756 



THE EQUIANGULAR SPIRAL 



[CH. 



ellipses, would have been shot off in spiral orbits from the sun, the 

 equiangular spiral being one case thereof.* 



A singular instance of the same spiral is given by the route which 

 certain insects follow towards a candle. Owing to the structure 

 of their compound eyes, these insects do not look straight ahead 

 but make for a light which they see abeam, at a certain angle. 

 As. they continually adjust their path to this constant angle, a spiral 

 pathway brings them to their destination at last|. 



In mechanical structures, curvature is essentially a mechanical 

 phenomenon. It is found in flexible structures as the result of 



Fig. 351. Spiral path of an insect, 

 as it draws towards a light. 

 From Wigglesworth (after van 

 Buddenbroek). 



Fig. 352. Dynamical aspect 

 of the equiangular spiral. 



bending, or it may be introduced into the construction for the 

 purpose of resisting such a bending-moment. But neither shell nor 

 tooth nor claw are flexible structures ; they have not been bent into 

 their pecuhar curvature, they have grown into it. 



We may for a moment, however, regard the equiangular or logarithmic 

 spiral of our shell from the dynamical point of view, by looking on growth 

 itself as the force concerned. In the growing structure, let growth at any 

 point P be resolved into a force F acting along the line joining P to a pole 0, 

 and a force T acting in a direction perpendicular to OP; and let the magnitude 

 of these forces (or of these rates of growth) remain constant. It follows that 



* Principia, I, 9; ii, 15, On these "Cotes's spirals" see Tait and Steele, p. 147. 

 t Cf. W. Buddenbroek, Sitzungsber. Heidelb. Akad., 1917; V. H. Wigglesworth, 

 Insect Physiology, 1839, p. 167. 



