somewhat curious, however, that this ratio seems to fall off a little
in the third year of growth, the animal continuing to grow in height to
a marked degree after growth in length has become very slow. The ratio
between height and weight is by much the most variable of our three
ratios; the coefficient _W_/_H_^3 steadily increases, and is more than
twice as great at three years old as it was at birth. This illustrates
the important, but obvious fact, that the coefficient _k_ is most
variable in the case of that dimension which grows most uniformly, that
is to say most nearly in proportion to the general bulk of the animal.
In short, the successive values of _k_, as determined (at successive
epochs) for one dimension, are a measure of the _variability_ of the
others.
――――――――――
From the whole of the foregoing discussion we see that a certain
definite rate of growth is a characteristic or specific phenomenon,
deep-seated in the physiology of the organism; and that a very large
part of the specific morphology of the organism depends upon the
fact that there is not only an average, or aggregate, rate of growth
common to the whole, but also a variation of rate in different parts
of the organism, tending towards a specific rate characteristic of
each different part or organ. The smallest change in the relative
magnitudes of these partial or localised velocities of growth will be
soon manifested in more and more striking differences of form. This
is as much as to say that the time-element, which is implicit in the
idea of growth, can never (or very seldom) be wholly neglected in our
consideration of form[132]. It is scarcely necessary to enlarge here
upon our statement, for not only is the truth of it self-evident,
but it will find illustration again and again throughout this book.
Nevertheless, let us go out of our way for a moment to consider it in
reference to a particular case, and to enquire whether it helps to
remove any of the difficulties which that case appears to present.
{104}
[Illustration: Fig. 23. Variability of length of tail-forceps in a
sample of Earwigs. (After Bateson, _P. Z. S._ 1892, p. 588.)]
In a very well-known paper, Bateson shewed that, among a large number
of earwigs, collected in a particular locality, the males fell into two
groups, characterised by large or by small tail-forceps, with very few
instances of intermediate magnitude. This distribution into two groups,
according to magnitude, is illustrated in the accompanying diagram
(Fig. 23); and the phenomenon was described, and has been often quoted,
as one of dimorphism, or discontinuous variation. In this diagram the
time-element does not appear; but it is certain, and evident, that
it lies close behind. Suppose we take some organism which is born
not at all times of the year (as man is) but at some one particular
season (for instance a fish), then any random sample will consist of
individuals whose _ages_, and therefore whose _magnitudes_, will form
a discontinuous series; and by plotting these magnitudes on a curve in
relation to the number of individuals of each particular magnitude, we
obtain a curve such as that shewn in Fig. 24, the first practical use
of which is to enable us to analyse our sample into its constituent
“age-groups,” or in other words to determine approximately the age,
or ages of the fish. And if, instead of measuring the whole length of
our fish, we had confined ourselves to particular parts, such as head,
or {105} tail or fin, we should have obtained discontinuous curves
of distribution, precisely analogous to those for the entire animal.
Now we know that the differences with which Bateson was dealing were
entirely a question of magnitude, and we cannot help seeing that the
discontinuous distributions of magnitude represented by his earwigs’
tails are just such as are illustrated by the magnitudes of the older
and younger fish; we may indeed go so far as to say that the curves
are precisely comparable, for in both cases we see a characteristic
feature of detail, namely that the “spread” of the curve is greater in
the second wave than in the first, that is to say (in the case of the
fish) in the older as well as larger series. Over the reason for this
phenomenon, which is simple and all but obvious, we need not pause.
[Illustration: Fig. 24. Variability of length of body in a sample of
Plaice.]
It is evident, then, that in this case of “dimorphism,” the tails of
the one group of earwigs (which Bateson calls the “high males”) have
either grown _faster_, or have been growing for a longer period of
time, than those of the “low males.” If we could be certain that the
whole random sample of earwigs were of one and the same age, then we
should have to refer the phenomenon of dimorphism to a physiological
phenomenon, simple in kind (however remarkable and unexpected); viz.
that there were two alternative {106} values, very different from
one another, for the mean velocity of growth, and that the individual
earwigs varied around one or other of these mean values, in each case
according to the law of probabilities. But on the other hand, if we
could believe that the two groups of earwigs were _of different ages_,
then the phenomenon would be simplicity itself, and there would be no
more to be said about it[133].
――――――――――
Before we pass from the subject of the relative rate of growth of
different parts or organs, we may take brief note of the fact that
various experiments have been made to determine whether the normal
ratios are maintained under altered circumstances of nutrition, and
especially in the case of partial starvation. For instance, it has been
found possible to keep young rats alive for many weeks on a diet such
as is just sufficient to maintain life without permitting any increase
of weight. The rat of three weeks old weighs about 25 gms., and under a
normal diet should weigh at ten weeks old about 150 gms., in the male,
or 115 gms. in the female; but the underfed rat is still kept at ten
weeks old to the weight of 25 gms. Under normal diet the proportions
of the body change very considerably between the ages of three and ten
weeks. For instance the tail gets relatively longer; and even when the
_total_ growth of the rat is prevented by underfeeding, the _form_
continues to alter so that this increasing length of the tail is still
manifest[134]. {107}
_Full-fed Rats._
Age in Length of Length of Total
weeks body (mm.) tail (mm.) length % of tail
0 48·7 16·9 65·6 25·8
1 64·5 29·4 93·9 31·3
3 90·4 59·1 149·5 39·5
6 128·0 110·0 238·0 46·2
10 173·0 150·0 323·0 46·4
_Underfed Rats._
6 98·0 72·3 170·3 42·5
10 99·6 83·9 183·5 45·7
Again as physiologists have long been aware, there is a marked
difference in the variation of weight of the different organs,
according to whether the animal’s total weight remain constant, or
be caused to diminish by actual starvation; and further striking
differences appear when the diet is not only scanty, but ill-balanced.
But these phenomena of abnormal growth, however interesting from
the physiological view, are of little practical importance to the
morphologist.
_The effect of temperature[135]._
The rates of growth which we have hitherto dealt with are based on
special investigations, conducted under particular local conditions.
For instance, Quetelet’s data, so far as we have used them to
illustrate the rate of growth in man, are drawn from his study of the
population of Belgium. But apart from that “fortuitous” individual
variation which we have already considered, it is obvious that the
normal rate of growth will be found to vary, in man and in other
animals, just as the average stature varies, in different localities,
and in different “races.” This phenomenon is a very complex one, and
is doubtless a resultant of many undefined contributory causes; but
we at least gain something in regard to it, when we discover that the
rate of growth is directly affected by temperature, and probably by
other physical {108} conditions. Réaumur was the first to shew, and
the observation was repeated by Bonnet[136], that the rate of growth or
development of the chick was dependent on temperature, being retarded
at temperatures below and somewhat accelerated at temperatures above
the normal temperature of incubation, that is to say the temperature
of the sitting hen. In the case of plants the fact that growth is
greatly affected by temperature is a matter of familiar knowledge; the
subject was first carefully studied by Alphonse De Candolle, and his
results and those of his followers are discussed in the textbooks of
Botany[137].
That variation of temperature constitutes only one factor in
determining the rate of growth is admirably illustrated in the case
of the Bamboo. It has been stated (by Lock) that in Ceylon the rate
of growth of the Bamboo is directly proportional to the humidity
of the atmosphere: and again (by Shibata) that in Japan it is
directly proportional to the temperature. The two statements have
been ingeniously and satisfactorily reconciled by Blackman[138], who
suggests that in Ceylon the temperature-conditions are all that can be
desired, but moisture is apt to be deficient: while in Japan there is
rain in abundance but the average temperature is somewhat too low. So
that in the one country it is the one factor, and in the other country
it is the other, which is _essentially_ variable.
