. 128. OF THE CONNEXION OF FORMS. 137 



her of their faces would not be sufficient to include a space 

 from all sides ; or because none of the methods mentioned 

 is applicable to them. The first is the reason why the 

 hexahedron, the other why the dodecahedron, have no 

 halves. Besides, it is not every form that can be resolved 

 by every one of the above-mentioned methods ; but cer- 

 tain properties of the form are required to render one of 

 these methods applicable. 



Thejirst method supposes the faces of the form which is 

 to be resolved not to touch the terminal points of two rhom- 

 bohedral axes ; or, which is the same, not to touch a prin- 

 cipal point and a subordinate one at the same time. For 

 as it is required to effect the reverse on one of those points, 

 from what has been done on the other, it would thus be 

 requisite, that one and the same face should at the same 

 time be made to enlarge itself and disappear. For this 

 reason, the hexahedral trigonal-icositetrahedron cannot be 

 resolved according to the first method. 



The second and third method supposes the number of 

 faces at the rhombohedral solid angles to be such as will 

 render it possible to enlarge the alternating ones. This 

 cannot take place, if the solid angles are formed of three 

 faces. In this case, the resolution too is impossible ; and 

 therefore, the two methods require the rhombohedral solid 

 angles to consist of six faces. The third method requires 

 moreover the condition of the first method to take place, 

 else it would be necessary to enlarge all the faces ; and con- 

 sequently no resolution at all could take place. By this 

 last condition, the hexahedral trigonal-icositetrahedron is 

 excluded, and the method becomes applicable only to the 

 tetracontaoctahedron, which, however, can be resolved ac- 

 cording to both the other methods. 



The axes undergo very remarkable changes by the reso- 

 lution. The rhombohedral ones remain unaltered ; the 

 prismatic axes disappear entirely in all the halves ; the 

 changes in the pyramidal axes are various. If the third 

 method has been applied, they remain constant like the 

 rhombohedral axes ; they are changed into prismatic axes 



