The Congruence of figures can make influential configurations more than knowability

Proposition 4

The congruence of figures is capable of making configurations influential more than knowability.

It is proved from the circumstances of the sublunary Soul and also of the faculties of the human soul which perceive the aspects.

These, then, are all inferior to the contemplative faculty and the understanding. They have closer affinity with the sensitive faculty and the one which controls the operations of the senses.

The instinct of the former, as we have stated in Chapter 3, is as much duller than the human instinct as the body of the Earth is cruder than the body of Man.

But congruence is later in order than knowability, and is, as it were, presented outwards to some operation which has the idea of the operations of the senses.

It is therefore right for it to be believed that these animal faculties are moved and affected by congruence rather than by the knowability of the figures.

Congruence and knowability were therefore opposites in the same kind of configuration.

Two figures will be opposed to each other, first with reference to congruence alone, after that with reference to configurations.

Proposition 5

Congruence is a property of the figure at the circumference rather than at the center?

For it has the more important position in the figure which can be whole, on account of the position from which it takes its name, because congruence belongs to whole figures, as is apparent from Book II. But of the central figure not more than one angle can stand at the center, by Definition I. However, at the circumference the whole of the circumferential figure can exist. The prop­ osition follows.

Proposition 6

Of the 2 figures which each aspect claims for itself, that at the circumference is more important than that at the center. For congruence is more important in this business than knowability, by IV.

But congruence is more important in the figure at the circumference, by V.

Hence, that which has superior power also makes the very thing in which it is have superior power, that is to say the figure at the circumference.

The same is also proved in the following way from the inmost properties of the soul, touched on in Chapter 3.

For as it is the soul which adapts its own formal essence to the harmonies of the configurations, certainly by the same distinction by which the soul is either a circle, or a point which is the center of a circle, so also will the figures which belong to its household, those at the circumference and at the center, be distinguished.

In fact, every soul bears some idea of a circle, in fact of a circle abstracted not only from matter but also in a sense from magnitude, as was stated in Chapter 3, and so the circle and its center in this case almost coincide, and the soul is either a potential circle or a point in which directions are distinguished, and thus it can be said in a sense to have qualities.

Yet it seems that the division must be observed between those faculties of the soul which should be considered rather as a circle, and those which should be considered rather as a point.

For just as a circle cannot be imagined without a center, and on the other hand every point has round it space for drawing a circle, so also in the case of the soul there is no operation without an image-forming impression, and on the other hand every internal reception or meditation is on account of an external motion, and very interior faculty of the mind is on account of those which are rather exterior.

What is the chief and supreme faculty of the soul, called the mind, if not a center?

What is the reasoning faculty if not a circle?

For just as the center is inside, and the circle more outside, so the mind keeps itself to itself, and reasoning weaves a sort of exterior web; and just as the center is the basis, source, and origin of a circle, so the mind is of reasoning.

Again, every such faculty of the soul, both the understanding, the contemplative, and indeed the sensitive faculty, are a sort of center. But the faculties of the soul which produce motion are a circle, because again, as a circle on the outside is placed about its center, so operation is towards the outside, and cognition and meditation are performed within; and as a circle is to a point, so in a way external action is to internal contemplation, ani­ mate motion to sensation.

For a point, because it is opposite to the circumference in all directions, is born suitable for representing passive experience; andfor what is the sensitive soul.

even as in this case when it is that which perceives radiations, except for sensing and perceiving what it experiences passively^ That is, because it is moved by its objects.

Also by comparing each point of comparison: as the center is the same in either case, so also the form of cognition is in a way the same, the chief mental one, and the sensitive, or, analogous to it, that which perceives radiations: neither uses contemplation in itself, insofar as it is what it is, but it recognizes without contemplation.

Hence, as the former, I mean sublunary Nature or even the sensitive faculty, is a sort of faint image of the latter, the chieffaculty of the human mind, so the former, contemplation by the reason, is an image of these actions and operations of the soul, a circle in each case.

Therefore, insofar as souls perceive celestial radiations, and so are to speak moved within themselves by them, let them be for us points; but on the other hand insofar as they are movers, that is, insofar as they transfer perceived har­ monies between the radiations into their own operations, and are stimulated by them to action, they should be considered as circle.

It therefore follows that to the extent to which it is recognizing the harmonies of the rays, it is concerned chiefly with the central figure; whereas to the extent to which it is operating, arousing occurrences in the sky (and similar ones in Man), it fits itself to the one at the circumference.

