CONICS I.

for if it were possible, let it not be, and let the straight line that generated the surface be the line ΔE, and the circle along which EΔ moves be the circle EZ. If, then, while the point A remains fixed, the straight line ΔE is moved along the circumference of the circle EZ, it will also pass through point B, and two straight lines will have the same endpoints: which is absurd.

Therefore, it is not the case that the straight line joined from A to B is not in the surface; therefore, it is in the surface.

Corollary.

And it is clear that if a straight line is joined from the vertex to any point inside the surface, it will fall inside the conic surface, and if it is joined to any of the points outside, it will be outside the surface.

II.

If two points are taken on either of the surfaces lying vertically opposite, and the straight line joining the points does not tend toward the vertex, it will fall inside the surface, and the extension of it will fall outside.

Let there be a conic surface whose vertex is point A, and let the circle along which the straight line generating the surface moves be BΓ, and let two points Δ, E be taken on either of the vertically opposite surfaces, and let the joined line ΔE not tend toward point A. I say that ΔE will be inside the surface, and its extension will be outside.

Let AE and AΔ be joined and extended; they will then fall upon the circumference of the circle. Let them fall...

2. καθ’] cv; κα- faded V, replaced in margin by a later hand. 10. Corollary] omitted in V.