nearest neighbour in the floor above may be vertically over him, but we must not confine our attention to assemblages thus rectangularly grouped in vertical lines. § 2. Consider now any particular person _C_ (Fig. 1) on any intermediate floor, _D_ and _D′_ his nearest neighbours, _E_ and _E′_ his next nearest neighbours all on his own floor. His next next nearest neighbours on that floor will be in the positions _F_ and _F′_ in the diagram. Thus we see that each person _C_ is surrounded by six persons, _DD′_, _EE′_ and _FF′_, being his nearest, his next nearest, and his next next nearest neighbours on his own floor. Excluding for simplicity the special cases of rectangular grouping, we see that the angles of the six equal and similar triangles _CDE_, _CEF_, &c., are all acute: and because the six triangles are equal and similar we see that the three pairs of mutually remote sides of the hexagon _DEFD′E′F′_ are equal and parallel. [Illustration: FIG. 1] § 3. Let now _A_, _A′_, _A″_, &c., denote places of persons of the homogeneous assemblage on the floor immediately above, and _B_, _B′_, _B″_, &c. on the floor immediately below, the floor of _C_. In the diagram let _a_, _a′_, _a″_ be points in which the floor of _CDE_ is cut by perpendiculars to it through _A_, _A′_, _A″_ of the floor above, and _b_, _b′_, _b″_ by perpendiculars from _B_, _B′_, _B″_ of the floor below. Of all the perpendiculars from the floors immediately above and below, just two, one from each, cut the area of the parallelogram _CDEF_: and they cut it in points similarly situated in respect to the oppositely oriented triangles into which it is divided by either of its diagonals. Hence if _a_ lies in the triangle _CDE_, the other five triangles of the hexagon must be cut in the corresponding points, as shown in the diagram. Thus, if we think only of the floor of _C_ and of the floor immediately above it, we have points _A_, _A′_, _A″_ vertically above _a_, _a′_, _a″_. Imagine now a triangular pyramid, or tetrahedron, standing on the base _CDE_ and having _A_ for vertex: we see that each of its sides _ACD_, _ADE_, _AEC_, is an acute angled triangle, because, as we have already seen, _CDE_ is an acute angled triangle, and because the shortest of the three distances, _CA_, _DA_, _EA_, is (§ 2) greater than _CE_ (though it may be either greater than or less than _DE_). Hence the tetrahedron _CDEA_ has all its angles acute; not only the angles of its triangular faces, but the six angles between the planes of its four faces. This important theorem regarding homogeneous assemblages was given by Bravais, to whom we owe the whole doctrine of homogeneous assemblages in its most perfect simplicity and complete generality. Similarly we see that we have equal and similar tetrahedrons on the bases _D′CF_, _E′F′C_; and three other tetrahedrons below the floor of _C_, having the oppositely oriented triangles _CD′E′_, &c. for their bases and _B_, _B′_, _B″_ for their vertices.