thus, for example, the normals to a surface are bitangents of the
surface of centres, and in the case of Dupin’s cyclide this surface
degenerates into two conics.
In the discussion of congruences it soon becomes necessary to introduce another number r, called the rank, which expresses the number of plane pencils each of which contains an arbitrary line and two lines of the congruence. The order of the focal surface is 2m(n − 1) − 2r, and its class is m(m − 1) − 2r. Our knowledge of congruences is almost exclusively confined to those in which either m or n does not exceed two. We give a brief account of those of the second order without singular lines, those of order unity not being especially interesting. A congruence generally has singular points through which an infinite number of lines pass; a singular point is said to be of order r when the lines through it lie on a cone of the rth degree. By means of formulae connecting the number of singular points and their orders with the class m of quadratic congruence Kümmer proved that the class cannot exceed seven. The focal surface is of degree four and class 2m; this kind of quartic surface has been extensively studied by Kümmer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at least one Reye complex; and so also does the most important class of (2, 6) congruences which includes all the above as special cases. The congruence (2, 2) belongs to a linear complex and forty different Reye complexes; as above remarked, the singular surface is Kümmer’s sixteen-nodal quartic, and the same surface is focal for six different congruences of this variety. The theory of (2, 2) congruences is completely analogous to that of the surfaces called cyclides in three dimensions. Further particulars regarding quadratic congruences will be found in Kümmer’s memoir of 1866, and the second volume of Sturm’s treatise. The properties of quadratic congruences having singular lines, i.e. degenerate focal surfaces, are not so interesting as those of the above class; they have been discussed by Kümmer, Sturm and others.
Since a ruled surface contains only ∞¹ elements, this theory is practically the same as that of curves. If a linear complex contains more than n generators of a ruled surface of the nth degree, it contains all the generators, hence for n = 2 there are Ruled surfaces. three linearly independent complexes, containing all the generators, and this is a well-known property of quadric surfaces. In ruled cubics the generators all meet two lines which may or may not coincide; these two cases correspond to the two main classes of cubics discussed by Cayley and Cremona. As regards ruled quartics, the generators must lie in one and may lie in two linear complexes. The first class is equivalent to a quartic in four dimensions and is always rational, but the latter class has to be subdivided into the elliptic and the rational, just like twisted quartic curves. A quintic skew may not lie in a linear complex, and then it is unicursal, while of sextics we have two classes not in a linear complex, viz. the elliptic variety, having thirty-six places where a linear complex contains six consecutive generators, and the rational, having six such places.
The general theory of skews in two linear complexes is identical with that of curves on a quadric in three dimensions and is known. But for skews lying in only one linear complex there are difficulties; the curve now lies in four dimensions, and we represent it in three by stereographic projection as a curve meeting a given plane in n points on a conic. To find the maximum deficiency for a given degree would probably be difficult, but as far as degree eight the space-curve theory of Halphen and Nöther can be translated into line geometry at once. When the skew does not lie in a linear complex at all the theory is more difficult still, and the general theory clearly cannot advance until further progress is made in the study of twisted curves.
References.—The earliest works of a general nature are Plücker, Neue Geometrie des Raumes (Leipzig, 1868); and Kümmer, “Über die algebraischen Strahlensysteme,” Berlin Academy (1866). Systematic development on purely synthetic lines will be found in the three volumes of Sturm, Liniengeometrie (Leipzig, 1892, 1893, 1896); vol. i. deals with the linear and Reye complexes, vols. ii. and iii. with quadratic congruences and complexes respectively. For a highly suggestive review by Gino Loria see Bulletin des sciences mathématiques (1893, 1897). A shorter treatise, giving a very interesting account of Klein’s coordinates, is the work of Koenigs, La Géométrie réglée et ses applications (Paris, 1898). English treatises are C. M. Jessop, Treatise on the Line Complex (1903); R. W. H. T. Hudson, Kümmer’s Quartic (1905). Many references to memoirs on line geometry will be found in Hagen, Synopsis der höheren Mathematik, ii. (Berlin, 1894); Loria, Il passato ed il presente delle principali teorie geometriche (Milan, 1897); a clear résumé of the principal results is contained in the very elegant volume of Pascal, Repertorio di mathematiche superiori, ii. (Milan, 1900). Another treatise dealing extensively with line geometry is Lie, Geometrie der Berührungstransformationen (Leipzig, 1896). Many memoirs on the subject have appeared in the Mathematische Annalen; a full list of these will be found in the index to the first fifty volumes, p. 115. Perhaps the two memoirs which have left most impression on the subsequent development of the subject are Klein, “Zur Theorie der Liniencomplexe des ersten und zweiten Grades,” Math. Ann. ii.; and Lie, “Über Complexe, insbesondere Linien- und Kugelcomplexe,” Math. Ann. v. (J. H. Gr.)
