ON MAGNITUDE
To terms of magnitude, and of direction, must we refer all our
conceptions of form. For the form of an object is defined when we know
its magnitude, actual or relative, in various directions; and growth
involves the same conceptions of magnitude and direction, with this
addition, that they are supposed to alter in time. Before we proceed
to the consideration of specific form, it will be worth our while to
consider, for a little while, certain phenomena of spatial magnitude,
or of the extension of a body in the several dimensions of space[24].
We are taught by elementary mathematics that, in similar solid figures,
the surface increases as the square, and the volume as the cube, of the
linear dimensions. If we take the simple case of a sphere, with radius
_r_, the area of its surface is equal to 4π_r_^2, and its volume to
(4/3)π_r_^3; from which it follows that the ratio of volume to surface,
or _V_/_S_, is (1/3)_r_. In other words, the greater the radius (or
the larger the sphere) the greater will be its volume, or its mass (if
it be uniformly dense throughout), in comparison with its superficial
area. And, taking _L_ to represent any linear dimension, we may write
the general equations in the form
_S_ ∝ _L_^2, _V_ ∝ _L_^3,
or _S_ = _k_ ⋅ _L_^2, and _V_ = _k′_ ⋅ _L_^3;
and _V_/_S_ ∝ _L_.
From these elementary principles a great number of consequences follow,
all more or less interesting, and some of them of great importance.
In the first place, though growth in length (let {17} us say) and
growth in volume (which is usually tantamount to mass or weight) are
parts of one and the same process or phenomenon, the one attracts
our _attention_ by its increase, very much more than the other. For
instance a fish, in doubling its length, multiplies its weight by no
less than eight times; and it all but doubles its weight in growing
from four inches long to five.
In the second place we see that a knowledge of the correlation between
length and weight in any particular species of animal, in other words
a determination of _k_ in the formula _W_ = _k_ ⋅ _L_^3, enables us
at any time to translate the one magnitude into the other, and (so to
speak) to weigh the animal with a measuring-rod; this however being
always subject to the condition that the animal shall in no way have
altered its form, nor its specific gravity. That its specific gravity
or density should materially or rapidly alter is not very likely; but
as long as growth lasts, changes of form, even though inappreciable
to the eye, are likely to go on. Now weighing is a far easier and far
more accurate operation than measuring; and the measurements which
would reveal slight and otherwise imperceptible changes in the form of
a fish—slight relative differences between length, breadth and depth,
for instance,—would need to be very delicate indeed. But if we can make
fairly accurate determinations of the length, which is very much the
easiest dimension to measure, and then correlate it with the weight,
then the value of _k_, according to whether it varies or remains
constant, will tell us at once whether there has or has not been a
tendency to gradual alteration in the general form. To this subject we
shall return, when we come to consider more particularly the rate of
growth.
But a much deeper interest arises out of this changing ratio of
dimensions when we come to consider the inevitable changes of physical
relations with which it is bound up. We are apt, and even accustomed,
to think that magnitude is so purely relative that differences of
magnitude make no other or more essential difference; that Lilliput and
Brobdingnag are all alike, according as we look at them through one
end of the glass or the other. But this is by no means so; for _scale_
has a very marked effect upon physical phenomena, and the effect of
scale constitutes what is known as the principle of similitude, or of
dynamical similarity. {18}
This effect of scale is simply due to the fact that, of the physical
forces, some act either directly at the surface of a body, or otherwise
in _proportion_ to the area of surface; and others, such as gravity,
act on all particles, internal and external alike, and exert a force
which is proportional to the mass, and so usually to the volume, of the
body.
The strength of an iron girder obviously varies with the cross-section
of its members, and each cross-section varies as the square of a linear
dimension; but the weight of the whole structure varies as the cube of
its linear dimensions. And it follows at once that, if we build two
bridges geometrically similar, the larger is the weaker of the two[25].
It was elementary engineering experience such as this that led Herbert
Spencer[26] to apply the principle of similitude to biology.
The same principle had been admirably applied, in a few clear
instances, by Lesage[27], a celebrated eighteenth century physician
of Geneva, in an unfinished and unpublished work[28]. Lesage argued,
for instance, that the larger ratio of surface to mass would lead in a
small animal to excessive transpiration, were the skin as “porous” as
our own; and that we may hence account for the hardened or thickened
skins of insects and other small terrestrial animals. Again, since
the weight of a fruit increases as the cube of its dimensions, while
the strength of the stalk increases as the square, it follows that
the stalk should grow out of apparent due proportion to the fruit; or
alternatively, that tall trees should not bear large fruit on slender
branches, and that melons and pumpkins must lie upon the ground. And
again, that in quadrupeds a large head must be supported on a neck
which is either {19} excessively thick and strong, like a bull’s, or
very short like the neck of an elephant.
But it was Galileo who, wellnigh 300 years ago, had first laid down
this general principle which we now know by the name of the principle
of similitude; and he did so with the utmost possible clearness, and
with a great wealth of illustration, drawn from structures living and
dead[29]. He showed that neither can man build a house nor can nature
construct an animal beyond a certain size, while retaining the same
proportions and employing the same materials as sufficed in the case
of a smaller structure[30]. The thing will fall to pieces of its own
weight unless we either change its relative proportions, which will at
length cause it to become clumsy, monstrous and inefficient, or else
we must find a new material, harder and stronger than was used before.
Both processes are familiar to us in nature and in art, and practical
applications, undreamed of by Galileo, meet us at every turn in this
modern age of steel.
Again, as Galileo was also careful to explain, besides the questions
of pure stress and strain, of the strength of muscles to lift an
increasing weight or of bones to resist its crushing stress, we have
the very important question of _bending moments_. This question enters,
more or less, into our whole range of problems; it affects, as we shall
afterwards see, or even determines the whole form of the skeleton, and
is very important in such a case as that of a tall tree[31].
Here we have to determine the point at which the tree will curve
under its own weight, if it be ever so little displaced from the
perpendicular[32]. In such an investigation we have to make {20} some
assumptions,—for instance, with regard to the trunk, that it tapers
uniformly, and with regard to the branches that their sectional area
varies according to some definite law, or (as Ruskin assumed[33]) tends
to be constant in any horizontal plane; and the mathematical treatment
is apt to be somewhat difficult. But Greenhill has shewn that (on such
assumptions as the above), a certain British Columbian pine-tree, which
yielded the Kew flagstaff measuring 221 ft. in height with a diameter
at the base of 21 inches, could not possibly, by theory, have grown
to more than about 300 ft. It is very curious that Galileo suggested
precisely the same height (_dugento braccia alta_) as the utmost limit
of the growth of a tree. In general, as Greenhill shews, the diameter
of a homogeneous body must increase as the power 3/2 of the height,
which accounts for the slender proportions of young trees, compared
with the stunted appearance of old and large ones[34]. In short, as
Goethe says in _Wahrheit und Dichtung_, “Es ist dafür gesorgt dass
die Bäume nicht in den Himmel wachsen.” But Eiffel’s great tree of
steel (1000 feet high) is built to a very different plan; for here
the profile of the tower follows the logarithmic curve, giving _equal
strength_ throughout, according to a principle which we shall have
occasion to discuss when we come to treat of “form and mechanical
efficiency” in connection with the skeletons of animals.
