APOLLONIUS OF PERGA

THEOREM I. PROPOSITION I.

Straight lines which are drawn from the vertex of a conic surface to points that are on the surface, will be within the surface itself.

Let there be a conic surface, whose vertex is a: and having taken some point b upon it, let a straight line a c b be joined. I say that a c b is in the surface. For if it were possible, let it not be in the surface: and let the straight line which describes the surface be d e: and the circle in which this d e is carried, let it be e f. Therefore, if d e is carried in e f while a remains fixed, the circumference of the circle will pass through point b: and there will be the same endpoints for two lines, which is absurd. Therefore, the line drawn from point a to b is not outside the surface. Therefore, it will be in the surface itself.

From which it is clear that if a straight line is drawn from the vertex to any point of those which are within the surface, it falls within; and if to any of those which are outside, it falls outside the surface.

A geometrical diagram shows three conic sections or cones with vertices labeled 'a'. The diagrams illustrate lines drawn from the vertex to points on the surface and base circles, with points labeled a, b, c, d, e, f, and g. The central figure shows a double cone meeting at vertex 'a'.

Regarding differing figures, or the cases of theorems, one ought to know that a case exists when those things which are given in the proposition are given in position, for the differing transformation of these, while the same conclusion remains, creates a case. Similarly, a case also arises from a transposed construction. Since, therefore, theorems may have many cases, one and the same demonstration suits them all, and the same elements: except in a few specific instances, as we will explain hereafter. For the first theorem immediately has three cases, because point b is sometimes taken on the lower surface, and this in two ways, either above the circle or below: but sometimes in that which is at the vertex. The first theorem therefore proposes to show that a straight line joining any two points is not in the surface, unless it pertains to the vertex itself. The reason for this is that the conic surface is effected by a straight line which has a fixed endpoint at the vertex. That this is clearly so is demonstrated in the second theorem.

THEOREM II. PROPOSITION II.

If in either of the surfaces which are at the vertex, two points are taken: and the straight line which joins the points does not pertain to the vertex, it will fall within the surface: but that which is in a straight line with it, will fall outside.