dodecahedron and icosahedron, simple as they are from the mathôÙeôÙmatôÙiôÙcal point of view, never occur. Not only do these latter never occur in CrysôÙtalôÙlogôÙraphy, but (as is explained in text-books of that science) it has been shewn that they cannot occur, owing to the fact that their indices (or numbers expressing the relation of the faces to the three primary axes) involve an irrational quantity: whereas it is a fundamental law of crysôÙtalôÙlogôÙraphy, involved in the whole theory of space-partitioning, that ãthe indices of any and every face of a crystal are small whole numbers85.ã At the same time, an imperfect pentagonal dodecahedron, whose pentagonal sides are non-equilateral, is common among crystals. If we may safely judge from Haeckelãs figures, the pentagonal dodecahedron of the Radiolarian is perfectly regular, and we must presume, accordingly, that it is not brought about by principles of space-partitioning similar to those which manifest themselves in the phenomenon of cryôÙstalôÙliôÙsaôÙtion. It will be observed that in all these radiolarian polyhedral shells, the surface of each external facet is formed of a minute hexagonal network, whose probable origin, in relation to a vesicular structure, is such as we have already discussed.
In certain allied Radiolaria (Fig. 232), which, like the dodecahedral form figured in Fig. 231, 5, have twenty radial spines, these latter are commonly described as being arranged in a certain very singular way. It is stated that their arrangement may be referred {481} to a series of five parallel circles on the sphere, corôÙreôÙsponôÙding to the equator (c), the tropics (b, d) and the polar circles (a, e); and that beginning with four equidistant spines in the equator, we have alternating whorls of four, radiating outwards from the sphere in each of the other parallel zones. This rule was laid down by the celebrated Johannes Mû¥ller, and has ever since been used and quoted as Mû¥llerãs law. The chief point in this alleged arrangement which strikes us at first sight as very curious, is that there is said to be no spine at either pole; and when we come to examine carefully the figure of the organism, we find that the received
description does not do justice to the facts. We see, in the first place, from such figures as Figs. 232, 234, that here, unlike our former cases, the radial spines issue through the facets (and through all the facets) of the polyhedron, instead of through its solid angles; and accordingly, that our