114 On Numerical Proportions 



situated as the four summits of a tetrahedron equal to the four* 

 vvhidi we wish to join, and we shall conceive that at each of 

 these points is placed one of the summits of each tetrahedron ; 

 whereas the three other summits of the same tetrahedron are in 

 the plan which passes Ijy the three other points, and correspond 

 to the middles of the intervals which they leave between them. 

 I shall give to the polvhedron resulting from this comI)ination of 

 four tetrahedrons so uniterl, the name of tetra-tetrahedron. 

 This polvhedron has sixteen summits and twenty-eight tri- 

 angular faces ; four of which are equilateral and twenty-four 

 isosceles. 



We shall easily demonstrate, that if we prolong the plans of 

 the twelve isosceles faces adjacent to the four equilateral faces of 

 tlie side where they join to those faces, the prolongation of those 

 j)hins will meet by threes outside of the tetrahedron, in four 

 points corresponding to the middles of its four e([uilateral faces, 

 and which will be the summits of a fifth tetrahedron equal to 

 the four preceding ones : bv uniting it with them, we have the 

 twenty summits of the polyhedron which I have called penta- 

 tctrahedron, and which has twenty-four faces ; viz. twelve qua- 

 drilaterals and twelve isosceles triangles. 



If we again consider twelve points situated with regard to 

 each other as the middles of the twelve ridges of a cube, and 

 place a tetrahedron so that, its centre of gravity being at the 

 same point with that of the cube, two of its ridges opposite pass 

 by four of these points ; and if we do the same thing in succes- 

 sion with respect to five other tetrahedrons, in order that the 

 number of the summits should he the same in all directions, we 

 t-hall obti'Jn a polyhedron with 24 summits and 14 faces, six 

 squares, and eight hexagons, which I shall call hexa-tetrahe- 

 dron. ■ 



These hexagons, all equal to each other, will have each three 

 sides greater and three smaller, which will be to each other as 

 1 : VY-l. 



This polyhedron is evidently an octohedron only, of which 

 the summits are truncated by plans perpendicular to its three 

 axes : its combinations with other representative forms are more 

 numerous than those of any of the preceding polyhedrons. 



We may at first combine it with an octohedron situated in 

 such a way that, having its centre of gravity at the same point, 

 all the faces and all the ridges of this octohedron shall be pa- 

 rallel to those of the octohedron ; from which we may conceive 

 that the hexa-tetrahcdron is the result of truncalures, by being 

 solely subjected t j the condition of its dimensions being less than 

 tiiose of the latter, in order that the polyhedron thus formed 



may 



