522 



THE LOGARITHMIC SPIRAL 



[CH. 



(Fig. 264, 1), which traces have the form of curved Unes in 

 Turbo, and of straight hnes in (e.g.) Nerita (Fig. 264, 2) ; that 

 is to say, apart from the side constituting the outer edge of the 

 operculum (which side is always and of necessity curved) the 

 successive increments constitute curvilinear triangles in the one 

 case, and rectilinear triangles in the other. The sides of these 

 triangles are tangents to the spiral hne of the operculum, and 

 may be supposed to generate it by their consecutive intersections. 

 In a number of such opercula, Moseley measured the breadths 

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



Fig. 264. Opercula of (1) Turbo, (2) Nerita. (After Moseley.) 



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

 here is one example of his results. 



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

 whorls, measured from the pole. 



* 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^ , PP^, 

 which (in Fig. 264, 1) are radii vectores drawn to the points where they meet the 

 common boundary. 



