The Conics of Apollonius

a 15. 11.
b 4. of this.
c 10. 11.
d cor. 13. & 16. 6.
e 1. 6.
f hypoth.
g 23. 6.
h 4. 6.
k 2. 6. 19. 5.

If a cone is cut by a plane (D F E), intersecting the base of the cone along a straight line (D E), which is perpendicular to the base (B C) of the triangle through the axis, and the diameter of the section (F G) is equidistant to one (A C) of the sides of the triangle through the axis; the straight line (K L) which is drawn from the section equidistant to the section (D E) of the intersecting plane and the base of the cone, up to the diameter of the section (F G), will be able to form a space equal to the content of the line (F L), which is intercepted from the diameter between it (K L) and the vertex of the section (F), and a certain other line (F H) which, being placed in relation to the line (A F) intercepted between the cone's angle (A) and the vertex of the section (F), has the same proportion as the square of the base (B C) of the triangle through the axis has to that which is contained by the other two sides of the triangle (A B, A C). Let a section of this kind be called a PARABOLA.

Through L, let M N be drawn parallel to B C; and the section of the plane through M N, KL (parallel to B D C Ea) is a circleb; and KL is perpendicular to MNc; whence KL sq. = ML * LNd. Furthermore, FL * HF : FL * FAe

BC sq. : AC * ABg = BC : AC. (ML : FL)h + BC : AB (ML : FM, or LN : FA)k = ML : FL + LN : FAg =) ML * LN : FL * FAl. Therefore ML * LN (d KL sq.) = FL * HF. Q. E. D.

Prop. XII.

Fig. 17.

If a cone is cut by a plane (A B C) through the axis; and is also cut by another plane (D F E) intersecting the base of the cone along a straight line (D E) which is perpendicular to the base (B C) of the triangle through the axis; and the diameter of the section (F G) produced, meets one side (A C) of the triangle through the axis outside the vertex of the cone at (H): the straight line (M N), which is drawn from the section equidistant to the common section (D E) of the intersecting plane and the base of the cone, up to the diameter of the section, will be able to form a space (F N X P) adjacent to the line (F L), to which that (F H), which is constituted in direct alignment with the diameter of the section and is subtended by the angle (F A H) outside the triangle, has the same proportion as the square of the line (A K), which is drawn equidistant to the diameter (F G) from the vertex (A) of the section to the base (B C) of the triangle, has to the rectangle contained by the parts of the base (B K, K C) which are created by it; having as width the line (F N), which is cut off from the diameter between it (M N) and the vertex of the section (F), and exceeding by a figure (F N O L) similar and similarly placed to that which is contained by the line (H F) subtended outside the angle and that (F L) alongside which those that are applied to the diameter (F G) are able to be formed. Let a section of this kind be called a Hyperbola.