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 one of its straighter congeners, to compare (for example) an {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 öÝ. 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 secã₤öÝ, is equal in length to the spiral arc OP. And this is practically obvious: for PPÿ£¢ãýã₤ãã₤PRÿ£¢ãý =ã₤dsã₤ãã₤dr =ã₤secã₤öÝ, and therefore secã₤öÝ =ã₤sã₤ãã₤r, or the ratio of arc to radius vector.
