CONIC SECTIONS. 
Alternando, DRX RE:ISxSK:: 
MRxRN:HSxSG. Therefore, as is 
obvious from what has already been shewn, 
PR;PS:;DR X RE;IS X SK. 
And if S be without the cone, and the 
line drawn through it touch the conic sur- 
face instead of cutting it, the reasoning is 
still the same when the square of the tan- 
gent is taken in place of the rectangle un- 
der the segments of the secant. 
PROP. IX. 
Fig. 1 1. Let a scalene cone be cut by 
a plane drawn through the axis perpendi- 
cular to the plane of the base, making the 
triangular section V A B ; and let V D, 
cutting A B produced in D, be drawn so 
as to make the angle B V D equal to tlie 
angle V A B, and draw M N in the plane 
of the base, perpendicular to AD; then 
every section of the cone, as P S Q, made 
by a plane parallel to the plane V M N 
(called a subcontrary section) is a circle ; 
and every circular section of the cone, 
which is not parallel to the base, is a sub- 
contrary section. 
Draw T S in the plane of the section 
parallel to M N,, which is plainly possible, 
because the two planes P Q and V M N 
are parallel: because T S is parallel to 
M N, a line in the plane of the base, there- 
fore every plane drawn through ST will 
cut tlie base in a line parallel to S T (16. 
11. E.) : therefore L O K, the common sec- 
tion of the base, and a plane drawn tlirough 
V and S T is parallel to ST and M N (9. 
11. E.): therefore KOL is perpendicular 
to A B, and it is bisected in O ; therefore 
S T is bisected in R. Again, the line PQ 
is parallel to V D, therefore V D^: P R X 
RQ::AD X DB:TR x RS(6,) But 
if a circle be described about the triangle 
A V B, D V will be a tangent of that circle 
( 32. 3; E.) ; therefore V D^ = A D x 
D B, .and consequently, R R X R Q = 
T R X R S, or R 'D (36. 3. E.). Be- 
cause the plane A V D is perpendicular to 
the base (/t y p), and M N is perpendi- 
cular to A D : therefore M N is perpendi- 
cular to the plane A V D : therefore T R, 
parallel to M N, is perpendicular to the 
same plane, and to P Q. And, hence, from 
what has already been shew'n, the section 
P Q is a circle. 
Next let P Q be a circular section, not 
parallel to the base of the cone : draw a 
plane through the vertex, parallel to the 
plane P Q, and let it cut the base in tlie 
line M N : draw A D tlirough the centre 
of the base perpendicular to M N, and lets 
plane drawn through V and A D cut the 
parallel planes in the lines P Q and V D, 
and the conic surface in the lines A V and 
V B : draw tlie plane V T L R S through 
S T parallel to M N, as before. It is shewn, 
as above, that T S is bisected in R : and, in 
like manner, it may be proved that any other 
line, as G H, parallel to M N is bisected- 
Because P Q, a line in a circle, bisects 
two 01 ’ more parallels, it is a diameter of tlie 
circle, and it cuts all the parallels at right 
angles. Because TS is perpendicular to 
P Q, therefore M N is perpendicular to 
DV (parallel to P Q): but MN is also 
perpendicular to DA: therefore it is 
perpendicular to the plane DAV (4. 
11. E.) : therefore A V.B is a section of the 
cone through the axis at right angles to tlie 
base (18. 11. E.). Again, because the 
section is a circle, therefore P R x R Q = 
S R X R T : consequently V D^ = A D x 
D B (Pr. 6.) ; Therefore V D is a tangent 
of the circle described about tlie triangle 
A V B, and the angle D V B is equal to the 
angle A V B (32. 3. E.). Therefore the cir- 
cular section is a subcontrary one. 
Cor. No other than a parallel and a 
subcontrary section of a cone is a circle. 
Fig. 12, IS, 14. If a cone 'be cut by a 
•plane P Q which neither passes through 
the vertex, nor is parallel to the base, then 
•a plane, as V M N being drawn through 
the vertex parallel to the cutting plane, it 
will necessarily meet the plane of the base 
of the cone. The line of common section 
of the parallel plane, and the base of the 
cone M N, may have one or other of these 
three different positions, viz. 
1. It may be without the base of the 
cone. 
2. It may touch the periphery of the 
base. 
3. It may cut the periphery of the base. 
These three different cases offer three 
sections for our consideration, that are very 
different from one another, and possess 
m-any properties peculiar to each, while 
they have many common to all the 
three. 
Def. 5. Fig. 12. If the line of common 
section M N be without the base of the 
cone, then the plane V M N drawn through 
the vertex will be entirely between, tlie 
two conic surfaces, not meeting either of 
them. In this case the cutting plane P Q 
will meet every line drawn in one of the 
conic surfaces, and the curve line of com- 
mon section will surround that conic sur- 
