788 



THE EQUIANGULAR SPIEAL 



[CH. 



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 = 6 log a, is no longer con- 

 stant but varies shghtly, and in accordance with some simple law. 

 Some ' writers, such as Naumann* and Grabau, maintained that 

 the moUuscan spiral was no true logarithmic spiral, but differed 

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

 They said that the logarithmic spiral originates in a mathematical 

 point, while the molluscan shell starts with a Httle embryonic, shell, 

 or central chamber (the "protoconch" of the conchologists), around 

 which the spiral is subsequently wrapped. But this need not affect 

 the logarithmic law of the shell as a whole ; indeed we have already 

 allowed for it by writing our equation in the form r = ma^. And 

 Grabauf, while he clung to Naumann's conchospiral against 

 Moseley's logarithmic spiral, confessed that they were so much ahke 

 that ordinary measurements would seldom shew a difference between 

 them. 



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 

 structure, of simple form, representing the more or 

 less globular embyyonic 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 must have some 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 mag- 

 nitude of the spa^ce 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 



Fig. 376. 



* C. F. Naumann, Beitrag zur Konchyliometrie, Poggend. Ann. l, p. 223, 1840; 

 Ueber die Spiralen der Ammoniten, ibid, li, p. 245, 1840; ibid, liv, p. 541, 1845; etc. 

 (See also p. 755.) Cf. also Lehmann, Die von Seyfriedsche Konchyliensammlung 

 und das Windungsgesetz von einigen Planorben, Constanz, 1855. 



f A. H. Grabau, Ueber die Naumannsche Conchospirale, und ihre Bedeutung 

 fiir die Conchyliometrie, Inauguraldiss., Leipzig, 1872; Ueber die Spiralen der 

 Conchylien, etc., Leipzig Progr. No. 502, 1880; cf. Sb. naturf. Gesellsch. Leipzig, 

 1881, pp. 23-32. 



