CURVE

, a line whose several parts p[rdot]oceed bowing, or tend different ways; in opposition to a straight line, all whose parts have the same course or direction.

The doctrine of curves, and of the figures and solids generated from them, constitute what is called the higher geometry.

In a curve, the line AD, which bisects all the parallel lines MN, is called a diameter; and the point A, where the diameter meets the curve, is called the vertex: if AD bisect all the parallels at right angles, it is called the axis. The parallel lines MN are called ordinates, or applicates; and their halves, PM, or PN, semi-ordinates. The portion of the diameter AP, between the vertex, or any other fixed point, and an ordinate, is called the absciss; also the concourse of all the diameters, if they meet all in one point, is the centre. This 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 ordinates; 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 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 of it.

Curve lines are distinguished into algebraical or geometrical, and transcendental or mechanical.

Algebraical or Geometrical Curves, are those in which the relation of the abscisses AP, to the ordinates PM, can be expressed by a common algebraic equation

And Transcendental or Mechanical Curves, are such as cannot be so defined or expressed by an algebraical equation. See Transcendental Curve.

Thus, suppose, for instance, the curve be the circle; and that the radius AC = r, the absciss AP = x, and the ordinate PM = y; then, because the nature of the circle is such, that the rectangle AP X PB is always = PM², therefore the equation is , defining this curve, which is therefore an algebraical or geometrical line. Or, suppose CP = x; then is , that is ; 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 algebraical curves geometrical ones; as admitting none else into the construction of problems, nor consequently into geometry. But Newton, 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 defined by a more simple equation, but for its being more easily described.

Algebraical or geometrical lines are best distinguished into orders according to the number of dimensions of the equation expressing the relation between its ordinates and abscisses, 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 2d order of lines, or the first order of curves, are the circle and conic sections, being 4 species only, viz, the circle, the ellipse, the hyperbola, and dx = y² the parabola: the lines of the 3d order, or curves of the 2d order, are expressed by an equation of the 3d degree, having three roots; and so on. Of these lines of the 3d order, Newton wrote an express treatise, under the title of Enumeratio Linearum Tertii 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. Bernoulli and Mr. Stone added two more.

Curves of the 2d and other higher kinds, Newton observes, have parts and properties similar to those of the 1st kind: Thus, as the conic sections have diameters and axes; the lines bisected by these are ordinates; and the intersection of the curve and diameter, the vertex: so, in curves of the 2d kind, any two parallel right lines being drawn to meet the curve in 3 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 3d part terminated by the curve on the other side, will cut, in the same manner, all other right lines parallel to these, and that meet the curve in three points, that is, so as that the sum of the two parts on one side, will still be equal to the 3d part on the other side. These three parts therefore thus equal, may be called ordinates, or applicates; the cutting line, the dia- | meter; and where it cuts the ordinates at right angles, the axis; the intersection of the diameter and the curve, the vertex; and the concourse of two diameters, the centre; also the concourse of all the diameters, the common or general centre.

Again, as an hyperbola of the first kind has two asymptotes; that of the 2d has 3; that of the 3d has 4; &c: and as the parts of any right line between the conic hyperbola and its two asymptotes, are equal on either side; so, in hyperbolas of the 2d kind, any right line cutting the curve and its three asymptotes in three points; the sum of the two parts of that right line, extended from any two asymptotes, the same way, to two points of the curve, will be equal to the 3d part extended from the 3d asymptote, the contrary way, to the 3d point of the curve.

Again, as in the conic sections that are not parabolical, the square of an ordinate, i. e. the rectangle of the ordinates drawn on the contrary sides of the diameter, is to the rectangle of the parts of the diameter terminated at the vertices of an ellipse or hyperbola, in the same proportion as a given line called the latus rectum, is to that part of the diameter which lies between the vertices, and called the latus transversum: so, in curves of the 2d kind, not parabolical, the parallelopiped under three ordinates, is to the parallelopiped under the parts of the diameter cut off at the ordinates and the three vertices of the figure, in a given ratio: in which, if there be taken three right lines situate at the three parts of the diameter between the vertices of the figure, each to each; 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 a conic parabola, which has only one vertex to one and the same diameter, the rectangle under the ordinates is equal to the rectangle under the part of the diameter cut off at the ordinates and vertex, and a given right line called the latus rectum: so, in curves of the 2d kind, which have only two vertices to the same diameter, the parallelopiped under three ordinates, is equal to the parallelopiped under two parts of the diameter cut off at the ordinates and the two vertices, and a given right line, which may therefore be called the latus transversum.

