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POPULAR SCIENCE REVIEW. 
over, a solid is formed, three of which form a rhombohedron. Four such 
rhombohedrons form the dodecahedron with rhombic faces, and seven of 
these rhombohedrons afford a prolonged dodecahedron, which is that of the 
cell of the bee. 
Mr. Willich lias subsequently discovered some new subdivisions of the 
cube, one of which is especially interesting. It is produced by carrying 
through the alternate solid angles of the cube six planes meeting at its 
centre ; these planes divide the cube into four equal and similar parts. 
The solids thus produced have precisely the same solid angle as that of the 
roof of the bee’s cell. Consequently if these solids be grouped back to back, 
one-half of a rhomboidal dodecahedron is produced. Each of these solids, 
equal to a fourth part of the cube, is formed of two triedral pyramids 
united together ; the angle of one is 109° 28' 1G", that of the other 90°. 
If each of these four double pyramids be divided into two other triangular 
pyramids, it will be found that the four pyramids having an angle of 109° 
28' 16", form the regular tetrahedron, the solid content of which is, conse- 
quently, one-third that of the cube from which it is derived ; that the four 
other pyramids, the angle of which is 90°, turned over and placed back to 
back, constitute a pyramid with four equilateral faces, equal to half the 
regular octahedron, and equal in consequence to two-thirds of the primary 
cube. Thus, the same analogy will be found to subsist between these three 
bodies as there is between the cylinder, the sphere, and the cone ; in the 
sense that the cube, the quadrangular pyramid derived from it, and 
the regular tetrahedron, are precisely in the ratio of the three numbers 
3, 2, 1. 
By another kind of section which consists in carrying the intersecting 
planes through the diagonals of the six faces of the cube, four solids are cut 
off, leaving in the centre a regular tetrahedron. These four solids are’similar 
to four of the solids previously obtained, and form similarly a quadran- 
gular pyramid which is half a regular octahedron. 
The solids resulting from the section of the cube have yielded to 
Mr. Willich, amongst many others, two polyhedrons very deserving of 
attention — one, with eight summits, results from the reunion of two tetra- 
hedrons, and its volume compared with that of the cube from which it is 
derived is 2f. The second has twelve summits, and its volume is four 
cubes. The former polyhedron with eight summits is reduced by the 
simple removal of its summits to the regular octahedron having equilateral 
triangles for the faces of the octahedron, the volume of which is one and 
a third times the volume of the cube ; it is thus the volume of the whole 
of the eight tetrahedrons which have been removed. 
The second polyhedron is formed, as already stated, of four cubes, and 
is reduced to the rhomboidal dodecahedron by the removal of its summits. 
These twelve summits removed, have the volume of two cubes, as has 
likewise the remaining dodecahedron. The sum of Mr. Willich’s re- 
searches renders the conclusion highly probable that the symmetrical 
polyhedrons which result from the union of the sections of the cube, or of 
the elementary solids of which it is composed, comprehend not only all the 
polyhedrons of geometry, but nearly all regular crystals. (Les Mondes , 
No. 4, pp. 55-57.) II. W. B. 