The annexed diagram (Fig. 25), shewing the growth in length of the
roots of some common plants during an identical period of forty-eight
hours, at temperatures varying from about 14° to 37° C., is a
sufficient illustration of the phenomenon. We see that in all cases
there is a certain optimum temperature at which the rate of growth is
a maximum, and we can also see that on either side of this optimum
temperature the acceleration of growth, positive or negative, with
increase of temperature is rapid, while at a distance from the optimum
it is very slow. From the data given by Sachs and others, we see
further that this optimum temperature is very much the same for all the
common plants of our own climate which have as yet been studied; in
them it is {109} somewhere about 26° C. (or say 77° F.), or about the
temperature of a warm summer’s day; while it is found, very naturally,
to be considerably higher in the case of plants such as the melon or
the maize, which are at home in warmer regions that our own.
――――――――――
[Illustration: Fig. 25. Relation of rate of growth to temperature in
certain plants. (From Sachs’s data.)]
In a large number of physical phenomena, and in a very marked degree in
all chemical reactions, it is found that rate of action is affected,
and for the most part accelerated, by rise of temperature; and this
effect of temperature tends to follow a definite “exponential” law,
which holds good within a considerable range of temperature, but is
altered or departed from when we pass beyond certain normal limits. The
law, as laid down by van’t Hoff for chemical reactions, is, that for
an interval of _n_ degrees the velocity varies as _x_^{_n_}, _x_ being
called the “temperature coefficient”[139] for the reaction in question.
{110}
Van’t Hoff’s law, which has become a fundamental principle of chemical
mechanics, is likewise applicable (with certain qualifications)
to the phenomena of vital chemistry; and it follows that, on very
much the same lines, we may speak of the “temperature coefficient”
of growth. At the same time we must remember that there is a very
important difference (though we can scarcely call it a _fundamental_
one) between the purely physical and the physiological phenomenon, in
that in the former we study (or seek and profess to study) one thing
at a time, while in the latter we have always to do with various
factors which intersect and interfere; increase in the one case (or
change of any kind) tends to be continuous, in the other case it tends
to be brought to arrest. This is the simple meaning of that _Law of
Optimum_, laid down by Errera and by Sachs as a general principle of
physiology: namely that _every_ physiological process which varies
(like growth itself) with the amount or intensity of some external
influence, does so according to a law in which progressive increase
is followed by progressive decrease; in other words the function has
its _optimum_ condition, and its curve shews a definite _maximum_. In
the case of temperature, as Jost puts it, it has on the one hand its
accelerating effect which tends to follow van’t Hoff’s law. But it has
also another and a cumulative effect upon the organism: “Sie schädigt
oder sie ermüdet ihn, und je höher sie steigt, desto rascher macht sie
die Schädigung geltend und desto schneller schreitet sie voran.” It
would seem to be this double effect of temperature in the case of the
organism which gives us our “optimum” curves, which are the expression,
accordingly, not of a primary phenomenon, but of a more or less complex
resultant. Moreover, as Blackman and others have pointed out, our
“optimum” temperature is very ill-defined until we take account also
of the _duration_ of our experiment; for obviously, a high temperature
may lead to a short, but exhausting, spell of rapid growth, while
the slower rate manifested at a lower temperature may be the best in
the end. {111} The mile and the hundred yards are won by different
runners; and maximum rate of working, and maximum amount of work done,
are two very different things[140].
――――――――――
In the case of maize, a certain series of experiments shewed that
the growth in length of the roots varied with the temperature as
follows[141]:
Temperature Growth in 48 hours
°C. mm.
18·0 1·1
23·5 10·8
26·6 29·6
28·5 26·5
30·2 64·6
33·5 69·5
36·5 20·7
Let us write our formula in the form
_V__{(_t_+_n_)}/_V__{_t_} = _x_^{_n_}.
Then choosing two values out of the above experimental series (say the
second and the second-last), we have _t_ = 23·5, _n_ = 10, and _V_,
_V′_ = 10·8 and 69·5 respectively.
Accordingly 69·5/10·8 = 6·4 = _x_^{10}.
Therefore (log 6·4)/10, or ·0806 = log _x_.
And, _x_ = 1·204 (for an interval of 1° C.).
This first approximation might be considerably improved by taking
account of all the experimental values, two only of which we have as
yet made use of; but even as it is, we see by Fig. 26 that it is in
very fair accordance with the actual results of observation, _within
those particular limits_ of temperature to which the experiment is
confined. {112}
For an experiment on _Lupinus albus_, quoted by Asa Gray[142], I have
worked out the corresponding coefficient, but a little more carefully.
Its value I find to be 1·16, or very nearly identical with that we have
just found for the maize; and the correspondence between the calculated
curve and the actual observations is now a close one.
[Illustration: Fig. 26. Relation of rate of growth to temperature in
Maize. Observed values (after Köppen), and calculated curve.]
Since the above paragraphs were written, new data have come to hand.
Miss I. Leitch has made careful observations of the rate of growth
of rootlets of the Pea; and I have attempted a further analysis of
her principal results[143]. In Fig. 27 are shewn the mean rates of
growth (based on about a hundred experiments) at some thirty-four
different temperatures between 0·8° and 29·3°, each experiment lasting
rather less than twenty-four hours. Working out the mean temperature
coefficient for a great many combinations of these values, I obtain
a value of 1·092 per C.°, or 2·41 for an interval of 10°, and a mean
value for the whole series showing a rate of growth of just about 1
mm. per hour at a temperature of 20°. My curve in Fig. 27 is drawn
from these determinations; and it will be seen that, while it is by
no means exact at the lower temperatures, and will of course fail
us altogether at very high {113} temperatures, yet it serves as a
very satisfactory guide to the relations between rate and temperature
within the ordinary limits of healthy growth. Miss Leitch holds that
the curve is _not_ a van’t Hoff curve; and this, in strict accuracy,
we need not dispute. But the phenomenon seems to me to be one into
which the van’t Hoff ratio enters largely, though doubtless combined
with other factors which we cannot at present determine or eliminate.
[Illustration: Fig. 27. Relation of rate of growth to temperature in
rootlets of Pea. (From Miss I. Leitch’s data.)]
While the above results conform fairly well to the law of the
temperature coefficient, it is evident that the imbibition of water
plays so large a part in the process of elongation of the root or stem
that the phenomenon is rather a physical than a chemical one: and
on this account, as Blackman has remarked, the data commonly given
for the rate of growth in plants are apt to be {114} irregular,
and sometimes (we might even say) misleading[144]. The fact also,
which we have already learned, that the elongation of a shoot tends
to proceed by jerks, rather than smoothly, is another indication
that the phenomenon is not purely and simply a chemical one. We have
abundant illustrations, however, among animals, in which we may study
the temperature coefficient under circumstances where, though the
phenomenon is always complicated by osmotic factors, true metabolic
growth or chemical combination plays a larger role. Thus Mlle. Maltaux
and Professor Massart[145] have studied the rate of division in a
certain flagellate, _Chilomonas paramoecium_, and found the process
to take 29 minutes at 15° C., 12 at 25°, and only 5 minutes at 35°
C. These velocities are in the ratio of 1 : 2·4 : 5·76, which ratio
corresponds precisely to a temperature coefficient of 2·4 for each rise
of 10°, or about 1·092 for each degree centigrade.
By means of this principle we may throw light on the apparently
complicated results of many experiments. For instance, Fig. 28 is an
illustration, which has been often copied, of O. Hertwig’s work on the
effect of temperature on the rate of development of the tadpole[146].
From inspection of this diagram, we see that the time taken to attain
certain stages of development (denoted by the numbers III–VII) was as
follows, at 20° and at 10° C., respectively.
At 20° At 10°
Stage III 2·0 6·5 days
Stage IV 2·7 8·1 days
Stage V 3·0 10·7 days
Stage VI 4·0 13·5 days
Stage VII 5·0 16·8 days
Total 16·7 55·6 days
That is to say, the time taken to produce a given result at {115} 10°
was (on the average) somewhere about 55·6/16·7, or 3·33, times as long
as was required at 20°.
[Illustration: Fig. 28. Diagram shewing time taken (in days), at
various temperatures (°C.), to reach certain stages of development
in the Frog: viz. I, gastrula; II, medullary plate; III, closure
of medullary folds; IV, tail-bud; V, tail and gills; VI, tail-fin;
VII, operculum beginning; VIII, do. closing; IX, first appearance of
hind-legs. (From Jenkinson, after O. Hertwig, 1898.)]
