Prof. De Morgan on certain Relations in a solid Figure. 323 



lution of which being added to the teriodide of chromium 

 gives rise to iodide of lead and chromate of the oxide of lead, 

 which are readily separable, the iodide being soluble in boiling 

 water, while the chromate is insoluble; but sulphate of lead 

 is formed, which^is also insoluble. It unfortunately happens 

 that the degree of heat required for the formation of the ter- 

 iodide always causes a small quantity of sulphuric acid to be 

 carried over with it. 



The proof, however, which we have that this compound is 

 indeed a teriodide of chromium, is afforded by the fact of its 

 being resolved into the chromic and hydriodic acids by the 

 action of water. 



I am, Gentlemen, yours, &c., 



Edinburgh, Feb. 5, 1838. HERBERT GiRAUD. 



L. On the Belation between the Number' ofFaces, Edges, and 

 Corners in a Solid Polyhedron. By Augustus De Morgan, 

 Professor of Mathematics in U?iiversity College.* 

 T^HE remarkable relation which exists between the number 

 -*■ of edges, faces, and corners (or solid angles) in a solid 

 figure, namely, that the number of faces and corners together 

 always exceeds the number of edges by two, is usually de- 

 monstrated by reference to the celebrated expression for the 

 area of a spherical triangle. The theorem was given by Euler, 

 in the Petersburg Acts for ] 758 ; but not having access to 

 that work, I cannot tell whether he employed the method just 

 alluded to, or not. However, since Legendre has derived the 

 theorem by means of the spherical triangle, as well as every 

 other elementary writer with whom I am acquainted ; and 

 since an equally simple relation which exists among the edges, 

 corners, and faces of a portion of a solid figure has been 

 little if at all noticed, I conjecture that the following demon- 

 stration is new. At any rate, it is more simple, and derived 

 from more elementary principles, than the one commonly 

 given, and is therefore worthy of notice. 



Let there be a number of polygons, so placed that each 

 has a common edge with one or more of the others. Let 

 every angular point be called a corner (whatever may be the 

 number of lines which meet there) ; every line joining two 

 corners, an edge ; every unsubdivided portion of space, a face. 

 Then the number of faces and corners together will always 

 exceed the number of edges by one. For this is evidently 

 true of a single polygon, while for every polygon which is 

 added, the number ot new edges is one more than the number 



* Communicated by the Author. 

 2H2 



