are still intact in the broken shell.
But the use of the helicometer soon shewed that it was by no means universally the case that one and the same right cone was tangent to all the turbinate whorls; in other words, there was not always one specific apical angle which held good for the entire system. In the great majority of cases, it is true, the same tangent touches all the whorls, and is a straight line. But in others, as in the large Cerithium nodosum, such a line is slightly convex to the axis of the shell; and in the short spire of Dolium, for instance, the convexity is marked, and the apex of the spire is a distinct cusp. On the other hand, in Pupa and Clausilia, the common tangent is concave to the axis of the shell.
So also is it, as we shall presently see, among the Ammonites: where there are some species in which the ratio of whorl to whorl remains, to all appearance, perfectly constant; others in which it gradually, though only slightly increases; and others again in which it slightly and gradually falls away. It is obvious that, among the manifold possibilities of growth, such conditions as these are very easily conceivable. It is much more remarkable that, among these shells, the relative velocities of growth in various dimensions should be as constant as it is, than that there should be an occasional departure from perfect regularity. In such cases as these latter, the logarithmic law of growth is only apôÙproxôÙiôÙmateôÙly true. The shell is no longer to be represented as a right cone which has been rolled up, but as a cone which had grown trumpet-shaped, or conversely whose mouth had narrowed in, and which in section is a curvilinear instead of a rectilinear triangle. But all that has happened is that a new factor, usually