774 



THE EQUIANGULAR SPIRAL 



[CH. 



Lastly, we may if we please, in this simple case, reduce the whole 

 matter to arithmetic, and, dividing the width of each whorl by 

 that of the next, see that these quotients are nearly identical, and 

 that their mean value, or common ratio, is precisely that which 

 we have already found. 



We may shew, in the same simple fashion, by measurements 

 of Terebra (Fig. 397), how the relative widths of successive whorls 

 fall into a geometric progression, the criterion of a logarithmic 

 spiral. ^ 



Measurements of a large specimen (15-5 cm.) of Terebra maculata, 

 along three several tangents (a, 6, c) to the whorls. {After Chr. 

 Peterson, 1921.) 



Width 



Ratio 



Width 



Ratio 



Mean 



1-29 



1-32 



1-33 



Mean ratio, b31 



The logarithmic spiral is not only very beautifully manifested in 

 the molluscan shell*, but also, in certain cases, in the little Hd or 

 "operculum" by which the entrance to the tubular shell is closed 

 after the animal has withdrawn itself withinf. In the spiral shell 

 of Turbo, for instance, the operculum is a thick calcareous structure, 

 with a beautifully curved outhne, which grows by successive incre- 

 ments applied to one portion of its edge, and shews, accordingly, 

 a spiral line of growth upon its surface. The successive increments 

 leave their traces on the surface of the operculum (Fig. 370), which 

 traces have the form of curved lines in Turbo, and of straight hnes 



* It has even been proposed to use a logarithmic spiral in place of a table of 

 logarithms. Cf. Ant. Favaro, Statique graphique, Paris, 1885; Hele-Shaw, in 

 Brit. Ass. Rep. 1892, p. 403. 



t Cf. Fred. Haussay, Hecherches sur Vopercule, Diss., Paris, 1884. 



