XI] IN ITS DYNAMICAL ASPECT 757 



the resultant of the forces F and T (as PQ) makes a constant angle with 

 the radius vector. But a constant angle between tangent and radius vector 

 is a fundamental property of the "equiangular" spiral: the very property 

 with which Descartes started his investigation, and that which gives its 

 alternative name to the curve. 



In such a spiral, radial growth and growth in the direction of the curve 

 bear a constant ratio to one another. For, if we consider a consecutive 

 radius vector, 0P\ whose increment as compared with OP is dr, while ds is 

 the small arc PP', then dr/ds = cos a = constant. 



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 laws is that which Nature tends 

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

 but 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 equiangular spiral*. 



Such a. definition, though not commonly used by mathematicians, 

 has been occasionally employed; and it is one from which the other 

 properties of the curve can be deduced with great ease and sim- 

 plicity. 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 any two radii at 

 a given angle to one another is always similar to itself, is called an 

 equiangular, or logarithmic, spiral." 



In this definition, we have the most fundamental and "intrinsic" 

 property of the curve, namely the property of continual similarity, 

 and the very property by reason of which it is associated with 

 organic growth in such structures as the horn or the shell. For it 

 is peculiarly characteristic of the spiral shell, for instance, that it 

 does not alter as it grows ; each increment is similar to its predecessor, 

 and the whole, after every spurt of growth, is just Uke what it was 

 before. We feel no surprise when the animal which secretes 

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

 metrical expansion as to preserve- its form unchanged; though even 

 there, as we have already seen, the unchanging form denotes a nice 

 balance between the rates of growth in various directions, which is 



* See an interesting paper by W. A. Whitworth, The equiangular spiral, its chief 

 properties proved geometrically, Messenger of Mathematics (1), i, p. 5, 1862. The 

 celebrated Christian Wiener gave an explanation on these lines of the logarithmic 

 spiral of the shell, in his highly original Grundzuge der Weltordnung, 1863. 



