CONIC SECTIONS. 
42l 
cone be perpendicular to its base, it is caller/ 
a right cone, as in tig. 2. ; if the axis be in- 
clined to the base, it is called a scalene or 
oblique cone, such as that in fig. 3. ; and a 
right cone is always understood, when the 
contrary is not expressed. 
If this leg or axis be greater than half the 
base, the solid produced is an acute-angled 
cone; if less, it is an obtuse-angled cone; 
and if equal, a right-angled cone. Thus the 
cone BAC (fig. 4.) is less acute than the 
cone BDC, because the angle BDF is less 
than the angle BAF. 
Properties of the cone. 1. Cones and py- 
ramids having the same bases and altitudes, 
are equal to each other. It is shewn that 
every triangular prism may be divided into 
three equal pyramids ; and therefore that a 
triangular pyramid is one-third of a prism 
standing on the same base, and having the 
same altitude. Hence, since every multan- 
gular body may be resolved into triangular 
ones, every pyramid is the third part of a 
prism, standing upon the same base, and 
having the same altitude; and as a cone may 
be esteemed an infinite-angular pyramid ; and 
a cylinder an infinite-angular prism, a cone 
is a third part of a cylinder which has the 
same base and altitude. Hence we have a 
method of measuring the solidity and surface 
of a cone and pyramid. Thus, find the so- 
lidity of a prism or cylinder, having the same 
base with the cone or pyramid ; which found, 
divide by 3, the quotient will be the solidity 
of the cone or pyramid: or the solidity of 
any cone is equal to the area of the base 
multiplied into one third part of its altitude. 
As for the surfaces, that of a right cone, not 
taking in the base, is equal to a triangle 
whose base is the periphery, and altitude the 
side of the cone ; therefore the surface of a 
right cone is had by multiplying the peri- 
phery of the base into half of the side, and 
adding the product to that of the base. 
2. The altitudes of similar cones are as the 
radii of the bases ; and the axes likewise are 
as the radii of the bases, and form the same 
angle with them. 
3. Cones are to one another in a ratio 
compounded of their bases and altitudes. 
4. Similar cones are in a triplicate ratio of 
their homologous sides, and likewise of their 
altitudes. 
5. Of all cones standing upon the same 
base, and having the same altitude, the su- 
perficies of that which is most oblique is the 
greatest, and so the superficies of the right 
cone is the least; but the proportion of the 
superficies of an oblique cone to that of a 
right one, or which is the same thing, the 
comparison thereof to a circle, or the conic 
sections, has not yet been determined. 
If a cone be cut by a plane through the 
vertex, the section will be a triangle ABC, 
Plate fig. 6. 
If a cone be cut bv a plane parallel to its 
base, the section will be a circle. If it be 
cut by a plane DEF, fig. 6. in such a. direc- 
tion that the side AC of a triangle passing 
through the vertex, and having its base BC 
perpendicular to EF, may be parallel to DP, 
the section is a parabola; if.it be cut by a 
plane DR, fig. 7, meeting AC, the section is 
an ellipse; and if it be cut by a plane DMO, 
fig. 8, which would meet AC extended be- 
yond A, it is an hyperbola. 
If any line PIG, fig. 6, be drawn in a para- 
bola perpendicular to DP, the square of HG curve meets the axis, is called the ■vertex. 
will be to the square of EP as DG to DP; 
for let LIIK be a section parallel to the base, 
and therefore a circle, the rectangle LGK 
will be equal to the square of IiG, and the 
rectangle BPC equal to the square of EP ; 
therefore these squares will be to each other 
as their rectangles, that is, as BP to LG, that 
is, DP to DG. 
There are three modes of investigating the 
properties of these curves. 1 . By taking the 
demonstrations from the sections of a cone, 
which, from the many intersections of planes 
with planes, and planes with solids, is apt to 
perplex the learner, and is now seldom 
adopted. 2. By taking some general pro- 
perty of the figures on a plane, from which 
all the rest may be determined geometrically ; 
or by taking the general property of each 
curve singly, and from thence deducing geo- 
metrically all the other properties of that 
curve. 3. By taking the equation to all the 
curves, from whence their respective proper- 
ties may be discovered algebraically ; or by 
taking the equation to each single curve, 
from whence its properties may be discovered 
algebraically. 
By whichever of these three methods the 
properties of conic sections are investigated, 
(7) A right line LST, drawn through the fo- 
cus parallel to the directrix, and terminated 
by the curve in the points E, 1 , is called 
the principal parameter, or the latus rec- 
tum. 
Cor. 1. SP being greater than PE in the 
hyperbola, two curves will be described, one 
on each side of the directrix; winch are 
called opposite hyperbolas. 
