CONIC SECTIONS. 



the properties of the curves from a de- 

 scription of them on a plane. Since this 

 time authors have been much divided as 

 to the best way of defining the curves, 

 and demonstrating their elementary pro- 

 perties ; many, in imitation of the ancient 

 geometricians, making the cone the 

 groundwork of their theories ; while 

 others have followed the example of Dr. 

 Wallis. 



OF THE COUE AND ITS SECTIONS. 



Definitions. 



Let ADB be a circle (Fig. 1, Plate I. 

 Conic Sections) and V a fixed point with- 

 out the plane of the circle ; then if a 

 right line passing continually through 

 the point V, be carried round the whole 

 periphery of the circle ADB, that right 

 line being extended indefinitely on the 

 same side of V as the circle, will describe 

 a conic surface ; and if it be likewise ex- 

 tended indefinitely on the other side of 

 V, it will describe two opposite conic sur- 

 faces. 



Cor. A straight line drawn from the 

 vertex to any point in a conic surface, 

 being- produced indefinitely, is wholly in 

 the opposite surfaces. 



For a line so drawn will coincide with 

 the line that generates the conic surfaces, 

 when this line, by being carried round 

 the circumference of the base, comes to 

 the proposed point. 



II. The solid figure, contained by the 

 conic surface and the circle ADB, is call- 

 ed a cone. The point V is named the 

 vertex of the cone ; the line, CV, draun 

 to the centre of the circle, i he axis of the 

 cone ; and the circle ADB, the base of 

 the cone. 



III. A right cone is when the axis 

 is perpendicular to the plane of the 

 base ; otherwise it is a scalene, or oblique 

 cone. 



IV. A right line that meets a conic sur- 

 face in one point only, and is every 

 where else without that surface, is called 

 a tangent. 



Fig. 1. The common intersection of a 

 conic surface and a plane, V D E, that 

 passes through the vertex, and cuts 

 the base of the cone, is a rectilineal 

 triangle. 



For the common section of the plane 

 of the base, and the plane drawn through 

 the vertex (which is a right line 3. 11. E) 

 will cut the periphery of the base in two 



points, D and E, and in these two points 

 only : then, having drawn DV and EV 

 to the vertex of the cone, these lines 

 will be both in the conic surface (Cor. 

 Def. 1.) and also in the plane surface ; 

 and there are no points, excepting in 

 these lines indefinitely produced, which 

 are common to both the surfaces. There- 

 fore the figure D V E, which is the 

 common intersection of the cone and a 

 plane through the vertex, is a rectilineal 

 triangle. 



Fig. 2 If a point E, be assumed in a 

 conic surface, and a line, PQ, be drawn 

 through !t so as to be parallel to a right 

 line, VB, passing through the vertex, 

 and contained in the conic surfaces : then 

 ihe right line P Q, will not meet either of 

 the opposite surfaces in another point, 

 but it will fall within the surface in which 

 the assumed point E is, on the one side, 

 and it will be wholly without both sur- 

 faces on the other side. 



For if a plane be conceived to be 

 drawn through the line VB and the 

 point E, the line P Q, parallel to V B, 

 will be wholly in that plane, 7. 11 E ; 

 and the common sections of the plane 

 and the conic surfaces will be the line 

 V B and the line V E C, drawn through 

 the vertex and the point E, Pr. I. Now 

 the line, QP, does not meet either of 

 the lines V B or V C in another point 

 different from E. Also QE, the part 

 of the line that is contained in the angle 

 B V C, is within the cone ; and P E, the 

 part of it that is contained in the angle 

 CVN, is without both the opposite sur- 

 faces. 



PROP. nr. 



Fig 3. If a plane be drawn through 

 the vertex of a cone and a tangent of the 

 conic surface GH, it will meet the conic 

 surface only in the line V D, drawn 

 through the vertex of the cone and the 

 point of contact of the tangent. 



For, because the point D and the ver- 

 tex V are common both to the plane 

 surface and to the conic surface, there- 

 fore the line VD, indefinitely produced, 

 is likewise common to both surfaces. 

 And because G H meets the conic sur- 

 face only in the point D, and is every 

 where else without the surface, there- 

 fore any line (different from V D) as 

 VF, drawn in one of the conic surfaces, 

 is contained on one side of the plane ; 

 and the same line continued in the oppo- 



