that make the contour of a fish. But we must rather look for it wherever the organism retains for us, and still presents to us at a single view, the successive phases of preceding growth, the successive magnitudes attained, the successive outlines occupied, as the organism or a part thereof pursued the even tenour of its growth, year by year and day by day. And it easily follows from this, that it is in the hard parts of organisms, and not the soft, fleshy, actively growing parts, that this spiral is commonly and charôÙacôÙterisôÙtiôÙcally found; not in the fresh mobile tissues whose form is constrained merely by the active forces of the moment; but in things like shell and tusk, and horn and claw, where the object is visibly composed of parts {516} successively, and permanently, laid down. In the main, the logarithmic spiral is charôÙacôÙterôÙisôÙtic, not of the living tissues, but of the dead. And for the same reason, it will always or nearly always be accompanied, and adorned, by a pattern formed of ãlines of growth,ã the lasting record of earlier and successive stages of form and magnitude.
It is evident that the spiral curve of the shell is, in a sense, a vector diagram of its own growth; for it shews at each instant of time, the direction, radial and tangential, of growth, and the unchanging ratio of velocities in these directions. Regarding the actual velocity of growth in the shell, we know very little (or practically nothing), by way of experimental measurement; but if we make a certain simple assumption, then we may go a good deal further in our description of the logarithmic spiral as it appears in this concrete case.
Let us make the assumption that similar increments are added to the shell in equal times; that is to say, that the amount of growth in unit time is