ON THE THEORY OF TRANSFORMATIONS, OR THE COMPARISON OF RELATED
FORMS[641]
In the foregoing chapters of this book we have attempted to study the
inter-relations of growth and form, and the part which certain of the
physical forces play in this complex interaction; and, as part of
the same enquiry, we have tried in comparatively simple cases to use
mathematical methods and mathematical terminology in order to describe
and define the forms of organisms. We have learned in so doing that
our own study of organic form, which we call by Goethe’s name of
Morphology, is but a portion of that wider Science of Form which deals
with the forms assumed by matter under all aspects and conditions, and,
in a still wider sense, with forms which are theoretically imaginable.
The study of form may be descriptive merely, or it may become
analytical. We begin by describing the shape of an object in the simple
words of common speech: we end by defining it in the precise language
of mathematics; and the one method tends to follow the other in strict
scientific order and historical continuity. Thus, for instance, the
form of the earth, of a raindrop or a rainbow, the shape of the
hanging chain, or the path of a stone thrown up into the air, may all
be described, however inadequately, in common words; but when we have
learned to comprehend and to define the sphere, the catenary, or the
parabola, we have made a wonderful and perhaps a manifold advance. The
mathematical definition of a “form” has a quality of precision which
was quite lacking in our earlier stage of mere description; it is
expressed in few words, or in still briefer symbols, and these {720}
words or symbols are so pregnant with meaning that thought itself
is economised; we are brought by means of it in touch with Galileo’s
aphorism (as old as Plato, as old as Pythagoras, as old perhaps as
the wisdom of the Egyptians), that “the Book of Nature is written in
characters of Geometry.”
Next, we soon reach through mathematical analysis to mathematical
synthesis; we discover homologies or identities which were not obvious
before, and which our descriptions obscured rather than revealed: as
for instance, when we learn that, however we hold our chain, or however
we fire our bullet, the contour of the one or the path of the other is
always mathematically homologous. Lastly, and this is the greatest gain
of all, we pass quickly and easily from the mathematical conception
of form in its statical aspect to form in its dynamical relations: we
pass from the conception of form to an understanding of the forces
which gave rise to it; and in the representation of form and in the
comparison of kindred forms, we see in the one case a diagram of forces
in equilibrium, and in the other case we discern the magnitude and
the direction of the forces which have sufficed to convert the one
form into the other. Here, since a change of material form is only
effected by the movement of matter, we have once again the support of
the schoolman’s and the philosopher’s axiom, “_Ignorato motu, ignoratur
Natura_.”
――――――――――
In the morphology of living things the use of mathematical methods
and symbols has made slow progress; and there are various reasons for
this failure to employ a method whose advantages are so obvious in the
investigation of other physical forms. To begin with, there would seem
to be a psychological reason lying in the fact that the student of
living things is by nature and training an observer of concrete objects
and phenomena, and the habit of mind which he possesses and cultivates
is alien to that of the theoretical mathematician. But this is by no
means the only reason; for in the kindred subject of mineralogy, for
instance, crystals were still treated in the days of Linnaeus as wholly
within the province of the naturalist, and were described by him after
the simple methods in use for animals and plants: but as soon as Haüy
showed the application of mathematics to {721} the description and
classification of crystals, his methods were immediately adopted and a
new science came into being.
A large part of the neglect and suspicion of mathematical methods
in organic morphology is due (as we have partly seen in our opening
chapter) to an ingrained and deep-seated belief that even when we seem
to discern a regular mathematical figure in an organism, the sphere,
the hexagon, or the spiral which we so recognise merely resembles, but
is never entirely explained by, its mathematical analogue; in short,
that the details in which the figure differs from its mathematical
prototype are more important and more interesting than the features in
which it agrees, and even that the peculiar aesthetic pleasure with
which we regard a living thing is somehow bound up with the departure
from mathematical regularity which it manifests as a peculiar attribute
of life. This view seems to me to involve a misapprehension. There is
no such essential difference between these phenomena of organic form
and those which are manifested in portions of inanimate matter[642]. No
chain hangs in a perfect catenary and no raindrop is a perfect sphere:
and this for the simple reason that forces and resistances other than
the main one are inevitably at work. The same is true of organic form,
but it is for the mathematician to unravel the conflicting forces which
are at work together. And this process of investigation may lead us
on step by step to new phenomena, as it has done in physics, where
sometimes a knowledge of form leads us to the interpretation of forces,
and at other times a knowledge of the forces at work guides us towards
a better insight into form. I would illustrate this by the case of the
earth itself. After the fundamental advance had been made which taught
us that the world was round, Newton showed that the forces at work upon
it must lead to its being imperfectly spherical, and in the course of
time its oblate spheroidal shape was actually verified. But now, in
turn, it has been shown that its form is still more complicated, and
the next step will be to seek for the forces that have deformed the
oblate spheroid. {722}
The organic forms which we can define, more or less precisely, in
mathematical terms, and afterwards proceed to explain and to account
for in terms of force, are of many kinds, as we have seen; but
nevertheless they are few in number compared with Nature’s all but
infinite variety. The reason for this is not far to seek. The living
organism represents, or occupies, a field of force which is never
simple, and which as a rule is of immense complexity. And just as in
the very simplest of actual cases we meet with a departure from such
symmetry as could only exist under conditions of _ideal_ simplicity, so
do we pass quickly to cases where the interference of numerous, though
still perhaps very simple, causes leads to a resultant which lies far
beyond our powers of analysis. Nor must we forget that the biologist
is much more exacting in his requirements, as regards form, than the
physicist; for the latter is usually content with either an ideal or a
general description of form, while the student of living things must
needs be specific. The physicist or mathematician can give us perfectly
satisfying expressions for the form of a wave, or even of a heap of
sand; but we never ask him to define the form of any particular wave of
the sea, nor the actual form of any mountain-peak or hill[643]. {723}
For various reasons, then, there are a vast multitude of organic forms
which we are unable to account for, or to define, in mathematical
terms; and this is not seldom the case even in forms which are
apparently of great simplicity and regularity. The curved outline of a
leaf, for instance, is such a case; its ovate, lanceolate, or cordate
shape is apparently very simple, but the difficulty of finding for it a
mathematical expression is very great indeed. To define the complicated
outline of a fish, for instance, or of a vertebrate skull, we never
even seek for a mathematical formula.
But in a very large part of morphology, our essential task lies in the
comparison of related forms rather than in the precise definition of
each; and the _deformation_ of a complicated figure may be a phenomenon
easy of comprehension, though the figure itself have to be left
unanalysed and undefined. This process of comparison, of recognising
in one form a definite permutation or _deformation_ of another, apart
altogether from a precise and adequate understanding of the original
“type” or standard of comparison, lies within the immediate province of
mathematics, and finds its solution in the elementary use of a certain
method of the mathematician. This method is the Method of Co-ordinates,
on which is based the Theory of Transformations.
I imagine that when Descartes conceived the method of co-ordinates, as
a generalisation from the proportional diagrams of the artist and the
architect, and long before the immense possibilities of this analysis
could be foreseen, he had in mind a very simple purpose; it was perhaps
no more than to find a way of translating the _form_ of a curve into
_numbers_ and into _words_. This is precisely what we do, by the
method of co-ordinates, every time we study a statistical curve; and
conversely, we translate numbers into form whenever we “plot a curve”
to illustrate a table of mortality, a rate of growth, or the daily
variation of temperature or barometric pressure. In precisely the same
way it is possible to inscribe in a net of rectangular co-ordinates the
outline, for instance, of a fish, and so to translate {724} it into a
table of numbers, from which again we may at pleasure reconstruct the
curve.
But it is the next step in the employment of co-ordinates which is of
special interest and use to the morphologist; and this step consists in
the alteration, or “transformation,” of our system of co-ordinates and
in the study of the corresponding transformation of the curve or figure
inscribed in the co-ordinate network.
Let us inscribe in a system of Cartesian co-ordinates the outline
of an organism, however complicated, or a part thereof: such as a
fish, a crab, or a mammalian skull. We may now treat this complicated
figure, in general terms, as a function of _x_, _y_. If we submit
our rectangular system to “deformation,” on simple and recognised
lines, altering, for instance, the direction of the axes, the ratio
of _x_/_y_, or substituting for _x_ and _y_ some more complicated
expressions, then we shall obtain a new system of co-ordinates, whose
deformation from the original type the inscribed figure will precisely
follow. In other words, we obtain a new figure, which represents the
old figure _under strain_, and is a function of the new co-ordinates
in precisely the same way as the old figure was of the original
co-ordinates _x_ and _y_.
The problem is closely akin to that of the cartographer who transfers
identical data to one projection or another; and whose object is to
secure (if it be possible) a complete correspondence, _in each small
unit of area_, between the one representation and the other. The
morphologist will not seek to draw his organic forms in a new and
artificial projection; but, in the converse aspect of the problem, he
will inquire whether two different but more or less obviously related
forms can be so analysed and interpreted that each may be shown to be
a transformed representation of the other. This once demonstrated, it
will be a comparatively easy task (in all probability) to postulate the
direction and magnitude of the force capable of effecting the required
transformation. Again, if such a simple alteration of the system of
forces can be proved adequate to meet the case, we may find ourselves
able to dispense with many widely current and more complicated
hypotheses of biological causation. For it is a maxim in physics that
an effect ought not to be ascribed to {725} the joint operation of
many causes if few are adequate to the production of it: _Frustra fit
per plura, quod fieri potest per pauciora._
――――――――――
It is evident that by the combined action of appropriate forces any
material form can be transformed into any other: just as out of a
“shapeless” mass of clay the potter or the sculptor models his artistic
product; or just as we attribute to Nature herself the power to
effect the gradual and successive transformation of the simplest into
the most complex organism. In like manner it is possible, at least
theoretically, to cause the outline of any closed curve to appear
as a projection of any other whatsoever. But we need not let these
theoretical considerations deter us from our method of comparison of
_related_ forms. We shall strictly limit ourselves to cases where the
transformation necessary to effect a comparison shall be of a simple
kind, and where the transformed, as well as the original, co-ordinates
shall constitute an harmonious and more or less symmetrical system. We
should fall into deserved and inevitable confusion if, whether by the
mathematical or any other method, we attempted to compare organisms
separated far apart in Nature and in zoological classification. We are
limited, not by the nature of our method, but by the whole nature of
the case, to the comparison of organisms such as are manifestly related
to one another and belong to the same zoological class.
Our inquiry lies, in short, just within the limits which Aristotle
himself laid down when, in defining a “genus,” he showed that (apart
from those superficial characters, such as colour, which he called
“accidents”) the essential differences between one “species” and
another are merely differences of proportion, of relative magnitude, or
(as he phrased it) of “excess and defect.” “Save only for a difference
in the way of excess or defect, the parts are identical in the case of
such animals as are of one and the same genus; and by ‘genus’ I mean,
for instance, Bird or Fish.” And again: “Within the limits of the same
genus, as a general rule, most of the parts exhibit differences ... in
the way of multitude or fewness, magnitude or parvitude, in short,
in the way of excess or defect. For ‘the more’ and ‘the less’ may be
represented as {726} ‘excess’ and ‘defect[644].’ ” It is precisely
this difference of relative magnitudes, this Aristotelian “excess and
defect” in the case of form, which our co-ordinate method is especially
adapted to analyse, and to reveal and demonstrate as the main cause of
what (again in the Aristotelian sense) we term “specific” differences.
The applicability of our method to particular cases will depend
upon, or be further limited by, certain practical considerations or
qualifications. Of these the chief, and indeed the essential, condition
is, that the form of the entire structure under investigation should
be found to vary in a more or less uniform manner, after the fashion
of an approximately homogeneous and isotropic body. But an imperfect
isotropy, provided always that some “principle of continuity” run
through its variations, will not seriously interfere with our method;
it will only cause our transformed co-ordinates to be somewhat less
regular and harmonious than are those, for instance, by which the
physicist depicts the motions of a perfect fluid or a theoretic field
of force in a uniform medium.
Again, it is essential that our structure vary in its entirety, or
at least that “independent variants” should be relatively few. That
independent variations occur, that localised centres of diminished or
exaggerated growth will now and then be found, is not only probable but
manifest; and they may even be so pronounced as to appear to constitute
new formations altogether. Such independent variants as these Aristotle
himself clearly recognised: “It happens further that some have parts
that others have not; for instance, some [birds] have spurs and others
not, some have crests, or combs, and others not; but, as a general
rule, most parts and those that go to make up the bulk of the body are
either identical with one another, or differ from one another in the
way of contrast and of excess and defect. For ‘the more’ and ‘the less’
may be represented as ‘excess’ or ‘defect.’ ”
If, in the evolution of a fish, for instance, it be the case that its
several and constituent parts—head, body, and tail, or this fin and
that fin—represent so many independent variants, then our co-ordinate
system will at once become too complex to be intelligible; we shall
be making not one comparison but several {727} separate comparisons,
and our general method will be found inapplicable. Now precisely
this independent variability of parts and organs—here, there, and
everywhere within the organism—would appear to be implicit in our
ordinary accepted notions regarding variation; and, unless I am greatly
mistaken, it is precisely on such a conception of the easy, frequent,
and normal independent variability of parts that our conception of
the process of natural selection is fundamentally based. For the
morphologist, when comparing one organism with another, describes
the differences between them point by point, and “character” by
“character[645].” If he is from time to time constrained to admit the
existence of “correlation” between characters (as a hundred years ago
Cuvier first showed the way), yet all the while he recognises this fact
of correlation somewhat vaguely, as a phenomenon due to causes which,
except in rare instances, he can hardly hope to trace; and he falls
readily into the habit of thinking and talking of evolution as though
it had proceeded on the lines of his own descriptions, point by point,
and character by character[646].
But if, on the other hand, diverse and dissimilar fishes can
be referred as a whole to identical functions of very different
co-ordinate systems, this fact will of itself constitute a proof
that variation has proceeded on definite and orderly lines, that a
comprehensive “law of growth” has pervaded the whole structure in its
integrity, and that some more or less simple and recognisable system of
forces has been at work. It will not only show how real and deep-seated
is the phenomenon of “correlation,” in regard to form, but it will also
demonstrate the fact that a correlation which had seemed too complex
for analysis or {728} comprehension is, in many cases, capable of very
simple graphic expression. This, after many trials, I believe to be
in general the case, bearing always in mind that the occurrence of
independent or localised variations must often be considered.
We are dealing in this chapter with the forms of related organisms,
in order to shew that the differences between them are as a general
rule simple and symmetrical, and just such as might have been brought
about by a slight and simple change in the system of forces to which
the living and growing organism was exposed. Mathematically speaking,
the phenomenon is identical with one met with by the geologist, when
he finds a bed of fossils squeezed flat or otherwise symmetrically
deformed by the pressures to which they, and the strata which contain
them, have been subjected. In the first step towards fossilisation,
when the body of a fish or shellfish is silted over and buried, we
may take it that the wet sand or mud exercises, approximately, a
hydrostatic pressure—that is to say a pressure which is uniform in
all directions, and by which the form of the buried object will not
be appreciably changed. As the strata consolidate and accumulate, the
fossil organisms which they contain will tend to be flattened by the
vast superincumbent load, just as the stratum which contains them
will also be compressed and will have its molecular arrangement more
or less modified[647]. But the deformation due to direct vertical
pressure in a horizontal stratum is not nearly so striking as are the
deformations produced by the oblique or shearing stresses to which
inclined and folded strata have been exposed, and by which their
various “dislocations” have been brought about. And especially in
mountain regions, where these dislocations are especially numerous and
complicated, the contained fossils are apt to be so curiously and yet
so symmetrically deformed (usually by a simple shear) that they may
easily be interpreted as so many distinct and separate “species[648].”
A great number of described species, and here and there a new genus
(as the genus Ellipsolithes for an obliquely deformed Goniatite or
Nautilus) are said to rest on no other foundation[649].
――――――――――
[Illustration: Fig. 352.]
[Illustration: Fig. 353.]
[Illustration: Fig. 354.]
[Illustration: Fig. 355.]
If we begin by drawing a net of rectangular equidistant co-ordinates
(about the axes _x_ and _y_), we may alter or _deform_ this {729}
network in various ways, several of which are very simple indeed.
Thus (1) we may alter the dimensions of our system, extending it
along one or other axis, and so converting each little square into a
corresponding and directly proportionate oblong (Fig. 353). It follows
that any figure which we may have inscribed in the original net, and
which we transfer to the new, will thereby be _deformed_ in strict
proportion to the deformation of the entire configuration, being still
defined by corresponding points in the network and being throughout
in conformity with the original figure. For instance, a circle
inscribed in the original “Cartesian” net will now, after extension
in the _y_-direction, be found elongated {730} into an ellipse. In
elementary mathematical language, for the original _x_ and _y_ we
have substituted _x__{1} and _c_ _y__{1}, and the equation to our
original circle, _x_^2 + _y_^2 = _a_^2, becomes that of the ellipse,
_x__{1}^2 + _c_^2 _y__{1}^2 = _a_^2.
If I draw the cannon-bone of an ox (Fig. 354, A), for instance, within
a system of rectangular co-ordinates, and then transfer the same
drawing, point for point, to a system in which for the _x_ of the
original diagram we substitute _x′_ = 2_x_/3, we obtain a drawing (B)
which is a very close approximation to the cannon-bone of the sheep. In
other words, the main (and perhaps the only) difference between the two
bones is simply that that of the sheep is elongated, along the vertical
axis, as compared with that of the ox in the relation of 3/2. And
similarly, the long slender cannon-bone of the giraffe (C) is referable
to the same identical type, subject to a reduction of breadth, or
increase of length, corresponding to _x″_ = _x_/3.
