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 measured by the areas subtended by equal angles. Thus, in the outer whorl of a spiral shell a definite area marked out by ridges, tubercles, etc., has very different linear dimensions to the cor­re­spon­ding areas of the inner whorl, but the symmetry of the figure implies that it subtends an equal angle with these; and it is reasonable to suppose that the successive regions, marked out in this way by successive natural boundaries or patterns, are produced in equal intervals of time.

If this be so, the radii measured from the pole to the boundary of the shell will in each case be proportional to the velocity of growth at this point upon the circumference, and at the time when it corresponded with the outer lip, or region of active growth; and while the direction of the radius vector corresponds with the direction of growth in thickness of the animal, so does the tangent to the curve correspond with the direction, for the time being, of the animal’s growth in length. The successive radii are a measure of the acceleration of growth, and the spiral curve of the shell itself is no other than the hodograph of the growth of the contained organism. {517}

So far as we have now gone, we have studied the elementary properties of the logarithmic spiral, including its fundamental property of continued similarity; and we have accordingly learned that the shell or the horn tends necessarily to assume the form of this math­e­mat­i­cal figure, because in these structures growth proceeds by successive increments, which are always similar in form, similarly situated, and of constant relative magnitude one to another. Our chief objects in enquiring further into the math­e­mat­i­cal properties of the logarithmic spiral will be: (1) to find means of confirming and verifying the fact that the shell (or other organic curve) is actually a logarithmic spiral; (2) to learn how, by the properties of the curve, we may further extend our knowledge or simplify our descriptions of the shell; and (3) to understand the factors by which the char­ac­ter­is­tic form of any particular logarithmic spiral is determined, and so to comprehend the nature of the specific or generic characters by which one spiral shell is found to differ from another.

Of the elementary properties of the logarithmic spiral, so far as we have now enumerated them, the following are those which we may most easily in­ves­ti­gate in the concrete case, such as we have to do with in the molluscan shell: (1) that the polar radii of points whose vectorial angles are in arithmetical progression, are themselves in geometrical progression; and (2) that the tangent at any point of a logarithmic spiral makes a constant angle (called the angle of the spiral) with the polar radius vector.

Fig. 261.

The former of these two propositions may be written in what is, perhaps, a simpler form, as follows: radii which form equal angles about the pole of the logarithmic spiral, are themselves continued proportionals. That is to say, in Fig. [261], when the angle ROQ is equal to the angle QOP, then OR : OQ :: OQ : OP.

A particular case of this proposition is when the equal angles are each angles of 360°: that is to say when in each case the radius vector makes a complete revolution, and when, therefore P, Q and R all lie upon the same radius. {518}