XI] 



OF VARIOUS CEPHALOPODS 



551 



Ammonite with an Orthoceras, it is necessary to estimate the 

 length of the right cone which has, so to speak, been coiled up 

 into the spiral shell. Our problem then is. To find the length of 

 a plane logarithmic spiral, in terms of the radius and the constant 

 angle a. In the annexed diagram, if OP be a radius vector, OQ 

 a line of reference perpendicular to OP, and PQ a tangent to the 

 curve, PQ, or sec a, is equal in length to the spiral arc OP. And 

 this is practically obvious : for PP'/PR' = dsjdr = sec a, and 

 therefore sec a = s/r, or the ratio of arc to radius vector. 



Accordingly, the ratio of I, the total length, to r, the radius 



Fig. 282. 



vector up to which the total length is to be measured, is expressed 

 by a simple table of secants; as follows: 



Putting the same table inversely, so as to shew the total 



