AN'ALYT'IC GEOMETRY. Geometry treat ed by means of algebra. Geometric conditions are expressed by equations which, after certain transformations, are interpreted again in geo metric concepts. The powerful algebraic method is thus made use of for discovering and demon strating in a simple and easy manner the most complicated relations existing between quantities in space.
The interpretation of geometric relations in algebraic terms is effected by the use of some sys tem of coordinates (q.v.). The primitive system of cofirdinates, called rectangular coordinates, is due to Descartes (Lat. Cartesian), from which fact they are often called Cartesian. In this sys tem the position of a point (as P„ in the figure) is determined by its distance from the fixed axes in the plane, called .axes of coordinates, which intersect at right angles in a point called the origin. The distance x, of P, from is called the abscissa. of P„ and the distance y, from XX', is called the ordinate. The two lines x„ y,. are called the coordinates of P,. Similarly, the coordinates of P, are x2, 3/2- Pi, P,, or the points (x, (x,, y,) are sufficient to deter mine the straight line AB. The algebraic func tion y= ax + b, a, b, being constants, will have different values according to the va rious values given to x. The various values of x, as x„ x„—taken with the corresponding values of y, as represent a series of points (x„ y,), y,), y,), lying in a straight line. That is, an algebraic equation of the first degree is represented by a straight line. In a similar manner a function of the second degree is represented by a curve. In the figure, c is a circle whose equation is + = r, r be ing the radius of the circle. This is evident by reference to the figure, since the coordinates of any point (r, y) form the sides of a right-angle triangle of hypotenuse r, so that + Here the function of x is since y= The curve e is an ellipse whose equa tion is + = a being the semi-major axis and b the semi-minor axis. The curve h is an hyperbola whose equation is If the equations + and — are solved for x, y, their roots are the coordi nates of the points of intersection of the curves c, h. These values may be real or imaginary; if real, the curves cut in real points, as in the ease of c. It: if imaginary, the curves are said to cut in imaginary points, as in the case of c, h.
The practical work of plotting a curve may be explained by referring to a particular exam ple; thus, to represent graphically the equation — 10. Rearranging and solving the
equation for g, y 5). Therefore, by giving x various values (noticing that > 5 for real values of y) we have corresponding values of y as follows: x = ± ± V6, ± ± ?8, t1/9.
Taking the approximate square roots, and laying off the abscissas and ordinates as indicated, and then connecting the successive points, the graph is the hyperbola h, shown in the figure.
The power of the analytic forms to express geometric relations may be seen from the follow ing: Let = 0 and z„, = 0 represent the equa tions + = 0 and 0. Any values of x, p satisfying these two equa tions will evidently satisfy the equation + = 0, k being any constant. But this equation is z, — kz.,= 0. Hence, if z = 0, = 0 are the equa tions of any two curves, any point common to the two satisfies the equation z,— 0, and, therefore, this is the equation of the curve pass ing through all intersections of the given curves. In the same way, equations of any degree may be represented and discussed.
The position of a point in space of three dimen sions may be expressed in terms of its distances from three fixed planes. In this way the prop erties of spheres, ellipsoids, and other solids are expressed by equations. In space of four dimen sions the coordinates of a point are (.r, y, z, w), and in space of n dimensions (r. y, z quantities) , although we cannot draw the figures.
The ellipse, hyperbola, end parabola being sec tions of a right circular cone, are known as conic sections (q.v.). They were chiefly investigated by purely geometric methods until the appearance of Descartes's Discours (1(i37). In the exten sive development of analytic geometry since Descartes, a large number of coordinate systems have been introduced, the most important being the polar, generalized, homogeneous, Lagrangian, Eulerian, baryeentric, and trilinear coordinates.
The most comprehensive English works are those by Salmon, Treatise on the Conic Sections (Dublin, 1869); higher Plane CUM'S (1873) ; Treatise on the Analytic Geometry of Three Dimensions (Dublin. 1874). Other noteworthy works are: B. F. A. Clebsch, Vo•lesungen fiber Geometric (Leipzig. 1876) ; 7)1. Chasles, Traite de (Paris, 1880) ; and among recent elementary works are those of Steiner, Briot, Bouquet, Townsend, and Scott. For a further discussion, see GEOMETRY and COORDINATES.