Proposition 1

The affinity between the rays and a circle and its arcs is greater than was that between the consonances.

The bases of what was previously stated in Book HI were already justifiably taken over into Book 4 as an axiom. However, it is proved as follows:

1. Consonances are between sounds; sounds consist in motions; their heights or depths in pitch, by which the consonances are expressed, arise from the speed and slowness of the motions, by what has been demonstrated in Book 3.

But the quick and slow sounds are elicited from the striking of strings which are stretched, not only if the stretching is in a circle, but also, and much more, if it is in a straight line.

Therefore, the consonances do not relate imme- iately to the circle and its arcs, on account of their circular shape, but on account of the length of Figure their parts, that is their mutual proportion.

They have what they have from the circle even when the circle has been destroyed and stretched out into a straight line.

Aspects on the contrary are by Definition I angles, which the circle measures along with its arcs only if it remains what it is said to be, that is insofar as it both has its circular shape and retains it completely

2. The consonances were not all propagated with an equnlly close relationship to the circle and its parts; for some of them related their origin to parts of the circle, insofar as they had some property not as parts of the circle but as straight lines, that is the same division as the whole circle, as was shown in Book III. It is the opposite in the case of the aspects; for the measure of what corresponds with the circle cannot in any way be related to straightness.

IV.

Proposition 2

The affinity between the rays and the regular figures is greater than was that of the consonances. It is proved first by the figure at the circumference.

For where the circle is complete, in that case the regular figure is also complete.

But the circle is more complete in measuring the angles of rays, by I.

Then the figure also can be considered more as com­ plete in relation to the rays.

It is the opposite in the case of consonances: as the circle and its parts could be stretched out straight, while preserving the consonances, so also all the sides of a figure could be stretched out into one and the same straight line, and could make a consonance with one straight side of the figure. On that basis, indeed, just as the circle loses its configuration, so also does the recti­ linear figure, so that it is no longer a figure.

It is proved second by the figure at the center. Angles are the elements of figures.

In this case two rays make the angle at the center. If that is repeated a certain number of times it completes the figure, as is apparent from these diagrams. However, that was not the case with the origin of the consonances: for there was no relationship with the angle at the center in that case. Therefore, thefigures are more closely akin to the aspects than to the consonances.

Proposition 3

The congruence of figures is capable of more in establishing influential configurations than in the case of consonances. Many arguments for this point are forthcoming.

1. Congruence is a property of a figure insofar as it is a whole figure and has an appearance. But a figure, insofar as the whole of it has this appearance, first, in itself on its own account has more affinity with configurations than with consonances, by Proposition

II. Second, it divides the circle as a whole harmonically; but the circle also has more affinity with configurations than with consonances, by Proposition

I. Hence on both showings, on that of the figure independently and that of the figure and the circle in common, the force of the congruence of figures is also greater in configurations than in consonances.

2. From the number of figures. For we have taken it, on account of the axioms previously stated, that figures are influential on account of their properties.

Then where the number of the things influenced corresponds more, there is a greater affinity between cause and effect, at least probably.

But as the congruent figures are few, so also the aspects are few, as experience testifies. For if they were not few, there would be a great confusion of them, and a great multitude, so that individual aspects could not be observed separately on their own days.

But they can be observed: therefore, they are not infinite in number. On the contrary, the consonances can be infinite,

by augmentation of the intervals by diapasons, just as the knowable figures are infinite.

3. From the essence of the terms, in which the proportions in either direction have their existence.

Motions, of which sounds are a disposition, are considered as Becoming, insofar as they take time; radiations more as instantaneous Being. For just as a body exists at this moment, so also a radiation exists at this moment; whereas of motion what has passed no longer is, what follows is not yet, in a moment it is nothing.

Congruence, however, seems to be among things which are, rather than among those which become. For the sides or wall of a house are congruent so that the house may exist, not so that now and perpetually it may be built.

4. From the affinity of congruence, as cause, with configurations. For the latter are angles; but congruence also exists in figures on account of their angles.

So far consonances and configurations have been opposite to each other. In what follows there will be another opposition among configurations alone, between congruence and knowability.