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 efl'ected by the use of some sys- tem of coordinates (q.v. ). The primitive system of coordinates, called rectangular coordinates, is due to Descartes (Lat. Cartesius), from which fact they are often called Cartesian. In this sys- tem the position of a point (as P„ in the figure) an hyperbola whose equation is b^'jr — a,-^- = k'. It the equations a;^ + y = »~ and b^-x- — a'y- =: A-,^ are solved for x, y, their roots are the coordi- nates of the points of intersection of tlie curves c, h. These values may be real or imaginary; if real, the curves cut in real points, as in the case of c, h; 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 2ar — 3i/-=:10. Rearranging and solving the equation for y, y= ± I/3 V 6 ( ar — 5 ) . Therefore, by giving x various values (noticing that a;- > ^ for real values of y) values of y as follow-s: X = ± i/5, ± /W, ± y= 0, ± i Vey ±
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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 YY' is called the abscissa of P,, and the distance 3/1 from XX', is called the ordinate. The two lines a-,, ;/i, are called the coordinates of F^. Similarly, the coordinates of i'. are aj,, 1/2. Pi, P^, or the points (x,, y,), (a?,, 1/2) are sufficient to determine the straight line AB. The algebraic function (q.v.), y = ax+b, a, b, being constants, will have different values according to the various values given to a;. The various values of x, as a?], X2, X3 — taken with the corresponding values of y, as 1/j, 2/2. 2/3 — will represent a series of points (a;,, j/i), {X,, y~), (x,, y,) , lying in a straight line. That is, an algebraic equation of the first degi'ee is represented by a straight line. In a similar manner a function of the second degi'ee is represented by a curve. In the figure, c is a circle whose equation is «; + »/ = »-, r be- ing the radius of the circle. This is evident by reference to the figure, since the coordinates of any point (x, y) form the sides of a right-angle triangle of hypotenuse r, so that x- -^ y-^=: r'. Hero the function of x is Vr" — x'-, since i/ = V r- — •jr'. The curve e is an ellipse whose equa- tion is b'.ir + a'y' = k', a being the semi-major axis and 6 the semi-minor axis. The curve h is we have corresponding |/3] ± /2', ± f ye.
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- mg; Let r, = and i, = represent the equa- tions a-iX" + b^y- — Cj = and a.jX- -- b^y,- — c, = 0. Any values of x, y satisfying these two equa- tions will evidently satisfy the equation ( a,a^ -)- fell/' — Ci) — k (a^- -- bjf — cA = 0, fc being any constant. But this equation is z^ — fcaa = 0. Hence, if 3 = 0, e. = are the equa- tions of any two curves, any point common to the two satisfies the equation z^ — /,~2 = 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 (.t, y, z, w) , and in space of n dimensions (,r. _!/. ~ n quantities) , although we cannot draw the figures. The ellipse, hyperbola, a.nd 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 (1637). 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, barycentric, and trilinear coordinates. The most comprehensive English works are those by Salmon, Treatise on the Conic Sections (Dublin, 1869); Higher Plane Currcs (1873); Treatise on the Analytic Geometry of Three Dimensions (Dublin. 1874). Other noteworthy w"orks are: R. F. A. riebsch, Vorlcsiingen iibcr Geometric (Leipzig, 1876) ; M. Chasles, Traitd de geomitrie supirieure (Paris, 18S0) ; and among recent elementary works are those of Steiner, Briot, Bouquet, Townsend, and Scott. For a further discussion, see Geoiietrt and Coordinates.
ANALYTIC JUDGEMENT. In Kantian philosophy, a judgment in which the predicate is the definition (q.v.) or part of the definition of the subject. All other judgments are synthetic. The distinction between analytic and