We may then put our equation again in the simple form, {116}
_x_^{10} = 3·33.
Or, 10 log _x_ = log 3·33 = ·52244.
Therefore log _x_ = ·05224,
and _x_ = 1·128.
That is to say, between the intervals of 10° and 20° C., if it take
_m_ days, at a certain given temperature, for a certain stage of
development to be attained, it will take _m_ × 1·128^{_n_} days, when
the temperature is _n_ degrees less, for the same stage to be arrived
at.
[Illustration: Fig. 29. Calculated values, corresponding to preceding
figure.]
Fig. 29 is calculated throughout from this value; and it will be seen
that it is extremely concordant with the original diagram, as regards
all the stages of development and the whole range of temperatures
shewn: in spite of the fact that the coefficient on which it is based
was derived by an easy method from a very few points in the original
curves. {117}
Karl Peter[147], experimenting chiefly on echinoderm eggs, and also
making use of Hertwig’s experiments on young tadpoles, gives the normal
temperature coefficients for intervals of 10° C. (commonly written
_Q__{10}) as follows.
Sphaerechinus 2·15,
Echinus 2·13,
Rana 2·86.
These values are not only concordant, but are evidently of the same
order of magnitude as the temperature-coefficient in ordinary chemical
reactions. Peter has also discovered the very interesting fact that
the temperature-coefficient alters with age, usually but not always
becoming smaller as age increases.
Sphaerechinus; Segmentation _Q_^{10} = 2·29,
Later stages _Q_^{10} = 2·03.
Echinus; Segmentation _Q_^{10} = 2·30,
Later stages _Q_^{10} = 2·08.
Rana; Segmentation _Q_^{10} = 2·23,
Later stages _Q_^{10} = 3·34.
Furthermore, the temperature coefficient varies with the temperature,
diminishing as the temperature rises,—a rule which van’t Hoff has shewn
to hold in ordinary chemical operations. Thus, in Rana the temperature
coefficient at low temperatures may be as high as 5·6: which is
just another way of saying that at low temperatures development is
exceptionally retarded.
――――――――――
In certain fish, such as plaice and haddock, I and others have found
clear evidence that the ascending curve of growth is subject to
seasonal interruptions, the rate during the winter months being always
slower than in the months of summer: it is as though we superimposed
a periodic, annual, sine-curve upon the continuous curve of growth.
And further, as growth itself grows less and less from year to year,
so will the difference between the winter and the summer rate also
grow less and less. The fluctuation in rate {118} will represent a
vibration which is gradually dying out; the amplitude of the sine-curve
will gradually diminish till it disappears; in short, our phenomenon is
simply expressed by what is known as a “damped sine-curve.” Exactly the
same thing occurs in man, though neither in his case nor in that of the
fish have we sufficient data for its complete illustration.
We can demonstrate the fact, however, in the case of man by the help
of certain very interesting measurements which have been recorded by
Daffner[148], of the height of German cadets, measured at half-yearly
intervals.
_Growth in height of German military Cadets, in half-yearly periods._
(_Daffner._)
Height in cent. Increment in cm.
Number ─────────────────────── Winter Summer
observed Age October April October ½-year ½-year Year
12 11–12 139·4 141·0 143·3 1·6 2·3 3·9
80 12–13 143·0 144·5 147·4 1·5 2·9 4·4
146 13–14 147·5 149·5 152·5 2·0 3·0 5·0
162 14–15 152·2 155·0 158·5 2·5 3·5 6·0
162 15–16 158·5 160·8 163·8 2·3 3·0 5·3
150 16–17 163·5 165·4 167·7 1·9 2·3 4·2
82 17–18 167·7 168·9 170·4 1·2 1·5 2·7
22 18–19 169·8 170·6 171·5 0·8 0·9 1·7
6 19–20 170·7 171·1 171·5 0·4 0·4 0·8
In the accompanying diagram (Fig. 30) the half-yearly increments are
set forth, from the above table, and it will be seen that they form two
even and entirely separate series. The curve joining up each series of
points is an acceleration-curve; and the comparison of the two curves
gives a clear view of the relative rates of growth during winter and
summer, and the fluctuation which these velocities undergo during the
years in question. The dotted line represents, approximately, the
acceleration-curve in its continuous fluctuation of alternate seasonal
decrease and increase.
――――――――――
In the case of trees, the seasonal fluctuations of growth[149] admit
{119} of easy determination, and it is a point of considerable
interest to compare the phenomenon in evergreen and in deciduous trees.
I happen to have no measurements at hand with which to make this
comparison in the case of our native trees, but from a paper by Mr
Charles E. Hall[150] I have compiled certain mean values for growth in
the climate of Uruguay.
[Illustration: Fig. 30. Half-yearly increments of growth, in cadets of
various ages. (From Daffner’s data.)]
_Mean monthly increase in Girth of Evergreen and Deciduous Trees,
at San Jorge, Uruguay._ (_After C. E. Hall._) _Values expressed as
percentages of total annual increase._
Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec.
Evergreens 9·1 8·8 8·6 8·9 7·7 5·4 4·3 6·0 9·1 11·1 10·8 10·2
Deciduous
trees 20·3 14·6 9·0 2·3 0·8 0·3 0·7 1·3 3·5 9·9 16·7 21·0
The measurements taken were those of the girth of the tree, in mm.,
at three feet from the ground. The evergreens included species of
Pinus, Eucalyptus and Acacia; the deciduous trees included Quercus,
Populus, Robinia and Melia. I have merely taken mean values for these
two groups, and expressed the monthly values as percentages of the mean
annual increase. The result (as shewn by Fig. 31) is very much what we
might have expected. The growth of the deciduous trees is completely
arrested in winter-time, and the arrest is all but complete over {120}
a considerable period of time; moreover, during the warm season, the
monthly values are regularly graded (approximately in a sine-curve)
with a clear maximum (in the southern hemisphere) about the month of
December. In the evergreen trees, on the other hand, the amplitude
of the periodic wave is very much less; there is a notable amount of
growth all the year round, and, while there is a marked diminution in
rate during the coldest months, there is a tendency towards equality
over a considerable part of the warmer season. It is probable that some
of the species examined, and especially the pines, were definitely
retarded in growth, either by a temperature above their optimum, or by
deficiency of moisture, during the hottest period of the year; with
the result that the seasonal curve in our diagram has (as it were) its
region of maximum cut off.
[Illustration: Fig. 31. Periodic annual fluctuation in rate of growth
of trees (in the southern hemisphere).]
In the case of trees, the seasonal periodicity of growth is so well
marked that we are entitled to make use of the phenomenon in a converse
way, and to draw deductions as to variations in {121} climate during
past years from the record of varying rates of growth which the tree,
by the thickness of its annual rings, has preserved for us. Mr. A. E.
Douglass, of the University of Arizona, has made a careful study of
this question[151], and I have received (through Professor H. H. Turner
of Oxford) some measurements of the average width of the successive
annual rings in “yellow pine,” 500 years old, from Arizona, in which
trees the annual rings are very clearly distinguished. From the year
1391 to 1518, the mean of two trees was used; from 1519 to 1912, the
mean of five; and the means of these, and sometimes of larger numbers,
were found to be very concordant. A correction was applied by drawing
a long, nearly straight line through the curve for the whole period,
which line was assumed to represent the slowly diminishing mean width
of ring accompanying the increase of size, or age, of the tree; and the
actual growth as measured was equated with this diminishing mean. The
figures used give, accordingly, the ratio of the actual growth in each
year to the mean growth corresponding to the age or magnitude of the
tree at that epoch.
It was at once manifest that the rate of growth so determined shewed a
tendency to fluctuate in a long period of between 100 and 200 years. I
then smoothed in groups of 100 (according to Gauss’s method) the yearly
values, so that each number thus found represented the mean annual
increase during a century: that is to say, the value ascribed to the
year 1500 represented the _average annual growth_ during the whole
period between 1450 and 1550, and so on. These values give us a curve
of beautiful and surprising smoothness, from which we seem compelled
to draw the direct conclusion that the climate of Arizona, during the
last 500 years, has fluctuated with a regular periodicity of almost
precisely 150 years. Here again we should be left in doubt (so far
as these {123} observations go) whether the essential factor be a
fluctuation of temperature or an alternation of moisture and aridity;
but the character of the Arizona climate, and the known facts of recent
years, encourage the belief that the latter is the more direct and more
important factor.
