CONICORUM LIBER I.

For if it is possible, let it not be, and let $Δ E$ be the straight line describing the surface, and let the circle through which it moves be $EZ$. Therefore, if while point $A$ remains fixed, the straight line $Δ E$ moves along the circumference of the circle $EZ$, it will also arrive at point $B$, and two straight lines will have the same endpoints; which cannot happen.

Therefore, it cannot be that the straight line drawn from $A$ to $B$ is not in the surface. Therefore, it is in the surface.

Corollarium.

And it is manifest that if a straight line is drawn from the vertex to some point of those which are within the surface, it will fall within the conic surface, and if it is drawn to some of those which are outside, it will fall outside the surface.

II.

If in either of the surfaces which are placed vertically opposite to each other, two points are taken, and the straight line joining those points does not fall at the vertex, it will fall inside the surface, but when produced in a straight line, it will fall outside.

Let there be a conic surface whose vertex is $A$, and let the circle through which the straight line describing the surface moves be $BΓ$, and in either of the surfaces which are opposite each other at the vertex, let two points $Δ, E$ be taken, and let the drawn line $Δ E$ not fall at point $A$. I say that $Δ E$ is inside the surface, but when produced in a straight line, it is outside.

Let $AE$ and $AΔ$ be drawn and produced; they will therefore fall upon the circumference of the circle [Proposition I]. Let them fall at $B$ and $Γ$, and let $BΓ$ be drawn; $BΓ$ will therefore be inside the circle; wherefore it is also inside the conic surface. Now, let some point $Z$ be taken on $Δ E$, and let the drawn line $AZ$ be produced; it will therefore fall...