OF TURBINATED AND DISCOID SHELLS. 359 



Why the Molliisks who inhabit turbinated and discoid shells should, in the pro- 

 gressive increase of their spiral dwellings, affect the particular law of the logarithmic 

 spiral, is easily to be understood. Providence has subjected the instinct which shapes 

 out each, to a rigid uniformity of operation. 



This uniformity manifests itself in turbinated shells in respect to their axes. Now 

 the law of the logarithmic spiral, considered under its more general form of a curve 

 of double curvature, is the only one according to which the MoUusk can wind its 

 spiral dwelling in an uniform direction through the space round its axis, in respect to 

 that axis. Under this general form it may be geometrically defined as the curve 

 whose tangent retains always the same angular position in respect to its axis*, and 

 in respect to a line drawn from the point where it touches the curve perpendicular to 

 the axis ; or in other words, which traverses the space round the axis always in the 

 same direction in respect to it. 



A second property of the logarithmic spiral, equally referring itself to the uniformity 

 of the animal's operations about the axis of its shell, is this ; that it has everywhere the 

 same geometrical curvature, and is the only curve except the circle-f- which possesses 

 this property. 



Certain physiological facts having reference to the growth of the Mollusk are de- 

 ducible from the geometrical description of its shell. If it be a land shell, its capacity 

 may be supposed (reasoning from that principle of economy which is an observable 

 law in Nature) to be precisely sufficient for the reception of the animal who built it. 

 If it be an aquatic shell, it serves the animal at once as a habitation and as a float ; 

 enabling it to vary its buoyancy according as it leaves a greater or a less portion of 

 the narrower extremity of its chamber unoccupied, and thus to ascend or descend in 

 the water, at will. Now that its buoyancy, and therefore the facility of thus varying its 

 position, may remain the same at every period of its growth, it is necessary that the 

 increment of the capacity of its float should bear a constant ratio to the corresponding 

 increment of its body, a ratio which always assigns a greater amount to the increment 

 of the capacity of the shell than to the corresponding increments of the animal's bulk. 

 Thus the chamber of the aquatic shell is increased, not only, as is the land shell, so 

 that it may contain the greater bulk of the Mollusk, but so that more and more of it 

 may be left unoccupied. Now the capacity of the shell and the dimensions of the 

 animal began together, and they increase thus in a constant ratio ; the whole bulk of 

 the animal bears therefore a constant ratio, of greater inequality, to the whole capacity 

 of the shell, in aquatic shells : in land shells, it is probably equal to it. 



Now let the generating curve of a shell be conceived to describe, as it revolves 

 round its axis, a series of successive equal angles, represented each by A 0. Corre- 

 sponding to these equal increments of the angle of revolution of the generating 



* So that moved parallel to itself until it intersected the axis, it would always intersect it at the same angle. 



t The circle may, in fact, be considered a logarithmic spiral, the constant inclination of whose tangent to its 



radius vector is a right angle. Of all curves, this spiral, considered as thus including the circle, is the simplest. 



