383 
it has therefore eight solid angles, 0„ 0 t , &c., 0 S , each angle 
being formed by the union of three planes; and twelve 
equal edges, such as 0,0,, 0 2 0„ &c. The inclination of any 
lace to another is measured by the angle contained between 
two perpendiculars drawn from any point in the edge made 
by the intersection of the two faces, each on one of the adjacent 
faces. In the cube this inclination of two adjacent faces is 90°. 
Ihe facial angles, or the angles between two edges of a face, 
such as 0^0,0^ are always 90°. 
11 The Octahedron (fig. 7, Plate I.) is bounded by eight 
equal faces each face, such as 0,0,0,, shown on a plane surftce 
(tig. oo, Plate IV.), being an equilateral triangle. It has six 
solid angles, O x , C 2 , &c., C 6 , each formed by the union of four 
planes, and twelve equal edges; the inclination of adjacent faces 
is 60o a “ g ° f 109328/5 and the facial snch as C 1 O i O s , 
12. The Rhombic Dodecahedron (fig. 5, Plate I) is 
bounded by twelve equal faces ; each face, such as oAo,C , 
’ d a ^ e ^ •)> 18 a geometrical rhomb bounded by four 
equal lines, o x G 2 being parallel to o 5 O s , and o x O s to o-fij The 
greater angles of the rhomb 0 % o,0 % and O 3 o 5 O,'being°109 o 28' 
and the lesser, 0,0,0, and 0l 0 3 o 5 , 70° 32'. ' It has twenty-four 
equal edges, such as 0,/),, C x o 2 , &c., eight solid angles, o„ o,_, 
*c., o 8 , formed by the union of three planes, and six solid 
angles, C„ 0 2 , &c., 0 6 , formed by the union of four plaues. 
Ihe inclination of adjacent faces is 120°. This form is called 
by some. German writers the granatoedron, as being a cha- 
racteristic form of the garnet. 
13, These three forms, the cuhe, octahedron, and rhombic 
dodecahedron, are called invariable forms, as, though differing in 
size, they always have similar faces and angles; that of & the 
cu e eiug a square, that of the octahedron an equilateral 
triangle, and that of the rhombic dodecahedron a rhomb 
whose larger angle is 1 09° 28'. 
14 The four other forms (figs. 2, 3, 4, and 6, Plate I.) are 
^ ec . vaiaable J each presenting an infinite variety of species 
differing from each other in their angles of inclination and 
those of their faces. 
l-o. The Three-faced Octahedron (fig. 6, Plate I.) is 
bounded by 24 equal faces, each being an isosceles triangle, 
/ W2 '? Plate IV.). These faces are so grouped 
ogether as to form a solid having eight solid angles, formed 
anedpq r° n ° f ,!' hre ® P lanes ' ° 2 , ^Sbo., o 8 (fig. 6) ; the plane 
*• ° -i-i 1D ? largest of the isosceles triangles; and 
J s° it angles, 0„ C 2 , &c., 0 6 , each formed bv the union 
ot eight of the equal angles of the isosceles triangles. 
2 e 2 8 
