The special metric that arises on taking S to be a degenerate surface passes over into ordinary Euclidean geometry when the degen erate S is (as locus) the locus (counted twice) of the Desarguesian points of space and (as envelope) is the imaginary circle at infinity. See GEOMETRY, NON-EUCLIDEAN; GEOMETRY, PROJECTIVE.
Bolyai, 'The Absolute Sci ence of space' ; Cayley, 'Sixth Memoir on Quantities' ; Chasles, Apercu historique' (3d ed., Paris 1889); id., 'Rapport sur le progres de la geometric (ib. 1870) ; id., de geometric supereicure (ib. 1880) ; Clebsch, R. F. A., 'Vorlesungen fiber Geometric' (Leipzig 1876); Coolidge, J. L., 'Non-Euclidean Geom etry' (Oxford 1909) ; Descartes, 'Discours' (1637) ; Enestrom (editor), Bibliotheca Math ematica (Leipzig) ; Halsted, 'Bibliography' (Amer. Journal of Mathematics, Vol. 1) ; Helmholz, Weber die Thatsachcn die der Geo metric zum Grunde Liegen' • Klein, 'Zur Nicht Euklidische Geometric' (Afathematische An nalen, Vols. IV, VI, VII and XXXVII), and lithographed 'Vorlesungen iiber Nicht-Eukli dische Geometric' ; Laguerre, 'Nouvelles An nales de Mathematique' (1859); Lobachevski, 'Geometrical Researches on the Theory of Parallels' ; Mach, Ernst, 'Space and Geom etry in the Light of Physiological, Psychologi cal and Physical Inquiry' (1906) (this repre sents what may be called the biological as dis tinguished from the logical movement in mod ern mathematical thought, and his for its aim to trace pure geometry back and down by a legitimate genealogy to a physical and physio logical parentage, thus reattaching mathemat ics to experience and reality from which the logical movement has more and more detached it) ; 'Merriman and Woodward, 'Higher Mathe matics' (New York 18%) ; Poincare, 'La sci ence et l'hypothese' ; the same in English by Halsted; Poncelet, (Traite des proprietes pro jectivcs des figures' (Paris 1822) ; Riemann, 'Hypotheses that lie at the Foundation of Ge ometry' ; Scott, C. A., 'An Introductory Ac
count of Certain Modern Ideas and Methods in Plane Analytical Geometry' ; id., article in Bulletin of the American Mathematical Society (Vol. III) ; Smith, W. B., 'Introductory Mod ern Geometry of the Point, Ray and Circle' ; Woods, 'The Boston Colloquium.' Consult also 'Encyklopidie der mathematischen Wis senschaf ten' (Leipzig 1898—) ; International Catalogue of Scientific Literature: Mathemat ics (London 1902, and annually). Records of recent publications touching the subject will be found in Johrbuch fiber die Fortschritte der Mathematik (Berlin 1871 —).