 1. The sum of these parts (7+2+1)
is 10, which is less than 14, the original number.
Perfect numbers are such
as have the sum of their fractional parts equal to themselves. For example:
1/2 of 28 = 14; 1/4 = 7; 1/7 = 4; 1/14 = 2; and 1/28 = 1. The sum of these
parts (14+7+4+2+1) is equal to 28.
The perfect numbers are
extremely rare. There is only one between 1 and 10, namely, 6; one between 10
and 100, namely, 28; one between 100 and 1,000, namely, 496; and one between
1,000 and 10,000, namely, 8,128. The perfect numbers are found by the
following rule: The first number of the evenlyeven series of numbers (1, 2,
4, 8, 16, 32, and so forth) is added to the second number of the series, and
if an incomposite number results it is multiplied by the last number of the
series of evenlyeven numbers whose sum produced it. The product is the first
perfect number. For example: the first and second evenlyeven numbers are 1
and 2. Their sum is 3, an incomposite number. If 3 be multiplied by 2, the
last number of the series of evenlyeven numbers used to produce it, the
product is 6, the first perfect number. If the addition of the evenlyeven
numbers does not result in an incomposite number, the next evenlyeven number
of the series must be added until an incomposite number results. The second
perfect number is found in the following manner: The sum of the evenlyeven
numbers 1, 2, and 4 is 7, an incomposite number. If 7 be multiplied by 4 (the
last of the series of evenlyeven numbers used to produce it) the product is
28, the second perfect number. This method of calculation may be continued to
infinity.
Perfect numbers when multiplied
by 2 produce superabundant numbers, and when divided by 2 produce deficient
numbers.
The Pythagoreans evolved their
philosophy from the science of numbers. The following quotation from Theoretic
Arithmetic is an excellent example of this practice:
"Perfect numbers, therefore,
are beautiful images of the virtues which are certain media between excess and
defect, and are not summits, as by some of the ancients they were supposed to
be. And evil indeed is opposed to evil, but both are opposed to one good.
Good, however, is never opposed to good, but to two evils at one and the same
time. Thus timidity is opposed to audacity, to both [of] which the want of
true courage is common; but both timidity and audacity are opposed to
fortitude. Craft also is opposed to fatuity, to both [of] which the want of
intellect is common; and both these are opposed to prudence. Thus, too,
profusion is opposed to avarice, to both [of] which illiberality is common;
and both these are opposed to liberality. And in a similar manner in the other
virtues; by all [of] which it is evident that perfect numbers have a great
similitude to the virtues. But they also resemble the virtues on another
account; for they are rarely found, as being few, and they are generated in a
very constant order. On the contrary, an infinite multitude of superabundant
and diminished numbers may be found, nor are they disposed in any orderly
series, nor generated from any certain end; and hence they have a great
similitude to the vices, which are numerous, inordinate, and indefinite."
THE TABLE OF THE TEN
NUMBERS
(The following outline of the
Pythagorean numbers is a paraphrase of the writings of Nicomachus, Theon of
Smyrna, Proclus, Porphyry, Plutarch, Clement of Alexandria, Aristotle, and
other early authorities.)
Monad1is so called
because it remains always in the same conditionthat is, separate from
multitude. Its attributes are as follows: It is called mind, because the mind
is stable and has preeminence; hermaphrodism, because it is both male and
female; odd and even, for being added to the even it makes odd, and to the
odd, even; God, because it is the beginning and end of all, but itself has
neither beginning nor end; good, for such is the nature of God; the receptacle
of matter, because it produces the duad, which is essentially material.
By the Pythagoreans monad was called
chaos, obscurity, chasm, Tartarus, Styx, abyss, Lethe, Atlas, Axis, Morpho (a
name for Venus), and Tower or Throne of Jupiter, because of the great power
which abides in the center of the universe and controls the circular motion of
the planers about itself. Monad is also called germinal reason, because it is
the origin of all the thoughts in the universe. Other names given to it were:
Apollo, because of its relation to the sun; Prometheus, because he brought man
light; Pyralios, one who exists in fire; geniture, because without it no
number can exist; substance, because substance is primary; cause of truth; and
constitution of symphony: all these because it is the primordial one.