(2) The second type is that where extension is not equal or uniform
at all distances from the origin: but grows greater or less, as,
for instance, when we stretch a _tapering_ elastic band. In such
cases, as I have represented it in Fig. 355, the ordinate increases
logarithmically, and for _y_ we substitute ε^{_y_}. It is obvious that
this logarithmic extension may involve both abscissae and ordinates,
_x_ becoming ε^{_x_}, while _y_ becomes ε^{_y_}. The circle in our
original figure is now deformed into some such shape as that of Fig.
356. This method of deformation is a common one, and will often be of
use to us in our comparison of organic forms.
(3) Our third type is the “simple shear,” where the rectangular
co-ordinates become “oblique,” their axes being inclined to one another
at a certain angle ω. Our original rectangle now becomes such a figure
as that of Fig. 357. The system may now be described in terms of the
oblique axes _X_, _Y_; or may be directly referred to new rectangular
co-ordinates ξ, η by the simple transposition _x_ = ξ − η cot ω, _y_
= η cosec ω.
[Illustration: Fig. 356.]
[Illustration: Fig. 357.]
[Illustration: Fig. 358.]
(4) Yet another important class of deformations may be represented
by the use of radial co-ordinates, in which one set of lines are
represented as radiating from a point or “focus,” while the other set
are transformed into circular arcs cutting the radii orthogonally.
These radial co-ordinates are especially applicable {731} to cases
where there exists (either within or without the figure) some part
which is supposed to suffer no deformation; a simple illustration is
afforded by the diagrams which illustrate the flexure of a beam (Fig.
358). In biology these co-ordinates will be especially applicable in
cases where the growing structure includes a “node,” or point where
growth is absent or at a minimum; and about which node the rate of
growth may be assumed to increase symmetrically. Precisely such a
case is furnished us in a leaf of an ordinary dicotyledon. The leaf
of a {732} typical monocotyledon—such as a grass or a hyacinth, for
instance—grows continuously from its base, and exhibits no node or
“point of arrest.” Its sides taper off gradually from its broad base
to its slender tip, according to some law of decrement specific to the
plant; and any alteration in the relative velocities of longitudinal
and transverse growth will merely make the leaf a little broader or
narrower, and will effect no other conspicuous alteration in its
contour. But if there once come into existence a node, or “locus of no
growth,” about which we may assume the growth—which in the hyacinth
leaf was longitudinal and transverse—to take place radially and
transversely to the radii, then we shall
[Illustration: Fig. 359.]
at once see, in the first place, that the sloping and slightly
curved sides of the hyacinth leaf suffer a transformation into what we
consider a more typical and “leaf-like” shape, the sides of the figure
broadening out to a zone of maximum breadth and then drawing inwards
to the pointed apex. If we now alter the ratio between the radial and
tangential velocities of growth—in other words, if we increase the
angles between corresponding radii—we pass successively through the
various configurations which the botanist describes as the lanceolate,
the ovate, and finally the cordate leaf. These successive changes may
to some extent, and in appropriate cases, be traced as the individual
leaf grows {733} to maturity; but as a much more general rule, the
balance of forces, the ratio between radial and tangential velocities
of growth, remains so nicely and constantly balanced that the leaf
increases in size without conspicuous modification of form. It is
rather what we may call a long-period variation, a tendency for the
relative velocities to alter from one generation to another, whose
result is brought into view by this method of illustration.
[Illustration: Fig. 360. _Begonia daedalea._]
There are various corollaries to this method of describing the form
of a leaf which may be here alluded to, for we shall not return again
to the subject of radial co-ordinates. For instance, the so-called
unsymmetrical leaf[650] of a begonia, in which one side of the leaf
may be merely ovate while the other has a cordate outline, is seen to
be really a case of _unequal_, and not truly asymmetrical, growth on
either side of the midrib. There is nothing more mysterious in its
conformation than, for instance, in that of a forked twig in which
one limb of the fork has grown longer than the other. The case of the
begonia leaf is of sufficient interest to deserve illustration, and in
Fig. 360 I have outlined a leaf of the large _Begonia daedalea_. On the
smaller left-hand side of the leaf I have taken at random three points,
_a_, _b_, _c_, and have measured the angles, _AOa_, etc., which the
radii from the hilus of the leaf to these points make with the median
axis. On the other side of the leaf I have marked the points _a′_,
_b′_, _c′_, such that the radii drawn to this margin of the leaf are
equal to the former, _Oa′_ to _Oa_, etc. Now if the two sides of the
leaf are {734} mathematically similar to one another, it is obvious
that the respective angles should be in continued proportion, i.e.
as _AOa_ is to _AOa′_, so should _AOb_ be to _AOb′_. This proves to
be very nearly the case. For I have measured the three angles on one
side, and one on the other, and have then compared, as follows, the
calculated with the observed values of the other two:
_AOa_ _AOb_ _AOc_ _AOa′_ _AOb′_ _AOc′_
Observed values 12° 28.5° 88° — — 157°
Calculated values — — — 21.5° 51.1° —
Observed values — — — 20 52 —
The agreement is very close, and what discrepancy there is may be amply
accounted for, firstly, by the slight irregularity of the sinuous
margin of the leaf; and secondly, by the fact that the true axis or
midrib of the leaf is not straight but slightly curved, and therefore
that it is curvilinear and not rectilinear triangles which we ought
to have measured. When we understand these few points regarding
the peripheral curvature of the leaf, it is easy to see that its
principal veins approximate closely to a beautiful system of isogonal
co-ordinates. It is also obvious that we can easily pass, by a process
of shearing, from those cases where the principal veins start from the
base of the leaf to those, as in most dicotyledons, where they arise
successively from the midrib.
It may sometimes happen that the node, or “point of arrest,” is at the
upper instead of the lower end of the leaf-blade; and occasionally
there may be a node at both ends. In the former case, as we have it
in the daisy, the form of the leaf will be, as it were, inverted, the
broad, more or less heart-shaped, outline appearing at the upper end,
while below the leaf tapers gradually downwards to an ill-defined base.
In the latter case, as in _Dionaea_, we obtain a leaf equally expanded,
and similarly ovate or cordate, at both ends. We may notice, lastly,
that the shape of a solid fruit, such as an apple or a cherry, is a
solid of revolution, developed from similar curves and to be explained
on the same principle. In the cherry we have a “point of arrest” at the
base of the berry, where it joins its peduncle, and about this point
the fruit (in imaginary section) swells out into a cordate outline;
while in the {735} apple we have two such well-marked points of
arrest, above and below, and about both of them the same conformation
tends to arise. The bean and the human kidney owe their “reniform”
shape to precisely the same phenomenon, namely, to the existence of
a node or “hilus,” about which the forces of growth are radially and
symmetrically arranged.
――――――――――
Most of the transformations which we have hitherto considered (other
than that of the simple shear) are particular cases of a general
transformation, obtainable by the method of conjugate functions and
equivalent to the projection of the original figure on a new plane.
Appropriate transformations, on these general lines, provide for the
cases of a coaxial system where the Cartesian co-ordinates are replaced
by coaxial circles, or a confocal system in which they are replaced by
confocal ellipses and hyperbolas.
[Illustration: Fig. 361.]
Yet another curious and important transformation, belonging to the
same class, is that by which a system of straight lines becomes
transformed into a conformal system of logarithmic spirals: the
straight line _Y_ − _AX_ = _c_ corresponding to the logarithmic
spiral θ − _A_ log _r_ = _c_ (Fig. 361). This beautiful and simple
transformation lets us at once convert, for instance, the straight
conical shell of the Pteropod or the _Orthoceras_ into the logarithmic
spiral of the Nautiloid; it involves a mathematical symbolism which is
but a slight extension of that which we have employed in our elementary
treatment of the logarithmic spiral.
These various systems of coordinates, which we have now briefly
considered, are sometimes called “isothermal co-ordinates,” from
the fact that, when employed in this particular branch of physics,
they perfectly represent the phenomena of the conduction of heat,
the contour lines of equal temperature appearing, under appropriate
conditions, as the orthogonal lines of the co-ordinate system. And it
follows that {736} the “law of growth” which our biological analysis
by means of orthogonal co-ordinate systems presupposes, or at least
foreshadows, is one according to which the organism grows or develops
along _stream lines_, which may be defined by a suitable mathematical
transformation.
When the system becomes no longer orthogonal, as in many of the
following illustrations—for instance, that of _Orthagoriscus_ (Fig.
382),—then the transformation is no longer within the reach of
comparatively simple mathematical analysis. Such departure from the
typical symmetry of a “stream-line” system is, in the first instance,
sufficiently accounted for by the simple fact that the developing
organism is very far from being homogeneous and isotropic, or, in
other words, does not behave like a perfect fluid. But though under
such circumstances our co-ordinate systems may be no longer capable of
strict mathematical analysis, they will still indicate _graphically_
the relation of the new co-ordinate system to the old, and conversely
will furnish us with some guidance as to the “law of growth,” or play
of forces, by which the transformation has been effected.
――――――――――
Before we pass from this brief discussion of transformations in
general, let us glance at one or two cases in which the forces applied
are more or less intelligible, but the resulting transformations are,
from the mathematical point of view, exceedingly complicated.
The “marbled papers” of the bookbinder are a beautiful illustration of
visible “stream lines.” On a dishful of a sort of semi-liquid gum the
workman dusts a few simple lines or patches of colouring matter; and
then, by passing a comb through the liquid, he draws the colour-bands
into the streaks, waves, and spirals which constitute the marbled
pattern, and which he then transfers to sheets of paper laid down
upon the gum. By some such system of shears, by the effect of unequal
traction or unequal growth in various directions and superposed on
an originally simple pattern, we may account for the not dissimilar
marbled patterns which we recognise, for instance, on a large serpent’s
skin. But it must be remarked, in the case of the marbled paper, that
though the method of application of the forces is simple, yet in the
aggregate the system of forces set up by the many {737} teeth of the
comb is exceedingly complex, and its complexity is revealed in the
complicated “diagram of forces” which constitutes the pattern.
To take another and still more instructive illustration. To turn one
circle (or sphere) into two circles would be, from the point of view
of the mathematician, an extraordinarily difficult transformation;
but, physically speaking, its achievement may be extremely simple.
The little round gourd grows naturally, by its symmetrical forces of
expansive growth, into a big, round, or somewhat oval pumpkin or melon.
But the Moorish husbandman ties a rag round its middle, and the same
forces of growth, unaltered save for the presence of this trammel,
now expand the globular structure into two superposed and connected
globes. And again, by varying the position of the encircling band, or
by applying several such ligatures instead of one, a great variety of
artificial forms of “gourd” may be, and actually are, produced. It
is clear, I think, that we may account for many ordinary biological
processes of development or transformation of form by the existence of
trammels or lines of constraint, which limit and determine the action
of the expansive forces of growth that would otherwise be uniform and
symmetrical. This case has a close parallel in the operations of the
glassblower, to which we have already, more than once, referred in
passing[651]. The glassblower starts his operations with a _tube_,
which he first closes at one end so as to form a hollow vesicle, within
which his blast of air exercises a uniform pressure on all sides; but
the spherical conformation which this uniform expansive force would
naturally tend to produce is modified into all kinds of forms by the
trammels or resistances set up as the workman lets one part or another
of his bubble be unequally heated or cooled. It was Oliver Wendell
Holmes who first shewed this curious parallel between the operations
of the glassblower and those of Nature, when she starts, as she so
often does, with a simple tube[652]. The alimentary canal, {738} the
arterial system including the heart, the central nervous system of the
vertebrate, including the brain itself, all begin as simple tubular
structures. And with them Nature does just what the glassblower does,
and, we might even say, no more than he. For she can expand the tube
here and narrow it there; thicken its walls or thin them; blow off a
lateral offshoot or caecal diverticulum; bend the tube, or twist and
coil it; and infold or crimp its walls as, so to speak, she pleases.
Such a form as that of the human stomach is easily explained when it is
regarded from this point of view; it is simply an ill-blown bubble, a
bubble that has been rendered lopsided by a trammel or restraint along
one side, such as to prevent its symmetrical expansion—such a trammel
as is produced if the glassblower lets one side of his bubble get cold,
and such as is actually present in the stomach itself in the form of a
muscular band.
――――――――――
We may now proceed to consider and illustrate a few permutations or
transformations of organic form, out of the vast multitude which are
equally open to this method of inquiry.
[Illustration: Fig. 362.]
We have already compared in a preliminary fashion the metacarpal or
cannon-bone of the ox, the sheep, and the giraffe (Fig. 354); and we
have seen that the essential difference in form between these three
bones is a matter of relative length and breadth, such that, if we
reduce the figures to an identical standard of length (or identical
values of _y_), the breadth (or value of _x_) will be approximately
two-thirds that of the ox in the case of the sheep and one-third
that of the ox in the case of the giraffe. We may easily, for the
sake of closer comparison, determine these ratios more accurately,
for instance, if it be our purpose to compare the different racial
varieties within the limits of a single species. And in such cases, by
the way, as when we compare with one another various breeds or races of
cattle or of horses, the ratios {739} of length and breadth in this
particular bone are extremely significant[653].
If, instead of limiting ourselves to the cannon-bone, we inscribe the
entire foot of our several Ungulates in a co-ordinate system, the
same ratios of _x_ that served us for the cannon-bones still give us
a first approximation to the required comparison; but even in the
case of such closely allied forms as the ox and the sheep there is
evidently something wanting in the comparison. The reason is that the
relative elongation of the several parts, or individual bones, has not
proceeded equally or proportionately in all cases; in other words,
that the equations for _x_ will not suffice without some simultaneous
modification of the values of _y_ (Fig. 362). In such a case it may be
found possible to satisfy the varying values of _y_ by some logarithmic
or other formula; but, even if that be possible, it will probably be
somewhat difficult of discovery or verification in such a case as the
present, owing to the fact that we have too few well-marked points
of correspondence between the one object and the other, and that
especially along the shaft of such long bones as the cannon-bone of
the ox, the deer, the llama, or the giraffe there is a complete lack
of easily recognisable corresponding points. In such a case a brief
tabular statement of apparently corresponding values of _y_, or of
those obviously corresponding values which coincide with the boundaries
of the several bones of the foot, will, as in the following example,
enable us to dispense with a fresh equation.
_a_ _b_ _c_ _d_
_y_ (Ox) 0 18 27 42 100
_y′_ (Sheep) 0 10 19 36 100
_y″_ (Giraffe) 0 5 10 24 100
This summary of values of _y′_, coupled with the equations for the
{740} value of _x_, will enable us, from any drawing of the ox’s foot,
to construct a figure of that of the sheep or of the giraffe with
remarkable accuracy.
[Illustration: Fig. 363.]
That underlying the varying amounts of extension to which the parts or
segments of the limb have been subject there is a law, or principle of
continuity, may be discerned from such a diagram as the above (Fig.
363), where the values of _y_ in the case of the ox are plotted as a
straight line, and the corresponding values for the sheep (extracted
from the above table) are seen to form a more or less regular and
even curve. This simple graphic result implies the existence of a
comparatively simple equation between _y_ and _y′_.
[Illustration: Fig. 364. (After Albert Dürer.)]
An elementary application of the principle of co-ordinates to the
study of proportion, as we have here used it to illustrate the
varying proportions of a bone, was in common use in the sixteenth and
seventeenth centuries by artists in their study of the human form. The
method is probably much more ancient, and may even be classical[654];
it is fully described and put in practice by Albert Dürer in his
_Geometry_, and especially in his _Treatise on Proportion_[655].
In this latter work, the manner in which the {741} human figure,
features, and facial expression are all transformed and modified by
slight variations in the relative magnitude of the parts is admirably
and copiously illustrated (Fig. 364).
[Illustration: Fig. 365.]
In a tapir’s foot there is a striking difference, and yet at the same
time there is an obvious underlying resemblance, between the middle
toe and either of its unsymmetrical lateral neighbours. Let us take
the median terminal phalanx and inscribe its outline in a net of
rectangular equidistant co-ordinates (Fig. 365, _a_). Let us then make
a similar network about axes which are no longer at right angles,
but inclined to one another at an angle of about 50° (_b_). If into
this new network we fill in, point for point, an outline precisely
corresponding to our original drawing of the middle toe, we shall find
that we have already represented the main features of the adjacent
lateral one. We shall, however, perceive that our new diagram looks a
little too bulky on one side, the inner side, of the lateral toe. If
now we substitute for our equidistant ordinates, ordinates which get
gradually closer and closer together as we pass towards the median
side of the toe, then we shall obtain a diagram which differs in no
essential respect from an actual outline copy of the lateral toe (_c_).
In short, the difference between the outline of the middle toe of the
tapir and the next lateral toe may be almost completely expressed
by saying that if the one be represented by rectangular equidistant
co-ordinates, the other will be represented by oblique co-ordinates,
whose axes make an angle of 50°, and in which the abscissal interspaces
decrease in a certain logarithmic ratio. We treated our original
complex curve or projection of the tapir’s toe as a function of the
form _F_ (_x_, _y_) = 0. The figure of the tapir’s lateral {742} toe
is a precisely identical function of the form _F_ (_e_^{_x_}, _y__{1})
= 0, where _x__{1}, _y__{1} are oblique co-ordinate axes inclined to
one another at an angle of 50°.
[Illustration: Fig. 366. (After Albert Dürer.)]
Dürer was acquainted with these oblique co-ordinates also, and I have
copied two illustrative figures from his book[656].
――――――――――
[Illustration: Fig. 367. _Oithona nana._]
[Illustration: Fig. 368. _Sapphirina._]
In Fig. 367 I have sketched the common Copepod _Oithona nana_,
{743} and have inscribed it in a rectangular net, with abscissae
three-fifths the length of the ordinates. Side by side (Fig. 368) is
drawn a very different Copepod, of the genus _Sapphirina_; and about
it is drawn a network such that each co-ordinate passes (as nearly as
possible) through points corresponding to those of the former figure.
