272 
ON THE REGULAR SOLIDS. 
[679 
the remaining 4 points B are then the centres of the faces, and the mid-points of 
the sides are points ©: there are thus 5x4,= 20, tetrahedra having 20 x 4 summits 
which are the 20 points B each 4 times; 20 x 4 centres of faces which are the 20 
points B each 4 times; and 20 x 6 mid-points of sides which are the 30 points © 
each 4 times. 
It thus appears that, as mentioned above, the five regular figures depend only 
on the points A, B, ©, and <f>. 
We might take as poles two opposite points 4, 5, 0, or $; and in each case 
determine in reference to these the positions of the other points; but for brevity I 
consider only the case in which we take as poles two opposite points A. We have 
the following table: 
Poles two opposite points A. 
A 0 
5 A 1 
N. P. D. 
0° 
63° 26' 
Longitudes. 
0°, 72°, 144°, 216°, 288° 
54 2 
116° 34' 
36°, 
b 
00 
I—t 
00 
o 
252°, 324° 
A 3 
180° 
— 
5B 1 
37° 22' 
36°, 
108°, 
, 324° 
5B 2 
79° 12' 
36°, 
108°, 
, 324° 
5B 3 
100° 48' 
0°, 
72°, 
, 288° 
5B 4 
142° 38' 
0°, 
72°, 
, 288° 
5®! 
31° 43' 
0°, 
72°, 
, 288° 
5® 2 
58° 77' 
36°, 
108°, 
, 324° 
10® 3 
90° 
( 0°, 
72°, 
, 288°)+ 18° 
o® 4 
121° 43' 
0°, 
72°, 
, 288° 
5® 5 
148° 17' 
36°, 
108°, 
, 324° 
54»! 
13° 16' 
36°, 
108°, 
, 324° 
10* 2 
52° 52' 
( 0°, 
72°, 
288°) + 9° 44' 
104> 3 
68° 10' 
( 0°, 
72°, 
, 288°)+ 13° 35' 
5* 4 
76° 42' 
0°, 
72°, 
, 288° 
5*5 
103° 18' 
36°, 
108°, 
, 324° 
io* 6 
111 0 50' 
(36°, 
108°, 
, 324°)+ 13° 35' 
10* 7 
127° 8' 
(36°, 
108°, 
, 324°)+ 9° 44' 
5* 8 
166° 44' 
0°, 
72°, 
, 108°.