Among animals, we may see in a general way, without the help of
mathematics or of physics, that exaggerated bulk brings with it a
certain clumsiness, a certain inefficiency, a new element of risk
and hazard, a vague preponderance of disadvantage. The case was
well put by Owen, in a passage which has an interest of its own as
a premonition (somewhat like De Candolle’s) of the “struggle for
existence.” Owen wrote as follows[35]: “In proportion to the bulk of a
species is the difficulty of the contest which, as a living organised
whole, the individual of such species {21} has to maintain against
the surrounding agencies that are ever tending to dissolve the vital
bond, and subjugate the living matter to the ordinary chemical and
physical forces. Any changes, therefore, in such external conditions
as a species may have been originally adapted to exist in, will
militate against that existence in a degree proportionate, perhaps in
a geometrical ratio, to the bulk of the species. If a dry season be
greatly prolonged, the large mammal will suffer from the drought sooner
than the small one; if any alteration of climate affect the quantity
of vegetable food, the bulky Herbivore will first feel the effects of
stinted nourishment.”
But the principle of Galileo carries us much further and along more
certain lines.
The tensile strength of a muscle, like that of a rope or of our girder,
varies with its cross-section; and the resistance of a bone to a
crushing stress varies, again like our girder, with its cross-section.
But in a terrestrial animal the weight which tends to crush its limbs
or which its muscles have to move, varies as the cube of its linear
dimensions; and so, to the possible magnitude of an animal, living
under the direct action of gravity, there is a definite limit set.
The elephant, in the dimensions of its limb-bones, is already shewing
signs of a tendency to disproportionate thickness as compared with
the smaller mammals; its movements are in many ways hampered and its
agility diminished: it is already tending towards the maximal limit of
size which the physical forces permit. But, as Galileo also saw, if
the animal be wholly immersed in water, like the whale, (or if it be
partly so, as was in all probability the case with the giant reptiles
of our secondary rocks), then the weight is counterpoised to the extent
of an equivalent volume of water, and is completely counterpoised if
the density of the animal’s body, with the included air, be identical
(as in a whale it very nearly is) with the water around. Under these
circumstances there is no longer a physical barrier to the indefinite
growth in magnitude of the animal[36]. Indeed, {22} in the case of the
aquatic animal there is, as Spencer pointed out, a distinct advantage,
in that the larger it grows the greater is its velocity. For its
available energy depends on the mass of its muscles; while its motion
through the water is opposed, not by gravity, but by “skin-friction,”
which increases only as the square of its dimensions; all other things
being equal, the bigger the ship, or the bigger the fish, the faster it
tends to go, but only in the ratio of the square root of the increasing
length. For the mechanical work (_W_) of which the fish is capable
being proportional to the mass of its muscles, or the cube of its
linear dimensions: and again this work being wholly done in producing a
velocity (_V_) against a resistance (_R_) which increases as the square
of the said linear dimensions; we have at once
_W_ = _l_^3,
and also _W_ = _R_ _V_^2 = _l_^2 _V_^2.
Therefore _l_^3 = _l_^2 _V_^2, and _V_ = √_l_.
This is what is known as Froude’s Law of the _correspondence of
speeds_.
But there is often another side to these questions, which makes them
too complicated to answer in a word. For instance, the work (per
stroke) of which two similar engines are capable should obviously vary
as the cubes of their linear dimensions, for it varies on the one hand
with the _surface_ of the piston, and on the other, with the _length_
of the stroke; so is it likewise in the animal, where the corresponding
variation depends on the cross-section of the muscle, and on the space
through which it contracts. But in two precisely similar engines,
the actual available horse-power varies as the square of the linear
dimensions, and not as the cube; and this for the obvious reason that
the actual energy developed depends upon the _heating-surface_ of
the boiler[37]. So likewise must there be a similar tendency, among
animals, for the rate of supply of kinetic energy to vary with the
surface of the {23} lung, that is to say (other things being equal)
with the _square_ of the linear dimensions of the animal. We may of
course (departing from the condition of similarity) increase the
heating-surface of the boiler, by means of an internal system of tubes,
without increasing its outward dimensions, and in this very way nature
increases the respiratory surface of a lung by a complex system of
branching tubes and minute air-cells; but nevertheless in two similar
and closely related animals, as also in two steam-engines of precisely
the same make, the law is bound to hold that the rate of working must
tend to vary with the square of the linear dimensions, according to
Froude’s law of _steamship comparison_. In the case of a very large
ship, built for speed, the difficulty is got over by increasing the
size and number of the boilers, till the ratio between boiler-room
and engine-room is far beyond what is required in an ordinary small
vessel[38]; but though we find lung-space increased among animals where
greater rate of working is required, as in general among birds, I do
not know that it can be shewn to increase, as in the “over-boilered”
ship, with the size of the animal, and in a ratio which outstrips
that of the other bodily dimensions. If it be the case then, that
the working mechanism of the muscles should be able to exert a force
proportionate to the cube of the linear bodily dimensions, while the
respiratory mechanism can only supply a store of energy at a rate
proportional to the square of the said dimensions, the singular result
ought to follow that, in swimming for instance, the larger fish ought
to be able to put on a spurt of speed far in excess of the smaller
one; but the distance travelled by the year’s end should be very much
alike for both of them. And it should also follow that the curve of
fatigue {24} should be a steeper one, and the staying power should be
less, in the smaller than in the larger individual. This is the case of
long-distance racing, where the big winner puts on his big spurt at the
end. And for an analogous reason, wise men know that in the ’Varsity
boat-race it is judicious and prudent to bet on the heavier crew.
Leaving aside the question of the supply of energy, and keeping to
that of the mechanical efficiency of the machine, we may find endless
biological illustrations of the principle of similitude.
In the case of the flying bird (apart from the initial difficulty of
raising itself into the air, which involves another problem) it may be
shewn that the bigger it gets (all its proportions remaining the same)
the more difficult it is for it to maintain itself aloft in flight. The
argument is as follows:
In order to keep aloft, the bird must communicate to the air a downward
momentum equivalent to its own weight, and therefore proportional
to _the cube of its own linear dimensions_. But the momentum so
communicated is proportional to the mass of air driven downwards, and
to the rate at which it is driven: the mass being proportional to the
bird’s wing-area, and also (with any given slope of wing) to the speed
of the bird, and the rate being again proportional to the bird’s speed;
accordingly the whole momentum varies as the wing-area, i.e. as _the
square of the linear dimensions, and also as the square of the speed_.
Therefore, in order that the bird may maintain level flight, its speed
must be proportional to _the square root of its linear dimensions_.
