C O NIC SEC T I O N S. 
77 
definition only extends to a right cone, that is, to a cone 
vvhofe axis is perpendicular or at right angles to its bale ; 
and not to oblique ones, in which the axis is oblique to 
the bale, the general definition, or de- 
feription of which may be this: If a 
line VA continually pafs through the 
point V, turning upon that point as a 
joint, and the lower part of it be car¬ 
ried round the circumference ABC 
of a circle ; then the fpace inclofed 
between that circle and the path of 
the line, is a cone. The circle ABC 
is the bafe of the cone ; V is its ver¬ 
tex ; and the line VD, from the ver¬ 
tex to the center of the bafe, is the 
axis of the cone. Alfo the other part 
of the revolving line, produced above 
V, will deferibe another cone Vacb, called the oppofite 
cone, and having the fame common axis produced D V d, 
and vertex V. 
The area or furface of every right cone, exclufive of 
its bafe, is equal to a triangle vvhofe bale is the peri¬ 
phery, and its height the llant fide of the cone. Or, the 
curve fuperficies of a right cone, is to the area of its cir¬ 
cular bafe, as the llant fide is to the radius of the bafe. 
And therefore the fame curve furface of the cone is 
equal to the fedlor of a circle whofe radius is the flant 
fide, and its arch equal to the circumference of the bafe 
oi the cone. Every cone, whether right or oblique, is 
equal to one-third part of a cylinder of equal bafe and 
altitude ; and therefore the folid content is found by 
multiplying the bafe by the altitude, and taking one- 
third ot the produft ■ and hence alfo all cones of the 
fame or equal bafe and altitude, are.equal. 
Although the folidity of an oblique cone be obtained 
in the fame manner with that of a right one, it is other- 
wife with regard to the furface, fince this cannot be re¬ 
duced to the meafure of a fe£tor of a circle, becaufe all 
the lines drawn from the vertex to the bafe are not equal. 
Dr. Barrow has demonftrated, in his LcElioncs Geometrical 
that the folidity of a cone with an elliptic bafe, forming 
part of a right cone, is equal to the produft of its furface 
by a third part of one of the perpendiculars drawn from 
the point in which the axis of the right cone interfebts 
the ellipfe ; and that it is alfo equal to one-third of the 
height of the cone multiplied by the elliptic bafe ; con- 
fequently that the perpendicular is to the height of the 
cone, as the elliptic bafe is to the curve furface. 
According to the different pofitions of the cutting- 
plane, there arife five different figures or fedftions, viz. a 
triangle, a circle, an ellipfe, a parabola, and an hyper¬ 
bola : the lafl three of which only are peculiarly called 
conic fe&ions. 
1. If the cutting-plane pafs through the 
vertex of the cone, and any 
part of the bafe, the feftion 
will evidently be a triangle ; 
as VAB, in the annexed fi¬ 
gure. 
2. If the plane cut theV 
cone parallel to the bafe, or 
make no angle with it, the fedtion will be a circle, as 
ABD. 
3. The fedtion DAB is an ellipfe, 
when the cone is cut obli¬ 
quely through both fides, 
or when the plane is in¬ 
clined to the bafe in a lefs 
angle than the fide of the 
coneis,asobvioufly fhewn 
in the third figure. 
4. The fedtion is a parabola, when the 
cone is cut by a plane parallel to the fide, E 
or when the cutting-plane and the fide of the cone make 
equal angles with the bafe, in fig. 4. 
Vol. V. No, 235. 
5. The fedtion is an hyperbola, when the cutting-plane 
makes a greater angle with the bafe than 
the fide of the cone makes. And if all the 
fides of the cone be continued through 
the vertex forming an oppofite equal cone, 
and alfo continued to cut the oppofi'e 
cone, this latter fedtion will be the oppo¬ 
fite hyperbola to the former; as dBc, in 
this figure. The vertices of any fedtion, 
are the points where the cutting-plane 
meets the oppofite fides of the cone, or 
the fides of the verticle triangular fedtion ; 
as A and B. Hence, the ellipfe and the 
oppofite hyperbolas have each two verti¬ 
ces ; but the parabola only one, unlefs 
we confider the other as at an infinite dif-1 
tance. The axis, or tranfverfe diameter 
of a conic fedtion, is the line or diflance 
A B between the vertices. Hence the axis of a parabola 
is infinite in length, Ab being only a fmall part of it. 
Ellipfe. 
Oppofite Hyperbola. Parabola. 
The center C, in thefe figures of the ellipfe and hy¬ 
perbola, is the middle of the axis. Hence the centre of 
a parabola is infinitely diftant from the vertex. And of 
an ellipfe, the axis and center lie within the curve ; but 
of an hyperbola without. The diameter is any right 
line, as AB or DE, drawn through the center, and ter¬ 
minated on each fide by the curve ; and the extremities 
of the diameter, or its interfedtions with the curve, are 
its vertices. Hence all the diameters of a parabola are 
parallel to the axis, and infinite in length. And hence 
alfo every diameter of the ellipfe and hyperbola have two 
vertices; but of the parabola only one, unlefs we con¬ 
fider the other as at an infinite diftance. 
The conjugate to any diameter, is the line drawn 
through the center, and parallel to the tangent of the 
curve at the vertex of the diameter. So F G, parallel 
to the tangent at D, is the conjugate to D E ; and HI, 
parallel to the tangent at A, is the conjugate to A B. 
Hence, the conjugate HI, of the axis A B, is perpendi¬ 
cular to it. An ordinate to any diameter, is a line pa¬ 
rallel to its conjugate, or to the tangent at its vertex, and 
terminated by the diameter and curve. So D K and EL 
are ordinates to the axis A B ; and M N and N O ordi¬ 
nates to the diameter D E. Hence, the ordinates of the 
axis are perpendicular to it. 
An ablcifs is a part of any diameter, contained between 
its vertex and an ordinate to it; as A K or B K, and DN 
or EN. Hence, in the ellipfe and hyperbola, every or¬ 
dinate has two abfeifies; but in the parabola, Only one ; 
the other vertex of the diameter being infinitely diftant. 
The latus retfum,’ or parameter, of any diameter, is a third 
proportional to that diameter and its conjugate. 
The focus is the point in the axis where the ordinate 
is equal to half the parameter ; as K and I,, where D K or 
E L is equal to the femiparameter. Hence the ellipfe and 
hyperbola have each two foci; but the parabola only one. 
If DAE, and FBG, in the annexed figure, be con- 
