Circle Geometry of In this beauti ful and growing theory, principally due to the French mathematicians E. Cosserat, C. Stepha nos, and G. Koenigs, the circle is employed as primary or generating element of space. In this element, space is 6-dimensional, like point 4-space in lines or planes. A circle is deter mined as the intersection of two spheres, as 0,----- 0 (j= 1, . 5). It is equally determined by any two spheres of the pencil or range, arnnxi-=- 0, of spheres containing it, and, in particular, by any two of the included five of which each is orthogonal to one of the fundamental spheres. The equa tions of those special spheres correspond to the five A-values that render the coefficients mii-Xml=0 in succession. For the sake of symmetry, the ten coefficients are taken as homogeneous co-ordinates of the circle. That the ten are equivalent to the necessary and sufficient number six of inde pendents is seen in the facts that only their rations are essential and that they satisfy five (equivalent to three independent) quadratic identities of the form wo(P)== POPe7 +pilep/d)=----0. The circle geometry of space is not parallel to the line geometry of ordinary space, but it is parallel, in a fact-to-fact fashion, to the line and the plane geometries of point 4-space.
Theory of Circles Orthogonal to Sphere. —Two spheres mkandme(k---1, ...., 5) are or thogonal when and only whcnInikmki=---0; hence there are col spheres orthogonal to a given sphere. A circle is orthogonal to a sphere when and only when any two (and hence all) of its generating spheres are orthogonal to the sphere. There are, accordingly, oo' circles orthogonal to a given sphere. A correlation subsists between such circles and the lines of space. If, in the assemblage of spheres orthogo nal to a given sphere, four mutually orthogonal spheres he taken as fundamental or co-ordinate sphere any equation Mmkxk=---0 (k =1, • .. , 4)
will represent a sphere of the assemblage, and conversely. Hence a pair of such equa tions will define a circle orthogonal to the fixed sphere, and conversely. It is imme diately plain that the co-ordinates of the circle regarded as element of the assemblage of circles orthogonal to a given sphere are analytically precisely the same as the line co-ordinates of space. Hence the geometry of such a circle assemblage is analytically identical with line geometry. The first chapters of such a circle geometry arc found in
The literature of line geometry and allied theories is extensive and is rapidly increasing. In addition to the fore going citations, may he mentioned the follow ing works, which together with further cita tions contained in them constitute a complete bibliography of the subject: Cayley