498 THE LOGARITHMIC SPIRAL [ch. 



the actual force of growth will be acting at a tangent. So long 

 as the two forces continue to act, the curve will approach, but 

 will never attain, the direction of AB, perpendicular to the original 

 direction OA. If the external force be constant in amount the 

 curve will approximate to the form of a hyperbola ; and, at any 

 rate, it is obvious that it will never tend to assume a spiral 

 form. 



In Uke manner, if we consider a horizontal beam, fixed at one 

 end, the imposition of a weight at the other will bend the beam 

 into a curve, which, as the beam elongates or the weight increases, 

 will bring the weighted end nearer and nearer to the vertical. 

 But such a force, constant in direction, will obviously never curve 

 the beam into a spiral, — a fact so patent and obvious that it would 

 be superfluous to state it, were it not that some naturahsts have 

 been in the habit of invoking gravity as the force to which may be 

 attributed the spiral flexure of the shell. 



But if, on the other hand, the deflecting force be inherent in 

 the growing body, or so connected with it in a system that its 

 direction (instead of being constant, as in the former case) changes 

 with the direction of growth, and is perpendicular (or inclined at 

 some constant angle) to this changing direction of the growing 

 force, then it is plain that there is no such limit to the deflection 

 from the normal, but the growing curve will tend to wind round 

 and round its point of origin. In the typical case of the snail- 

 shell, such an intrinsic force is manifestly present in the action 

 of the columellar muscle. 



Many other simple illustrations can be given of a spiral course 

 being impressed upon what is primarily rectilinear motion, by 

 any steady deflecting force which the moving body carries, so 

 to speak, along with it, and which continually gives a lop-sided 

 tendency to its forward movement. For instance, we have been told 

 that a man or a horse, travelling over a great prairie, is very apt 

 to find himself, after a long day's journey, back again near to his 

 starting point. Here some small and imperceptible bias, such as 

 might for instance be caused by one leg being in a minute degree 

 longer or stronger than the other, has steadily deflected the forward 

 movement to one side; and has gradually brought the traveller 

 back, perhaps in a circle to the very point from which he set out. 