Now the rate at which the bird, in steady flight, has to work in order
to drive itself forward, is the rate at which it communicates energy to
the air; and this is proportional to _m_ _V_^2, i.e. to the mass and
to the square of the velocity of the air displaced. But the mass of
air displaced per second is proportional to the wing-area and to the
speed of the bird’s motion, and therefore to the power 2½ of the linear
dimensions; and the speed at which it is displaced is proportional
to the bird’s speed, and therefore to the square root of the linear
dimensions. Therefore the energy communicated per second (being
proportional to the mass and to the square of the speed) is jointly
proportional to the power 2½ of the linear dimensions, as above, and
to the first power thereof: {25} that is to say, it increases in
proportion _to the power_ 3½ _of the linear dimensions_, and therefore
faster than the weight of the bird increases.
Put in mathematical form, the equations are as follows:
(_m_ = the mass of air thrust downwards; _V_ its velocity, proportional
to that of the bird; _M_ its momentum; _l_ a linear dimension of the
bird; _w_ its weight; _W_ the work done in moving itself forward.)
_M_ = _w_ = _l_^3.
But _M_ = _m_ _V_, and _m_ = _l_^2 _V_.
Therefore _M_ = _l_^2 _V_^2,
and _l_^2 _V_^2 = _l_^3,
or _V_ = √_l_.
But, again, _W_ = _m_ _V_^2 = _l_^2 _V_ × _V_^2
= _l_^2 × √_l_ × _l_
= _l_^{3½}.
The work requiring to be done, then, varies as the power 3½ of the
bird’s linear dimensions, while the work of which the bird is capable
depends on the mass of its muscles, and therefore varies as the cube
of its linear dimensions[39]. The disproportion does not seem at first
sight very great, but it is quite enough to tell. It is as much as
to say that, every time we double the linear dimensions of the bird,
the difficulty of flight is increased in the ratio of 2^3 : 2^{3½},
or 8 : 11·3, or, say, 1 : 1·4. If we take the ostrich to exceed the
sparrow in linear dimensions as 25 : 1, which seems well within the
mark, we have the ratio between 25^{3½} and 25^3, or between 5^7 : 5^6;
in other words, flight is just five times more difficult for the larger
than for the smaller bird[40].
The above investigation includes, besides the final result, a number
of others, explicit or implied, which are of not less importance. Of
these the simplest and also the most important is {26} contained in
the equation _V_ = √_l_, a result which happens to be identical with
one we had also arrived at in the case of the fish. In the bird’s case
it has a deeper significance than in the other; because it implies here
not merely that the velocity will tend to increase in a certain ratio
with the length, but that it _must_ do so as an essential and primary
condition of the bird’s remaining aloft. It is accordingly of great
practical importance in aeronautics, for it shews how a provision of
increasing speed must accompany every enlargement of our aeroplanes. If
a given machine weighing, say, 500 lbs. be stable at 40 miles an hour,
then one geometrically similar which weighs, say, a couple of tons must
have its speed determined as follows:
_W_ : _w_ :: _L_^3 : _l_^3 :: 8 : 1.
Therefore _L_ : _l_ :: 2 : 1.
But _V_^2 : _v_^2 :: _L_ : _l_.
Therefore _V_ : _v_ :: √2 : 1 = 1·414 : 1.
That is to say, the larger machine must be capable of a speed equal
to 1·414 × 40, or about 56½ miles per hour.
It is highly probable, as Lanchester[41] remarks, that Lilienthal
met his untimely death not so much from any intrinsic fault in
the design or construction of his machine, but simply because his
engine fell somewhat short of the power required to give the speed
which was necessary for stability. An arrow is a very imperfectly
designed aeroplane, but nevertheless it is evidently capable, to a
certain extent and at a high velocity, of acquiring “stability” and
hence of actual “flight”: the duration and consequent range of its
trajectory, as compared with a bullet of similar initial velocity,
being correspondingly benefited. When we return to our birds, and
again compare the ostrich with the sparrow, we know little or nothing
about the speed in flight of the latter, but that of the swift is
estimated[42] to vary from a minimum of 20 to 50 feet or more per
second,—say from 14 to 35 miles per hour. Let us take the same lower
limit as not far from the minimal velocity of the sparrow’s flight
also; and it {27} would follow that the ostrich, of 25 times the
sparrow’s linear dimensions, would be compelled to fly (if it flew at
all) with a _minimum_ velocity of 5 × 14, or 70 miles an hour.
The same principle of _necessary speed_, or the indispensable relation
between the dimensions of a flying object and the minimum velocity at
which it is stable, accounts for a great number of observed phenomena.
It tells us why the larger birds have a marked difficulty in rising
from the ground, that is to say, in acquiring to begin with the
horizontal velocity necessary for their support; and why accordingly,
as Mouillard[43] and others have observed, the heavier birds, even
those weighing no more than a pound or two, can be effectively “caged”
in a small enclosure open to the sky. It tells us why very small birds,
especially those as small as humming-birds, and _à fortiori_ the still
smaller insects, are capable of “stationary flight,” a very slight and
scarcely perceptible velocity _relatively to the air_ being sufficient
for their support and stability. And again, since it is in all cases
velocity relative to the air that we are speaking of, we comprehend the
reason why one may always tell which way the wind blows by watching the
direction in which a bird _starts_ to fly.
It is not improbable that the ostrich has already reached a magnitude,
and we may take it for certain that the moa did so, at which flight by
muscular action, according to the normal anatomy of a bird, has become
physiologically impossible. The same reasoning applies to the case of
man. It would be very difficult, and probably absolutely impossible,
for a bird to fly were it the bigness of a man. But Borelli, in
discussing this question, laid even greater stress on the obvious fact
that a man’s pectoral muscles are so immensely less in proportion than
those of a bird, that however we may fit ourselves with wings we can
never expect to move them by any power of our own relatively weaker
muscles; so it is that artificial flight only became possible when
an engine was devised whose efficiency was extraordinarily great in
comparison with its weight and size.
Had Leonardo da Vinci known what Galileo knew, he would not have spent
a great part of his life on vain efforts to make to himself wings.
Borelli had learned the lesson thoroughly, and {28} in one of his
chapters he deals with the proposition, “Est impossible, ut homines
propriis viribus artificiose volare possint[44].”
But just as it is easier to swim than to fly, so is it obvious that,
in a denser atmosphere, the conditions of flight would be altered, and
flight facilitated. We know that in the carboniferous epoch there lived
giant dragon-flies, with wings of a span far greater than nowadays they
ever attain; and the small bodies and huge extended wings of the fossil
pterodactyles would seem in like manner to be quite abnormal according
to our present standards, and to be beyond the limits of mechanical
efficiency under present conditions. But as Harlé suggests[45],
following upon a suggestion of Arrhenius, we have only to suppose that
in carboniferous and jurassic days the terrestrial atmosphere was
notably denser than it is at present, by reason, for instance, of its
containing a much larger proportion of carbonic acid, and we have at
once a means of reconciling the apparent mechanical discrepancy.
