Fig. 286. Trochus niloticus, L.
Dolium and some Cones, is much the commoner of the two. And such tendency to decrease may lead to θ becoming a negative angle; in which case we have a depressed spire, as in the Cypraeae.
When we find a “reversed shell,” a whelk or a snail for instance whose spire winds to the left instead of to the right, we may describe it mathematically by the simple statement that the angle θ has changed sign. In the genus Ampullaria, or Apple-snails, inhabiting tropical or sub-tropical rivers, we have a remarkable condition; for in certain “species” the spiral turns to the right, in others to the left, and in others again we have a flattened {561} “discoid” shell; and furthermore we have numerous intermediate stages, on either side, shewing right and left-handed spirals of varying degrees of acuteness[527]. In this case, the angle θ may be said to vary, within the limits of a genus, from somewhere about 35° to somewhere about 125°.
The angle of retardation (β) is very small in Dentalium and Patella; it is very large in Haliotis. It becomes infinite in Argonauta and in Cypraea. Connected with the angle of retardation are the various possibilities of contact or separation, in various degrees, between adjacent whorls in the discoid, and between both adjacent and opposite whorls in the turbinated shell. But with these phenomena we have already dealt sufficiently.
Of Bivalve Shells.
Hitherto we have dealt only with univalve shells, and it is in these that all the mathematical problems connected with the spiral, or helico-spiral, are best illustrated. But the case of the bivalve shell, of Lamellibranchs or of Brachiopods, presents no essential difference, save only that we have here to do with two conjugate spirals, whose two axes have a definite relation to one another, and some freedom of rotatory movement relatively to one another.
The generating curve is particularly well seen in the bivalve, where it simply constitutes what we call “the outline of the shell.” It is for the most part a plane curve, but not always; for there are forms, such as Hippopus, Tridacna and many Cockles, or Rhynchonella and Spirifer among the Brachiopods, in which the edges of the two valves interlock, and others, such as Pholas, Mya, etc., where in part they fail to meet. In such cases as these the generating curves are conjugate, having a similar relation, but of opposite sign, to a median plane of reference. A great variety of form is exhibited by these generating curves among the bivalves. In a good many cases the curve is approximately circular, as in Anomia, Cyclas, Artemis, Isocardia; it is nearly semi-circular in Argiope. It is approximately elliptical in Orthis and in Anodon; it may be called semi-elliptical in Spirifer. It is a nearly rectilinear {562} triangle in Lithocardium, and a curvilinear triangle in Mactra. Many apparently diverse but more or less related forms may be shewn to be deformations of a common type, by a simple application of the mathematical theory of “Transformations,” which we shall have to study in a later chapter. In such a series as is furnished, for instance, by Gervillea, Perna, Avicula, Modiola, Mytilus, etc., a “simple shear” accounts for most, if not all, of the apparent differences.
Upon the surface of the bivalve shell we usually see with great clearness the “lines of growth” which represent the successive margins of the shell, or in other words the successive positions assumed during growth by the growing generating curve; and we have a good illustration, accordingly, of how it is characteristic of the generating curve that it should constantly increase, while never altering its geometric similarity.
Underlying these “lines of growth,” which are so characteristic of a molluscan shell (and of not a few other organic formations), there is, then, a “law of growth” which we may attempt to enquire into and which may be illustrated in various ways. The simplest cases are those in which we can study the lines of growth on a more or less flattened shell, such as the one valve of an oyster, a Pecten or a Tellina, or some such bivalve mollusc. Here around an origin, the so-called “umbo” of the shell, we have a series of curves, sometimes nearly circular, sometimes elliptical, and often asymmetrical; and such curves are obviously not “concentric,” though we are often apt to call them so, but are always “co-axial.” This manner of arrangement may be illustrated by various analogies. We might for instance compare it to a series of waves, radiating outwards from a point, through a medium which offered a resistance increasing, with the angle of divergence, according to some simple law. We may find another, and perhaps a simpler illustration as follows: