792] 
585 
792. 
LOCUS. 
[From the Encyclopaedia Britannica, Ninth Edition, voi. xiv. (1882), pp. 764, 765.] 
Locus, in Greek tottos, a geometrical term, the invention of the notion of which 
is attributed to Plato. It occurs in such statements as these:—the locus of the points 
which are at the same distance from a fixed point, or of a point which moves so as 
to be always at the same distance from a fixed point, is a circle ; conversely a circle 
is the locus of the points at the same distance from a fixed point, or of a point 
moving so as to be always at the same distance from a fixed point ; and so, in general, 
a curve of any given kind is the locus of the points which satisfy, or of a point 
moving so as always to satisfy, a given condition. The theory of loci is thus identical 
with that of curves ; and it is in fact in this very point of view that a curve is 
considered in the article Curve, [785]; see that article, and also Geometry (Analytical), 
[790]. It is only necessary to add that the notion of a locus is useful as regards deter 
minate problems or theorems : thus, to find the centre of the circle circumscribed about 
a given triangle ABC, we see that the circumscribed circle must pass through the two 
vertices A, B, and the locus of the centres of the circles which pass through these two 
points is the straight line at right angles to the side AB at its mid-point ; similarly the 
circumscribed circle must pass through A, C, and the locus of the centres of the circles 
through these two points is the line at right angles to the side AG at its mid-point; 
thus we get the ordinary construction, and also the theorem that the lines at right 
angles to the sides, at their mid-points respectively, meet in a point. The notion of 
a locus applies, of course, not only to plane but also to solid geometry. Here the locus 
of the points satisfying a single (or onefold) condition is a surface ; the locus of the 
points satisfying two conditions (or a twofold condition) is a curve in space, which is 
in general a twisted curve or curve of double curvature. 
C. XI. 
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