Further, as in the conic sections, where two parallels, terminated on each side by a curve, are cut by two other parallels terminated on each side by a curve, the 1st by the 3d, and the 2d by the 4th; the rectangle of the parts of the 1st is to the rectangle of the parts of the 3d, as that of the 2d is to that of the 4th: so, when four such right lines occur in a curve of the 2d kind, each in three points; the parallelopiped of the parts of the 1st, will be to that of the parts of the 3d, as that of the 2d to that of the 4th.

Lastly, the legs of curves, both of the 1st, 2d, and higher kinds, are either of the parabolic or hyperbolic kind: an hyperbolic leg being that which approaches infinitely towards some asymptote; and a parabolic one, 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 hyperbolic leg will coincide with the asymptote; and that of the parabolic leg, recede infinitely, and vanish. Therefore the asymptote of any leg is found, by seeking the tangent of that leg to a point infinitely distant; and the direction of an infinite leg is found, by seeking the position of a right line parallel to the tangent, where the point of contact is infinitely remote, for this line tends that way towards which the infinite leg is directed.

Newton reduces all curves of the 2d kind to four cases of equations, expressing the relation between the ordinate and absciss, viz, in the 1st case, ; in the 2d, " ; in the 3d, " ; in the 4th, " . See Newton's Enumeratio, sect. 3; and Stirling's Lineæ, &c, pa. 83.

Under these four cases, the author brings a great number of different forms of curves, to which he gives different names. An hyperbola lying wholly within the angle of the asymptotes, like a conic hyperbola, he calls an inscribed hyperbola; that which cuts the asymptotes, and contains the parts cut off within its own periphery, a circumscribed hyperbola; that which has one of its infinite legs inscribed and the other circumscribed, he calls ambigenal; that whose legs look towards each other, and are directed the same way, converging; that where they look contrary ways, diverging; that where they are convex different ways, crosslegged; that applied to its asymptote with a concave vertex, and diverging legs, conchoidal; that which cuts its asymptote with contrary flexures, and is produced each way into contrary legs, anguineous, or snake-like; that which cuts its conjugate across, cruciform; that which returning around cuts itself, nodated; that whose parts concur in the angle of contact, and there terminate, cuspidated; that whose conjugate is oval, and infinitely small, i. e. a point, pointed; that which, from the impossibility of two roots, is without either oval, node, cusp, or point, pure. And in the same manner he denominates a parabola converging, diverging, cruciform, &c. Also when the number of hyperbolic legs exceeds that of the conic hyperbola, that is more than two, he calls the hyperbola redundant.

Under those 4 cases the author enumerates 72 different curves: of these, 9 are redundant hyperbolas, without diameters, having three asymptotes including a triangle; the first consisting of three hyperbolas, one inscribed, another circumscribed, and the third ambigenal, with an oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; the 5th and 6th, pure; the 7th and 8th, cruciform; the 9th or last, anguineal. There are 12 redundant hyperbolas, having only one diameter: the 1st, oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; 5th, 6th, 7th, and 8th, pure; the 9th and 10th, cruciform; the 11th and 12th, conchoidal. And to this class Stirling adds 2 more. There are 2 redundant hyperbolas, with three diameters. There are 9 redundant hyperbolas, with three asymptotes converging to a common point; the 1st being formed of the 5th and 6th redundant parabolas, | whose asymptotes include a triangle; the 2d formed of the 7th and 8th; the 3d and 4th, of the 9th; the 5th is formed of the 5th and 7th of the redundant hyperbolas, with one diameter; the 6th, of the 6th and 7th; the 7th, of the 8th and 9th; the 8th, of the 10th and 11th; the 9th, of the 12th and 13th: all which conversions are effected, by diminishing the triangle comprehended between the asymptotes, till it vanish into a point.

Six are defective parabolas, having no diameters: the 1st, oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; the 5th, pure; &c.

Seven are defective hyperbolas, having diameters: the 1st and 2d, conchoidal, with an oval; the 3d, nodated; the 4th, cuspidated, which is the cissoid of the ancients; the 5th and 6th, pointed; the 7th, pure.

Seven are parabolic hyperbolas, having diameters: the 1st, oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; the 5th, pure; the 6th, cruciform; the 7th, anguineous.

Four are parabolic hyperbolas: four are hyperbolisms of the hyperbola: three, hyperbolas of the ellipsis: two, hyperbolisms of the parabola.

