THE SHAPES OF HORNS, AND OF TEETH OR TUSKS: WITH A NOTE ON TORSION
We have had so much to say on the subject of shell-spirals that we must
deal briefly with the analogous problems which are presented by the
horns of sheep, goats, antelopes and other horned quadrupeds; and all
the more, because these horn-spirals are on the whole less symmetrical,
less easy of measurement than those of the shell, and in other ways
also are less easy of investigation. Let us dispense altogether in this
case with mathematics; and be content with a very simple account of the
configuration of a horn.
There are three types of horn which deserve separate consideration:
firstly, the horn of the rhinoceros; secondly the horns of the sheep,
the goat, the ox or the antelope, that is to say, of the so-called
hollow-horned ruminants; and thirdly, the solid bony horns, or
“antlers,” which are characteristic of the deer.
The horn of the rhinoceros presents no difficulty. It is
physiologically equivalent to a mass of consolidated hairs, and,
like ordinary hair, it consists of non-living or “formed” material,
continually added to by the living tissues at its base. In section,
that is to say in the form of its “generating curve,” the horn is
approximately elliptical, with the long axis fore-and-aft, or, in some
species, nearly circular. Its longitudinal growth proceeds with a
maximum velocity anteriorly, and a minimum posteriorly; and the ratio
of these velocities being constant, the horn curves into the form of
a logarithmic spiral in the manner that we have already studied. The
spiral is of small angle, but in the longer-horned species, such as the
great white rhinoceros (Ceratorhinus), the spiral form is distinctly
to be recognised. As the horn {613} occupies a median position on the
head,—a position, that is to say, of symmetry in respect to the field
of force on either side,—there is no tendency towards a lateral twist,
and the horn accordingly develops as a _plane_ logarithmic spiral. When
two horns coexist, the hinder one is much the smaller of the two: which
is as much as to say that the force, or rate, of growth diminishes as
we pass backwards, just as it does within the limits of the single
horn. And accordingly, while both horns have _essentially_ the same
shape, the spiral curvature is less manifest in the second one, simply
by reason of its comparative shortness.
The paired horns of the ordinary hollow-horned ruminants, such as the
sheep or the goat, grow under conditions which are in some respects
similar, but which differ in other and important respects from the
conditions under which the horn grows in the rhinoceros. As regards
its structure, the entire horn now consists of a bony core with
a covering of skin; the inner, or dermal, layer of the latter is
richly supplied with nutrient blood-vessels, while the outer layer,
or epidermis, develops the fibrous or chitinous material, chemically
and morphologically akin to a mass of cemented or consolidated hairs,
which constitutes the “sheath” of the horn. A zone of active growth at
the base of the horn keeps adding to this sheath, ring by ring, and
the specific form of this annular zone is, accordingly, that of the
“generating curve” of the horn. Each horn no longer lies, as it does
in the rhinoceros, in the plane of symmetry of the animal of which it
forms a part; and the limited field of force concerned in the genesis
and growth of the horn is bound, accordingly, to be more or less
laterally asymmetrical. But the two horns are in symmetry one with
another; they form “conjugate” spirals, one being the “mirror-image”
of the other. Just as in the hairy coat of the animal each hair, on
either side of the median “parting,” tends to have a certain definite
direction of its own axis, inclined away from the median axial plane of
the whole system, so is it both with the bony core of the horn and with
the consolidated mass of hairs or hair-like substance which constitutes
its sheath; the primary axis of the horn is more or less inclined to,
and may even be nearly perpendicular to, the axial plane of the animal.
The growth of the horny sheath is not continuous, but more or {614}
less definitely periodic: sometimes, as in the sheep, this periodicity
is particularly well-marked, and causes the horny sheath to be composed
of a series of all but separate rings, which are supposed to be formed
year by year, and so to record the age of the animal[560].
Just as we sought for the true generating curve in the orifice, or
“lip,” of the molluscan shell, so we might be apt to assume that in the
spiral horn the generating curve corresponded to the lip or margin of
one of the horny rings or annuli. This annular margin, or boundary of
the ring, is usually a sinuous curve, not lying in a plane, but such as
would form the boundary of an anticlastic surface of great complexity:
to the meaning and origin of which phenomenon we shall return
presently. But, as we have already seen in the case of the molluscan
shell, the complexities of the lip itself, or of the corresponding
lines of growth upon the shell, need not concern us in our study of
the development of the spiral: inasmuch as we may substitute for
these actual boundary lines, their “trace,” or projection on a plane
perpendicular to the axis—in other words the simple outline of a
transverse section of the whorl. In the horn, this transverse section
is often circular or nearly so, as in the oxen and many antelopes: it
now and then becomes of somewhat complicated polygonal outline, as
in a highland ram; but in many antelopes, and in most of the sheep,
the outline is that of an isosceles, or sometimes nearly equilateral
triangle, a form which is typically displayed, for instance, in _Ovis
Ammon_. The horn in this latter case is a trihedral prism, whose three
faces are, (1) an upper, or frontal face, in continuation of the plane
of the frontal bone; (2) an outer, or orbital, starting from the upper
margin of the orbit; and (3) an inner, or “nuchal,” abutting on the
parietal bone[561]. Along these three faces, and their corresponding
angles or edges, we can trace in the fibrous substance of the horn
a series of homologous spirals, such as we {615} have called in
a preceding chapter the “_ensemble_ of generating spirals” which
constitute the surface.
