The Arcs of a Circle

Axiom 3

The arcs of a circle, of which the figures are powerful in more, and more important, degrees of congruence and knowability, also take the more influential configurations.

If the first two axioms are generally agreed,so will this be, because on account of the fact that every individual one is like that, if the former is intensified, the latter will also be more like that.

Understand this, however, in the following way: that in the figure at the circumference the comparison of the degrees of congruence takes precedence, in that at the center comparison of the degrees of knowability, and in fact the part of the figure at the circumference is more important.

Proposition 9

The influential configurations are those which intercept the following arcs of the zodiac circle: 180°: opposition from the diameter of the circle, as in Figure 1.

• 90°: the quartile □ from the tetragon, as in Figure 2.
• 120°: the trine A , and 60°: the sextile )(, from the triangle and the hexagon, as in Figures 3 and 4.
• 45°: the octile or semiquartile, and 135°: the trioctile or quartile-and-a-half from the octagon and its star, as in Figures 5 and 6.
• 30°: the semisextile H, and 150°: the quincunx (5/12ths), from the dodecagon and its star, as in Figures 7 and 8.
• 72°: the quintile and 108°: the tridecile or quintile-and-a-half, from the pentagon and the decagonal star, as in Figures 9 and 10.
• 144°: the biquintile and 36°: the half-quintile or decile, from the pentagonal star and the decagon, as in Figures 11 and 12.

These shapes are knowable and constructible has been shown in Book I: that they are also congruent, in Book II. However, that the configurations of arcs expressed by such are influential is in the Axioms I and II previously stated.

Proposition 10

The influentiality of the aspects the first and strongest degree is that of conjunction and opposition For in conjunction the two rays are congruent on the same line, and descend from the same side: in opposition ^ they descend from different sides indeed, but are nonetheless parts of one continuous line.

This in fact is the most perfect congruence and a kind of basic principle of all congruence. Thus as conjunction is represented by a point marked on the circumference of the circle, but opposition by the diameter, these are certainly basic principles.

The former is also the measure of all knowledge in this class, as all knowledge of a straight line in a circle is contained in the constrmtible demarcation of it by means of the length or square of the diameter, as was made clear in Book I. Therefore, entiality is also in these aspects.

Proposition 11

The second degree of influentiality of aspects is that of the quartile □ .

For in the quartile many privileges coincide, the first of which is that the figure at its center is similar to that at its circumference.

Hence whatever degrees it holds in congruence and knowability are understood to be in a sense double relatively to the other aspects. For just as the quartile is the first, after opposition, to be unfolded from the narrowness of its line to some breadth or surface extension of its tetragonal area,^^ so the remaining aspects descend from the identity of the figures of the quartile aspect to some difference in their figures.

Therefore, as elsewhere in natural philosophy, excellence which is unified is stronger, in this ideal and objective portrayal the degree of strength will be greater where figures in distinct positions, that is one at the center and one at the circumference, are found to be the same in kind.

Second, as far as congruence is concerned, in the square it is most perfect and of all kinds, for this figure is itself congruent with itself in the solid to form a cube, which is the measure of all solidity; and is congruent in the simplest way, taking in only three angles; and is itself congruent with itself in the plane, with four angles. Again, it is congruent in the solid with the triangle, pentcigon.

hexagon, octagon, and decagon in different ways, to form solid figures; and is congruent with all of those, and in addition with the dodecagon, and the icosagon to a certain extent, to lay out a flat surface. In this property it is surpassed by no other.

Third, the area of the square is expressible, which is the basic principle of a certain singular and outstanding congruence in the plane, so that a definite number of areas of this figure take up a definite number of squares on the diameter, and thus the figures are not only in themselves congruent with each other in their angles and sides, but in a way, that is in definite lines of theirs, also with the sides of the square on the diameter. This property the quartile aspect shares partly with the half-sextile alone. See Book 2.

Fourth, the degree of knowledge of the side is also not ignoble, as it is expressible in the square. In this degree it is pre-eminent over all the other figures, except the hexagon. However, it does not on that account give way to it, since knowability is not to be compared with congruence, as has been explained above, and indeed the accumulation of privileges is valid for increasing the influence, by Axiom III of this Chapter.

Proposition 12

The third degree of influentiality is that of the trine A , sextile % and semi-sextile That I place the trine, sextile, and half sixth in the same degree is not due to identity of properties but to equal power.

First, their principal figures transmit shared operations to congruent planes; for they meet each other, and others, such as the square, in various ways. In fact, the triangle and hexagon are pre-eminent in this case, because the individual kinds are congruent also with themselves.

The hexagon is pre-eminent over the triangle, because it holds the most perfect congruence in the plane, that is in its three angles alone. Both are preeminent over the dodecagon, because they are congruent with other figures even in the solid, which the dodecagon cannot be. Yet on the other hand the dodecagon is pre-eminent over the others in the expressibility of its area, whereas their areas are medial and so more ignoble.

This difference between the areas, as I stated just now, redounds to the perfection of its congruence. Similarly the triangle also is again pre-eminent over the hexagon, in that the triangular kind is itself congruent with itself in the solid invarious ways, and begets three regular solids:

The hexagon is congruent only with other figures. Thus on balancing the weights of the different properties against each other, congruence, which is the first and chief element in influence, in the case of these 3 very nearly produces equilibrium.

In knowability the hexagon holds first place, as its side is expressible; second place is held by the triangle, for it takes the same degree as the tetragon, having its side expressible in the square, but in a more humble proportion. The dodecagon is last in this instance, having an inexpressible side.

However, knowability is not the chief indication of influence, and is not considered in the most important figure, that is the one at the circumference, but only in the one at the center which is less important.

If it has any effect, it renders the trine a little more influential than the sextile, because the trine is formed by an angle of the hexagon at the center, and the half-sixth a little less influential than either, as it is measured out by an angle of the dodecagonal star at the center.

However, knowledge of the half-sixth is more noble than the rest which follow, because the side of the figure at its center is of the most noble kind among the inexpressible, that is the binomial, and in their twofold subdivision it always holds the superior place, to the extent that along with its partner the side of the figure at the circumference it forms an expressible triangle.

That is the mark of almost absolute perfection, and also in fact makes this figure in knowability a contender with the triangle and the hexagon, because it weighs in the balance against its inexpressibility, very heavily.