e. From this the Pythagoreans proceed to the ten, another form of this tetrad. As the four is the perfect form of three, this fourfold, thus perfected and developed so that all its moments shall be accepted as real differences, is the number ten (δεκάς), the real tetrad. Sextus (adv. Math. IV. 3; VII. 94, 95) says: “Tetraktus means the number which, comprising within itself the four first numbers, forms the most perfect number, that is the number ten; for one and two and three and four make ten. When we come to ten, we again consider it as a unity and begin once more from the beginning. The tetraktus, it is said, has the source and root of eternal nature within itself, because it is the Logos of the universe, of the spiritual and of the corporeal.” It is an important work of thought to show the moments not merely to be four units, but complete numbers; but the reality in which the determinations are laid hold of, is here, however, only the external and superficial one of number; there is no Notion present although the tetraktus does not mean number so much as idea. One of the later philosophers, Proclus, (in Timæum, p. 269) says, in a Pythagorean hymn:—
“The divine number goes on,”...
“Till from the still unprofaned sanctuary of the Monad
It reaches to the holy Tetrad, which creates the mother of all that is;
Which received all within itself, or formed the ancient bounds of all,
Incapable of turning or of wearying; men call it the holy Dekad.”
What we find about the progression of the other numbers is more indefinite and unsatisfying, and the Notion loses itself in them. Up to five there may certainly be a kind of thought in numbers, but from six onwards they are merely arbitrary determinations.
2. Application of the System to the Universe. This simple idea and the simple reality contained therein, must now, however, be further developed in order to come to reality as it is when put together and expanded. The question now meets us as to how, in this relation, the Pythagoreans passed from abstract logical determinations to forms which indicate the concrete use of numbers. In what pertains to space or music, determinations of objects formed by the Pythagoreans through numbers, still bear a somewhat closer relation to the thing, but when they enter the region of the concrete in nature and in mind, numbers become purely formal and empty.
a. To show how the Pythagoreans constructed out of numbers the system of the world, Sextus instances (adv. Math. X. 277-283), space relations, and undoubtedly we have in them to do with such ideal principles, for numbers are, in fact, perfect determinations of abstract space. That is to say, if we begin with the point, the first negation of vacuity, “the point corresponds to unity; it is indivisible and the principle of lines, as the unity is that of numbers. While the point exists as the monad or One, the line expresses the duad or Two, for both become comprehensible through transition; the line is the pure relationship of two points and is without breadth. Surface results from the threefold; but the solid figure or body belongs to the fourfold, and in it there are three dimensions present. Others say that body consists of one point” (i.e. its essence is one point), “for the flowing point makes the line, the flowing line, however, makes surface, and this surface makes body. They distinguish themselves from the first mentioned, in that the former make numbers primarily proceed from the monad and the undetermined duad, and then points and lines, plane surfaces and solid figures, from numbers, while they construct all from one point.” To the first, distinction is opposition or form set forth as duality; the others have form as activity. “Thus what is corporeal is formed under the directing influence of numbers, but from them also proceed the definite bodies, water, air, fire, and the whole universe generally, which they declare to be harmonious. This harmony is one which again consists of numeral relations only, which constitute the various concords of the absolute harmony.”
We must here remark that the progression from the point to actual space also has the signification of occupation of space, for “according to their fundamental tenets and teaching,” says Aristotle (Metaph. I. 8), “they speak of sensuously perceptible bodies in nowise differently from those which are mathematical.” Since lines and surfaces are only abstract moments in space, external construction likewise proceeds from here very well. On the other hand, the transition from the occupation of space generally to what is determined, to water, earth, &c., is quite another thing and is more difficult; or rather the Pythagoreans have not taken this step, for the universe itself has, with them, the speculative, simple form, which is found in the fact of being represented as a system of number-relations. But with all this, the physical is not yet determined.