Six are diverging parabolas; one, a trident; the 2d, oval; the 3d, nodated; the 4th, pointed; the 5th, cuspidated (which is Neil's parabola, usually called the semi-cubical parabola); the 6th, pure.

Lastly, one, commonly called the cubical parabola.

Mr. Stirling and Mr. Stone have shewn that this enumeration is imperfect, the former having added four new species of curves to the number, and the latter two, or rather these two were first noticed by Mr. Nic. Bernoulli. Also Mr. Murdoch and Mr. Geo. Sanderson have found some new species; though some persons dispute the realíty of them. See the Genesis Curvarum per umbras, and the Ladies' Diary 1788 and 1789, the prize question.

Organical Description of Curves.—Sir Isaac Newton shews that curves may be generated by shadows. He says, if upon an infinite plane, illuminated from a lucid point, the shadows of figures be projected; the shadows of the conic sections will always be conic sections; those of the curves of the 2d kind, will always be curves of the 2d kind; those of the curves of the 3d kind, will always be curves of the 3d kind; and so on ad infinitum

And, like as the projected shadow of a circle generates all the conic sections, so the 5 diverging parabolas, by their shadows, will generate and exhibit all the rest of the curves of the 2d kind: and thus some of the most simple curves of the other kinds may be found, which will form, by their shadows upon a plane, projected from a lucid point, all the other curves of that same kind. And in the French Memoirs may be seen a demonstration of this projection, with a specimen of a few of the curves of the 2d order, which may be generated by a plane cutting a solid formed from the motion of an infinite right line along a diverging parabola, having an oval, always passing through a given or fixed point above the plane of that parabola. The above method of Newton has also been pursued and illustrated with great elegance by Mr. Murdoch, in his treatise entitled Newtoni Genesis Curvarum per umbras, seu Perspectivæ Universalis Elementa.

Mr. Maclaurin, in his Geometria Organica, shews how to describe several of the species of curves of the 2d order, especially those having a double point, by the motion of right lines and angles: but a good commodious description by a continued motion of those curves which have no double point, is ranked by Newton among the most difficult problems. Newton gives also other methods of description, by lines or angles revolving above given poles; and Mr. Brackenridge has given a general method of describing curves, by the intersection of right lines moving about points in a given plane. See Philos. Trans. No. 437, or Abr vol. 8, pa. 58; and some particular cases are demonstrated in his Exerc. Geometrica de Curvarum Descriptione.

Curves above the 2d Order. The number of species in the higher orders of curves increase amazingly, those of the 3d order only it is thought amounting to some thousands, all comprehended under the following ten particular equations, viz, .

Those who wish to see how far this doctrine has been advanced, with regard to curves of the higher orders, as well as those of the 1st and 2d orders, may consult Mr. Maclaurin's Geometria Organica, and Brackenridge's Exerc. Geom.

All geometrical lines of the odd orders, viz, the 3d, 5th, 7th, &c, have at least one leg running on infinitely; because all equations of the odd dimensions have at least one real root. But vast numbers of the lines of the even orders are only ovals; among which there are several having very pretty figures, some being like single hearts, some double ones, some resembling fiddles, and others again single knots, double knots, &c.

Two geometrical lines of any order will cut one another in as many points, as are denoted by the product of the two numbers expressing those orders.

The theory of curves forms a considerable branch of the mathematical sciences. Those who are curious of advancing beyond the knowledge of the circle and the conic sections, and to consider geometrical curves of a higher nature, and in a general view, will do well to study Cramer's Introduction à l'Analyse des Lignes Courbes Algebraiques, which the learned and ingenious author composed for the use of beginners. There is an excellent posthumous piece too of Maclaurin's, printed as an Appendix to his Algebra, and entitled De Linearum Geometricarum Proprietatibus Generalibus. The same author, at a very early age, gave a remarkable specimen of his genius and knowledge in his Geometria Organica; and he carried these speculations farther afterwards, as may be seen in the theorems he | has given in the Philos. Trans. See Abr. vol. 8, pa. 62. Other writings on this subject, beside the Treatises on the Conic Sections, are Archimedes de Spiralibus; Des Cartes Geometria; Dr. Barrow's Lectiones Geometricæ; Newton's Enumeratio Linearum Tertii Ordinis; Stirling's Illustratio Tractatûs Newtoni de Lineis Tertii Ordinis; Maclaurin's Geometria Organica; Brackenridge's Descriptio Linearum Curvarum; M. De Gua's Usages de l'Analyse de Des Cartes; beside many other Tracts on Curves in the Memoirs of several Academies &c.

