XI] AND THE SPIRAL OF ARCHIMEDES 753 



by the whole number of whorls, (or more strictly speaking) multiplied 

 by the whole angle (6) through which it has revolved: so that 

 r = ad. And it is also plain that the radius meets the curve (or 

 its tangent) at an angle which changes slowly but continuously, 

 and which tends towards a right angle as the whorls increase in 

 number and become more and more nearly circular. 



But, in contrast to this, in the equiangular spiral of the Nautilus 

 or the snail-shell or Globigerina, the whorls continually increase 

 in breadth, and do so in a steady and unchanging ratio. Our 

 definition is as follows: "If, instead of travelhng with a uniform 

 velocity, our point move along the radius vector with a velocity 

 increasing as its distance from the pole, then the path described is 



Fig. 349. The spiral of Archimedes. 



called an equiangular spiral." Each whorl which the radius vector 

 intersects will be broader than its predecessor in a definite ratio; 

 the radius vector will increase in length in geometrical progression, as 

 it sweeps through successive equal angles ; and the equation to the 

 spiral will be r = a^. As the spiral of Archimedes, in our example of 

 the coiled rope, might be looked upon as a coiled cyfinder, so (but 

 equally roughly) may the equiangular spiral, in the case of the shell, 

 be pictured as a cone coiled upon itself; and it is the conical shape 

 of the elephant's trunk or the chameleon's tail which makes them coil 

 into a rough simulacrum of an equiangular spiral. 



While the one spiral was known in ancient times, and was 

 investigated if not discovered by Archimedes, the other was first 



