The Fourth Degree of the Configurations

Proposition 13

The fourth degree of the configurations in influentiality is that of the quintile, biquintile, and quincunx.

For these have in common congruence of the whole of the primary figures in the plane, though not of the individual kinds each with it self, but of the first two figures mutually with each other, and of the last with the others which are akin to it.

The first two aspects are preeminent in virtue of the fact that their figures, the pentagon and its star, are congruent in the solid as well, and make two regular solid figures.

By this nobility they almost bring their aspects to the level of the trine and quadrant.

The dodecagonal star is not congruent in the solid. Yet on the otherhand the dodecagonal star is also pre-eminent by its plane congruence, which in it is capable of being continued to infinity, whereas the former figures cannot be continuedfar without an admixture of irregularity. See all this in Book 2.

As far as the knowability of the sides in the figures at the centers is concerned, here also the sides of the decagon, tridecile, and dodecagon stand in the middle position, and they are central in this class, between the side of the triangle which precedes and the sides of the pentagon and the pentagonal star which are the central figures in the following class.

For it was demonstrated in Book I that in knowledge the dodecagon’s side takes precedence over the pentagon’s, the tridecile’s over the pentagonal star’s.

Thus, knowability leads in the same direction as congruence too, by Proposition 7.

This had to be stated in advance, chiefly for the sake of this demonstration, in case the decile or tridecile should be given preference over the quintile and biquintile.

If, however, someone should wish to dismiss thefigure at the center and seek knowability rather in the one at the circumference, no less than congruence, although it must be admitted that on that basis the decile would be preferred to the quintile, and the tridecile to the biquintile, he should nevertheless remember that the parts of congruence are the most important, as we have shown in Proposition IV. Therefore, it is a greater thing, and has more effect on influentiality, to create a solid figure (which is like a sort of mathematical idea of physical influentiality) than to have a side which is knowable in a more perfect degree.

The side of the dodecagon indeed on this basis brings its aspect into the same class as the chords subtended by a tenth part of a circle, and by three tenths, because they contend with each other for pre-eminence in knowability. For just as those 2 chords are associated with each other, and the smaller is part of the greater in the divine proportion of the extreme and mean, so also the side of the dodecagon and the side of its star are associated, and that in respect of a division and combination, though not proportional.

In fact, this latter pair falls into the first kind of inexpressibles, which embraces the binomials and apotomes; but on the other hand the former pair acquires a new property of division according to extreme and mean, as may be seen in Book 1.

Hence these degrees are not just balanced, but the side of the decagon even has a little the better of it. It was therefore correctly done that I placed the aspect of the quincunx, or 150, in the same degree as the quintile, 72, and the biquintile, 144, though giving the chief seat to these last.

Proposition 14

The fifth, lowest, and feeblest degree of the aspects is that of the decile and the tridecile, the octile, and the trioctile.

I have made the fifth place for the decile and the tridecile (Maestlin calls them the half-quintile and the quintile -and-a-half) which until now I left out in the Ephemerides.

With them 1 have associated the octile and trioctile, or half-quadrant and quadrant-and-a-half which the compilers of almanacs, at my suggestion indeed and to a large extent on the authority of Ptolemy, have seized on, but with too much enthusiasm and too little consideration.

Therefore, both points must be tested, first that these 4 are feebler than the quintile and biquin­ tile, and second that the decile and tridecile are stronger than the octile and trioctile, by a very little.

Since, then, our propositions place the greatest weight for determining influence on the congruence of the chief figure, that is the one at the circumference, it is evident

that the pentagon and its star are congruent each with the figures of its own kind, to form a perfect solid, as has just been stated, and are also splendidly congruent with each other to lay out a plane.

On the other hand, the decagon and the octagon along with their stars cannot be congruent in the solid, each with the others of its own kind.

The decagon and the octagon are congruent, but with others which are not all of their own kind.

The stars in fact start on a certain congruence in the solid but do not complete it. Even in the plane their congruence is more ignoble, because they do not deliver shared operations, each independently with its ozvn star, as the pentagon does with its own;

But with those stars of their own, and the octagon with the tetragon, they come into partnership in an alien congruence, and the decagon itself prevents the possibility of that’s being continued.

Its star also produces a split congruence in the intermediate spaces which are intercepted. However, the octagon and its star rejoice in an alternating continuation of congruence, with an admixture of squares. The congruence occurs in a variety offorms.

Thus these 4 are almost on a par in plane congruence, especially as each set of figures has inexpressible areas. But in knowability the pentagonal group is greatly pre-eminent.

First, if we were to consider the figures at the center, which in this case are in fact the pentagon and its star, their sides indeed fall under the same class of inexpressibles as the sides of the octagon and its star, being elasson and mizon.

But if we consider the figures at the circumferences, which in this case are the sides of the decagon and its star, these are not onlyfrom the nobler kind of binomials and apotomes, whereas the octagons lines are from the fourth kind, which is that of the mizons and elassons, but also all the sides of the pentagonal group acquire the noblest property of division in the proportion of extreme and mean, which clearly does not apply at all to the octagons lines.

But if the octagonal group seemed to be somewhat pre-eminent in congruence, in this case on the other hand it is much more strongly pushed down by the pentagonal group. I was therefore right to