XI] ITS MATHEMATICAL PROPERTIES 507 



Furthermore we see how any slight change in any one of the 

 forces P, T, L will tend to modify the angle a, and produce a slight 

 departure from the absolute regularity of the logarithmic spiral. 

 Such slight departures from the absolute simplicity and uniformity 

 of the theoretic law we shall not be surprised to find, more or less 

 frequently, in Nature, in the complex system of forces presented 

 by the living organism. 



In the growth of a shell, we can conceive no simpler law than 

 this, namely, that it shall widen and lengthen in the same unvarying 

 proportions : and this simplest of law^s is that which Nature tends 

 to follow. The shell, like the creature within it, grows in size 

 hut does not change its shape ; and the existence of this constant 

 relativity of growth, or constant similarity of form, is of the essence, 

 and may be made the basis of a definition, of the logarithmic 

 spiral. 



Such a definition, though not commonly used by mathe- 

 maticians, has been occasionally employed ; and it is one from 

 which the other properties of the curve can be deduced with 

 great ease and simplicity. In mathematical language it would run 

 as follows : " Any [plane] curve proceeding from a fixed point 

 (which is called the pole), and such that the arc intercepted between 

 this point and any other whatsoever on the curve is always similar 

 to itself, is called an equiangular, or logarithmic, spiral*." 



In this definition, we have what is probably the most funda- 

 mental and "intrinsic" property of the curve,. namely the property 

 of continual similarity : and this is indeed the very property by 

 reason of which it is peculiarly associated with organic growth in 

 such structures as the horn or the shell, or the scorpioid cyme 

 which is described on p. 502. For it is peculiarly characteristic 

 of the spiral of a shell, for instance, that (under all normal circum- 

 stances) it does not alter its shape as it grows ; each increment is 

 geometrically similar to its predecessor, and the whole, at any 

 epoch, is similar to what constituted the whole at another and an 

 earUer epoch. We feel no surprise when the animal which secretes 

 the shell, or any other animal whatsoever, grows by such sym- 



* See an interesting paper by Whitworth, W. A., "The Equiangular Spiral, 

 its chief properties proved geometrically," in the Messenger of Mathematics (1), 

 I, p. 5, 1862. 



