And ·07207 is the logarithm of 1·1805.
Fig. 263. Operculum of Turbo.
The logarithmic spiral is not only very beautifully manifested in the molluscan shell, but also, in certain cases, in the little lid or “operculum” by which the entrance to the tubular shell is closed after the animal has withdrawn itself within. In the spiral shell of Turbo, for instance, the operculum is a thick calcareous structure, with a beautifully curved outline, which grows by successive increments applied to one portion of its edge, and shews, accordingly, a spiral line of growth upon its surface. The successive increments leave their traces on the surface of the operculum {522} (Fig. [264], 1), which traces have the form of curved lines in Turbo, and of straight lines in (e.g.) Nerita (Fig. [264], 2); that is to say, apart from the side constituting the outer edge of the operculum (which side is always and of necessity curved) the successive increments constitute curvilinear triangles in the one case, and rectilinear triangles in the other. The sides of these triangles are tangents to the spiral line of the operculum, and may be supposed to generate it by their consecutive intersections.
Fig. 264. Opercula of (1) Turbo, (2) Nerita. (After Moseley.)
In a number of such opercula, Moseley measured the breadths of the successive whorls along a radius vector[508], just in the same way as he did with the entire shell in the foregoing cases; and here is one example of his results.
| Distance | Ratio | Distance | Ratio | Distance | Ratio | Distance | Ratio |
|---|---|---|---|---|---|---|---|
| ·24 | ·16 | ·2 | ·18 | ||||
| 2·28 | 2·31 | 2·30 | 2·30 | ||||
| ·55 | ·37 | ·6 | ·42 | ||||
| 2·32 | 2·30 | 2·30 | 2·24 | ||||
| 1·28 | ·85 | 1·38 | ·94 |
{523}
The ratio is approximately constant, and this spiral also is, therefore, a logarithmic spiral.