In each particular case the nature of the logarithmic spiral, as defined by its constant angle, will be chiefly determined by the rate of growth; that is to say by the particular ratio in which each new chamber exceeds its predecessor in magnitude. But shells having the same constant angle (α) may still differ from one another in many ways—in the general form and relative position of the chambers, in their extent of overlap, and hence in the actual contour and appearance of the shell; and these variations must correspond to particular distributions of energy within the system, which is governed as a whole by the law of minimum potential. {602}

Our problem, then, becomes reduced to that of investigating the possible con­fi­gur­a­tions which may be derived from the successive symmetrical apposition of similar bodies whose magnitudes are in continued proportion; and it is obvious, math­e­mat­i­cally speaking, that the various possible arrangements all come under the head of the logarithmic spiral, together with the limiting cases which it includes. Since the difference between one such form and another depends upon the numerical value of certain coefficients of magnitude, it is plain that any one must tend to pass into any other by small and continuous gradations; in other words, that a clas­si­fi­ca­tion of these forms must (like any clas­si­fi­ca­tion whatsoever of logarithmic spirals or of any other math­e­mat­i­cal curves), be theoretic or “artificial.” But we may easily make such an artificial clas­si­fi­ca­tion, and shall probably find it to agree, more or less, with the usual methods of clas­si­fi­ca­tion recognised by biological students of the Foraminifera.

Firstly we have the typically spiral shells, which occur in great variety, and which (for our present purpose) we need hardly describe further. We may merely notice how in certain cases, for instance Globigerina, the individual chambers are little removed from spheres; in other words, the area of contact between the adjacent chambers is small. In such forms as Cyclammina and Pulvinulina, on the other hand, each chamber is greatly overlapped by its successor, and the spherical form of each is lost in a marked asymmetry. Furthermore, in Globigerina and some others we have a tendency to the development of a helicoid spiral in space, as in so many of our univalve molluscan shells. The math­e­mat­i­cal problem of how a shell should grow, under the assumptions which we have made, would probably find its most general statement in such a case as that of Globigerina, where the whole organism lives and grows freely poised in a medium whose density is little different from its own.

The majority of spiral forms, on the other hand, are plane or discoid spirals, and we may take it that in these cases some force has exercised a controlling influence, so as to keep all the chambers in a plane. This is especially the case in forms like Rotalia or Discorbina (Fig. [316]), where the organism lives attached to a rock or a frond of sea-weed; for here (just as in the case of {603} the coiled tubes which little worms such as Serpula and Spirorbis make, under similar conditions) the spiral disc is itself asymmetrical, its whorls being markedly flattened on their attached surfaces.

Fig. 316. Discorbina bertheloti, d’Orb.

We may also conceive, among other conditions, the very curious case in which the protoplasm may entirely overspread the surface of the shell without reaching a position of equi­lib­rium; in which case a new shell will be formed enclosing the old one, {604} whether the old one be in the form of a single, solitary chamber, or have already attained to the form of a chambered or spiral shell. This is precisely what often happens in the case of Orbulina, when within the spherical shell we find a small, but perfectly formed, spiral “Globigerina[549].”

The various Miliolidae (Fig. [311]), only differ from the typical spiral, or rotaline forms, in the large angle subtended by each chamber, and the consequent abruptness of their inclination to each other. In these cases the outward appearance of a spiral tends to be lost; and it behoves us to recollect, all the more, that our spiral curve is not necessarily identical with the outline of the shell, but is always a line drawn through cor­re­spon­ding points in the successive chambers of the latter.

Fig. 317. A, Tertularia trochus, d’Orb. B, T. concava, Karrer.