Use of Curves in the Construction of Equations. One great use of curves in Geometry is, by means of their intersections, to give the solution of problems. See Construction.

Suppose, ex. gr. it were required to construct the following equation of 9 dimensions, : assume the equation to a cubic parabola x³=y; then, by writing y for x³, the given equation will become ; an equation to another curve of the 2d kind, where m or f may be assumed = 0 or any thing else: and by the descriptions and intersections of these curves will be given the roots of the equation to be constructed. It is sufficient to describe the cubic parabola once. When the equation to be constructed, by omitting the two last terms hx 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, gx²+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 three dimensions, we shall fall upon Wallis's construction by the cubic parabola and right line.

Rectification, Inflection, Quadrature, &c of Curves. See the respective terms.

Curve of a Double Curvature, is such a curve as has not all its parts in the same plane.

M. Clairaut has published an ingenious treatise on curves of a double curvature. See his Recherches sur les Courbes à Double Courbure. Mr. Euler has also treated this subject in the Appendix to his Analysis Infinitorum, vol. 2, pa. 323.

Family of Curves, is an assemblage of several curves of different kinds, all defined by the same equation of an indeterminate degree; but differently, according to the diversity of their kind. For example, Suppose an equation of an indeterminate degree, : if m=2, then will ax=y²; if m=3, then will a²x=y³; if m=4, then is a³x=y⁴; &c: all which curves are said to be of the same family or tribe.

The equations by which the families of curves are defined, are not to be confounded with the transcendental ones: for though with regard to the whole family, they be of an indeterminate degree; yet with respect to each several curve of the family, they are determinate; whereas transcendental equations are of an indefinite degree with respect to the same curve.

All Algebraical curves therefore compose a certain family, consisting of innumerable others, each of which comprehends infinite kinds. For the equations by which curves are defined involve only products, either of powers of the abscisses and ordinates by constant coefficients; or of powers of the abscisses by powers of the ordinates; or of constant, pure, and simple quantities by one another. Moreover, every equation to a curve may have 0 for one member or side of it; for example, ax = y², by transposition becomes . Therefore the equation for all algebraic curves will be

Catacaustic, and Diacaustic Curves. See CATACAUSTIC, and Diacaustic.

Exponential Curve, is that which is defined by an exponential equation, as ax^z=y, &c.

Curves by the Light, or Courbes a la Lumiere, a name given to certain curves by M. Kurdwanowski, a Polish gentleman. He observed that any line, straight or curved, exposed to the action of a luminous point, received the light differently in its different parts, according to their distance from the light. These different effects of the light upon each point of the line, may be represented by the ordinates of some curve, which will vary precisely with these effects. Priestley's Hist. of Vision, pa. 752.

Logarithmic Curve. See Logarithmic Curve.

Curve Reflectoire, so called because it is the appearance of the plane bottom of a bason covered with water, to an eye perpendicularly over it. In this position, the bottom of the bason will appear to rise upwards, from the centre outwards; but the curvature will be less and less, and at last the surface of the water will be an asymptote to it. M. Mairan, who first conceived this idea from the phenomena of light, found also several kinds of these curves; and he gives a geometrical deduction of their properties, shewing their analogy to caustics by refraction. Mem. Ac. 1740; Priestley's Hist. of Vision, pa. 752.

Radical Curves, a name given by some authors to curves of the spiral kind, whose ordinates, if they may be so called, do all terminate in the centre of the including circle, and appear like so many radii of that circle: whence the name.

Regular Curves, are such as have their curvature turning regularly and continually the same way; in opposition to such as bend contrary ways, by having points of contrary flexure, which are called irregular curves.

Characteristic Triangle of a Curve, is the differential or elementary right-angled triangle whose three sides are, the fluxions of the absciss, ordinate, and curve; the fluxion of the curve being the hypothenuse. So, if pq be parallel to, and indesi- nitely near to the ordinate PQ, and Qr parallel to the absciss AP; then Qr is the fluxion of the absciss AP, and qr the fluxion of the ordinate PQ, and Qq the fluxion of the curve AQ; hence the elementary triangle Qqr is the characteristic triangle of the curve AQ; and the three sides are x^., y^., z^.; in which .

CURVILINEAR Angle, Figure, Superficies, &c, | are such as are formed or bounded by curves; in opposition to rectilinear ones, which are formed by straight lines or planes.