776 THE EQUIANGULAR SPIRAL [ch. 



of tlie successive whorls along a radius vector*, just in the same 



way as he did with the entire shell in the foregoing cases; 

 here is one example of his results. 



Operculum of Turbo sp. ; breadth {in inches) of successive 

 whorls, mea^redfrom the pole 



and 



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 advances into its new position, and comes to 

 occupy a new-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 



* As the successive increments evidently constitute similar figures, similarly 

 related to the pole (P), it follows that their linear dimensions are to one another 

 as the radii vectores drawn to similar points in them: for instance as PPj, PP2> 

 which (in Fig. 370) are radii vectores drawn to the points where they meet the 

 common boundary. 