VI. Non-Euclidean Geometry
The various metrical geometries are concerned with the properties of the various types of congruence-groups, which are defined in the study of the axioms of geometry and of their immediate consequences. But this point of view of the subject is the outcome of recent research, and historically the subject has a different origin. Non-Euclidean geometry arose from the discussion, extending from the Greek period to the present day, of the various assumptions which are implicit in the traditional Euclidean system of geometry. In the course of these investigations it became evident that metrical geometries, each internally consistent but inconsistent in many respects with each other and with the Euclidean system, could be developed. A short historical sketch will explain this origin of the subject, and describe the famous and interesting progress of thought on the subject. But previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given, assuming the standpoint arrived at below under VII. Axioms of Geometry.
First assume the equation to the absolute (cf. loc. cit.) to be w2 − x2 − y2 − z2 = 0. The absolute is then real, and the geometry is hyberbolic.
The distance (d12) between the two points (x1, y1, z1, w1) and (x2, y2, z2, w2) is given by
| cosh (d12/γ) = (w1w2 − x1x2 − y1y2 − z1z2) / {(w12 − x12 − y12 − z12)
(w22 − x22 − y22 − z22)}12 |
(1) |
The only points to which the metrical geometry applies are those within the region enclosed by the quadric; the other points are “improper ideal points.” The angle (θ12) between two planes, l1x + m1y + n1z + r1w = 0 and l2x + m2y + n2z + r2w = 0, is given by
| cos θ12 = (l1l2 + m1m2 + n1n2 − r1r2) / {(l12 + m12 + n12 − r12)
(l22 + m22 + n22 − r22)}12 |
(2) |
These planes only have a real angle of inclination if they possess a line of intersection within the actual space, i.e. if they intersect. Planes which do not intersect possess a shortest distance along a line which is perpendicular to both of them. If this shortest distance is δ12, we have
| cosh (δ12/γ) = (l1l2 + m1m2 + n1n2 − r1r2) / {(l12 + m12 + n12 − r12)
(l22 + m22 + n22 − r22)}12 |
(3) |
 |
| Fig. 67. |
Thus in the case of the two planes one and only one of the two, θ12 and δ12, is real. The same considerations hold for coplanar straight lines (see VII. Axioms of Geometry). Let O (fig. 67) be the point (0, 0, 0, 1), OX the line y = 0, z = 0, OY the line z = 0, x = 0, and OZ the line x = 0, y = 0. These are the coordinate axes and are at right angles to each other. Let P be any point, and let ρ be the distance OP, θ the angle POZ, and φ the angle between the planes ZOX and ZOP. Then the coordinates of P can be taken to be
sin φ, sinh (ρ/γ) cos θ, cosh (ρ/γ).
If ABC is a triangle, and the sides and angles are named according to the usual convention, we have
| sinh (a/γ) / sin A = sinh (b/γ) / sin B = sinh (c/γ) / sin C, | (4) |
and also
| cosh (a/γ) = cosh (b/γ) cosh (c/γ) − sinh (b/γ) sinh (c/γ) cos A, | (5) |
 |
| Fig. 68. |
with two similar equations. The sum of the three angles of a triangle is always less than two right angles. The area of the triangle ABC is λ2(π − A − B − C). If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line. The whole theory of similarity is inapplicable; two triangles are either congruent, or their angles are not equal two by two. Thus the elements of a triangle are determined when its three angles are given. By keeping A and B and the line BC fixed, but by making C move off to infinity along BC, the lines BC and AC become parallel, and the sides a and b become infinite. Hence from equation (5) above, it follows that two parallel lines (cf. Section VII. Axioms of Geometry) must be considered as making a zero angle with each other. Also if B be a right angle, from the equation (5), remembering that, in the limit,
