(0) 
(0 
s it just possible 
whole numbers 
ved the problem 
■educes it. The 
id by Amthor. 1 
tions is 
52 n, 
4 n, 
'7 n, 
50 n, 
i0 n, 
6 n, 
3 n, 
10 n. 
that W+ X = a 
i 2 = 44567491 2 , 
to make Y + Z 
form <7 (<?+!)• 
an ’ equation 
! eight unknown 
0 digits! 
i. 
îdra of a certain 
i.edra in question 
beral and equi- 
ON SEMI-REGULAR POLYHEDRA 99 
angular, but not similar, polygons; those discovered by 
Archimedes were 13 in number. If we for convenience 
designate a polyhedron contained by m regular polygons 
of a sides, n regular polygons of (3 sides, &c., by (m a , %...), 
the thirteen Archimedean polyhedra, which we will denote by 
P v P 2 ... P 13 , are as follows: 
Figure wi4h 8 faces: ^ = (4.5, 4 6 ), 
Figures with 14 faces: P 2 = (8 3 , 6 4 ), P 3 = (6 4 , 8 6 ), 
Figures with 26 faces: 
Figures with 32 faces: 
Figure with 38 faces : 
Figures with 62 faces: 
p t = (83, «„). 
P, = (83, I84), P,= (12„ 8,. 6 S ). 
P, = (20 3 , 12 5 ), P 8 =( 12 b. 20 6 ), 
P, = ( 20 3 , 12,„). 
P 10 =(323, 6 t ). 
P n =(20 3 , 30 4 , 12 5 ), 
P, 2 =(30 4 ,20 s ,12 10 ). 
Figure with 92 faces: P 13 =(80 3 , 12 5 ). 
Kepler 1 showed how these figures can be obtained. A 
method of obtaining some of them is indicated in a fragment 
of a scholium to the Vatican MS. of Pappus. If a solid 
angle of one of the regular solids be cut off symmetrically by 
a plane, i. e. in such a way that the plane cuts off the same 
length from each of the edges meeting at the angle, the 
section is a regular polygon which is a triangle, square or 
pentagon according as the solid angle is formed of three, four, 
or five plane angles. If certain equal portions be so cut off 
from all the solid angles respectively, they will leave regular 
polygons inscribed in the faces of the solid; this happens 
(A) when the cutting planes bisect the sides of the faces and 
so leave in each face a polygon of the same kind, and (B) when 
the cutting planes cut off a smaller portion from each angle in 
such a way that a regular polygon is left in each face which 
has double the number of sides (as when we make, say, an 
octagon out of a square by cutting off the necessary portions, 
1 Kepler, Harmonice mundi in Opera (1864), v, pp. 123-6. 
H 2 
xxv. (1880), pp-