Suppose we are situated at some point P of I. Suppose further that we are capable of only such motions as are furnished by displace ments, transformations, that is, that leave the absolute unchanged. Just as in ordinary ge ometry, so here, since the total angle about P is finite, a finite time would suffice to make a complete rotation about P by turning at any finite rate, however small. Again, just as in ordinary geometry, so here we could travel on any straight 'line in either direction (sense) at any finite velocity however great, without, in any finite time, however great, reaching, much less passing, the absolute conic C. The dwellers within C could not escape from the region I by any combination of displacements. For those inhabitants their plane, the region I, is strictly limited by the absolute. By no ex perience possible to them could they assure themselves that C existed, or, C being granted, that there is a region on beyond, outside of C. If we set el = 4, then the geometry within I becomes a detailed picture of the Lobachev skian, or hyperbolic, geometry so far as con cerns the plane. For example, one of the L. propositions is that through any point of the plane two lines can be drawn parallel to a given line. This proposition is matched or pictured by the fact that through any point of I two and but two lines can be drawn meeting any line within C where the line cuts C. These two lines are parallel to the third line, meeting it at its infinite points. Another L. proposition is that as a point recedes from a line, the angle of the two parallels through the point to the line increases and approaches the value r as the distance becomes infinite. The like holds within C, for when the point common to the parallels is a point of C, the parallels to the given line become parallel to each other, one of their angles being zero and the other rt. Again, the sum of the angle of a triangle within C is less than it and decreases as the triangle increases, becoming zero when the vertices are infinitely apart, i.e., are on C: a fact agreeing with the L. geometry. Once more, in the L. geometry, two Imes perpendicu lar to a third do not intersect. Now two lines within C, to be perpendicular to a third line within C, must belong to the pencil vertexed at the pole of the third line. But this pole is outside of C. Hence the perpendicular, as re garded by geometers within C, do not inter sect, for, as we have seen, the region E does not exist for such geometers. They might in deed posit such a region but then it would have only ideal as distinguished from intuition al or experiential existence. Still again, in the L. geometry we have the proposition that a circle of infinite radius is not a straight line. Now, in theory of the region I, any circle of infinite radius and infinite centre is, as before shown, the absolute itself and so not a straight line. If the radius be kept co and the centre be taken on C, then the circle is again not a straight line but a conic having 4-point contact with C. Is it, then, impossible to conceive a straight line of the region I as a circle? It is, unless the line be regarded as a circle having i its centre in E, at the ideal pole of the given line. In such case the radius would be imagi nary.
Special Metric in the To obtain this we take for absolute C a degenerate conic which regarded as a locus is a pair of coinci dent straight lines and which regarded as an envelope is a pair of points (pencils) of the line. It is easy to see how a proper conic can degenerate into a conic of the sort described. Thus let a' y' = be an ellipse.
Call its vertices A (a, o) and A' (—a, o).For
all values of in the line y= mx V b' is tangent to the ellipse. Now let b ap proach zero. The ellipse degenerates (as a locus) into the double line y =0, and (as an envelope) into the points A and A' enveloped by y= m (x ± a) as m ranges through the ensemble of real numbers. Such a degraded Conic being taken for absolute C, every pencil of the plane will contain two distinct absolute lines, or lines at 00, viz., those that pass through A and A'. These absolute lines will be real or imaginary according as A and A' are real or imaginary. Accordingly the special metric so far as angles are concerned is analogous to the general metric. Not so, however, in case of distances. For the absolute points of any range are coincident, namely, the two points common to the range and the double line AA'. To render the distance between two points non-zero, it is necessary to take c infinite, and then, in agreement with ordinary (parabolic) geometry, the distance becomes an algebraic instead of a transcendental function of the co-ordinates. The distance between two points of a range containing A or A' is zero, for such a range is tangent to the absolute. Similarly the angle of any two lines intersecting on AA' is zero: such lines are parallel. Every conic through A and A' is to be called a circle, and two circles tangent to each other at A and A' are to be regarded as concentric circles: In every system of concentric circles there is one of infinite radius, namely, the double line AA', which is, accordingly, the locus of points at co. Here, as in ordinary geometry, the principle of duality fails. For example, the envelope of a line making a constant angle with a given line is not a proper conic but a point.
The special (parabolic) geometry here indi cated may be derived from ordinary projective geometry, when the double line AA becomes the locus (counted twice) of the Desarguesian points of the plane and the points A and A' be come the circular points at co.
Generalized Metric in absolute will naturally be a surface S of second order (and class). The distance between two points is an arbitrary constant c times the logarithm of the anharmonic ratio of the given pair of points and the pair of (absolute) points cut from S by the line joining the given points. Reciprocally, the angle of two planes is an arbitrarily chosen constant c' times the anhar monic ratio of the two planes to the pair of (absolute) planes, i.e., the two planes contain ing the line common to the given planes and tangent to S.
There are co' collineations of space that con vert S into itself. These fall into two classes according as they do not or do exchange the two systems of generatices of S. The colline ations of the first class — called displacements — constitute a closed system, or group. The second class is not a group.
The locus of a point at a given distance from a fixed point is a second order surface — gener alized sphere—tangent to S along the conic cut from S by the polar plane of the given point, the sphere centre. S itself is a sphere, viz., the locus of all points at ao , of all points, that is, that, in the metric system in hand, are infinitely distant from any point not on S.
If S is imaginary, all lines are of finite length. The sum of the angles of an axial pen cil of planes is finite. This general theory passes into the Elliptic geometry if we set = i / 2, the sum of the angles of au axial pen cil being then 7.
If S is real (and not ruled), the geometry of the interior of S includes the Hyperbolic geometry as a special case, namely, when c i/ 2.