422 
G0N1C SECTIONS. 
irate to that diameter. (21) The segments 
of any diameter of a conic section, which is 
intercepted between an ordinate and the ver- 
tex, is called an abscissa. (22) A diameter 
which is parallel to the tangent at the vertex 
of any diameter of the ellipse or hyperbola, 
is c alled a conjugate diameter. (23) A line 
which is a third proportional to any diameter 
of the ellipse or hyperbola and its conjugates, 
is called a parameter to that diameter. (24) 
If a line be drawn through the focus of a pa- 
rabola, parallel to the ordinates of any dia- 
meter, which is terminated both ways by the 
curve, it is called a parameter to that dia- 
meter. 
From these fundamental properties all the 
others are derived, and the curves may be 
described mechanically. This description 
depends for the parabola, on the property 
that a line from any point in the curve to the 
focus is equal to the line drawn from the 
same point perpendicularly on the directrix ; 
lor the ellipse, on the property that the sum 
of the lines drawn from the foci to any point 
is equal to the major axis ; for the lu/perbola, 
that the difference of the lines drawn from 
the foci to any point in the curve is equal to 
the major axis. Hence to draw the para- 
bola. 
Description of conic sections on a plane. 
1. Parabola. Let AB (fig. 14.) be any 
right line, and C any point without it, and 
DKF a ruler, which let be placed in the 
same plane in which the right line and point 
are, in such a manner that one side of it, as 
DK, be applied to the right line AB, and the 
other side KF coincide with the point C ; 
and at F, the extremity of the side KF, let 
be fixed one end of the thread FNC, whose 
length is equal to KF, and the other extre- 
mity of it at the point C ; and let part of the 
thread, as FN, be brought close to the side 
K F by a small pin N ; then let the square 
DKF be removed from B towards A, so that 
all the while its side DK be applied close to 
the line BA ; and in the mean time the thread 
being extended, will always be applied to 
the side KF, being stopt from going from it 
by means of the small pin ; and by the mo- 
tion of the small pin N there will be described 
a certain curve, which is called a semi-para- 
bola. And if the square be brought to its 
first given position, and in the same manner 
be moved along the line AB, from B to- 
wards II, the other semi-parabola will be de- 
scribed. 
2. Ellipse. If any two points, as A and 
P> (fig. 15.), be taken in any plane, and in 
them are fixed the extremities of a thread, 
whose length is greater than the distance be- 
tween the points, and the thread extended 
by means of a small pin C ; and if the pin he 
moved round from any point until it return 
to the place from whence it began to move, 
the thread being extended during the whole 
time of the revolution, the figure which the 
small pin by this revolution describes is called 
an ellipse. 
3. Hyperbola. If to the point A (fig. 16.) 
in any plane, one end of the ruler AB be 
placed in such a manner, that about that 
point, as a centre, it may freely move ; and 
if to the other end B of the ruler AB be fixed 
the extremity of the thread BDC, whose 
length is smaller than the ruler AB, and the 
either efid of the thread being fixed in the 
point C, coinciding with the side of the ruler 
AB which is in fine same plane with the 
given point A ; let part of the thread, as BD, 
be brought close to the side of the ruler AB, 
by means of a small pin D ; then let the ruler 
be moved about the point A, from C to- 
wards T, the thread all the while being ex- 
tended, and the remaining part coinciding 
with the side of the ruler being stopt from 
going from it by means of the small pin ; and 
by the motion of the small pin D, a certain 
figure is described, which is called the semi- 
hyperbola. 
The ellipse returns into itself. The para- 
bola and hyperbola may be extended with- 
out limit. 
Every line perpendicular to the directrix 
of a parabola meets it in one point, and falls 
afterwards within it ; and every line drawn 
from the focus meets it in one point, and falls 
afterwards without it. And every line that 
passes through a parabola, not perpendicular 
to the directrix, will meet it again, but only 
once: 
Every line passing through the centre of 
an ellipse is bisected by it ; the transverse 
axis is the greatest of all these lines, the less- 
er axis the least, and those nearer the trans- 
verse axis greater than those more remote. 
In the hyperbola, every line passing thro’ 
tire centre is bisected by the opposite hyper- 
bola, and the transverse axis is the least of 
all these lines ; also the second axis is the 
least of all the second diameters. Every line 
drawn from the centre within the angle con- 
tained by the asymptotes, meets at once, 
and falls afterwards within it ; and every line 
drawn through the centre without the angle, 
never meets it; and a line which cuts one of 
the asymptotes, and cuts the other extended 
beyond the centre, will meet both the oppo- 
site hyperbolas in one point. 
