The Conics of Apollonius (1675) - Page 23

a 1. 6.
b 4. 6.
c hyp.
d 23. 6.
e 9. 5.
f 4. of this, & cor. 13, ac 16. 6.

Through N, let R S be drawn parallel to B C. And F N * H N : F N * N Xa

(H N : N X)b

H F : F Lc

A K sq. : B K K Cd = A K B K (b F G G B, or F N N R) + A K K G (b A G G C, or H N N S) = F N N R + H N N Sd =) F N H N : N R N S. Therefore F N N Xe = (N R N Sf =) N M sq. Q. E. D.
### Prop. XIII.
Fig. 18.
If a cone is cut by a plane (A B C) through the axis, and is cut by another plane (E L D) meeting both sides of the triangle through the axis, which is neither equidistant to the base of the cone nor positioned subcontrarily; and the plane in which the base of the cone (B C) lies and the cutting plane meet along a straight line (F G) which is perpendicular either to the base (B C) of the triangle through the axis, or to that line (B C K) which is constituted in direct alignment with it; the straight line (L M) which is drawn from the section of the cone, equidistant to the common section (F G) of the planes, up to the diameter of the section (E D), will be able to form a space (E O X M) adjacent to the line (E H), to which the diameter of the section (E D) has the same proportion as the square of the line (A K) drawn equidistant to the diameter (E D) from the vertex (A) of the cone to the base (B C) of the triangle, has to the rectangle contained by the parts of the base (B K, C K) which are intercepted between it (A K) and the straight lines of the triangle (A B, A C); having as width the line (E M), which is cut off from the diameter (E D) by it to the vertex of the section (E): and deficient by a figure (O H N X) ? similar and similarly placed to that which is contained by the diameter (E D) and the line (E H) alongside which they are able to be formed. Let a section of this kind be called an Ellipse.
Through M, let P M R be drawn parallel to B C. And E M D M : E M M Xa

(D M : M Xb

D E : E Hc

A K sq. : B K * K Cd = A K * B K (b E G * G B, or E M * M P) + A K * K C (b D G * G C, or D M * M R) = E M * M P + D M * M Rd =) E M * D M : M P * M R. Therefore E M * M Xe = (M P * M Rf =) M L sq. Q. E. D.

Prop. XIV.

Fig. 19.

If surfaces (B C A X O), which are at the vertex (A), are cut by a plane not through the vertex, there will be in both surfaces a section (D E F and G H K) which is called a Hyperbola. And the two sections will have the same diameter (M E, H N), and the lines (E P, H R), alongside which the ordinates to the diameter are able to be formed, equidistant to that which is in the base of the cone, will be equal to one another.