This calculation deals, not only with the complete facets, but with the areas of the broken hexagons at the periphery of the circle. If we neglect these latter, and consider our whole field as consisting of successive rings of hexagons about a central one, we may obtain a still simpler rule[365]. For obviously, around our central hexagon there stands a zone of six, and around these a zone of twelve, and around these a zone of eighteen, and so on. And the total number, excluding the central hexagon, is accordingly: For one zone 6 = 2 × 3 = 3 × 1 × 2, For two zones 18 = 3 × 6 = 3 × 2 × 3, For three zones 36 = 4 × 9 = 3 × 3 × 4, For four zones 60 = 5 × 12 = 3 × 4 × 5, For five zones 90 = 6 x 15 = 3 × 5 × 6, and so forth. If _N_ be the number of zones, and if we add one to the above numbers for the odd central hexagon, the rule evidently is, that the total number, _H_, = 3_N_(_N_ + 1) + 1. Thus, if in a preparation of a fly’s cornea, I can count twenty-five facets in a line from a central one, the total number in the entire circular field is (3 × 25 × 26) + 1 = 1951[366]. ―――――――――― The same principles which account for the development of hexagonal symmetry hold true, as a matter of course, not only of individual _cells_ (in the biological sense), but of any close-packed bodies of uniform size and originally circular outline; and the hexagonal pattern is therefore of very common occurrence, under widely different circumstances. The curious reader may consult Sir Thomas Browne’s quaint and beautiful account, in the _Garden of Cyrus_, of hexagonal (and also of quincuncial) symmetry in plants and animals, which “doth neatly declare how nature Geometrizeth, and observeth order in all things.” {325} We have many varied examples of this principle among corals, wherever the polypes are in close juxtaposition, with neither empty space nor accumulations of matrix between their adjacent walls. _Favosites gothlandica_, for instance, furnishes us with an excellent example. In the great genus Lithostrotion we have some species that are “massive” and others that are “fasciculate”; in other words in some the long cylindrical corallites are in close contact with one another, and in others they are separate and loosely bundled (Fig. 127). Accordingly in the former the corallites are [Illustration: Fig. 127. _Lithostrotion Martini._ (After Nicholson.)] [Illustration: Fig. 128. _Cyathophyllum hexagonum._ (From Nicholson, after Zittel.)]