We have made use in the last chapter of the mathôÙeôÙmatôÙiôÙcal principle of Geodetics (or Geodesics) in order to explain the conformation of a certain class of sponge-spicules; but the principle is of much wider application in morphology, and would seem to deserve attention which it has not yet received.
Defining, meanwhile, our geodetic line (as we have already done) as the shortest distance between two points on the surface of a solid of revolution, we find that the geodetics of the cylinder give us one of the simplest of cases. Here it is plain that the geodetics are of three kinds: (1) a series of annuli around the cylinder, that is to say, a system of circles, in planes parallel to one another and at right angles to the axis of the cylinder (Fig. 236, a); (2) a series of straight lines parallel to the axis; and (3) a series of spiral curves winding round the wall of the cylinder (b, c). These three systems are all of frequent occurrence, and are all illustrated in the local thickenings of the wall of the cylindrical cells or vessels of plants.
The spiral, or rather helicoid, geodetic is particularly common in cylindrical structures, and is beautifully shewn for instance in the spiral coil which stiffens the tracheal tubes of an insect, or the so-called ãtracheidesã of a woody stem. A similar {489} phenomenon is often witnessed in the splitting of a glass tube. If a crack appear in a thin tube, such as a test-tube, it has a tendency to be prolonged in its own