[Illustration: Fig. 32. Long-period fluctuation in rate of growth of
Arizona trees (smoothed in 100-year periods), from A.D. 1390–1490 to
A.D. 1810–1910.]
It has been often remarked that our common European trees, such for
instance as the elm or the cherry, tend to have larger leaves the
further north we go; but in this case the phenomenon is to be ascribed
rather to the longer hours of daylight than to any difference of
temperature[152]. The point is a physiological one, and consequently of
little importance to us here[153]; the main point for the morphologist
is the very simple one that physical or climatic conditions have
greatly influenced the rate of growth. The case is analogous to the
direct influence of temperature in modifying the colouration of
organisms, such as certain butterflies. Now if temperature affects the
rate of growth in strict uniformity, alike in all directions and in all
parts or organs, its direct effect must be limited to the production
of local races or varieties differing from one another in actual
magnitude, as the Siberian goldfinch or bullfinch, for instance, differ
from our own. But if there be even ever so little of a discriminating
action in the enhancement of growth by temperature, such that it
accelerates the growth of one tissue or one organ more than another,
then it is evident that it must at once lead to an actual difference of
racial, or even “specific” form.
It is not to be doubted that the various factors of climate have
some such discriminating influence. The leaves of our northern trees
may themselves be an instance of it; and we have, {124} probably, a
still better instance of it in the case of Alpine plants[154], whose
general habit is dwarfed, though their floral organs suffer little or
no reduction. The subject, however, has been little investigated, and
great as its theoretic importance would be to us, we must meanwhile
leave it alone.
_Osmotic factors in growth._
The curves of growth which we have now been studying represent
phenomena which have at least a two-fold interest, morphological and
physiological. To the morphologist, who recognises that form is a
“function” of growth, the important facts are mainly these: (1) that
the rate of growth is an orderly phenomenon, with general features
common to very various organisms, while each particular organism has
its own characteristic phenomena, or “specific constants”; (2) that
rate of growth varies with temperature, that is to say with season
and with climate, and with various other physical factors, external
and internal; (3) that it varies in different parts of the body,
and according to various directions or axes; such variations being
definitely correlated with one another, and thus giving rise to the
characteristic proportions, or form, of the organism, and to the
changes in form which it undergoes in the course of its development.
But to the physiologist, the phenomenon suggests many other important
considerations, and throws much light on the very nature of growth
itself, as a manifestation of chemical and physical energies.
To be content to shew that a certain rate of growth occurs in a certain
organism under certain conditions, or to speak of the phenomenon as
a “reaction” of the living organism to its environment or to certain
stimuli, would be but an example of that “lack of particularity[155]”
in regard to the actual mechanism of physical cause and effect with
which we are apt in biology to be too easily satisfied. But in the case
of rate of growth we pass somewhat {125} beyond these limitations; for
the affinity with certain types of chemical reaction is plain, and has
been recognised by a great number of physiologists.
A large part of the phenomenon of growth, both in animals and still
more conspicuously in plants, is associated with “turgor,” that is to
say, is dependent on osmotic conditions; in other words, the velocity
of growth depends in great measure (as we have already seen, p. 113)
on the amount of water taken up into the living cells, as well as on
the actual amount of chemical metabolism performed by them[156]. Of the
chemical phenomena which result in the actual increase of protoplasm we
shall speak presently, but the rôle of water in growth deserves also a
passing word, even in our morphological enquiry.
It has been shewn by Loeb that in Cerianthus or Tubularia, for
instance, the cells in order to grow must be turgescent; and this
turgescence is only possible so long as the salt water in which the
cells lie does not overstep a certain limit of concentration. The
limit, in the case of Tubularia, is passed when the salt amounts to
about 5·4 per cent. Sea-water contains some 3·0 to 3·5 p.c. of salts;
but it is when the salinity falls much below this normal, to about 2·2
p.c., that Tubularia exhibits its maximal turgescence, and maximal
growth. A further dilution is said to act as a poison to the animal.
Loeb has also shewn[157] that in certain eggs (e.g. those of the
little fish _Fundulus_) an increasing concentration of the sea-water
(leading to a diminishing “water-content” of the egg) retards the rate
of segmentation and at length renders segmentation impossible; though
nuclear division, by the way, goes on for some time longer.
Among many other observations of the same kind, those of
Bialaszewicz[158], on the early growth of the frog, are notable. He
shews that the growth of the embryo while still _within the {126}
vitelline membrane_ depends wholly on the absorption of water; that
whether rate of growth be fast or slow (in accordance with temperature)
the quantity of water absorbed is constant; and that successive changes
of form correspond to definite quantities of water absorbed. The
solid residue, as Davenport has also shewn, may actually and notably
diminish, while the embryo organism is increasing rapidly in bulk and
weight.
On the other hand, in later stages and especially in the higher
animals, the percentage of water tends to diminish. This has been shewn
by Davenport in the frog, by Potts in the chick, and particularly by
Fehling in the case of man[159]. Fehling’s results are epitomised as
follows:
Age in weeks 6 17 22 24 26 30 35 39
Percentage of water 97·5 91·8 92·0 89·9 86·4 83·7 82·9 74·2
And the following illustrate Davenport’s results for the frog:
Age in weeks 1 2 5 7 9 14 41 84
Percentage of water 56·3 58·5 76·7 89·3 93·1 95·0 90·2 87·5
To such phenomena of osmotic balance as the above, or in other words to
the dependence of growth on the uptake of water, Höber[160] and also
Loeb are inclined to refer the modifications of form which certain
phyllopod crustacea undergo, when the highly saline waters which
they inhabit are further concentrated, or are abnormally diluted.
Their growth, according to Schmankewitsch, is retarded by increase of
concentration, so that the individuals from the more saline waters
appear stunted and dwarfish; and they become altered or transformed
in other ways, which for the most part suggest “degeneration,” or
a failure to attain full and perfect development[161]. Important
physiological changes also ensue. The rate of multiplication is
increased, and parthenogenetic reproduction is encouraged. Male
individuals become plentiful in the less saline waters, and here
the females bring forth {127} their young alive; males disappear
altogether in the more concentrated brines, and then the females lay
eggs, which, however, only begin to develop when the salinity is
somewhat reduced.
The best-known case is the little “brine-shrimp,” _Artemia salina_,
found, in one form or another, all the world over, and first discovered
more than a century and a half ago in the salt-pans at Lymington. Among
many allied forms, one, _A. milhausenii_, inhabits the natron-lakes of
Egypt and Arabia, where, under the name of “loul,” or “Fezzan-worm,”
it is eaten by the Arabs[162]. This fact is interesting, because it
indicates (and investigation has apparently confirmed) that the tissues
of the creature are not impregnated with salt, as is the medium in
which it lives. The fluids of the body, the _milieu interne_ (as
Claude Bernard called them[163]), are no more salt than are those of
any ordinary crustacean or other animal, but contain only some 0·8 per
cent. of NaCl[164], while the _milieu externe_ may contain 10, 20, or
more per cent. of this and other salts; which is as much as to say
that the skin, or body-wall, of the creature acts as a “semi-permeable
membrane,” through which the dissolved salts are not permitted
to diffuse, though water passes through freely: until a statical
equilibrium (doubtless of a complex kind) is at length attained.
