GEOMETRY, NON-EUCLIDEAN. In the follow ing, acquaintance with the elements of ordinary (Cartesian) geometry will he presupposed.
One-dimensional Spaces: Range and Pencil; Elements at Infinity.— Any geometric entity in a given space may be taken as generat ing element of the space, which is then regarded as the assemblage of all the elements of the chosen kind. A space being assumed, its di mensionality depends upon the choice of gene rating element and is the number of independ ent parameters, or co-ordinates, necessary for the determination of the element as in some sense a continuous function of them. This is what is meant, to take the most familiar examples, by saying that any surface, say a plane, is two-dimensional, and that ordinary space is three-dimensional, in points. Any space being assumed, it is always possible to select as element an infinity of different kinds of entities for any one (kind) of which the space shall have prescribed dimensionality k. Thus the plane is two-dimensional in lines (see below), its dimensionality is 3 in circles, 4 in parabolas, 5 in conics, . . ., while the dimensionality of ordinary space is 3 in planes, 4 in lines or in spheres (see LINE GEOMETRY), 6 in circles, etc. A plane curve may in general be conceived either as a locus, assemblage of its points, or as an envelope, assemblage of its (tangent) lines. In either view the curve appears as a one-dimensional space, of points in the former view, of lines in the latter. Of such one-fold spaces, the simplest, and hence in a sense the most important, varieties are the range and the pencil, the former being the straight line regarded as the locus or assemblage of its points, and the later being the point regarded as the envelope or assem blage of its lines (the lines through it). Com monly the line is called the base of its range, and the point is called the vertix of its pencil. In passing it may be pointed out that if a pair or triplet, . . . of points (lines) be taken as element of the line (point), the line (point) appears as a space of 2, or 3, . .. dimensions in such pairs, triplets. . . .
Let V and b respectively be any pencil and range. The plane being supposed Euclidean in respect to parallels (see GEOMETRY, Non Eucunr.AN), V contains a single line parallel to b. Plainly, through any (finite) point of b there passes one and but one line of V ; and, conversely, every line of V, except the mentioned parallel, passes through a (finite) point of b. In order to avoid the exception and render the one-to-one correspondence complete, a convention is made, namely, that every range shall be regarded as having one and but one infinitely distant point P., called the infinite point of the range, and that the infinite point of any range is identical with that of any parallel range. Accordingly any infinite point of the plane is the vertex of a pencil of parallel lines, and the system of lines parallel to a given one constitute a pencil vertexed at m . The notion of parallel lines meeting at m had occurred to Kepler, but the systematic introduction of the convention was made by Gerard Desargues (1593-1662), chief among the founders of modern pure geometry. From that convention it readily
follows, by the theory of similar triangles, that the natural assumption concerning the (infinite) distances from any two finite points of a range to its infinite point is that they are equal. The locus of the infinite points of the plane is a straight line, called the infinite line of the plane. As for space, the locus of its infinite points is a plane. In general the locus of the infinite points in a point-space of n dimensions is a point-space of n —1 dimensions. If a range rotate (in a plane) about one of its finite points, every other point of the range will generate a circle; the path of the infinite point being a straight line, the latter appears as a circle of infinite radius; a perfectly natural phenomenon, for the curve ture, 1 :r, of a circle of radius r, vanishes for Non-homogeneous and Homogeneous Co ordinates of Point and Line of Range and In a range choose a point 0 for origin of distances. Denote by d the distance from 0 of an arbitrary point P of the range. Let x=pd, where the factor p may have any chosen value whatever. To any value of x there corresponds a position of P, and con versely. Hence x may serve as co-ordinate of the elements of the range. If a pencil be paired with a range as above, x will equally serve for co-ordinate of the lines of the pencil; or, in the latter case, d may be taken to repre sent the tangent of the angle made by a vary ing line of the pencil with a fixed line o, called origin of angles. Any point (line) of .a range (pencil) will be represented by a linear equa tion ax b =0, the co-ordinate of the element being — b :a. Conversely any element is defined by such an equation. In general n elements will give rise to an equation of nth degree in x, and any such equation will represent n elements. These (points or lines) will be real or imagi nary elements of the range or pencil according to the corresponding character of the roots of the equation. All the equations can be rendered homogeneous by replacing .r by the ratio xi:x2 and clearing of fractions. The quantities ox, and ax,, / being any chosen finite quantity called pro portionality 'factor, are described as homogene ous co-ordinates of the point (line) of the range (pencil). The position of the element depends on the 'ratio of the quantities, which is the same as the ratio of the x's, and the element is accordingly spoken of as the point or line (xi, x2). One obvious advantage of the homo geneity thus introduced lies in the artistic quality, notably the symmetry, which it lends to the analysis; for example, the equation of a point assumes the form coil aix2=0; in par ticular, the equations of the origin and P. are respectively xi= 0 and x,-= 0. Obvious analogous interpretations hold for the pencil. Indeed it is at once evident that the geometry of the range and that of the pencil are analyt ically one. The algebra remaining the same, either geometry passes over into the other on a mere exchange of notions: point (line) for line (point), pencil (range) for range (pencil).