[Illustration: Fig. 318. Diagram of Ram’s horns. (After Sir Vincent
Brooke, from _P.Z.S._) _a_, frontal; _b_, orbital; _c_, nuchal surface.]
In some few cases, of which the male musk ox is one of the most
notable, the horn is not developed in a continuous spiral curve. It
changes its shape as growth proceeds; and this, as we have seen, is
enough to show that it does not constitute a logarithmic spiral. The
reason is that the bony exostoses, or horn-cores, about which the
horny sheath is shaped and moulded, neither grow continuously nor even
remain of constant size after attaining their full growth. But as the
horns grow heavy the bony core is bent downwards by their weight, and
so guides the growth of the horn in a new direction. Moreover as age
advances, the horn-core is further weakened and to a great extent
absorbed: and the horny sheath or horn proper, deprived of its support,
continues to grow, but in a flattened curve very different from its
original spiral[562]. The chamois is a somewhat analogous case. Here
the terminal, or oldest, part of the horn is curved; it tends to assume
a spiral form, though from its comparative shortness it seems merely to
be bent into a hook. But later on, the bony core within, as it grows
and strengthens, stiffens the horn, and guides it into a straighter
course or form. The same phenomenon {616} of change of curvature,
manifesting itself at the time when, or the place where, the horn is
freed from the support of the internal core, is seen in a good many
other antelopes (such as the hartebeest) and in many buffaloes; and the
cases where it is most manifest appear to be those where the bony core
is relatively short, or relatively weak.
[Illustration: Fig. 319. Head of Arabian Wild Goat, _Capra sinaitica_.
(After Sclater, from _P.Z.S._)]
But in the great majority of horns, we have no difficulty in
recognising a continuous logarithmic spiral, nor in referring it, as
before, to an unequal rate of growth (parallel to the axis) on two
opposite sides of the horn, the inequality maintaining a constant
ratio as long as growth proceeds. In certain antelopes, such as the
gemsbok, the spiral angle is very small, or in other words the horn
is very nearly straight; in other species of the same genus Oryx,
such as the Beisa antelope and the Leucoryx, a gentle {617} curve
(not unlike though generally less than that of a Dentalium shell) is
evident; and the spiral angle, according to the few measurements I have
made, is found to measure from about 20° to nearly 40°. In some of the
large wild goats, such as the Scinde wild goat, we have a beautiful
logarithmic spiral, with a constant angle of rather less than 70°;
and we may easily arrange a series of forms, such for example as the
Siberian ibex, the moufflon, _Ovis Ammon_, etc., and ending with the
long-horned Highland ram: in which, as we pass from one to another,
we recognise precisely homologous spirals, with an increasing angular
constant, the spiral angle being, for instance, about 75° or rather
less in _Ovis Ammon_, and in the Highland ram a very little more. We
have already seen that in the neighbourhood of 70° or 80° a small
change of angle makes a marked difference in the appearance of the
spire; and we know also that the actual length of the horn makes a very
striking difference, for the spiral becomes especially conspicuous to
the eye when a horn or shell is long enough to shew several whorls, or
at least a considerable part of one entire whorl.
Even in the simplest cases, such as the wild goats, the spiral is never
(strictly speaking) a plane or discoid spiral: but in greater or less
degree there is always superposed upon the plane logarithmic spiral a
helical spiral in space. Sometimes the latter is scarcely apparent,
for the helical curvature is comparatively small, and the horn (though
long, as in the said wild goats) is not nearly long enough to shew a
complete convolution: at other times, as in the ram, and still better
in many antelopes, such as the koodoo, the helicoid or corkscrew curve
of the horn is its most characteristic feature.
Accordingly we may study, as in the molluscan shell, the helicoid
component of the spire—in other words the variation in what we have
called (on p. 555) the angle _θ_. This factor it is which, more than
the constant angle of the logarithmic spiral, imparts a characteristic
appearance to the various species of sheep, for instance to the various
closely allied species of Asiatic wild sheep, or Argali. In all of
these the constant angle of the logarithmic spiral is very much the
same, but the shearing component differs greatly. And thus the long
drawn out horns of {618} _Ovis Poli_, four feet or more from tip to
tip, differ conspicuously from those of _Ovis Ammon_ or _O. hodgsoni_,
in which a very similar logarithmic spiral is wound (as it were) round
a much blunter cone.
