562 THE LOGARITHMIC SPIRAL [ch. 



triangle in Lithocardium, and a curvilinear triangle in Mactra. 

 Many apparently diverse but more or less related forms may be 

 shewn to be deformations of a common type, by a simple applica- 

 tion of the mathematical theory of "Transformations/' which we 

 shall have to study in a later chapter. In such a series as is 

 furnished, for instance, by Gervillea, Perna, Avicula, Modiola, 

 Mytilus, etc., a "simple shear" accounts for most, if not all, of 

 the apparent differences. 



Upon the surface of the bivalve shell we usually see with great 

 clearness the "lines of growth" which represent the successive 

 margins of the shell, or in other words the successive positions 

 assumed during growth by the growing generating curve ; and 

 we have a good illustration, accordingly, of how it is characteristic 

 of the generating curve that it should constantly increase, while 

 never altering its geometric similarity. 



Underlying these "Hues of growth," which are so characteristic 

 of a molluscan shell (and of not a few other organic formations), 

 there is, then, a " law of growth " which we may attempt to enquire 

 into and which may be illustrated in various ways. The simplest 

 cases are those in which we can study the lines of growth on a 

 more or less flattened shell, such as the one valve of an oyster, 

 a Pecten or a Tellina, or some such bivalve mollusc. Here around 

 an origin, the so-called "umbo" of the shell, we have a series of 

 curves, sometimes nearly circular, sometimes elliptical, and often 

 asymmetrical; and such curves are obviously not "concentric," 

 though we are often apt to call them so, but are always " co-axial." 

 This manner of arrangement may be illustrated by various 

 analogies. We might for instance compare it to a series of waves, 

 radiating outwards from a point, through a medium which offered 

 a resistance increasing, with the angle of divergence, according to 

 some simple law. We may find another, and perhaps a simpler 

 illustration as follows : 



In a very simple and beautiful theorem, Galileo shewed that, 

 if we imagine a number of inclined planes, or gutters, sloping 

 downwards (in a vertical plane) at various angles from a common 

 starting-point, and if we imagine a number of balls rolling each 

 down its own gutter under the influence of gravity (and without 

 hindrance from friction), then, at any given instant, the locus of 



