109 
the angles of crystals. 
The plane angles of the tetrahedron are 6o° ; and hence, to 
find its dihedral angles, we have to find the angle of an 
equilateral spherical triangle whose sides are 6o°. If a be 
this angle, we have 
cos. a = cotan. Go . tan. 30 = tan."" 30 = — . 
3 
Let d be the angle of the planes (o, 1, 1 ) ( 1, o, 1), and 
we have by the formula 
— cos. 0 = ~ ~ ^ = o because cos. = 
2 — 2 cos. a 3 
Hence the angle of the planes is a right angle. And in 
the same manner the angles made by the other planes will 
be right angles. The figure will be a cube bounded by the 
3 planes (o, 1, 1) twice repeated. 
Irregular Tetrahedrons and Octahedrons. 
29. If we have an octahedron composed of two right 
quadrilateral pyramids, similar and equal, set base to base, 
we shall call this a right octahedron ; and it will be termed 
square, rectangular, or rhombic, when the base is so. The 
tetrahedron, from which the right rectangular octahedron is 
derived, may be called the direct symmetrical tetrahedron ; and 
that from which the right rhombic octahedron is derived, 
may be called the inverse symmetrical tetrahedron, on account 
of properties which will be explained immediately. Also, 
all the planes which can be derived from the octahedrons, 
may be derived more simply from the corresponding tetra- 
hedrons ; and we shall find the coexistent planes, and the 
angles made by the faces, in the same manner as in the 
previous cases. 
