ALLIED THEMES the study of whose connections and general comparative anatomy is one of the most in structive and fascinating chapters in the de velomnent of modern geometry.
Plane Geometry of the Point in Four Space that is 4-dimensional in points is also 4-dimensional in lineoids (ordinary 3-di mensional spaces). It is 6-dimensional in lines and also in planes. Hence in 4-space the point and the lineoid are dual (reciprocal) elements, and so the the plane and the line. The lineoid contains co lines; the point, 004 planes. The lineoid contains co' points and as many planes; the point contains ao ° lineoids and as many lines. Hence in 4-space, the point, plane and line geometries of the lineoid are respectively dual to the lineoid, line and plane theories of the point. Between any two of these pairs of reciprocal geometries there is a fact-to-fact correspondence, and the algebras of any such pair are identical. The emphasis here falls upon the fact thit the line geometry of the lineoid (i.e., ordinary line geometry) is pre cisely the same analytically as the geometry of the 4-space point regarded as the assemblage of its (generating) planes. For an introduc tory detailed account of the elements of the latter theory, and of the mentioned parallel ism, consult 'The Plane Geometry ofythe Point in Point-space of Four Dimensions) ( 'Ameri can Jour. of Math.,) Vol. XXV).
Geometry of (Ordinary) Space in Penta spherical Co-ordinates- The square of the tangent-distance from a point to a sphere is named the power of the point with respect to the sphere. Denote by xx(k =1, , 5) the powers of a point with respect to five fixed mutually orthogonal spheres. The xi, satisfy the identity Ixk'= O. To any set of values of their ratios there corresponds a definite point and conversely. The quantities Xxk are called pentaspherical point co-ordinates. Their dis covery and introduction into geometry are mainly ascribable to Gaston Darboux (cf. his memoir 'Sur une classe remarquable de courbes et de surfaces algebraiques,) 1873), but in part also to Felix Klein and Sophus Lie (cf. 'Mathe matische Annalen,) Vol. V). In these co-ordi nates the equation of a sphere is linear, viz., nik rk =0 (k =1, ... 5); conversely, every
such equation represents a sphere. The radius is p = (V Ime) : I(mk Rk), where the Rk are the radii of the fundamental spheres. Certain analytic correspondences between line geom etry (in Klien co-ordinates) and point geometry in pentaspherical point co-ordinates are imme diately obvious. For example: in the former 0 (j=-- 1, . , 6) is the identity satisfied by the line co-ordinates xi ; in the latter, Zxks =0 (k =1, ..., 5) is the identity connecting the pentaspherical point co-ordinates; in the former, Zmixi =0 represents a linear complex; in the latter, Zmkxk =•=0 represents a sphere; in the former, means that the com plex is special; in the latter, Zink =0 signifies that the sphere is a point; and so on and on.
Sphere Geometry of In this doc trine, due to Sophus Lie, the sphere is taken as primary element. To pick out a sphere from among all the spheres of space, it is necessary and sufficient to know four independent things about it, as the (three) co-ordinates of its centre and the length of its radius. Hence the sphere like the line, has four independent co ordinates, it has four degrees of freedom, and sphere geometry, like line geometry, is 4-di mensional. We have seen that every equation in pentaspherical point co-ordinates xk represents a sphere, and conversely; hence the five coefficients mk may be taken as homo geneous sphere their ratios being equivalent to four independents. The system may be rendered homologous to that of the six line co-ordinates by introducing a sixth sphere co-ordinate me by the definition, Ime, where i 1,/ —1 and .., 5). The six homogeneous, sphere co-ordinates mi (j = 1, ..., 6) satisfy the quadratic identity =0, identical in form with that con necting the Klien line co-ordinates. The condi tion that the spheres in and fps' shall be tangent 0, which is precisely like the con dition, 0, that the lines x and x' shall in tersect, a most interesting and fruitful principle of correspondence discovered by Lie in his brilliant memoir, 'Mier Complexe, in besondere Linienund Kugel-Complexe, mit Anwandung auf die Theorie partieller Differentialgleichungen) ('Mathematische Annalen,' Vol. V, 1871).