CURVE

In Analytical Geometry, a line of which no three consecutive points are in the same direction or straight line. The general theory of curve lines, and of the figures bounded by them, forms an extensive and important part of mathematical science, and in fact properly constitutes what is called the high or transcendent geometry. It will easily be understood, however, that the curves which form the object of geometrical speculation are those only of which the succe ion of points observes a regular law, capable of being expre ed by a mathematical formula; not those which are formed irregularly by the hand, for example, like letters in writing. Although the ancient geometricians had bestowed some attention on the subject of curve lines, and in addition to the conic sections had investigated some of the properties of a few other curves, as the ci oid, the conchoid, particular kinds of spirals, and perhaps others whose names may have been forgotten; yet their researches had only led them to the knowledge of a small number of particular propositions deduced from a laborious consideration of the circumstances of each individual case, and not admitting of extended application. The general methods of investigation which the modern geometer can apply with so much greater advantage are owing to the progre of algebra, and the happy invention by Des cartes of the method of defining curves by algebraic equations. Let M N be a curve, and A B, A C two straight lines in the same plane, to which all its points are referred; then A B and A C are the axes of the curve. Take a point P in the curve, and draw P Q parallel to A C; the line AQ is called the absci of the point P, and Is de30S CURVE. noted by*; and PQ the ordinate, and denoted by y. Now if for any given value of A Q y. a corresponding value of P Q can be a igned, that is, if P Q can be expre ed in terms of A Q and known quantities, that expre ion is called the equation of the curve, and all Us properties can be thence deduced by X Q "~B means of algebraic transformations without any reference to the diagram. The first use of this method of defining curve lines by means of an equation between the absci and ordinate is to enable us to divide them into cla es, of which all the individuals have some properties in common. Des cartes divided them into two great cla es, — geometrical and mechanical. It is now usual to indicate the same distinctions by the terms algebraic and transcendental. Algebraic curves are those in which the relation between the absci and the ordinate is expre ed by an algebraic equation: transcendental curves are those in which the relation between x and i/ is not expre ed by an algebraic, but by a differential equation; that is, by an equation between dx and di/. There is still a cla which may be regarded as intermediate between these two; namely, exponential curves, or those in the equation of which one or both of the unknown quantities enters as an exponent. Such, for example, is the equation ^ = a t. In order to form an idea of a curve of which the equation is given, it is nece ary to suppose the equation resolved; that is to say, that the value of y is found in terms of x and given numbers. This being found, we take all the positive values of x from 0 to + infinity, and also all the negative values to — infinity. The corresponding ordinate a or values of y will give all the points of the curve; the positive ordinates being taken on one side of the axis cf the absci a, and the negative on the opposite side. Algebraic curves are divided into different cla es or orders, according to the degree of the equation which expre es the relation between their co-ordinates. Straight lines are denominated lines of theirs/ order, because the equation of a straight line being of the form 0 = A -(- B ar + Cy, is only of one dimension; that is to say, it contains no powers or products of the variables x and y. Lines of the second order are those of which the equation rises to two dimensions. The general form of this equation is, 0 = A + Ba: + Cy -I- D«2 + Exy + Fy2; and the curves which it includes are the conic sections; that is, the circle, the ellipse, the hyperbola, and the parabola. These curve lines are the simplest of all, becjiuse lines of the first order are not curves; hence they are sometimes called curves of the first order. Following out the analogy, lines of the third, or those whose equation rises to the third degree, are curves of the second order, and so on. Tlie dimension of the equation of a curve line is not altered by changing the origin or position of its co-ordinates, or in making any trans fon nation of its axes. It follows, therefore, that a line of the nth order can never be intersected by a straight line in more than n points, because the ordinates can never have more than n real v.alues. This follows from the general theory of equations; and in fact all the properties of any curve line being deducible from its equation, a complete theory of algebraic equations of any degree includes the theory of lines of that order. In this way it is proved that all lines of any order of which the number is uneven have nece arily at least two infinite branches; for in this case one at least of the co-ordinates is raised to an uneven power in the equation — for example, the 3d, the 5th, .; and therefore will have at least one real value, whatever value (which maybe increased in infinitum, positively and negatively) may be given to the other co-ordinate. We have mentioned that the lines of the sec(^id order include only the circle and the conic sections. Newton, in his Enumeratio Line arum Tertii Ordinis, reckons 72 species of lines of the third order, or curves of the 2d degree. As the order is more elevated the number of getiera and species becomes more numerous. The subdivisions into genera and species are, however, founded on some arbitrary properties, and consequently are not made uniformly by different authors. Cramer, in his Analyse des Lignes Courbes, found fourteen different genera of curves of the third order; and Euler (Introduclio in An a lysin In finito rum) sixteen. The whole number of curves belonging to this order has been supposed to amount to some you sands. Curves of Double Curvature.—Yi\t\\ we have spoken only of curves on a plane; but if they are traced on surfaces which are not plane, they will have a double curvature; that which belongs to the line itself, and that of the surface on which it is traced. In order to investigate the properties of a curve of this sort, it must be supposed to be projected on two different planes perpendicular to each other; the projections will be two ordinarj^ curves having a common axis and different ordinates. One of these curves will be defined by an equation between x and y, and the other by an equation between x and x; so that the [s. 321]