Mr. A. Cayley on Poinsot's /b«/- neiv Regular Solids. 125 



shown by fig. 3. There are, as in the ordinary dodecahedron, three 

 angles at each vertex, and the pj^ 2 



sum of these is simply four 

 right angles, or e = l. On 

 account of the stellation, 

 e' = 2. Each of the project- 

 ing parts of the face is equal 

 ^ of the face of the ordinary 

 icosahedron; and if we reckon 

 the area of the stellated pen- 

 tagon to be that of the inte- 

 rior pentagon plus the pro- 

 jecting parts, the area of the 

 face will be 5 + 1, or M of 

 the face of the ordinary 

 icosahedron ; and the sum 

 of the faces will be four 

 times the spherical surface, and accordingly Poinsot writes E =4. 

 If, however, what seems preferable, we reckon the area of the 

 stellated pentagon as five times the triangle, having for its vertex 

 the centre of the face and standing upon a side (or what is the 

 same thing, reckon the stellated pentagon as twice the interior 

 pentagon pluc the projecting parts), then the area of the face 

 will be 10-!- 1 or ^- of' the face of the ordinary icosahedron, and 

 the sum of the faces will be seven times the spherical surface, 

 or D = 7. 



4. The small stellated Dodecahedron.— ^ach face is formed by 

 stellating a face of the ordinary dodecahedron, as shown by 

 fig. 4. There are five angles at each vertex ; and the sum of 

 these is four right angles, or e = l. 

 On account of the stellation, <?' = 2. 

 The area of each of the projecting 

 parts is } of the interior pentagon 

 or face of the ordinary dodecahe- 

 dron; and, according to the first 

 mode of measurement, the area of 

 the stellated face is twice that of 

 the face of the ordinary dodecahe- 

 dron, and the sum of the faces is 

 twice the sj)]ierical surface, and 

 accordingly Poinsot writes E = 2. 

 But according to the second mode 

 of measurement, the area of the stellated ])entagon is three 

 times that of the face of tiie ordinary dodecahedron, and the 

 sum of the faces is three times the spherical surface, or we have 

 D = 3. 



