530 THE LOGARITHMIC SPIRAL [ch. 



the mouth) could be restored from a broken fragment. For if we 

 draw our tangents to the cone, it follows from the symmetry 

 of the figure that we can continue the projection of the sutural 

 line, and so mark off the successive whorls, by simply drawing 

 a series of consecutive parallels, and by then filling into the 

 quadrilaterals so marked off a series of curves similar to one 

 another, and to the whorls which 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 approximately 

 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 of small or all 

 but imperceptible magnitude, has been introduced into the case; 

 so that the ratio, log r = d log a, is no longer constant, but varies; 

 sUghtly, and in accordance with some simple law. 