Very similar problems, involving in various ways the principle of
dynamical similitude, occur all through the physiology of locomotion:
as, for instance, when we see that a cockchafer can carry a plate,
many times his own weight, upon his back, or that a flea can jump many
inches high.
Problems of this latter class have been admirably treated both by
Galileo and by Borelli, but many later writers have remained ignorant
of their work. Linnaeus, for instance, remarked that, if an elephant
were as strong in proportion as a stag-beetle, it would be able to pull
up rocks by the root, and to level mountains. And Kirby and Spence have
a well-known passage directed to shew that such powers as have been
conferred upon the insect have been withheld from the higher animals,
for the reason that had these latter been endued therewith they would
have “caused the early desolation of the world[46].” {29}
Such problems as that which is presented by the flea’s jumping powers,
though essentially physiological in their nature, have their interest
for us here: because a steady, progressive diminution of activity with
increasing size would tend to set limits to the possible growth in
magnitude of an animal just as surely as those factors which tend to
break and crush the living fabric under its own weight. In the case
of a leap, we have to do rather with a sudden impulse than with a
continued strain, and this impulse should be measured in terms of the
velocity imparted. The velocity is proportional to the impulse (_x_),
and inversely proportional to the mass (_M_) moved: _V_ = _x_/_M_. But,
according to what we still speak of as “Borelli’s law,” the impulse
(i.e. the work of the impulse) is proportional to the volume of the
muscle by which it is produced[47], that is to say (in similarly
constructed animals) to the mass of the whole body; for the impulse is
proportional on the one hand to the cross-section of the muscle, and
on the other to the distance through which it contracts. It follows at
once from this that the velocity is constant, whatever be the size of
the animals: in other words, that all animals, provided always that
they are similarly fashioned, with their various levers etc., in like
proportion, ought to jump, not to the same relative, but to the same
actual height[48]. According to this, then, the flea is not a better,
but rather a worse jumper than a horse or a man. As a matter of fact,
Borelli is careful to point out that in the act of leaping the impulse
is not actually instantaneous, as in the blow of a hammer, but takes
some little time, during which the levers are being extended by which
the centre of gravity of the animal is being propelled forwards; and
this interval of time will be longer in the case of the longer levers
of the larger animal. To some extent, then, this principle acts as a
corrective to the more general one, {30} and tends to leave a certain
balance of advantage, in regard to leaping power, on the side of the
larger animal[49].
But on the other hand, the question of strength of materials comes in
once more, and the factors of stress and strain and bending moment make
it, so to speak, more and more difficult for nature to endow the larger
animal with the length of lever with which she has provided the flea or
the grasshopper.
To Kirby and Spence it seemed that “This wonderful strength of insects
is doubtless the result of something peculiar in the structure and
arrangement of their muscles, and principally their extraordinary power
of contraction.” This hypothesis, which is so easily seen, on physical
grounds, to be unnecessary, has been amply disproved in a series of
excellent papers by F. Plateau[50].
A somewhat simple problem is presented to us by the act of walking.
It is obvious that there will be a great economy of work, if the leg
swing at its normal _pendulum-rate_; and, though this rate is hard to
calculate, owing to the shape and the jointing of the limb, we may
easily convince ourselves, by counting our steps, that the leg does
actually swing, or tend to swing, just as a pendulum does, at a certain
definite rate[51]. When we walk quicker, we cause the leg-pendulum to
describe a greater arc, but we do not appreciably cause it to swing, or
vibrate, quicker, until we shorten the pendulum and begin to run. Now
let two individuals, _A_ and _B_, walk in a similar fashion, that is
to say, with a similar _angle_ of swing. The _arc_ through which the
leg swings, or the _amplitude_ of each step, will therefore vary as the
length of leg, or say as _a_/_b_; but the time of swing will vary as
the square {31} root of the pendulum-length, or √_a_/√_b_. Therefore
the velocity, which is measured by amplitude/time, will also vary as
the square-roots of the length of leg: that is to say, the average
velocities of _A_ and _B_ are in the ratio of √_a_ : √_b_.
The smaller man, or smaller animal, is so far at a disadvantage
compared with the larger in speed, but only to the extent of the ratio
between the square roots of their linear dimensions: whereas, if the
rate of movement of the limb were identical, irrespective of the size
of the animal,—if the limbs of the mouse for instance swung at the same
rate as those of the horse,—then, as F. Plateau said, the mouse would
be as slow or slower in its gait than the tortoise. M. Delisle[52]
observed a “minute fly” walk three inches in half-a-second. This was
good steady walking. When we walk five miles an hour we go about 88
inches in a second, or 88/6 = 14·7 times the pace of M. Delisle’s
fly. We should walk at just about the fly’s pace if our stature were
1/(14·7)^2, or 1/216 of our present height,—say 72/216 inches, or
one-third of an inch high.
But the leg comprises a complicated system of levers, by whose various
exercise we shall obtain very different results. For instance, by
being careful to rise upon our instep, we considerably increase the
length or amplitude of our stride, and very considerably increase
our speed accordingly. On the other hand, in running, we bend and
so shorten the leg, in order to accommodate it to a quicker rate of
pendulum-swing[53]. In short, the jointed structure of the leg permits
us to use it as the shortest possible pendulum when it is swinging, and
as the longest possible lever when it is exerting its propulsive force.
Apart from such modifications as that described in the last
paragraph,—apart, that is to say, from differences in mechanical
construction or in the manner in which the mechanism is used,—we have
now arrived at a curiously simple and uniform result. For in all the
three forms of locomotion which we have attempted {32} to study, alike
in swimming, in flight and in walking, the general result, attained
under very different conditions and arrived at by very different modes
of reasoning, is in every case that the velocity tends to vary as the
square root of the linear dimensions of the organism.
From all the foregoing discussion we learn that, as Crookes once upon
a time remarked[54], the form as well as the actions of our bodies
are entirely conditioned (save for certain exceptions in the case of
aquatic animals, nicely balanced with the density of the surrounding
medium) by the strength of gravity upon this globe. Were the force of
gravity to be doubled, our bipedal form would be a failure, and the
majority of terrestrial animals would resemble short-legged saurians,
or else serpents. Birds and insects would also suffer, though there
would be some compensation for them in the increased density of the
air. While on the other hand if gravity were halved, we should get a
lighter, more graceful, more active type, requiring less energy and
less heat, less heart, less lungs, less blood.
Throughout the whole field of morphology we may find examples of a
tendency (referable doubtless in each case to some definite physical
cause) for surface to keep pace with volume, through some alteration
of its form. The development of “villi” on the inner surface of the
stomach and intestine (which enlarge its surface much as we enlarge
the effective surface of a bath-towel), the various valvular folds
of the intestinal lining, including the remarkable “spiral fold” of
the shark’s gut, the convolutions of the brain, whose complexity is
evidently correlated (in part at least) with the magnitude of the
animal,—all these and many more are cases in which a more or less
constant ratio tends to be maintained between mass and surface, which
ratio would have been more and more departed from had it not been for
the alterations of surface-form[55]. {33}
In the case of very small animals, and of individual cells, the
principle becomes especially important, in consequence of the molecular
forces whose action is strictly limited to the superficial layer.
