Page:EB1911 - Volume 11.djvu/745

The angle A is called by N. I. Lobatchewsky the “angle of parallelism.”

The whole theory of lines and planes at right angles to each other is simply the theory of conjugate elements with respect to the absolute, where ideal lines and planes are introduced.

Thus if l and l′ be any two conjugate lines with respect to the absolute (of which one of the two must be improper, say l′), then any plane through l′ and containing proper points is perpendicular to l. Also if p is any plane containing proper points, and P is its pole, which is necessarily improper, then the lines through P are the normals to P. The equation of the sphere, centre (x₁, y₁, z₁, w₁) and radius ρ, is

$(w_{1}^{2}-x_{1}^{2}-y_{1}^{2}-z_{1}^{2})(w^{2}-x^{2}-y^{2}-z^{2})\cosh ^{2}(\rho /\gamma ){=}(w_{1}w-x_{1}x-y_{1}y-z_{1}z)^{2}$.

(7).

The equation of the surface of equal distance (σ) from the plane lx + my + nz + rw = 0 is

(l2 + m2 + n2 − r2) (w2 − x2 − y2 − z2) sinh2 (σ/γ) = (rw + lx + my + nz)2

(8).

A surface of equal distance is a sphere whose centre is improper; and both types of surface are included in the family

k2 (w2 − x2 − y2 − z2) = (ax + by + cz + dw)2

(9).

But this family also includes a third type of surfaces, which can be looked on either as the limits of spheres whose centres have approached the absolute, or as the limits of surfaces of equal distance whose central planes have approached a position tangential to the absolute. These surfaces are called limit-surfaces. Thus (9) denotes a limit-surface, if d² − a² − b² − c² = 0. Two limit-surfaces only differ in position. Thus the two limit-surfaces which touch the plane YOZ at O, but have their concavities turned in opposite directions, have as their equations

w² − x² − y² − z² = (w ± x)².

The geodesic geometry of a sphere is elliptic, that of a surface of equal distance is hyperbolic, and that of a limit-surface is parabolic (i.e. Euclidean). The equation of the surface (cylinder) of equal distance (δ) from the line OX is

(w² − x²) tanh² (δ/γ) − y² − z² = 0.

This is not a ruled surface. Hence in this geometry it is not possible for two straight lines to be at a constant distance from each other.

Secondly, let the equation of the absolute be x² + y² + z² + w² = 0. The absolute is now imaginary and the geometry is elliptic.

The distance (d₁₂) between the two points (x₁, y₁, z₁, w₁) and (x₂, y₂, z₂, w₂) is given by

cos (d12/γ) = ± (x1x2 + y1y2 + z1z2 + w1w2) / {(x12 + y12 + z12 + w12) (x22 + y22 + z22 + w22)}1/2

(10).

Thus there are two distances between the points, and if one is d₁₂, the other is πγ-d₁₂. Every straight line returns into itself, forming a closed series. Thus there are two segments between any two points, together forming the whole line which contains them; one distance is associated with one segment, and the other distance with the other segment. The complete length of every straight line is πγ.

The angle between the two planes l₁x + m₁y + n₁z + r + ₁w = 0 and l₂x + m₂y + n₂z + r₂w = 0 is

cos θ12 = (l1l2 + m1m2 + n1n2 + r1r2) / {(l12 + m12 + n12 +r12) (l22 + m22 + n22 + r22)}1/2

(11).

The polar plane with respect to the absolute of the point (x₁, y₁, z₁, w₁) is the real plane x₁x + y₁y + z₁z + w₁w = 0, and the pole of the plane l₁x + m₁y + n₁z + r₁w = 0 is the point (l₁, m₁, n₁, r₁). Thus (from equations 10 and 11) it follows that the angle between the polar planes of the points (x₁, ...) and (x₂, ...) is d₁₂/γ, and that the distance between the poles of the planes (l₁, ...) and (l₂, ...) is γθ₁₂. Thus there is complete reciprocity between points and planes in respect to all properties. This complete reign of the principle of duality is one of the great beauties of this geometry. The theory of lines and planes at right angles is simply the theory of conjugate elements with respect to the absolute. A tetrahedron self-conjugate with respect to the absolute has all its intersecting elements (edges and planes) at right angles. If l and l′ are two conjugate lines, the planes through one are the planes perpendicular to the other. If P is the pole of the plane p, the lines through P are the normals to the plane p. The distance from P to p is 1/2πγ. Thus every sphere is also a surface of equal distance from the polar of its centre, and conversely. A plane does not divide space; for the line joining any two points P and Q only cuts the plane once, in L say, then it is always possible to go from P to Q by the segment of the line PQ which does not contain L. But P and Q may be said to be separated by a plane p, if the point in which PQ cuts p lies on the shortest segment between P and Q. With this sense of “separation,” it is possible^[1] to find three points P, Q, R such that P and Q are separated by the plane p, but P and R are not separated by p, nor are Q and R.

