If the planes intersect, the ideal line is termed proper, otherwise
it is improper. It can be proved that any two planes, with which
any two of the ideal points are both coherent, will serve as the
guiding planes used in the definition. The ideal planes are
defined as in projective geometry, and all the other definitions
(for segments, order, &c.) of projective geometry are applied
to the ideal elements. If an ideal plane contains some proper
ideal points, it is called proper, otherwise it is improper. Every
ideal plane contains some improper ideal points.
It can now be proved that all the axioms of projective geometry hold of the ideal elements as thus obtained; and also that the order of the ideal points as obtained by the projective method agrees with the order of the proper ideal points as obtained from that of the associated points of the descriptive geometry. Thus a projective space has been constructed out of the ideal elements, and the proper ideal elements correspond element by element with the associated descriptive elements. Thus the proper ideal elements form a region in the projective space within which the descriptive axioms hold. Accordingly, by substituting ideal elements, a descriptive space can always be considered as a region within a projective space. This is the justification for the ordinary use of the “points at infinity” in the ordinary Euclidean geometry; the reasoning has been transferred from the original descriptive space to the associated projective space of ideal elements; and with the Euclidean parallel axiom the improper ideal elements reduce to the ideal points on a single improper ideal plane, namely, the plane at infinity.[1]
Congruence and Measurement.—The property of physical space which is expressed by the term “measurability” has now to be considered. This property has often been considered as essential to the very idea of space. For example, Kant writes,[2] “Space is represented as an infinite given quantity.” This quantitative aspect of space arises from the measurability of distances, of angles, of surfaces and of volumes. These four types of quantity depend upon the two first among them as fundamental. The measurability of space is essentially connected with the idea of congruence, of which the simplest examples are to be found in the proofs of equality by the method of superposition, as used in elementary plane geometry. The mere concepts of “part” and of “whole” must of necessity be inadequate as the foundation of measurement, since we require the comparison as to quantity of regions of space which have no portions in common. The idea of congruence, as exemplified by the method of superposition in geometrical reasoning, appears to be founded upon that of the “rigid body,” which moves from one position to another with its internal spatial relations unchanged. But unless there is a previous concept of the metrical relations between the parts of the body, there can be no basis from which to deduce that they are unchanged.
It would therefore appear as if the idea of the congruence, or metrical equality, of two portions of space (as empirically suggested by the motion of rigid bodies) must be considered as a fundamental idea incapable of definition in terms of those geometrical concepts which have already been enumerated. This was in effect the point of view of Pasch.[3] It has, however, been proved by Sophus Lie[4] that congruence is capable of definition without recourse to a new fundamental idea. This he does by means of his theory of finite continuous groups (see Groups, Theory of), of which the definition is possible in terms of our established geometrical ideas, remembering that coordinates have already been introduced. The displacement of a rigid body is simply a mode of defining to the senses a one-one transformation of all space into itself. For at any point of space a particle may be conceived to be placed, and to be rigidly connected with the rigid body; and thus there is a definite correspondence of any point of space with the new point occupied by the associated particle after displacement. Again two successive displacements of a rigid body from position A to position B, and from position B to position C, are the same in effect as one displacement from A to C. But this is the characteristic “group” property. Thus the transformations of space into itself defined by displacements of rigid bodies form a group.
Call this group of transformations a congruence-group. Now according to Lie a congruence-group is defined by the following characteristics:—
1. A congruence-group is a finite continuous group of one-one transformations, containing the identical transformation.
2. It is a sub-group of the general projective group, i.e. of the group of which any transformation converts planes into planes, and straight lines into straight lines.
3. An infinitesimal transformation can always be found satisfying the condition that, at least throughout a certain enclosed region, any definite line and any definite point on the line are latent, i.e. correspond to themselves.
4. No infinitesimal transformation of the group exists, such that, at least in the region for which (3) holds, a straight line, a point on it, and a plane through it, shall all be latent.
The property enunciated by conditions (3) and (4), taken together, is named by Lie “Free mobility in the infinitesimal.” Lie proves the following theorems for a projective space:—
1. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (α) must be the region enclosed by a real closed quadric, or (β) must be the whole of the projective space with the exception of a single plane. In case (α) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (β) an imaginary conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent.
2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent.
By a proper choice of non-homogeneous co-ordinates the equation of any quadrics of the types considered, either in theorem 1 (α), or in theorem 2, can be written in the form 1 + c(x2 + y2 + z2) = 0, where c is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations:
| dx/dt = u − ω3y + ω2z + cx (ux + vy + wz), | (A) |
| dy/dt = v − ω1z + ω3x + cy (ux + vy + wz), | |
| dz/dt = w − ω2x + ω1y + cz (ux + vy + wz). |
In the ease considered in theorem 1 (β), with the proper choice of co-ordinates the three equations defining the general infinitesimal transformation are:
| dx/dt = u − ω3y + ω2z, | (B) |
| dy/dt = v − ω1z + ω3x, | |
| dz/dt = w − ω2x + ω1y. |
In this case the latent plane is the plane for which at least one of x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and the latent conic is the conic in which the cone x2 + y2 + z2 = 0 intersects the latent plane.
It follows from theorems 1 and 2 that there is not one unique congruence-group, but an indefinite number of them. There is one congruence-group corresponding to each closed real quadric, one to each imaginary quadric with a real equation, and one to each imaginary conic in a real plane and with a real equation. The quadric thus associated with each congruence-group is called the absolute for that group, and in the degenerate case of 1 (β) the absolute is the latent plane together with the latent imaginary conic. If the absolute is real, the congruence-group is hyperbolic; if imaginary, it is elliptic; if the absolute is a plane and imaginary conic, the group is parabolic. Metrical geometry is simply the theory of the properties of some particular congruence-group selected for study.
The definition of distance is connected with the corresponding congruence-group by two considerations in respect to a range of five points (A1, A2, P1, P2, P3), of which A1 and A2 are on the absolute.
Let {A1P1A2P2} stand for the cross ratio (as defined above) of the range (A1P1A2P2), with a similar notation for the other ranges. Then
and
(2), if the points A1, A2, P1, P2 are transformed into A′1, A′2, P′1, P′2 by any transformation of the congruence-group, (α) {A1P1A2P2} = {A′1P′1A′2P′2}, since the transformation is projective, and (β) A′1, A′2 are on the absolute since A1 and A2 are on it. Thus if we define
