XI] 



OF NAUMANN'S CONCHOSPIRAL 



531 



Some writers, such as Naumann and Grabau, maintained that 

 the molluscan spiral was no true logarithmic spiral, but differed 

 from it specifically, and they gave to it the name of Conchospiral. 

 They pointed out that the logarithmic spiral originates in a 

 mathematical point, while the molluscan shell starts with a little 

 embryonic shell, or central chamber (the "protoconch" of the 

 conchologists), around which the spiral is subsequently wrapped. 

 It is plain that this undoubted and obvious- fact need not 

 affect the logarithmic law of the shell as a whole; we have 

 only to add a small constant to our equation, which becomes 

 r = m + a^. 



There would seem, by the way, to be considerable confusion 

 in the books with regard to the so-called "protoconch." In many 

 cases it is a definite fitructure, of simple form, representing the 

 more or less globular embryonic shell before it began to elongate 

 into its conical or spiral form. But in many cases what is described 

 as the "protoconch" is merely an empty space in the middle of 

 the spiral coil, resulting from the fact that the actual spiral shell 

 has a definite magnitude to begin with, and that we cannot follow 

 it down to its vanishing point in infinity. For instance, in the 

 accompanying figure, the large space a 

 is styled the protoconch, but it is the 

 little bulbous or hemispherical chamber 

 within it, at the end of the spire, 

 which is the real beginning of the 

 tubular shell. The form and magni- 

 tude of the space a are determined by 

 the "angle of retardation," or ratio of 

 rate of growth between the inner and 

 outer curves of the spiral shell. They 

 are independent of the shape and size of 

 the embryo, and depend only (as we shall 

 see better presently) on the direction and relative rate of growth 

 of the double contour of the shell. 



Fig. 268. 



Now that we have dealt, in a very general way, with some of 

 the more obvious properties of the logarithmic spiral, let us 

 consider certain of them a little more particularly, keeping in 



34—2 