In fact, in the case of an aspect, we give prior concern for its influence than for the means by which it is perceived by the operative soul, and therefore, also prior attention to the figure at the circumference rather than to the figure at the center.

In this case, figure was compared with figure in one and the same aspect. In the following couple, supposing the figure is the same, its congruence will be opposed to its knowability.

Proposition 7

In the figure at the circumference, congruence takes precedence over the knowability of the side. In the figure at the center on the other hand, the knowability of the side takes precedence over the congruence of the figure.

This proposition pertains to the completion of the matter which was left half finished in Proposition III. For although congruence is more important in 4 than in 3, constructibility was also more important in Book 3.

Yet this should not be completely separated from the constitution of the aspects, since no congruence is without a knowledge-producing construction;

On it depends that of both the side and particularly of the area bounded by the sides of the figure. For it is from the angles, in which suitability for congruence resides, that the construction is derived.

It seems, therefore, that as far as this proposition is concerned its contrary is true, both in the one case and in the other.

For as far as the figure at the center is concerned, one angle of it has in actual fact been expressed through the rays; but of the figure at the circumference no angle is expressed, but only its side, up to a point. However, congruence belongs to angles. Therefore, it seems that we must take most notice of that in the central figure. For if sublunary

Nature perceives the quantity of an angle, which two rays form at the Earth, it seems that it can also perceive the fitness of that angle, along with the others, for congruence.

The fact that on the other hand we should take more notice in the figure at the circumference of the knowability of its side than of its congruence seems evident from the following. For the knowledge by which a figure is known, as was demonstrated in Book I, is based on the equality either of a side to an expressible part of the diameter, or of the square of a side to an expressible part of the square of the diameter, or of the area of thefigure to the same, or on another linkage of either the side, or of its square and area, to the diameter or its square, or determination by it. Suppose, therefore, that sublunary Nature has sensation of the zodiac circle, a circle which is of course sensible, and stands outside it, which it examines against the idea of an abstract intellectual circle, which it has within itself and born or created along with itself.

In that case it certainly follows that in the natural order it first senses the size of the arc of the zodiac marked off between two planets, the size of the chord which it subtends, its kind, whether it is expressible in length or only in square, whether it makes an expressible sum of squares in combination with any other, and an expressible rectangle, and on what property the expressibility of its area is founded.

These things, I say, seem to become known to sublunary Nature first in the natural order, because a side is prior to a figure which is described by a repetition of the side; and it is only afterwards when the whole figure has been described on the zodiac circle, that the angles become evident, and their quantity, and whether they are among the congruent, and whether the figure converges with all its angles to the same kind of congruence, or whether the congruence is continuous.

In short, congruence is a feature of angles, knowability of sides. Therefore, when the angle of a figure becomes known before its side, in that case the parts of congruence seem to be earlier and more important than those of knowability; when it is known after, they are later.

But it is the angle of the figure which reaches the center which becomes known first, whereas the side of the figure described at the surface (I mean between only two planets) becomes known first.

Therefore, it seems to be right if in the figure at the center consideration of congruence is held to be more important, but in that at the surface that of knowability.

We must therefore rebut these points which have been adduced to support the contrary of our proposition, and in the same operation we must reinforce by true arguments the order of the properties of congruence and knowability.

First, then, although it is true in respect of the central figure that one angle is formed by the rays of two planets, yet it does not follow from that that the mind when it perceives the quantity of the angle perceives first in the natural order the congruence of the figure to which the angle is going to belong.

The reason is evident, that congruence, insofar as it belongs to one angle, and several equal to it, placed in a single plane, is too general.

For there are infinite forms of angles which are congruent in that way, and they are always greater in number in proportion as the individual angles are smaller. This, then, is not the congruence which we discuss in Book II which is a property not of angles individually but of complete figures on account of their angles, and not of single figures but of several in combination.

Thus, the objection is not only rebutted but turned against itself For the assumption it had made about the figure at the circumference, we can with equal justice make about that at the center, that congruence, as was stated in our proposition here, is in it later than knowability, so that the part it plays should be more important than that of the former, by our adversary’s own admission.

For a figure must come into being before the whole of it can be congruent.

However, in this instance unless the side of the figure is knowable, the figure cannot come into being. For although it is true that if in a figure one angle is given, which the rays of two planets form at the center, the number of all of them is given, and through them the fitness of the whole figure for congruence, and the nature of the side does not enter into this demonstration, yet that one angle of the figure is not given, that is, it is not recognized as an angle of a congruent figure, except through a knowable side.

