CONIC SECTIONS. 



site conic surface, as V K, is contained on 

 the other side of the plane. 



Cor. 1. Any straight line drawn in the 

 plane \ GH, so as to meet the line VD, 

 is a tangent of the conic surfaces. 



Cor. 2. No other plane, besides the 

 plane VGH, can be drawn so as to touch 

 the conic surfaces in the line VD, with- 

 out cutting them. 



For RS, the common section of the 

 plane VGH, and the plane of the base, 

 is a tangent to the periphery of the 

 base, Cor. 1. And if there were two 

 such planes, there would likewise be 

 two tangents of a circle drawn through 

 the same point of the periphery, which 

 is absurd. 



PHOP. iv. 



Fig. 4. A right line drawn through 

 a point of a conic surface, so as neither 

 to be a tangent, nor to be parallel to 

 aright line contained in the conic sur- 

 face, will meet either the same, or the 

 opposite, conic surface again in another 

 pomt. 



Let a plane be drawn through the ver- 

 tex of the cone and the right line (DB 

 or DC) then that plane will cut the 

 cone ; for if it did not, the right line (DB 

 or DC,) would be a tangent contrary to 

 the hypothesis. Let VGand VH be the 

 common sections of the plane and the 

 conic surface ; then the right line (DB or 

 DC) will not be parallel to VH con- 

 tained in the conic surface (hyp), 

 therefore it will meet VH either in the 

 same conic surface (as DB), or when 

 produced in the opposite conic surface 

 (as DC.) 



Fig. 5. If either of two opposite conic 

 surfaces be cut by a plane parallel to the 

 base of the cone, the section is a cir- 

 cle, having its centre in the axis of the 

 cone. 



Through V C,the axis of the cone, let 

 two planes be drawn, cutting the base in 

 the lines CD and CE, and the plane 

 parallel to the base in the lines GH and 

 GL, and the conic surfaces in the lines 

 V H D and V L E : then because the 

 base is parallel to the cutting plane, 

 therefore C D is parallel to G H, and 

 C E to . G L, 16. 11. E. Therefore, 

 on account of equiangular triangles, 4. 

 6. E. 



DC : CV : : HG : GV 



CV : CE : : GV : GL 



Ex aquo DC : CE : : HG : GL 



But DC = CE, therefore HG = GL 

 And in like manner it may be shown that, 

 any right line drawn from G to a point in 

 the intersection of the plane, and the co- 

 nic surface, is equal toGH; therefore 

 the section is a circle. 



Cor. If through a point situated with- 

 in or without a conic surface, two straight 

 lines, both parallel to the plane of the 

 base of the cone, (that is parallel to 

 straight lines in that plane,) be drawn to 

 cut or touch the conic surface : then the 

 rectangle contained by the two segments 

 (between the point and the conic sur- 

 face,) of one of the lines when it cuts, or 

 the square of its segments when it touches 

 the conic surface, is equal to the rectan- 

 gle contained by the two segments of the 

 other line when it cuts, or to the square 

 of its segment when it touches the conic 

 surface. 



For a plane drawn through the two 

 lines will be parallel to the plane of the 

 base, 15. 16. E ; and it will intersect the 

 conic surface of the periphery of a cir- 

 cle ; whence the corollory is manifest, 35 

 and 36. 3. E. 



When a straight line drawn through a 

 point, situated within or without a cone, 

 meets one or both of the conic surfaces 

 in two points, it is called a secant ; and 

 the two parts of such a line, between the 

 point through which it is drawn, and the 

 conic surface or surfaces, are called the 

 segments of the secant. And when a line, 

 drawn from a point without a cone, 

 touches one of the conic surfaces, that 

 part of it between the point from which 

 it is drawn and the conic surface is denot- 

 ed by the word tangent, ia the following- 

 propositions. 



Fig. 6,7, and 8. If a straight line be 

 drawn from the vertex of a cone, to a 

 point, as B, in the plane of the base, but 

 not in the periphery of the base ; and, 

 through any point, as P, situated without 

 or within the cone, another straight line, 

 parallel to the former, be drawn to cut 

 or touch the conic surface or opposite 

 surfaces ; then the square of the line 

 drawn from the vertex of the cone to the 

 point B is to the rectangle under the 

 segments of the secant, or to the square 

 of the tangent, drawn from the point P, 

 as the rectangle under the segments of 

 any line drawn from B, to cut the base of 

 the cone, is to the rectangle under the 

 segments of any line parallel to the base 