It will be seen that two differences are apparent. (1) The values of
_y_ in Fig. 368 are large in the upper part of the figure, and diminish
rapidly towards its base. (2) The values of _x_ are very large in the
neighbourhood of the origin, but diminish rapidly as we pass towards
either side, away from the median vertical axis; and it is probable
that they do so according to a definite, but somewhat complicated,
ratio. If, instead of seeking for an actual equation, we simply
tabulate our values of _x_ and _y_ in the second figure as compared
with the first (just as we did in comparing the feet of the Ungulates),
we get the dimensions of a net in which, by simply projecting the
figure of _Oithona_, we obtain that of _Sapphirina_ without further
trouble, e.g.:
_x_ (_Oithona_) 0 3 6 9 12 15 —
_x′_ (_Sapphirina_) 0 8 10 12 13 14 —
_y_ (_Oithona_) 0 5 10 15 20 25 30
_y′_ (_Sapphirina_) 0 2 7 3 23 32 40
In this manner, with a single model or type to copy from, we may
record in very brief space the data requisite for the production of
approximate outlines of a great number of forms. For instance the
difference, at first sight immense, between the attenuated body of a
_Caprella_ and the thick-set body of a _Cyamus_ is obviously little,
and is probably nothing, more than a difference of relative magnitudes,
capable of tabulation by numbers and of complete expression by means of
rectilinear co-ordinates.
The Crustacea afford innumerable instances of more complex
deformations. Thus we may compare various higher Crustacea with one
another, even in the case of such dissimilar forms as a lobster and
a crab. It is obvious that the whole body of the former is elongated
as compared with the latter, and that the crab is relatively broad
in the region of the carapace, while it tapers off rapidly towards
its attenuated and abbreviated tail. In a general way, the elongated
rectangular system of co-ordinates {744} in which we may inscribe the
outline of the lobster becomes a shortened triangle in the case of
the crab. In a little more detail we may compare the outline of the
carapace in various crabs one with another: and the comparison will
be found easy and significant, even, in many cases, down to minute
details, such as the number and situation of the marginal spines,
though these are in other cases subject to independent variability.
[Illustration: Fig. 369. Carapaces of various crabs. 1, _Geryon_; 2,
_Corystes_; 3, _Scyramathia_; 4, _Paralomis_; 5, _Lupa_; 6, _Chorinus_.]
If we choose, to begin with, such a crab as _Geryon_ (Fig. 369, 1),
and inscribe it in our equidistant rectangular co-ordinates, we shall
see that we pass easily to forms more elongated in a transverse
{745} direction, such as _Matuta_ or _Lupa_ (5), and conversely, by
transverse compression, to such a form as _Corystes_ (2). In certain
other cases the carapace conforms to a triangular diagram, more or less
curvilinear, as in Fig. 4, which represents the genus _Paralomis_. Here
we can easily see that the posterior border is transversely elongated
as compared with that of _Geryon_, while at the same time the anterior
part is longitudinally extended as compared with the posterior. A
system of slightly curved and converging ordinates, with orthogonal and
logarithmically interspaced abscissal lines, as shown in the figure,
appears to satisfy the conditions.
In an interesting series of cases, such as the genus _Chorinus_,
or _Scyramathia_, and in the spider-crabs generally, we appear to
have just the converse of this. While the carapace of these crabs
presents a somewhat triangular form, which seems at first sight more
or less similar to those just described, we soon see that the actual
posterior border is now narrow instead of broad, the broadest part of
the carapace corresponding precisely, not to that which is broadest
in _Paralomis_, but to that which was broadest in _Geryon_; while the
most striking difference from the latter lies in an antero-posterior
lengthening of the forepart of the carapace, culminating in a great
elongation of the frontal region, with its two spines or “horns.” The
curved ordinates here converge posteriorly and diverge widely in front
(Figs. 3 and 6), while the decremental interspacing of the abscissae is
very marked indeed.
We put our method to a severer test when we attempt to sketch an entire
and complicated animal than when we simply compare corresponding
parts such as the carapaces of various Malacostraca, or related bones
as in the case of the tapir’s toes. Nevertheless, up to a certain
point, the method stands the test very well. In other words, one
particular mode and direction of variation is often (or even usually)
so prominent and so paramount throughout the entire organism, that one
comprehensive system of co-ordinates suffices to give a fair picture
of the actual phenomenon. To take another illustration from the
Crustacea, I have drawn roughly in Fig. 370, 1 a little amphipod of the
family Phoxocephalidae (_Harpinia_ sp.). Deforming the co-ordinates
of the figure into the {746} curved orthogonal system in Fig. 2,
we at once obtain a very fair representation of an allied genus,
belonging to a different family of amphipods, namely _Stegocephalus_.
As we proceed further from our type our co-ordinates will require
greater deformation, and the resultant figure will usually be somewhat
less accurate. In Fig. 3 I show a network, to which, if we transfer
our diagram of _Harpinia_ or of _Stegocephalus_, we shall obtain a
tolerable representation of the aberrant genus _Hyperia_, with its
narrow abdomen, its reduced pleural lappets, its great eyes, and its
inflated head.
[Illustration: Fig 370. 1. _Harpinia plumosa_ Kr. 2. _Stegocephalus
inflatus_ Kr. 3. _Hyperia galba_.]
――――――――――
The hydroid zoophytes constitute a “polymorphic” group, within which a
vast number of species have already been distinguished; and the labours
of the systematic naturalist are constantly adding to the number. The
specific distinctions are for the most part based, not upon characters
directly presented {747} by the living animal, but upon the form,
size and arrangement of the little cups, or “calycles,” secreted and
inhabited by the little individual polypes which compose the compound
organism. The variations, which are apparently infinite, of these
conformations are easily seen to be a question of relative magnitudes,
and are capable of complete expression, sometimes by very simple,
sometimes by somewhat more complex, co-ordinate networks.
[Illustration: Fig. 371. _a_, _Campanularia macroscyphus_, Allm.; _b_,
_Gonothyraea hyalina_, Hincks; _c_, _Clytia Johnstoni_, Alder.]
For instance, the varying shapes of the simple wineglass-shaped cups of
the Campanularidae are at once sufficiently represented and compared
by means of simple Cartesian co-ordinates (Fig. 371). In the two
allied families of Plumulariidae and Aglaopheniidae the calycles are
set unilaterally upon a jointed stem, and small cup-like structures
(holding rudimentary polypes) are associated with the large calycles
in definite number and position. These small calyculi are variable in
number, but in the great majority of cases they accompany the large
calycle in groups of three—two standing by its upper border, and one,
which is especially variable in form and magnitude, lying at its base.
The stem is liable to flexure and, in a high degree, to extension or
compression; and these variations extend, often on an exaggerated
scale, to the related calycles. As a result we find that we can draw
various systems of curved or sinuous co-ordinates, which express,
all but completely, the configuration of the various {748} hydroids
which we inscribe therein (Fig. 372). The comparative smoothness or
denticulation of the margin of the calycle, and the number of its
denticles, constitutes an independent variation, and requires separate
description; we have already seen (p. 236) that this denticulation is
in all probability due to a particular physical cause.
[Illustration: Fig. 372. _a_, _Cladocarpus crenatus_, F.; _b_,
_Aglaophenia pluma_, L.; _c_, _A. rhynchocarpa_, A.; _d_, _A cornuta_,
K.; _e_, _A. ramulosa_, K.]
――――――――――
[Illustration: Fig. 373. _Argyropelecus Olfersi._]
[Illustration: Fig. 374. _Sternoptyx diaphana._]
Among the fishes we discover a great variety of deformations, some of
them of a very simple kind, while others are more striking and more
unexpected. A comparatively simple case, involving a simple shear, is
illustrated by Figs. 373 and 374. Fig. 373 represents, within Cartesian
co-ordinates, a certain little oceanic fish known as _Argyropelecus
Olfersi_. Fig. 474 represents precisely the same outline, transferred
to a system of oblique co-ordinates whose {749} axes are inclined
at an angle of 70°; but this is now (as far as can be seen on the
scale of the drawing) a very good figure of an allied fish, assigned
to a different genus, under the name of _Sternoptyx diaphana_. The
deformation illustrated by this case of _Argyropelecus_ is precisely
analogous to the simplest and commonest kind of deformation to which
fossils are subject (as we have seen on p. 553) as the result of
shearing-stresses in the solid rock.
[Illustration: Fig. 375. _Scarus_ sp.]
[Illustration: Fig. 376. _Pomacanthus._]
Fig. 375 is an outline diagram of a typical Scaroid fish. Let us
deform its rectilinear co-ordinates into a system of (approximately)
coaxial circles, as in Fig. 376, and then filling into the new system,
space by space and point by point, our former diagram of _Scarus_,
we obtain a very good outline of an allied fish, belonging to a
neighbouring family, of the genus _Pomacanthus_. This case is all the
more interesting, because upon the body of our _Pomacanthus_ there are
striking colour bands, which correspond in direction very closely to
the lines of our new curved ordinates. In like manner, the still more
bizarre outlines of other fishes of the same family of Chaetodonts
will be found to correspond to very slight modifications of similar
co-ordinates; in other words, to small variations in the values of the
constants of the coaxial curves.
In Figs. 377–380 I have represented another series of Acanthopterygian
fishes, not very distantly related to the foregoing. If we start
this series with the figure of _Polyprion_, in Fig. 377, we see that
the outlines of _Pseudopriacanthus_ (Fig. 378) and of _Sebastes_ or
_Scorpaena_ (Fig. 379) are easily derived by substituting a system of
triangular, or radial, co-ordinates for the rectangular ones in {750}
which we had inscribed _Polyprion_. The very curious fish _Antigonia
capros_, an oceanic relative of our own “boar-fish,” conforms closely
to the peculiar deformation represented in Fig. 380.
[Illustration: Fig. 377. _Polyprion._]
[Illustration: Fig. 378. _Pseudopriacanthus altus._]
[Illustration: Fig. 379. _Scorpaena_ sp.]
[Illustration: Fig. 380. _Antigonia capros._]
[Illustration: Fig. 381. _Diodon._]
[Illustration: Fig. 382. _Orthagoriscus._]
Fig. 381 is a common, typical _Diodon_ or porcupine-fish, and in
Fig. 382 I have deformed its vertical co-ordinates into a system of
concentric circles, and its horizontal co-ordinates into a system of
curves which, approximately and provisionally, are made to resemble a
system of hyperbolas[657]. The old outline, transferred {751} in its
integrity to the new network, appears as a manifest representation of
the closely allied, but very different looking, sunfish, _Orthagoriscus
mola_. This is a particularly instructive case of deformation or
transformation. It is true that, in a mathematical sense, it is not a
perfectly satisfactory or perfectly regular deformation, for the system
is no longer isogonal; but nevertheless, it is symmetrical to the eye,
and obviously approaches to an isogonal system under certain conditions
of friction or constraint. And as such it accounts, by one single
integral transformation, for all the apparently separate and distinct
external differences between the two fishes. It leaves the parts
near to the origin of the system, the whole region of the head, the
opercular orifice and the pectoral fin, practically unchanged {752} in
form, size and position; and it shews a greater and greater apparent
modification of size and form as we pass from the origin towards the
periphery of the system.
In a word, it is sufficient to account for the new and striking contour
in all its essential details, of rounded body, exaggerated dorsal and
ventral fins, and truncated tail. In like manner, and using precisely
the same co-ordinate networks, it appears to me possible to shew the
relations, almost bone for bone, of the skeletons of the two fishes;
in other words, to reconstruct the skeleton of the one from our
knowledge of the skeleton of the other, under the guidance of the same
correspondence as is indicated in their external configuration.
――――――――――
The family of the crocodiles has had a special interest for the
evolutionist ever since Huxley pointed out that, in a degree only
second to the horse and its ancestors, it furnishes us with a close
and almost unbroken series of transitional forms, running down in
continuous succession from one geological formation to another. I
should be inclined to transpose this general statement into other
terms, and to say that the Crocodilia constitute a case in which,
with unusually little complication from the presence of independent
variants, the trend of one particular mode of transformation is
visibly manifested. If we exclude meanwhile from our comparison a few
of the oldest of the crocodiles, such as _Belodon_, which differ more
fundamentally from the rest, we shall find a long series of genera
in which we can refer not only the changing contours of the skull,
but even the shape and size of the many constituent bones and their
intervening spaces or “vacuities,” to one and the same simple system
of transformed co-ordinates. The manner in which the skulls of various
Crocodilians differ from one another may be sufficiently illustrated by
three or four examples.
[Illustration: Fig. 383. A, _Crocodilus porosus_. B, _C. americanus_.
C, _Notosuchus terrestris_.]
Let us take one of the typical modern crocodiles as our standard of
form, e.g. _C. porosus_, and inscribe it, as in Fig. 383, _a_, in the
usual Cartesian co-ordinates. By deforming the rectangular network
into a triangular system, with the apex of the triangle a little
way in front of the snout, as in _b_, we pass to such a form as _C.
americanus_. By an exaggeration of the same process we at once get
an approximation to the form of one of the sharp-snouted, {753} or
longirostrine, crocodiles, such as the genus _Tomistoma_; and, in
the species figured, the oblique position of the orbits, the arched
contour of the occipital border, and certain other characters suggest a
certain amount of curvature, such as I have represented in the diagram
(Fig. 383, _b_), on the part of the horizontal co-ordinates. In the
still more elongated skull of such a form as the Indian Gavial, the
whole skull has undergone a great longitudinal extension, or, in other
words, the ratio of _x_/_y_ is greatly diminished; and this extension
is not uniform, but is at a maximum in the region of the nasal and
maxillary bones. This especially elongated region is at the same time
narrowed in an exceptional degree, and its excessive narrowing is
represented by a curvature, convex towards the median axis, on the
part of the vertical ordinates. Let us take as a last illustration
one of the Mesozoic crocodiles, the little _Notosuchus_, from the
Cretaceous formation. This little crocodile is very different from
our type in the proportions of its skull. The region of the snout,
in front of and including the frontal bones, is greatly shortened;
from constituting fully two-thirds of the whole length of the
skull in _Crocodilus_, it now constitutes less than half, or, say,
three-sevenths of the whole; and the whole skull, and especially its
posterior part, is curiously compact, broad, and squat. The orbit is
unusually large. If in the diagram of this skull we select a number of
points obviously corresponding {754} to points where our rectangular
co-ordinates intersect particular bones or other recognisable features
in our typical crocodile, we shall easily discover that the lines
joining these points in _Notosuchus_ fall into such a co-ordinate
network as that which is represented in Fig. 383, _c_. To all intents
and purposes, then, this not very complex system, representing
one harmonious “deformation,” accounts for _all_ the differences
between the two figures, and is sufficient to enable one at any time
to reconstruct a detailed drawing, bone for bone, of the skull of
_Notosuchus_ from the model furnished by the common crocodile.
[Illustration: Fig. 384. Pelvis of (A) _Stegosaurus_; (B)
_Camptosaurus_.]
The many diverse forms of Dinosaurian reptiles, all of which manifest
a strong family likeness underlying much superficial diversity,
furnish us with plentiful material for comparison by the method of
transformations. As an instance, I have figured the pelvic bones of
_Stegosaurus_ and of _Camptosaurus_ (Fig. 384, _a_, _b_) to show that,
when the former is taken as our Cartesian type, a slight curvature
and an approximately logarithmic extension of the _x_-axis brings us
easily to the configuration of the other. In the original specimen of
_Camptosaurus_ described by Marsh[658], the anterior portion of the
iliac bone is missing; and in Marsh’s restoration this part of the
bone is drawn as though it came somewhat abruptly to a sharp point.
In my figure I {755} have completed this missing part of the bone
in harmony with the general co-ordinate network which is suggested
by our comparison of the two entire pelves; and I venture to think
that the result is more natural in appearance, and more likely to be
correct than was Marsh’s conjectural restoration. It would seem, in
fact, that there is an obvious field for the employment of the method
of co-ordinates in this task of reproducing missing portions of a
structure to the proper scale and in harmony with related types. To
this subject we shall presently return.
[Illustration: Fig. 385. Shoulder-girdle of _Cryptocleidus_. _a_,
young; _b_, adult.]
[Illustration: Fig. 386. Shoulder-girdle of _Ichthyosaurus_.]
In Fig. 385, _a_, _b_, I have drawn the shoulder-girdle of
_Cryptocleidus_, a Plesiosaurian reptile, half-grown in the one case
and full-grown in the other. The change of form during growth in
this region of the body is very considerable, and its nature is well
brought out by the two co-ordinate systems. In Fig. 386 I have drawn
the shoulder-girdle of an Ichthyosaur, referring it to _Cryptocleidus_
as a standard of comparison. The interclavicle, which is present in
_Ichthyosaurus_, is minute and hidden in _Cryptocleidus_; but the
numerous other differences between the two {756} forms, chief among
which is the great elongation in _Ichthyosaurus_ of the two clavicles,
are all seen by our diagrams to be part and parcel of one general and
systematic deformation.
[Illustration: Fig. 387. _a_, Skull of _Dimorphodon_. _b_, Skull of
_Pteranodon_.]
Before we leave the group of reptiles we may glance at the very
strangely modified skull of _Pteranodon_, one of the extinct flying
reptiles, or Pterosauria. In this very curious skull the region of
the jaws, or beak, is greatly elongated and pointed; the occipital
bone is drawn out into an enormous backwardly-directed crest; the
posterior part of the lower jaw is similarly produced backwards; the
orbit is small; and the quadrate bone is strongly inclined downwards
and forwards. The whole skull has a configuration which stands,
apparently, in the strongest possible contrast to that of a more normal
Ornithosaurian such as _Dimorphodon_. But if we inscribe the latter in
Cartesian coordinates (Fig. 387, _a_), and refer our _Pteranodon_ to
a system of oblique co-ordinates (_b_), in which the two co-ordinate
systems of parallel lines become each a pencil of diverging rays, we
make manifest a correspondence which extends uniformly throughout all
parts of these very different-looking skulls.
