824 THE EQUIANGULAR SPIRAL [ch. 



to be said for withdrawing them, as brachiopods and others have 

 been withdrawn, from Cuvier's great class of the Mollusca. But 

 whether bivalves and univalves be near relations or no is not the 

 question. Both of them secrete a shell, and in both the shell 

 grows by the successive addition of similar parts, gnomon after 

 gnomon; so that in both the equiangular spiral makes, and is 

 bound to make, its appearance. There is a mathematical analogy 

 between the two; but it has no more bearing on zoological classi- 

 fication than has the still closer Hkeness between Nautilus and the 

 nautiloid Foraminifera. 



The generating curve is particularly well seen in the bivalve, 

 where it simply constitutes what we call "the outline of the shell." 

 It is for the most part a plane curve, but not always ; for there are 

 forms such as Hippopus, Tridacna and many Cockles, or Rhynchonella 

 and Spirifer among the Brachiopods, in which the edges of the two 

 valves interlock, and others, such as Pholas, Mya, etc., where they 

 gape asunder. In such cases as these the generating curves, though 

 not plafie, are still conjugate, having a similar relation, but of 

 opposite sign, to a median plane of reference or of projection. There 

 are a few exceptional cases, e.g. Area (Parallelepipedon) tortuosa, where 

 there is no median plane of symmetry, but the generating curve, 

 and therefore the outline of the shell itself, is a tortuous curve in 

 three dimensions. 



A great variety of form is exhibited among the bivalves by these 

 generating curves. In many cases the curve or outline is all but 

 circular, as in Anomia, Sphaerium, Artemis, Isocardia] it is nearly 

 semicircular in Argiope; it is approximately elliptical in Anodon, 

 Lutraria, Orthis; it may be called semi-elliptical in Spirifer; it is 

 a nearly rectilinear 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 application of the mathematical theory of "trans- 

 formations," 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 