Therefore, the soul knows the side before (in natural order) it recognizes that it is given a congruent angle.

But if we Compare both figures in this case, the side or area of the figure center is given less, in fact, by the actual radiation, than the side of the figure at the circumference. For the latter is always bounded by the actual rays,the latter not always, but only in certain figures, as in the triangle, because the individual sides of the angle at the circumference subtend an arc equal to the intercept.

Therefore, the figure at the center is more remote from the act of knowing than that at the circumference. Therefore, the recognition of congruence is also more remote. However, the objection rested on the contrary, as if the congruence of that at the center was more accessible to recognition than the congruence of that at the circumference.

Further, if the quantity of the angle is perceived, by what will it be perceived if not by its own measure, that is, by the arc of that circle which is drawn from the point of the angle in question, that is the Earth?

But not at first by an arc of that circle which is drawn round the central figure, and which passes through the Earth. Hence in the perception of the quantity of an angle in the central figure, it must as a soul show that power by which it is a circle, not that by which it is a tiny point, to which the angle extends. But by the same character of its circular essence it also perceives the side of the figure at the circumference, and its arc, and that at first.

Only after that, with the doubling of this arc, does the arc of the smaller circle also emerge, which is drawn round the central figure, and passes through the center of the previous one.

This arc assists the inscription of thefigure at the center in the circle. For the order which occurs in reasoning is also the same in instinct. Again, then, a way is demonstrated to the perception of the central figure, a longer way, and so also to its congruence.

Hence the objection reverses itself, as it based its superior force on priority in perception.

To the reasoning for the other branch of the argument the following reply has to be made. In fact even in the figure at the circumference the knowability of the side is in truth prior to the congruence of the whole figure, on account of the arguments stated, which are also valid in this case.

But it does not follow that of two things, one of which is the cause of the other, the one which is the cause also in addition moves a third one more strongly.

For in relation to the grasp of the soul, which has to be moved, often for that purpose the cause has less force than the effect.

So in this case the sublunary Soul, insofar as it is indeed perceptive, is more moved by the knowability of the figure at the center; whereas insofar as it is operative, it is more moved by the congruence of the figure at the circumference.

However, my own arguments to prove the proposition are the following.

We first took it in Proposition VI that the center is a sort of idea of the speculative mind, or understanding, the circumference of the practical or operative faculty, because as the center is the basis and origin of a circle, so meditation is of action.

Then the figure which offers its angle to the center, that is to the Earth, where the soul, the perceiver of the figure, has its seat, offers itself, so to speak, for knowing and judging, since the center represents the judgment seat of knowledge.

Therefore, in the figure at the center more notice should be taken of knowledge, notwithstanding the fact that it is obtained by means of the circle as if by an instrument, as was stated a little earlier.

On the other hand, thefigure which arranges the angles at the circumference applies itself more to imitation and to expression in the operation of the soul, as if working towards the idea of the operations.

But it is congruence rather than knowability which has the idea of sensible operations and constructions, because it is a consequence of whole figures, while the side by which the figure is known is merely an element of it. Hence in the figure at the circumference we must have regard to congruence rather than to knowability.

Another argument for this second part depends on the same consideration of the soul. It prevails on account of what the others are. But configurations are perceived on account of the operation of sublunary Nature, and indeed of the inferior faculties of the human soul.

They are perceived for the purpose of expressing them operationally. The status of the motor faculty is therefore greater in this business. But the knowability of the figure at the circumference assists perception, and its congruence assists operation, as hitherto.

Therefore, the congruence of the figure at the circumference also prevails over the knowability of the same.

Proposition VHP® The arc of a circle which is established by an incongruent figure imparts no influence to the rays of two planets which delimit the arc.

For if congruence is the most important cause of influence, by Propositions 3, 4, and 7, therefore if that is lacking, knowability will not be sufficient as a more humble substitute cause.

The latter is more powerful than congruence in the shape at the center, by the other part of Proposition 7. Yet on the other hand the figure at the circumference is more important than the one at the center, by 6.

In that congruence is more powerful, by the first part of Proposition 7. Therefore, the congruence of the figure at the circumference is still more powerful than the knowability of the figure at the center.

This is why the aspects are few, even if the knowable shapes are infinite and of various degrees.