CONIC SECTIONS. 
face, and will completely inclose a space. 
In this position of the cutting plane, the line 
of common section, unless when it is a 
circle, is called an ellipse. 
Def. 6. Fig. 13. If the line of common 
section M N, touch the periphery of the 
base of the cone, then the plane drawn 
through the vertex will touch the conic 
surlaces (Pr. 3), and the opposite surflices 
will be on opposite sides of it. In this 
case tlie cutting plane will meet every line 
drawn from the vertex in one of the conic 
surfaces, excepting only the line V B, in 
which the touching plane meets the conic 
surface; and as the cutting plane is inde- 
tinitely extended along the touching plane 
without meeting it, it is obvious that the 
rairve line, formed by the common section 
of the cutting plane and the conic surface, 
does not return into itself so as to inclose 
a space, but it is open on the side opposite 
to the vertex of the cone. In this position 
of the cutting plane, the conic section is 
called a parabola. 
Cor. 1. Every right line, drawn in the 
plane of a parabola, which meets the curve 
ifi one point, but neither touches the curve, 
(see Def. 8), nor is parallel to the line V B 
in the conic surface, will meet the para- 
bola again in another point. This is mani- 
fest from Prop. IV. 
Cor. 2. All right lines drawn in the plane 
of a parabola, which meet the curve in one 
point only, but are not tangents, are pa- 
rallel to one another. For they are all pa- 
rallel to the line V B in the conic surface. 
(Cor. 1 ) 
Def. 7. Fig. 14. If the line of common 
section M N cut the periphei-y of the base, 
then the plane drawn through the vertex 
will divide each of the opposite conic sur- 
faces into two parts lying on opposite sides 
of it. In this case the cutting plane being 
indefinitely extended, will meet every line 
drawn from tlie vertex in those parts of the 
two conic surfaces that lie on the same side 
of the plane through the vertex, as the 
cutting plane itself ; and thus twm curves 
will be formed by the common intersection 
of the cutting plane, and the two opposite 
conic surfaces. It is obvious that these 
curve lihes may be indefinitely extended, 
and that they do not return into themselves 
so as to inclose a space. In this position 
of the cutting plane, the conic section 
formed by its intersection with one of tiie 
conic surfaces, is called a hyperbola ; and 
the two conic sections formed by its inter- 
section with the two opposite conic sur- 
faces, are called opposite hyperbolas, or 
opposite sections. 
Cor. 1. LetmVm be the common sec- 
tion of the cone, and a plane drawn through 
tlie vertex parallel to the plane of the two 
hyperbolas : then every right line drawn 
through a point of one of the hyperbolas, 
so as to be parallel to either of the two 
lines V m or V n, will not meet either of 
the two curves again in another point. 
(Pr. 2.) 
Cor. 2. Every right line drawn in the 
plane of the hyperbolas, which meets one 
of the curves, but is not a tangent, nor pa- 
rallel to V m nor V n, will meet the same, 
or the opposite hyperbola again in another 
point. (Pr. 4.) If it be parallel to V O, a 
line contained in the angle mV n, it will 
meet the opposite hyperbola : but if it be 
parallel to R V S, witliout the angle mV n, 
it will meet the same hyperbola again. 
Def. 8. A right line drawn in the plane 
of a conic section, so as to meet the curve 
of the section in one point only, and which, 
being produced both w'ays, is contained on 
one and the same side of the section, is 
called a tangent of the section. 
Cor. 1. A tangent of a conic section is a 
tangent of the conic surface. For it can 
meet the conic surface only in the point in 
which it meets the section. 
Cor. 2. There cannot be more than one 
tangent of a conic section at the same point 
of the curve. For if there tvere two tanr 
gents, then two planes drawn through them 
and the vertex of the cone would meet the 
conic surface in the same right line without 
cutting the conic surface, which is absurd. 
(Cor. 2. Pr. 3.) 
PROP. X. 
If a point be assumed without, or within 
a conic section, and two straight lines be 
drawn through it to cut the section, or op- 
posite sections, and so as to be parallel 
to two lines given by position: then the 
rectangle under the segments of the secant, 
or the square of the tangent, parallel to one 
of the lines given by position, will have to 
the rectangle under the segments of the 
secant, or to the square of the tangent, pa- 
rallel to the other line given by position, a 
ratio that is always the same, wherever 
the point (through which the line is drawn) 
is assumed without or within the section. 
For secants and tangents of a conic section 
are secants and tangents of a conic sur- 
face : and thus this proposition is included 
in Proposition VIE 
A 2 