Between greater and lesser the monad is
equal; between intention and remission it is middle; in multitude it is mean;
and in time it is now, because
p. 72
eternity knows neither past nor
future. It is called Jupiter, because he is Father and head of the gods; Vesta,
the fire of the home, because it is located in the midst of the universe and
remains there inclining to no side as a dot in a circle; form, because it
circumscribes, comprehends, and terminates; love, concord, and piety, because
it is indivisible. Other symbolic names for the monad are ship, chariot,
Proteus (a god capable of changing his form), Mnemosyne, and Polyonymous
(having many names).
The following symbolic names were given
to the duad2because it has been divided, and is two rather than one; and
when there are two, each is opposed to the other: genius, evil, darkness,
inequality, instability, movability, boldness, fortitude, contention, matter,
dissimilarity, partition between multitude and monad, defect, shapelessness,
indefiniteness, indeterminate ness, harmony, tolerance, root, feet of
fountainabounding idea, top, Phanes, opinion, fallacy, alterity, diffidence,
impulse, death, motion, generation, mutation, division, longitude,
augmentation, composition, communion, misfortune, sustentation, imposition,
marriage, soul, and science.
In his book, Numbers, W. Wynn
Westcott says of the duad: "it was called 'Audacity,' from its being the
earliest number to separate itself from the Divine One; from the 'Adytum of
Godnourished Silence,' as the Chaldean oracles say."
As the monad is the father, so the duad
is the mother; therefore, the duad has certain points in common with the
goddesses Isis, Rhea (Jove's mother), Phrygia, Lydia, Dindymene (Cybele), and
Ceres; Erato (one of the Muses); Diana, because the moon is forked; Dictynna,
Venus, Dione, Cytherea; Juno, because she is both wife and sister of Jupiter;
and Maia, the mother of Mercury.
While the monad is the symbol of wisdom,
the duad is the symbol of ignorance, for in it exists the sense of
separatenesswhich sense is the beginning of ignorance. The duad, however, is
also the mother of wisdom, for ignoranceout of the nature of
itselfinvariably gives birth to wisdom.
The Pythagoreans revered the monad but
despised the duad, because it was the symbol of polarity. By the power of the
duad the deep was created in contradistinction to the heavens. The deep
mirrored the heavens and became the symbol of illusion, for the below was
merely a reflection of the above. The below was called maya, the
illusion, the sea, the Great Void, and to symbolize it the Magi of Persia
carried mirrors. From the duad arose disputes and contentions, until by
bringing the monad between the duad, equilibrium was reestablished by the
SaviorGod, who took upon Himself the form of a number and was crucified
between two thieves for the sins of men.
The triad3is the first number
actually odd (monad not always being considered a number). It is the first
equilibrium of unities; therefore, Pythagoras said that Apollo gave oracles
from a tripod, and advised offer of libation three times. The keywords to the
qualities of the triad are friendship, peace, justice, prudence, piety,
temperance, and virtue. The following deities partake of the principles of the
triad: Saturn (ruler of time), Latona, Cornucopiæ, Ophion (the great serpent),
Thetis, Hecate, Polyhymnia (a Muse), Pluto, Triton, President of the Sea,
Tritogenia, Achelous, and the Faces, Furies, and Graces. This number is called
wisdom, because men organize the present, foresee the future, and benefit by
the experiences of the fast. It is cause of wisdom and understanding. The
triad is the number of knowledgemusic, geometry, and astronomy, and the
science of the celestials and terrestrials. Pythagoras taught that the cube of
this number had the power of the lunar circle.
The sacredness of the triad and its
symbolthe triangleis derived from the fact that it is made up of the monad
and the duad. The monad is the symbol of the Divine Father and the duad of the
Great Mother. The triad being made of these two is therefore androgynous and
is symbolic of the fact that God gave birth to His worlds out of Himself, who
in His creative aspect is always symbolized by the triangle. The monad passing
into the duad was thus capable of becoming the parent of progeny, for the duad
was the womb of Meru, within which the world was incubated and within which it
still exists in embryo.
The tetrad4was esteemed by the
Pythagoreans as the primogenial number, the root of all things, the fountain
of Nature and the most perfect number. All tetrads are intellectual; they have
an emergent order and encircle the world as the Empyreum passes through it.
Why the Pythagoreans expressed God as a tetrad is explained in a sacred
discourse ascribed to Pythagoras, wherein God is called the Number of Numbers.
This is because the decad, or 10, is composed of 1, 2, 3, and 4. The number 4
is symbolic of God because it is symbolic of the first four numbers. Moreover,
the tetrad is the center of the week, being halfway between 1 and 7. The
tetrad is also the first geometric solid.
