might be looked upon as a coiled cylinder, so may the logarithmic spiral, in the case of the shell, be pictured as a cone coiled upon itself.
Now it is obvious that if the whorls increase very slowly indeed, the logarithmic spiral will come to look like a spiral of Archimedes, with which however it never becomes identical; for it is incorrect to say, as is sometimes done, that the Archimedean spiral is a ãlimiting caseã of the logarithmic spiral. The Nummulite is a case in point. Here we have a large number of whorls, very narrow, very close together, and apparently of equal breadth, which give rise to an appearance similar to that of our coiled rope. And, in a case of this kind, we might actually find that the whorls were of equal breadth, being produced (as is apparently the case in the Nummulite) not by any very slow and gradual growth in thickness of a continuous tube, but by a succession of similar cells or chambers laid on, round and round, determined as to their size by constant surface-tension conditions and therefore of unvarying dimensions. But even in this case we should have no Archimedean spiral, but only a logarithmic spiral in which the constant angle approximated to 90ô¯. {505}
For, in the logarithmic spiral, when öÝ tends to 90ô¯, the expression r =ã₤ a ÿ£¢ ö¡ã₤cotã₤öÝ tends to r =ã₤ a (1ã₤+ã₤ö¡ã₤cotã₤öÝ); while the equation to the Archimedean spiral is r =ã₤ b ö¡. The nummulite must always have a central core, or initial cell, around which the coil is not only wrapped, but out of which it springs ; and this initial chamber corresponds to our aÿ£¢ãý in the expression r =ã₤ aÿ£¢ãý ã₤+ã₤ a ö¡ã₤cotã₤öÝ. The outer whorls resemble those of an Archimedean spiral, because of the other term a ö¡ã₤cotã₤öÝ in the same expression. It follows from