CUR 
C U K 
CUR 
Tight angles, it is called the axis. The pa- 
rallel lines MN tye called ordinates, or ap- 
plicates, and tiieir halves P M, or P N, 
semi-ordinates. The portion of the diame- 
ter A P, between the vertex, or any other 
fixed point, and an ordinate, is called the 
absciss ; also the concourse of all the dia- 
meters, if they meet ail ui one point, is the 
centre, 't his definition of the diameter, 
as bisecting the parallel ordinates, respects 
only the conic sections, or such curves as 
are cut only in two points by the ordi- 
nates ; but in the lines of the 3d order, which 
may be cut in three points by the ordinates, 
then the diameter is that line which cuts the 
ordinates so, that the sum of the two parts 
that lie on the one side of it, shall be equal to 
the part- on the other . side ; and so on for 
curves of higher orders; the sum of the parts 
of : the ordinates on one side of the diameter, 
being always equal to the sum of the parts on 
the other side ot it. 
Curve lilies are distinguished into algebra- 
ical or geometrical, and transcendental or 
mechanical. 
Algebraical or geometrical curves are those 
in which the relation of the abscisses AT, to 
the ordinates P M, can be expressed by a 
common algebraic equation. 
And transcendental or mechanical curves 
are such as cannot be so defined or express- 
ed by an algebraical equation. 
I bus suppose, for instance, the curve be 
the circle, (PI. Mis. fig. 2S) and that the radius 
= r, the absciss A P— r, and the ordinate 
PM = //; then, because the nature of the 
circle is such, that the rectangle APx P B is 
alw ays — P M 2 , therefore the equation is 
a’ - 2 r—j =z>/ 2 , or 2rx— x 7 =f } defining this 
curve, which is therefore an algebraical or 
geometrical line. Or suppose CP — x, then 
is C M 2 — C P 2 = P M J , that is, r 2 — x 2 — i / 2 ; 
which is another form of the equation of the 
curve. 
The doctrine of curve lines in general, as 
expressed by algebraical equations, was first 
introduced by Des Cartes, who called alge- 
braical curves geometrical ones, as admitting 
none else into the construction of problems, 
nor consequently into geometry. But New- 
ton, and' after him Leibnitz and Wolfius, are 
of another opinion; and think, that in the 
construction of a problem, one curve is not 
to be preferred to another for its being de- 
fined by a more simple equation, but for its 
being more easily described. 
Algebraical or geometrical lines are best dis- 
tinguished into orders according to the num- 
ber ot dimensions of the equation expressing 
the relation between its ordinates and ab- 
scisses, or, which is the same thing, according 
to the number of points in which they may 
be cut by a right line. And curves' of the 
same kind or order are those whose equations 
rise to the same dimension. Hence, of the 
first, order, there is the right line only ; of the 
second order of lines, or the first order of 
curves, are the circle and conic sections, be- 
ing four species only, viz. dx — x 2 = f the 
c 2 _ s 
circle - — . dx — x* — y 2 the ellipse, — ,dx-\-x 2 
=f the hyperbola, and dr — y 1 the parabola ; 
the lines of the third order, or curves of the 
second order, are expressed by an equation of 
the third degree, having three roots; and so 
on. Of these lines of the third order, Newton 
s.'Vol. I. ' 
wrote an express treatise, under the title of 
Kninneratio LinearumTertii Ordinis, shewing 
their distinctive characters and properties, to 
the number of 72 different species of curves; 
but Mr. Stirling afterwards added four more 
to. that number, and Mr. Nic. Bernouilli and 
Mr. Stone added two more. 
As to the curves of the second order, sir 
Isaac Newton observes, 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. ordinates, 
and the intersection of the curve and diame- 
ter, 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 third 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 that the sum of the two 
parts on one side will he still equal to the 
third part on the other side. 
