44 i 
*[12oo]; 
CO 0 2 B 
~ 2 ~ j 
7T 2 1 0 ; ^b 2 ,in cobaltine, cubane, fah- 
lerz, gersdorfitte, and pyrite. 
i[l % °°] > — 3 7r 3 1 0; £6 3 , in hauerite, pyrite, and 
sal ammoniac. 
y[ 1 4 oo] j — QJi • 7r 4 1 0 ; £ 5 4 , in cobaltine and fahlerz. 
141. Platonic bodies .— There are five solid bodies described 
by the ancient geometers as regular solids. From their 
mathematical properties having been investigated by Plato 
and his followers, they are called the Platonic bodies. They have 
all their faces, edges, and angles, whether plane or solid, equal 
for each body. 
They are the tetrahedron , bounded by four equal faces, each 
being an equilateral triangle ; the cube , bounded by six equal 
squares ; the octahedron, bounded by eight equal faces, each 
being an equilateral triangle; the pentagonal dodecahedron , 
bounded by twelve equal and equilateral pentagons ; and the 
icosahedron, by twenty equal faces, each being an equilateral 
triangle. 
The first three, described by Plato himself, have been 
observed in natural crystals. The last two, described after 
his death, have not been observed in nature. 
The regular pentagonal dodecahedron is that particular case 
of the pentagonal dodecahedron, where the unequal edge, such 
as $ 2 3 4 (fig. 23, Plate III.), is equal to the other four S 2 o p o^ v 
V 4 , an d 
In this case m=cot A 3 
— 1 + a/5 _ 
2 
P618034, 
but cot 31° 43'= P618085. 
Hence A 3 = 31° 43' true to minutes. 
The value of m is generally determined by continued 
fractions. 
Thus P619046 and cot 31° 42'=P61914 
m = -#=1-625 cot 31° 36'= 1-62548 
m= f=l-6 cot 32° 0'= 1-60033 
The regular icosahedron is derived from the particular 
pentagonal dodecahedron in which the edge S^^a line joining 
the points ^ and S 2 . In this case 
to = cot A 3 = 3+ ^ =2-61803=cot 20° 54', 
2 
where the ratio for m expressed in its lowest terms is 
In this particular pentagonal dodecahedron each solid angle 
