CONIC SECTIONS. 
base of tiie roue, and a straight line (as 
PQR), parallel to the line V B that joins 
the point B and the vertex of the cone, be 
drawn to cut the opposite surfaces through 
a point P, situated without or witliin the 
cone : the proposition may be demonstrated 
in this case, in the very same words as in 
the former case. 
And if the point P (fig. 8.) be without the 
cone as well as the line V B, and P S, pa- 
rallel to V B, be drawn to touch the conic 
sinface, instead of cutting it ; then tlie plane 
P V B will meet the conic surface in a line 
V S M j and B M will touch the base of 
the cone, and P N, parallel to B M, will 
touch the conic surface. And because the 
two triangles S P N and V B M are equian- 
gular, therefore 
VB : PS BM : PN 
' AndVB^: PS":: B M" : PN" 
But B M" is equal to the rectangle under 
the segments of any line drawn from B to 
cut the base of the cone ; and P N" is equal 
to the rectangle under the segments of any 
line, parallel to the base of the cone, drawn 
from P to cut the conic surface j and hence 
the proposition is manifest in this case 
also. 
PROP. VII. 
jRg; 9. If a point be assumed without or 
within a cone, and two lines be drawm 
through it to meet a conic surface, or op- 
posite surfaces, and so as to be parallel to 
two straight lines given by position ; then 
tlie rectangle under the segments of the se- 
cant, or the square of the tangent, parallel 
to one of the lines given by position, has to 
the rectangle under the segments of the se- 
cant, or to the square of the tangent, paral- 
lel to the other line given by position, a ra- 
tio that is constantly the same, wherever 
the point (from whicli the lines are drawn) 
is assumed without or within the cone. 
Let V B and V C be two straight line.s, 
(fig. 9.) drawn from the vertex of a cone to 
the plane of the base, and given by position 
(or parallel to lines given by position) ; and 
let PQ and MN be two straight lines 
drawn tlirough any assumed point, as R, to 
cut the conic surface, and so as to be re- 
spectively parallel to C V and V B : and as 
C V" is to the rectangle C K X C L (con- 
tained by the segments of any line drawn 
from C to cut the base of the cone), so let 
D, any assumed line, or magnitude, be to E; 
and as VB" is to B G x B H (the rectan- 
gle contained by the segments of any line 
drawn from B to cut the base of the code), 
VOL. II. 
so let F be to E j and draw ST parallel to 
the base of the cone through the point R ; 
then, Pr. 6. 
(CV" : CK X C L, or) D : E :: PR X 
RQ:SR X RT, and BV":BG X BH, 
or) F : E :: M R X R N : S R X R T. 
Therefore invertendo and ex aequo, 
D:F::PR X RQ:MR X RN. 
And, as the same reasoning applies where- 
ever the point R is assumed, therefore the 
ratio of the rectangles P R X R Q) and 
M R X R N is the same with, or equal to, 
the constant ratio of D to F, wherever the 
point R is assumed. 
And, in like manner, may the proposition 
be demonstrated in all other cases, or in all 
positions of the lines P Q, and M N, whe- 
ther they cut, or touch, the same or oppo- 
site surfaces. 
PROP. VIII. 
Fig. 10. If a right line, as PT, drawn 
through a point P in the surface of a cone, 
so as to be parallel to a right line V B con- 
tained in the conic surface, meet two paral 
lei lines (in the points R and S) that cut or 
touch the conic smface or opposite surfaces : 
then PR is to PS as the rectangle under 
the segments of the secant, or the square of 
the tangent, drawn through the point R, is 
to the rectangle under the segments of tlie 
secant, or to the square of the tangent, 
drawn through the point S. 
Through the two parallels P T and V B 
(fig. 10.) draw a plane cutting the conic 
surface again in the line V A, and the plane 
of the base in the line B A ; and, through 
R and S, draw M N and H G parallel to 
A B. Because P T is parallel to V B, and 
R N to S G, therefore R N G S is a paral- 
lelogram ; and R N is = G S. It is obvious 
that the triangles PMR and PHS are 
equiangular: thereforte PR is to PS as 
M R is to H S, 4. 6. E, or as M R x R N 
istoHS X SG, I.6.E. ButMRxRN 
and H S X S G are respectively equal to 
the rectangles contained by the segments 
of any two lines, parallel to the base of the 
cone, drawn through R and S to cut the 
conic surface. Cor. Pr. 5, and hence the 
proposition is manifest, when PT meets 
two lines parallel to the plane of the base. 
And if P T meet two parallel lines D E 
and I K, not parallel to the plane of the 
base ; then, let the same construction be 
made as before : and because D E is pa- 
rallel to I K, and M N to GH ; therefore, 
DRxRE;MRxRN;;ISxSK:HS 
X SGj 
A a 
