Geometrical Forms of Tw^binated aiid Discoid Shells, 465 



From the known properties of the logarithmic spiral the author 

 concludes that the law of the geometrical description of turbinated 

 shells is, that they are generated by the revolution about a fixed 

 axis, (namely, the axis of the shell,) of a curve, which continually 

 varies its dimensions according to the law, that each linear incre- 

 ment shall vary as the existing dimensions of the line of which it is 

 the increment. If such be the law of nature, the whorls of the shell, 

 as well as the spires on the operculum, must have the form of the 

 logarithmic spiral ; and that this is likewise the case is shown by 

 the almost perfect accordance of numerical results, deduced from the 

 property of that curve, with those deduced from a great variety of 

 careful measurements made of the distances between successive 

 whorls on radii vectores drawn on shells of the Turbo duplicatus. 

 Turbo phasianus, Buccinum subulatum, and in a fine section of a 

 Nautilus pompilius. The author further states that, besides the results 

 given in the paper, a great number of measurements were similarly 

 made upon other shells of the genera Trochus, Strombus, and Murex, 

 all confirmatory of the law in question. 



One of the interesting deductions which the author has derived 

 from the prevalence of this law in the generation of the shells of a 

 large class of mollusca, is that a distinction maybe expected to arise 

 with regard to the growth of land and of aquatic shells, the latter 

 serving both as a habitation and as a float to the animal which forms 

 it ; and that, although the facility of varying its position at every 

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

 enlargement of the capacity of the 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 increment of the bulk of the animal. 



Another conclusion deducible from the law of formation here con- 

 sidered is, that the growth of the animal, corresponding to a given 

 increment in the angle of the generating curve, will always be pro- 

 portional to the bulk it has then attained : and if the physical vital 

 energies of the animal be proportional to its actual bulk, its growth, 

 in any given time, will be proportional to its growth up to that time. 

 Hence the whole angle of revolution of the curve generating the 

 shell will be proportional to the whole corresponding time of the 

 animal's growth ; and therefore, the whole number of whorls and 

 parts of whorls will, at any period, be proportional to its age. 



The form of the molluscous animal remaining always similar to 

 itself, the surface of the organ by which it deposits its shell will 

 vary as the square of the linear dimensions ; but as the deposition 

 of its shell must vary as the cube of the same dimensions, there must 

 be an increased functional activity of the organ, varying as the sim- 

 ple linear dimensions. 



Since to each species of shell there must correspond a particular 

 number expressing the ratio of the geometrical progression of the 

 similar successive linear dimensions of the whorls ; and since the 

 constant angle of the particular logarithmic spiral, which is aflfected 

 by that species of shell, is deducible from this number, the author 



Phil. Mag. S. 3. Vol. 13. No. 84. Dec. 1838. 2 H 



