Fig. 246.
In such a spiral, radial growth and growth in the direction of the curve bear a constant ratio to one another. For, if we consider a consecutive radius vector, OP′, whose increment as compared with OP is dr, while ds is the small arc PP′, then
dr ⁄ ds = cos α = constant.
In the concrete case of the shell, the distribution of forces will be, originally, a little more complicated than this, though by resolving the forces in question, the system may be reduced to this simple form. And furthermore, the actual distribution of forces will not always be identical; for example, there is a distinct difference between the cases (as in the snail) where a columellar muscle exerts a definite traction in the direction of the pole, and those (such as Nautilus) where there is no columellar muscle, and where some other force must be discovered, or postulated, to account for the flexure. In the most frequent case, we have, as in Fig. [247], three forces to deal with, acting at a point, p : L, acting
Fig. 247.
in the direction of the tangent to the curve, and representing the force of longitudinal growth; T, perpendicular to L, and representing the organism’s tendency to grow in breadth; and P, the traction exercised, in the direction of the pole, by the columellar muscle. Let us resolve L and T into components along P (namely A′, B′), and perpendicular to P (namely A, B); we have now only two forces to consider, viz. P − A′ − B′, and A − B. And these two latter we can again resolve, if we please, so as to deal only with forces in the direction of P and T. Now, the ratio of these forces remaining constant, the locus of the point p is an equiangular spiral. {507}
Furthermore we see how any slight change in any one of the forces P, T, L will tend to modify the angle α, and produce a slight departure from the absolute regularity of the logarithmic spiral. Such slight departures from the absolute simplicity and uniformity of the theoretic law we shall not be surprised to find, more or less frequently, in Nature, in the complex system of forces presented by the living organism.
In the growth of a shell, we can conceive no simpler law than this, namely, that it shall widen and lengthen in the same unvarying proportions: and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size but does not change its shape; and the existence of this constant relativity of growth, or constant similarity of form, is of the essence, and may be made the basis of a definition, of the logarithmic spiral.