CURVE. 
'Transcendental curve, iS such as -when 
Expressed by an equation, one of the terms 
thereof is a variable quantity. 
Geometrical lines or curves are divided 
into orders, according to the number Of 
dimensibns of the equation expressing the 
relation between the ordinates and abscissesj 
br according to the number of points by 
which they may be cut by a right line. So 
that a line of the first order will be only A 
right line expressed by the equation y -f- 
ux-{-h = 0. A line of the second, or qua- 
dratic order, will be th'e conic sections and 
circle whose most general equation is y^ 
a x-\- h X y-\-c !<?' -\-dx-\-ei=.0. A line 
of the third order is that whose equation 
has thfee dimensions, or may be cut by a 
right line in three points, whose most gene- 
ral equation is y’ -|- a x -|- 6 X 
d x-\-e X y -^-f S X -f- = 0. 
A line of the fourth order, is that whose 
equation has four dimensions, or which may 
be tiit in four points by a right line, whose 
most general equation is y‘^ a x b X 
y'^-\-cx^-\-dx-\-eX -f-/' + S' 
h X k X y i x'* m tl ^ -f- 
5 = 0. And so on. 
And a curve of the first kind (for a right 
line is not to be reckoned among curVes) 
is the same with a line of the second order ; 
and a curve of the second order, the same 
as a line of the third ; and a line of an in- 
finite order, is that which a right line can 
cut in an infinite number of points, such as 
a spiralj quadratrixj cycloid, the figures of 
the sines, tangents, secants, and every line 
which is generated by the infinite revolu- 
tions of a circle or wheel. 
As to the curves of the second orderj Sir 
Isaac Newton obseiwes they have parts and 
properties similar to those of the first ;.thus 
as the conic sections have diameters and 
axes, the lines cut by these are called ordi- 
nates, and the intersection of the curve and 
diameter, the vertex; so in Curves of the 
second order, any two parallel lines being 
drawn so as to meet the curve in three 
points, a right line cutting these parallels so 
as that the sum of the two parts between 
the secant and the curve on one side is 
equal to the tliird part terminated by the 
curve on the other side^ will cut in the same 
manner all other right lines parallel to these ^ 
and meet the curve in three parts, so as 
tiiat tlie sum of the two parts on one side 
will be still equal to the third part on the 
other side. 
These three parts, therefore, thus equal, 
may be called ordinates or applicates : the 
secant may be stiled the diameter ; the in- 
tersection of the diameter and the curve the 
vertex ; and the point of concourse of any 
two diameters the centre. And if the dia- 
meter be normal to the ordinates, it may be 
called axis ; and that point where all the 
diameters terminate the general centre. 
Again, as an hyperbola of the first order 
has two assymptotes ; that of the second 
three ; that of the third four, &c. : and as 
the parts of any right line lying between 
the conic hyperbola and its two assymp- 
totes are every where equal ; so in the hy- 
perbola of the second order, if any right 
line be drawn cutting both the curve and 
its three assymptotes in three points, the 
sum of the two parts of that right line being 
drawn the same way from any two assymp- 
totes to two points of the curve, will be 
equal to a third part drawn a contrary way 
from the third assymptote to a third point 
of the curve. Again, as in conic sections 
not parabolical, the square of the ordinate, 
that is the rectangle under the ordinates 
drawn to contrary sides of the diameter, is 
to the rectangle of the parts of the diameter 
which are terriiinated at the vertices of the 
ellipsis or hyperbola, as the latus rectum is 
to the latus transversum ; so in non-parabo- 
lic curves of the second order, a parallelo- 
piped under the three ordinates is to a 
paralleJopiped under the parts of the dia- 
meter, terminated at the ordinates, and the 
three vertices of the figure, in a certain 
given ratio ; in which ratio, if you take three 
right lines situated at tlie three parts of the 
diameter between the vertices of the figure,' 
one answering to another, then these three 
right lines may be called the latera recta of 
the figure, and the parts of the diameter, be- 
tween the vertices, the latera trans versa. 
And as in the conic parabola, having to one 
and the same, diameter but one only vertex, 
the rectangle under the ordindtes is equal 
to that under the. part of the diameter cut 
off between the ordinates and the. vertex, 
and the latus rectum; so in curves of tlie 
second order, which have but two vertices 
to the same diameter, the parallelopiped 
under three ordinates, is equal ,to the paral- 
lelepiped ynder the two parts of the diame- 
ter, cut oil' between the ordinates and those 
two vertices and a given right line, which 
therefore may be called the latus rectum. 
Moreover, as in the conic sections, when 
two parallels terminated on each side of the 
curve are cut by two other parallels termi- 
nated on each by the curve, the first by the 
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