in Chemical Combinations. 113 



twentv summits of this polyhedron Avill be those of a new poly- 

 hedron which will have SO faces ; viz. six rectangular parallelo- 

 grams and 24 isosceles triangles: I shall give it the name of tri- 

 acontahedron. 



It is easy to see from this construction that the diagonal of 

 the cube is equal to that of the prism, and that in this way 

 all the summits of the triacontahedron are in one and the same 

 spherical surface. 



ft will be in vain to endeavour to form otiier combinations 

 presenting some regularity by combining two of the foregoing 

 polyhedrons. Let us proceed to another mode of combination. 

 If we consider twelve points placed with regard to each other 

 as the middles of the tuehe ridges of a cube, tliese points will 

 be situated by fours in three rectangular plans : hence it fol- 

 lows, that if we place at the first four the foiu- angles of the 

 square base common to the two pyramids of which one of the 

 octohedrons is composed, to the other four the four angles of 

 the base of a second ootohedron, and to tlie other four those 

 of a third octohedron, the summits of the three octohedrons 

 will be two and two in the intersections of the three rectangular 

 plans, and these 18 summits will be those of a polyhedron with 

 ^2 triangular faces, eight of which will be equilateral and 24 iso- 

 sceles : I shall give to this polyhedron the name of trioctohe- 

 dron, which refers to its generation. 



The trioctohedron may, like the octohedron, be combined 

 with two tetrahedrons forming a cube : for this purpose we shall 

 prolong the plans of its triangular isosceles faces from the side at 

 which they are joined with the equilateral faces until these plans 

 meet by threes o\itside of the polyhedron opposite those last 

 faces. The eight points thus determined are evidently situated 

 with respect to each other like the eight summits of a cube : 

 hence it follows tiiat we might thereon place the eight summits 

 of two tetrahedrons, the union of which with the trioctohedron 

 will form a polyhedron with 26 summits and 24 equal quadrilateral 

 faces. The trapezoidal form of the mineralogists is a particular 

 ease of this form, which results from a certain proportion between 

 the axis and the sides of the square bases "of the straight octo- 

 hedrons, of which we may conceive the trioctohedron to be 

 formed. I shall in general preserve the name of trapezoidal, 

 as expressing a property always belonging to it, whatever are the 

 dimensions of these octohedrons. 



It is not with tetrahedrons as with octohedrons : we cannot 

 unite three of them in a polyhedron which presents some regu- 

 larity ; but there exists one formed by the combination of four 

 ;etrahedrons. In order to obtain it, we shall consider four points 



Vol. 45. No. 202. /VZ. 1815. H situated 



