Pappus' Process

In Truth, what these authors have discovered does not establish that it is possible to carry out all kinds of division so as to obtain geometrical knowledge.

To make this clear I shall first explain Pappus’ mechanical procedures for trisection: then I shall draw attention to what differentiates them from constructions that give knowledge.

First, Pappus himself, in the preamble before proposition 31, divides prob trisection of (which in a more general sense of the word he calls Geometrical, whereas an angle. r i i ^ jor us the word Geometrical has a more restricted significance) into Plane, Solid and Linear: and it is stated that Trisection of an angle cannot be carried out by Plane constructions (which for me are, in the restricted sense. Geometrical, rigorous, and of the degrees I have explained), and on this he exposes the ill- conceived attempt of ancient Geometers who here labored in vain. So he himself carries out his trisection by Solid constructions, and for sec­ tion in general uses curves. The method of trisection is this. An angle having been proposed for trisec­ tion, from a point on one of the lines enclosing the angle he draws a line per­ pendicular to the other such line, which perpendicular line is understood to determine the lengths of the lines enclosing the angle. And having drawn lines parallel to the shorter line and to the perpendicular, the former from the first point and the latter from the proposed angle, so that they meet and also form a right angle: now through the foot of the perpendicular he causes there to pass the surface of a Cone, a solidfigure; then he inclines this applied Cone, or makes it nod, until with this same surface it defines the section known as a hyperbola in the plane, such that the two parallels just drawn are Asymptotes to it: then taking as center the point that is the foot of the perpendicular, with radius twice the length of the first line enclosing the angle, he draws in the plane an arc, to cut the line of the Conic section; and joining the center of the arc with that point of intersection, he draws a line parallel to this [i.e. the join just drawn] from the proposed angle; having done this he shows that the part cut offfrom the angle is one third.

Pappus makes this problem a Solid one because he used a Cone, a solid figure. But insofar as between given Asymptotes (drawn perpendicular to one arwther) making a right angle, through a given point lying between them, it is possible to draw the Conic section called a Hyperbolaj^^^ in the planes even without using a Cone; the problem seems equally to be classifiable as Linear. For such a line is generated by Geometrical motion, and a continuous change in distances, that is, it is represented by a collection of points, of indeterminate number; and this is no less true [of this curve] than of the Quadratrix and the Spiral, the lines which he [Pappus] uses in Proposition 35 to carry out Trisection and General division [of an angle[.

This is Pappus’ mechanical procedure.

Surely between the given Asymptotes and through the given point only one Hyperbola can be drawn, whether this be done by adjusting the inclination of a Cone or by the infinite continuation of points?

Surely there is only one point of intersection of circle and Hyperbola on one side. Surely there is only one, definite, angle made between the line that connects the points of the Hyperbola and the diameter of the figure?

I affirm that all these things are necessary and certain if the Hyperbola has actually been drawn. For earlier also, in Biirgi’s analytical trisection, the third part obtained on the chord bore a certain and necessary length or proportion in regard to the chord of the whole arc. But because we are not investigating what it will be, once the construction is carried out, but rather by what means, in order to give it existence, a thing not yet constructed is to be constructed: accordingly, we get nothing more from the Solid and Linear Problems of the ancients, as far as obtaining knowledge of the required line is concerned, than we got before from the Analytical method of the moderns.

There is clearly only one line of a Hyperbola [that lies] between the given Asymptotes, passes through the given point, and can be drawn in their plane.

But when it is not yet drawn, I am required to adjust the inclination of the Cone over the point of application until it [the hyperbola] comes into being and is drawn: alternatively, not using the Cone, lam required to change the construction lines that plot the Hyperbola by repeatedly finding points, until the curve is long enough: and the parts that lie between the points I have plotted I am required to suppose to have been plotted also: in either case, I am required to pass over by a single act or motion something which potentially involves infinite division; so that by this passage something may be attained which is concealed in that potential infinity, without the light of perfect knowledge, which the problems the ancients dubbed Plane do have.

This kind of postulate is used frequently by Francois Viete, a Frenchman, and Dutch (Belgici) Geometers of our dayj in solutions of their problems,which by their very nature are not soluble except in a way that goes against the rules of the art^‘^ such as numerically or by Geometrical motions whose changes need to be guided by some kind of infinity.

Now, in a case where everything is available that was considered conducive to achieving certainty we shall arrive at a result which is either a little greater or a little smaller than the required value, and always [becomes] closer to it; as we also said before in relation to the Analytic method of trisection.

That what I say about this solid problem of Trisection is true is, as it were, suggested by the very word “solid”.

For if a proportion between solids is not given in a form such as [a ratio between] two cubic numbers: we cannot, as an intellectual procedure, measure the proposed solid in terms of the other one: because two intermediate proportionals cannot be constructed exactly in the plane: though they may be present in the cubes, yet there is no passage from the plane figures to form any of those cubes without the two means: [it is] as if the bridge were broken.

For finding 2 mean proportionals, some give instructions to use Geometrical motion, thereby ordering one to do something that is useless for achieving certainty through an appropriate Geometrical act:

Pappus gives instructions that use Conic sections, to be produced with the help of 2 [mean] proportionals, although the Cone itself is a solid.

So we are always assuming what it is required to prove; and the bridge lies on the other bank.