――――――――――
The ram’s horn then, like the snail’s shell, is a curve of double
curvature, in which one component has imposed upon the structure a
plane logarithmic spiral, and the other has produced a continuous
displacement, or “shear,” proportionate in magnitude to, and
perpendicular or otherwise inclined in direction to, the axis of the
former spiral curvature. The result is precisely analogous to that
which we have studied in the snail and other spiral univalves; but
while the form, and therefore the resultant forces, are similar, the
original distribution of force is not the same: for we have not here,
as we had in the snail-shell, a “columellar” muscle, to introduce the
component acting in the direction of the axis. We have, it is true, the
central bony core, which in part performs an analogous function; but
the main phenomenon here is apparently a complex distribution of rates
of growth, perpendicular to the plane of the generating curve.
Let us continue to dispense with mathematics, for the mathematical
treatment of a curve of double curvature is never very simple, and let
us deal with the matter by experiment. We have seen that the generating
curve, or transverse section, of a typical ram’s horn is triangular in
form. Measuring (along the curve of the horn) the length of the three
edges of the trihedral structure in a specimen of _Ovis Ammon_, and
calling them respectively the outer, inner, and hinder edges (from
their position at the base of the horn, relatively to the skull),
I find the outer edge to measure 80 cm., the inner 74 cm., and the
posterior 45 cm.; let us say that, roughly, they are in the ratio of
9 : 8 : 5. Then, if we make a number of little cardboard triangles,
equip each with three little legs (I make them of cork), whose relative
lengths are as 9 : 8 : 5, and pile them up and stick them all together,
we straightway build up a curve of double curvature precisely analogous
to the ram’s horn: except only that, in this first approximation,
we have not allowed for the gradual increment (or decrement) of the
triangular surfaces, that is to say, for the _tapering_ of the horn due
to the growth in its own plane of the generating curve. {619}
In this case then, and in most other trihedral or three-sided horns,
one of the three components, or three unequal velocities of growth, is
of relatively small magnitude, but the other two are nearly equal one
to the other. It would involve but little change for these latter to
become precisely equal; and again but little to turn the balance of
inequality the other way. But the immediate consequence of this altered
ratio of growth would be that the horn would appear to wind the other
way, as it does in the antelopes, and also in certain goats, e.g. the
markhor, _Capra falconeri_.
For these two opposite directions of twist Dr Wherry has introduced a
convenient nomenclature. When the horn winds so that we follow it from
base to apex in the direction of the hands of a watch, it is customary
to call it a “left-handed” spiral. Such a spiral we have in the horn
on the left-hand side of a ram’s head. Accordingly, Dr Wherry calls
the condition _homonymous_, where, as in the sheep, a right-handed
spiral is on the right side of the head, and a left-handed spiral on
the left side; while he calls the opposite condition _heteronymous_,
as we have it in the antelopes, where the right-handed twist is on the
left side of the head, and the left-handed twist on the right-hand
side. Among the goats, we may have either condition. Thus the
domestic and most of the wild goats agree with the sheep; but in the
markhor the twisted horns are heteronymous, as in the antelopes. The
difference, as we have seen, is easily explained; and (very much as in
the case of our opposite spirals in the apple-snail, referred to on p.
560), it has no very deep importance.
Summarised then, in a very few words, the argument by which we account
for the spiral conformation of the horn is as follows: The horn
elongates by dint of continual growth within a narrow zone, or annulus,
at its base. If the rate of growth be identical on all sides of this
zone, the horn will grow straight; if it be greater on one side than
on the other, the horn will become curved: and it probably _will_
be greater on one side than on the other, because each single horn
occupies an unsymmetrical field with reference to the plane of symmetry
of the animal. If the maximal and minimal velocities of growth be
precisely at opposite sides of the zone of growth, the resultant spiral
will be a plane spiral; but if they be not precisely or diametrically
opposite, then the spiral will be a spiral in space, with a winding
or helical component; and it is by no means likely that the maximum
and minimum _will_ occur at precisely opposite ends of a diameter, for
{620} no such plane of symmetry is manifested in the field of force to
which the growing annulus corresponds or appertains.
Now we must carefully remember that the rates of growth of which we are
here speaking are the net rates of longitudinal increment, in which
increment the activity of the living cells in the zone of growth at the
base of the horn is only one (though it is the fundamental) factor.
