Classes of Bisections

The proper form of bisection is used in the first class of figures.

• It is common both to the second and to the third classes

The first class exists according to a rule which differs from that obtaining for the 2 remaining classes: in such a way that the first class is related to each of the two remaining ones; but the latter are distinguished one from another, so that, for figures which have proper constructions there are in a way only two Kinds.

The Tetragon and the Octagon quickly apply themselves in almost all the figures of the Trigon’s sect because a sixth of a circle and a twelfth join up to make a quarter, a twelfth and a twentyfourth combine to make an eighth.

The Tetragon fits itself into the Pentagon’s sect to some extent; because a fifth part of a circle, added to a twentieth, constitutes a fourth part.

The reason is that the Numbers 3 and 5 can be decomposed into numbers in continuous double proportion For the number 3 can be separated into 1 and 2, and 5 into 1 and 4.

There is no such common ground between the classes Three and Five. For although a sixth part of a circle added to a 30th constitutes a 5th.

Nonetheless, the 30th belongs to the class of the Pentehaedecagon, which does not have a proper demonstration. In the same way, a tenth of a circle, added to a 15th (notice the admixture with class 4), make a 6th.

Because of this duality in hinds, the characteristic numbers are, of the first 12, of the second 20, or half of it, 10. These matters will be referred to below in Book III and applied to the distinctions between the kinds of melody.

L Comparison of the Figures or divisions of the circle

1. The diameter

It is expressible in length.

2. The Hexagon side, equal to the semidiameter, and thus Expressible in length.

3. The Tetragon and the Trigon

They have sides Expressible only in square. In the fourth rank come the sides of the Dodecagon, Decagon, and the associated sides of their stars; for they are inexpressible in square, and are Composites of the first species, namely they are Binomials and Apotomes, the Dodecagon side of the Sixth [kind] and the Decagon in fact of the Fourth [kind].2’3

There follow, in the Fifth place, the sides of the Pentagon and its star, which, like the sides of the Octagon and its star, belong to the fourth species of composites, called Mizon and Elasson.

Let it not be supposed that what has been said in favor of the Decagon diminishes the Pentagon: nor that, in belonging to the same species, the side of the Octagon makes its figure equal to the Pentagon or the Decagon; the pentagon gains new power from its origin; because throughout this sect having features in common with the Decagon, the Divine proportion reigns everywhere: being immediately present in the sides of the Pentagon and of its star; but it does not occur in the Decagon and its star, except if the side of the Hexagon is an intermediary; it does not occur at all in the Octagon.

Beyond these properties of the sides, there is another indicator of nobility, because the figures are differentiated by the aptness and perfection of the areas they enclose.

Here, after the diameter (whose area is zero, and which, as Ptolemy notes, only divides the area of the circle into two equal parts, as it also does the circumference) the highest place is given to the Tetragon and the Dodecagon, which have Expressible areas, and the Tetragon indeed is of the greatest distinction; because its area is the same as the square of its side, for the class of area to which it belongs is to be square: so it encloses half the square of the diameter: thus the Dodecagon stands lower than the first, enclosing three quarters of the square of the diameter. In the next place there follow the Trigon, Hexagon and Octagon, whose area is Medial in species, while the concepts [of the characters] of the areas of the Pentagon and Decagon have no names.