540 



THE LOGARITHMIC SPIRAL 



[CH, 



intuslabiatus ; these measurements Grabaii gives for every 45° of 

 arc, but I have only set forth one quarter of these measurements, 

 that is to say, the breadths of successive whorls measured along 

 one diameter on both sides of the pole. The ratio between 

 alternate measurements is therefore the same ratio as Moseley 

 adopted, namely the ratio of breadth between contiguous whorls 

 along a radius vector. I have then added to these observed 

 values the corresponding calculated values of the angle a, as 

 obtained from our usual formula. 



There is considerable irregularity in the ratios derived from 

 these measurements, but it will be seen that this irregularity only 

 imphes a variation of the angle of the spiral between about 85° 

 and 87° ; and the values fluctuate pretty regularly about the 

 mean, which is 86° 15'. Considering the difficulty of measuring 

 the whorls, especially towards the centre, and in particular the 

 difl&culty of determining with precise accuracy the position of the 

 pole, it is clear that in such a case as this we are scarcely justified 

 in asserting that the law of the logarithmic spiral is departed from. 



In some cases, however, it is undoubtedly departed from. 

 Here for instance is another table from Grabau, shewing the 

 corresponding ratios in an Ammonite of the group of Arcestes 

 tornatus. In this case we see a distinct tendency of the ratios to 



Ammonites tornatus. 



Breadth of whorls 

 (180" apart) 



0-25 mm, 



0-30 



0-35 



0-50 



0-70 



100 



1-40 



210 



3-05 



4-70 



7-60 

 12-10 

 19-35 



Ratio of breadth of 

 successive whorls 

 (3fi0° apart) 



1-400 

 1-667 

 2-000 

 2-000 

 2-000 

 2-100 

 2-179 

 2-238 

 2-492 

 2-574 

 2-546 



The spiral 

 angle (a) as 

 calculated 



86° 56' 



85 21 



83 42 



83 42 



83 42 



83 16 



82 56 



82 42 



81 44 



81 27 



81 33 



Mean 83° 22' 



