XI] OF CERTAIN OPERCULA 523 



The ratio is approximately constant, and this spiral also is, 

 therefore, a logarithmic spiral. 



But here comes in a very beautiful illustration of that property 

 of the logarithmic spiral which causes its whole shape to remain 

 unchanged, in spite of its apparently unsymmetrical, or unilateral, 

 mode of growth. For the mouth of the tubular shell, into which 

 the operculum has to fit, is growing or widening on all sides : 

 while the operculum is increasing, not by additions made at the 

 same time all round its margin, but by additions made only on 

 one side of it at each successive stage. One edge of the operculum 

 thus remains unaltered as it is advanced into each new position, 

 and as it is placed in a newly formed section of the tube, similar 

 to but greater than the last. Nevertheless, the two apposed 

 structures, the chamber and its plug, at all times fit one another 

 to perfection. The mechanical problem (by no means an easy 

 one), is thus solved : " How to shape a tube of a variable section, 

 so that a piston driven along it shall, by one side of its margin, 

 coincide continually with its surface as it advances, provided only 

 that the piston be made at the same time continually to revolve 

 in its own plane." 



As Moseley puts it : " That the same edge which fitted a portion 

 of the first less section should be capable of adjustment, so as to 

 fit a portion of the next similar but greater section, supposes 

 a geometrical provision in the curved form of the chamber of 

 great apparent complication and difficulty. But God hath 

 bestowed upon this humble architect the practical skill of a 

 learned geometrician, and he makes this provision with admirable 

 precision in that curvature of the logarithmic spiral which he 

 gives to the section of {he 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." 



But in many, and indeed more numerous Gastropod mollusca, 

 the operculum does not grow in this remarkable spiral fashion, 

 but by the apparently much simpler process of accretion by 

 concentric rings. This suggests to us another mathematical 



