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 heart[493]. {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 locomotion[494]. 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.
CHAPTER XI THE LOGARITHMIC SPIRAL
The very numerous examples of spiral conformation which we meet with in our studies of organic form are peculiarly adapted to mathematical methods of investigation. But ere we begin to study them, we must take care to define our terms, and we had better also attempt some rough preliminary classification of the objects with which we shall have to deal.
In general terms, a Spiral Curve is a line which, starting from a point of origin, continually diminishes in curvature as it recedes from that point; or, in other words, whose radius of curvature continually increases. This definition is wide enough to include a number of different curves, but on the other hand it excludes at least one which in popular speech we are apt to confuse with a true spiral. This latter curve is the simple Screw, or cylindrical Helix, which curve, as is very evident, neither starts from a definite origin, nor varies in its curvature as it proceeds. The “spiral” thickening of a woody plant-cell, the “spiral” thread within an insect’s tracheal tube, or the “spiral” twist and twine of a climbing stem are not, mathematically speaking, spirals at all, but screws or helices. They belong to a distinct, though by no means very remote, family of curves. Some of these helical forms we have just now treated of, briefly and parenthetically, under the subject of Geodetics.
Of true organic spirals we have no lack[495]. We think at once of the beautiful spiral curves of the horns of ruminants, and of the still more varied, if not more beautiful, spirals of molluscan shells. Closely related spirals may be traced in the arrangement {494} of the florets in the sunflower; a true spiral, though not, by the way, so easy of investigation, is presented to us by the outline of a cordate leaf; and yet again, we can recognise typical though transitory spirals in the coil of an elephant’s trunk, in the “circling {495} spires” of a snake, in the coils of a cuttle-fish’s arm, or of a monkey’s or a chameleon’s tail.
Fig. 237. The shell of Nautilus pompilius, from a radiograph: to shew the logarithmic spiral of the shell, together with the arrangement of the internal septa. (From Messrs Green and Gardiner, in Proc. Malacol. Soc. II, 1897.)
Among such forms as these, and the many others which we might easily add to them, it is obvious that we have to do with things which, though mathematically similar, are biologically speaking fundamentally different. And not only are they biologically remote, but they are also physically different, in regard to the nature of the forces to which they are severally due. For in the first place, the spiral coil of the elephant’s trunk or of the chameleon’s tail is, as we have said, but a transitory configuration, and is plainly the result of certain muscular forces acting upon a structure of a definite, and normally an essentially different, form. It is rather a position, or an attitude, than a form, in the sense in which we have been using this latter term; and, unlike most of the forms which we have been studying, it has little or no direct relation to the phenomenon of Growth.