We have made use in the last chapter of the math­e­mat­i­cal 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 “tracheides” 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 direction, and the more perfectly homogeneous and isotropic be the glass the more evenly will the split tend to follow the straight course in which it began. As a result, the crack in our test-tube is often seen to continue till the tube is split into a continuous spiral ribbon.

In a right cone, the spiral geodetic falls into closer and closer coils as the diameter of the cone narrows; and a very beautiful geodetic of this kind is exemplified in the sutural line of a spiral shell, such as Turritella, or in the striations which run parallel with the spiral suture. Similarly, in an ellipsoidal surface, we have a spiral geodetic, whose coils get closer together as we approach the ends of the long axis of the ellipse; in the splitting of the integument of an Equisetum-spore, by which are formed the spiral “elaters” of the spore, we have a case of this kind, though the spiral is not sufficiently prolonged to shew all its features in detail.

We have seen in these various cases, that our original definition of a geodetic requires to be modified; for it is only subject to conditions that it is “the shortest distance between two points on the surface of the solid,” and one of the commonest of these restricting conditions is that our geodetic may be constrained to go twice, or many times, round the surface on its way. In short, we must redefine our geodetic, as a curve drawn upon a surface, such that, if we take any two adjacent points on the curve, the curve gives the shortest distance between them. Again, in the geodetic systems which we meet with in morphology, it sometimes happens that we have two opposite systems of geodetic spirals separate and distinct from one another, as in Fig. 236, c; and it is also common to find the two systems interfering with one another, and forming a criss-cross, or reticulated arrangement. This is a very common source of reticulated patterns.

Among the ciliated Infusoria, we have in the spiral lines along which their cilia are arranged a great variety of beautiful geodetic curves; though it is probable enough that in some complicated cases these are not simple geodetics, but projections of curves other than a straight line upon the surface of the solid. {490}

Lastly, a very instructive case is furnished by the arrangement of the muscular fibres on the surface of a hollow organ, such as the heart or the stomach. Here we may consider the phenomenon from the point of view of mechanical efficiency, as well as from that of purely descriptive or objective anatomy. In fact we have an a priori right to expect that the muscular fibres covering such hollow or tubular organs will coincide with geodetic lines, in the sense in which we are now using the term. For if we imagine a contractile fibre, or elastic band, to be fixed by its two ends upon a curved surface, it is obvious that its first effort of contraction will tend to expend itself in accommodating the band to the form of the surface, in “stretching it tight,” or in other words in causing it to assume a direction which is the shortest possible line upon the surface between the two extremes: and it is only then that further contraction will have the effect of constricting the tube and so exercising pressure on its contents. Thus the muscular fibres, as they wind over the curved surface of an organ, arrange themselves automatically in geodesic curves: in precisely the same manner as we also automatically construct complex systems of geodesics whenever we wind a ball of wool or a spindle of tow, or when the skilful surgeon bandages a limb. In these latter cases we see the production of those “figures-of-eight,” to which, in the case for instance of the heart-muscles, Pettigrew and other anatomists have ascribed peculiar importance. In the case of both heart and stomach we must look upon these organs as developed from a simple cylindrical tube, after the fashion of the glass-blower, as is further discussed on p. 737 of this book, the modification of the simple cylinder consisting of various degrees of dilatation and of twisting. In the primitive undistorted cylinder, as in an artery or in the intestine, the muscular fibres run in geodetic lines, which as a rule are not spiral, but are merely either annular or longitudinal; these are the ordinary “circular and longitudinal coats,” which form the normal musculature of all tubular organs, or of the body-wall of a cylindrical worm492. If we consider each muscular fibre as an elastic strand, imbedded in the elastic membrane which constitutes the wall of the organ, it {491} is evident that, whatever be the distortion suffered by the entire organ, the individual fibre will follow the same course, which will still, in a sense, be a geodetic. But if the distortion be considerable, as for instance if the tube become bent upon itself, or if at some point its walls bulge outwards in a diverticulum or pouch, it is obvious that the old system of geodetics will only mark the shortest distance between two points more or less ap­prox­i­mate to one another, and that new systems of geodetics will tend to appear, peculiar to the new surface, and linking up points more remote from one another. This is evidently the case in the human stomach. We still have the systems, or their unobliterated remains, of circular and longitudinal muscles; but we also see two new systems of fibres, both obviously geodetic (or rather, when we look more closely, both parts of one and the same geodetic system), in the form of annuli encircling the pouch or diverticulum at the cardiac end of the stomach, and of oblique fibres taking a spiral course from the neighbourhood of the oesophagus over the sides of the organ.

In the heart we have a similar, but more complicated phenomenon. Its musculature consists, in great part, of the original simple system of circular and longitudinal muscles which enveloped the original arterial tubes, which tubes, after a process of local thickening, expansion, and especially twisting, came together to constitute the composite, or double, mammalian heart; and these systems of muscular fibres, geodetic to begin with, remain geodetic (in the sense in which we are using the word) after all the twisting to which the primitive cylindrical tube or tubes have been subjected. That is to say, these fibres still run their shortest possible course, from start to finish, over the complicated curved surface of the organ; and it is only because they do so that their contraction, or longitudinal shortening, is able to produce its direct effect, as Borelli well understood, in the contraction or systole of the heart493. {492}

As a parenthetic corollary to the case of the spiral pattern upon the wall of a cylindrical cell, we may consider for a moment the spiral line which many small organisms tend to follow in their path of locomotion494. The helicoid spiral, traced around the wall of our cylinder, may be explained as a composition of two velocities, one a uniform velocity in the direction of the axis of the cylinder, the other a uniform velocity in a circle perpendicular to the axis. In a somewhat analogous fashion, the smaller ciliated organisms, such as the ciliate and flagellate Infusoria, the Rotifers, the swarm-spores of various Protists, and so forth, have a tendency to combine a direct with a revolving path in their ordinary locomotion. The means of locomotion which they possess in their cilia are at best somewhat primitive and inefficient; they have no apparent means of steering, or modifying their direction; and, if their course tended to swerve ever so little to one side, the result would be to bring them round and round again in an ap­prox­i­mate­ly circular path (such as a man astray on the prairie is said to follow), with little or no progress in a definite longitudinal direction. But as a matter of fact, either through the direct action of their cilia or by reason of a more or less unsymmetrical form of the body, all these creatures tend more or less to rotate about their long axis while they swim. And this axial rotation, just as in the case of a rifle-bullet, causes their natural swerve, which is always in the same direction as regards their own bodies, to be in a continually changing direction as regards space: in short, to make a spiral course around, and more or less ap­prox­i­mate to, a straight axial line.