The Essence of the Harmonic Proportions, Both Sensible and Intelligible

On numbers, I should not contest the view that Aristotle rightly refuted the Pythagoreans; for the numbers are at second remove, in a sense, or even at third, and fourth, and beyond any limit I can state, and they have in them nothing which they have not got either from quantities, or from other true and real entities, or even from various products of Mind.

Thus, I do not attribute anything either to the Platonic numbers, which are attributed to the change in Republics, which Bodin adopts in his Historical Method, nor to those called Climacteric, that is on their own account, except insofar as they num- revolutions and configurations of the stars, as I indicated not long ago openly enough in the prolegomena to the Ephemerides.

But as far as continuous quantities are concerned, I entirely agree with Proclus, though his rhetoric sweeps on like a torrent, overflowing its banks and concealing the hidden shoals and whirlpools of doubt, while his mind, full of the majesty of such great matters, struggles in the constraints of language, and his conclusion, never satisfying itself with the flood of words, oversteps the simplicity of his propositions.

I therefore believe that if I now append my reasons why I establish the intellectual circle and its parts as the terms for the insensible harmonies (reasons which I had conceived before I had read Proclus), I shall not only be saying what agrees with Proclus but also making a kind of summary of the passage quoted, as far indeed as it serves my purpose.

I shall not state among my reasons that as the objects which are The terms of the compared had been first outside, and next in the sense, they have been jnsensibie V 1 •! 1 • harmonies are abstracted finally from things and from the sensible emanations of abstract things, a sense which the term “abstracting” properly engenders; for quantities, as has been stated above, this process of abstraction relates to the sensible harmonies of sounds and rays, and comes into the reckoning only when, just as the good or evil deeds of citizens are adjudged by laws which were passed long ago, sounds and rays are adjudged by the archetypal harmony which is already present within.

But this is why the quantity which provides the terms for the harmonic proportions is said to be intellectual, because that quantity must be capable of by far the most subtle construction.

Yet that construction is never drawn from sensible things in a diagram, though it is assisted by them; and it does not arise from the assembling of many individual sensible things into one axiom, but it is obtained a priori.

The valid objection which Proclus made to Aristotle, generally understood in this way, I can thoroughly confirm in particular with absolutely clear arguments taken from Book I. For of the figures which cut off a harmonic arc of a circle the specific difference, by which, as a part of their definition, their essence is exhibited, is the fact that they must be knowable.

Yet what after all is knowability without Mind, which is capable of knowledge? And do not say that it can be the case that knowledge is not knowledge of something; for knowledge consists in comparison, as when the side of a figure is equal to the semidiameter.

What equality may then be without a mind, especially in the case of things which are independent of place, is unintelligible; and we return to the argument already adduced above for harmonies which are also sensible.

Also they must be not only knowable but also known, so that the archetypal harmony may in fact shine forth within the mind. For the possibility of knowing is not sufficient for us as a criterion of sensible harmonies.

Therefore, if a part of the essence of something is within the mind, and thus in its operation or activity, that thing must be established within, that is to say the terms of the harmonies, the circle and its part.

You may ask how knowledge of a thing can be possessed, when The soul has the mind has never learnt it, and perhaps cannot learn it, if it is de- knowledge of mathematics by prived of sensation of external things? To this Proclus replies above instinct, in words which are commonplace in his philosophy: we today, if I am not mistaken, quite correctly use the term “instinct.”

To the human mind and to other minds quantity is known by instinct, even if for this purpose it is deprived of all sensation. Of itself it understands a straight line, of itself an equal distance from a given point, of itself it forms for itself from these an image of a circle. If so, it can much more readily find the construction by means of that, and so perform the function of the eye in seeing the diagram (if there is nevertheless a need for one).

Certainly the mind itself, if it never had the use of an eye at all, would demand an eye for itself for the comprehension of things which are placed outside it, and would lay down laws for its structure which were drawn from itself (if in fact it were pure and sound and without hindrance, that is, if it were only what it Proclus says is). For the recognition of quantities, which is innate in the mind, dictates what the nature of the eye must be; and therefore, the eye has almost the same

a little after the been made as it is because the mind is as it is, and not the other way passage just quoted: that the round.

Why waste words?

