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)}1/2
(1)