tube of a variable section, so that a piston driven along it shall, by one side of its margin, coincide continually with its surface as it advances, provided only that the piston be made at the same time continually to revolve in its own plane.ã
As Moseley puts it: ãThat the same edge which fitted a portion of the first less section should be capable of adjustment, so as to fit a portion of the next similar but greater section, supposes a geometrical provision in the curved form of the chamber of great apparent complication and difficulty. But God hath bestowed upon this humble architect the practical skill of a learned geometrician, and he makes this provision with admirable precision in that curvature of the logarithmic spiral which he gives to the section of the shell. This curvature obtaining, he has only to turn his operculum slightly round in its own plane as he advances it into each newly formed portion of his chamber, to adapt one margin of it to a new and larger surface and a different curvature, leaving the space to be filled up by increasing the operculum wholly on the other margin.ã
But in many, and indeed more numerous Gastropod mollusca, the operculum does not grow in this remarkable spiral fashion, but by the apparently much simpler process of accretion by concentric rings. This suggests to us another mathôÙeôÙmatôÙiôÙcal {524} feature of the logarithmic spiral. We have already seen that the logarithmic spiral has a number of ãlimiting cases,ã apparently very diverse from one another. Thus the right cone is a logarithmic spiral in which the revolution of the radius vector is infinitely slow; and, in the same sense, the straight line itself is a limiting case of the logarithmic spiral. The spiral of Archimedes, though not a limiting case of the logarithmic spiral, closely resembles one in which the