-270 
[679 
679. 
ON THE REGULAR SOLIDS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 127—131.] 
In a regular solid, or say in the spherical figure obtained by projecting such 
solid, by lines from the centre, on the surface of a concentric sphere, we naturally 
consider 1° the summits, 2° the centres of the faces, 3° the mid-points of the sides. 
But, imagining the five regular figures drawn in proper relation to each other on 
the same spherical surface, the only points which have thus to be considered are 12 
points A, 20 points B, 30 points 0, and 60 points These may be, in the first 
instance, described by reference to the dodecahedron; viz. the points A are the 
centres of the faces, the points B are the summits, the points 0 are the mid-points 
of the sides, and the points <I> are the mid-points of the diagonals of the faces 
(viz. there are thus 5 points $ in each face of the dodecahedron, or in all 60 
points <I>). But reciprocally we may describe them in reference to the icosahedron; 
viz. the points A are the summits, the points B the centres of the faces, the points 
0 the mid-points of the sides, (viz. each point © is the common mid-point of a 
side of the dodecahedron and a side of the icosahedron, which sides there intersect 
at right angles), and the points are points lying by 3’s on the faces of the 
icosahedron, each point <I> of the face being given as the intersection of a perpendicular 
A© of the face by a line BB, joining the centres of two adjacent faces and inter 
secting A© at right angles. 
The points A lie opposite to each other in pairs in such wise that, taking any 
two opposite points as poles, the relative situation is as follows: 
A 
Longitudes. 
1 
— 
? 
5 
0°, 
72°, 144°, 216°, 
288°, 
5 
36°, 
108°, 180°, 252°, 
324°, 
1 
— 
? 
where the points A in the same horizontal line form a zone of points equidistant 
from the point taken as the North Pole. And the points B lie also opposite to