The Conics of Apollonius

Coroll. 2. Straight lines drawn from the vertex to points which are on the surface will meet the circumference of the base, if need be, produced.

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Fig. 6.

Prop. II.

If two points (D, E) are taken on either of the surfaces which are at the vertex, and the straight line (DE) joining the points does not belong to the vertex, it falls within the surface, but the line which is in direct alignment (EF) will fall outside.

a 2 Cor. 1. 2 of this.
b 2. 3.
c 2. 11.
d 10. 3.
e 1. Cor. 1 of this.

Let the straight lines AD and AE be joined, meeting the points B and G of the circumference of the base, which the straight line BG connects; this will fall within the circle, and therefore within the conical surface; therefore the plane of the triangle ABG is within the conical surface; therefore the straight line DE situated on it is within the same. Furthermore, the straight line AF falls on BG extended outside the circle; and therefore outside the conical surface; therefore point F is also outside it.

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Fig. 7.

Prop. III.

If a cone (ABC) is cut by a plane through the vertex (A), the section (ABC) will be a triangle.

a 1. of this.
b 3. 11.
c 20. def. 1.

For AB and AC are straight lines: likewise BC is a straight line. Therefore ABC is a triangle.

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Fig. 8.

Prop. IV.

If either of the surfaces which are at the vertex is cut by a plane (DHE) equidistant to the circle (BKC), through which the straight line describing the surface is carried, the plane (DHE) which is enclosed by the surface will be a circle, having a center (G) on the axis (AF); but the figure (ADE) contained by the circle (DHE) and that part of the conical surface which is intercepted between the cutting plane (DHE) and the vertex (A) will be a cone.

a 3. of this.
b 3. 11.
c 2. Cor. 1. of this.
d hyp. & 16. 11.
e 4. 6.
f hyp. & 15. def. 2.
g 14. 5.

Let the plane through the axis AF make a triangle ABC; and let its common section with the plane DHE be the straight line DF. In the section DHE, let a point H be taken at random, and let the straight line AHK be drawn, meeting the circumference of the base in K, and let GH and FK be connected. And because DE, BC, and GH, FK are parallel, it will be FB . GD

) FK . GH. Hence since FB and FK are equal, GD and GH will also be equal. And the same can be shown regarding the remaining straight lines drawn from G to the section. When-