In other words, if the horny sheath were continually being added to
with equal rapidity all round its zone of active growth, but at the
same time had its elongation more retarded on one side than the other
(prior to its complete solidification) by varying degrees of adhesion
or membranous attachment to the bone core within, then the net result
would be a spiral curve precisely such as would have arisen from
initial inequalities in the rate of growth itself. It seems highly
probable that this is a very important factor, and sometimes even the
chief factor in the case. The same phenomenon of attachment to the bony
core, and the consequent friction or retardation with which the sheath
slides over its surface, will lead to various subsidiary phenomena:
among others to the presence of transverse folds or corrugations upon
the horn, and to their unequal distribution upon its several faces or
edges. And while it is perfectly true that nearly all the characters
of the horn can be accounted for by unequal velocities of longitudinal
growth upon its different sides, it is also plain that the actual field
of force is a very complicated one indeed. For example, we can easily
see that (at least in the great majority of cases) the direction of
growth of the horny fibres of the sheath is by no means parallel to
the axis of the core within; accordingly these fibres will tend to
wind in a system of helicoid curves around the core, and not only this
helicoid twist but any other tendency to spiral curvature on the part
of the sheath will tend to be opposed or modified by the resistance of
the core within. But on the other hand living bone is a very plastic
structure, and yields easily though slowly to any forces tending to its
deformation; and so, to a considerable extent, the bony core itself
will tend to be modelled by the curvature which the growing sheath
assumes, and the final result will be determined by an equilibrium
between these two systems of forces. {621}
While it is not very safe, perhaps, to lay down any general rule as to
what horns are more, and what are less spirally curved, I think it may
be said that, on the whole, the thicker the horn, the greater is its
spiral curvature. It is the slender horns, of such forms as the Beisa
antelope, which are gently curved, and it is the robust horns of goats
or of sheep in which the curvature is more pronounced. Other things
being the same, this is what we should expect to find; for it is where
the transverse section of the horn is large that we may expect to find
the more marked differences in the intensity of the field of force,
whether of active growth or of retardation, on opposite sides or in
different sectors thereof.
[Illustration: Fig. 320. Head of _Ovis Ammon_, shewing St Venant’s
curves.]
But there is yet another and a very remarkable phenomenon which we may
discern in the growth of a horn, when it takes the form of a curve
of double curvature, namely, an effect of torsional strain; and this
it is which gives rise to the sinuous “lines of growth,” or sinuous
boundaries of the separate horny rings, of which we have already
spoken. It is not at first sight obvious that a mechanical strain of
torsion is necessarily involved in the growth of the horn. In our
experimental illustration (p. 618), we built up a twisted coil of
separate elements, and no torsional strain attended the development
of the system. So would it be if the horny sheath grew by successive
annular increments, free save for their relation to one another, and
having no attachment to the solid core within. But as a matter of fact
there is {622} such an attachment, by subcutaneous connective tissue,
to the bony core; and accordingly a torsional strain will be set up
in the growing horny sheath, again provided that the forces of growth
therein be directed more or less obliquely to the axis of the core; for
a “couple” is thus introduced, giving rise to a strain which the sheath
would not experience were it free (so to speak) to slip along, impelled
only by the pressure of its own growth from below. And furthermore,
the successive small increments of the growing horn (that is to say,
of the horny sheath) are not instantaneously converted from living to
solid and rigid substance; but there is an intermediate stage, probably
long-continued, during which the new-formed horny substance in the
neighbourhood of the zone of active growth is still plastic and capable
of deformation.
Now we know, from the celebrated experiments of St Venant[563], that
in the torsion of an elastic body, other than a cylinder of circular
section, a very remarkable state of strain is introduced. If the body
be thus cylindrical (whether solid or hollow), then a twist leaves
each circular section unchanged, in dimensions and in figure. But in
all other cases, such as an elliptic rod or a prism of any particular
sectional form, forces are introduced which act parallel to the axis of
the structure, and which warp each section into a complex anticlastic
surface. Thus in the case of a triangular and equilateral prism, such
as is shewn in section in Fig. 321, if the part of the rod represented
in the section be twisted by a force acting in the direction of the
arrow, then the originally plane section will be warped as indicated in
the diagram:—where the full contour-lines represent elevation above,
and the dotted lines represent depression below, the original level.
On the external surface of the prism, then, contour-lines which were
originally parallel and horizontal, will be found warped into sinuous
curves, such that, on each of the three faces, the curve will be convex
upwards on one half, and concave upwards on the other half of the face.
The ram’s horn, and still better that of _Ovis Ammon_, is comparable
to such a prism, save that in section it is not quite equilateral,
and that its three faces are not plane. The warping is therefore not
precisely identical on the three faces {623} of the horn; but, in the
general distribution of the curves, it is in complete accordance with
theory. Similar anticlastic curves are well seen in many antelopes; but
they are conspicuous by their absence in the _cylindrical_ horns of
oxen.
[Illustration: Fig. 321.]
[Illustration: Fig. 322.]
The better to illustrate this phenomenon, the nature of which is indeed
obvious enough from a superficial examination of the horn, I made a
plaster cast of one of the horny rings in a horn of _Ovis Ammon_, so
as to get an accurate pattern of its sinuous edge: and then, filling
the mould up with wet clay, I modelled an anticlastic surface, such
as to correspond as nearly as possible with the sinuous outline[564].
