Apollonius: Conic Sections, i.33-34

Apollonius of Perga

Source: On Conics, in Great Books of the Western World, vol. 2, Robert Maynard Hutchins, ed., Encyclopedia Britannica, 1952, pp. 640-641.

Proposition 33. If in a parabola some point is taken, and from it an ordinate 1is dropped to the diameter2 and to the straight line cut off by it on the diameter from the vertex, a straight line in the same straight line from its extremity is made equal, then the straight line joined from the point thus resulting to the point taken will touch the section.

    Let there be a parabola whose diameter is the straight line AB, and let the straight line CD be dropped ordinatewise, and let the straight line AE be made equal to the straight line ED, and let the straight line AC be joined.
    I say that the straight line AC produced will fall outside the section.3
    For if possible, let it fall within, as the straight line CF 4, and let the straight line GB be dropped ordinatewise. And since

sq.BG : sq. CD > sq. FB : sq. CD 5 ,

but

sq. FB : sq. CD : : sq. BA : sq. AD 6,

and

sq. BG : sq. CD : : BE : DE (i.20 7),

therefore

BE : DE > sq. BA : sq. AD.

But

BE : DE : : 4 rect. BE, EA : 4 rect. DE, EA;

therefore also

4 rect. BE, EA : 4 rect. DE, EA > sq. AB : sq. AD.

Therefore alternately

4 rect. BE, EA : sq. AB > 4 rect. DE, EA : sq. AD;

and this is absurd; for since AE = DE, hence

4 rect. DE, EA = sq. AD8.

But

4 rect. BE, EA < sq. AB;

for E is not the midpoint of AB (Eucl. vi.27; ii.5)9. Therefore the straight line AC does not fall within the section; therefore it touches it.

Proposition 34. If on a hyperbola or ellipse or circumference of a circle some point is taken, and from it a straight line is dropped ordinatewise to the diameter, and whatever ratio the straight lines cut off by the ordinate from the ends of the figure's transverse side 10 have to each other, that ratio have the segments of the transverse side to each other so that the segments from the vertex are corresponding, then the straight line joining the point taken on the transverse side and that taken on the section will touch the section.

Let there be an hyperbola or ellipse or circumference of a circle whose diameter is the straight line AB, and let some point C be taken on the section, and from C let the straight line CD be drawn ordinatewise, and let it be contrived that

BD : DA : : BE : EA 11 ,

and let the straight line EC be joined.
I say that the straight line CE touches the section. 12

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