CURVES, 
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 parallelepiped under 
the parts of the first right line, be to that 
under the parts of the third ; as the paral- 
Jelopiped 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 eitlier of the parabolic 
or hyperbolic kind : an hyperbolic leg be- 
ing that which approaches infinitely towards 
some assymptote; a parabolic that which 
has no assymptote. These legs are best 
distinguished by their tangents ; for if the 
point of contact go off to an infinite dis- 
tance, the tangent of the hyperbolic leg will 
coincide with the assymptote ; and that of 
the parabolic leg recede infinitely and va- 
nish. The assymptote, tlierefore, of any 
leg is found by seeking the tangent of that 
leg to a point infinitely distant; and the 
bearing of an infinite leg is found by seek- 
ing the position of a right line parallel to 
the tangent, when the point of contact is 
.infinitely remote: for this line tends the 
same way towards which the infinite leg is 
directed. For the other properties of 
curves of the second order, w'e refer the 
reader to Mr. Maclaurin’s treatise “ De 
Linearum geometricamm Proprietatibus 
generalibus.” 
Sir Isaac Newton reduces all curves of 
the second order to the four following par- 
ticular equations, still expressing them all. 
In the first, the relation between the ordi- 
nate and the abscisse, making tlie abscisse 
x and the ordinate y, assumes this form xy^ 
-[- c y = a x’ -j- ft -j- c a: d. In the 
second case, the equation takes , this form 
xy = ax^+bx‘ + cx -j- d. In the third 
case, the equation is y^ = a x’ -|- ft x^ -J- 
cx -f- d. And in the fourth case the equa- 
tion is of this form, y = ax^ ^ cx 
d. Under these four cases the same 
author enumerates seventy-two different 
forms of curves, to which he gives different 
names, as ambigenal, cuspidated, nodated, 
&c. 
Curves, genesis of, of the second order by 
shadows. If (says Sir Isaac Newton) upon 
an infinite plane illuminated from a lucid 
point the shadows of figures be projected, 
the shadows of tlie conic sections will be 
always conic sections ; those of the curves 
of the second kind will be always curves of 
the second kind ; those of the curves of the 
third kind will be always curves of the 
third kind, and so on in infinitum. And as 
a circle by projecting its shadow generates 
all the conic sections, so the five diverging 
parabolas by their shadows will generate 
and exhibit all the rest of the curves of 
the second kind : and so some of the most 
simple curves of the other kinds may be 
found which will form by their shadows 
upon a plane, projecting from a lucid point, 
all the rest of the curves of that same 
kind. 
Curves of the second order having double 
points. As curves of the second order may 
be cut by a right line in three points ; and 
as two of these points are sometimes coinci- 
dent, these coincident intersections, whether 
at a finite or an infinite distance, are called 
the double point. 
Curves, use of, in the construction of 
equations. One great use of curves in geo- 
metry is, by means of their intersections, to 
give the solution of problems. See Equa- 
tions. 
Suppose, ex. gr, it were required to con- 
struct the following equation of 9 dimen- 
sions, 
x^ -{• b x’’ c xfi dx^ tx^ ■\-m-^f.x^ 
+ gx^ + hx k — Oi assume the equa- 
tion to a cubic parabola x® =r y ; then, by 
writing y for x’, the given equation will be- 
come y’-|-ftxy^-|-ey^-j-dx^y-(-<7xy-[- 
my -\- f x’ -f-gx^ ftx -f- fe = 0 ; an equa- 
tion to another curve of the second kind, 
where m or^ may be assumed = 0, or any 
thing else : and by the descriptions and in- 
tersections of these curves will be given the 
roots of the equation to be constructed. It 
is sufficient to describe tlie cubic parabola 
once. When the equation to be construct- 
ed, by omitting the two last terms ft x and k, 
is reduced to 7 dimensions ; the other curve, 
by expunging m, will have the double point 
in the beginning of the absciss, and may be 
easily described as above: if it be reduced 
to 6 dimensions, by omitting the last three 
terms, g x^ + hx + k ; the other curve, by 
expunging f, will become a conic section. 
And if, by omitting the last three terms, 
the equation be reduced to 3 dimensions, 
we shall fall upon Wallis’s construction by 
the cubic parabola and right line. 
CuRVES,/o»Mly 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, 
according to the diversity of their kinds. 