――――――――――
[Illustration: Fig. 388. Pelvis of _Archaeopteryx_.]
[Illustration: Fig. 389. Pelvis of _Apatornis_.]
We have dealt so far, and for the most part we shall continue to
deal, with our co-ordinate method as a means of comparing one known
structure with another. But it is obvious, as I have said, {757}
that it may also be employed for drawing hypothetical structures,
on the assumption that they have varied from a known form in some
definite way. And this process may be especially useful, and will be
most obviously legitimate, when we apply it to the particular case of
representing intermediate stages between two forms which are actually
known to exist, in other words, of reconstructing the transitional
stages through which the course of evolution must have successively
travelled if it has brought about the change from some ancestral type
to its presumed descendant. Some little time ago I sent to my friend,
Mr Gerhard Heilmann of Copenhagen, a few of my own rough co-ordinate
diagrams, including some in which the pelves of certain ancient and
primitive birds were compared one with another. Mr Heilmann, who is
both a skilled draughtsman and an able morphologist, returned me a set
of diagrams which are a vast improvement on my own, {758} and which
are reproduced in Figs. 388–393. Here we have, as extreme cases, the
pelvis of _Archaeopteryx_, the most ancient of known birds, and that of
_Apatornis_, one of the fossil “toothed”
[Illustration: Fig. 390. The co-ordinate systems of Figs. 388 and 389,
with three intermediate systems interpolated.]
[Illustration: Fig. 391. The first intermediate co-ordinate network,
with its corresponding inscribed pelvis.]
birds from the North American Cretaceous formations—a bird shewing
some resemblance to the modern terns. The pelvis of _Archaeopteryx_
is taken as our type, and referred accordingly to {759} Cartesian
co-ordinates (Fig. 388); while the corresponding coordinates of the
very different pelvis of _Apatornis_ are represented in Fig. 389. In
Fig. 390 the outlines of these two co-ordinate systems are superposed
upon one another, and those of three intermediate and equidistant
co-ordinate systems are interpolated between them. From each of these
latter systems, so determined by direct interpolation, a complete
co-ordinate diagram is drawn, and the corresponding outline of a pelvis
is found from each of these systems of co-ordinates, as in Figs. 391,
392. Finally, in Fig. 393 the complete series is represented, beginning
with the known pelvis of _Archaeopteryx_, and leading up by our three
intermediate hypothetical types to the known pelvis of _Apatornis_.
[Illustration: Fig. 392. The second and third intermediate co-ordinate
networks, with their corresponding inscribed pelves.]
――――――――――
Among mammalian skulls I will take two illustrations only, one drawn
from a comparison of the human skull with that of the higher apes, and
another from the group of Perissodactyle {760} Ungulates, the group
which includes the rhinoceros, the tapir, and the horse.
[Illustration: Fig. 393. The pelves of _Archaeopteryx_ and of
_Apatornis_, with three transitional types interpolated between them.]
Let us begin by choosing as our type the skull of _Hyrachyus agrarius_,
Cope, from the Middle Eocene of North America, as figured by Osborn in
his Monograph of the Extinct Rhinoceroses[659] (Fig. 394).
[Illustration: Fig. 394. Skull of _Hyrachyus agrarius_. (After
Osborn.)]
[Illustration: Fig. 395. Skull of _Aceratherium tridactylum_. (After
Osborn.)]
The many other forms of primitive rhinoceros described in the monograph
differ from _Hyrachyus_ in various details—in the characters of the
teeth, sometimes in the number of the toes, and so forth; and they also
differ very considerably in the general {761} appearance of the skull.
But these differences in the conformation of the skull, conspicuous
as they are at first sight, will be found easy to bring under the
conception of a simple and homogeneous transformation, such as would
result from the application of some not very complicated stress. For
instance, the corresponding co-ordinates of _Aceratherium tridactylum_,
as shown in Fig. 395, indicate that the essential difference between
this skull and the former one may be summed up by saying that the long
axis of the skull of _Aceratherium_ has undergone a slight double
curvature, while the upper parts of the skull have at the same time
been {762} subject to a vertical expansion, or to growth in somewhat
greater proportion than the lower parts. Precisely the same changes, on
a somewhat greater scale, give us the skull of an existing rhinoceros.
[Illustration: Fig. 396. Occipital view of the skulls of various
extinct rhinoceroses (_Aceratherium_ spp.). (After Osborn.)]
Among the species of _Aceratherium_, the posterior, or occipital,
view of the skull presents specific differences which are perhaps
more conspicuous than those furnished by the side view; and these
differences are very strikingly brought out by the series of conformal
transformations which I have represented in Fig. 396. In this case it
will perhaps be noticed that the correspondence is not always quite
accurate in small details. It could easily have been made much more
accurate by giving a slightly sinuous curvature to certain of the
co-ordinates. But as they stand, the correspondence indicated is very
close, and the simplicity of the figures illustrates all the better the
general character of the transformation.
By similar and not more violent changes we pass easily to such allied
forms as the Titanotheres (Fig. 397); and the well-known series of
species of _Titanotherium_, by which Professor Osborn has {763}
illustrated the evolution of this genus, constitutes a simple and
suitable case for the application of our method.
But our method enables us to pass over greater gaps than these, and
to discern the general, and to a very large extent even the detailed,
resemblances between the skull of the rhinoceros and those of the
tapir or the horse. From the Cartesian co-ordinates in which we have
begun by inscribing the skull of a primitive rhinoceros, we pass to
the tapir’s skull (Fig. 398), firstly, by converting the rectangular
into a triangular network, by which we represent the depression of the
anterior and the progressively increasing elevation of the posterior
part of the skull; and secondly, by giving to the vertical ordinates a
curvature such as to bring about a certain longitudinal compression, or
condensation, in the forepart of the skull, especially in the nasal and
orbital regions.
[Illustration: Fig. 397. _Titanotherium robustum_.]
[Illustration: Fig. 398. Tapir’s skull.]
The conformation of the horse’s skull departs from that of our
primitive Perissodactyle (that is to say our early type of rhinoceros,
_Hyrachyus_) in a direction that is nearly the opposite of that taken
by _Titanotherium_ and by the recent species of rhinoceros. For we
perceive, by Fig. 399, that the horizontal co-ordinates, which in these
latter cases became transformed into curves with the concavity upwards,
are curved, in the case of the horse, in the opposite direction. And
the vertical ordinates, which are also curved, somewhat in the same
fashion as in the tapir, are very nearly equidistant, instead of being,
as in that animal, crowded together anteriorly. Ordinates and abscissae
form an oblique {764} system, as is shown in the figure. In this case
I have attempted to produce the network beyond the region which is
actually required to include the diagram of the horse’s skull, in order
to show better the form of the general transformation, with a part only
of which we have actually to deal.
[Illustration: Fig. 399. Horse’s skull.]
[Illustration: Fig. 400. Rabbit’s skull.]
It is at first sight not a little surprising to find that we can pass,
by a cognate and even simpler transformation, from our Perissodactyle
skulls to that of the rabbit; but the fact that we can easily do so is
a simple illustration of the undoubted affinity which exists between
the Rodentia, especially the family of the Leporidae, and the more
primitive Ungulates. For my part, I would go further; for I think there
is strong reason to believe that the Perissodactyles are more closely
related to the Leporidae than the former are to the other Ungulates,
or than the Leporidae are to the rest of the Rodentia. Be that as it
may, it is obvious from Fig. 400 that the rabbit’s skull conforms
to a system of {765} co-ordinates corresponding to the Cartesian
co-ordinates in which we have inscribed the skull of _Hyrachyus_,
with the difference, firstly, that the horizontal ordinates of the
latter are transformed into equidistant curved lines, approximately
arcs of circles, with their concavity directed downwards; and
secondly, that the vertical ordinates are transformed into a pencil
of rays approximately orthogonal to the circular arcs. In short,
the configuration of the rabbit’s skull is derived from that of our
primitive rhinoceros by the unexpectedly simple process of submitting
the latter to a
[Illustration: Fig. 401. _A_, outline diagram of the Cartesian
co-ordinates of the skull of _Hyracotherium_ or _Eohippus_, as shewn
in Fig. 402, A. _H_, outline of the corresponding projection of the
horse’s skull. _B_–_G_, intermediate, or interpolated, outlines.]
strong and uniform flexure in the downward direction (cf. Fig. 358,
p. 731). In the case of the rabbit the configuration of the individual
bones does not conform quite so well to the general transformation as
it does when we are comparing the several Perissodactyles one with
another; and the chief departures from conformity will be found in the
size of the orbit and in the outline of the immediately surrounding
bones. The simple fact is that the relatively enormous eye of the
rabbit constitutes an independent variation, which cannot be brought
into the general and fundamental transformation, but must be dealt with
{768} separately. The enlargement of the eye, like the modification
in form and number of the teeth, is a separate phenomenon, which
supplements but in no way contradicts our general comparison of the
skulls taken in their entirety.
[Illustration: 〈two printed pages〉
Fig. 402. _A_, skull of _Hyracotherium_, from the Eocene, after W. B.
Scott; _H_, skull of horse, represented as a co-ordinate transformation
of that of _Hyracotherium_, and to the same scale of magnitude;
_B_–_G_, various artificial or imaginary types, reconstructed as
intermediate stages between _A_ and _H_; _M_, skull of _Mesohippus_,
from the Oligocene, after Scott, for comparison with _C_; _P_, skull of
_Protohippus_, from the Miocene, after Cope, for comparison with _E_;
_Pp_, lower jaw of _Protohippus placidus_ (after Matthew and Gidley),
for comparison with _F_; _Mi_, _Miohippus_ (after Osborn), _Pa_,
_Parahippus_ (after Peterson), shewing resemblance, but less perfect
agreement, with _C_ and _D_.]
――――――――――
Before we leave the Perissodactyla and their allies, let us look a
little more closely into the case of the horse and its immediate
relations or ancestors, doing so with the help of a set of diagrams
which I again owe to Mr Gerard Heilmann[660]. Here we start afresh,
with the skull (Fig. 402, _A_) of _Hyracotherium_ (or _Eohippus_),
inscribed in a simple Cartesian network. At the other end of the
series (_H_) is a skull of Equus, in its own corresponding network;
and the intermediate stages (_B_–_G_) are all drawn by direct and
simple interpolation, as in Mr Heilmann’s former series of drawings of
_Archaeopteryx_ and _Apatornis_. In this present case, the relative
magnitudes are shewn, as well as the forms, of the several skulls.
Alongside of these reconstructed diagrams, are set figures of certain
extinct “horses” (Equidae or Palaeotheriidae), and in two cases,
viz. _Mesohippus_ and _Protohippus_ (_M_, _P_), it will be seen that
the actual fossil skull coincides in the most perfect fashion with
one of the hypothetical forms or stages which our method shews to be
implicitly involved in the transition from _Hyracotherium_ to _Equus_.
In a third case, that of _Parahippus_ (_Pa_), the correspondence (as
Mr Heilmann points out) is by no means exact. The outline of this
skull comes nearest to that of the hypothetical transition stage
_D_, but the “fit” is now a bad one; for the skull of _Parahippus_
is evidently a longer, straighter and narrower skull, and differs in
other minor characters besides. In short, though some writers have
placed _Parahippus_ in the direct line of descent between _Equus_ and
_Eohippus_, we see at once that there is no place for it there, and
that it must, accordingly, represent a somewhat divergent branch or
offshoot of the Equidae[661]. It may be noticed, especially in the
case of _Protohippus_ {769} (_P_), that the configuration of the
angle of the jaw does not tally quite so accurately with that of our
hypothetical diagrams as do other parts of the skull. As a matter of
fact, this region is somewhat variable, in different species of a
genus, and even in different individuals of the same species; in the
small figure (_Pp_) of _Protohippus placidus_ the correspondence is
more exact.
[Illustration: Fig. 403. Human scapulae (after Dwight). _A_, Caucasian;
_B_, Negro; _C_, North American Indian (from Kentucky Mountains).]
In considering this series of figures we cannot but be struck, not
only with the regularity of the succession of “transformations,” but
also with the slight and inconsiderable differences which separate
the known and recorded stages, and even the two extremes of the whole
series. These differences are no greater (save in regard to actual
magnitude) than those between one human skull and another, at least
if we take into account the older or remoter races; and they are
again no greater, but if anything less, than the range of variation,
racial and individual, in certain other human bones, for instance the
scapula[662].
The variability of this latter bone is great, but it is neither {770}
surprising nor peculiar; for it is linked with all the considerations
of mechanical efficiency and functional modification which we dealt
with in our last chapter. The scapula occupies, as it were, a focus
in a very important field of force; and the lines of force converging
on it will be very greatly modified by the varying development of the
muscles over a large area of the body and of the uses to which they are
habitually put.
[Illustration: Fig. 404. Human skull.]
[Illustration: Fig. 405. Co-ordinates of chimpanzee’s skull, as a
projection of the Cartesian co-ordinates of Fig. 404.]
Let us now inscribe in our Cartesian co-ordinates the outline of a
human skull (Fig. 404), for the purpose of comparing it with the
skulls of some of the higher apes. We know beforehand that the main
differences between the human and the simian types depend upon the
enlargement or expansion of the brain and braincase in man, and the
relative diminution or enfeeblement of his jaws. Together with these
changes, the “facial angle” increases from an oblique angle to nearly a
right angle in man, {771} and the configuration of every constituent
bone of the face and skull undergoes an alteration. We do not know to
begin with, and we are not shewn by the ordinary methods of comparison,
how far these various changes form part of one harmonious and congruent
transformation, or whether we are to look, for instance, upon the
changes undergone by the frontal, the occipital, the maxillary, and
the mandibular regions as a congeries of separate modifications or
independent variants. But as soon as we have marked out a number of
points in the gorilla’s or chimpanzee’s skull, corresponding with
those which our co-ordinate network intersected in the human skull,
we find that these corresponding points may be at once linked up by
smoothly curved lines of intersection, which form a new system of
co-ordinates and constitute a simple “projection” of our human skull.
The network represented in Fig. 405 constitutes such a projection of
the human skull on what we may call, figuratively speaking, the “plane”
of the chimpanzee; and the full diagram in Fig. 406 demonstrates the
correspondence. In Fig. 407 I have shewn the similar deformation in
the case of a baboon, and it is obvious that the transformation is of
precisely the same order, and differs only in an increased intensity or
degree of deformation.
[Illustration: Fig. 406. Skull of chimpanzee.]
[Illustration: Fig. 407. Skull of baboon.]
In both dimensions, as we pass from above downwards and from behind
forwards, the corresponding areas of the network are seen to increase
in a gradual and approximately logarithmic order in the lower as
compared with the higher type of skull; and, in short, it becomes
at once manifest that the modifications of jaws, braincase, and the
regions between are all portions of one continuous and integral
process. It is of course easy to draw the {772} inverse diagrams,
by which the Cartesian co-ordinates of the ape are transformed into
curvilinear and non-equidistant co-ordinates in man.
From this comparison of the gorilla’s or chimpanzee’s with the
human skull we realise that an inherent weakness underlies the
anthropologist’s method of comparing skulls by reference to a small
number of axes. The most important of these are the “facial” and
“basicranial” axes, which include between them the “facial angle.”
But it is, in the first place, evident that these axes are merely the
principal axes of a system of co-ordinates, and that their restricted
and isolated use neglects all that can be learned from the filling in
of the rest of the co-ordinate network. And, in the second place, the
“facial axis,” for instance, as ordinarily used in the anthropological
comparison of one human skull with another, or of the human skull with
the gorilla’s, is in all cases treated as a straight line; but our
investigation has shewn that rectilinear axes only meet the case in
the simplest and most closely related transformations; and that, for
instance, in the anthropoid skull no rectilinear axis is homologous
with a rectilinear axis in a man’s skull, but what is a straight line
in the one has become a certain definite curve in the other.
Mr Heilmann tells me that he has tried, but without success, to obtain
a transitional series between the human skull and some prehuman,
anthropoid type, which series (as in the case of the Equidae) should
be found to contain other known types in direct linear sequence.
It appears impossible, however, to obtain such a series, or to
pass by successive and continuous gradations through such forms as
Mesopithecus, Pithecanthropus, _Homo neanderthalensis_, and the lower
or higher races of modern man. The failure is not the fault of our
method. It merely indicates that no one straight line of descent, or
of consecutive transformation, exists; but on the contrary, that among
human and anthropoid types, recent and extinct, we have to do with a
complex problem of divergent, rather than of continuous, variation.
And in like manner, easy as it is to correlate the baboon’s and
chimpanzee’s skulls severally with that of man, and easy as it is to
see that the chimpanzee’s skull is much nearer to the human type than
is the baboon’s, it is also not difficult to perceive that the series
is not, {773} strictly speaking, continuous, and that neither of
our two apes lies _precisely_ on the same direct line or sequence of
deformation by which we may hypothetically connect the other with man.
[Illustration: Fig. 408. Skull of dog, compared with the human skull of
Fig. 404.]
As a final illustration I have drawn the outline of a dog’s skull (Fig.
408), and inscribed it in a network comparable with the Cartesian
network of the human skull in Fig. 404. Here we attempt to bridge over
a wider gulf than we have crossed in any of our former comparisons.