In the cases just mentioned, action is _facilitated_ by increase of
surface: diffusion, for instance, of nutrient liquids or respiratory
gases is rendered more rapid by the greater area of surface; but
there are other cases in which the ratio of surface to mass may
make an essential change in the whole condition of the system. We
know, for instance, that iron rusts when exposed to moist air, but
that it rusts ever so much faster, and is soon eaten away, if the
iron be first reduced to a heap of small filings; this is a mere
difference of degree. But the spherical surface of the raindrop and
the spherical surface of the ocean (though both happen to be alike in
mathematical form) are two totally different phenomena, the one due
to surface-energy, and the other to that form of mass-energy which
we ascribe to gravity. The contrast is still more clearly seen in
the case of waves: for the little ripple, whose form and manner of
propagation are governed by surface-tension, is found to travel with
a velocity which is inversely as the square root of its length; while
the ordinary big waves, controlled by gravitation, have a velocity
directly proportional to the square root of their wave-length. In like
manner we shall find that the form of all small organisms is largely
independent of gravity, and largely if not mainly due to the force of
surface-tension: either as the direct result of the continued action of
surface tension on the semi-fluid body, or else as the result of its
action at a prior stage of development, in bringing about a form which
subsequent chemical changes have rendered rigid and lasting. In either
case, we shall find a very great tendency in small organisms to assume
either the spherical form or other simple forms related to ordinary
inanimate surface-tension phenomena; which forms do not recur in the
external morphology of large animals, or if they in part recur it is
for other reasons. {34}
Now this is a very important matter, and is a notable illustration of
that principle of similitude which we have already discussed in regard
to several of its manifestations. We are coming easily to a conclusion
which will affect the whole course of our argument throughout this
book, namely that there is an essential difference in kind between
the phenomena of form in the larger and the smaller organisms. I have
called this book a study of _Growth and Form_, because in the most
familiar illustrations of organic form, as in our own bodies for
example, these two factors are inseparably associated, and because we
are here justified in thinking of form as the direct resultant and
consequence of growth: of growth, whose varying rate in one direction
or another has produced, by its gradual and unequal increments, the
successive stages of development and the final configuration of
the whole material structure. But it is by no means true that form
and growth are in this direct and simple fashion correlative or
complementary in the case of minute portions of living matter. For in
the smaller organisms, and in the individual cells of the larger, we
have reached an order of magnitude in which the intermolecular forces
strive under favourable conditions with, and at length altogether
outweigh, the force of gravity, and also those other forces leading to
movements of convection which are the prevailing factors in the larger
material aggregate.
However we shall require to deal more fully with this matter in our
discussion of the rate of growth, and we may leave it meanwhile, in
order to deal with other matters more or less directly concerned with
the magnitude of the cell.
The living cell is a very complex field of energy, and of energy of
many kinds, surface-energy included. Now the whole surface-energy of
the cell is by no means restricted to its _outer_ surface; for the cell
is a very heterogeneous structure, and all its protoplasmic alveoli and
other visible (as well as invisible) heterogeneities make up a great
system of internal surfaces, at every part of which one “phase” comes
in contact with another “phase,” and surface-energy is accordingly
manifested. But still, the external surface is a definite portion of
the system, with a definite “phase” of its own, and however little we
may know of the distribution of the total energy of the system, it
is at least plain that {35} the conditions which favour equilibrium
will be greatly altered by the changed ratio of external surface to
mass which a change of magnitude, unaccompanied by change of form,
produces in the cell. In short, however it may be brought about, the
phenomenon of division of the cell will be precisely what is required
to keep approximately constant the ratio between surface and mass,
and to restore the balance between the surface-energy and the other
energies of the system. When a germ-cell, for instance, divides or
“segments” into two, it does not increase in mass; at least if there be
some slight alleged tendency for the egg to increase in mass or volume
during segmentation, it is very slight indeed, generally imperceptible,
and wholly denied by some[56]. The development or growth of the egg
from a one-celled stage to stages of two or many cells, is thus a
somewhat peculiar kind of growth; it is growth which is limited to
increase of surface, unaccompanied by growth in volume or in mass.
In the case of a soap-bubble, by the way, if it divide into two
bubbles, the volume is actually diminished[57] while the surface-area
is greatly increased. This is due to a cause which we shall have to
study later, namely to the increased pressure due to the greater
curvature of the smaller bubbles.
An immediate and remarkable result of the principles just described is
a tendency on the part of all cells, according to their kind, to vary
but little about a certain mean size, and to have, in fact, certain
absolute limitations of magnitude.
Sachs[58] pointed out, in 1895, that there is a tendency for each
nucleus to be only able to gather around itself a certain definite
amount of protoplasm. Driesch[59], a little later, found that, by
artificial subdivision of the egg, it was possible to rear dwarf
sea-urchin larvae, one-half, one-quarter, or even one-eighth of their
{36} normal size; and that these dwarf bodies were composed of
only a half, a quarter or an eighth of the normal number of cells.
Similar observations have been often repeated and amply confirmed.
For instance, in the development of _Crepidula_ (a little American
“slipper-limpet,” now much at home on our own oyster-beds), Conklin[60]
has succeeded in rearing dwarf and giant individuals, of which the
latter may be as much as twenty-five times as big as the former. But
nevertheless, the individual cells, of skin, gut, liver, muscle, and
of all the other tissues, are just the same size in one as in the
other,—in dwarf and in giant[61]. Driesch has laid particular stress
upon this principle of a “fixed cell-size.”
We get an excellent, and more familiar illustration of the same
principle in comparing the large brain-cells or ganglion-cells, both of
the lower and of the higher animals[62].
[Illustration: Fig. 1. Motor ganglion-cells, from the cervical spinal
cord. (From Minot, after Irving Hardesty.)]
In Fig. 1 we have certain identical nerve-cells taken from various
mammals, from the mouse to the elephant, all represented on the same
scale of magnification; and we see at once that they are all of much
the same _order_ of magnitude. The nerve-cell of the elephant is about
twice that of the mouse in linear dimensions, and therefore about
eight times greater in volume, or mass. But making some allowance for
difference of shape, the linear dimensions of the elephant are to
those of the mouse in a ratio certainly not less than one to fifty;
from which it would follow that the bulk of the larger animal is
something like 125,000 times that of the less. And it also follows,
the size of the nerve-cells being {37} about as eight to one, that, in
corresponding parts of the nervous system of the two animals, there
are more than 15,000 times as many individual cells in one as in
the other. In short we may (with Enriques) lay it down as a general
law that among animals, whether large or small, the ganglion-cells
vary in size within narrow limits; and that, amidst all the great
variety of structural type of ganglion observed in different classes
of animals, it is always found that the smaller species have simpler
ganglia than the larger, that is to say ganglia containing a smaller
number of cellular elements[63]. The bearing of such simple facts as
this upon the cell-theory in general is not to be disregarded; and the
warning is especially clear against exaggerated attempts to correlate
physiological processes with the visible mechanism of associated cells,
rather than with the system of energies, or the field of force, which
is associated with them. For the life of {38} the body is more than
the _sum_ of the properties of the cells of which it is composed: as
Goethe said, “Das Lebendige ist zwar in Elemente zerlegt, aber man kann
es aus diesen nicht wieder zusammenstellen und beleben.”
