XI] 



ITS MATHEMATICAL PROPERTIES 



789 



are independent of the shape and size of the embryo, and depend only (as we 

 shall see better presently) on the direction and relative rate of growth of the 

 double contour of the shell*. 



rd0 



Now that we have dealt, m a. general way, with some of the more 

 obvious properties of the equiangular or logarithmic spiral, let us 

 consider certain of them a httle more particularly, keeping in view 

 as our chief object of study the range of variation of the molluscan 

 shell. 



There is yet another equation to the logarithmic 

 spiral, very commonly employed, and without the 

 help of which we cannot get far. It is as follows : 



This follows directly from the fact that the angle 

 a (the angle between the radius vector and the 

 tangent to the curve) is constant. 



For then, 



tan a (= tan (f>) = rdd/dr; 

 therefore dr/r = dO cot a, 



and, integrating, log r = ^ cot a. 



or 



Fig. 377. 



It is easy to see (we might indeed have noted it before) that the 

 logarithmic spiral is but a plotting in polar coordinates of increase 

 by compound interest. For if A be the "amount" of £1 in one year 

 (A = 1 + a, where a is the rate of interest), and PA the amount 

 of P in one year, then the whole amount, M, in t years is M = PA^\ 

 this, provided that interest is payable once a year. But, as we are 

 taught by algebra, and as we have seen in our study of growth, 

 this formula becomes Pe"* when the intervals of time between the 

 payments of interest decrease without limit, that is to say, wh^n we 

 may consider growth to be continuous. And this formula PeP-^ is 

 precisely that of our logarithmic spiral, when we represent the time 



* J. F. Blake (cf. infra, p. 793) says of Naumann's formula: "By such a 

 modification he hoped to bring the measurements of actual shells more into 

 harmony with calculation. The errors of observation, however, are always greater 

 than this change would correct — if founded on fact, which is doubtful; and all 

 practical advantage is lost by the compUcation of the equations." 