But, nevertheless, it is obvious that our method still holds good, in
spite of the fact that there are various specific differences, such as
the open or closed orbit, etc., which have to be separately described
and accounted for. We see that the chief essential differences in plan
between the dog’s skull and the man’s lie in the fact that, relatively
speaking, the former tapers away in front, a triangular taking the
place of a rectangular conformation; secondly, that, coincident with
the tapering off, there is a progressive elongation, or pulling out,
of the whole forepart of the skull; and lastly, as a minor difference,
that the straight vertical ordinates of the human skull become
curved, with their convexity directed forwards, in the dog. While
the net result is that in the dog, just as in the chimpanzee, the
brain-pan is smaller and the jaws are larger than in man, it is now
conspicuously evident that the co-ordinate network of the ape is by no
means intermediate between those which fit the other two. The mode of
deformation is on different lines; and, while it may be correct to say
that the chimpanzee and the baboon are more brute-like, it would be
by no means accurate to assert that they are more dog-like, than man.
{774}
In this brief account of co-ordinate transformations and of their
morphological utility I have dealt with plane co-ordinates only, and
have made no mention of the less elementary subject of co-ordinates in
three-dimensional space. In theory there is no difficulty whatsoever
in such an extension of our method; it is just as easy to refer the
form of our fish or of our skull to the rectangular co-ordinates _x_,
_y_, _z_, or to the polar co-ordinates ξ, η, ζ, as it is to refer
their plane projections to the two axes to which our investigation has
been confined. And that it would be advantageous to do so goes without
saying; for it is the shape of the solid object, not that of the mere
drawing of the object, that we want to understand; and already we have
found some of our easy problems in solid geometry leading us (as in
the case of the form of the bivalve and even of the univalve shell)
quickly in the direction of co-ordinate analysis and the theory of
conformal transformations. But this extended theme I have not attempted
to pursue, and it must be left to other times, and to other hands.
Nevertheless, let us glance for a moment at the sort of simple cases,
the simplest possible cases, with which such an investigation might
begin; and we have found our plane co-ordinate systems so easily and
effectively applicable to certain fishes that we may seek among them
for our first and tentative introduction to the three-dimensional field.
It is obvious enough that the same method of description and analysis
which we have applied to one plane, we may apply to another: drawing
by observation, and by a process of trial and error, our various
cross-sections and the co-ordinate systems which seem best to
correspond. But the new and important problem which now emerges is to
_correlate_ the deformation or transformation which we discover in
one plane with that which we have observed in another: and at length,
perhaps, after grasping the general principles of such correlation, to
forecast approximately what is likely to take place in the other two
planes of reference when we are acquainted with one, that is to say, to
determine the values along one axis in terms of the other two.
Let us imagine a common “round” fish, and a common “flat” fish, such
as a haddock and a plaice. These two fishes are not as nicely adapted
for comparison by means of plane co-ordinates as {775} some which
we have studied, owing to the presence of essentially unimportant,
but yet conspicuous differences in the position of the eyes, or in
the number of the fins,—that is to say in the manner in which the
continuous dorsal fin of the plaice appears in the haddock to be cut
or scolloped into a number of separate fins. But speaking broadly, and
apart from such minor differences as these, it is manifest that the
chief factor in the case (so far as we at present see) is simply the
broadening out of the plaice’s body, as compared with the haddock’s,
in the dorso-ventral direction, that is to say, along the _y_ axis;
in other words, the ratio _x_/_y_ is much less, (and indeed little
more than half as great), in the haddock than in the plaice. But we
also recognise at once that while the plaice (as compared with the
haddock) is expanded in one direction, it is also flattened, or thinned
out, in the other: _y_ increases, but _z_ diminishes, relatively to
_x_. And furthermore, we soon see that this is a common or even a
general phenomenon. The high, expanded body in our Antigonia or in
our sun-fish is at the same time flattened or _compressed_ from side
to side, in comparison with the related fishes which we have chosen
as standards of reference or comparison; and conversely, such a fish
as the skate, while it is expanded from side to side in comparison
with a shark or dogfish, is at the same time flattened or _depressed_
in its vertical section. We proceed then, to enquire whether there
be any simple relation of _magnitude_ discernible between these twin
factors of expansion and compression; and the very fact that the two
dimensions tend to vary _inversely_ already assures us that, in the
general process of deformation, the _volume_ is less affected than
are the _linear dimensions_. Some years ago, when I was studying the
length-weight co-efficient in fishes (of which we have already spoken
in Chap. III, p. 98), that is to say the coefficient _k_ in the formula
_W_ = _k_ _L_^3, or _k_ = _W_/_L_^3, I was not a little surprised to
find that _k_ was all but identical in two such different looking
fishes as our haddock and our plaice: thus indicating that these two
fishes, little as they resemble one another externally (though they
belong to two closely related families), have approximately the same
_volume_ when they are equal in _length_; or, in other words, that the
extent to which the plaice’s body has become expanded or broadened is
_just about {776} compensated for_ by the extent to which it has also
got flattened or thinned. In short, if we could permit ourselves to
conceive of a haddock being directly transformed into a plaice, a very
large part of the change would be simply accounted for by supposing the
former fish to be “rolled out,” as a baker rolls a piece of dough. This
is, as it were, an extreme case of the _balancement des organes_, or
“compensation of parts.”
Simple Cartesian co-ordinates will not suffice very well to compare
the haddock with the plaice, for the deformation undergone by the
former in comparison with the latter is more on the lines of that
by which we have compared our Antigonia with our Polyprion; that is
to say, the expansion is greater towards the middle of the fish’s
length, and dwindles away towards either end. But again simplifying our
illustration to the utmost, and being content with a rough comparison,
we may assert that, when haddock and plaice are brought to the same
standard of length, we can inscribe them both (approximately) in
rectangular co-ordinate networks, such that _Y_ in the plaice is about
twice as great as _y_ in the haddock. But if the volumes of the two
fishes be equal, this is as much as to say that _xyz_ in the one case
(or rather the summation of all these values) is equal to _XYZ_ in the
other; and therefore (since _X_ = _x_, and _Y_ = 2_y_), it follows
that _Z_ = _z_/2. When we have drawn our vertical transverse section
of the haddock (or projected that fish in the _yz_ plane), we have
reason accordingly to anticipate that we can draw a similar projection
(or section) of the plaice by simply doubling the _y_’s and halving
the _z_’s: and, very approximately, this turns out to be the case. The
plaice is (in round numbers) just about twice as broad and also just
about half as thick as the haddock; and therefore the ratio of breadth
to thickness (or _y_ to _z_) is just about four times as great in the
one case as in the other.
It is true that this simple, or simplified, illustration carries us
but a very little way, and only half prepares us for much greater
complications. For instance, we have no right or reason to presume that
the equality of weights, or volumes, is a common, much less a general
rule. And again, in all cases of more complex deformation, such as
that by which we have compared Diodon with the sunfish, we must be
prepared for very much more {777} recondite methods of comparison and
analysis, leading doubtless to very much more complicated results.
In this last case, of Diodon and the sunfish, we have seen that the
vertical _expansion_ of the latter as compared with the former fish,
increases rapidly as we go backwards towards the tail; but we can by no
means say that the lateral _compression_ increases in like proportion.
If anything, it would seem that the said expansion and compression tend
to vary inversely; for the Diodon is very thick in front and greatly
thinned away behind, while the flattened sunfish is more nearly of the
same thickness all the way along. Interesting as the whole subject is
we must meanwhile leave it alone; recognising, however, that if the
difficulties of description and representation could be overcome, it is
by means of such co-ordinates in space that we should at last obtain an
adequate and satisfying picture of the processes of deformation and of
the directions of growth[663].
{778}
EPILOGUE.
In the beginning of this book I said that its scope and treatment were
of so prefatory a kind that of other preface it had no need; and now,
for the same reason, with no formal and elaborate conclusion do I bring
it to a close. The fact that I set little store by certain postulates
(often deemed to be fundamental) of our present-day biology the reader
will have discovered and I have not endeavoured to conceal. But it is
not for the sake of polemical argument that I have written, and the
doctrines which I do not subscribe to I have only spoken of by the
way. My task is finished if I have been able to shew that a certain
mathematical aspect of morphology, to which as yet the morphologist
gives little heed, is interwoven with his problems, complementary to
his descriptive task, and helpful, nay essential, to his proper study
and comprehension of Form. _Hic artem remumque repono._
And while I have sought to shew the naturalist how a few mathematical
concepts and dynamical principles may help and guide him, I have
tried to shew the mathematician a field for his labour,—a field which
few have entered and no man has explored. Here may be found homely
problems, such as often tax the highest skill of the mathematician, and
reward his ingenuity all the more for their trivial associations and
outward semblance of simplicity.
That I am no skilled mathematician I have had little need to confess,
but something of the use and beauty of mathematics I think I am able to
understand. I know that in the study of material things, number, order
and position are the threefold clue to exact knowledge; that these
three, in the mathematician’s hands, furnish the “first outlines for
a sketch of the Universe”; that by square and circle we are helped,
like Emile Verhaeren’s carpenter, to conceive “Les lois indubitables et
fécondes Qui sont la règle et la clarté du monde.”
For the harmony of the world is made manifest in Form and Number, and
the heart and soul and all the poetry of Natural {779} Philosophy
are embodied in the concept of mathematical beauty. A greater than
Verhaeren had this in mind when he told of “the golden compasses,
prepared In God’s eternal store.” A greater than Milton had magnified
the theme and glorified Him “who sitteth upon the circle of the earth,”
saying: He measureth the waters in the hollow of his hand, he meteth
out the heavens with his span, he comprehendeth the dust of the earth
in a measure.
Moreover the perfection of mathematical beauty is such (as Maclaurin
learned of the bee), that whatsoever is most beautiful and regular is
also found to be most useful and excellent.
The living and the dead, things animate and inanimate, we dwellers
in the world and this world wherein we dwell,—πάντα γα μὰν τὰ
γιγνωσκόμενα,—are bound alike by physical and mathematical law.
“Conterminous with space and coeval with time is the kingdom of
Mathematics; within this range her dominion is supreme; otherwise than
according to her order nothing can exist, and nothing takes place in
contradiction to her laws.” So said, some forty years ago, a certain
mathematician; and Philolaus the Pythagorean had said much the same.
But with no less love and insight has the science of Form and Number
been appraised in our own day and generation by a very great Naturalist
indeed:—by that old man eloquent, that wise student and pupil of
the ant and the bee, who died but yesterday, and who in his all but
saecular life tasted of the firstfruits of immortality; who curiously
conjoined the wisdom of antiquity with the learning of to-day; whose
Provençal verse seems set to Dorian music; in whose plainest words is a
sound as of bees’ industrious murmur; and who, being of the same blood
and marrow with Plato and Pythagoras, saw in Number “la clef de la
voûte,” and found in it “le comment et le pourquoi des choses.”
NOTES:
[1] These sayings of Kant and of Du Bois, and others like to them,
have been the text of many discourses: see, for instance, Stallo’s
_Concepts_, p. 21, 1882; Höber, _Biol. Centralbl._ XIX, p. 284, 1890,
etc. Cf. also Jellett, _Rep. Brit. Ass._ 1874, p. 1.
[2] “Quum enim mundi universi fabrica sit perfectissima, atque a
Creatore sapientissimo absoluta, nihil omnino in mundo contingit in quo
non maximi minimive ratio quaepiam eluceat; quamobrem dubium prorsus
est nullum quin omnes mundi effectus ex causis finalibus, ope methodi
maximorum et minimorum, aeque feliciter determinari queant atque ex
ipsis causis efficientibus.” _Methodus inveniendi_, etc. 1744 (_cit._
Mach, _Science of Mechanics_, 1902, p. 455).
[3] Cf. Opp. (ed. Erdmann), p. 106, “Bien loin d’exclure les causes
finales..., c’est de là qu’il faut tout déduire en Physique.”
[4] Cf. p. 162. “La force vitale dirige des phénomènes qu’elle ne
produit pas: les agents physiques produisent des phénomènes qu’ils ne
dirigent pas.”
[5] It is now and then conceded with reluctance. Thus Enriques, a
learned and philosophic naturalist, writing “della economia di sostanza
nelle osse cave” (_Arch. f. Entw. Mech._ XX, 1906), says “una certa
impronta di teleologismo quà e là è rimasta, mio malgrado, in questo
scritto.”
[6] Cf. Cleland, On Terminal Forms of Life, _J. Anat. and Phys._ XVIII,
1884.
[7] Conklin, Embryology of Crepidula, _Journ. of Morphol._ XIII, p.
203, 1897; Lillie, F. R., Adaptation in Cleavage, _Woods Holl Biol.
Lectures_, pp. 43–67, 1899.
[8] I am inclined to trace back Driesch’s teaching of Entelechy to no
less a person than Melanchthon. When Bacon (_de Augm._ IV, 3) states
with disapproval that the soul “has been regarded rather as a function
than as a substance,” R. L. Ellis points out that he is referring to
Melanchthon’s exposition of the Aristotelian doctrine. For Melanchthon,
whose view of the peripatetic philosophy had long great influence in
the Protestant Universities, affirmed that, according to the true view
of Aristotle’s opinion, the soul is not a substance, but an ἑντελέχεια,
or _function_. He defined it as δύναμις _quaedam ciens actiones_—a
description all but identical with that of Claude Bernard’s “_force
vitale_.”
[9] Ray Lankester, _Encycl. Brit._ (9th ed.), art. “Zoology,” p. 806,
1888.
[10] Alfred Russel Wallace, especially in his later years, relied
upon a direct but somewhat crude teleology. Cf. his _World of Life, a
Manifestation of Creative Power, Directive Mind and Ultimate Purpose_,
1910.
[11] Janet, _Les Causes Finales_, 1876, p. 350.
[12] The phrase is Leibniz’s, in his _Théodicée_.
[13] Cf. (_int. al._) Bosanquet, The Meaning of Teleology, _Proc.
Brit. Acad._ 1905–6, pp. 235–245. Cf. also Leibniz (_Discours de
Métaphysique; Lettres inédites, ed._ de Careil, 1857, p. 354; _cit._
Janet, p. 643), “L’un et l’autre est bon, l’un et l’autre peut être
utile ... et les auteurs qui suivent ces routes différentes ne
devraient point se maltraiter: _et seq._”
[14] The reader will understand that I speak, not of the “severe and
diligent inquiry” of variation or of “fortuity,” but merely of the
easy assumption that these phenomena are a sufficient basis on which
to rest, with the all-powerful help of natural selection, a theory of
definite and progressive evolution.
[15] _Revue Philosophique._ XXXIII, 1892.
[16] This general principle was clearly grasped by Dr George Rainey (a
learned physician of St Bartholomew’s) many years ago, and expressed in
such words as the following: “......it is illogical to suppose that in
the case of vital organisms a distinct force exists to produce results
perfectly within the reach of physical agencies, especially as in many
instances no end could be attained were that the case, but that of
opposing one force by another capable of effecting exactly the same
purpose.” (On Artificial Calculi, _Q.J.M.S._ (_Trans. Microsc. Soc._),
VI, p. 49, 1858.) Cf. also Helmholtz, _infra cit._, p. 9.
[17] Whereby he incurred the reproach of Socrates, in the _Phaedo_.
[18] In a famous lecture (Conservation of Forces applied to Organic
Nature, _Proc. Roy. Instit._, April 12, 1861), Helmholtz laid it down,
as “the fundamental principle of physiology,” that “There may be other
agents acting in the living body than those agents which act in the
inorganic world; but those forces, as far as they cause chemical and
mechanical influence in the body, must be _quite of the same character_
as inorganic forces: in this at least, that their effects must be
ruled by necessity, and must always be the same when acting in the
same conditions; and so there cannot exist any arbitrary choice in the
direction of their actions.” It would follow from this, that, like the
other “physical” forces, they must be subject to mathematical analysis
and deduction. Cf. also Dr T. Young’s Croonian Lecture On the Heart and
Arteries, _Phil. Trans._ 1809, p. 1; _Coll. Works_, I, 511.
[19] _Ektropismus, oder die physikalische Theorie des Lebens_, Leipzig,
1910.
[20] Wilde Lecture, _Nature_, March 12, 1908; _ibid._ Sept. 6, 1900, p.
485; _Aether and Matter_, p. 288. Cf. also Lord Kelvin, _Fortnightly
Review_, 1892, p. 313.
[21] Joly, The Abundance of Life, _Proc. Roy. Dublin Soc._ VII, 1890;
and in _Scientific Essays_, etc. 1915, p. 60 _et seq._
[22] Papillon, _Histoire de la philosophie moderne_, I, p. 300.
[23] With the special and important properties of _colloidal_ matter we
are, for the time being, not concerned.
[24] Cf. Hans Przibram, _Anwendung elementarer Mathematik auf
Biologische Probleme_ (in Roux’s _Vorträge_, Heft III), Leipzig, 1908,
p. 10.
[25] The subject is treated from an engineering point of view by Prof.
James Thomson, Comparisons of Similar Structures as to Elasticity,
Strength, and Stability, _Trans. Inst. Engineers, Scotland_, 1876
(_Collected Papers_, 1912, pp. 361–372), and by Prof. A. Barr, _ibid._
1899; see also Rayleigh, _Nature_, April 22, 1915.
[26] Cf. Spencer, The Form of the Earth, etc., _Phil. Mag._ XXX, pp.
194–6, 1847; also _Principles of Biology_, pt. II, ch. I, 1864 (p. 123,
etc.).
[27] George Louis Lesage (1724–1803), well known as the author of one
of the few attempts to explain gravitation. (Cf. Leray, _Constitution
de la Matière_, 1869; Kelvin, _Proc. R. S. E._ VII, p. 577, 1872, etc.;
Clerk Maxwell, _Phil. Trans._ vol. 157, p. 50, 1867; art. “Atom,”
_Encycl. Brit._ 1875, p. 46.)
[28] Cf. Pierre Prévost, _Notices de la vie et des écrits de Lesage_,
1805; quoted by Janet, _Causes Finales_, app. III.