Among the structural changes which result from increased concentration
of the brine (partly during the life-time of the individual, but more
markedly during the short season which suffices for the development of
three or four, or perhaps more, successive generations), it is found
that the tail comes to bear fewer and fewer bristles, and the tail-fins
themselves tend at last to disappear; these changes corresponding
to what have been {128} described as the specific characters of _A.
milhausenii_, and of a still more extreme form, _A. köppeniana_;
while on the other hand, progressive dilution of the water tends to
precisely opposite conditions, resulting in forms which have also
been described as separate species, and even referred to a separate
genus, Callaonella, closely akin to Branchipus (Fig. 33). _Pari passu_
with these changes, there is a marked change in the relative lengths
of the fore and hind portions of the body, that is to say, of the
“cephalothorax” and abdomen: the latter growing relatively longer, the
salter the water. In other words, not only is the rate of growth of the
whole
[Illustration: Fig. 33. Brine-shrimps (Artemia), from more or less
saline water. Upper figures shew tail-segment and tail-fins; lower
figures, relative length of cephalothorax and abdomen. (After Abonyi.)]
animal lessened by the saline concentration, but the specific rates
of growth in the parts of its body are relatively changed. This latter
phenomenon lends itself to numerical statement, and Abonyi has lately
shewn that we may construct a very regular curve, by plotting the
proportionate length of the creature’s abdomen against the salinity,
or density, of the water; and the several species of Artemia, with all
their other correlated specific characters, are then found to occupy
successive, more or less well-defined, and more or less extended,
regions of the curve (Fig. 33). In short, the density of the water is
so clearly a “function” of the specific {129} character, that we may
briefly define the species _Artemia_ (_Callaonella_) _Jelskii_, for
instance, as the Artemia of density 1000–1010 (NaCl), or the typical
_A. salina_, or _principalis_, as the Artemia of density 1018–1025,
and so forth. It is a most interesting fact that these Artemiae, under
the protection of their semi-permeable skins, are capable of living
in waters not only of great density, but of very varied chemical
composition. The natron-lakes, for instance, contain large quantities
of magnesium
[Illustration: Fig. 34. Percentage ratio of length of abdomen to
cephalothorax in brine-shrimps, at various salinities. (After Abonyi.)]
sulphate; and the Artemiae continue to live equally well in
artificial solutions where this salt, or where calcium chloride, has
largely taken the place of sodium chloride in the more common habitat.
Furthermore, such waters as those of the natron-lakes are subject to
very great changes of chemical composition as concentration proceeds,
owing to the different solubilities of the constituent salts. It
appears that the forms which the Artemiae assume, and the changes which
they undergo, are identical or {130} indistinguishable, whichever of
the above salts happen to exist, or to predominate, in their saline
habitat. At the same time we still lack (so far as I know) the simple,
but crucial experiments which shall tell us whether, in solutions
of different chemical composition, it is _at equal densities_, or
at “_isotonic_” concentrations (that is to say, under conditions
where the osmotic pressure, and consequently the rate of diffusion,
is identical), that the same structural changes are produced, or
corresponding phases of equilibrium attained.
While Höber and others[165] have referred all these phenomena to
osmosis, Abonyi is inclined to believe that the viscosity, or
mechanical resistance, of the fluid also reacts upon the organism; and
other possible modes of operation have been suggested. But we may take
it for certain that the phenomenon as a whole is not a simple one;
and that it includes besides the passive phenomena of intermolecular
diffusion, some other form of activity which plays the part of a
regulatory mechanism[166].
_Growth and catalytic action._
In ordinary chemical reactions we have to deal (1) with a specific
velocity proper to the particular reaction, (2) with variations due
to temperature and other physical conditions, (3) according to van’t
Hoff’s “Law of Mass,” with variations due to the actual quantities
present of the reacting substances, and (4) in certain cases, with
variations due to the presence of “catalysing agents.” In the simpler
reactions, the law of mass involves a steady, gradual slowing-down of
the process, according to a logarithmic ratio, as the reaction proceeds
and as the initial amount of substance diminishes; a phenomenon,
however, which need not necessarily {131} occur in the organism, part
of whose energies are devoted to the continual bringing-up of fresh
supplies.
Catalytic action occurs when some substance, often in very minute
quantity, is present, and by its presence produces or accelerates an
action, by opening “a way round,” without the catalytic agent itself
being diminished or used up[167]. Here the velocity curve, though
quickened, is not necessarily altered in form, for gradually the law
of mass exerts its effect and the rate of the reaction gradually
diminishes. But in certain cases we have the very remarkable phenomenon
that a body acting as a catalyser is necessarily formed as a product,
or bye-product, of the main reaction, and in such a case as this the
reaction-velocity will tend to be steadily accelerated. Instead of
dwindling away, the reaction will continue with an ever-increasing
velocity: always subject to the reservation that limiting conditions
will in time make themselves felt, such as a failure of some necessary
ingredient, or a development of some substance which shall antagonise
or finally destroy the original reaction. Such an action as this we
have learned, from Ostwald, to describe as “autocatalysis.” Now we know
that certain products of protoplasmic metabolism, such as the enzymes,
are very powerful catalysers, and we are entitled to speak of an
autocatalytic action on the part of protoplasm itself. This catalytic
activity of protoplasm is a very important phenomenon. As Blackman
says, in the address already quoted, the botanists (or the zoologists)
“call it _growth_, attribute it to a specific power of protoplasm for
assimilation, and leave it alone as a fundamental phenomenon; but they
are much concerned as to the distribution of new growth in innumerable
specifically distinct forms.” While the chemist, on the other hand,
recognises it as a familiar phenomenon, and refers it to the same
category as his other known examples of _autocatalysis_. {132}
This very important, and perhaps even fundamental phenomenon of growth
would seem to have been first recognised by Professor Chodat of Geneva,
as we are told by his pupil Monnier[168]. “On peut bien, ainsi que
M. Chodat l’a proposé, considérer l’accroissement comme une réaction
chimique complexe, dans laquelle le catalysateur est la cellule
vivante, et les corps en présence sont l’eau, les sels, et l’acide
carbonique.”
Very soon afterwards a similar suggestion was made by Loeb[169], in
connection with the synthesis of _nuclein_ or nuclear protoplasm;
for he remarked that, as in an autocatalysed chemical reaction, the
velocity of the synthesis increases during the initial stage of
cell-division in proportion to the amount of nuclear matter already
synthesised. In other words, one of the products of the reaction, i.e.
one of the constituents of the nucleus, accelerates the production of
nuclear from cytoplasmic material.
The phenomenon of autocatalysis is by no means confined to living or
protoplasmic chemistry, but at the same time it is characteristically,
and apparently constantly, associated therewith. And it would seem
that to it we may ascribe a considerable part of the difference
between the growth of the organism and the simpler growth of the
crystal[170]: the fact, for instance, that the cell can grow in a
very low concentration of its nutritive solution, while the crystal
grows only in a supersaturated one; and the fundamental fact that the
nutritive solution need only contain the more or less raw materials of
the complex constituents of the cell, while the crystal grows only in a
solution of its own actual substance[171].
As F. F. Blackman has pointed out, the multiplication of an organism,
for instance the prodigiously rapid increase of a bacterium, {133}
which tends to double its numbers every few minutes, till (were it
not for limiting factors) its numbers would be all but incalculable
in a day[172], is a simple but most striking illustration of the
potentialities of protoplasmic catalysis; and (apart from the large
share taken by mere “turgescence” or imbibition of water) the same is
true of the growth, or cell-multiplication, of a multicellular organism
in its first stage of rapid acceleration.
It is not necessary for us to pursue this subject much further, for it
is sufficiently clear that the normal “curve of growth” of an organism,
in all its general features, very closely resembles the velocity-curve
of chemical autocatalysis. We see in it the first and most typical
phase of greater and greater acceleration; this is followed by a phase
in which limiting conditions (whose details are practically unknown)
lead to a falling off of the former acceleration; and in most cases
we come at length to a third phase, in which retardation of growth
is succeeded by actual diminution of mass. Here we may recognise the
influence of processes, or of products, which have become actually
deleterious; their deleterious influence is staved off for a while,
as the organism draws on its accumulated reserves, but they lead ere
long to the stoppage of all activity, and to the physical phenomenon
of death. But when we have once admitted that the limiting conditions
of growth, which cause a phase of retardation to follow a phase of
acceleration, are very imperfectly known, it is plain that, _ipso
facto_, we must admit that a resemblance rather than an identity
between this phenomenon and that of chemical autocatalysis is all that
we can safely assert meanwhile. Indeed, as Enriques has shewn, points
of contrast between the two phenomena are not lacking; for instance, as
the chemical reaction draws to a close, it is by the gradual attainment
of chemical equilibrium: but when organic growth draws to a close, it
is by reason of a very different kind of equilibrium, due in the main
to the gradual differentiation of the organism into parts, among whose
peculiar {134} and specialised functions that of cell-multiplication
tends to fall into abeyance[173].
