ilO On Numerical Fropor lions 



two other molecules united in the same manner, at first in one 

 and the same plan, so that the two lines mutually cut each other 

 into two equal parts ; and if wa afterwards remove them, by 

 keeping them always in a situation parallel to that which they 

 had on this plan, we shall obtain a tetrahedron, wliich will be 

 regular in the case onlv where the two lines were equally perpen- 

 dicular to each otlier, and where they have been removed from 

 each other to a distance, which is to their length as 1 : V 2. 



Let us now conceive three molecules joined by lines form- 

 ing any given triangle ; let us place in the same plan an- 

 other triangle equal to the first, and of which the situation is 

 such that the two triangles have their' centre of gravity at the 

 same point, and their equal sides resjiectively parallel. Separating 

 tliese two triangles, so that the three sides of each triangle 

 may remain constantly parallel to their primitive position, we 

 shall obtain six points placed as they ought to be to represent 

 the six summits of an octahedron, which will be regular only 

 in the case where we have thus joined two equilateral triangles, 

 and where we have separated them perpendicularly to their plan 

 from a quantity which is to one of their sides as -v/ 2 ^3. 



If we suppose in the case of the tetrahedron which we draw 

 by the two lines of which we have spoken, two plans parallel 

 bttween them, and we place in each of them a line which re- 

 presents the position in which will be found the line of the other 

 plan before they had been separated, the extremities of these 

 two new lines will be the four summits of a symmetrical tetra- 

 hedron at the first, which shall have its centre of gravity at the 

 same point, and the eight summits of those two tetrahedrons 

 joined in this manner will be those of a parallelopipedon. It is 

 thus that the parallelopipedon form results from the union of two 

 tetrahedrons. It is easy to see that when the two tetrahedrons 

 are regular the parallelopipedon becomes a cube ; a rhomboidal 

 parallelopipedon when the tetrahedrons are regular pyramids ; 

 a straight prism with rhomboidal bases when four ridges of every 

 tetrahedron are equal to each other ; and finall}', the base of 

 this prism becomes a square v/hcn to this condition is added the 

 equality of the two other ridges. In the case of the octahedron, 

 if we place in the same way in the plans of two triangles, of 

 which we have spoken, those which represent the position in 

 which will be found the triangle of the other plan before they 

 had been separated, the six angles of these two new triangles 

 will be the six summits of an octahedron symmetrical to the 

 first, which shall have its centre of gravity at the same points ; 

 and the twelve summits of those two octahedrons, thus joined, 

 will be those of a hexahedral prism : this form results, there- 

 fore^ 