Pythagoras maintained that the soul of
man consists of a tetrad, the four powers of the soul being mind, science,
opinion, and sense. The tetrad connects all beings, elements, numbers, and
seasons; nor can anything be named which does not depend upon the tetractys.
It is the Cause and Maker of all things, the intelligible God, Author of
celestial and sensible good, Plutarch interprets this tetractys, which he said
was also called the world, to be 36, consisting of the first four odd numbers
added to the first four even numbers, thus:
1 + 3 +5 +7 
= 16 
2 + 4 + 6 + 8 
= 20 

36 
Keywords given to the tetrad are
impetuosity, strength, virility, twomothered, and the key keeper of Nature,
because the universal constitution cannot be without it. It is also called
harmony and the first profundity. The following deities partook of the nature
of the tetrad: Hercules, Mercury, Vulcan, Bacchus, and Urania (one of the
Muses).
The triad represents the primary colors
and the major planets, while the tetrad represents the secondary colors and
the minor planets. From the first triangle come forth the seven spirits,
symbolized by a triangle and a square. These together form the Masonic apron.
The pentad5is the union of an odd
and an even number (3 and 2). Among the Greeks, the pentagram was a sacred
symbol of light, health, and vitality. It also symbolized the fifth
elementetherbecause it is free from the disturbances of the four lower
elements. It is called equilibrium, because it divides the perfect number 10
into two equal parts.
The pentad is symbolic of Nature, for,
when multiplied by itself it returns into itself, just as grains of wheat,
starting in the form of seed, pass through Nature's processes and reproduce
the seed of the wheat as the ultimate form of their own growth. Other numbers
multiplied by themselves produce other numbers, but only 5 and 6 multiplied by
themselves represent and retain their original number as the last figure in
their products.
The pentad represents all the superior
and inferior beings. It is sometimes referred to as the hierophant, or the
priest of the Mysteries, because of its connection with the spiritual ethers,
by means of which mystic development is attained. Keywords of the pentad are
reconciliation, alternation, marriage, immortality, cordiality, Providence,
and sound. Among the deities who partook of the nature of the pentad were
Pallas, Nemesis, Bubastia (Bast), Venus, Androgynia, Cytherea, and the
messengers of Jupiter.
The tetrad (the elements) plus the monad
equals the pentad. The Pythagoreans taught that the elements of earth, fire,
air, and water were permeated by a substance called etherthe basis of
vitality and life. Therefore, they chose the fivepointed star, or pentagram,
as the symbol of vitality, health, and interpenetration.
It was customary for the philosophers to
conceal the element of earth under the symbol of a dragon, and many of the
heroes of antiquity were told to go forth and slay the dragon. Hence, they
drove their sword (the monad) into the body of the dragon (the tetrad). This
resulted in the formation of the pentad, a symbol of the victory of the
spiritual nature over the material nature. The four elements are symbolized in
the early Biblical writings as the four rivers that poured out of Garden of
Eden. The elements themselves are under the control of the composite Cherubim
of Ezekiel.
The Pythagoreans held the hexad6to
represent, as Clement of Alexandria conceived, the creation of the world
according to both the prophets and the ancient Mysteries. It was called by the
Pythagoreans the perfection of all the parts. This number was particularly
sacred to Orpheus, and also to the Fate, Lachesis, and the Muse, Thalia. It
was called the form of forms, the articulation of the universe, and the maker
of the soul.
Among the Greeks, harmony and the soul
were considered to be similar in nature, because all souls are harmonic. The
hexad is also the symbol of marriage, because it is formed by the union of two
triangles, one masculine and the other feminine. Among the keywords given to
the hexad are: time, for it is the measure of duration; panacea, because
health is equilibrium, and the hexad is a balance number; the world, because
the world, like the hexad, is often seen to consist of contraries by harmony;
omnisufficient, because its parts are sufficient for totality (3 +2 + 1 = 6);
unwearied, because it contains the elements of immortality.
By the Pythagoreans the heptad7was
called "worthy of veneration." It was held to be the number of religion,
because man is controlled by seven celestial spirits to whom it is proper for
him to make offerings. It was called the number of life, because it was
believed that human creatures born in the seventh month of embryonic life
usually lived, but those born in the eighth month often died. One author
called it the Motherless Virgin, Minerva, because it was nor born of a mother
but out of the crown, or the head of the Father, the monad. Keywords of the
heptad are fortune, occasion, custody, control, government, judgment, dreams,
voices, sounds, and that which leads all things to their end. Deities whose
attributes were expressed by the heptad were Ægis, Osiris, Mars, and Cleo (one
of the Muses).