b. Another application or exhibition of the essential nature of the determination of numbers is to be found in the relations of music, and it is more especially in their case that number constitutes the determining factor. The differences here show themselves as various relations of numbers, and this mode of determining what is musical is the only one. The relation borne by tones to one another is founded on quantitative differences whereby harmonies may be formed, in distinction to others by which discords are constituted. The Pythagoreans, according to Porphyry (De vita Pyth. 30), treated music as something soul-instructing and scholastic [Psychagogisches und Pädagogisches]. Pythagoras was the first to discern that musical relations, these audible differences, are mathematically determinable, that what we hear as consonance and dissonance is a mathematical arrangement. The subjective, and, in the case of hearing, simple feeling which, however, exists inherently in relation, Pythagoras has justified to the understanding, and he attained his object by means of fixed determinations. For to him the discovery of the fundamental tones of harmony are ascribed, and these rest on the most simple number-relations. Iamblichus (De vita Pyth. XXVI. 115) says that Pythagoras, in passing by the workshop of a smith, observed the strokes that gave forth a particular chord; he then took into consideration the weight of the hammer giving forth a certain harmony, and from that determined mathematically the tone as related thereto.[44] And finally he applied the same, and experimented in strings, by which means there were three different relations presented to him—Diapason, Diapente, and Diatessaron. It is known that the tone of a string, or, in the wind instrument, of its equivalent, the column of air in a reed, depends on three conditions; on its length, on its thickness, and on the amount of tension. Now if we have two strings of equal thickness and length, a difference in tension brings about a difference in sound. If we want to know what tone any string has, we have only to consider its tension, and this may be measured by the weight depending from the string, by means of which it is extended. Pythagoras here found that if one string were weighted with twelve pounds, and another with six (λόγος διπλάσιος, 1 : 2) it would produce the musical chord of the octave (διὰ πασῶν); the proportion of 8 : 12, or of 2 : 3 (λόγος ἡμιόλιος) would give the chord of the fifth (διὰ πέντε); the proportion of 9 : 12, or 3 : 4 (λόγος ἐπίτριτος), the fourth (διὰ τεσσάρων).[45] A different number of vibrations in like times determines the height and depth of the tone, and this number is likewise proportionate to the weight, if thickness and length are equal. In the first case, the more distended string makes as many vibrations again as the other; in the second case, it makes three vibrations for the other’s two, and so it goes on. Here number is the real factor which determines the difference, for tone, as the vibration of a body, is only a quantitatively determined quiver or movement, that is, a determination made through space and time. For there can be no determination for the difference excepting that of number or the amount of vibrations in one time; and hence a determination made through numbers is nowhere more in place than here. There certainly are also qualitative differences, such as those existing between the tones of metals and catgut strings, and between the human voice and wind instruments; but the peculiar musical relation borne by the tone of one instrument to another, in which harmony is to be found, is a relationship of numbers.
From this point the Pythagoreans enter into further applications of the theory of music, in which we cannot follow them. The à priori law of progression, and the necessity of movement in number-relations, is a matter which is entirely dark; minds confused may wander about at will, for everywhere ideas are hinted at, and superficial harmonies present themselves and disappear again. But in all that treats of the further construction of the universe as a numerical system, we have the whole extent of the confusion and turbidity of thought belonging to the later Pythagoreans. We cannot say how much pains they took to express philosophic thought in a system of numbers, and also to understand the expressions given utterance to by others, and to put in them all the meaning possible. When they determined the physical and the moral universe by means of numbers, everything came into indefinite and insipid relationships in which the Notion disappeared. In this matter, however, so far as the older Pythagoreans are concerned, we are acquainted with the main principles only. Plato exemplifies to us the conception of the universe as a system of numbers, but Cicero and the ancients always call these numbers the Platonic, and it does not appear that they were ascribed to the Pythagoreans. It was thus later on that this came to be said; even in Cicero’s time they had become proverbially dark, and there is but little after all that is really old.