A PARENTHETIC NOTE ON GEODETICS
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.
[Illustration: Fig. 236. Annular and spiral thickenings in the walls of
plant-cells.]
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 worm[492]. 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 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.
{493}