TERMINOLOGY. . 



the subordinate ones which arc parallel to them, again, till 

 the rest disappear : or, 



3. the alternating faces from the principal points, and those 

 from the subordinate ones, which arc not parallel to them, 

 till the rest disappear. 



The enlarged faces, if they can geometrically include a 

 space by themselves, will produce a form of many axes, 

 which is a Half, if only one of the three methods has been 

 applied ; a Fourth, if two at the same time, or subsequent- 

 ly, have been employed in resolving the original form. 



If, in the first mode of resolution, instead of enlarging 

 the faces contiguous to the principal points, we enlarge 

 those from the subordinate ones, the result from the same 

 original form will be another half, equal and similar to the 

 first, but in a different situation from the other form. 

 The two halves can be brought into a parallel position, by 

 inverting the perpendicular axis of one of them. The posi- 

 tion now mentioned is called the inverse, in reference to the 

 other or normal position ; and one half of this kind is said 

 to be the Inverse of the other, which has been obtained in 

 the normal position. 



A similar result takes place, if, in applying the second 

 mode of resolution, those faces are made to disappear, 

 which produce the half in the normal position, while the 

 others are enlarged. Both these kinds of halves are re- 

 markable for the parallelism of their faces, which, however, 

 is a consequence of the method of resolution applied. 



The third method of resolution, if treated in the same 

 manner, enlarging those faces which had been made to dis- 

 appear before, and vice -versa, does not yield exactly the 

 same result. In respect to position, there is no difference 

 among the two halves ; but there is a difference according 

 to Right and Left, as mentioned in . . 67. and 76. The 

 same relation exists in the Fourths, which, like the halves 

 of the first and third method of resolution, have no paral- 

 lel faces. 



Some of the forms derived from the hexahedron, do not 

 allow of any resolution at all ; either because half the num- 



