considerably greater increase of the constant angle, say to 50ô¯ or more, the shell would only have the appearance of a gentle curve; (3) the charôÙacôÙterôÙisôÙtic close coils of the Nautilus or Ammonite would be typically represented only when the constant angle lies within a few degrees on either side of about 80ô¯. The coiled up spiral of a Nautilus, with a constant angle of about 80ô¯, is about six times the length of its radius vector, or rather more than three times its own diameter; while that of an Ammonite, with a constant angle of, say, from 85ô¯ to 88ô¯, is from about six to fifteen times as long as its own diameter. And (4) as we approach an angle of 90ô¯ (at which point the spiral vanishes in a circle), the length of the coil increases with enormous rapidity. Our spiral would soon assume the appearance of the close coils of a Nummulite, and the successive increments of breadth in the successive whorls would become inappreciable to the eye. The logarithmic spiral of high constant angle would, as we have already seen, tend to become inôÙdisôÙtinôÙguishôÙable, without the most careful measurement, from an Archimedean spiral. And it is obvious, moreover, that our ordinary methods of {553} determining the constant angle of the spiral would not in these cases be accurate enough to enable us to measure the length of the coil: we should have to devise a new method, based on the measurement of radii or diameters over a large number of whorls.
The geometrical form of the shell involves many other beautiful properties, of great interest to the mathematician, but which it is not possible to reduce to such simple expressions as we have been content to use. For instance, we may obtain an equation which shall express completely the surface of any shell, in terms of polar or of rectangular