Geometry, which before the origin of truth about the things was coeternal with the divine mind and is God himself(for what gods is adapted could there be in God which would not be God himself?), supplied to mathematical entities, and the God with patterns for the creation of the world, and passed over to Creator of the Man along with the image of God; and was not in fact taken in through whole universe used mathematical archetypes, the eyes.

Therefore, since quantities possess constructibility not by virtue coeternal with of the figures’ passing before the eyes, but in virtue of being clear to Himself, in the construction of the eyes of the mind, in virtue not so much of having been abstracted the world.

from sensible things but of never having been associated with them, therefore, we have rightly established abstract quantity as the terms for archetypal proportions, that is those which are constructible from the divisions of the circle. There is another reason why I choose abstract quantities, that the circle, which is a figure, that is a species of the fourth kind, although it is a quantity, yet in this connection is considered purely as a figure, without distinction between large and small, to such an extent that it is in a sense abstracted from quantity itself, as if from what is subject to it, and its nature can be recognized even in the narrow space of a point. This, I think, was what Proclus meant when he said that mathe­ matical things existed in the soul in an uncorporeal and unlocalized mode.

The symbolism

Finally there is a chief and supreme argument, that quantities pos of divine and sess a certain wonderful and obviously divine organization, and there human things in is a shared symbolic representation of divine and human things in quantities.

them. Of the semblance of the Holy Trinity in the spherical I have written in many places^^ in my Optics, in my Commentaries on Mars, and in the theory of the sphere; and I want to return to the subject here.

The straight line, by its extension from a point at the center to a single point at the surface sketches out the first rudiments of creation, and imitates the eternal begetting of the Son (represented and depicted by the departure from the center towards the infinite points of the whole surface, by infinite lines, subject to the most perfect equality in all respects).

This straight line is of course an element of a corporeal form. If this is spread out sideways, it now suggests a corporeal form, creating a plane; but a spherical shape cut by a plane gives the shape of a circle at its section, a true image of the created mind, which is in charge of ruling the body.

It is in the same proportion to the spherical as the human mind is to the divine, that is to say as a line to a surface, though each is circular.

But to the plane, in which it is also placed, it is as the curved to the straight, which are incompatible and incommensurable.

Also the circle exists splendidly both in the plane which cuts, circumscribing the spherical shape, and in the spherical shape which is cut, by the mutual concurrence of the two, just as the mind both exists in the body, giving form to it and to its connections with the corporeal form, like a kind of irradiation shed from the divine face onto the body and drawing thence its more noble nature.

Just as this is a confirmation from the harmonic proportions of the circle as the subject and the source of their terms, equally it is the strongest possible argument for abstraction, as the suggestion of the divinity of the mind exists neither in a circle of definite quantity, nor in an imperfect one, for they are material and sensible.

The main point is that the circle should be abstracted from corporeal and sensible things to the same extent as concepts of the curved, the symbol of the mind, are separated and, so to speak, abstracted from the straight, the shadow of bodies.

Therefore, we have sufficient support for seeking the terms for harmonic proportions, which are objects of the mind alone, chiefly in abstract quantities.

Therefore, to conclude this section we shall gather the chief points in a package. For the sensible harmonies have this in common with the archetypal ones, that they demand terms and comparison of them, an activity of the soul itself: the essence of both consists in this comparison.

But the terms of the sensible harmonies are sensible, and must be present outside the soul: the terms of the archetypal harmonies are present within the soul beforehand, Therefore, the sensible harmonies need in addition to be received by means of an emanation which they have emitted; and it is by the senses, the servants of the soul, that they are received.

Another comparison is also needed, of the individual; sensible terms with the individual archetypal ones, I mean with the circle and a knowable part of it; but the archetypal harmony has neither need, as its terms are present in the soul before hand, and inborn in it, and in fact are the soul itself, and they are not an image of their true pattern, but are in a sense their own pattern.

Thus only a simple comparison, which the soul sets up, of its own parts with each other, so to speak, completes the whole essence of the archetypal harmony.

Finally the soul itself, engaged in this activity, is the harmony with which we are concerned, just as, without reference to this activity, the circle and its part are, that is to say they are the terms of the harmony; and in the end harmony is wholly spiritualized, and so deified.