[B65]

If a number of equal ideal spheres be described, with their centres placed in two parallel the centre of each sphere at the distance (T) [e] from the the centres of six surrounding spheres in its own layer, & at the distance (T) [e] from the centres of the adjoining spheres in the other & parallel layer; then, if planes of intersection between the several spheres in both planes be formed flat intermediate surfaces, there will result figures, still spherical on their outer & free surfaces, but joined laterally to the other figures in the same layer by six equilateral rhombic faces, and joined to those in the other & parallel plane by three similar rhombic faces, united into a pyramid. All These rhombs will have the same two angles (viz. 70°. 31'. 44" and 109°. 28'. 16'), & be inclined to each other at the same angle (viz 120°) as in a rhombic dodecahedron. The whole compound figure will also exactly resemble the opposed cells in the comb of the Hive-Bee, except that in the latter, the edges of the six lateral rhombs (instead of being arched over by part of a sphere) have been produced into an hexagonal prism of indefinite length. As I shall have to allude to the statement several times I will for brevity sake call it our geometrical statement.