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°.