XI] ITS MATHEMATICAL PROPERTIES 793 



of the one before ; we cannot expect to see either of them under the materialised 

 conditions of the actual shell. Our shells of small constant angle and gentle 

 curvature, such as Dentalium, are accordingly as much as we can ever expect 

 to see of their respective spirals. 



The spiral whose constant angle is 45° is both a simple case and 

 a mathematical curiosity; for, since the tangent of 45° is unity, 

 we need merely write r = e^\ which is as much as to say that the 

 natural logarithms of the radii give us, without more ado, the 

 vector angles. In this spiral the ratio between the breadths of 

 two consecutive whorls becomes r = e^^ ■= e2x3-i4i6_ Reducing this 

 from Naperian to common logs, we have log r = 2-729; whicH tells 

 us (by our tables) that the radius vector is multiphed about 535J 

 times after a whole polar revolution; it is doubled after turning 

 through a polar angle of less than 40°. Spirals of so low an angle 

 as 45° are common enough in tooth and claw, but rare among 

 molluscan shells; but one or two of the more strongly curved 

 Dentaliums, like D. elejphantinum, come near the mark. It is not 

 easy to determine the pole, nor to measure the constant angle, in 

 forms like these. 



Let us return to the problem of how to ascertain, by direct 

 measurement, the spiral angle of any particular shell. The method 

 aL-eady employed is only applicable to complete spirals, that is to 

 say to those in which the angle of the spiral is large, and further- 

 more it is inapplicable to portions, or broken fragments, of a shell. 

 In the case of the broken fragment, it is plain that the determination 

 of the angle is not merely of theoretic interest; but may be of great 

 practical use to the conchologist as the one and only way by which 

 he may restore the outline of the missing portions. We have a con- 

 siderable choice of methods, which have been summarised by, and 

 are partly due to, a very careful student of the Cephalopoda, the 

 late Rev. J. F. Blake*. 



(1) When an equiangular spiral rolls on a straight line, the pole 

 traces another straight line at an ^ngle to the first equal to the 

 complement of the constant angle of the spiral; for the contact 

 point is the instantaneous centre of the rotational movement, and 

 the line joining, it to the pole of the spiral is normal to the roulette 

 path of that point. But the difficult^ of determining the pole 



* On the measurement of the curves formed by Cephalopods and other MoUusks, 

 Phil. Mag. (5), vi, pp. 241-263, 1878. 



