The Heptagon

Proposition 45

The Heptagon and all shapes the number of whose sides are Primes (so-called), and their stars, and the complete classes [of figures] derived from them, have no Geometrical description independent of the circle.

In the circle, although the quantity of the side is determinate, it is equally impossible to evaluate.’^®

This is a matter of importance, for it is on account of this result that the Heptagon and other figures of this kind were not employed by God in ordering the structure of the World, as He did employ the knowable figures explained in our preceding sections.

Let:

• the Heptagon be BCDEFGH
• all angles be joined with one another
• A be the center of the circle
• BAP a Diameter
• A be joined to E.

First of all, such figures do not possess any non intrinsic construction like that mentioned above for the number of their sides and angles is one of the primes.

But no pair of the previous figures divides the complete circle into parts that can be counted by any Prime Number: instead they [the resultant figures] correspond to a Number which is a Multiple of the Numbers [corresponding] to each figure.

But nor do figures of this kind have a proper construction through the number of their angles: because whatever can be extracted from this is vague and non unique and very ill-determined.

For let the Heptagon be divided up into its five triangles, two on the outside being isosceles and Obtuse-angled, namely triangles BDG and BGH, one on the inside being isosceles and Acute-angled, namely BEF, and two Scalene triangles lying in between, namely BED and BFG.

So since the [arc of the] circumference on which the sides containing the angles stand, the angles themselves being on the opposite part of the circumference, takes its measure from its angle, [we may note that] the angle BEF stands on three parts [i£. sevenths] of the circumference, BH, HG, GF; the angle BFE similarly [stands] on the three [parts] BG, GD, DE; while EBF is on one [part] EE Therefore BEF is a triangle such that each of its base angles is equal to three times the angle at its vertex.

Similarly we may show that the Scalene triangle BED has angles in continuous double proportion.

The simple angle is the one at B, the double at E, and the quadruple at D, being double the angle at E.

Thus if this figure [the heptagon] has a precise {certam) description independent of the circle, as did the pentagon above, it is required (as has already been pointed out by Campanus, Girolamo Cardano, and Foix de Candale)’^’^^ that first of all it must be possible to construct such triangles, as a triangle was constructed for the Pentagon having each of the angles at its base equal to twice the angle at the vertex.

But for that Pentagon Triangle we obtained from the angles a precise proportion for the sides; in this Heptagon triangle, we have no precise proportion.

For let I, K be the points in which BF is cut by EH, EG the trisectors of the angle BEE So in triangle FEI, because the angle FEI is bisected: so in it the ratio of FE to El is equal to the ratio of FK to KL But EF is equal to the whole of FI. For angle FEI is 4 sevenths of a right angle, and angle EFI is 6 sevenths, therefore EIF is also 4 sevenths. So the sides (crura) FE, FI opposite the equal angles are equal.

For the same reason El and IB are equal: so the ratio of FI to IB is equal to the ratio of FK to KI. Further, in triangle KEB, because angle KEB is bisected by the line EIH: therefore the ratio of KE to EB is equal to the ratio of KI to IB.

But KE and FE are equal, because triangle KEF is isosceles and similar to the triangle EBF indeed EF was equal to the line IF, and EB is equal to the line FB; so the ratio of IF to FB is equal to the ratio of KI to IB.

So, for the same line BF, the chord subtending three sevenths of the circle, we have found two proportionalities, of three parts: first that the ratio of the mean line, KI, to the least one, KF, is equal to the ratio of the greatest one, IB, to the line IF, composed [i.e. the sum] of the two smaller ones, that is to the line FE, the side of the heptagon (septanguli): second that the ratio of the greatest line, IB, to the mean one, IK, is equal to the ratio of the whole line, BF, to the line FI, composed [i.e. the sum] of the two smallest.

