84 
ON POINSOTS FOUR NEW REGULAR SOLIDS. 
[241 
E, viz. the faces make together E times the spherical surface, the area of a 
stellated face being reckoned (as by Poinsot), each portion being taken once only. 
D, viz. the faces make together D times the spherical surface, the area of a 
stellated face being reckoned as the sum of the triangles having their vertices at the 
centre of the face and standing on the sides. 
The Table is 
Designation. 
II. 
S. 
A. 
tl. 
n'. 
e. 
e'. 
D. 
E. 
Tetrahedron 
4 
4 
6 
3 
3 
1 
1 
1 
1 
[Hexahedron 
6 
8 
12 
4 
3 
1 
1 
1 
1 
^Octahedron 
8 
6 
12 
3 
4 
1 
1 
1 
1 
[Dodecahedron 
12 
20 
30 
5 
3 
1 
1 
1 
1 
[icosahedron 
20 
12 
30 
3 
5 
1 
1 
1 
1 
(Great stellated dodecahedron ... 
12 
20 
30 
5 
3 
1 
2 
7 
4 
(Great icosahedron 
20 
12 
30 
3 
5 
2 
1 
7 
7 
(Small stellated dodecahedron ... 
12 
12 
30 
5 
5 
1 
2 
3 
2 
(Great dodecahedron 
12 
12 
30 
5 
5 
2 
1 
3 
3 
where the figures which are polar reciprocals of each other are written in pairs: viz. 
as is well known, the tetrahedron is its own reciprocal, the hexahedron and octahedron 
are reciprocals, and the dodecahedron and icosahedron are reciprocals; moreover the 
great stellated dodecahedron and the great icosahedron are reciprocals, and the small 
stellated dodecahedron and the great dodecahedron are reciprocals. The number which 
I have called D is reciprocal to itself; this is not the case for Poinsots E; and I 
have not been able to define E in such a manner as to enable me to form the 
definition of a reciprocal number E': this may be possible, but in the mean time it 
seems better to discard E altogether, and use instead of it the number D. 
Euler’s well-known relation applying to ordinary polyhedra is 
S + H = A +2. 
Poinsot in his memoir has (by an extension of Legendre’s demonstration of Euler’s 
theorem) obtained the more general relation, 
eS + H = A +2E,