of small or all but imperceptible magnitude, has been introduced into the case; so that the ratio, logã₤r =ã₤ö¡ã₤logã₤öÝ, is no longer constant, but varies slightly, and in accordance with some simple law. {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 mathôÙeôÙmatôÙiôÙcal point, while the molluscan shell starts with a little embryonic shell, or central chamber (the ãprotoconchã 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 =ã₤mã₤+ã₤aÿ£¢ö¡ã₤.
There would seem, by the way, to be considerable confusion in the books with regard to the so-called ãprotoconch.ã In many cases it is a definite structure, 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 ãprotoconchã 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 magnitude of the space a are determined by the ãangle of retardation,ã or ratio of rate of growth between the inner and outer curves