relative to Polyhedrons. 443 



Proposition. 



In any polyhedron, the number of corners, together with 

 the number of faces, exceeds the number of edges by two. 

 Or, symbolically, let 



c — the number of corners, 

 y= the number of faces, 

 and e = the number of edges, 



then c +/=*?+ 2. 



Demonstration. 



1. We shall demonstrate that when you increase c by 1, 

 then the increase upon c +yis equal to the increase upon e. 



Let ABDGHKbea face in the original 

 polyhedron, and L the additional corner, 

 which, by drawing L A, L B, &c, forms the a< 

 new polyhedron, in which c is greater by 1. 



Suppose that L is not in the plane pro- 

 duced of any of the faces that meet A D in 

 the edges A B, B D, &c. Let n = the num- 

 ber of sides in the polygon A D. 



Then it is evident that n faces, with their common vertex 

 L, are added, while A D ceases to be a face, or 1 face is sub- 

 tracted, so that the real increase in the number of faces or in 

 f'\s — n— 1, and since there is the additional corner L, the 

 total increase on c -rf\s = 1 +n — l=7i. 



Again, all the lines L A, L B, &c. form new edges, so that 

 the total increase in the number of edges or upon e is = n. 



Hence the increase upon c +J"is = the increase upon e. 



Suppose now that L is in the plane of the face produced 

 that intersects A D in K H. 



Then L K H ceases to be an additional face, and is only 

 part of a face of the original polyhedron ; hence the number 

 of faces added, instead of », is only »— 1, while, as before, A 

 D ceases to be a face, or 1 face is subtracted, so that the real 

 increase in the number of faces is =n — 2. Hence the increase 

 on c +f'\s =l+n—2 = n—\. 



Again, the number of edges added is, as before, = n ; but 

 K H, which was an edge in the original polyhedron, ceases 

 to be an edge in the new polyhedron, or 1 edge is subtracted, 

 hence the actual increase on e is =n — 1. 



Hence, as before, the increase on c +,/is = the increase 

 on e. 



2. It follows that, if the proposition be true, or if c +f 

 =e+2 in the original polyhedron, it must also be true in the 

 new polyhedron. 



2 H 2 



