244 G. H. KNIBBS. 



24. Is elliptic and hyperbolic space of n-dimensions representable 



in space of (n + 1) dimensions ? 



25. Elliptic and hyperbolic space merely a specialised region in a 



homaloid. 



26. Space of n-dimensions as the boundary of (n+ 1) dimensional space. 



27. Relativity of geometrical forms and figures. 



28. Complex generation of geometrical figures of uniform intensity. 



29. Involutional, evolutional, pedal, modular and umbilical generation. 



30. Generation of figures of non-uniform intensity. 



31. The total intensity-volume of seolotropic space. 



32. Conversion of seolotropic into isotropic space. 



33. Conformal representation of functional dependence. 



34. Eiemann surfaces. 



35. Connectivity of space. 



36. Conception of n-ply extended magnitudes. 



37. Illimitability of operative schemes for generation of geometrical 



figures. 



38. Pangeometry. 



0. Introductory. — The discussion of the elementary principles 

 of geometry, has of late years received so much attention, in con- 

 nection with what has been called non-euclidean geometry, and 

 with the theory of space generally, 1 that further contribution 

 thereto might be thought supererogatory. It has been imagined 

 that the spatial interpretation of the "purely abstract methods of 

 analytical geometry" 2 require a more generalised theory of space 

 than that which is implicitly contained in euclidean geometry, 

 and in this more generalised theory, space is not only to be con- 



1 The following are some of the leading contributors to non-euclidean 

 geometry: — Lambert 1786; Lobatchewsky 1826 ; Gauss '28; J. Bolyai '32; 

 Perronet Thompson '33; Jacobi '34; Sohncke '37; Grassmann '44; Cayley 

 '45; Grassmann '47; Eiemann '54; Cayley, Hoffmann '59; Delboeuf, 

 Young '60; Hoiiel, Beltrami, Helmholtz '66; Witte '67; Pluck er, Baltzer, 

 Battaglini, Helmholtz, Geiser '68; Kronecker, Christoffel, Clifford, 

 Lipschitz '69; Schlaefli, Flye '70; Beez, Eosares, Lie, Klein '71; Saleta, 

 Konig, Jordan, Frischauf, Kober '72; Hoffmann, Freye, Cassani, Frahm 

 '73; Grassman '74; Halphen, Erschenck, Spottiswoode, Lewes, Hoiiel, 

 Funcke '76; Zollner, Frank, Engel, Liebmann, Luroth, Giinther '76; 

 Eethy, Frankland, Erdmann, Mehler, Cantor, Newcomb, Tannery '77; 

 Weisserborn, Land, Krause, Land '78; Schlegel '79; Smith, Ball, Chrystal 

 '80; Veronese, Story '82 ; Killing '85; Dixson'88; Segre'90?; Gerard '93; 

 Mansion, Schubert, Halstead '94; Staekel, Veronese '95; Eussell '97; 

 Poincaie, Ball '98; Stringham '99. 



2 On the origin and significance of geometrical axioms, v. Helmholtz, 

 1870. 



