LOCUS. This word, or the Greek Toros, signifying simply place, was used by the first geometers to denote a lino or surface over which a point may travel, so as always to be in a position which satisfies some given condition. Thus, suppose it required to find the position of a point at which a given line subtends a right angle : the answer is, that the number of such points is infinite ; for that any point whatso ever upon the surface of a sphere which has the given line for its diameter is such a point as was required to be found. This would be expressed as follows ;—the locus of the point at which a given line sub tends a right angle, is the sphere described on the given line as a diameter. If, however, the point were required to be in a given plane, its locus would no longer be the whole sphere, but only that circle which is the common section of the sphere and the given plane.
The following assertions are really nothing more than common pro positions of geometry, stated in such a manner as to introduce the term locus. (1.) The locus of the vertex of an isosceles triangle described upon a given base is the straight line which bisects the base at right angles. (2.) The locus of the vertex of a triangle which has a given base and a given area is a pair of straight lines parallel to, but on different sides of, the base. (3.) The locus of the vertex of a tri angle which has a given base and a given vertical angle, and which lies on a given side of the base, is an arc of a circle of which the given base is the chord ; and so on.
The geometrical analysis of the Greeks depended much upon the investigation of loci, and the method of using them will sufficiently appear by one instance. Suppose, for example, it is required to de
scribe a triangle of given area and given vertical angle upon a given base. Laying down the given base, it is easy to draw the parallel which is the containing line, or locus, of the vertices of all the triangles which have the given area ; and also, upon the same side, the arc of the circle which is the locus of the vertices of all the triangles having the given vertical angle. If then the parallel and the arc of the circle intersect, the point or points of intersection are obviously the vertices of triangles which satisfy all the required conditions; if they do not Intersect, the problem is impossible. When the locus of all the points satisfying a given condition cannot be ascertained by elementary geo metry, and when this locus is therefore taken for granted, we have the species of solution which was called mechanical. An instance of this will appear in the article TRISECTION OF TUE ANGLE.
It is to be understood that no curve whatever is called the locus of a point, unless any point whatsoever of that curve may be taken as the point in question. Thus, if each of six points should satisfy certain conditions, all lying upon a given circle, and if no other point of the circle should satisfy those conditions, that circle would not be called the locus of the points.