Among certain lower and microscopic organisms, such for instance as
the Rotifera, we are still more palpably struck by the small number of
cells which go to constitute a usually complex organ, such as kidney,
stomach, ovary, etc. We can sometimes number them in a few units, in
place of the thousands that make up such an organ in larger, if not
always higher, animals. These facts constitute one among many arguments
which combine to teach us that, however important and advantageous the
subdivision of organisms into cells may be from the constructional, or
from the dynamical point of view, the phenomenon has less essential
importance in theoretical biology than was once, and is often still,
assigned to it.
Again, just as Sachs shewed that there was a limit to the amount of
cytoplasm which could gather round a single nucleus, so Boveri has
demonstrated that the nucleus itself has definite limitations of size,
and that, in cell-division after fertilisation, each new nucleus has
the same size as its parent-nucleus[64].
In all these cases, then, there are reasons, partly no doubt
physiological, but in very large part purely physical, which set limits
to the normal magnitude of the organism or of the cell. But as we have
already discussed the existence of absolute and definite limitations,
of a physical kind, to the _possible_ increase in magnitude of an
organism, let us now enquire whether there be not also a lower limit,
below which the very existence of an organism is impossible, or
at least where, under changed conditions, its very nature must be
profoundly modified.
Among the smallest of known organisms we have, for instance,
_Micromonas mesnili_, Bonel, a flagellate infusorian, which measures
about ·34 _µ_, or ·00034 mm., by ·00025 mm.; smaller even than this
we have a pathogenic micrococcus of the rabbit, _M. progrediens_,
Schröter, the diameter of which is said to be only ·00015 mm. or
·15 _µ_, or 1·5 × 10^{−5} cm.,—about equal to the thickness of {39}
the thinnest gold-leaf; and as small if not smaller still are a few
bacteria and their spores. But here we have reached, or all but reached
the utmost limits of ordinary microscopic vision; and there remain
still smaller organisms, the so-called “filter-passers,” which the
ultra-microscope reveals, but which are mainly brought within our ken
only by the maladies, such as hydrophobia, foot-and-mouth disease, or
the “mosaic” disease of the tobacco-plant, to which these invisible
micro-organisms give rise[65]. Accordingly, since it is only by the
diseases which they occasion that these tiny bodies are made known to
us, we might be tempted to suppose that innumerable other invisible
organisms, smaller and yet smaller, exist unseen and unrecognised by
man.
[Illustration: Fig. 2. Relative magnitudes of: A, human blood-corpuscle
(7·5 µ in diameter); B, _Bacillus anthracis_ (4 – 15 µ × 1 µ); C,
various Micrococci (diam. 0·5 – 1 µ, rarely 2 µ); D, _Micromonas
progrediens_, Schröter (diam. 0·15 µ).]
To illustrate some of these small magnitudes I have adapted the
preceding diagram from one given by Zsigmondy[66]. Upon the {40} same
scale the minute ultramicroscopic particles of colloid gold would be
represented by the finest dots which we could make visible to the naked
eye upon the paper.
A bacillus of ordinary, typical size is, say, 1 µ in length. The length
(or height) of a man is about a million and three-quarter times as
great, i.e. 1·75 metres, or 1·75 × 10^6 µ; and the mass of the man is
in the neighbourhood of five million, million, million (5 × 10^{18})
times greater than that of the bacillus. If we ask whether there may
not exist organisms as much less than the bacillus as the bacillus is
less than the dimensions of a man, it is very easy to see that this
is quite impossible, for we are rapidly approaching a point where the
question of molecular dimensions, and of the ultimate divisibility of
matter, begins to call for our attention, and to obtrude itself as a
crucial factor in the case.
Clerk Maxwell dealt with this matter in his article “Atom[67],” and, in
somewhat greater detail, Errera discusses the question on the following
lines[68]. The weight of a hydrogen molecule is, according to the
physical chemists, somewhere about 8·6 × 2 × 10^{−22} milligrammes; and
that of any other element, whose molecular weight is _M_, is given by
the equation
(_M_) = 8·6 × _M_ × 10^{−22}.
Accordingly, the weight of the atom of sulphur may be taken as
8·6 × 32 × 10^{−22} mgm. = 275 × 10^{−22} mgm.
The analysis of ordinary bacteria shews them to consist[69] of about
85% of water, and 15% of solids; while the solid residue of vegetable
protoplasm contains about one part in a thousand of sulphur. We may
assume, therefore, that the living protoplasm contains about
1/1000 × 15/100 = 15 × 10^{−5}
parts of sulphur, taking the total weight as = 1.
But our little micrococcus, of 0·15 µ in diameter, would, if it were
spherical, have a volume of
π/6 × 0·15^3 µ = 18 × 10^{−4} cubic microns; {41}
and therefore (taking its density as equal to that of water), a weight
of
18 × 10^{−4} × 10^{−9} = 18 × 10^{−13} mgm.
But of this total weight, the sulphur represents only
18 × 10^{−13} × 15 × 10^{−5} = 27 × 10^{−17} mgm.
And if we divide this by the weight of an atom of sulphur, we have
(27 × 10^{−17}) ÷ (275 × 10^{−22}) = 10,000, or thereby.
According to this estimate, then, our little _Micrococcus
progrediens_ should contain only about 10,000 atoms of sulphur,
an element indispensable to its protoplasmic constitution; and it
follows that an organism of one-tenth the diameter of our micrococcus
would only contain 10 sulphur-atoms, and therefore only ten chemical
“molecules” or units of protoplasm!
It may be open to doubt whether the presence of sulphur be really
essential to the constitution of the proteid or “protoplasmic”
molecule; but Errera gives us yet another illustration of a similar
kind, which is free from this objection or dubiety. The molecule of
albumin, as is generally agreed, can scarcely be less than a thousand
times the size of that of such an element as sulphur: according to
one particular determination[70], serum albumin has a constitution
corresponding to a molecular weight of 10,166, and even this may be
far short of the true complexity of a typical albuminoid molecule. The
weight of such a molecule is
8·6 × 10166 × 10^{−22} = 8·7 × 10^{−18} mgm.
Now the bacteria contain about 14% of albuminoids, these constituting
by far the greater part of the dry residue; and therefore (from
equation (5)), the weight of albumin in our micrococcus is about
14/100 × 18 × 10^{−13} = 2·5 × 10^{−13} mgm.