Finally, after making a plaster cast of this sectional surface, I drew
its contour-lines (as shewn in Fig. 322), with the help of a simple
form of spherometer. It will be seen that in great part this diagram
is precisely similar to St Venant’s diagram of the cross-section of
a twisted triangular prism; and this is especially the case in the
neighbourhood of the sharp angle of our prismatic section. That in
parts the diagram is somewhat asymmetrical is not to be wondered
at: and (apart from inaccuracies due to the somewhat rough means
by which it was made) this asymmetry can be sufficiently accounted
for by anisotropy of the material, by inequalities in thickness of
different parts of the horny sheath, and especially (I think) by
unequal distributions of rigidity due to the presence of the smaller
corrugations of the {624} horn. It is apparently on account of these
minor corrugations that, in such horns as the Highland ram’s, where
they are strongly marked, the main St Venant effect is not nearly so
well shewn as in the smoother horns such as those of _O. Ammon_ and its
immediate congeners[565].
_A further Note upon Torsion._
The phenomenon of torsion, to which we have been thus introduced, opens
up many wide questions in connection with form. Some of the associated
phenomena are admirably illustrated in the case of climbing plants; but
we can only deal with these still more briefly and parenthetically.
The subject of climbing plants has been elaborately dealt with not
only in Darwin’s books[566], but also by a very large number of
earlier and later writers. In “twining” plants, which constitute the
greater number of “climbers,” the essential phenomenon is a tendency
of the growing shoot to revolve about a vertical axis—a tendency long
ago discussed and investigated by such writers as Palm, H. von Mohl
and Dutrochet[567]. This tendency to revolution—“circumvolution,”
as Darwin calls it, “revolving nutation,” as Sachs puts it—is very
closely comparable to the process by which an antelope’s horn (such as
the koodoo’s) grows into its spiral or rather helicoid form; and it
is simply due, in like manner, to inequalities in the rate of growth
on different sides of the growing stem. There is only this difference
between the two cases, that in the antelope’s horn the zone of active
growth is confined to the base of the horn, while in the climbing stem
the same phenomenon is at work throughout the whole length of the
growing structure. This growth is in the main due to “turgescence,”
that is to the extension, or elongation, of ready-formed cells through
the imbibition of water; it is a phenomenon due to osmotic pressure.
The particular stimuli to which these movements (that is to say, these
inequalities of growth) have been {625} ascribed, such as contact
(thigmotaxis), exposure to light (heliotropism), and so forth, need not
be discussed here[568].
A simple stem growing upright in the dark, or in uniformly diffused
light, would be in a position of equilibrium to a field of force
radially symmetrical about its vertical axis. But this complete
radial symmetry will not often occur; and the radial anomalies may
be such as arise intrinsically from structural peculiarities in the
stem itself, or externally to it by reason of unequal illumination or
through various other localised forces. The essential fact, so far
as we are concerned, is that in twining plants we have a very marked
tendency to inequalities in longitudinal growth on different aspects
of the stem—a tendency which is but an exaggerated manifestation of
one which is more or less present, under certain conditions, in all
plants whatsoever. Just as in the case of the ruminants’ horns so
we find here, that this inequality may be, so to speak, positive or
negative, the maximum lying to the one side or the other of the twining
stem; and so it comes to pass that some climbers twine to the one side
and some to the other: the hop and the honeysuckle following the sun,
and the field-convolvulus twining in the reverse direction; there are
also some, like the woody nightshade (_Solanum Dulcamara_) which twine
indifferently either way.
Together with this circumnutatory movement, there is very generally
to be seen an actual _torsion_ of the twining stem—a twist, that is
to say, about its own axis; and Mohl made the curious observation,
confirmed by Darwin, that when a stem twines around a smooth
cylindrical stick the torsion does not take place, save “only in
that degree which follows as a mechanical necessity from the spiral
winding”: but that stems which had climbed around a rough stick were
all more or less, and generally much, twisted. Here Darwin did not
refrain from introducing that teleological argument which pervades
his whole train of reasoning: “The stem,” he says, “probably gains
rigidity by being twisted (on the same principle that a much twisted
rope {626} is stiffer than a slackly twisted one), and is thus
indirectly benefited so as to be able to pass over inequalities in its
spiral ascent, and to carry its own weight when allowed to revolve
freely.” The mechanical explanation would appear to be very simple, and
such as to render the teleological hypothesis unnecessary. In the case
of the roughened support, there is a temporary adhesion or “clinging”
between it and the growing stem which twines around it; and a system of
forces is thus set up, producing a “couple,” just as it was in the case
of the ram’s or antelope’s horn through direct adhesion of the bony
core to the surrounding sheath. The twist is the direct result of this
couple, and it disappears when the support is so smooth that no such
force comes to be exerted.
