CONIC SECTIONS. 
be a tangent contrary to the hypothesis. 
Let V G and VH be the common sections 
of the plane and the conic surface ; then 
the right line (D B or D C) will not be pa- 
rallel to V H contained in the conic surface 
(hyp), therefore it will meet VH either in 
the same conic surface (as D B), or, when 
produced in the opposite conic surface 
(as D C). 
PROP. V. 
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 circle, 
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 C D and C E, and the plane parallel to 
the base in the lines G H and G L, and the 
conic surfaces in the lines V H D and VL E ; 
then, because the base is parallel to the cut- 
ting plane, therefore CD is parallel to GH, 
and CE to GL, 16. 11. E. Therefore, on 
account of equiangular triangles. 4. 6. E. 
DC:CV::HG:GV 
CV:CE::GV:GL 
Ex eequo D C : CE :: HG : GL 
But D C = C E, therefore H G = G L, 
And in like manner it may be shewn that 
any right line drawn from G to a point in 
the intersection of the plane, and the conic 
surface is equal to G H ; tlierefore tire sec- 
tion is a circle. 
Cor. If, through a point situated within 
or without a conic surface, two straight 
lines, both parallel to the plane of tire 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 surface), of one of the lines 
when it cuts, or the square of its segment 
when it touclres the conic surface, is equal 
to the rectangle contained by the two seg- 
ments of the other line when it cuts, or to 
the square of its segment when it touclres 
the conic surface. 
For a plane drawn through the two lines 
will be parallel to the plane of the base, to. 
16. E; and it will intersect the conic sur- 
face in the periphery of a circle : whence 
the corollary is manifest, 35 aird 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, betw'een the point 
through which it is drawn, and the conic 
surface or surfaces, are called the segments 
of the secant. And when a line, drawil 
from a point without a cone, touches one 
of the conic surfaces j that part of it be- 
tween the point from which it is drawn and 
the conic surface is denoted by tire word 
tangent in the following propositions. 
PROP, VI. 
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 tlie rectangle un- 
der 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 seg- 
ments of any line, parallel to the base of the 
cone, drawn through the point P, to cut the 
conic surface. 
Fig. 6. Let the point B be without the 
base of the cone, and let QR, drawn 
through P without or within the conic sur- 
face, be parallel to V B, and let it cut the 
conic surface in Q and R : through P and 
the line V B draw a plane cutting the conic 
surface in the lines V G and V H, and the 
plane of the base in the line B G H ; and 
through P draw L K parallel to G H. Be- 
cause V B and P R Q are parallel, therefore 
the line PRQ is contained in the plane 
B VP, 7. 11. E; and the points Q and R 
are in the lines VH and V G, the common 
sections of the plane and the conic surface. 
Because Q P is parallel to V B, and L K to 
G H, therefore the triangle Q P L is equi- 
angular to the triangle V B H, and the tri- 
angle PKR to the triangle VGB; there- 
fore 4. 6. E. 
V B : P R B G : P K 
VB : PQ :: BH : PL 
Consequently, V B^ : P R X P Q :: B G 
X B H : P K X P L, 23. 6. E. But the 
rectangle B G x B H is equal to the rect- 
angle under the segments of any other line 
drawn from B to cut the base of the cone, 
35 and 36 . ,3. E ; and the rectangle P K x 
K L is equal to the rectangle under the 
segments of any other line, parallel to the 
plane of the base, drawn from P to cut the 
conic surface, Cor. Pr. 5 ; and hence the 
proportion is manifest in this case. 
Fig. 7. And if the point B be within the 