[29] Discorsi e Dimostrazioni matematiche, intorno à due nuove
scienze, attenenti alla Mecanica, ed ai Movimenti Locali: appresso gli
Elzevirii, MDCXXXVIII. _Opere_, ed. Favaro, VIII, p. 169 seq. Transl.
by Henry Crew and A. de Salvio, 1914, p. 130, etc. See _Nature_, June
17, 1915.
[30] So Werner remarked that Michael Angelo and Bramanti could not have
built of gypsum at Paris on the scale they built of travertin in Rome.
[31] Sir G. Greenhill, Determination of the greatest height to which
a Tree of given proportions can grow, _Cambr. Phil. Soc. Pr._ IV, p.
65, 1881, and Chree, _ibid._ VII, 1892. Cf. Poynting and Thomson’s
_Properties of Matter_, 1907, p 99.
[32] In like manner the wheat-straw bends over under the weight of
the loaded ear, and the tip of the cat’s tail bends over when held
upright,—not because they “possess flexibility,” but because they
outstrip the dimensions within which stable equilibrium is possible in
a vertical position. The kitten’s tail, on the other hand, stands up
spiky and straight.
[33] _Modern Painters._
[34] The stem of the giant bamboo may attain a height of 60 metres,
while not more than about 40 cm. in diameter near its base, which
dimensions are not very far short of the theoretical limits (A. J.
Ewart, _Phil. Trans._ vol. 198, p. 71, 1906).
[35] _Trans. Zool. Soc._ IV, 1850, p. 27.
[36] It would seem to be a common if not a general rule that marine
organisms, zoophytes, molluscs, etc., tend to be larger than the
corresponding and closely related forms living in fresh water. While
the phenomenon may have various causes, it has been attributed
(among others) to the simple fact that the forces of growth are less
antagonised by gravity in the denser medium (cf. Houssay, _La Forme
et la Vie_, 1900, p. 815). The effect of gravity on outward _form_ is
illustrated, for instance, by the contrast between the uniformly upward
branching of a sea-weed and the drooping curves of a shrub or tree.
[37] The analogy is not a very strict one. We are not taking account,
for instance, of a proportionate increase in thickness of the
boiler-plates.
[38] Let _L_ be the length, _S_ the (wetted) surface, _T_ the tonnage,
_D_ the displacement (or volume) of a ship; and let it cross the
Atlantic at a speed _V_. Then, in comparing two ships, similarly
constructed but of different magnitudes, we know that _L_ = _V_^2,
_S_ = _L_^2 = _V_^4, _D_ = _T_ = _L_^3 = _V_^6; also _R_ (resistance)
= _S_ ⋅ _V_^2 = _V_^6; _H_ (horse-power) = _R_ ⋅ _V_ = _V_^7; and the
coal (_C_) necessary for the voyage = _H_/_V_ = _V_^6. That is to
say, in ordinary engineering language, to increase the speed across
the Atlantic by 1 per cent. the ship’s length must be increased 2 per
cent., her tonnage or displacement 6 per cent., her coal-consumpt also
6 per cent., her horse-power, and therefore her boiler-capacity, 7 per
cent. Her bunkers, accordingly, keep pace with the enlargement of the
ship, but her boilers tend to increase out of proportion to the space
available.
[39] This is the result arrived at by Helmholtz, Ueber ein Theorem
geometrisch ähnliche Bewegungen flüssiger Körper betreffend, nebst
Anwendung auf das Problem Luftballons zu lenken, _Monatsber. Akad.
Berlin_, 1873, pp. 501–14. It was criticised and challenged (somewhat
rashly) by K. Müllenhof, Die Grösse der Flugflächen, etc., _Pflüger’s
Archiv_, XXXV, p. 407, XXXVI, p. 548, 1885.
[40] Cf. also Chabrier, Vol des Insectes, _Mém. Mus. Hist. Nat. Paris_,
VI–VIII, 1820–22.
[41] _Aerial Flight_, vol. II (_Aerodonetics_), 1908, p. 150.
[42] By Lanchester, _op. cit._ p. 131.
[43] Cf. _L’empire de l’air; ornithologie appliquée à l’aviation_. 1881.
[44] _De Motu Animalium_, I, prop. cciv, ed. 1685, p. 243.
[45] Harlé, On Atmospheric Pressure in past Geological Ages, _Bull.
Geol. Soc. Fr._ XI, pp. 118–121; or _Cosmos_, p. 30, July 8, 1911.
[46] _Introduction to Entomology_, 1826, II, p. 190. K. and S., like
many less learned authors, are fond of popular illustrations of the
“wonders of Nature,” to the neglect of dynamical principles. They
suggest, for instance, that if the white ant were as big as a man, its
tunnels would be “magnificent cylinders of more than three hundred
feet in diameter”; and that if a certain noisy Brazilian insect were
as big as a man, its voice would be heard all the world over: “so that
Stentor becomes a mute when compared with these insects!” It is an easy
consequence of anthropomorphism, and hence a common characteristic of
fairy-tales, to neglect the principle of dynamical, while dwelling on
the aspect of geometrical, similarity.
[47] I.e. the available energy of muscle, in ft.-lbs. per lb. of
muscle, is the same for all animals: a postulate which requires
considerable qualification when we are comparing very different _kinds_
of muscle, such as the insect’s and the mammal’s.
[48] Prop. clxxvii. Animalia minora et minus ponderosa majores saltus
efficiunt respectu sui corporis, si caetera fuerint paria.
[49] See also (_int. al._), John Bernoulli, _de Motu Musculorum_,
Basil., 1694; Chabry, Mécanisme du Saut, _J. de l’Anat. et de
la Physiol._ XIX, 1883; Sur la longueur des membres des animaux
sauteurs, _ibid._ XXI, p. 356, 1885; Le Hello, De l’action des organes
locomoteurs, etc., _ibid._ XXIX, p. 65–93, 1893, etc.
[50] Recherches sur la force absolue des muscles des Invertébrés,
_Bull. Acad. E. de Belgique_ (3), VI, VII, 1883–84; see also _ibid._
(2), XX, 1865, XXII, 1866; _Ann. Mag. N. H._ XVII, p. 139, 1866, XIX,
p. 95, 1867. The subject was also well treated by Straus-Dürckheim,
in his _Considérations générales sur l’anatomie comparée des animaux
articulés_, 1828.
[51] The fact that the limb tends to swing in pendulum-time was first
observed by the brothers Weber (_Mechanik der menschl. Gehwerkzeuge_,
Göttingen, 1836). Some later writers have criticised the statement
(e.g. Fischer, Die Kinematik des Beinschwingens etc., _Abh. math. phys.
Kl. k. Sächs. Ges._ XXV–XXVIII, 1899–1903), but for all that, with
proper qualifications, it remains substantially true.
[52] Quoted in Mr John Bishop’s interesting article in Todd’s
_Cyclopaedia_, III, p. 443.
[53] There is probably also another factor involved here: for in
bending, and therefore shortening, the leg we bring its centre of
gravity nearer to the pivot, that is to say, to the joint, and so the
muscle tends to move it the more quickly.
[54] _Proc. Psychical Soc._ XII, pp. 338–355, 1897.
[55] For various calculations of the increase of surface due to
histological and anatomical subdivision, see E. Babak, Ueber die
Oberflächenentwickelung bei Organismen, _Biol. Centralbl._ XXX, pp.
225–239, 257–267, 1910. In connection with the physical theory of
surface-energy, Wolfgang Ostwald has introduced the conception of
_specific surface_, that is to say the ratio of surface to volume, or
_S_/_V_. In a cube, _V_ = _l_^3, and _S_ = 6_l_^2; therefore _S_/_V_
= 6/_l_. Therefore if the side _l_ measure 6 cm., the ratio _S_/_V_
= 1, and such a cube may be taken as our standard, or unit of specific
surface. A human blood-corpuscle has, accordingly, a specific surface
of somewhere about 14,000 or 15,000. It is found in physical chemistry
that surface energy becomes an important factor when the specific
surface reaches a value of 10,000 or thereby.
[56] Though the entire egg is not increasing in mass, this is not to
say that its living protoplasm is not increasing all the while at the
expense of the reserve material.
[57] Cf. Tait, _Proc. R.S.E._ V, 1866, and VI, 1868.
[58] _Physiolog. Notizen_ (9), p. 425, 1895. Cf. Strasbürger, Ueber die
Wirkungssphäre der Kerne und die Zellgrösse, _Histolog. Beitr._ (5),
pp. 95–129, 1893; J. J. Gerassimow, Ueber die Grösse des Zellkernes,
_Beih. Bot. Centralbl._ XVIII, 1905; also G. Levi and T. Terni, Le
variazioni dell’ indice plasmatico-nucleare durante l’intercinesi,
_Arch. Ital. di Anat._ X, p. 545, 1911.
[59] _Arch. f. Entw. Mech._ IV, 1898, pp. 75, 247.
[60] Conklin, E. G., Cell-size and nuclear-size, _J. Exp. Zool._ XII.
pp. 1–98, 1912.
[61] Thus the fibres of the crystalline lens are of the same size in
large and small dogs; Rabl, _Z. f. w. Z._ LXVII, 1899. Cf. (_int. al._)
Pearson, On the Size of the Blood-corpuscles in Rana, _Biometrika_,
VI, p. 403, 1909. Dr Thomas Young caught sight of the phenomenon,
early in last century: “The solid particles of the blood do not by
any means vary in magnitude in the same ratio with the bulk of the
animal,” _Natural Philosophy_, ed. 1845, p. 466; and Leeuwenhoek and
Stephen Hales were aware of it a hundred years before. But in this
case, though the blood-corpuscles show no relation of magnitude to the
size of the animal, they do seem to have some relation to its activity.
At least the corpuscles in the sluggish Amphibia are much the largest
known to us, while the smallest are found among the deer and other
agile and speedy mammals. (Cf. Gulliver, _P.Z.S._ 1875, p. 474, etc.)
This apparent correlation may have its bearing on modern views of the
surface-condensation or adsorption of oxygen in the blood-corpuscles,
a process which would be greatly facilitated and intensified by the
increase of surface due to their minuteness.
[62] Cf. P. Enriques, La forma come funzione della grandezza: Ricerche
sui gangli nervosi degli Invertebrati, _Arch. f. Entw. Mech._ XXV, p.
655, 1907–8.
[63] While the difference in cell-volume is vastly less than that
between the volumes, and very much less also than that between
the surfaces, of the respective animals, yet there _is_ a certain
difference; and this it has been attempted to correlate with the
need for each cell in the many-celled ganglion of the larger
animal to possess a more complex “exchange-system” of branches,
for intercommunication with its more numerous neighbours. Another
explanation is based on the fact that, while such cells as continue to
divide throughout life tend to uniformity of size in all mammals, those
which do not do so, and in particular the ganglion cells, continue to
grow, and their size becomes, therefore, a function of the duration of
life. Cf. G. Levi, Studii sulla grandezza delle cellule, _Arch. Ital.
di Anat. e di Embryolog._ V, p. 291, 1906.
[64] Boveri. _Zellen-studien, V. Ueber die Abhängigkeit der Kerngrösse
und Zellenzahl der Seeigellarven von der Chromosomenzahl der
Ausgangszellen._ Jena, 1905.
[65] Recent important researches suggest that such ultra-minute
“filter-passers” are the true cause of certain acute maladies commonly
ascribed to the presence of much larger organisms; cf. Hort, Lakin and
Benians, The true infective Agent in Cerebrospinal Fever, etc., _J.
Roy. Army Med. Corps_, Feb. 1910.
[66] _Zur Erkenntniss der Kolloide_, 1905, p. 122; where there will be
found an interesting discussion of various molecular and other minute
magnitudes.
[67] _Encyclopaedia Britannica_, 9th edit., vol. III, p. 42, 1875.
[68] Sur la limite de petitesse des organismes, _Bull. Soc. R. des
Sc. méd. et nat. de Bruxelles_, Jan. 1903; _Rec. d’œuvres_ (_Physiol.
générale_), p. 325.
[69] Cf. A. Fischer, _Vorlesungen über Bakterien_, 1897, p. 50.
[70] F. Hofmeister, quoted in Cohnheim’s _Chemie der Eiweisskörper_,
1900, p. 18.
[71] McKendrick arrived at a still lower estimate, of about 1250
proteid molecules in the minutest organisms. _Brit. Ass. Rep._ 1901, p.
808.
[72] Cf. Perrin, _Les Atomes_, 1914, p. 74.
[73] Cf. Tait, On Compression of Air in small Bubbles, _Proc. R. S. E._
V, 1865.
[74] _Phil. Mag._ XLVIII, 1899; _Collected Papers_, IV, p. 430.
[75] Carpenter, _The Microscope_, edit. 1862, p. 185.
[76] The modern literature on the Brownian Movement is very large,
owing to the value which the phenomenon is shewn to have in determining
the size of the atom. For a fuller, but still elementary account, see
J. Cox, _Beyond the Atom_, 1913, pp. 118–128; and see, further, Perrin,
_Les Atomes_, pp. 119–189.
[77] Cf. R. Gans, Wie fallen Stäbe und Scheiben in einer reibenden
Flüssigkeit? _Münchener Bericht_, 1911, p. 191; K. Przibram, Ueber die
Brown’sche Bewegung nicht kugelförmiger Teilchen, _Wiener Ber._ 1912,
p. 2339.
[78] Ueber die ungeordnete Bewegung niederer Thiere, _Pflüger’s
Archiv_, CLIII, p. 401, 1913.
[79] Sometimes we find one and the same diagram suffice, whether the
intervals of time be great or small; and we then invoke “Wolff’s
Law,” and assert that the life-history of the individual repeats, or
recapitulates, the history of the race.
[80] Our subject is one of Bacon’s “Instances of the Course,” or
studies wherein we “measure Nature by periods of Time.” In Bacon’s
_Catalogue of Particular Histories_, one of the odd hundred histories
or investigations which he foreshadowed is precisely that which we are
engaged on, viz. a “History of the Growth and Increase of the Body, in
the whole and in its parts.”
[81] Cf. Aristotle, _Phys._ vi, 5, 235 _a_ 11, ὲπεὶ γὰρ ἅπασα κίνησις
ἐν χρόνῳ, κτλ. Bacon emphasised, in like manner, the fact that “all
motion or natural action is performed in time: some more quickly, some
more slowly, but all in periods determined and fixed in the nature of
things. Even those actions which seem to be performed suddenly, and (as
we say) in the twinkling of an eye, are found to admit of degree in
respect of duration.” _Nov. Org._ XLVI.
[82] Cf. (e.g.) _Elem. Physiol._ ed. 1766, VIII, p. 114, “Ducimur
autem ad evolutionem potissimum, quando a perfecto animale retrorsum
progredimur, et incrementorum atque mutationum seriem relegimus. Ita
inveniemus perfectum illud animal fuisse imperfectius, alterius figurae
et fabricae, et denique rude et informe: et tamen idem semper animal
sub iis diversis phasibus fuisse, quae absque ullo saltu perpetuos
parvosque per gradus cohaereant.”
[83] _Beiträge zur Entwickelungsgeschichte des Hühnchens im Ei_, p. 40,
1817. Roux ascribes the same views also to Von Baer and to R. H. Lotze
(_Allg. Physiologie_, p. 353, 1851).
[84] Roux, _Die Entwickelungsmechanik_, p. 99, 1905.
[85] _Op. cit._ p. 302, “Magnum hoc naturae instrumentum, etiam in
corpore animato evolvendo potenter operatur; etc.”
[86] _Ibid._ p. 306. “Subtiliora ista, et aliquantum hypothesi mista,
tamen magnum mihi videntur speciem veri habere.”
[87] Cf. His, On the Principles of Animal Morphology, _Proc. R. S.
E._ XV, 1888, p. 294: “My own attempts to introduce some elementary
mechanical or physiological conceptions into embryology have not
generally been agreed to by morphologists. To one it seemed ridiculous
to speak of the elasticity of the germinal layers; another thought
that, by such considerations, we ‘put the cart before the horse’:
and one more recent author states, that we have better things to
do in embryology than to discuss tensions of germinal layers and
similar questions, since all explanations must of necessity be of
a phylogenetic nature. This opposition to the application of the
fundamental principles of science to embryological questions would
scarcely be intelligible had it not a dogmatic background. No other
explanation of living forms is allowed than heredity, and any which
is founded on another basis must be rejected ....... To think that
heredity will build organic beings without mechanical means is a piece
of unscientific mysticism.”
[88] Hertwig, O., _Zeit und Streitfragen der Biologie_, II. 1897.
[89] Cf. Roux, _Gesammelte Abhandlungen_, II, p. 31, 1895.
[90] _Treatise on Comparative Embryology_, I, p. 4, 1881.
[91] Cf. Fick, _Anal. Anzeiger_, XXV, p. 190, 1904.
[92] 1st ed. p. 444; 6th ed. p. 390. The student should not fail
to consult the passage in question; for there is always a risk of
misunderstanding or misinterpretation when one attempts to epitomise
Darwin’s carefully condensed arguments.
[93] “In omni rerum naturalium historia utile est _mensuras definiri
et numeros_,” Haller, _Elem. Physiol._ II, p. 258, 1760. Cf. Hales,
_Vegetable Staticks_, Introduction.
[94] Brussels, 1871. Cf. the same author’s _Physique sociale_, 1835,
and _Lettres sur la théorie des probabilités_, 1846. See also, for the
general subject, Boyd, R., Tables of weights of the Human Body, etc.
_Phil. Trans._ vol. CLI, 1861; Roberts, C., _Manual of Anthropometry_,
1878; Daffner, F., _Das Wachsthum des Menschen_ (2nd ed.), 1902, etc.
[95] Dr Johnson was not far wrong in saying that “life declines from
thirty-five”; though the Autocrat of the Breakfast-table, like Cicero,
declares that “the furnace is in full blast for ten years longer.”
[96] Joly, _The Abundance of Life_, 1915 (1890), p. 86.