If we divide this weight by that which we have arrived at as the
weight of an albumin molecule, we have
(2·5 × 10^{−13}) ÷ (8·7 × 10^{−18}) = 2·9 × 10^{−4},
in other words, our micrococcus apparently contains something less
than 30,000 molecules of albumin. {42}
According to the most recent estimates, the weight of the hydrogen
molecule is somewhat less than that on which Errera based his
calculations, namely about 16 × 10^{−22} mgms. and according to
this value, our micrococcus would contain just about 27,000 albumin
molecules. In other words, whichever determination we accept, we see
that an organism one-tenth as large as our micrococcus, in linear
dimensions, would only contain some thirty molecules of albumin; or, in
other words, our micrococcus is only about thirty times as large, in
linear dimensions, as a single albumin molecule[71].
We must doubtless make large allowances for uncertainty in the
assumptions and estimates upon which these calculations are based; and
we must also remember that the data with which the physicist provides
us in regard to molecular magnitudes are, to a very great extent,
_maximal_ values, above which the molecular magnitude (or rather
the sphere of the molecule’s range of motion) is not likely to lie:
but below which there is a greater element of uncertainty as to its
possibly greater minuteness. But nevertheless, when we shall have made
all reasonable allowances for uncertainty upon the physical side, it
will still be clear that the smallest known bodies which are described
as organisms draw nigh towards molecular magnitudes, and we must
recognise that the subdivision of the organism cannot proceed to an
indefinite extent, and in all probability cannot go very much further
than it appears to have done in these already discovered forms. For,
even, after giving all due regard to the complexity of our unit (that
is to say the albumin-molecule), with all the increased possibilities
of interrelation with its neighbours which this complexity implies, we
cannot but see that physiologically, and comparatively speaking, we
have come down to a very simple thing.
While such considerations as these, based on the chemical composition
of the organism, teach us that there must be a definite lower limit
to its magnitude, other considerations of a purely physical kind
lead us to the same conclusion. For our discussion of the principle
of similitude has already taught us that, long before we reach these
almost infinitesimal magnitudes, the {43} diminishing organism will
have greatly changed in all its physical relations, and must at length
arrive under conditions which must surely be incompatible with anything
such as we understand by life, at least in its full and ordinary
development and manifestation.
We are told, for instance, that the powerful force of surface-tension,
or capillarity, begins to act within a range of about 1/500,000 of an
inch, or say 0·05 µ. A soap-film, or a film of oil upon water, may be
attenuated to far less magnitudes than this; the black spots upon a
soap-bubble are known, by various concordant methods of measurement, to
be only about 6 × 10^{−7} cm., or about ·006 µ thick, and Lord Rayleigh
and M. Devaux[72] have obtained films of oil of ·002 µ, or even ·001 µ
in thickness.
But while it is possible for a fluid film to exist in these almost
molecular dimensions, it is certain that, long before we reach them,
there must arise new conditions of which we have little knowledge and
which it is not easy even to imagine.
It would seem that, in an organism of ·1 µ in diameter, or even rather
more, there can be no essential distinction between the interior
and the surface layers. No hollow vesicle, I take it, can exist of
these dimensions, or at least, if it be possible for it to do so,
the contained gas or fluid must be under pressures of a formidable
kind[73], and of which we have no knowledge or experience. Nor, I
imagine, can there be any real complexity, or heterogeneity, of its
fluid or semi-fluid contents; there can be no vacuoles within such a
cell, nor any layers defined within its fluid substance, for something
of the nature of a boundary-film is the necessary condition of the
existence of such layers. Moreover, the whole organism, provided that
it be fluid or semi-fluid, can only be spherical in form. What, then,
can we attribute, in the way of properties, to an organism of a size as
small as, or smaller than, say ·05 µ? It must, in all probability, be
a homogeneous, structureless sphere, composed of a very small number
of albuminoid or other molecules. Its vital properties and functions
must be extraordinarily limited; its specific outward characters, even
if we could see it, must be _nil_; and its specific properties must
be little more than those of an ion-laden corpuscle, enabling it to
perform {44} this or that chemical reaction, or to produce this or
that pathogenic effect. Even among inorganic, non-living bodies, there
must be a certain grade of minuteness at which the ordinary properties
become modified. For instance, while under ordinary circumstances
crystallisation starts in a solution about a minute solid fragment or
crystal of the salt, Ostwald has shewn that we may have particles so
minute that they fail to serve as a nucleus for crystallisation,—which
is as much as to say that they are too minute to have the form and
properties of a “crystal”; and again, in his thin oil-films, Lord
Rayleigh has noted the striking change of physical properties which
ensues when the film becomes attenuated to something less than one
close-packed layer of molecules[74].
Thus, as Clerk Maxwell put it, “molecular science sets us face to face
with physiological theories. It forbids the physiologist from imagining
that structural details of infinitely small dimensions [such as Leibniz
assumed, one within another, _ad infinitum_] can furnish an explanation
of the infinite variety which exists in the properties and functions
of the most minute organisms.” And for this reason he reprobates, with
not undue severity, those advocates of pangenesis and similar theories
of heredity, who would place “a whole world of wonders within a body
so small and so devoid of visible structure as a germ.” But indeed it
scarcely needed Maxwell’s criticism to shew forth the immense physical
difficulties of Darwin’s theory of Pangenesis: which, after all, is as
old as Democritus, and is no other than that Promethean _particulam
undique desectam_ of which we have read, and at which we have smiled,
in our Horace.
There are many other ways in which, when we “make a long excursion
into space,” we find our ordinary rules of physical behaviour entirely
upset. A very familiar case, analysed by Stokes, is that the viscosity
of the surrounding medium has a relatively powerful effect upon bodies
below a certain size. A droplet of water, a thousandth of an inch (25
µ) in diameter, cannot fall in still air quicker than about an inch and
a half per second; and as its size decreases, its resistance varies
as the diameter, and not (as with larger bodies) as the surface of
the {45} drop. Thus a drop one-tenth of that size (2·5 µ), the size,
apparently, of the drops of water in a light cloud, will fall a hundred
times slower, or say an inch a minute; and one again a tenth of this
diameter (say ·25 µ, or about twice as big, in linear dimensions, as
our micrococcus), will scarcely fall an inch in two hours. By reason
of this principle, not only do the smaller bacteria fall very slowly
through the air, but all minute bodies meet with great proportionate
resistance to their movements in a fluid. Even such comparatively large
organisms as the diatoms and the foraminifera, laden though they are
with a heavy shell of flint or lime, seem to be poised in the water of
the ocean, and fall in it with exceeding slowness.
The Brownian movement has also to be reckoned with,—that remarkable
phenomenon studied nearly a century ago (1827) by Robert Brown,
_facile princeps botanicorum_. It is one more of those fundamental
physical phenomena which the biologists have contributed, or helped to
contribute, to the science of physics.
The quivering motion, accompanied by rotation, and even by
translation, manifested by the fine granular particles issuing from
a crushed pollen-grain, and which Robert Brown proved to have no
vital significance but to be manifested also by all minute particles
whatsoever, organic and inorganic, was for many years unexplained.