It would seem to follow, as a natural consequence, from what has been
said, that we could without much difficulty reduce our curves of
growth to logarithmic formulae[174] akin to those which the physical
chemist finds applicable to his autocatalytic reactions. This has
been diligently attempted by various writers[175]; but the results,
while not destructive of the hypothesis itself, are only partially
successful. The difficulty arises mainly from the fact that, in the
life-history of an organism, we have usually to deal (as indeed we
have seen) with several recurrent periods of relative acceleration
and retardation. It is easy to find a formula which shall satisfy the
conditions during any one of these periodic phases, but it is very
difficult to frame a comprehensive formula which shall apply to the
entire period of growth, or to the whole duration of life.
But if it be meanwhile impossible to formulate or to solve in precise
mathematical terms the equation to the growth of an organism, we have
yet gone a very long way towards the solution of such problems when we
have found a “qualitative expression,” as Poincaré puts it; that is to
say, when we have gained a fair approximate knowledge of the general
curve which represents the unknown function.
――――――――――
As soon as we have touched on such matters as the chemical phenomenon
of catalysis, we are on the threshold of a subject which, if we were
able to pursue it, would soon lead us far into the special domain of
physiology; and there it would be necessary to follow it if we were
dealing with growth as a phenomenon in itself, instead of merely as a
help to our study and comprehension of form. For instance the whole
question of _diet_, of overfeeding and underfeeding, would present
itself for discussion[176]. But without attempting to open up this
large subject, we may say a {135} further passing word upon the
essential fact that certain chemical substances have the power of
accelerating or of retarding, or in some way regulating, growth, and of
so influencing directly the morphological features of the organism.
Thus lecithin has been shewn by Hatai[177], Danilewsky[178] and others
to have a remarkable power of stimulating growth in various animals;
and the so-called “auximones,” which Professor Bottomley prepares
by the action of bacteria upon peat appear to be, after a somewhat
similar fashion, potent accelerators of the growth of plants. But by
much the most interesting cases, from our point of view, are those
where a particular substance appears to exert a differential effect,
stimulating the growth of one part or organ of the body more than
another.
It has been known for a number of years that a diseased condition of
the pituitary body accompanies the phenomenon known as “acromegaly,”
in which the bones are variously enlarged or elongated, and which is
more or less exemplified in every skeleton of a “giant”; while on the
other hand, disease or extirpation of the thyroid causes an arrest of
skeletal development, and, if it take place early, the subject remains
a dwarf. These, then, are well-known illustrations of the regulation of
function by some internal glandular secretion, some enzyme or “hormone”
(as Bayliss and Starling call it), or “harmozone,” as Gley calls it in
the particular case where the function regulated is that of growth,
with its consequent influence on form.
Among other illustrations (which are plentiful) we have, for instance
the growth of the placental decidua, which Loeb has shewn to be due
to a substance given off by the corpus luteum of the ovary, giving to
the uterine tissues an abnormal capacity for growth, which in turn is
called into action by the contact of the ovum, or even of any foreign
body. And various sexual characters, such as the plumage, comb and
spurs of the cock, are believed in like manner to arise in response to
some particular internal secretion. When the source of such a secretion
is removed by castration, well-known morphological changes take place
in various animals; and when a converse change takes place, the female
acquires, in greater or less degree, characters which are {136} proper
to the male, as in certain extreme cases, known from time immemorial,
when late in life a hen assumes the plumage of the cock.
There are some very remarkable experiments by Gudernatsch, in which
he has shewn that by feeding tadpoles (whether of frogs or toads) on
thyroid gland substance, their legs may be made to grow out at any
time, days or weeks before the normal date of their appearance[179]. No
other organic food was found to produce the same effect; but since the
thyroid gland is known to contain iodine[180], Morse experimented with
this latter substance, and found that if the tadpoles were fed with
iodised amino-acids the legs developed precociously, just as when the
thyroid gland itself was used. We may take it, then, as an established
fact, whose full extent and bearings are still awaiting investigation,
that there exist substances both within and without the organism which
have a marvellous power of accelerating growth, and of doing so in such
a way as to affect not only the size but the form or proportions of the
organism.
――――――――――
If we once admit, as we are now bound to do, the existence of such
factors as these, which, by their physiological activity and apart from
any direct action of the nervous system, tend towards the acceleration
of growth and consequent modification of form, we are led into wide
fields of speculation by an easy and a legitimate pathway. Professor
Gley carries such speculations a long, long way: for he says[181] that
by these chemical influences “Toute une partie de la construction des
êtres parait s’expliquer d’une façon toute mécanique. La forteresse,
si longtemps inaccessible, du vitalisme est entamée. Car la notion
morphogénique était, suivant le mot de Dastre[182], comme ‘le dernier
réduit de la force vitale.’ ”
The physiological speculations we need not discuss: but, to take a
single example from morphology, we begin to understand the possibility,
and to comprehend the probable meaning, of the {137} all but sudden
appearance on the earth of such exaggerated and almost monstrous
forms as those of the great secondary reptiles and the great tertiary
mammals[183]. We begin to see that it is in order to account, not for
the appearance, but for the disappearance of such forms as these that
natural selection must be invoked. And we then, I think, draw near to
the conclusion that what is true of these is universally true, and that
the great function of natural selection is not to originate, but to
remove: _donec ad interitum genus id natura redegit_[184].
The world of things living, like the world of things inanimate, grows
of itself, and pursues its ceaseless course of creative evolution.
It has room, wide but not unbounded, for variety of living form and
structure, as these tend towards their seemingly endless, but yet
strictly limited, possibilities of permutation and degree: it has room
for the great and for the small, room for the weak and for the strong.
Environment and circumstance do not always make a prison, wherein
perforce the organism must either live or die; for the ways of life may
be changed, and many a refuge found, before the sentence of unfitness
is pronounced and the penalty of extermination paid. But there comes a
time when “variation,” in form, dimensions, or other qualities of the
organism, goes farther than is compatible with all the means at hand of
health and welfare for the individual and the stock; when, under the
active and creative stimulus of forces from within and from without,
the active and creative energies of growth pass the bounds of physical
and physiological equilibrium: and so reach the limits which, as again
Lucretius tells us, natural law has set between what may and what may
not be,
“et quid quaeque queant per foedera naturai
quid porro nequeant.”
Then, at last, we are entitled to use the customary metaphor, and to
see in natural selection an inexorable force, whose function {138} is
not to create but to destroy,—to weed, to prune, to cut down and to
cast into the fire[185].
_Regeneration, or growth and repair._
The phenomenon of regeneration, or the restoration of lost or
amputated parts, is a particular case of growth which deserves
separate consideration. As we are all aware, this property is
manifested in a high degree among invertebrates and many cold-blooded
vertebrates, diminishing as we ascend the scale, until at length, in
the warm-blooded animals, it lessens down to no more than that _vis
medicatrix_ which heals a wound. Ever since the days of Aristotle, and
especially since the experiments of Trembley, Réaumur and Spallanzani
in the middle of the eighteenth century, the physiologist and the
psychologist have alike recognised that the phenomenon is both
perplexing and important. The general phenomenon is amply discussed
elsewhere, and we need only deal with it in its immediate relation to
growth[186].
Regeneration, like growth in other cases, proceeds with a velocity
which varies according to a definite law; the rate varies with the
time, and we may study it as velocity and as acceleration.
Let us take, as an instance, Miss M. L. Durbin’s measurements of the
rate of regeneration of tadpoles’ tails: the rate being here measured
in terms, not of mass, but of length, or longitudinal increment[187].
