The Conics of Apollonius (1675) - Page 29

Prop. XVIII.

Fig. 27.

If a straight line (A F B) meets a section (at F) and is extended in either direction so that it falls outside the section, and if a point (C) is taken within the section, and a line (C D) is drawn through that point parallel to the line (A B) which meets the section: then the drawn line (C D), when extended, will meet the section on both sides.

Let any point E be taken on the section, and let E F be connected. This line will meet C D. If it meets between the points E and F, it is manifest that it meets the section; if it meets outside, then it will meet the section sooner. By a similar argument, it will meet the section on the side of A F.

Prop. XIX.

Fig. 28.

In every conic section, a straight line (B C) drawn from the diameter (A B) parallel to the ordinate, will meet the section.

a 10. of this.

Let some point D be taken on the section, and let A D be joined. This will meet the ordinate at A, and therefore the parallel to it, A C. If it meets between points A and D, then B C when extendeda will meet the section; if outside, it will meet it sooner.

Prop. XX.

Fig. 29.

If in a parabola two straight lines (C E, D F) are applied as ordinates from the section to the diameter (A B), the lines (A E, A F) cut off from the diameter to the vertex will be as the squares of the ordinates to one another.

a 11. of this.
b 7. 5.
c 3. 6.

Let A G be the latus rectum upright side. Therefore, the square of C E a = A E * A G, and the square of D F = A F * A G. Therefore, the square of C E : the square of D F b

(A E A G : A F A G c

) A E : A F. Q. E. D.

Conversely. If the square of C E : the square of D F :: A E : A F, then the points C and D are on the parabola.

Corollary. A D F = C E.

This is the first and primary property of the parabola, emerging from its definition.

Prop. XXI.

Fig. 30.

If in a hyperbola, or an ellipse, or the circumference of a circle, straight lines (D E, F G) are applied as ordinates to the diameter (A B), the squares of these lines will be 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 latus rectum upright side (A C) is to the transverse side (A B); and they will be to one another as the areas contained by the aforementioned intercepted lines.