112 On Numerical Proportions 



also in this way that I have obtained, by combining other num- 

 bers, tetrahedrons and octahedrons, the various representative 

 forms required lor the explanation upon the same principles of 

 all the combinations in a determinate ratio which are known to 

 me. 



On attempting to join tetrahedrons and octahedrons in all 

 possible ways, we find that there result from most of them re- 

 presentative forms in which the various molecules are arranged 

 in an irregular manner, and that there are some in one direction, 

 without there being any in another direction corresponding to 

 the first. All these forms ought to be rejected, and we observe 

 in fact, that the proportions which they suppose in chemical 

 combinations are not met with in nature. If we try, for in- 

 stance, to combine tetrahedrons and octahedrons, so as that the 

 number of the former shall be the half of that of the latter, we 

 find only awkward forms which do not present any regularity, 

 or any proportion between the relative sizes of their different 

 faces. Hence we ought to conclude that a body A, the particles 

 of which have for their representative form tetrahedrons, and a 

 body B, of which the particles are represented by octahedrons, 

 will not unite so as that there shall be in the combination one 

 proportion of A and two proportions of B : on the contrary, 

 this combination will be easy between two proportions of A and 

 one of B, since two tetrahedrons a-nd one octahedron form by 

 their junction a dodecahedron. In the same case, A and B will 

 unite in equal proportions by means of two forms which I shall 

 describe, and in which the number of the tetrahedrons is equal 

 to that of the octahedrons. 



1. An octahedron may be joined with a tetrahedron, by 

 placing the summits of the octahedron on the prolongations of 

 the lines which, issuing from tlia centre of gravity of the tetra- 

 hedron, pass by the middles of its six ridges : we thus form a 

 polyhedron with ten summits and sixteen triangular faces, four 

 equilateral and twelve isbsceles, to which I shall give the name 

 of hexadecahedron. 



2. Two octahedrons joined in a hexahedral prism may be 

 joined with two tetrahedrons forming a cube, in a manner ana- 

 logous to that in which an octahedron is united to a cube in the 

 dodecahedron. In order to form a clear idea of this combina- 

 tion, we must consider one of the diagonals of the cube as the 

 aKis of this polyhedron, and elevate for it a plan perpendicular 

 passing by the centre of the cube. This plan will cut six of 

 its ridges into two equal parts, the points of division being si- 

 tuated like the six angles of a regular hexagon, by placing 

 thereon the middles of the six vertical ridges of a hexahedral 

 prism formed by the junction of two regular octahedrons : the 



twenty 