[97] “_Lou pes, mèstre de tout_ [Le poids, maître de tout], _mèstre
sènso vergougno, Que te tirasso en bas de sa brutalo pougno_,” J. H.
Fabre, _Oubreto prouvençalo_, p. 61.
[98] The continuity of the phenomenon of growth, and the natural
passage from the phase of increase to that of decrease or decay, are
admirably discussed by Enriques, in “La morte,” _Riv. di Scienza_,
1907, and in “Wachsthum und seine analytische Darstellung,” _Biol.
Centralbl._ June, 1909. Haller (_Elem_. VII, p. 68) recognised
_decrementum_ as a phase of growth, not less important (theoretically)
than _incrementum_: “_tristis, sed copiosa, haec est materies_.”
[99] Cf. (_int. al._), Friedenthal, H., Das Wachstum des
Körpergewichtes ... in verschiedenen Lebensältern, _Zeit. f. allg.
Physiol._ IX, pp. 487–514, 1909.
[100] As Haller observed it to do in the chick (_Elem._ VIII, p. 294):
“Hoc iterum incrementum miro ordine ita distribuitur, ut in principio
incubationis maximum est: inde perpetuo minuatur.”
[101] There is a famous passage in Lucretius (v. 883) where he compares
the course of life, or rate of growth, in the horse and his boyish
master: _Principio circum tribus actis impiger annis Floret equus, puer
hautquaquam_, etc.
[102] Minot, C. S., Senescence and Rejuvenation, _Journ. of Physiol._
XII, pp. 97–153, 1891; The Problem of Age, Growth and Death, _Pop.
Science Monthly_ (June–Dec.), 1907.
[103] Quoted in Vierordt’s _Anatomische ... Daten und Tabellen_, 1906.
p. 13.
[104] _Unsere Körperform_, Leipzig, 1874.
[105] No such point of inflection appears in the curve of weight
according to C. M. Jackson’s data (On the Prenatal Growth of the Human
Body, etc., _Amer. Journ. of Anat._ IX, 1009, pp. 126, 156), nor in
those quoted by him from Ahlfeld, Fehling and others. But it is plain
that the very rapid increase of the monthly weights, approximately in
the ratio of the cubes of the corresponding lengths, would tend to
conceal any such breach of continuity, unless it happened to be very
marked indeed. Moreover in the case of Jackson’s data (and probably
also in the others) the actual age of the embryos was not determined,
but was estimated from their lengths. The following is Jackson’s
estimate of average weights at intervals of a lunar month:
Months 0 1 2 3 4 5 6 7 8 9 10
Wt in gms. ·0 ·04 3 36 120 330 600 1000 1500 2200 3200
[106] G. Kraus (after Wallich-Martius), _Ann. du Jardin bot. de
Buitenzorg_, XII, 1, 1894, p. 210. Cf. W. Ostwald, _Zeitliche
Eigenschaften_, etc. p. 56.
[107] Cf. Chodat, R., et Monnier, A., Sur la courbe de croissance des
végétaux, _Bull. Herb. Boissier_ (2), V, pp. 615, 616, 1905.
[108] Cf. Fr. Boas, Growth of Toronto Children, _Rep. of U.S. Comm.
of Education_, 1896–7, pp. 1541–1599, 1898; Boas and Clark Wissler,
Statistics of Growth, _Education Rep._ 1904, pp. 25–132, 1906; H. P.
Bowditch, _Rep. Mass. State Board of Health_, 1877; K. Pearson, On the
Magnitude of certain coefficients of Correlation in Man, _Pr. R. S._
LXVI, 1900.
[109] _l.c._ p. 42, and other papers there quoted.
[110] See, for an admirable résumé of facts, Wolfgang Ostwald, _Ueber
die Zeitliche Eigenschaften der Entwickelungsvorgänge_ (71 pp.),
Leipzig, 1908 (Roux’s _Vorträge_, Heft V): to which work I am much
indebted. A long list of observations on the growth-rate of various
animals is also given by H. Przibram, _Exp. Zoologie_, 1913, pt. IV
(_Vitalität_), pp. 85–87.
[111] Cf. St Loup, Vitesse de croissance chez les Souris, _Bull. Soc.
Zool. Fr._ XVIII, 242, 1893; Robertson, _Arch. f. Entwickelungsmech._
XXV, p. 587, 1908; Donaldson. _Boas Memorial Volume_, New York, 1906.
[112] Luciani e Lo Monaco, _Arch. Ital. de Biologie_, XXVII, p. 340,
1897.
[113] Schaper, _Arch. f. Entwickelungsmech._ XIV, p. 356, 1902. Cf.
Barfurth, Versuche über die Verwandlung der Froschlarven, _Arch. f.
mikr. Anat._ XXIX, 1887.
[114] Joh. Schmidt, Contributions to the Life-history of the Eel,
_Rapports du Conseil Intern. pour l’exploration de la Mer_, vol. V, pp.
137–274, Copenhague, 1906.
[115] That the metamorphoses of an insect are but phases in a process
of growth, was firstly clearly recognised by Swammerdam, _Biblia
Naturae_, 1737, pp. 6, 579 etc.
[116] From Bose, J. C., _Plant Response_, London, 1906, p. 417.
[117] This phenomenon, of _incrementum inequale_, as opposed to
_incrementum in universum_, was most carefully studied by Haller:
“Incrementum inequale multis modis fit, ut aliae partes corporis aliis
celerius increscant. Diximus hepar minus fieri, majorem pulmonem,
minimum thymum, etc.” (_Elem._ VIII (2), p. 34).
[118] See (_inter alia_) Fischel, A., Variabilität und Wachsthum des
embryonalen Körpers, _Morphol. Jahrb._ XXIV, pp. 369–404, 1896. Oppel,
_Vergleichung des Entwickelungsgrades der Organe zu verschiedenen
Entwickelungszeiten bei Wirbelthieren_, Jena, 1891. Faucon, A., _Pesées
et Mensurations fœtales à différents âges de la grossesse_. (Thèse.)
Paris, 1897. Loisel, G., Croissance comparée en poids et en longueur
des fœtus mâle et femelle dans l’espèce humaine, _C. R. Soc. de
Biologie_, Paris, 1903. Jackson, C. M., Pre-natal growth of the human
body and the relative growth of the various organs and parts, _Am. J.
of Anat._ IX, 1909; Post-natal growth and variability of the body and
of the various organs in the albino rat, _ibid._ XV, 1913.
[119] _l.c._ p. 1542.
[120] Variation and Correlation in Brain-weight, _Biometrika_, IV, pp.
13–104, 1905.
[121] _Die Säugethiere_, p. 117.
[122] _Amer. J. of Anatomy_, VIII, pp. 319–353, 1908. Donaldson
(_Journ. Comp. Neur. and Psychol._ XVIII, pp. 345–392, 1908) also
gives a logarithmic formula for brain-weight (_y_) as compared
with body-weight (_x_), which in the case of the white rat is _y_
= ·554 − ·569 log(_x_ − 8·7), and the agreement is very close. But the
formula is admittedly empirical and as Raymond Pearl says (_Amer. Nat._
1909, p. 303), “no ulterior biological significance is to be attached
to it.”
[123] _Biometrika_, IV, pp. 13–104, 1904.
[124] Donaldson, H. H., A Comparison of the White Rat with Man in
respect to the Growth of the entire Body, _Boas Memorial Vol._, New
York, 1906, pp. 5–26.
[125] Besides many papers quoted by Dubois on the growth and weight of
the brain, and numerous papers in _Biometrika_, see also the following:
Ziehen, Th., _Das Gehirn: Massverhältnisse_, in Bardeleben’s _Handb.
der Anat. des Menschen_, IV, pp. 353–386, 1899. Spitzka, E. A.,
Brain-weight of Animals with special reference to the Weight of the
Brain in the Macaque Monkey, _J. Comp. Neurol._ XIII, pp. 9–17, 1903.
Warneke, P., Mitteilung neuer Gehirn und Körpergewichtsbestimmungen
bei Säugern, nebst Zusammenstellung der gesammten bisher beobachteten
absoluten und relativen Gehirngewichte bei den verschiedenen Species,
_J. f. Psychol. u. Neurol._ XIII, pp. 355–403, 1909. Donaldson, H. H.,
On the regular seasonal Changes in the relative Weight of the Central
Nervous System of the Leopard Frog, _Journ. of Morph._ XXII, pp.
663–694, 1911.
[126] Cf. Jenkinson, Growth, Variability and Correlation in Young
Trout, _Biometrika_, VIII, pp. 444–455, 1912.
[127] Cf. chap. xvii, p. 739.
[128] “ ...I marked in the same manner as the Vine, young Honeysuckle
shoots, etc....; and I found in them all a gradual scale of unequal
extensions, those parts extending most which were tenderest,”
_Vegetable Staticks_, Exp. cxxiii.
[129] From Sachs, _Textbook of Botany_, 1882, p. 820.
[130] Variation and Differentiation in Ceratophyllum, _Carnegie Inst.
Publications_, No. 58, Washington, 1907.
[131] Cf. Lämmel, Ueber periodische Variationen in Organismen, _Biol.
Centralbl._ XXII, pp. 368–376, 1903.
[132] Herein lies the easy answer to a contention frequently raised
by Bergson, and to which he ascribes great importance, that “a mere
variation of size is one thing, and a change of form is another.” Thus
he considers “a change in the form of leaves” to constitute “a profound
morphological difference.” _Creative Evolution_, p. 71.
[133] I do not say that the assumption that these two groups of earwigs
were of different ages is altogether an easy one; for of course,
even in an insect whose metamorphosis is so simple as the earwig’s,
consisting only in the acquisition of wings or wing-cases, we usually
take it for granted that growth proceeds no more after the final stage,
or “adult form” is attained, and further that this adult form is
attained at an approximately constant age, and constant magnitude. But
even if we are not permitted to think that the earwig may have grown,
or moulted, after once the elytra were produced, it seems to me far
from impossible, and far from unlikely, that prior to the appearance
of the elytra one more stage of growth, or one more moult took place
in some cases than in others: for the number of moults is known to be
variable in many species of Orthoptera. Unfortunately Bateson tells
us nothing about the sizes or total lengths of his earwigs; but his
figures suggest that it was bigger earwigs that had the longer tails;
and that the rate of growth of the tails had had a certain definite
ratio to that of the bodies, but not necessarily a simple ratio of
equality.
[134] Jackson, C. M., _J. of Exp. Zool._ XIX, 1915, p. 99; cf. also
Hans Aron, Unters. über die Beeinflüssung der Wachstum durch die
Ernährung, _Berl. klin. Wochenbl._ LI, pp. 972–977, 1913, etc.
[135] The temperature limitations of life, and to some extent of
growth, are summarised for a large number of species by Davenport,
_Exper. Morphology_, cc. viii, xviii, and by Hans Przibram, _Exp.
Zoologie_, IV, c. v.
[136] Réaumur: _L’art de faire éclore et élever en toute saison des
oiseaux domestiques, foit par le moyen de la chaleur du fumier_, Paris,
1749.
[137] Cf. (_int. al._) de Vries, H., Matériaux pour la connaissance
de l’influence de la température sur les plantes, _Arch. Néerl._ V,
385–401, 1870. Köppen, Wärme und Pflanzenwachstum, _Bull. Soc. Imp.
Nat. Moscou._ XLIII, pp. 41–110, 1870.
[138] Blackman, F. F., _Ann. of Botany_, XIX, p. 281, 1905.
[139] For various instances of a “temperature coefficient” in
physiological processes, see Kanitz, _Zeitschr. f. Elektrochemie_,
1907, p. 707; _Biol. Centralbl._ XXVII, p. 11, 1907; Hertzog, R.
O., Temperatureinfluss auf die Entwicklungsgeschwindigkeit der
Organismen, _Zeitschr. f. Elektrochemie_, XI, p 820, 1905; Krogh,
Quantitative Relation between Temperature and Standard Metabolism,
_Int. Zeitschr. f. physik.-chem. Biologie_, I, p. 491, 1914; Pütter,
A., Ueber Temperaturkoefficienten, _Zeitschr. f. allgem. Physiol._
XVI, p. 574, 1914. Also Cohen, _Physical Chemistry for Physicians and
Biologists_ (English edition), 1903; Pike, F. H., and Scott. E. L., The
Regulation of the Physico-chemical Condition of the Organism, _American
Naturalist_, Jan. 1915, and various papers quoted therein.
[140] Cf. Errera, L., _L’Optimum_, 1896 (_Rec. d’Oeuvres, Physiol.
générale_, pp. 338–368, 1910); Sachs, _Physiologie d. Pflanzen_, 1882,
p. 233; Pfeffer, _Pflanzenphysiologie_, ii, p. 78, 1904; and cf. Jost,
Ueber die Reactionsgeschwindigkeit im Organismus, _Biol. Centralbl._
XXVI, pp. 225–244, 1906.
[141] After Köppen, _Bull. Soc. Nat. Moscou_, XLIII, pp. 41–110, 1871.
[142] _Botany_, p. 387.
[143] Leitch, I., Some Experiments on the Influence of Temperature on
the Rate of Growth in _Pisum sativum, Ann. of Botany_, XXX, pp. 25–46,
1916. (Cf. especially Table III, p. 45.)
[144] Blackman, F. F., Presidential Address in Botany, _Brit. Ass._
Dublin, 1908.
[145] _Rec. de l’Inst. Bot. de Bruxelles_, VI, 1906.
[146] Hertwig, O., Einfluss der Temperatur auf die Entwicklung von
_Rana fusca_ und _R. esculenta_, _Arch. f. mikrosk. Anat._ LI, p.
319, 1898. Cf. also Bialaszewicz, K., Beiträge z. Kenntniss d.
Wachsthumsvorgänge bei Amphibienembryonen, _Bull. Acad. Sci. de
Cracovie_, p. 783, 1908; Abstr. in _Arch. f. Entwicklungsmech._ XXVIII,
p. 160, 1909.
[147] Der Grad der Beschleunigung tierischer Entwickelung durch
erhöhte Temperatur, _A. f. Entw._ Mech. XX. p. 130, 1905. More
recently, Bialaszewicz has determined the coefficient for the rate of
segmentation in Rana as being 2·4 per 10° C.
[148] _Das Wachstum des Menschen_, p. 329, 1902.
[149] The _diurnal_ periodicity is beautifully shewn in the case of
the Hop by Joh. Schmidt (_C. R. du Laboratoire de Carlsberg_, X, pp.
235–248, Copenhague, 1913).
[150] _Trans. Botan. Soc. Edinburgh_, XVIII, 1891, p. 456.
[151] I had not received, when this was written, Mr Douglass’s paper,
On a method of estimating Rainfall by the Growth of Trees, _Bull.
Amer. Geograph. Soc._ XLVI, pp. 321–335, 1914. Mr Douglass does not
fail to notice the long period here described; but he lays more stress
on the occurrence of shorter cycles (of 11, 21 and 33 years), well
known to meteorologists. Mr Douglass is inclined (and I think rightly)
to correlate the variations in growth directly with fluctuations
in rainfall, that is to say with alternate periods of moisture and
aridity; but he points out that the temperature curves (and also the
sunspot curves) are markedly similar.
[152] It may well be that the effect is not due to light after all; but
to increased absorption of heat by the soil, as a result of the long
hours of exposure to the sun.
[153] On growth in relation to light, see Davenport, _Exp. Morphology_,
II, ch. xvii. In some cases (as in the roots of Peas), exposure to
light seems to have no effect on growth; in other cases, as in diatoms
(according to Whipple’s experiments, quoted by Davenport, II, p.
423), the effect of light on growth or multiplication is well-marked,
measurable, and apparently capable of expression by a logarithmic
formula. The discrepancy would seem to arise from the fact that, while
light-energy always tends to be absorbed by the chlorophyll of the
plant, converted into chemical energy, and stored in the shape of
starch or other reserve materials, the actual rate of growth depends
on the rate at which these reserves are drawn on: and this is another
matter, in which light-energy is no longer directly concerned.
[154] Cf. for instance, Nägeli’s classical account of the effect of
change of habitat on Alpine and other plants: _Sitzungsber. Baier.
Akad. Wiss._ 1865, pp. 228–284.
[155] Cf. Blackman, F. F., Presidential Address in Botany, _Brit. Ass._
Dublin, 1908. The fact was first enunciated by Baudrimont and St Ange,
Recherches sur le développement du fœtus, _Mém. Acad. Sci._ XI, p. 469,
1851.
[156] Cf. Loeb, _Untersuchungen zur physiol. Morphologie der Thiere_,
1892; also Experiments on Cleavage, _J. of Morph._ VII, p. 253, 1892;
Zusammenstellung der Ergebnisse einiger Arbeiten über die Dynamik des
thierischen Wachsthum, _Arch. f. Entw. Mech._ XV, 1902–3, p. 669;
Davenport, On the Rôle of Water in Growth, _Boston Soc. N. H._ 1897;
Ida H. Hyde, _Am. J. of Physiol._ XII, 1905, p. 241, etc.
[157] _Pflüger’s Archiv_, LV, 1893.
[158] Beiträge zur Kenntniss der Wachstumsvorgänge bei
Amphibienembryonen, _Bull. Acad. Sci. de Cracovie_, 1908, p. 783; cf.
_Arch. f. Entw. Mech._ XXVIII, p. 160, 1909; XXXIV, p. 489, 1912.
[159] Fehling, H., _Arch. für Gynaekologie_, XI, 1877; cf. Morgan,
_Experimental Zoology_, p. 240, 1907.
[160] Höber, R., Bedeutung der Theorie der Lösungen für Physiologie und
Medizin, _Biol. Centralbl._ XIX, 1899; cf. pp. 272–274.