If a line GM (fig. 14.) be drawn from a 
point in a parabola perpendicular to the 
axis, it will be an 'ordinate to the axis, and 
its square will be equal to the rectangle under 
the abscissa MI and latus rectum; for be- 
cause GMC is a right angle, GM 2 is equal 
to the difference of GC 2 and CM 2 ; but GC 
is equal to GE, which is equal to MB ; there- 
fore GM 2 is equal to BM 2 — CM 2 ; which, 
because Cl and IB are equal, is (8 I)uc. 2.) 
equal to four times the rectangle MI and IB, 
or equal to the rectangle under MI and the 
latus rectum. 
Hence it follows, that if different ordinates 
be drawn to the axis, their squares, being 
each equal to the rectangle under the abscissa 
and latus rectum, will be to each other in 
the proportion of the abscissas ; which is the 
same property as takes place in the parabola 
cut from the cone, and proves those curves 
to be the same. 
This property is extended also to the or- 
dinates of other diameters, whose squares are 
equal to the rectangle under the abscissas 
and parameters of their respective diameters. 
In the ellipse, the square of the ordinate 
is to the rectangle under the segments of the 
diameter, as the square of the diameter pa- 
rallel to the ordinate is to the square of the di- 
ameter to which it is drawn, or as the first 
diameter to its latus rectum; that is, LK 2 
(fig. 15.) is to EK x KF as EF 2 to Gil 2 . 
In the hyperbola, (fig. 16,) the square of the 
ordinate is to the rectangle contained under 
the segments of the diameters betwixt its ver- 
tices, as tiie square of the diameter parallel fr> 
the ordinate, to the square of the diameter to 
which it is drawn, or as the first diameter to 
its latus rectum ; that is, SX 2 is to EX x ,XK 
as MN 2 to KE 2 . 
Or if an ordinate be drawn to a second dia- 
meter, its square will be to the sum of the 
squares of the second diameter, and of the 
line intercepted betwixt the ordinate and 
centre, in the same proportion ; that is, ]fiv’ 
(fig. 16.) is to ZG 2 added to GM 2 as JkE 2 to 
MN 2 . ‘ . 
These are the most important properties' 
of the conic sections ; and by means of these, 
it is demonstrated, that the* figures arc the 
same described on a plane as cut with a 
cone ; which we have demonstrated in the 
case of the parabola. 
From the genesis of the sections it may be 
observed how one section degenerates into 
another. For an ellipsis being that plane of 
any section of the cone which is between the 
circle and parabola, it will bit easy to con- 
ceive that there may be a great variety of el- 
lipses produced from the same cone ; and 
when the section comes to be exactly paral- 
lel to one side of the cone, then the' ellipsis 
degenerates into a parabola. Now a pa- 
rabola being that section whose plane is al- 
ways exactly parallel to the side of the 
cone, cannot vary as the ellipsis may ; for as 
soon as ever it begins to move out of that 
position of being parallel to the side of the 
cone, it degenerates either into an ellipsis or 
hyperbola : that is, if the section inclines to- 
wards the plane of the cone’s base, it becomes 
an ellipsis ; but if it- incline towards the cone’s 
vertex, it then becomes an hyperbola, which 
is the plane of any section that falls between 
the parabola and the triangle ; and therefore 
there may be as many varieties of hyperbola 
produced" from one and the same cone as 
there may be ellipses. 
In short, a circle may change into an el- 
lipsis, the ellipsis into a parabola, the parabola 
into an hyperbola, and the hyperbola into a 
plain isosceles triangle. And the centre of 
the circle, which is its focus, divides itself 
into two focuses, as soon as ever the circle 
begins to degenerate into an ellipsis; but 
when the ellipsis changes into a parabola, 
one end of it flies open, one of its foci va- 
nishes, and the remaining focus goes along 
with the parabola when it degenerates into 
an hyperbola. And when the hyperbola 
degenerates into a plain isosceles triangle, 
this focus becomes the vertical point of the 
triangle, namely, the vertex of the cone. So 
that the centre of the cone’s base may be 
truly said to pass gradually through all the 
sections until it arrives at the vertex of the 
conp, still carrying its latus rectum along 
with it. For the diameter of a circle being 
that right line which passes through its centre 
or focus, and by which all other right lines 
drawn within the circle are regulated and 
valued, may he called the circle’s latus rec- 
tum ; and though it lose the name of diame- 
ter when the circle degenerates into an ellip- 
sis, yet it retains the name of latus rectum 
with its first properties in all the sections, 
gradually shortening as the focus carries it 
along from one section to another, until at 
last both it and the focus become coincident, 
and terminate in the vertex of the cone. 
Equations of the conic sections are derived 
from the above properties. The equation of 
