. 127. OF THE CONNEXION OF FORMS. 133 



The hexahedral Trigonal-icositetrahedrons are forms in- 

 termediate between the hexahedron and the dodecahe- 

 dron. The moveable plane passes through the edge of the 

 hexahedron ; the limits to which its inclination approaches 

 will therefore be on one side a plane perpendicular to the 

 section of the edge, on the other a plane perpendicular to 

 the section of the face, of which the first produces the 

 dodecahedron (. 122.) ; and the other the hexahedron 

 ( 124.). 



The varieties are determined according to the mutual 

 inclination of the faces at the place of the edge of the hex- 

 ahedron. Every different inclination, greater than 90 

 and less than 180, yields a particular hexahedral Trigonal- 

 icositetrahedron . 



. 127. THE TETRACONTAOCTAHEDRON. DESIGNA- 

 TION OF THE TESSULAR FORMS. 



In the seventh situation, the moveable plane is 

 the face of a Tetracontaoctahedron (. 77.). 



No two faces coincide in a single plane. The form there- 

 fore is contained under forty-eight faces, which are scalene 

 triangles, on account of their intersection with two faces 

 from the same, and with one face from the adjacent solid 

 angle of the hexahedron. The rhombohedral solid angles 

 are formed by six faces, the pyramidal solid angles by 

 eight, and the prismatic solid angles by four. All of them 

 are equiangular and digrammic. Hence the form will be a 

 Tetracontaoctahedron. 



It may here be observed, that the seven forms thus con- 

 nected with the hexahedron, the hexahedron itself being 

 one of the number, perfectly agree with each other in re- 

 spect to the kind, number, and situation of their axes, 

 which is an immediate consequence from the method of 

 derivation employed. The same property does not extend 

 to the rest of the forms of several axes; and thus the num- 

 ber of the different kinds of axes in their peculiar situation, 



