50 



Tetrahedron, 

 Cube, 



Octahedron, 

 Dodecahedron, 



6 edges, 

 6 faces, 

 6 angles, 

 6 angles of 

 four plane 

 angles, 



4 faces. 

 12 edges, 

 12 edges. 

 12 faces, 24 edgear. 



4 angles, 

 8 angles, 

 8 faces, 

 8 angles of 

 three plane 

 angles, 



The faces of either of these solids must necessarily be produced 

 through the replacement, by tangent planes, (these planes being 

 continued down to the obliteration of the primary,) of the edges or 

 angles of either of the other solids, provided those edges or angles 

 equal the number of its faces. 



For example, the twelve faces of the Dodecahedron may be de- 

 rived either from the tangent replacement of the twelve edges of 

 the regular Octahedron, (Fig. 62,) or from a similar replacement of 

 the twelve edges of the Cube, (Fig. 57.)* 



The eight faces of the Octahedron may be the result either of the 

 tangent replacement of the eight angles of the Cube, (Fig. 56,) or 

 of the eight obtuse solid angles of the Dodecahedron, (Fig. 68.) 



The six faces of the Cube may be derived from the truncation of 

 the six edges of the Tetrahedron, (Fig. 53,) or from the six acute 

 solid angles of the Dodecahedron, (Fig. 67.) 



The Tetrahedron cannot be derived in the 

 same manner with the other solids; but may 

 result from the Octahedron, by the suppres- 

 sion of half of its eight faces ; or, in other 

 words, by the enlargement of one of the two 

 parallel faces, until the other is made to dis- 

 appear. Fig. 119 shews the planes of the 

 Tetrahedron in the position it occupied in 

 the Octahedron.! 



Fig. 119. 



* The student in crystallography who has not 

 attended to the connections subsisting between 

 the different solids, by which one form may be 

 transformed into another, is recommended to 

 verify some of the changes here described, by 

 shaving pieces of wax, or some other soft sub- 

 stance, with a knife. 



t The Tetrahedron thus derived, passes again 

 to the Octahedron, by the truncation of its solid 

 angles, as seen in Fig. 120. 



