The foundation of Pythagorean Mathematics was as follows:
The first natural division of Numbers is into EVEN and ODD, an EVEN number being one which is divisible into two equal parts, without leaving a monad between them. The ODD number, when divided into two equal parts, leaves the monad in the middle between the parts.
All even numbers also (except the dyad—two—which is simply two unities) may be divided into two equal parts, and also into two unequal parts, yet so that in neither division will either parity be mingled with imparity, nor imparity with parity. The binary number two cannot be divided into two unequal parts.
Thus io divides into 5 and 5, equal parts, also into 3 and 7, both imparities, and into 6 and 4, both parities; and 8 divides into 4 and 4, equals and parities, and into 5 and 3, both imparities.
But the ODD number is only divisible into uneven parts, and one part is also a parity and the other part an imparity; thus 7 into 4 and 3, or 5 and 2, in both cases unequal, and odd and even.
The ancients also remarked the monad to be "odd," and to be the first "odd number," because it cannot be divided into two equal numbers. Another reason they saw was that the monad, added to an even number, became an odd number, but if evens are added to evens the result is an even number.
Aristotle, in his Pythagoric treatise, remarks that the monad partakes also of the nature of the even number, because when added to the odd it makes the even, and added to the even the odd is formed.
Hence it is called "evenly odd." Archytas of Tarentum was of the same opinion.
The Monad, then, is the first idea of the odd number; and so the Pythagoreans speak of the "two" as the "first idea of the indefinite dyad," and attribute the number 2 to that which is indefinite, unknown, and inordinate in the world; just as they adapt the monad to all that is definite and orderly. They noted also that in the series of numbers from unity, the terms are increased each by the monad once added, and so their ratios to each other are lessened; thus 2 is 1 + 1, or double its predecessor; 3 is not double 2, but 2 and the monad, sesquialter; 4 to 3 is 3 and the monad, and the ratio is sesquitertian; the sesquiquintan 6 to 5 is less also than its forerunner, the sesquiquartan 5 and 4, and so on through the series.
They also noted that every number is one half of the total of the numbers about it, in the natural series; thus 5 is half of 6 and 4. And also of the sum of the numbers again above and below this pair; thus 5 is also half of 7 and 3, and so on till unity is reached; for the monad alone has not two terms, one below and one above; it has one above it only, and hence it is said to be the "source of all multitude."
"Evenly even" is another term applied anciently to one sort of even numbers. Such are those which divide into two equal parts, and each part divides evenly, and the even division is continued until unity is reached; such a number is 64. These numbers form a series, in a duple ratio from unity; thus 1, 2, 4, 8, 16, 32. "Evenly odd," applied to an even number, points out that like 6, so, 14, and 28, when divided into two equal parts, these are found to be indivisible into equal parts. A series of these numbers is formed by doubling the items of a series of odd numbers, thus:
1, 3, 5, 7, 9 produce 2, 6, 10, 14, 18.
Unevenly even numbers may be parted into two equal divisions, and these parts again equally divided, but the process does not proceed until unity is reached; such numbers are 24 and 28.
Odd numbers also are susceptible of being looked upon from three points of view, thus:
"First and incomposite"; such are 3, 5, 7, 11, 13, 19, 23, 29, 31; no other number measures them but unity; they are not composed of other numbers, but are generated from unity alone.
"Second and composite" are indeed "odd," but contain and are composed from other numbers; such are 9, 15, 21, 25, 27, 33, and 39. These have parts which are denominated from a foreign number, or word, as well as proper unity, thus 9 has a third part which is 3; 15 has a third part which is 5; and a fifth part 3; hence as containing a foreign part, it is called second, and as containing a divisibility, it is composite.
The Third Variety of odd numbers is more complex, and is of itself second and composite, but with reference to another is first and incomposite; such are 9 and 25. These are divisible, each of them that is second and composite, yet have no common measure; thus 3 which divides the 9 does not divide the 25.
Odd numbers are sorted out into these three classes by a device, called the "Sieve of Eratosthenes," which is of too complex a nature to form part of a monograph so discursive as this must be.
Even numbers have also been divided by the ancient sages into Perfect, Deficient and Superabundant.
Superperfect or Superabundant are such as 12 and 24.
Deficient are such as 8 and 14.
Perfect are such as 6 and 28; equal to the number of their parts; as 28—half is 14, a fourth is 7, a seventh is 4, a fourteenth part is 2, and the twenty-eighth is 1, which quotients added together are 28.
