The Extant Greek Works of Apollonius of Perga, Vol. I

and let them meet at B, Γ, and let BΓ be joined; therefore BΓ will be inside the circle, and thus also inside the conic surface. Now, let an arbitrary point Z be taken on ΔE, and let the joined line AZ be extended. It will fall upon the straight line BΓ, for the triangle BΓA is in one plane. Let it fall at H. Since, therefore, H is inside the conic surface, AH is also inside the conic surface; thus Z is also inside the conic surface. Similarly, it will be shown that all points on ΔE are inside the surface; therefore ΔE is inside the surface.

Now, let ΔE be extended to Θ. I say that it will fall outside the conic surface. For if it is possible, let some part of it, such as Θ, not be outside the conic surface, and let the joined line AΘ be extended; it will fall either upon the circumference of the circle or inside, which is impossible; for it falls upon the extended BΓ as at K. Therefore, EΘ is outside the surface.

Therefore, ΔE is inside the conic surface, and the line in a straight line with it is outside.

III.

If a cone is cut by a plane through the vertex, the section is a triangle.

Let there be a cone whose vertex is point A and whose base is the circle BΓ, and let it be cut by some plane through point A, making as sections on the surface the lines AB, AΓ, and in the base the straight line BΓ. I say that ABΓ is a triangle.

1. ἄρα (therefore)] cv; euan. V, rep. mg. m. rec. 16. περιφέρειαν (circumference) V (in alt. φ inc. fol. 3^u), corr. m. rec. ἀδύνατον (impossible)] cv, -τον euan. V. 20. ἐκτός (outside)] ἐκτός : — V. 28. ΑΒΓ] p, ΑΓ V, corr. m. 2 v.