170 RECIPROCAL FIGURES, FRAMES, 



Each of the diagrams must fulfil the conditions of being a plane projection 

 of a plane-sided polyhedron, for if any of the sides of the polyhedron of which 

 it is the projection are not plane, there will be as many points corresponding 

 to that side as there are different planes passing through three points of the 

 side, and the other diagram will be indefinite. 



Relation between the Number of Edges, Summits, and Faces of Polyhedra. 



It is manifest that after a closed surface has been divided into separate 

 faces by lines drawn upon it, every new line drawn from a point in the system. 

 either introduces one new point into the system, or divides a face into two 

 l>arts, according as it is drawn to an isolated point, or to a point already 

 connected with the system. Hence the sum of points and faces is increased 

 by one for every new line. If the closed surface is acyclic, or simply connected*, 

 like that of a solid body without any passage through it, then, if from any 

 point we draw a closed curve on the surface, we divide the surface into two 

 faces. We have here one line, one point, and two faces. Hence, if c be the 

 number of lines, s the number of points, and / the number of faces, then in 

 general 



e s f m 



* See Riemann, Crelle's Journal, 1857, Lehrsdlte aus der analysis situs, for space of two dimen- 

 sions; .also Cayley on the Partitions of a Close, Phil. Mag. 1861; Helmholtz, Crelle's Journal, 1858, 

 Wirbclbfwguiig, for the application of the idea of multiple continuity to space of three dimensions ; 

 J. B. Listing, Gottingen Trans., 1861, Der Censut R&umlicher Complexe, a complete treatise on the 

 Kubject of Cyclosis and Periphraxy. 



On the importance of this subject see Gauss, Werke, v. 605, "Von der Geometria Situs ilir 

 Leibnitz abnte und in die nur einem Paar Geometern (Euler und Vandermonde) einen schwachen 

 Blick ru thun vergbnnt war, wissen und haben wir nach anderthalbhundert Jahren noch nicht vicl 

 nifhr wie nichtfl." 



Note added March 14, 1870. Since this was written, I have seen Listing's Census. In his 

 notation, the surface of an w-ly connected body (a body with n - 1 holes through it) is (2/t-ii) 

 cyclic. If 2n 2 = A', expresses the degree of cyclosis, then Listing's general equation is 



where is the number of points, e the number of lines, A 7 ", the number of endless curves, f the number 

 of faces, A' t the number of degrees of cyclosis of the faces, w t the number of periphractic or closed 

 faces, v the number of regions of space, K t the number of degrees of cyclosis, vr 3 their number of 

 degrees of periphraxy or the number of regions which they completely surround, and to is to be put 

 1 or 0, according as the system does or does not extend to infinity. 