This kind of proportionality seems to carry the implication that there is a unique precisely determinate proportion between the lines EF and FB; and Cardano, who, when he discussed this matter concerning the sides of the Scalene triangle BED, gave it the name Reflexive Proportion, boasted, falsely, that he had found the side of the heptagon (septanguli) F o r no precise quantity follows for either the line EF or IF; because what we think is new information given in the second relationship is the same as the information given in the first.

For, whatever 4 proportional quantities are related to one another in such a way that [the sum of] the first two is equal to the third; it also holds that the ratio of the first to the third, and of the second to the fourth, is equal to the ratio of the third to the quantity composed [i.e. the sum] of the third and the fourth, which composite quantity becomes a fifth member [of the series set up by the relationships].

So the number of Cases-^”^ is infinite, either in terms of commensurable quantities or in terms of incommensurable ones.

In fact the number of cases for commensurable terms is the same as that of superparticular proportions, that is the same as the number of uneven square Numbers.

The same as there are superpartient numbers BF. 9- or 25- or 49- or 8 i. or 1 2 1 . or 49- 64 . BI. 6 . 15 - 28 . 45- . 6 6 35- . 4 0 IK. . 6 . 12 . 20 . 3 0 . 2 10 . 15 - KF. 1 . 4- 9- 16 . 2 5 . etc. 4- . etc. 9 For the ratio of 15 to 9 is equal to the ratio of 40 to 24, the number that is the sum of 15 and 9. And the ratio of 40 to 15 is equal to the ratio of 64 (made up of 40, 15 and 9) to 24, the sum of 15 and 9.

This property is common to many proportional relationships, and it follows necessarily from the structure of the heptagon but, from only what has been given, it is not possible to construct the triangle belonging to the heptagon {triangulum septangulare).

The reason why in the Pentagon the proportion of the side can be precisely determined from the angles, even independently of the circle, while the same is not true for the Heptagon and other such figures, is easily seen from what has been said already.

In the triangle BFK pertaining to the Pentagon, bisection of the angle BFK at once gives the isosceles triangles BKT and KTF, two of its [the pentagon’s] elementary triangles, and it follows from the equality of their angles BFK, BKT, that the sides BK, KT, TF are equal; but in the case of the Heptagon triangle, trisection of the angle produces three elementary triangles, two isosceles triangles BEI, KEF and one scalene triangle, lEK, nor does it follow from the proportion between its [the scalene triangle’s] angles that there is any particular propor­ tion between the sides, as is known by Geometry.

Thus, since the angles of this figure have no significance independently of the circle; so the required triangle cannot be constructed independently of the circle.

So this figure cannot be inscribed in a circle, by means of anything prior to itself in regard to knowledge or description, but this vague proportion is narrowed down into a single result only by some procedure for inscription and thus we have a circular argument; for in order to find what is required to carry through the inscription we are instructed to make use of the inscription procedure itself, as if it were already possible.

So the ratio between the Side [of the heptagon], EF, and the side of the star, FB is latent; is latent, I mean, in quantitative matters, so that by reason of the relevant principle regarding quantities, that is [the method involving the use of an] indeterminate magnitude,^*

is in fact possible to construct the side of the heptagon in correct pro­ portion to the diameter of the circle: since let there be given a mag­ nitude that is certainly greater than the side of the heptagon, and one that is certainly less than it, in the same Circle:

Further, subdivision proceeding to infinity can always give magnitudes greater than the side EF or less than it: but, on account of the formal properties of quantities, it is simply impossible [to find such a procedure of subdivision], because the figure of the heptagon, and similar figures, are completely lacking in any mean quantities which might lead to demon strating or finding a proportional relation for the side of the figure [i.e. its relationship to the diameter of the circle] and thus to constructing it or demonstrating that it is knowable (noscibilis). Since this is so, it is not possible to inscribe a 14-sided figure in a circle with diameter AP, the side being EF, nor for two neighboring sides [of such a figure] to subtend a chord EF, which would be the side of the Heptagon inscribed in the circle: nor will it be possible for this side [i.e. of the 14-gon] to be compared with the diameter, since by its Nature its relationship to the Diameter is unknown.