Among many ancient nations the heptad is
a sacred number. The Elohim of the Jews were supposedly seven in number. They
were the Spirits of the Dawn, more commonly known as the Archangels
controlling the planets. The seven Archangels, with the three spirits
controlling the sun in its threefold aspect, constitute the 10, the sacred
Pythagorean decad. The mysterious Pythagorean tetractys, or four rows of dots,
increasing from 1 to 4, was symbolic of the stages of creation. The great
Pythagorean truth that all things in Nature are regenerated through the decad,
or 10, is subtly preserved in Freemasonry through these grips being effected
by the uniting of 10 fingers, five on the hand of each person.
The 3 (spirit, mind, and soul) descend
into the 4 (the world), the sum being the 7, or the mystic nature of man,
consisting of a threefold spiritual body and a fourfold material form. These
are symbolized by the cube, which has six surfaces and a mysterious seventh
point within. The six surfaces are the directions: north, east, south, west,
up, and down; or, front, back, right, left, above, and below; or again, earth,
fire, air, water, spirit, and matter. In the midst of these stands the 1,
which is the upright figure of man, from whose center in the cube radiate six
pyramids. From this comes the great occult axiom: "The center is the father of
the directions, the dimensions, and the distances."
The heptad is the number of the law,
because it is the number of the Makers of Cosmic law, the Seven Spirits before
the Throne.
The ogdoad8was sacred because it was
the number of the first cube, which form had eight corners, and was the only
evenlyeven number under 10 (1248421). Thus, the 8 is divided into two
4's, each 4 is divided into two 2's, and each 2 is divided into two 1's,
thereby reestablishing the monad. Among the keywords of the ogdoad are love,
counsel, prudence, law, and convenience. Among the divinities partaking of its
nature were Panarmonia, Rhea, Cibele, Cadmæa, Dindymene, Orcia, Neptune,
Themis, and Euterpe (a Muse).
The ogdoad was a mysterious number
associated with the Eleusinian Mysteries of Greece and the Cabiri. It was
called the little holy number. It derived its form partly from the twisted
snakes on the Caduceus of Hermes and partly from the serpentine motion of the
celestial bodies; possibly also from the moon's nodes.
The ennead9was the first square of
an odd number (3x3). It was associated with failure and shortcoming because it
fell short of the perfect number 10 by one. It was called the called the
number of man, because of the nine months of his embryonic life. Among its
keywords are ocean and horizon, because to the ancients these were boundless.
The ennead is the limitless number because there is nothing beyond it but the
infinite 10. It was called boundary and limitation, because it gathered all
numbers within itself. It was called the sphere of the air, because it
surrounded the numbers as air surrounds the earth, Among the gods and
goddesses who partook in greater or less degree of its nature were Prometheus,
Vulcan, Juno, the sister and wife of Jupiter, Pæan, and Aglaia, Tritogenia,
Curetes, Proserpine, Hyperion, and Terpsichore (a Muse).
The 9 was looked upon as evil, because
it was an inverted 6. According to the Eleusinian Mysteries, it was the number
of the spheres through which the consciousness passed on its way to birth.
Because of its close resemblance to the spermatozoon, the 9 has been
associated with germinal life.
The decad10according to the
Pythagoreans, is the greatest of numbers, not only because it is the tetractys
(the 10 dots) but because it comprehends all arithmetic and harmonic
proportions. Pythagoras said that 10 is the nature of number, because all
nations reckon to it and when they arrive at it they return to the monad. The
decad was called both heaven and the world, because the former includes the
latter. Being a perfect number, the decad was applied by the Pythagoreans to
those things relating to age, power, faith, necessity, and the power of
memory. It was also called unwearied, because, like God, it was tireless. The
Pythagoreans divided the heavenly bodies into ten orders. They also stated
that the decad perfected all numbers and comprehended within itself the nature
of odd and even, moved and unmoved, good and ill. They associated its power
with the following deities: Atlas (for it carried the numbers on its
shoulders), Urania, Mnemosyne, the Sun, Phanes, and the One God.
The decimal system can probably be
traced back to the time when it was customary to reckon on the fingers, these
being among the most primitive of calculating devices and still in use among
many aboriginal peoples.
Next: The Human Body in Symbolism