From a number of tadpoles, whose average length was 34·2 mm., their
tails being on an average 21·2 mm. long, about half the tail {139}
(11·5 mm.) was cut off, and the amounts regenerated in successive
periods are shewn as follows:
Days after operation 3 7 10 14 18 24 30
(1) Amount regenerated in mm. 1·4 3·4 4·3 5·2 5·5 6·2 6·5
(2) Increment during each period 1·4 2·0 0·9 0·9 0·3 0·7 0·3
(3) (?) Rate per day during
each period 0·46 0·50 0·30 0·25 0·07 0·12 0·05
The first line of numbers in this table, if plotted as a curve against
the number of days, will give us a very satisfactory view of the “curve
of growth” within the period of the observations: that is to say, of
the successive relations of length to time, or the _velocity_ of the
process. But the third line is not so satisfactory, and must not be
plotted directly as an acceleration curve. For it is evident that
the “rates” here determined do not correspond to velocities _at_ the
dates to which they are referred, but are the mean velocities over a
preceding period; and moreover the periods over which these means are
taken are here of very unequal length. But we may draw a good deal
more information from this experiment, if we begin by drawing a smooth
curve, as nearly as possible through the points corresponding to the
amounts regenerated (according to the first line of the table); and if
we then interpolate from this smooth curve the actual lengths attained,
day by day, and derive from these, by subtraction, the successive
daily increments, which are the measure of the daily mean _velocities_
(Table, p. 141). (The more accurate and strictly correct method would
be to draw successive tangents to the curve.)
In our curve of growth (Fig. 35) we cannot safely interpolate values
for the first three days, that is to say for the dates between
amputation and the first actual measurement of the regenerated part.
What goes on in these three days is very important; but we know
nothing about it, save that our curve descended to zero somewhere or
other within that period. As we have already learned, we can more or
less safely interpolate between known points, or actual observations;
but here we have no known starting-point. In short, for all that the
observations tell us, and for all that the appearance of the curve
can suggest, the curve of growth may have descended evenly to the
base-line, which it would then have reached about the end of the second
{140} day; or it may have had within the first three days a change of
direction, or “point of inflection,” and may then have sprung at once
from the base-line at zero. That is to say, there may
[Illustration: Fig. 35. Curve of regenerative growth in tadpoles’
tails. (From M. L. Durbin’s data.)]
have been an intervening “latent period,” during which no growth
[Illustration: Fig. 36. Mean daily increments, corresponding to Fig.
35.]
{141}
occurred, between the time of injury and the first measurement
of regenerative growth; or, for all we yet know, regeneration may
have begun at once, but with a velocity much less than that which
it afterwards attained. This apparently trifling difference would
correspond to a very great difference in the nature of the phenomenon,
and would lead to a very striking difference in the curve which we have
next to draw.
The curve already drawn (Fig. 35) illustrates, as we have seen, the
relation of length to time, i.e. _L_/_T_ = _V_. The second (Fig. 36)
represents the rate of change of velocity; it sets _V_ against _T_;
_The foregoing table, extended by graphic interpolation._
Total Daily
Days increment increment Logs of do.
1 —
— —
2 —
— —
3 1·40
·60 1·78
4 2·00
·52 1·72
5 2·52
·45 1·65
6 2·97
·43 1·63
7 3·40
·32 1·51
8 3·72
·30 1·48
9 4·02
·28 1·45
10 4·30
·22 1·34
11 4·52
·21 1·32
12 4·73
·19 1·28
13 4·92
·18 1·26
14 5·10
·17 1·23
15 5·27
·13 1·11
16 5·40
·14 1·15
17 5·54
·13 1·11
18 5·67
·11 1·04
19 5·78
·10 1·00
20 5·88
·10 1·00
21 5·98
·09 ·95
22 6·07
·07 ·85
23 6·14
·07 ·84
24 6·21
·08 ·90
25 6·29
·06 ·78
26 6·35
·06 ·78
27 6·41
·05 ·70
28 6·46
·04 ·60
29 6·50
·03 ·48
30 6·53
{142}
and _V_/_T_ or _L_/_T_^2, represents (as we have learned) the
_acceleration_ of growth, this being simply the “differential
coefficient,” the first derivative of the former curve.
[Illustration: Fig. 37. Logarithms of values shewn in Fig. 36.]
Now, plotting this acceleration curve from the date of the first
measurement made three days after the amputation of the tail (Fig.
36), we see that it has no point of inflection, but falls steadily,
only more and more slowly, till at last it comes down nearly to the
base-line. The velocities of growth are continually diminishing. As
regards the missing portion at the beginning of the curve, we cannot
be sure whether it bent round and came down to zero, or whether, as
in our ordinary acceleration curves of growth from birth onwards, it
started from a maximum. The former is, in this case, obviously the more
probable, but we cannot be sure.
As regards that large portion of the curve which we are acquainted
with, we see that it resembles the curve known as a rectangular
hyperbola, which is the form assumed when two variables (in this case
_V_ and _T_) vary inversely as one another. If we take the logarithms
of the velocities (as given in the table) and plot them against time
(Fig. 37), we see that they fall, approximately, into a straight line;
and if this curve be plotted on the {143} proper scale we shall find
that the angle which it makes with the base is about 25°, of which the
tangent is ·46, or in round numbers ½.
Had the angle been 45° (tan 45° = 1), the curve would have been
actually a rectangular hyperbola, with _V_ _T_ = constant. As it is,
we may assume, provisionally, that it belongs to the same family
of curves, so that _V_^{_m_} _T_^{_n_}, or _V_^{_m_/_n_} _T_, or
_V_ _T_^{_n_/_m_}, are all severally constant. In other words, the
velocity varies inversely as some power of the time, or _vice versa_.
And in this particular case, the equation _V_ _T_^2 = constant, holds
very nearly true; that is to say the velocity varies, or tends to vary,
inversely as the square of the time. If some general law akin to this
could be established as a general law, or even as a common rule, it
would be of great importance.
[Illustration: Fig. 38. Rate of regenerative growth in larger tadpoles.]
But though neither in this case nor in any other can the minute
increments of growth during the first few hours, or the first couple
of days, after injury, be directly measured, yet the most important
point is quite capable of solution. What the foregoing curve leaves
us in ignorance of, is simply whether growth starts at zero, with
zero velocity, and works up quickly to a maximum velocity from which
it afterwards gradually falls away; or whether after a latent period,
it begins, so to speak, in full force. The answer {144} to this
question-depends on whether, in the days following the first actual
measurement, we can or cannot detect a daily _increment_ in velocity,
before that velocity begins its normal course of diminution. Now
this preliminary ascent to a maximum, or point of inflection of the
curve, though not shewn in the above-quoted experiment, has been often
observed: as for instance, in another similar experiment by the author
of the former, the tadpoles being in this case of larger size (average
49·1 mm.)[188].
Days 3 5 7 10 12 14 17 24 28 31
Increment 0·86 2·15 3·66 5·20 5·95 6·38 7·10 7·60 8·20 8·40
Or, by graphic interpolation,
Total Daily
Days increment do.
1 ·23 ·23
2 ·53 ·30
3 ·86 ·33
4 1·30 ·44
5 2·00 ·70
6 2·78 ·78
7 3·58 ·80
8 4·30 ·72
9 4·90 ·60
10 5·29 ·39
11 5·62 ·33
12 5·90 ·28
13 6·13 ·23
14 6·38 ·25
15 6·61 ·23
16 6·81 ·20
17 7·00 ·19 etc.
The acceleration curve is drawn in Fig. 39.
Here we have just what we lacked in the former case, namely a visible
point of inflection in the curve about the seventh day (Figs. 38,
39), whose existence is confirmed by successive observations on the
3rd, 5th, 7th and 10th days, and which justifies to some extent our
extrapolation for the otherwise unknown period up to and ending with
the third day; but even here there is a short space near the very
beginning during which we are not quite sure of the precise slope of
the curve.
