Such a definition, though not commonly used by mathematicians, has been occasionally employed; and it is one from which the other properties of the curve can be deduced with great ease and simplicity. In mathematical language it would run as follows: “Any [plane] curve proceeding from a fixed point (which is called the pole), and such that the arc intercepted between this point and any other whatsoever on the curve is always similar to itself, is called an equiangular, or logarithmic, spiral[498].”
In this definition, we have what is probably the most fundamental and “intrinsic” property of the curve, namely the property of continual similarity: and this is indeed the very property by reason of which it is peculiarly associated with organic growth in such structures as the horn or the shell, or the scorpioid cyme which is described on p. [502]. For it is peculiarly characteristic of the spiral of a shell, for instance, that (under all normal circumstances) it does not alter its shape as it grows; each increment is geometrically similar to its predecessor, and the whole, at any epoch, is similar to what constituted the whole at another and an earlier epoch. We feel no surprise when the animal which secretes the shell, or any other animal whatsoever, grows by such {508} symmetrical expansion as to preserve its form unchanged; though even there, as we have already seen, the unchanging form denotes a nice balance between the rates of growth in various directions, which is but seldom accurately maintained for long. But the shell retains its unchanging form in spite of its asymmetrical growth; it grows at one end only, and so does the horn. And this remarkable property of increasing by terminal growth, but nevertheless retaining unchanged the form of the entire figure, is characteristic of the logarithmic spiral, and of no other mathematical curve.
Fig. 248.
We may at once illustrate this curious phenomenon by drawing the outline of a little Nautilus shell within a big one. We know, or we may see at once, that they are of precisely the same shape; so that, if we look at the little shell through a magnifying glass, it becomes identical with the big one. But we know, on the other hand, that the little Nautilus shell grows into the big one, not by uniform growth or magnification in all directions, as is (though only approximately) the case when the boy grows into the man, but by growing at one end only.
Though of all curves, this property of continued similarity is found only in the logarithmic spiral, there are very many rectilinear figures in which it may be observed. For instance, as we may easily see, it holds good of any right cone; for evidently, in Fig. [248], the little inner cone (represented in its triangular section) may become identical with the larger one either by magnification all round (as in a), or simply by an increment at one end (as in b); indeed, in the case of the cone, we have yet a third possibility, for the same result is attained when it increases all round, save only at the base, that is to say when the triangular section increases {509} on two of its sides, as in c. All this is closely associated with the fact, which we have already noted, that the Nautilus shell is but a cone rolled up; in other words, the cone is but a particular variety, or “limiting case,” of the spiral shell.
This property, which we so easily recognise in the cone, would seem to have engaged the particular attention of the most ancient mathematicians even from the days of Pythagoras, and so, with little doubt, from the more ancient days of that Egyptian school whence he derived the foundations of his learning[499]; and its bearing on our biological problem of the shell, though apparently indirect, is yet so close that it deserves our further consideration.
| Fig. 249. | Fig. 250. |