520 THE LOGARITHMIC SPIRAL ' [ch. 



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 fx 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 



r = (^ - ir." 



Accordingly, Moseley made the following measurements, 

 beginning from the second and third whorls respectively: 



"By the ratios of the two first admeasurements, the formula 

 gives 



r = (1-645)^ = 1-1804. 



By the mean of the ratios deduced from the second two admeasure- 

 ments, it gives 



r = (1-389)* = 1-1806. 



" It is scarcely possible to imagine a more accurate verification 

 than is deduced from these larger admeasurements, and we may 

 with safety annex to the species Turbo duplicatus the character- 

 istic number 1-18." 



By similar and equally concordant observations, Moseley found 

 for Ttirho phasianus the characteristic ratio, 1-75; and for Bucci- 

 num subulatum that of 1-13. 



From the table referring to Turbo duplicatus, on page 519, it 

 is perhaps worth while to illustrate the logarithmic statement of 

 the same facts: that is to say, the elementary corollary to the 

 fact that the successive radii are in geometric progression, that 

 their logarithms differ from one another by a constant amount. 