[161] Schmankewitsch has made other interesting observations on change
of size and form, after some generations, in relation to change of
density; e.g. in the flagellate infusorian _Anisonema acinus_, Bütschli
(_Z. f. w. Z._ XXIX, p. 429, 1877).
[162] These “Fezzan-worms,” when first described, were supposed to be
“insects’ eggs”; cf. Humboldt, _Personal Narrative_, VI, i, 8, note;
Kirby and Spence, Letter X.
[163] Cf. _Introd. à l’étude de la médecine expérimentale_, 1885, p.
110.
[164] Cf. Abonyi, _Z. f. w. Z._ CXIV, p. 134, 1915. But Frédéricq has
shewn that the amount of NaCl in the blood of Crustacea (_Carcinus
moenas_) varies, and all but corresponds, with the density of the water
in which the creature has been kept (_Arch. de Zool. Exp. et Gén._ (2),
III, p. xxxv, 1885); and other results of Frédéricq’s, and various
data given or quoted by Bottazzi (Osmotischer Druck und elektrische
Leitungsfähigkeit der Flüssigkeiten der Organismen, in Asher-Spiro’s
_Ergebn. d. Physiologie_, VII, pp. 160–402, 1908) suggest that the case
of the brine-shrimps must be looked upon as an extreme or exceptional
one.
[165] Cf. Schmankewitsch, _Z. f. w. Zool._ XXV, 1875, XXIX, 1877, etc.;
transl. in appendix to Packard’s _Monogr. of N. American Phyllopoda_,
1883, pp. 466–514; Daday de Deés, _Ann. Sci. Nat._ (_Zool._), (9),
XI, 1910; Samter und Heymons, _Abh. d. K. pr. Akad. Wiss._ 1902;
Bateson, _Mat. for the Study of Variation_, 1894, pp. 96–101; Anikin,
_Mitth. Kais. Univ. Tomsk_, XIV: _Zool. Centralbl._ VI, pp. 756–760,
1908; Abonyi, _Z. f. w. Z._ CXIV, pp. 96–168, 1915 (with copious
bibliography), etc.
[166] According to the empirical canon of physiology, that (as
Frédéricq expresses it) “L’être vivant est agencé de telle manière que
chaque influence perturbatrice provoque d’elle-même la mise en activité
de l’appareil compensateur qui doit neutraliser et réparer le dommage.”
[167] Such phenomena come precisely under the head of what Bacon called
_Instances of Magic_: “By which I mean those wherein the material or
efficient cause is scanty and small as compared with the work or effect
produced; so that even when they are common, they seem like miracles,
some at first sight, others even after attentive consideration. These
magical effects are brought about in three ways ... [of which one is]
by excitation or invitation in another body, as in the magnet which
excites numberless needles without losing any of its virtue, _or in
yeast and such-like_.” _Nov. Org._, cap. li.
[168] Monnier, A., Les matières minérales, et la loi d’accroissement
des Végétaux, _Publ. de l’Inst. de Bot. de l’Univ. de Genève_
(7), III, 1905. Cf. Robertson, On the Normal Rate of Growth of an
Individual, and its Biochemical Significance, _Arch. f. Entw. Mech._
XXV, pp. 581–614, XXVI, pp. 108–118, 1908; Wolfgang Ostwald, _Die
zeitlichen Eigenschaften der Entwickelungsvorgänge_, 1908; Hatai, S.,
Interpretation of Growth-curves from a Dynamical Standpoint, _Anat.
Record_, V, p. 373, 1911.
[169] _Biochem. Zeitschr._ II, 1906, p. 34.
[170] Even a crystal may be said, in a sense, to display
“autocatalysis”: for the bigger its surface becomes, the more rapidly
does the mass go on increasing.
[171] Cf. Loeb, The Stimulation of Growth, _Science_, May 14, 1915.
[172] _B. coli-communis_, according to Buchner, tends to double in
22 minutes; in 24 hours, therefore, a single individual would be
multiplied by something like 10^{28}; _Sitzungsber. München. Ges.
Morphol. u. Physiol._ III, pp. 65–71, 1888. Cf. Marshall Ward, Biology
of _Bacillus ramosus_, etc. _Pr. R. S._ LVIII, 265–468, 1895. The
comparatively large infusorian Stylonichia, according to Maupas, would
multiply in a month by 10^{43}.
[173] Cf. Enriques, Wachsthum und seine analytisehe Darstellung, _Biol.
Centralbl._ 1909, p. 337.
[174] Cf. (_int. al._) Mellor, _Chemical Statics and Dynamics_, 1904,
p. 291.
[175] Cf. Robertson, _l.c._
[176] See, for a brief resumé of this subject, Morgan’s _Experimental
Zoology_, chap. xvi.
[177] _Amer. J. of Physiol._, X, 1904.
[178] _C.R._ CXXI, CXXII, 1895–96.
[179] Cf. Loeb, _Science_, May 14, 1915.
[180] Cf. Baumann u. Roos, Vorkommen von Iod im Thierkörper, _Zeitschr.
für Physiol. Chem._ XXI, XXII, 1895, 6.
[181] Le Néo-Vitalisme, _Rev. Scientifique_, Mars 1911, p. 22 (of
reprint).
[182] _La vie et la mort_, p. 43, 1902.
[183] Cf. Dendy, _Evolutionary Biology_, 1912, p. 408; _Brit. Ass.
Report_ (Portsmouth), 1911, p. 278.
[184] Lucret. v, 877. “Lucretius nowhere seems to recognise the
possibility of improvement or change of species by ‘natural selection’;
the animals remain as they were at the first, except that the weaker
and more useless kinds have been crushed out. Hence he stands in marked
contrast with modern evolutionists.” Kelsey’s note, _ad loc._
[185] Even after we have so narrowed the scope and sphere of natural
selection, it is still hard to understand; for the causes of
_extinction_ are often wellnigh as hard to comprehend as are those of
the _origin_ of species. If we assert (as has been lightly done) that
Smilodon perished owing to its gigantic tusks, that Teleosaurus was
handicapped by its exaggerated snout, or Stegosaurus weighed down by
its intolerable load of armour, we may be reminded of other kindred
forms to show that similar conditions did not necessarily lead to
extermination, or that rapid extinction ensued apart from any such
visible or apparent disadvantages. Cf. Lucas, F. A., On Momentum in
Variation, _Amer. Nat._ xli, p. 46, 1907.
[186] See Professor T. H. Morgan’s _Regeneration_ (316 pp.), 1901 for
a full account and copious bibliography. The early experiments on
regeneration, by Vallisneri, Réaumur, Bonnet, Trembley, Baster, and
others, are epitomised by Haller, _Elem. Physiologiae_, VIII, p. 156
_seq._
[187] _Journ. Experim. Zool._ VII, p. 397, 1909.
[188] _Op. cit._ p. 406, Exp. IV.
[189] The experiments of Loeb on the growth of Tubularia in various
saline solutions, referred to on p. 125, might as well or better have
been referred to under the heading of regeneration, as they were
performed on cut pieces of the zoophyte. (Cf. Morgan, _op. cit._ p. 35.)
[190] _Powers of the Creator_, I, p. 7, 1851. See also _Rare and
Remarkable Animals_, II, pp. 17–19, 90, 1847.
[191] Lillie, F. R., The smallest Parts of Stentor capable of
Regeneration, _Journ. of Morphology_, XII, p. 239, 1897.
[192] Boveri, Entwicklungsfähigkeit kernloser Seeigeleier, etc., _Arch.
f. Entw. Mech._ II, 1895. See also Morgan, Studies of the partial
larvae of Sphaerechinus, _ibid._ 1895; J. Loeb, On the Limits of
Divisibility of Living Matter, _Biol. Lectures_, 1894, etc.
[193] Cf. Przibram, H., Scheerenumkehr bei dekapoden Crustaceen, _Arch.
f. Entw. Mech._ XIX, 181–247, 1905; XXV, 266–344, 1907. Emmel, _ibid._
XXII, 542, 1906; Regeneration of lost parts in Lobster, _Rep. Comm.
Inland Fisheries, Rhode Island_, XXXV, XXXVI, 1905–6; _Science_ (n.s.),
XXVI, 83–87, 1907. Zeleny, Compensatory Regulation, _J. Exp. Zool._ II,
1–102, 347–369, 1905; etc.
[194] Lobsters are occasionally found with two symmetrical claws:
which are then usually serrated, sometimes (but very rarely) both
blunt-toothed. Cf. Calman, _P.Z.S._ 1906, pp. 633, 634, and _reff._
[195] Wilson, E. B., Reversal of Symmetry in _Alpheus heterochelis_,
_Biol. Bull._ IV, p. 197, 1903.
[196] _J. Exp. Zool._ VII, p. 457, 1909.
[197] _Biologica_, III, p. 161, June. 1913.
[198] _Anatomical and Pathological Observations_, p. 3, 1845;
_Anatomical Memoirs_, II, p. 392, 1868.
[199] Giard, A., L’œuf et les débuts de l’évolution, _Bull. Sci. du
Nord de la Fr._ VIII, pp. 252–258, 1876.
[200] _Entwickelungsvorgänge der Eizelle_, 1876; _Investigations on
Microscopic Foams and Protoplasm_, p. 1, 1894.
[201] _Journ. of Morphology_, I, p. 229, 1887.
[202] While it has been very common to look upon the phenomena of
mitosis as sufficiently explained by the results _towards which_ they
seem to lead, we may find here and there a strong protest against this
mode of interpretation. The following is a case in point: “On a tenté
d’établir dans la mitose dite primitive plusieurs catégories, plusieurs
types de mitose. On a choisi le plus souvent comme base de ces systèmes
des concepts abstraits et téléologiques: répartition plus ou moins
exacte de la chromatine entre les deux noyaux-fils suivant qu’il y
a ou non des chromosomes (_Dangeard_), distribution particulière et
signification dualiste des substances nucléaires (substance kinétique
et substance générative ou héréditaire, _Hartmann et ses élèves_), etc.
Pour moi tous ces essais sont à rejeter catégoriquement à cause de leur
caractère finaliste; de plus, ils sont construits sur des concepts non
démontrés, et qui parfois représentent des généralisations absolument
erronées.” A. Alexeieff, _Archiv für Protistenkunde_, XIX, p. 344, 1913.
[203] This is the old philosophic axiom writ large: _Ignorato motu,
ignoratur natura_; which again is but an adaptation of Aristotle’s
phrase, ἡ ἀρχὴ τῆς κινήσεως, as equivalent to the “Efficient Cause.”
FitzGerald holds that “all explanation consists in a description of
underlying motions”; _Scientific Writings_, 1902, p. 385.
[204] As when Nägeli concluded that the organism is, in a certain
sense, “vorgebildet”; _Beitr. zur wiss. Botanik_, II, 1860. Cf. E. B.
Wilson, _The Cell, etc._, p. 302.
[205] “La matière arrangée par une sagesse divine doit être
essentiellement organisée partout ... il y a machine dans les parties
de la machine Naturelle à l’infini.” _Sur le principe de la Vie_,
p. 431 (Erdmann). This is the very converse of the doctrine of the
Atomists, who could not conceive a condition “_ubi dimidiae partis pars
semper habebit Dimidiam partem, nec res praefiniet ulla_.”
[206] Cf. an interesting passage from the _Elements_ (I, p. 445,
Molesworth’s edit.), quoted by Owen, _Hunterian Lectures on the
Invertebrates_, 2nd ed. pp. 40, 41, 1855.
[207] “Wir müssen deshalb den lebenden Zellen, abgesehen von der
Molekularstructur der organischen Verbindungen welche sie enthält, noch
eine andere und in anderer Weise complicirte Structur zuschreiben,
und diese es ist welche wir mit dem Namen _Organisation_ bezeichnen,”
Brücke, Die Elementarorganismen, _Wiener Sitzungsber._ XLIV, 1861, p.
386; quoted by Wilson, _The Cell_, etc. p. 289. Cf. also Hardy, _Journ.
of Physiol._ XXIV, 1899, p. 159.
[208] Precisely as in the Lucretian _concursus_, _motus_, _ordo_,
_positura_, _figurae_, whereby bodies _mutato ordine mutant naturam_.
[209] Otto Warburg, Beiträge zur Physiologie der Zelle, insbesondere
über die Oxidationsgeschwindigkeit in Zellen; in Asher-Spiro’s
_Ergebnisse der Physiologie_, XIV, pp. 253–337, 1914 (see p. 315). (Cf.
Bayliss, _General Physiology_, 1915, p. 590).
[210] Hardy, W. B., On some Problems of Living Matter (Guthrie
Lecture), _Tr. Physical Soc. London_, xxviii, p. 99–118, 1916.
[211] As a matter of fact both phrases occur, side by side, in Graham’s
classical paper on “Liquid Diffusion applied to Analysis,” _Phil.
Trans._ CLI, p. 184, 1861; _Chem. and Phys. Researches_ (ed. Angus
Smith), 1876, p. 554.
[212] L. Rhumbler, Mechanische Erklärung der Aehnlichkeit zwischen
Magnetischen Kraftliniensystemen und Zelltheilungsfiguren, _Arch. f.
Entw. Mech._ XV, p. 482, 1903.
[213] Gallardo, A., Essai d’interpretation des figures caryocinétiques,
_Anales del Museo de Buenos-Aires_ (2), II, 1896; La division de la
cellule, phenomène bipolaire de caractère electro-colloidal, _Arch. f.
Entw. Mech._ XXVIII, 1909, etc.
[214] _Arch. f. Entw. Mech._ III, IV, 1896–97.
[215] On various theories of the mechanism of mitosis, see (e.g.)
Wilson, _The Cell in Development_, etc., pp. 100–114; Meves,
_Zelltheilung_, in Merkel u. Bonnet’s _Ergebnisse der Anatomie_, etc.,
VII, VIII, 1897–8; Ida H. Hyde, _Amer. Journ. of Physiol._ XII, pp.
241–275, 1905; and especially Prenant, A., Theories et interprétations
physiques de la mitose, _J. de l’Anat. et Physiol._ XLVI, pp. 511–578,
1910.
[216] Hartog, M., Une force nouvelle: le mitokinétisme, _C.R._ 11 Juli,
1910; Mitokinetism in the Mitotic Spindle and in the Polyasters, _Arch.
f. Entw. Mech._ XXVII, pp. 141–145, 1909; cf. _ibid._ XL, pp. 33–64,
1914. Cf. also Hartog’s papers in _Proc. R. S._ (B), LXXVI, 1905;
_Science Progress_ (n. s.), I, 1907; _Riv. di Scienza_, II, 1908; _C.
R. Assoc. fr. pour l’Avancem. des Sc._ 1914, etc.
[217] The configurations, as obtained by the usual experimental
methods, were of course known long before Faraday’s day, and
constituted the “convergent and divergent magnetic curves” of
eighteenth century mathematicians. As Leslie said, in 1821, they were
“regarded with wonder by a certain class of dreaming philosophers,
who did not hesitate to consider them as the actual traces of an
invisible fluid, perpetually circulating between the poles of the
magnet.” Faraday’s great advance was to interpret them as indications
of _stress in a medium_,—of tension or attraction along the lines, and
of repulsion transverse to the lines, of the diagram.
[218] Cf. also the curious phenomenon in a dividing egg described as
“spinning” by Mrs G. F. Andrews, _J. of Morph._ XII, pp. 367–389, 1897.
[219] Whitman, _J. of Morph._ II, p. 40, 1889.
[220] “Souvent il n’y a qu’une séparation _physique_ entre le
cytoplasme et le suc nucléaire, comme entre deux liquides immiscibles,
etc.;” Alexeieff, Sur la mitose dite “primitive,” _Arch. f.
Protistenk._ XXIX, p. 357, 1913.
[221] The appearance of “vacuolation” is a result of endosmosis or the
diffusion of a less dense fluid into the denser plasma of the cell.
_Caeteris paribus_, it is less apparent in marine organisms than in
those of freshwater, and in many or most marine Ciliates and even
Rhizopods a contractile vacuole has not been observed (Bütschli, in
Bronn’s _Protozoa_, p. 1414); it is also absent, and probably for the
same reason, in parasitic Protozoa, such as the Gregarines and the
Entamoebae. Rossbach shewed that the contractile vacuole of ordinary
freshwater Ciliates was very greatly diminished in a 5 per cent.
solution of NaCl, and all but disappeared in a 1 per cent. solution of
sugar (_Arb. z. z. Inst. Würzburg_, 1872, cf. Massart, _Arch. de Biol._
LX, p. 515, 1889). _Actinophrys sol_, when gradually acclimatised
to sea-water, loses its vacuoles, and _vice versa_ (Gruber, _Biol.
Centralbl._ IX, p. 22, 1889); and the same is true of Amoeba (Zuelzer,
_Arch. f. Entw. Mech._ 1910, p. 632). The gradual enlargement of the
contractile vacuole is precisely analogous to the change of size of a
bubble until the gases on either side of the film are equally diffused,
as described long ago by Draper (_Phil. Mag._ (n. s.), XI, p. 559,
1837). Rhumbler has shewn that contractile or pulsating vacuoles may
be well imitated in chloroform-drops, suspended in water in which
various substances are dissolved (_Arch. f. Entw. Mech._ VII, 1898,
p. 103). The pressure within the contractile vacuole, always greater
than without, diminishes with its size, being inversely proportional
to its radius; and when it lies near the surface of the cell, as in
a Heliozoon, it bursts as soon as it reaches a thinness which its
viscosity or molecular cohesion no longer permits it to maintain.
[222] Cf. p. 660.
[223] The elongated or curved “macronucleus” of an Infusorian is
to be looked upon as a single mass of chromatin, rather than as an
aggregation of particles in a fluid drop, as in the case described. It
has a shape of its own, in which ordinary surface-tension plays a very
subordinate part.
[224] _Théorie physico-chimique de la Vie_, p. 73, 1910; _Mechanism of
Life_, p. 56, 1911.