These three parts, therefore, thus equal, 
may he called ordinates or applicates; the 
secant may he styled the diameter; the inter- 
section of the diameter and the curve the 
vertex ; and the point of concourse of any two 
diameters the centre. And if the diameter be 
normal to the ordinates, it may he called 
axis ; and that point where all the diameters 
terminate, the general centre. Again, as an 
hyperbola of the first order has two asymp- 
totes, 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 true asymptotes are every where equal, 
so in the hyberbola of the second order, if 
any right line he drawn cutting both the 
curve , and its three asymptotes in three 
points, the sum of the two parts of that right 
line being drawn the same way from any two 
asymptotes to two points of the curve, will 
be equal to a third part drawn a contrary way 
from the third asymptote 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 rect- 
angle of the parts of the diameter which are 
terminated at the vertices of the ellipsis or 
hyperbola, as the latus rectum is to the latus 
transversum ; so in nonparabolic curves of the 
second order, a parallelepiped under the 
three ordinates is to a parallelopiped under 
the parts of the diameter, terminated at the 
ordinates, and the three vertices of the li«m re, 
in a certain given ratio; in which rat?o, if 
you take three right lines situated at the 
three parts of the diameter between the ver- 
tices 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 between the vertices, the 
latent transversa. And, as in the conic para- 
bola, having to one. and the same diameter 
hut one only vertex, the rectangle under the 
ordinates is equal' to that under the part of the 
diameter cut off between the ordinates and 
t he vertex and the latus rectum ; so in curves 
of the second order, which have but two 
vertices to the same diameter, the parallelo- 
piped under three ordinates; is equal to the 
parallelopiped under the two parts of the di- 
ameter cut off between the ordinates and 
4/3 
those two vertices and a given right line, 
which therefore may be called tins latus rec- 
tum. Moreover, as in the conic sections, 
when two parallels terminated on each side of 
the curve, are cut by two other parallels 
terminated on -each by the curve, the first 
by the third, and the second by the fourth; 
as here the rectangle under the parts of the 
first is to the rectangle under the parts of the 
third, as the rectangle under the parts of the 
second is to that under the parts of the fourth; 
so when four such right lines occur in a curve 
of the second kind, each in three points, then 
shall the parallelopiped under the parts of 
the first right line, be to that under the parts 
ot the thifd, as the parallelopiped under the- 
parts of the second line, to that under the 
parts of the fourth. Lastly, the legs of curves, 
both of the first, second* and higher kinds, 
are either of the parabolic or hyperbolic 
kind; an hyperbolic leg being that which ap- 
proaches infinitely towards some asymptote ; 
a parabolic, that which has no asymptote. 
These legs are best distinguished by their 
tangents ; for if the point of contact go off to 
an infinite distance, the tangent of the hyper- 
bolic*" leg will coincide with the asympote; 
and that of the parabolic leg recede infinitely 
and vanish. The* asymptote, therefore, of 
any leg is found by seeking the tangent of 
that leg to a point infinitely distant; and the 
bearing of an infinite leg L found by seeking 
the position of a right line parallel to the tan- 
gent, when the point of contact is infinitely 
remote; for this line tends the same way to- 
wards which the infinite leg is directed. Fbr 
the other properties of curves of the second 
order, we reier the reader to Mr. Madam 
rin’s treatised De Linearum Geometricarmn 
Proprietatibus ge'n&ralibus. 
Sir Isaac Newton reduces all curves of the- 
second order to the four following particular 
equations, still expressing them all. In the 
first, the relation between the ordinate and 
the absciss, making the absciss x and the 
ordinary//, assumes this form, xy 2 -|- ey — ax* 
-f- bx 2 + + f /. ip the second case,, the 
equation takes this form, x y = ax 5 -f bx 2 -f- C x 
-\-d. In the third case, the equation is // 
~ax 2 bx 2 -\-cx-\- d. And in the fourth case, 
the equation is of this form, y= u' 3 -J- bx 2 -{- 
CX -\-d. 
Curves , family of, according to Wolfius, 
is a congeries of several curves of different 
kinds, all defined by the same- equation of an 
indeterminate degree, but differently, accord- 
ingto the diversity of their kinds. Forexample: 
let the equation of an’ indeterminate degree 
he a >>l 1 x=sy m . If m—1, ax will.be equal 
to // 2 . IL/n==3, then will a ! x~f. If 4, 
then will a'x — y\ &c. all which cun ey are 
said to he of the same family. The equations, 
however, by which the families of qifrv'es are 
defined, must not be confounded with .tran- 
scendental ones; though. with regard. tp. the 
whole family they be of an indeterminate de- 
gree, yet with respect to each several pine 
ot the family, they are determinate ; whereas 
transcendental equations are of an indefinite 
degree with respect to the same, curve. The 
genesis and properties of particular curves, 
as the conchoid, cycloid, c kc. see "under 
their proper heads C o x c h o jd,C. yclo id, Ac. 
.CLRULE efiah'j in Roman antiquity, a 
chair adorned with ivory,* wherein j lie great 
magistrates of Rome had a right to .sit, and be 
carried. * 
