LOCUS (Lat.. place). The place of all points satisfying a given condition. For example, the circumference of a circle whose centre is 0 and whose radius is r is the locus of all points satis fying the condition of being in the same plane and at a distance r from a fixed point tf in that plane. In space, the locus is the surface of a sphere of centre 0 and radius r. In proving a theorem concerning the locus of points, it is necessary and sufficient to prove two things: that any point on the supposed locus satisfies the given condition, and that any point not on the supposed locus does not satisfy the condition. A few important propositions of loci are: (1) The locus of points equidistant from two given is the perpendicular bisector of the line joining them; (2) the locus of points equidistant from two given lines consists of the bisectors of their included angles; (3) the locus of points equidis tant from three given points is the perpendicular to their plane passing through the centre of the circle on which they lie; (4) the locus of points the sum of whose distances from two fixed points is constant is an ellipse whose foci are the given points; (5) the locus of the vertices of constant angles subtended by a given line-segment is an arc of which that segment is the cord; (6) if a pair of variable quantities. x. y, are connected by
an equation, and each pair of values is repre sented by a point in a plane. these points will he on a definite curve called the locus or graph of the equation. (See COoRDINATES.) Thus the study of curves by analytic methods is a develop ment of the theory of loci. (See Ccave.s.) The method of loci in pure geometry was extensively used by the Greek geometers, particularly Apol lonius, Hippias. Eudoxus, Nicomedes. and Di oeles. The methods of Apollonius were restored by Robert Simson in his De Laois Planis (174g). but as an analytic instrument the theory of loci dates from Descartes (1637).