Another important class of climbers includes the so-called
“leaf-climbers.” In these, some portion of the leaf, generally the
petiole, sometimes (as in the fumitory) the elongated midrib, curls
round a support; and a phenomenon of like nature occurs in many, though
not all, of the so-called “tendril-bearers.” Except that a different
part of the plant, leaf or tendril instead of stem, is concerned in
the twining process, the phenomenon here is strictly analogous to
our former case; but in the resulting helix there is, as a rule,
this obvious difference, that, while the twining stem, for instance
of the hop, makes a slow revolution about its support, the typical
leaf-climber makes a close, firm coil: the axis of the latter is nearly
perpendicular and parallel to the axis of its support, while in the
twining stem the angle between the two axes is comparatively small.
Mathematically speaking, the difference merely amounts to this, that
the component in the direction of the vertical axis is large in the
one case, and the corresponding component is small, if not absent, in
the other; in other words, we have in the climbing stem a considerable
vertical component, due to its own tendency to grow in height, while
this longitudinal or vertical extension of the whole system is not
apparent, or little apparent, in the other cases. But from the fact
that the twining stem tends to run obliquely to its support, and the
coiling petiole of the leaf-climber tends to run transversely to the
axis of its support, there immediately follows this marked difference,
that the phenomenon {627} of _torsion_, so manifest in the former
case, will be absent in the latter.
――――――――――
There is one other phenomenon which meets us in the twining and twisted
stem, and which is doubtless illustrated also, though not so well, in
the antelope’s horn; it is a phenomenon which forms the subject of a
second chapter of St Venant’s researches on the effects of torsional
strain in elastic bodies. We have already seen how one effect of
torsion, in for instance a prism, is to produce strains parallel to the
axis, elevating parts and depressing other parts of each transverse
section. But in addition to this, the same torsion has the effect of
materially altering the form of the section itself, as we may easily
see by twisting a square or oblong piece of india-rubber. If we start
with a cylinder, such as a round piece of catapult india-rubber, and
twist it on its own long axis, we have already seen that it suffers no
other distortion; it still remains a cylinder, that is to say, it is
still in section everywhere circular. But if it be of any other shape
than cylindrical the case is quite different, for now the sectional
shape tends to alter under the strain of torsion. Thus, if our rod be
elliptical in section to begin with, it will, under torsion, become a
more elongated ellipse; if it be square, its angles will become more
prominent, and its sides will curve inwards, till at length the square
assumes the appearance of a four-pointed star, with rounded angles.
Furthermore, looking at the results of this process of modification, we
find experimentally that the resultant figures are more easily twisted,
less resistant to torsion, than were those from which we evolved
them; and this is a very curious physical or mathematical fact. So a
cylinder, which is especially resistant to torsion, is very easily bent
or flexed; while projecting ribs or angles, such as an engineer makes
in a bar or pillar of iron for the purpose of greatly increasing its
strength in the way of resistance to _bending_, actually make it much
weaker than before (for the same amount of metal per unit length) in
the way of resistance to _torsion_.
In the hop itself, and in a very considerable number of other twining
and twisting stems, the ribbed or channelled form of the stem is a
conspicuous feature. We may safely take it, (1) that {628} such
stems are especially susceptible of torsion; and (2) that the effect
of torsion will be to intensify any such peculiarities of sectional
outline which they may possess, though not to initiate them in an
originally cylindrical structure. In the leaf-climbers the case does
not present itself, for there, as we have seen, torsion itself is not,
or is very slightly, manifested. There are very distinct traces of the
phenomenon in the horns of certain antelopes, but the reason why it is
not a more conspicuous feature of the antelope’s horn or of the ram’s
is apparently a very simple one: namely, that the presence of the bony
core within tends to check that deformation which is perpendicular,
while it permits that which is parallel, to the axis of the horn.
_Of Deer’s Antlers._
But let us return to our subject of the shapes of horns, and consider
briefly our last class of these structures, namely the bony antlers
of the various species of elk and deer[569]. The problems which these
present to us are very different from those which we have had to do
with in the antelope or the sheep.
