The Conics of Apollonius

a Hypoth.

For because the diameter FG will never meet the side AC at the parts X, if it is produced at will, namely to H, and through H KL is drawn parallel to BC, and MN parallel to DE, the plane through KL and MN will be parallel to the plane BDCE, and on the produced surface of the cone it will effect a circle, to which if the plane DFE is protracted, it is clear that the cone, and the section DFE, etc., are increased.

Prop. IX.

Fig. 13. 14.

If a cone (ABC) is cut by a plane (DKE) meeting both sides (AB, AC) of the triangle through the axis, which is neither equidistant to the base (BC) nor positioned subcontrarily, the section (DKE) will not be a circle.

If it is possible, let DKE be a circle; and from H, the center of the base, let a perpendicular HG be drawn to FG (the common section of the base with the cutting plane), through which and the axis let the triangle ABC pass. Then let any point K be taken on the line DKE, through which let KML be drawn parallel to FG, meeting the straight line DEG (the common section of the cutting plane and the triangle ABC) at M. Whence KM = ML.

Furthermore, through M let NX be drawn parallel to BG. And because the plane through NX and KL is parallel to the plane BC, it therefore effects a circle. In which KL as a diameter is perpendicular to NX, it will be NM x MX = (KM sq. =) DM x ME. (because DKE is a circle). Therefore NM . BM :: ME . MX. Therefore the triangles NMD and EMX are similar: and angle MEX = (angle DNM =) angle ABC. Thus the section is subcontrary, contrary to the Hypothesis. Therefore the section DKE is not a circle. Q. E. D.

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Prop. X.

Fig. 15.

If two points (G, H) are taken in a conic section (FED); the straight line (GH), which joins such points, will fall within the section, and that which is constituted in direct alignment with it will fall outside.

a hyp.

For because the points GH are outside the side of the triangle (ABC) drawn through the axis, the straight line GH will not reach the vertex A; therefore it will fall within the cone, and therefore within the section; if it is produced, it will fall outside the cone, and therefore outside the section.

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Prop. XI.

Fig. 16.

If a cone is cut by a plane (ABC) through the axis; and is also cut by another plane