XI] OF VARIOUS CEPHALOPODS 809 



changes in the spiral angle may be detected even in ammonites which 

 present nothing abnormal to the eye. But let us suppose that the 

 spiral angle increases somewhat rapidly; we shall then get a spiral 

 with gradually narrowing whorls, which condition is characteristic of 

 Oekotraustes, a subgenus of Ammonites. If on the other hand, the 

 angle a gradually diminishes, and even falls away to zero, we shall 

 have the spiral curve opening out, as it does in Scaphites, Ancyloceras 



Fig. 393. An ammonitoid shell (Macroscaphites) to shew change of 

 curvature. 



and Lituites, until the spiral coil is replaced by a spiral curye so 

 gentle as to seem all but straight. Lastly, there are a few cases, 

 such as BelleropJion expansus and some Goniatites, where the outer 

 spiral does not perceptibly change, but the whorls become more 

 "embracing" or the whole shell more involute. Here it is the 

 angle of retardation, the ratio of growth between the outer and 

 inner parts of the whorl, which undergoes a gradual change. 



In order to understand the relation of a close-coiled shell to its 

 straighter congeners, to compare (for example) an 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 is, to find the length of a plane equiangular spiral, in 

 terms of the radius and the constant angle a. Then, 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. In other words, the arc measured from the pole is 

 equal to the polar tangent*. And this is practically obvious: for 



* Descartes made this discovery, and records it in a letter to Mersenne, 1638. 

 The equiangular spiral was thus the first transcendental curve to be "rectified." 