[Illustration: Fig. 323. Antlers of Swedish Elk. (After Lönnberg, from
_P.Z.S._)]
With regard to its structure, it is plain that the bony antler
corresponds, upon the whole, to the bony core of the antelope’s horn;
while in place of the hard horny sheath of the latter, we have the
soft “velvet,” which every season covers the new growing antler, and
protects the large nutrient blood-vessels by help of which the antler
grows[570]. The main difference lies in the fact that, in the one case,
the bony core, imprisoned within its sheath, is rendered incapable
of branching and incapable also of lateral expansion, and the whole
horn is only permitted to grow in length, while retaining a sectional
contour that is identical with (or but little altered from) that which
it possesses at its growing base: {629} but in the antler, on the
other hand, no such restraint is imposed, and the living, growing
fabric of bone may expand into a broad flattened plate over which
the blood-vessels run. In the immediate neighbourhood of the main
blood-vessels growth will be most active; in the interspaces between,
it may wholly fail: with the result that we may have great notches
cut out of the flattened plate, or may at length find it reduced to
the form of a simple branching structure. The main point, as it seems
to me, is that the “horn” is essentially an _axial rod_, while the
“antler” is essentially an outspread _surface_[571]. In other words,
I believe that the whole configuration of an antler is more easily
understood by conceiving it as a plate or a surface, more and more
notched and scolloped till but a slender skeleton may remain, than to
look upon it the other way, namely as an axial stem (or beam) giving
{630} off branches (or tines), the interspaces between which latter
may sometimes be filled up to form a continuous plate.
[Illustration: Fig. 324. Head and antlers of a Stag (_Cervus
Duvauceli_). (After Lydekker, from _P.Z.S._)]
In a sense it matters very little whether we regard the broad
plate-like antlers of the elk or the slender branching antlers of the
stag as the more primitive type; for we are not concerned here with
the question of hypothetical phylogeny. And even from the mathematical
point of view it makes little or no difference whether we describe
the plate as constituted by the interconnection of the branches, or
the branches derived by a process of notching or incision from the
plate. The important point for us is to recognise that (save for
occasional slight irregularities) the branching system in the one
_conforms_ essentially to the curved plate or surface which we see
plainly in the other. In short the arrangement of the branches is more
or less comparable to that of the veins in a leaf, or to that of the
blood-vessels as they course over the curved surface of an organ. It is
a process of ramification, not, like that of a tree, in various planes,
but strictly limited {631} to a single surface. And just as the veins
within a leaf are not necessarily confined (as they happen to be in
most ordinary leaves) to a _plane_ surface, but, as in the petal of a
tulip or the capsule of a poppy, may have to run their course within a
curved surface, so does the analogy of the leaf lead us directly to the
mode of branching which is characteristic of the antler. The surface to
which the branches of the antler tend to be confined is a more or less
spheroidal, or occasionally an ellipsoidal one; and furthermore, when
we inspect any well-developed pair of antlers, such as those of a red
deer, a sambur or a wapiti, we have no difficulty in seeing that the
two antlers make up between them _a single surface_, and constitute a
symmetrical figure, each half being the mirror-image of the other.
To put the case in another way, a pair of antlers (apart from
occasional slight irregularities) tends to constitute a figure such
that we could conceive an elastic sheet stretched over or round the
entire system, so as to form one continuous and even surface; and not
only would the surface curvature be on the whole smooth and even, but
the boundary of the surface would also tend to be an even curve: that
is to say the tips of all the tines would approximately have their
locus in a continuous curve.
It follows from this that if we want to make a simple model of a set
of antlers, we shall be very greatly helped by taking some appropriate
spheroidal surface as our groundwork or scaffolding. The best form of
surface is a matter for trial and investigation in each particular
case; but even in a sphere, by selecting appropriate areas thereof,
we can obtain sufficient varieties of surface to meet all ordinary
cases. With merely a bit of sculptor’s clay or plasticine, we should
be put hard to it to model the horns of a wapiti or a reindeer: but
if we start with an orange (or a round florence flask) and lay our
little tapered rolls of plasticine upon it, in simple natural curves,
it is surprising to see how quickly and successfully we can imitate
one type of antler after another. In doing so, we shall be struck by
the fact that our model may vary in its mode of branching within very
considerable limits, and yet look perfectly natural. For the same wide
range of variation is characteristic of the natural antlers themselves.
As Sir V. Brooke says (_op. cit._ p. 892), “No two antlers are ever
exactly alike; and the {632} variation to which the antlers are subject
is so great that in the absence of a large series they would be held
to be indicative of several distinct species[572].” But all these many
variations lie within a limited range, for they are all subject to our
general rule that the entire structure is essentially confined to a
single curved surface.
It is plain that in the curvatures both of the beam and of its tines,
in the angles by which these latter meet the beam, and in the contours
of the entire system, there are involved many elegant mathematical
problems with which we cannot at present attempt to deal. Nor must
we attempt meanwhile to enquire into the physical meaning or origin
of these phenomena, for as yet the clue seems to be lacking and we
should only heap one hypothesis upon another. That there is a complete
contrast of mathematical properties between the horn and the antler is
the main lesson with which, in the meantime, we must rest content.
