By the generality of the biographers of Pythagoras, it is said to be difficult to give a clear idea of his philosophy, as it is almost certain he never committed it to writing, and that it has been disfigured by the fantastic dreams and chimeras of later Pythagoreans. Diogenes Laërtius, however, whose “Lives of the Philosophers” was supposed to be written about the end of the second century of our era, says “there are three volumes extant written by Pythagoras. One on education, one on politics, and one on natural philosophy.” And adds, that there were several other books extant, attributed to Pythagoras, but which were not written by him. Also, in his “Life of Philolaus,” that Plato wrote to Dion to take care and purchase the books of Pythagoras.[4] But whether this great philosopher committed his discoveries to writing or not, his doctrines regarding the philosophy of beauty are well-known to be, that he considered numbers as the essence and the principle of all things, and attributed to them a real and distinct existence; so that, in his view, they were the elements out of which the universe was constructed, and to which it owed its beauty. Diogenes Laërtius gives the following account of this law:—“That the monad was the beginning of everything. From the monad proceeds an indefinite duad, which is subordinate to the monad as to its cause. That from the monad and indefinite duad proceeds numbers. That the part of science to which Pythagoras applied himself above all others, was arithmetic; and that he taught ‘that from numbers proceed signs, and from these latter, lines, of which plane figures consist; that from plane figures are derived solid bodies; that of all plane figures the most beautiful was the circle, and of all solid bodies the most beautiful was the sphere.’ He discovered the numerical relations of sounds on a single string; and taught that everything owes its existence and consistency to harmony. In so far as I know, the most condensed account of all that is known of the Pythagorian system of numbers is the following:—‘The monad or unity is that quantity, which, being deprived of all number, remains fixed. It is the fountain of all number. The duad is imperfect and passive, and the cause of increase and division. The triad, composed of the monad and duad, partakes of the nature of both. The tetrad, tetractys, or quaternion number is most perfect. The decad, which is the sum of the four former, comprehends all arithmetical and musical proportions.’”[5]

These short quotations, I believe, comprise all that is known, for certain, of the manner in which Pythagoras systematised the law of numbers. Yet, from the teachings of this great philosopher and his disciples, the harmonic law of nature, in which the fundamental principles of beauty are embodied, became so generally understood and universally applied in practice throughout all Greece, that the fragments of their works, which have reached us through a period of two thousand years, are still held to be examples of the highest artistic excellence ever attained by mankind. In the present state of art, therefore, a knowledge of this law, and of the manner in which it may again be applied in the production of beauty in all works of form and colour, must be of singular advantage; and the object of this work is to assist in the attainment of such a knowledge.

It has been remarked, with equal comprehensiveness and truth, by a writer[6] in the British and Foreign Medical Review, that “there is harmony of numbers in all nature—in the force of gravity—in the planetary movements—in the laws of heat, light, electricity, and chemical affinity—in the forms of animals and plants—in the perceptions of the mind. The direction, indeed, of modern natural and physical science is towards a generalization which shall express the fundamental laws of all by one simple numerical ratio. And we think modern science will soon shew that the mysticism of Pythagoras was mystical only to the unlettered, and that it was a system of philosophy founded on the then existing mathematics, which latter seem to have comprised more of the philosophy of numbers than our present.” Many years of careful investigation have convinced me of the truth of this remark, and of the great advantage derivable from an application of the Pythagorean system in the arts of design. For so simple is its nature, that any one of an ordinary capacity of mind, and having a knowledge of the most simple rules of arithmetic, may, in a very short period, easily comprehend its nature, and be able to apply it in practice.

The elements of the Pythagorean system of harmonic number, so far as can be gathered from the quotations I have given above, seem to be simply the indivisible monad (1); the duad (2), arising from the union of one monad with another; the triad (3), arising from the union of the monad with the duad; and the tetrad (4), arising from the union of one duad with another, which tetrad is considered a perfect number. From the union of these four elements arises the decad (10), the number, which, agreeably to the Pythagorean system, comprehends all arithmetical and harmonic proportions. If, therefore, we take these elements and unite them progressively in the following order, we shall find the series of harmonic numbers (2), (3), (5), and (7), which, with their multiples, are the complete numerical elements of all harmony, thus:—

1+1=2
1+2=3
2+3=5
3+4=7

In order to render an extended series of harmonic numbers useful, it must be divided into scales; and it is a rule in the formation of these scales, that the first must begin with the monad (1) and end with the duad (2), the second begin with the duad (2) and end with the tetrad (4), and that the beginning and end of all other scales must be continued in the same arithmetical progression. These primary elements will then form the foundation of a series of such scales.

I.(1)(2)
II.(2)(3)(4)
III.(4)(5)(6)(7)(8)
IV.(8)(9)(10)( )(12)( )(14)(15)(16)

The first of these scales has in (1) and (2) a beginning and an end; but the second has in (2), (3), and (4) the essential requisites demanded by Aristotle in every composition, viz., “a beginning, a middle, and an end;” while the third has not only these essential requisites, but two intermediate parts (5) and (7), by which the beginning, the middle, and the end are united. In the fourth scale, however, the arithmetical progression is interrupted by the omission of numbers 11 and 13, which, not being multiples of either (2), (3), (5), or (7), are inadmissible.

Such is the nature of the harmonic law which governs the progressive scales of numbers by the simple multiplication of the monad.