Experimental Mathematics. 



403 



Evolute and Involute. 



It is clear from the nature of the spiral that its evolute 

 must be some form of spiral which also merges ultimately in 

 the pole. 



Here is a logarithmic spiral cut out of wood, from which 

 the curve AP, fig. 6, has been drawn. An adjustable string is 



Fio;. 5. 



attached, and we find that of all the involutes which can be 

 drawn to this curve, with varying lengths of string, there is 

 one, viz. A'P^ which exactly fits the wooden model. There- 

 fore the spirals AP, A'P' are reciprocally evolute and in- 

 volute. And we have seen that they must have a common 

 pole; therefore the evolute of A'P' is the same curve turned 

 through an angle about 0. 



Again, let PP' be any position of the string ; then the 

 angle OPP' is complementary to OP'P, and therefore POP' 

 is a right angle. But the unwound portion of the string 

 PP' is obviously the length of the spiral from the pole to Po 

 Accordingly the length of the spiral at any point is the 

 intercept on the tangent between the radius vector and its 

 normal through the pole. This of course agrees with 



s =s i sec etdr = r sec a, 

 a being the angle of the spiral. 