In Deficient numbers, such as 14, the parts are surpassed
by the whole: one seventh is 2, a half is 7, a fourteenth is 1; the aggregate is 10, or less than 14.
In Superabundant, as 12, the whole surpasses the aggregate of its parts; thus the sixth is 2, a fourth is 3, a third is 4, a half is 6, and a twelfth is 1; and the aggregate is 16, or more than 12.
Superperfect numbers they looked on as similar to Briareus, the hundred-handed giant: his parts were too numerous; the deficient numbers resembled Cyclops, who had but one eye; whilst the perfect numbers have the temperament of a middle limit, and are the emulators of Virtue, a medium between excess and defect, not the summit, as some ancients falsely thought.
Evil is indeed opposed to evil, but both to one good. Good, however, is never opposed to good, but to two evils.
The Perfect numbers are also like the virtues, few in number; whilst the other two classes are like the vices—numerous, inordinate, and indefinite.
There is but one perfect number between 1 and 10, that is 6; only one between 10 and 100, that is 28; only one between 100 and 1000, that is 496; and between 1000 and 10,000 only one, that is 8128.
Odd numbers they called Gnomons, because, being added to squares, they keep the same figures as in Geometry: see Simplicius, liber 3.
A number which is formed by the multiplication of an odd and an even number together they called Hermaphrodite, or "arrenothelus."
In connection with these notes on parity and imparity, definite and indefinite numbers, it is to be noted that the old philosophers were deeply imbued with the union of numerical ideas with Nature—in its common acceptation, and also to the natures, essences or substrata of things.
The nature of good to them was definite, that of evil indefinite; and the more indefinite the nature of the evil, the worse it was. Goodness alone can define or bound the indefinite. In the human soul exists a certain vestige of
divine goodness (Buddhi); this bounds and moderates the indefiniteness and inequality of its desires.
It may be demonstrated that all inequality arises from equality, so that obtaining, as it were, the power of a mother and a root, she pours forth with exuberant fertility all sorts of inequality; and did space and time allow, it could be also shown that all inequality may be reduced to equality.
Iamblichus, in his treatise on the Arithmetic of Nicomachus, throws another light on numbers; he says some are like friends, they are Amicable numbers, as 284 and 220.
Pythagoras, being asked what a friend was, said ἐτερος εγω = "another I." Now this is demonstrated to be the case in these numbers; the parts of each are generative of each other, according to the nature of friendship.
Ozanam, a French mathematician, A.D. 1710, gives examples in his "Mathematical Recreations" of such Amicable Numbers. He remarks that 220 is equal to the sum of the aliquot parts of 284; thus 1 + 2 + 4 + 71 + 142 = 220: and 284 is equal to the sum of the aliquot parts of 220; thus 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.
Another such pair of numbers are 17,296 and 18,416.
Very curious speculations as to the relation between Numbers and marriage, and the character of offspring from it, are to be found scattered through the writings of the Philosophers. Plato, in his "Republic," has a passage concerning a geometric number, which, divinely generated, will be fortunate or unfortunate. Nicomachus also speaks of this same number, and he calls it the Nuptial Number; and he passes from it to state that from two good parents only good offspring can come; from two bad parents only bad; and from a good and a bad parent only bad; whence he warns the Republic against wedlock in a confused or disorderly manner, from which, the progeny being depraved, discord will result. Simplicius, in his commentary on the 2nd Book of Aristotle, "On the Heavens," remarks that Pythagoras and his followers claimed to have heard the Music of the Spheres, to have heard an harmonic sound produced by the motion of the planets, and from the sound
to have calculated by numbers the ratio of distance and size of the Sun, Moon, Venus and Mercury. To this Aristotle objected, but perhaps the difficulty might be solved: in this sublunary sphere all things are not commensurate, nor is everything sensible to every body alike. Animals can be scented, and their presence definitely known, by dogs when at great distances from them, and when man is in complete ignorance of their existence. Some of the ancients thought the soul had three vehicles—the terrestrial body, an aerial one in which it is punished, and an ethereal one, luminous and celestial, in which the soul abides when in a state of bliss. It may be that some one, by purification of the senses, by hereditary magical power, or by probity, or by the sacred operations of his religion, may perceive, with a terrestrial body laid aside, things imperceptible to us, and hear sounds inaudible to us still in bondage; or with mantle partly unfolded, some adept or truth-seeker may perceive, with eyes upraised, sights invisible to mortals, whilst yet his ears are deaf to the sounds beyond us both. For why do we see the stars, while yet we hear not their motion: