772 THE EQUIANGULAR SPIRAL [cH. 



ments were, as before, made with a fine pair of compasses and a 

 diagonal scale. The sight was assisted by a magnifying glass. 

 In a parallel colimin to the following admeasurements are the 

 terms of a geometric progression, whose first term is the width of 

 the widest whorl measured, and whose common ratio is 1-1804. 



Turritella duplicata 



Widths of successive Terms of a geometrical progression, 



whorls, measured in whose first term is the width of 



inches and parts the widest whorl, and whose 



of an inch common ratio is 1-1804 



1-31 1-310 



1-12 1110 



0-94 0-940 



0-80 0-797 



0-67 0-675 



0-57 0-572 



0-48 0-484 



0-41 0-410 



The close coincidence between the observed and the calculated 

 figures is very remarkable, and is amply sufiicient to justify the 

 conclusion that we are here deaUng with a true logarithmic spiral*. 



Nevertheless, in order to verify his conclusion still further, 

 and to get partially rid of the inaccuracies due to successive small 

 measurements, Moseley proceeded to investigate the same shell, 

 measuring not single whorls but groups of whorls taken several 

 at a time: making use of the following property of a geometrical 

 progression, that ''if /x represent the ratio of the sum of every 

 even number (m) of its terms to the sum of half that number of 

 terms, then the common ratio (r) of the series is represented by 

 the formula 



2 



r={fjL- 1)^." 



* Moseley, writing a hundred years ago, uses an obsolete nomenclature which 

 is apt to be very misleading. His Turbo duplicatus, of Linnaeus, is now Turritella 

 duplicata, the common large Indian Turritella, a slender, tapering shell with a 

 very beautiful spiral, about six or seven inches long. But the operculum which 

 he describes as that of Turbo does indeed belong to that genus, sensu stricto; it is 

 the well-known calcareous operculum or "eyestone" of some such common species 

 as Turbo petholatus. Turritella has a very different kind of operculum, a thin 

 chitinous disc in the form of a close spiral coil, not nearly fiUing up the aperture 

 of the shell. Moseley's Turbo phasianus is again no true Turbo, but is (to judge 

 from his figure) Phasianella bulimoides Lam. =P. australis (Gmelin); and his 

 Buccinum aubulatum is Terebra subulata (L.). 