_Of Teeth, and of Beak and Claw._
In a fashion similar to that manifested in the shell or the horn,
we find the logarithmic spiral to be implicit in a great many other
organic structures where the phenomena of growth proceed in a similar
way: that is to say, where about an axis there is some asymmetry
leading to unequal rates of longitudinal growth, and where the
structure is of such a kind that each new increment is added on as a
permanent and unchanging part of the entire conformation. Nail and
claw, beak and tooth, all come under this category. The logarithmic
spiral _always_ tends to manifest itself in such structures as these,
though it usually only attracts our attention in elongated structures,
where (that is to say) the radius vector has described a considerable
angle. When the canary-bird’s claws grow long from lack of use, or
when the incisor tooth of a rabbit or a rat grows long by reason of an
injury to the opponent tooth against which it was wont to bite, we know
that the tooth or claw tends to grow into a spiral curve, and we speak
of it as a malformation. But there has been no fundamental change of
form, save only an abnormal increase in length; {633} the elongated
tooth or claw has the selfsame curvature that it had when it was short,
but the spiral curvature becomes more and more manifest the longer it
grows. A curious analogous case is that of the New Zealand huia bird,
in which the beak of the female is described as being comparatively
short and straight, while that of the male is long and curved; it is
easy to see that there is a slight curvature also in the beak of the
female, and that the beak of the male shows nothing but the same curve
produced. In the case of the more curved beaks, such as those of an
eagle or a parrot, we may, if we please, determine the constant angle
of the logarithmic spiral, just as we have done in the case of the
Nautilus shell; and here again, as the bird grows older or the beak
longer, the spiral nature of the curve becomes more and more apparent,
as in the hooked beak of an old eagle, or as in the great beak of some
large parrot such as a hyacinthine macaw.
Let us glance at one or two instances to illustrate the spiral
curvature of teeth.
A dentist knows that every tooth has a curvature of its own, and that
in pulling the tooth he must follow the direction of the curve; but in
an ordinary tooth this curvature is scarcely visible, and is least so
when the diameter of the tooth is large compared with its length.
In the simply formed, more or less conical teeth, such as are those
of the dolphin, and in the more or less similarly shaped canines
and incisors of mammals in general, the curvature of the tooth is
particularly well seen. We see it in the little teeth of a hedgehog,
and in the canines of a dog or a cat it is very obvious indeed. When
the great canine of the carnivore becomes still further enlarged
or elongated, as in Machairodus, it grows into the strongly curved
sabre-tooth of that great extinct tiger. In rodents, it is the incisors
which undergo a great elongation; their rate of growth differs, though
but slightly, on the two sides, anterior and posterior, of the axis,
and by summation of these slight differences in the rapid growth of
the tooth an unmistakeable logarithmic spiral is gradually built up.
We see it admirably in the beaver, or in the great ground-rat, Geomys.
The elephant is a similar case, save that the tooth, or tusk, remains,
owing to comparative lack of wear, in a more perfect condition. In
the rodent (save only in those abnormal cases mentioned on the last
page) the {634} anterior, first-formed, part of the tooth wears
away as fast as it is added to from behind; and in the grown animal,
all those portions of the tooth near to the pole of the logarithmic
spiral have long disappeared. In the elephant, on the other hand,
we see, practically speaking, the whole unworn tooth, from point to
root; and its actual tip nearly coincides with the pole of the spiral.
If we assume (as with no great inaccuracy we may do) that the tip
actually coincides with the pole, then we may very easily construct
the continuous spiral of which the existing tusk constitutes a part;
and by so doing, we see the short, gently curved tusk of our ordinary
elephant growing gradually into the spiral tusk of the mammoth. No
doubt, just as in the case of our molluscan shells, we have a tendency
to variation, both individual and specific, in the constant angle of
the spiral; some elephants, and some species of elephant, undoubtedly
have a higher spiral angle than others. But in most cases, the angle
would seem to be such that a spiral configuration would become very
manifest indeed if only the tusk pursued its steady growth, unchanged
otherwise in form, till it attained the dimensions which we meet with
in the mammoth. In a species such as _Mastodon angustidens_, or _M.
arvernensis_, the specific angle is low and the tusk comparatively
straight; but the American mastodons and the existing species of
elephant have tusks which do not differ appreciably, except in
size, from the great spiral tusks of the mammoth, though from their
comparative shortness the spiral is little developed and only appears
to the eye as a gentle curve. Wherever the tooth is very long indeed,
as in the mammoth or the beaver, the effect of some slight and all
but inevitable lateral asymmetry in the rate of growth begins to shew
itself: in other words, the spiral is seen to lie not absolutely in a
plane, but to be a curve of double curvature, like a twisted horn. We
see this condition very well in the huge canine tusks of the Babirussa;
it is a conspicuous feature in the mammoth, and it is more or less
perceptible in any large tusk of the ordinary elephants.
The form of a molar tooth, which is essentially a branching or budding
system, and in which such longitudinal growth as gives rise to a spiral
curve is but little manifest, constitutes an entirely different problem
with which I shall not at present attempt to deal.
{635}