XI] OF CERTAIN OPERCULA 777 



apparent complication and difficulty. But God hath bestowed 

 upon this humble architect the practical skill of a learned geo- 

 metrician, and he makes this provision with admirable precision 

 in that curvature of the logarithmic spiral which he gives to the 

 section of the shell. This curvature obtaining, he has only to turn 

 his operculum shghtly round in its own plane as he advances it 

 into each newly formed portion of his chamber, to adapt one margin 

 of it to a new and larger surface and a different curvature, leaving 

 the space to be filled up by increasing the operculum wholly on 

 the other margin." The fact is that self-similar or gnomonic growth 

 is taking place both in the shell and its operculum; in both of them 

 growth is in reference to a fixed centre, and to a fixed axis through 

 that centre; and in both of them growth proceeds in geometric 

 progression from the centre while rotation takes place in arithmetic 

 progression about the axis. The same architecture which builds 

 the house constructs the door. Moreover, not only are house and 

 door governed by the same law of growth, but, growing together, 

 door and doorway adapt themselves to one another. 



The operculum of the gastropods varies from a more or less close -wound 

 spiral, as in Turritella, Trochus or Pleurotomaria, to cases in which accretion 

 takes place, by concentric (or more. or less excentric) rings, all round. But 

 these latter cases, so Mr Winckworth tells me, are not very common. Paludina 

 and Ampullaria come near to having a concentric operculum, and so do some 

 of the Murices, such as M. tribulus, and a few Turrids, and the genus Helicina; 

 but even these opercula probably begin as spirals, adding on their gnomonic 

 increments at one end or side, and only growing on all sides later on. There 

 would seem to be a truly concentric operculum in the Siphonium group of 

 Vermetus, where the spiral of the shell itself is lost, or nearly so; but it is 

 usually overgrown with Melobesia, and hard to see. 



Fig. 372. 



One more proposition, an all but self-evident one, we may make 

 passing mention of here: If upon any polar radius vector OP, 



