CONIC SECTIONS. 
Cor. A sti-aight line drawn from the ver- 
tex to any point in a conic surface, being 
produced indefinitely, is wholly in the op- 
posite surfaces. 
For a line, so drawn, will coincide witli 
the line that generates the conic surfaces, 
when this line, by being carried round the 
circumference of the base, comes to the pro- 
posed point. 
II. The solid figure, contained by the 
conic surface and the circle A D B, is called 
a cone. The point V is named tlie vertex 
of the cone; the line V C, drawn to the 
centre of the circle, the axis of the cone ; 
and the circle A D B, the base of the 
cone. 
III. A right cone is when the axis is per- 
pendicular to the plane of the base ; other- 
wise 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 tan- 
gent. 
PROP. I. 
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. l.),'and also 
in the plane surface; and there are no 
points, excepting in these lines indefinitely 
produced, which are common to both the 
surfaces. Therefore the figure D V E, 
which is the common intersection of the 
cone and a plane through the vertex, is a 
vectilineal triangle. 
PROP. II. 
Fig. 2. If a point, E, be assumed in a 
conic surface, and a line, PQ, be drawn 
through it so as to be parallel to a right 
line, V B, passing through the vertex, and 
contained in the conic sui'faces ; then tlie 
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 as- 
sumed point E is, on the one side, and it 
will be wholly witliout both surfaces on the 
other side. 
For if a plane be conceived to be drawn 
through tlie line V B and the point E, the 
line P Q, parallel to V B, will be wholly in 
that plane, 7. 11. E; and the common sec- 
tions of the plane and tlie conic surfaces 
will be the line V B and the line V E C 
drawn through the vertex and the point E, 
Pr. 1. Now the line, QP, does not meet 
either of the lines V B or V C in another 
point different trom E. Also Q E, 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. III. 
Fig. 3. If a plane be drawn through the 
vertex of a cone and a tangent of the conic 
surface G H, it will meet the conic surface 
only in the line VD, drawn through the 
vertex of the cone and the point of contact 
of the tangent. 
For, because the point D and the vertex 
V are common both to the plane surface 
and to the conic surface, therefore the line 
V I), indefinitely produced, is likewise com- 
mon to both surfaces. And because G H 
meets the conic surface pnly in the point 
D, and is every where else without the sur- 
face, therefore any line (different from VD) 
as V F, drawn in one of the conic surfaces, 
is contained on one side of the plane ; and 
the same line continued in the opposite 
conic surfece, as V K, is contained on the 
other side of the plane. 
Cor. 1. Any straight line drawn in the 
plane, V G H, so as to meet the line V D, 
is a tangent of the conic surfaces. 
Cor. 2. No other plane, besides the plane 
V G H, can be drawn so as to touch the 
conic surfaces in the line V D without cut- 
ting them. 
For, RS the common section of the plane 
V G H, and the plane of the base is a tan- 
gent to the periphery of the base. Cor. li 
And, if there were two such planes, there 
would likewise be two tangents of a circle 
drawn through the same point of the peri- 
phery, which is absurd. 
PROP. 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 a right line 
contained in the conic surface, will meet 
eithei- the same, or the opposite, conic sur- 
fece again in another point. 
Let a plane be drawn through the vertex 
of the cone and the right line(DB'orDC), 
tlien that plane will cut the cone ; for if it 
did not tlie right Ihie (D B or D C) would 
