contains them around their intersection. The two lines here referred to are of course those which are drawn through the two circular points at infinity. This paradox is therefore only a degraded form of the property of the tangents to the fundamental conic.
It can also be readily shown that, if a plane receive two small rotations round two points, then the total rotation produced could have been produced by a single rotation about a certain point on the line joining the two points.
Let A, B be the two points and P the pole of the line AB, then a rotation round A will displace B along the line PB to an adjacent point B′. The rotation around B will displace A to A′ along the line PA; but, if A′B′ intersects AB in O, then a single rotation about O would have effected the required displacement of A and B, and therefore of the whole line. For, as the point O in the line AB could only move by displacement into the line A′B′, while it can also only move in the direction OP, it must obviously remain unaltered.
We are now in a position to inquire how the magnitude of an angle is to be expressed in the present system of measurement. Our definition of the magnitude of an angle must be made consistent with the supposition that when the angle is carried round by rotation about the vertex the magnitude shall remain unaltered. As anharmonic ratios are unaltered by the rotation, it follows that the anharmonic ratio of the pencil formed by the two legs of the angle and the two tangents to the fundamental conic must remain unaltered. Remembering that the tangents do not move by the rotation, it is natural to choose a function of this anharmonic ratio as the appropriate measure of an angle. The question still remains as to what function should be chosen. The student of ordinary geometry is doubtless aware that the angle between two lines multiplied into 2i is equal to the logarithm of the anharmonic ratio of the pencil formed by joining the intersection of the two lines to the two imaginary circular points at infinity. This consideration suggests that the angle between the straight lines in the generalized sense may be appropriately measured by the logarithm of the anharmonic ratio of the pencil formed by the two legs of the angle and the two tangents drawn from their point of intersection to the fundamental conic. There is also a convenience in assuming the angle to be actually equal to c times the logarithm of the anharmonic ratio, where c is the same constant as is employed in the expression of the distance. In this case the angle between two lines is by a well-known theorem equal to the distance between their poles. There is here an analogy to a well-known theorem in spherical geometry.
It will now be obvious that, however the angle be situated, its magnitude is unchanged by any displacement of the plane; for, as we have already seen that the displacement does not alter the distance between the poles of the two lines forming the angle, it follows that the magnitude of the angle itself is unaltered.
Just as in the measurement of distance we find a pair of fundamental points on each straight line, so in the measurement of angles we find a pair of fundamental rays in each plane pencil. These rays are the two tangents from the vertex of the pencil to the fundamental conic. In ordinary geometry the two fundamental points on each straight line coalesce into the single point at infinity; but it is exceedingly interesting to observe that even in ordinary geometry the two fundamental rays on each pencil do not coincide. It should also be observed that in the degraded circumstances of ordinary geometry it would be impracticable to employ the same constant c for the purpose of both linear and angular measurement.
It is easy to see that the definition of a right angle in the generalized sense is embodied in the statement that “if two corresponding legs of an harmonic pencil touch the fundamental conic then the two other legs are at right angles.” We also see that all the perpendiculars to a given line pass through a point, i.e., the pole of the given line; and from a given point a perpendicular can be drawn to a given line by joining the point to the pole of the line. The common perpendicular to two lines is obtained by joining their poles.
The student of modern geometry is already accustomed to think of parallel lines as lines which intersect at infinity, or as lines whose inclination is zero. In speaking of the generalized geometry in a plane, we may define that two straight lines which intersect upon the fundamental conic are parallel. It thus follows that through any point two distinct parallels can be drawn to a given straight line. The only exception will arise in the case where the given line touches the fundamental conic. This is precisely the case in which the generalized system of measurement degrades to the ordinary system. It will follow that in the present theory of measurement the three angles of a triangle are together not equal to two right angles. In fact, to take an extreme case, we may suppose the three vertices of the triangle to lie upon the fundamental conic. In this case each of the three angles, and therefore their sum, is equal to zero.
A sphere in the generalized system of measurement is the locus of a point which moves at a constant distance from a fixed point. It can therefore be easily shown that a sphere is a quadric which touches the fundamental quadric along its intersection with the polar plane of the centre of the sphere.
In discussing the general case of the displacement of a rigid system it will simplify matters to suppose that the fundamental quadric has real rectilinear generators. It must, however, be understood that the results are not on that account less general. A displacement must not alter the quadric, and must not deform a straight line. Hence it follows that the only effect of a displacement upon a generator of the fundamental quadric will be to convey it to a position previously occupied by a different generator. We shall further suppose that the displacement is such that the two generators to which we have referred belong to the same system. Let A, B, C, D be four generators of the first system which by displacement are brought to coincide with four other generators A′, B′, C′, D′. Let X be one generator of the second system which the displacement brings to X′. Since the anharmonic ratio of the four points in which four fixed generators of the one system are cut by any generator of the other system is constant, we must have, using an obvious notation for anharmonic ratio,
X(ABCD) = X′(ABCD);
but, since anharmonic ratios are unaltered by displacement, we have
X(ABCD) = X′(A′B′C′D′),
whence
X′(ABCD) = X′(A′B′C′D′).
It therefore follows that the anharmonic ratio in which four generators cut a fixed generator X′ is equal to the anharmonic ratio in which the four generators after displacement cut the same generator X′.
If P be a generator which passes through one of the double points on X′ determined by the two systems of points in which X′ is cut by the four generators before and after displacement, we must have
X(A, B, G, P) = X′(A′, B′, C′, P′);
hence we see that the generator P will be unaltered by displacement. Similar reasoning applies to the generator which passes through the other double point, and of course to a pair of generators of the second system, and hence we have the following remarkable theorem:—
In the most general displacement of a rigid system two generators of each of the systems on the fundamental quadric remain unaltered.
These four fixed generators are the edges of a tetrahedron. Denoting the four faces of this tetrahedron by the equations
x = 0, y = 0, z = 0, w = 0,
the equation of the fundamental quadric is
xz + h²yw = 0.
If the quadric be unaltered by the transformation
x′ = αx, y′ = βy, z′ = γz, w′ = δw,
then we must have
αγ = βδ
When this condition is satisfied, then, whatever h may be, every quadric of the family
xz + hyw = 0
will remain unaltered.
The family of quadrics here indicated are analogous to the right circular cylinders which have for a common axis the screw along which any displacement of a rigid body in ordinary space may bo effected.
The two lines
x = 0, z = 0
and
y = 0, w = 0
are conjugate polars with respect to the fundamental quadric, and both these lines are unaltered by the displacement. Hence we see that in any displacement of a rigid system there are two right lines which remain unaltered, and these lines are conjugate polars with respect to the fundamental quadric.
Since the pole of a plane through one of these lines lies on the other line, it appears that a rotation of a rigid system about a straight line is identical with a translation of the system along its conjugate polar.
Clifford has pointed out the real nature of the lines which are to be called parallel in the generalized system of measurement. We have explained that in the plane two parallel lines intersect upon the fundamental conic; in a certain sense also we may consider two lines in space of three dimensions to be parallel whenever they intersect upon the fundamental quadric. This is the view of parallel lines to which we are conducted by simply generalizing the property that two parallel lines intersect at infinity. But we can take a different definition of two parallel lines. Let us, for example, call two lines parallel when they admit of an indefinitely large number of common perpendiculars. It is exceedingly interesting