Let A, B, C be any three non-collinear points, then four triangles are defined by these points. Thus if a, b, c and A, B, C are the elements of any one triangle, then the four triangles have as their elements:

(1)	a ,	b,	c,	A,	B,	C.
(2)	a,	πγ − b,	πγ − c,	A,	π − B,	π − C.
(3)	πγ − a,	b,	πγ − c,	π − A,	B,	π − C.
(4)	πγ − a,	πγ − b,	c,	π − A,	π − B,	C.

The formulae connecting the elements are

sin A/sin (a/γ) = sin B/sin (b/γ) = sin C/sin (c/γ),

(12).

and

cos (a/γ) = cos (b/γ) cos (c/γ) + sin (b/γ) sin (c/γ) cos A,

(13).

with two similar equations.

Two cases arise, namely (I.) according as one of the four triangles has as its sides the shortest segments between the angular points, or (II.) according as this is not the case. When case I. holds there is said to be a “principal triangle.”^[2] If all the figures considered lie within a sphere of radius 1/4πγ only case I. can hold, and the principal triangle is the triangle wholly within this sphere, also the peculiarities in respect to the separation of points by a plane cannot then arise. The sum of the three angles of a triangle ABC is always greater than two right angles, and the area of the triangle is γ²(A + B + C − π). Thus as in hyperbolic geometry the theory of similarity does not hold, and the elements of a triangle are determined when its three angles are given. The coordinates of a point can be written in the form

sin (ρ/γ) sin Φ cos φ, sin (ρ/γ) sin Φ sin φ, sin (ρ/γ) cos Φ, cos (ρ/γ),

where ρ, Φ and φ have the same meanings as in the corresponding formulae in hyperbolic geometry. Again, suppose a watch is laid on the plane OXY, face upwards with its centre at O, and the line 12 to 6 (as marked on dial) along the line YOY. Let the watch be continually pushed along the plane along the line OX, that is, in the direction 9 to 3. Then the line XOX being of finite length, the watch will return to O, but at its first return it will be found to be face downwards on the other side of the plane, with the line 12 to 6 reversed in direction along the line YOY. This peculiarity was first pointed out by Felix Klein. The theory of parallels as it exists in hyperbolic space has no application in elliptic geometry. But another property of Euclidean parallel lines holds in elliptic geometry, and by the use of it parallel lines are defined. For the equation of the surface (cylinder) of equal distance (δ) from the line XOX is

(x² + w²) tan² (δ/γ) − (y² + z²) = 0.

This is also the surface of equal distance, 1/2πγ−δ, from the line conjugate to XOX. Now from the form of the above equation this is a ruled surface, and through every point of it two generators pass. But these generators are lines of equal distance from XOX. Thus throughout every point of space two lines can be drawn which are lines of equal distance from a given line l. This property was discovered by W. K. Clifford. The two lines are called Clifford’s right and left parallels to l through the point. This property of parallelism is reciprocal, so that if m is a left parallel to l, then l is a left parallel to m. Note also that two parallel lines l and m are not coplanar. Many of those properties of Euclidean parallels, which do not hold for Lobatchewsky’s parallels in hyperbolic geometry, do hold for Clifford’s parallels in elliptic geometry. The geodesic geometry of spheres is elliptic, the geodesic geometry of surfaces of equal distance from lines (cylinders) is Euclidean, and surfaces of revolution can be found^[3] of which the geodesic geometry is hyperbolic. But it is to be noticed that the connectivity of these surfaces is different to that of a Euclidean plane. For instance there are only ∞² congruence transformations of the cylindrical surfaces of equal distance into themselves, instead of the ∞³ for the ordinary plane. It would obviously be possible to state “axioms” which these geodesics satisfy, and thus to define independently, and not as loci, quasi-spaces of these peculiar types. The existence of such Euclidean quasi-geometries was first pointed out by Clifford.^[4]

In both elliptic and hyperbolic geometry the spherical geometry, i.e. the relations between the angles formed by lines and planes passing through the same point, is the same as the “spherical trigonometry” in Euclidean geometry. The constant γ, which appears in the formulae both of hyperbolic and elliptic geometry, does not by its variation produce different types of geometry. There is only one type of elliptic geometry and one type of hyperbolic geometry; and the magnitude of the constant γ in each case simply depends upon the magnitude of the arbitrary unit of length in comparison with the natural unit of length

1. Cf. A. N. Whitehead, Universal Algebra, Bk. vi. (Cambridge, 1898).
2. Cf. A. N. Whitehead, loc. cit.
3. Cf. A. N. Whitehead, “The Geodesic Geometry of Surfaces in non-Euclidean Space,” Proc. Lond. Math. Soc. vol. xxix.
4. Cf. Klein, “Zur nicht-Euklidischen Geometrie,” Math. Annal. vol. xxxvii.