cases and after all the labours of modern crystallographers, why this or that position is actually a place of minimum potential. In the case of our little Foraminifer (just as in the case of the crystal), let us then be content to assert that each drop or bead of protoplasm takes up a position of minimum potential energy, in relation to all the circumstances of the case; and let us not attempt, in the present state of our knowledge, to define that position of minimum potential by reference to angle of contact or any other particular condition of equiôÙlibôÙrium. In most cases the whole exposed surface, on some portion of which the drop must come to rest, is an extremely complicated one, and the forces involved constitute a system which, in its entirety, is more complicated still; but from the symmetry of the case and the continuity of the whole phenomenon, we are entitled to believe that the conditions are just the same, or very nearly the same, time after time, from one chamber to another: as the one chamber is conformed so will the next tend to be, and as the one is situated relatively to the system so will its successor tend to be situated in turn. The physical law of minimum potential (including also the law of minimal area) is all that we need in order to explain, in general terms, the continued similarity of one chamber to another; and the physiological law of growth, by which a continued proportionality of size tends to run through the series of successive chambers, impresses upon this series of similar increments the form of a logarithmic spiral.
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 (öÝ) may still differ from one another in many waysãin 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 conôÙfiôÙgurôÙaôÙtions which may be derived from the successive symmetrical apposition of similar bodies whose magnitudes are in continued proportion; and it is obvious, mathôÙeôÙmatôÙiôÙcally 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 clasôÙsiôÙfiôÙcaôÙtion of these forms must (like any clasôÙsiôÙfiôÙcaôÙtion whatsoever of logarithmic spirals or of any other mathôÙeôÙmatôÙiôÙcal curves), be theoretic or ãartificial.ã But we may easily make such an artificial clasôÙsiôÙfiôÙcaôÙtion, and shall probably find it to agree, more or less, with the usual methods of clasôÙsiôÙfiôÙcaôÙtion 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 mathôÙeôÙmatôÙiôÙcal 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.