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Even numbers are divided into three classes: evenly-even, evenly-odd, and oddly-odd.

The evenly-odd numbers are all in duple ration from unity; thus 1,2 4, 8, 16, 32, 64, 128, 256, 512, and 1024. The proof of the perfect evenly-even number is that it can be halved and the halves again halved back to unity, as half of 64 = 32; half of 32 = 16; half of 16 = 8; half of 8 = 4; half of 4 = 2; half of 2 = 1; beyond unity it is impossible to go.

The evenly-even numbers posses 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 sun 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+3+4+8) equals the fifth term (16) minus one.

In a series of evenly-even numbers, the first mulitiplied by the last equals the last; te 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 evenly-odd 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 evenly-odd numbers 2, 6, 10, 14, 18, 22. Thus, every 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.

The evenly-odd numbers are also remarkable in that each term is one-half of the sum of the terms on either side of it. For example: 10 is one-half of the sum of 6 and 14; 18 is one-half the sum of 14 and 22; and 6 is one-half the sum of 2 and 10.

Even numbers are also divided into another three classes: superperfect, deficient, and perfect.

Superperfect or superabundant numbers are such as have the sum of their fractional parts greater than themselves. For example: half of 24 = 12; one fourth = 6; one third = 8; one sixth = 4; one twelveth = 2; and one twenty-fourth = 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: half of 14 = 7; one-seventh = 2; and one-fourteenth = 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: half of 28 = 14; one-fourth = 7; one-seventh = 4; one-fourteenth = 2; and one-twenty-eighth = 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 1000, namely 496; and one between 1000 and 10000, namely 8128.

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 similutude 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."

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The odd numbers are divided by a mathematical contrivance - called "the Sieve of Eratosthenes" - into three general classes: incomposite, composite, and incomposite-composite.

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 incomposite-composite 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 incomposite. Accordingly, the term incomposite-composite was created to describe their properties.

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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 maybe 5 +4, 3+6, 7+2, or 8+1. The Pythagoreans considered the odd number - of which the monad was the prototype - to be deinite and masculine. They were not all agreed, however, as to the nature of unity, or 1. Some delcared 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 evenly-odd. 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 number - of which the duad was the prototype - to be indefinite and feminine.

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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 be cast upon the meanings of the words number, monad, and one.

The monad signifies:

1. the all-including ONE. The Pythagoreans called the monad the "noble number, Sire of Gods and men".
2. 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.
3. The monad may also be likened 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.
4. By some Pythagoreans the monad is also considered synonymous with the One.

From the study of the mysterious Pythagorean monad, Leibnitz evolved his magnificent theory of the world atoms - a theory in perfect accord with the ancient teachings of the Mysteries, for Leibnitz himself was an initiate of a secret school.

Number is the term applied to all numerals and their combinations. (A strict interpretation of the term number by certain of the Pythagorean excludes 1 and 2). Pythagoras defined number to be the extension and energy of the spermatic reasons contained in the monad. The follwers 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.

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(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 later 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, 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 parts - magnitude which is stationary and magnitude which is movable, the stationary part having priority. Multitude is 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.

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