IX] OF EULER'S LAW 733 



hedral solids, using a binomial nomenclature based on the number 

 of their corners or vertices, and sides or faces. Thus, for example, 

 he called a figure with eight corners and seven faces Octogonum 

 hsptaedrum; and the analogy between this and Linnaeus's botanical 

 classification and nomenclature — e.g. Hexandria trigynia and the 

 rest — is very close and curious. 



A simple theorem, of which Euler was vastly proud and which 

 we still speak of as Euler 's Law*, is fundamental to the theory of 

 polyhedra. It tells us that in every polyhedron whatsoever, the 

 faces and corners together outnumber the edges by two t : 



C-E+F=2 (1). 



Another fundamental theorem follows. We know from Euchd 

 that the three angles of a triangle are equal to two right angles; 

 consequently, that in a polygon of C angles, the sum of the angles 

 -= 2 (C — 2) right angles. And there follows from this — but by no 

 means expectedly — the analogous and extremely simple relation 



* Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. Imp. 

 Petropol. IV, p. 109 seq. (ad annos 17.52 et 1753), 1758: "In omni solido hedris 

 planis inclusum, aggregatura ex numero angulorum solidorum et ex numero 

 hedraruiu binario excedit numerum acierum." For a proof, see {int. al.) De 

 Morgan, article Polyhedron in the Penny Cyclopaedia. There is reason to believe 

 that Descartes was acquainted with this theorem between 1672 and 1676; cf. 

 Foucher de Careil, (Euvres inedites de Descartes, Paris, ii, p. 214. Cf. Baltzer, 

 Monatsber. Berlin. Akad. 1861, p. 1043; and de Jonquieres, C.R. 1890, p. 261. 

 (The student will be struck by the resemblance between this formula and the phase 

 rule of Willard Gibbs.) 



t If we include, besides the corners, edges and faces (i.e. points, lines and 

 surfaces) the solid figure itself, Euler's Law becomes 



C-E + F-S = \. 



And in this form the theorem extends to n dimensions, as follows: 



With equal beauty and simplicity, the simplest figure in each ^-dimensional 

 space is given as follows: 



Kq rC% A^2 "'S "^4 ^^^» 



w = 1 " =1 (point) 



12-1 =1 (line) 



2 3-3 +1 =1 (triangle) 



3 4 -6 +4 -1 —1 (tetrahedron) 



4 5 -10 +10 -5 +1 =1 (pentahedroid) 

 etc. 



And, in a figure of /i- dimensions, the sum of the plane angles =2*"-^' (C-2) 

 right angles. 



