1 90 On Numerical Proportions 



agonal. As we cannot establish betvvieen the respective dimen- 

 sions of the two polyhedrons, with the view of diminishing the 

 number of faces, any relation vvhich is symmetrical with respect 

 to all the similar ridges, I shall suppose that they are such that 

 the same sphere can be circumscribed for them ; and the poly- 

 hedron with forty-foiu' summits which results from this supposi- 

 tion, having seventy faces, viz. four hexagon, two square, and 

 sixty-four triangles, I shall give it the name of eptaconta-hedron. 



Finally, in order to combine the octo-tetrahedron with the 

 tii-octohedron, it is sufficient to observe that each of these poly- 

 hedrons has as n;aiiy summits as the other has faces, and reci- 

 procally ; we shall soon find that the positions of these summits 

 and of these faces are precisely such, that by placing the six 

 smnmits with four faces of the tri-octohedron on the perpendi- 

 calars raised in the midst of the six square faces of the octo- 

 tetrahedron, all the summits of each polyhedron answer to the 

 fates of the other. 



if we determine the respective dimensions of the polyhedrons, 

 so as that the ridges of the tri-octohedron vvhich join at the six 

 fummits just mentioned may pass by the middles of the ridges 

 of the square faces of the octo-tetrahedron, there will result from 

 the junction of those two representative forms a new polyhedron 

 Tvhich will have fiftv summits and seventy-two faces; viz. twenty- 

 four quadrilateral and forty-eight isosceles triangles. This poly- 

 hedron mav be considered as a tri-octohedron, of which the thirty- 

 t'vvo faces shall have been covered by as many triangular pyra- 

 ir.ids : this is the reason why 1 designate it by the name of py- 

 ramidated tri-octo'iedron, 



I shall only point out three other representative forms, com- 

 posed of four, five, and seveff octohedrons, and to which I have 

 given the names of tetra-octol!edron,penta-cctohedron,and epta- 

 octohedron ; and for the sake of brevity, 1 shall not speak of the 

 combinations v,'!iich may be made of those three representative 

 forms with the preceding polyhedrons. 



If we take notice that, one octohedron being given, there are 

 four different positions in which another octohedron of the same 

 si:^e forms with the first a hexahedral prism, we shall easily 

 conceive that four octohedrons situated in those four positions 

 will have their centre of gravity at the same point, and will form 

 a combination into which they will all enter in the same way. 

 This combination is the tetra-octohedron, which has twenty- 

 four summits and fourteen faces, of which six are octagon and 

 the eight others are equilateral triangles : by adding thereto the 

 same octohedron which has served to determine the respective 

 positions of the four octohedrons which we have combined, we 

 shall have the penta-octohedronj the summits of which are thirty 



in 



