124! Mr. A. Cayley on Poin sot's /om7" new Regular Solids. 



lization will be best understood by conceiving, as he does, that 

 the polyhedron is projected on a concentric sphei'e, so that the 

 faces become spherical polygons. Then for the ordinary polyhe- 

 drons of geometry, the sum of the angles at a vertex = 4 right 

 angles ; but it may, according to the more general notion, be = e 

 times 4 right angles. In like manner for the ordinary polyhe- 

 drons, the sides of a face subtend at the centre angles, the sum 

 of which is = 4 right angles ; but accoi'ding to the more general 

 notion, this sum may be (viz. if the polygons are stellated) = e' 

 times four right angles. And finally, the sum of the spherical 

 polygons is ordinarily equal to the entire spherical surface ; but 

 according to the moi'c general notion, it may be = D times the 

 spherical surface, (e is Poinsot's e; e' does not occur in Poinsot ; 

 and, for a reason which will appear, I have written D for Poin- 

 sot's E.) 



The new polyhedra are constructed as follows : — 

 1. The great Icosahedron. — Each face is made up of seven 

 faces, or rather four faces and six half faces of the ordinaiy ico- 

 sahedron, in the manner shown 



by fig. 1. There are, as in the 

 ordinary icosahedron, five an- 

 gles at each vertex; but these 

 make up together, not four, but 

 eight right angles, or e = 2 ; 

 but, as in the ordinary poly- 

 hedra, e' = l; and the sum of 

 all the faces is obviously seven 

 times the spherical surface, or 

 D = 7. (AlsoE = 7.) 



2. The great Dodecahedron. — 

 Each face is made up of five 

 faces of the ordinary icosahedron 

 in the manner shown by the 

 figure 2. There are five angles 

 at each vertex, and these make 

 up together eight right angles, 

 or e = 2 ; but, as in ordinary 

 polyhedra, e' = l ; and the sum 

 of all the faces is obviously 

 12 X 2%, or three times the 

 spherical surface, or D = 3. 

 (Also E = 3.) 



3. The great stellated Dodeca- 

 hedron. — Each face is formed 

 by stellating a face of the great 

 dudccahedrun in the manner 



Fig. 1. 



Fig. 2. 