Cor. 2. When the revolving line SP comes 
into the position SAD ; SP, PE, will be equal 
to SA, AD; therefore, SA is to AD in the 
determining ratio. 
Cor. 3. When the line SP comes into the 
position SLor ST, the distance ot P from the 
directrix will be equal to SD, and ST- or S 1 
will be to SD in the determining ratio ; and 
therefore the latus rectum LT is bisected in 
S. 
Cor. 4. The latus rectum in the parabola 
is equal to tw ice the distance of the focus 
from the directrix, or to four times its dis- 
tance from the vertex: for SI- is equal to 
SD, and SA is equal to AD ; therefor L L 
is equal to twice SD, or to tour times SA. 
(8) The tangents DLQ, DT q (figs. 11, 
12.), which are drawn through the extremi- 
ties of tire latus rectum, are called focal tan- 
the general property is discovered, which ! gents. (9) J he right line AM, in the ellipse 
makes the bases of the other methods. If j and hyperbola, is called the transverse axis, 
they are considered as curves on a plane 
surface, they are shewn to have the same 
properties with those formed by the inter- 
sections of a plane with a cone: and again, 
the equation is produced, which makes the 
basis of the algebraical process. For a 
beginner, the geometrical method seems 
clearly to have the preference; and that 
which deduces from a common property the 
relations of each curve to the other, as well 
as the respective properties of each curve, 
seems to be better than that which considers 
each curve separately, and, after an exami- 
nation of its properties, enters into a compa- 
rison of each curve with the others. Bosco 
or the axis major. (10) If the transverse 
axis be bisected in C, the point C is called 
the centre of the ellipse or hyperbola. (1 1) 
If a line BC&, which is bisected in C, be 
drawn perpendicular to the transverse axis, 
and CB, C b, be each of them a mean pro- 
portional between SA, SM, the segments of 
the axis intercepted betw een the focus and 
the vertices BC b, is called the conjugate axis, 
or the axis minor. (12) A right line PNp, 
drawn through any point N in the axis 'pa- 
rallel (o the tangent KAG, or perpendicular 
to the axis, and terminated by the curve in 
the points P and p, is called an ordinate to 
the axis. (13) And the segment of this axis 
...h in a very elegant manner has deduced AN, intercepted between the ordinate and 
the properties of the three curves, from a pro- ! the vertex, in all the sections, as also the 
perty common to all; and his method has ! other segment NM in the ellipse and hyper- 
been made still easier for beginners lately by 
Mr. Newton, of Jesus-college, Cambridge; 
who, laying aside the musical proportion on 
which Boscovich founds his demonstrations, 
has, within a short compass, introduced every 
thing requisite for the study of the Principia v , „ . . . 
;md the hie her mathematics. The basis of j parabola parallel to the axis, is. called a dia- 
bola, is called an abscissa. (14) Any line 
passing through the centre of an ellipse or 
hyperbola, which is terminated both ways by 
the curve in the former, and by the opposite 
curves in the latter, is called a diameter. 
(15) Aline drawn through any point in the 
this method is the relation of two- lines to 
each other, drawn the one from a given 
point, the other perpendicular to a line given 
in position. 
(1) If any point S be assumed without the 
line DX (fig. 9, 10.), and, whilst the line SP 
revolves about S as a centre, a point P moves 
in it in such a manner that its distance from 
the point S shall always be to PE (its distance 
from the line DX) in a given ratio, the curve 
described by the point P is called a conic 
section, a parabola, an ellipse, or an hyper- 
bola, according as SP is equal to, less, or 
greater than, PE. (2) The indefinite right 
line DX is called the directrix. (3) The 
point S is called the focus. (4) The ratio of 
SP to PE is called the determining ratio. 
(5) If a line SD be drawn through the focus 
perpendicular to the directrix, which is pro- 
duced indefinitely', it is called the axis of the 
conic section. (6) The point A, where the 
meter to the parabola. (16) Any point 
where a diameter meets the curve is called a 
vertex to that diameter. (17) If from the 
centre C, at the distance C A (fig. 13,), half 
the transverse axis, a circle be described, 
cutting the directrix of the hyperbola m the 
points" H, h, and lines be draw n from the 
centre through the points of intersection, 
these lines are called the asymptotes. (18) 
If AM be the transverse axis, and P>b the 
conjugate axis, of any two opposite hyper- 
bolas ; and two other hyperbolas be described, 
of which the transverse axis is B b, and tire 
conjugate axis AM ; these hyperbolas are 
said to be conjugate to the former. (19) 
When the two axes are equal, the hyper- 
bolas are said to be equilateral. (20) If a 
right line be drawn through any point in the 
diameter of a conic section parallel to the 
tangent at its vertex, which is terminated 
botk ways by the curve, it is called an oixli- 
