The Conics of Apollonius (1675) - Page 30

of their squares to the areas contained by the lines (E B, E A, and G B, G A) which are intercepted between them and the vertices (A, B) of the transverse side of the figure, as the figure's latus rectum upright side (A C) is to the transverse latus side (A B); and they are to one another as the areas that are contained by the lines we mentioned as being intercepted.

a 4. 6.
b 1. 6.
c 12. of this.
d 11. 5.

Through E and G, let E H and G K be drawn parallel to A C, meeting the extended B C at H and K. C A : A B a

(H E : E B b

) H E * E A (c this is the square of D E) : E B * E A. By a similar argument, C A : A B

square of F G : G B G A. Therefore, square of D E : E B E A d

square of F G : G B * G A. And conversely, square of D E : square of F G

E B E A : G B G A.
Corollary. In a hyperbola, F E is less than D E (in an ellipse, this holds until the diameter conjugate to A B; for from there the ordinates begin to decrease). Sch.
Conversely; if the square of D E : E B * E A

R : T, or the square of D E : square of F G

E B E A : G B G A, the points D and F will be on one of these sections.
This is the first and primary property of these sections, resulting from their definition.
Corollary. Square of D E : E B * B A

square of T : square of M.

The path is now paved for investigating the tangents of the sections.

Prop. XXII.

Fig. 31.
32.

If a straight line (C D) cuts a parabola or a hyperbola at two points (C, D), and does not coincide with the diameter of the section (A B) within the section, then when extended it will meet the same diameter outside the section.

a cor. 20. &
b cor. 21. of this.

Let C E and D B be applied as ordinates: these are unequal, and a D B is smaller, b therefore the joined C D, when extended, will meet E A outside the section on the side of A. Q. E. D.

Prop. XXIII.

that is, conjugates.

If a straight line (E F) cuts an ellipse between two *diameters (A B, C D), then when extended it will meet each of them outside the section.

Fig. 33.
a 21. of this.
b 5. 2.
c 14. 5.

Let E G and F H be applied as ordinates. And because the square of E G : the square of F H a :: B G * G A : B H * H A, b and B G * G A is greater than B H * H A (for point G is closer to the center M than point H is); c the square of E G is greater than the square of F H. And E G is greater than F H. Therefore, E F, when extended, will meet the diameter B A on the side of