Measurements

Figures with an odd number of sides greater than 5 (except the Pentekaedecagon), and the chords subtended by any number of their sides, and the whole Classes associated with them,264 come into the same category as the Heptagon and other figures with a Prime number of sides.

For if the number of sides is odd and not a Prime: it is either the smallest multiple of two odd Primes; or the square of some Prime: or it is a multiple of one Prime and the square of another, or a multiple of squares with themselves or in combination.

So if these figures were describable and inscribable, and knowable; then they would also have a proper construction from their angles, or an improper one by taking into account figures of which they form part. But they do not have a proper construction; because they have not got a Prime number of sides, from which a construction might take shape: they do not have an improper construction, neither for those in the first group, such as the 21-angle, because the figures which contribute to them either both or either of them as here the Heptagon (after the Trigon and the Pentagon, which give the Pentekaedecagon), have no proper construction [of their own], by XLV above; nor [is there an im­ proper construction] for those in the second groupf^’^ such as the Nonangle [Enneagon], because there is no way of dividing a fractional arc, for example a third, into the same number of equal parts as the whole circle has been divided into, by XLVI above: nor [is there an improper construction] for those in the third group, or the fourth, because the earlier figures which form part of them are indemonstrable.

Concerning the Enneagon, whose number of sides, 9, is the square of the first odd Prime, namely three, there has been a contest among Geometers, with many exerting themselves to construct the side of this figure also. All of them did so in vain, nor would they ever have attacked this problem if they had paid attention to the difference between things that are knowable and those that are not.

Campanus wished to construct the Nonangle [enneagon] by Trisection of an angle,^^’^ which is shown to be unknowable in XLVI above.

Though it is trisected of necessity by the method of Pappus and Clavius, namely by using Geometrical motion, yet what has this to do with plane figures, with which we are dealing here, when solid figures are needed to make the lines which are aids to trisection, the Hyperbola, the Quadratrix, the Spiral and the Conchoid’? Indeed Campanus himself when attempting trisection does not tell us he is taking the third of an angle as if he were actually sure this was the case, which was what was really required. The place where he says this is, in my copy, towards the end of the Works of Euclid, folio 586 referring to the end of Book IV. Giordano Bruno of Nola, in a hexagon ABCDEF, draws perpendiculars GH, IK, tangent to the circle, to the opposite sides BC, E F [each] produced in both directions.

So, having drawn IH the diagonal of the parallelogram that has been constructed, he thought that the circle was cut [by this diagonal] in such a way that between A and D, the points of contact and M, N the points of intersection [of the diagonal with the circle] there are Ninths of the circle AN, DM. However, it can be shown from the diagram that since the square of the semidiagonal [of the parallelogram], LH, is expressible, namely 7 sixteenths of the square of the diameter [of the circle] (for [angle] ABH is 60 [degrees] therefore BH is half AB, or AL: and its square is therefore four times that of AL; so AH squared is three quarters the square of AL. But LH squared is equal to [the sum of] the squares of LA and AH),’^^^ so the sine of 40 degrees, that is half the chord subtended by two Ninths of the circle, would have to be Expressible in square, namely the root of three twenty-eighths of the square of the diameter. For having dropped a perpendic­ ular, AO, from A to LH, the ratio of the square of LH, 7 sixteenths, to the square of HA, 3 sixteenths: will be equal to the ratio of the square of LA, 4 sixteenths, to the square of AO, 12 sevenths and one sixteenth, that is, three twenty- eighths.

So this chord subtended by the angle of the enneagon would be nobler than some of the preceding ]chords], even though it is associated with them: however, since there is an odd number of sides, namely the number which is the square of the Prime number 3, there is nothing that is associated with the Tetragon and the Trigon, by bisection of arcs, though this degree of knowledge belongs to these figures.

Part 48: Corollary

Thus, the Concept, Knowledge, Determination, Description, and Construction of a figure sets up boundaries between the primary Orders to which the figures belong. The Classes of knowable figures are no more than 4.

3 figures that have proper demonstrations, among which are to be included the heads of families.

1. Tetragon, following the diameter of the circle, whose characteristic number is 2
2. Trigon, whose characteristic number is 3.
3. Pentagon, characterized by 5

The last class has improper demonstrations.

4. Improper demonstrations, whose characteristic number is the product of two factors, 3 and 5 namely 15

• The first figure in this class is the Pentekaedecagon.