394 
sin 54° 44' =44^==^+^ and cos 54° 44/= — _J_ 
A 0, a/3 
Hence Ao ,= — - 
a/ 2 
m \/ o 
AO, v/: 
V' i4~ m v / 2v / f 1 + 2m 
m 
1 + 2w& 
40 i; 
Or — L: 
40, 
I??/. 
1 
1+2 m 1 + 1+1 
m 
48. If we call the distances 1, 1, and ?n, at which each of the 
24 faces of the three-faced octahedron if produced would cut 
three of the semi-cubical axes at right angles to each other, 
indices; then the ratio of ^-1 = unitv divided by the sum of the 
AO, *" J 
reciprocals of the indices. Calling B this ratio, then when m — 2 
R=& m—3 B=f ; |=f J 2 =e. w — 4 12 = 4 . m= 7 . I?=+_ ; 
™=! J? =ni tad«i=$f 
49. When m=l, the three-faced octahedron becomes the 
octahedron, and its. three indices are 1, 1, and 1, and B—%. 
Taking 1 1 m as the symbol for the three-faced octahedron, 
111 must be taken as the symbol for the octahedron. 
50. For the octahedron Naumann’s symbol is 0; Miller’s, 
111; Brooke, Levy, and Des Cloizeau’s a 1 . 
51. When the third index becomes infinite, or, in other 
words, the face cuts two axes and is parallel to the third, then 
m— 77 = 00 , and — = 0 ; and the three-faced octahedron is then 
0 m 
the rhombic dodecahedron. 
52. The three indices' of the rhombic dodecahedron are, 
therefore, 1 , 1 , and oo ; and 1 1 oo becomes its symbol. Nau- 
mann’s symbol is oo 0 ; Miller’s, 110; Brooke’s, &c., b 1 . 
To inscribe the four -faced Cube in the Cube. 
53. (Fig. 37, Plate IY.) Describe the square AC ',D,C 2 equal 
one-fourth of the square 0,0 4 0 8 0 5 (fig. 27, Plate IY.), this 
being a face of the cube in which the four-faced cube is to be 
inscribed. Join AID, (fig. 37, Plate IY.). Produce JD,G, to 0,, 
and C 2 A to D s . Make 0,0, and AD^ — AD,. Join 0,D 5 . 
Produce AG, to M, and make AM=m, m being any whole 
number or fraction greater than unity. The particular value 
of m will determine the particular species of the four-faced 