Nearly fifty years after Brown wrote, it was said to be “due, either
directly to some calorical changes continually taking place in the
fluid, or to some obscure chemical action between the solid particles
and the fluid which is indirectly promoted by heat[75].” Very shortly
after these last words were written, it was ascribed by Wiener to
molecular action, and we now know that it is indeed due to the impact
or bombardment of molecules upon a body so small that these impacts do
not for the moment, as it were, “average out” to approximate equality
on all sides. The movement becomes manifest with particles of somewhere
about 20 µ in diameter, it is admirably displayed by particles of about
12 µ in diameter, and becomes more marked the smaller the particles
are. The bombardment causes our particles to behave just like molecules
of uncommon size, and this {46} behaviour is manifested in several
ways[76]. Firstly, we have the quivering movement of the particles;
secondly, their movement backwards and forwards, in short, straight,
disjointed paths; thirdly, the particles rotate, and do so the more
rapidly the smaller they are, and by theory, confirmed by observation,
it is found that particles of 1 µ in diameter rotate on an average
through 100° per second, while particles of 13 µ in diameter turn
through only 14° per minute. Lastly, the very curious result appears,
that in a layer of fluid the particles are not equally distributed, nor
do they all ever fall, under the influence of gravity, to the bottom.
But just as the molecules of the atmosphere are so distributed, under
the influence of gravity, that the density (and therefore the number
of molecules per unit volume) falls off in geometrical progression as
we ascend to higher and higher layers, so is it with our particles,
even within the narrow limits of the little portion of fluid under
our microscope. It is only in regard to particles of the simplest
form that these phenomena have been theoretically investigated[77],
and we may take it as certain that more complex particles, such as
the twisted body of a Spirillum, would show other and still more
complicated manifestations. It is at least clear that, just as the
early microscopists in the days before Robert Brown never doubted but
that these phenomena were purely vital, so we also may still be apt
to confuse, in certain cases, the one phenomenon with the other. We
cannot, indeed, without the most careful scrutiny, decide whether the
movements of our minutest organisms are intrinsically “vital” (in the
sense of being beyond a physical mechanism, or working model) or not.
For example, Schaudinn has suggested that the undulating movements of
_Spirochaete pallida_ must be due to the presence of a minute, unseen,
“undulating membrane”; and Doflein says of the same species that “sie
verharrt oft mit eigenthümlich zitternden Bewegungen zu einem Orte.”
Both movements, the trembling or quivering {47} movement described
by Doflein, and the undulating or rotating movement described by
Schaudinn, are just such as may be easily and naturally interpreted as
part and parcel of the Brownian phenomenon.
While the Brownian movement may thus simulate in a deceptive way
the active movements of an organism, the reverse statement also to
a certain extent holds good. One sometimes lies awake of a summer’s
morning watching the flies as they dance under the ceiling. It is a
very remarkable dance. The dancers do not whirl or gyrate, either in
company or alone; but they advance and retire; they seem to jostle and
rebound; between the rebounds they dart hither or thither in short
straight snatches of hurried flight; and turn again sharply in a new
rebound at the end of each little rush. Their motions are wholly
“erratic,” independent of one another, and devoid of common purpose.
This is nothing else than a vastly magnified picture, or simulacrum, of
the Brownian movement; the parallel between the two cases lies in their
complete irregularity, but this in itself implies a close resemblance.
One might see the same thing in a crowded market-place, always provided
that the bustling crowd had no _business_ whatsoever. In like manner
Lucretius, and Epicurus before him, watched the dust-motes quivering
in the beam, and saw in them a mimic representation, _rei simulacrum
et imago_, of the eternal motions of the atoms. Again the same
phenomenon may be witnessed under the microscope, in a drop of water
swarming with Paramoecia or suchlike Infusoria; and here the analogy
has been put to a numerical test. Following with a pencil the track
of each little swimmer, and dotting its place every few seconds (to
the beat of a metronome), Karl Przibram found that the mean successive
distances from a common base-line obeyed with great exactitude the
“Einstein formula,” that is to say the particular form of the “law of
chance” which is applicable to the case of the Brownian movement[78].
The phenomenon is (of course) merely analogous, and by no means
identical with the Brownian movement; for the range of motion of the
little active organisms, whether they be gnats or infusoria, is vastly
greater than that of the minute particles which are {48} passive under
bombardment; but nevertheless Przibram is inclined to think that even
his comparatively large infusoria are small enough for the molecular
bombardment to be a stimulus, though not the actual cause, of their
irregular and interrupted movements.
There is yet another very remarkable phenomenon which may come into
play in the case of the minutest of organisms; and this is their
relation to the rays of light, as Arrhenius has told us. On the waves
of a beam of light, a very minute particle (_in vacuo_) should be
actually caught up, and carried along with an immense velocity; and
this “radiant pressure” exercises its most powerful influence on bodies
which (if they be of spherical form) are just about ·00016 mm., or
·16 µ in diameter. This is just about the size, as we have seen, of
some of our smallest known protozoa and bacteria, while we have some
reason to believe that others yet unseen, and perhaps the spores of
many, are smaller still. Now we have seen that such minute particles
fall with extreme slowness in air, even at ordinary atmospheric
pressures: our organism measuring ·16 µ would fall but 83 metres in a
year, which is as much as to say that its weight offers practically no
impediment to its transference, by the slightest current, to the very
highest regions of the atmosphere. Beyond the atmosphere, however, it
cannot go, until some new force enable it to resist the attraction of
terrestrial gravity, which the viscosity of an atmosphere is no longer
at hand to oppose. But it is conceivable that our particle _may_ go yet
farther, and actually break loose from the bonds of earth. For in the
upper regions of the atmosphere, say fifty miles high, it will come in
contact with the rays and flashes of the Northern Lights, which consist
(as Arrhenius maintains) of a fine dust, or cloud of vapour-drops,
laden with a charge of negative electricity, and projected outwards
from the sun. As soon as our particle acquires a charge of negative
electricity it will begin to be repelled by the similarly laden auroral
particles, and the amount of charge necessary to enable a particle
of given size (such as our little monad of ·16 µ) to resist the
attraction of gravity may be calculated, and is found to be such as
the actual conditions can easily supply. Finally, when once set free
from the entanglement of the earth’s {49} atmosphere, the particle
may be propelled by the “radiant pressure” of light, with a velocity
which will carry it.—like Uriel gliding on a sunbeam,—as far as the
orbit of Mars in twenty days, of Jupiter in eighty days, and as far as
the nearest fixed star in three thousand years! This, and much more,
is Arrhenius’s contribution towards the acceptance of Lord Kelvin’s
hypothesis that life may be, and may have been, disseminated across the
bounds of space, throughout the solar system and the whole universe!
It may well be that we need attach no great practical importance to
this bold conception; for even though stellar space be shewn to be
_mare liberum_ to minute material travellers, we may be sure that those
which reach a stellar or even a planetary bourne are infinitely, or all
but infinitely, few. But whether or no, the remote possibilities of the
case serve to illustrate in a very vivid way the profound differences
of physical property and potentiality which are associated in the scale
of magnitude with simple differences of degree.
{50}