CURVE, in geometry, a line, which, running on continually in all directions, may be cut by one right line in more points than one. Curves are divided into algebraical or geometrical, and transcen dental. Geometrical or algebraical curves are those, whose ordinates and abscisses being right lines, the nature thereof can be expressed by a finite equation having those ordinates and abscisses in it.
Transcendental curve, is such as, when expressed by an equation, one of the terms thereof is a variable quantity.
Geometrical lines or curves are diirid ed into orders, according to the number of dimensions of the equation expressing the relation between the ordinates and abscisses, or according to the number of points by which they may be cut by a right line. So that a line of the first or der will be only a right line, expressed by the equation y + a x + A line of the second, or quadratic order, will be the conic sections and circle, whose most general equation is yI + ax+hxy+c x' + dx+ e = 0. A line of the third order, is that whose equations has three dimensions, or may be cut by a right line in three points, whose most general equation is y3 +ax+h X yz-1 + dx+eXy+f x3 + g x' + h x+ "k= 0. A line of the fourth order, is that whose equation has four dimensions, or which may be cut in four points by a right line, whose most general equation + f x3 + gx3 + hs +k Xy+ Les ntr3 + n x' ± p x ± y = 0. And so on.
And a curve of the first kind (for aright line is not to be reckoned among curves) is the same with a line of the second or der ; and a curve of the second order, the same as a line of the third ; and a line of an infinite order, is that which a right line can cut in an infinite number of points, such as a spiral, quadratrix, cy cloid, the figures of the sines, tangents, secants, and every line which is gene rated by the infinite revolutions of a eir de or wheel.
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 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 cut ting 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 mit 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 be still equal to the third part on the other side. These three parts, therefore, equal, may be called ordinates or appli cates : the secant may be styled the dia meter ; the intersection 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 be called axis and that point where all the diameters terminate the general centre. Again, as an hyperbola of the first order has two asymptotes; 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 asymptotes, are every where equal ; so in the hyper bola of the second order, if any right line be drawn, cutting both the curve and its three asymptotes in three points, the sum of the two parts of that right line, be ing drawn the same way from any two asymptotes to two points of the curve, will be equal to a third part drawn a con trary 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 un der the ordinates, drawn to contrary sides of the diameter, is to the rectangle of the parts of the diameter which are termi nated at the vertices of the ellipsis or hy perbola, as the latus rectum is to the latus transversum ; so in non-parabolic curves of the second order, a parallelopiped un der the three ordinates is to a parallelo piped under the parts of the diameter, 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 the 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, between the vertices, the latera transversa. And as in the conic
parabola, having to one and the same di ameter but one only vertex, the rectangle under the ordinates is equal to that under the part of the diameter cut off between the ordinates and the vertex, and the la tus rectum;` so in curves of the second order, which have hut two vertices to the same diameter, the parallelopiped under three ordinates is equal to the parallelo piped under the two parts of the diame ter, cut off 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 sec tions, 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 paral lelopiped under the parts of the first right line be to that under the parts of the third, 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 approaches infinitely towards some asymptote ; a parabolic that which has no asymptote. These legs are best distinguished by their tan gents ; for if the point of contact go off to an infinite distance, the tangent of the hyperbolic leg will coincide with the asymptote ; and that of the parabolic leg recede infinitely and vanish. The as symptote, 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 is found by seeking the position of a right line parallel to the tan gent, when the point of contact isinfinite ly remote : for this line tends the same way towards which the infinite leg is di rected. For the other properties of curves of the second order, we refer the reader to Mr. Maclattrin's treatise " De Linearum geometricarum Proptietatibus generalibus." Sir Isaac Newton reduceS all curves of the second order to the four following particular equations, still expressing thenf i all. In the first, the relation between , the ordinate and the abscisse, making the abscisse x and the ordinate y, assumes this form,xy'--1-ey=ax3-1-bx.-Fcx 1 -f-d. In the second case, the equation takes this form, x y ..-.-_- a x3 + b x. -1- c x - F d. In the third case, the equation is y'.---a x3 4- b x. + c x + d. And in the fourth case the equation is of this form, y = a x3 - 1 - b x. + c x + d. Under these four cases the same author enu merates seventy-two different forms of curves, to which he gives different names, as ambigenal, cuspidated, nodated, &c. 1 Canvas, genesis of, of the second order by shadows.. If (says Sir Isaac Newton) • upon an infinite plane illuminated from a I lucid point the shadows of figures bepro jected, the shadows of the conic sections will be always conic sections ; those of the curves of the second kind will he al ways curves of the second kind; those of the curves of the third kind will be al ways curves of the third kind, and so on in hefinitunt. And as a circle, by project ing its shadow, generates all the conic • sections, so the five diverging parabolas by their shadows will generate and ex hibit all the rest of the curves of the se cond kind ; and so some of the mostsim- , ple curves of the other kinds may be fbund, which will form by their shadows upon a plane, projecting from a lucid ' point, all the rest of the curves of that same kind.