――――――――――
We have now learned that, according to these experiments, with which
many others are in substantial agreement, the rate of growth in the
regenerative process is as follows. After a very short latent period,
not yet actually proved but whose existence is highly probable, growth
commences with a velocity which very {145} rapidly increases to a
maximum. The curve quickly,—almost suddenly,—changes its direction,
as the velocity begins to fall; and the rate of fall, that is, the
negative acceleration, proceeds at a slower and slower rate, which rate
varies inversely as some power of the time, and is found in both of the
above-quoted experiments to be very approximately as 1/_T_^2. But it
is obvious that the value which we have found for the latter portion
of the curve (however closely it be conformed to) is only an empirical
value; it has only a temporary usefulness, and must in time give place
to a formula which shall represent the entire phenomenon, from start to
finish.
[Illustration: Fig. 39. Daily increment, or amount regenerated,
corresponding to Fig. 38.]
While the curve of regenerative growth is apparently different from
the curve of ordinary growth as usually drawn (and while this apparent
difference has been commented on and treated as valid by certain
writers) we are now in a position to see that it only looks different
because we are able to study it, if not from the beginning, at least
very nearly so: while an ordinary curve of growth, as it is usually
presented to us, is one which dates, not {146} from the beginning of
growth, but from the comparatively late, and unimportant, and even
fallacious epoch of birth. A complete curve of growth, starting from
zero, has the same essential characteristics as the regeneration curve.
Indeed the more we consider the phenomenon of regeneration, the more
plainly does it shew itself to us as but a particular case of the
general phenomenon of growth[189], following the same lines, obeying
the same laws, and merely started into activity by the special
stimulus, direct or indirect, caused by the infliction of a wound.
Neither more nor less than in other problems of physiology are we
called upon, in the case of regeneration, to indulge in metaphysical
speculation, or to dwell upon the beneficent purpose which seemingly
underlies this process of healing and restoration.
――――――――――
It is a very general rule, though apparently not a universal one, that
regeneration tends to fall somewhat short of a _complete_ restoration
of the lost part; a certain percentage only of the lost tissues is
restored. This fact was well known to some of those old investigators,
who, like the Abbé Trembley and like Voltaire, found a fascination in
the study of artificial injury and the regeneration which followed
it. Sir John Graham Dalyell, for instance, says, in the course of
an admirable paragraph on regeneration[190]: “The reproductive
faculty ... is not confined to one portion, but may extend over many;
and it may ensue even in relation to the regenerated portion more than
once. Nevertheless, the faculty gradually weakens, so that in general
every successive regeneration is smaller and more imperfect than the
organisation preceding it; and at length it is exhausted.”
In certain minute animals, such as the Infusoria, in which the
capacity for “regeneration” is so great that the entire animal may be
restored from the merest fragment, it becomes of great interest to
discover whether there be some definite size at which the fragment
ceases to display this power. This question has {147} been studied by
Lillie[191], who found that in Stentor, while still smaller fragments
were capable of surviving for days, the smallest portions capable
of regeneration were of a size equal to a sphere of about 80 µ in
diameter, that is to say of a volume equal to about one twenty-seventh
of the average entire animal. He arrives at the remarkable conclusion
that for this, and for all other species of animals, there is a
“minimal organisation mass,” that is to say a “minimal mass of definite
size consisting of nucleus and cytoplasm within which the organisation
of the species can just find its latent expression.” And in like
manner, Boveri[192] has shewn that the fragment of a sea-urchin’s
egg capable of growing up into a new embryo, and so discharging the
complete functions of an entire and uninjured ovum, reaches its limit
at about one-twentieth of the original egg,—other writers having found
a limit at about one-fourth. These magnitudes, small as they are,
represent objects easily visible under a low power of the microscope,
and so stand in a very different category to the minimal magnitudes in
which life itself can be manifested, and which we have discussed in
chapter II.
A number of phenomena connected with the linear rate of regeneration
are illustrated and epitomised in the accompanying diagram (Fig. 40),
which I have constructed from certain data given by Ellis in a paper
on the relation of the amount of tail _regenerated_ to the amount
_removed_, in Tadpoles. These data are summarised in the next Table.
The tadpoles were all very much of a size, about 40 mm.; the average
length of tail was very near to 26 mm., or 65 per cent. of the whole
body-length; and in four series of experiments about 10, 20, 40 and 60
per cent. of the tail were severally removed. The amount regenerated in
successive intervals of three days is shewn in our table. By plotting
the actual amounts regenerated against these three-day intervals of
time, we may interpolate values for the time taken to regenerate
definite percentage amounts, 5 per cent., 10 per cent., etc. of the
{148}
[Illustration: Fig. 40. Relation between the percentage amount of
tail removed, the percentage restored, and the time required for its
restoration. (From M. M. Ellis’s data.)]
_The Rate of Regenerative Growth in Tadpoles’ Tails._
(_After M. M. Ellis, J. Exp. Zool._ VII, _p._ 421, 1909.)
Body Tail Amount Per cent. % amount regenerated in days
length length removed of tail ────────────────────────────
Series† mm. mm. mm. removed 3 6 9 12 15 18 32
_O_ 39·575 25·895 3·2 12·36 13 31 44 44 44 44 44
_P_ 40·21 26·13 5·28 20·20 10 29 40 44 44 44 44
_R_ 39·86 25·70 10·4 40·50 6 20 31 40 48 48 48
_S_ 40·34 26·11 14·8 56·7 0 16 33 39 45 48 48
† Each series gives the mean of 20 experiments.
amount removed; and my diagram is constructed from the four sets of
values thus obtained, that is to say from the four sets of experiments
which differed from one another in the amount of tail amputated.
To these we have to add the general result of a fifth series of
experiments, which shewed that when as much as 75 per cent. of the
tail was cut off, no regeneration took place at all, but the animal
presently died. In our diagram, then, each {149} curve indicates the
time taken to regenerate _n_ per cent. of the amount removed. All the
curves converge towards infinity, when the amount removed (as shewn by
the ordinate) approaches 75 per cent.; and all of the curves start from
zero, for nothing is regenerated where nothing had been removed. Each
curve approximates in form to a cubic parabola.
The amount regenerated varies also with the age of the tadpole and
with other factors, such as temperature; in other words, for any given
age, or size, of tadpole and also for various specific temperatures, a
similar diagram might be constructed.
――――――――――
The power of reproducing, or regenerating, a lost limb is particularly
well developed in arthropod animals, and is sometimes accompanied by
remarkable modification of the form of the regenerated limb. A case in
point, which has attracted much attention, occurs in connection with
the claws of certain Crustacea[193].
In many Crustacea we have an asymmetry of the great claws, one being
larger than the other and also more or less different in form. For
instance, in the common lobster, one claw, the larger of the two, is
provided with a few great “crushing” teeth, while the smaller claw
has more numerous teeth, small and serrated. Though Aristotle thought
otherwise, it appears that the crushing-claw may be on the right or
left side, indifferently; whether it be on one or the other is a
problem of “chance.” It is otherwise in many other Crustacea, where the
larger and more powerful claw is always left or right, as the case may
be, according to the species: where, in other words, the “probability”
of the large or the small claw being left or being right is tantamount
to certainty[194].
The one claw is the larger because it has grown the faster; {150} it
has a higher “coefficient of growth,” and accordingly, as age advances,
the disproportion between the two claws becomes more and more evident.
Moreover, we must assume that the characteristic form of the claw is a
“function” of its magnitude; the knobbiness is a phenomenon coincident
with growth, and we never, under any circumstances, find the smaller
claw with big crushing teeth and the big claw with little serrated
ones. There are many other somewhat similar cases where size and form
are manifestly correlated, and we have already seen, to some extent,
that the phenomenon of growth is accompanied by certain ratios of
velocity that lead inevitably to changes of form. Meanwhile, then, we
must simply assume that the essential difference between the two claws
is one of magnitude, with which a certain differentiation of form is
inseparably associated.
If we amputate a claw, or if, as often happens, the crab “casts it
off,” it undergoes a process of regeneration,—it grows anew, and
evidently does so with an accelerated velocity, which acceleration
will cease when equilibrium of the parts is once more attained: the
accelerated velocity being a case in point to illustrate that _vis
revulsionis_ of Haller, to which we have already referred.
With the help of this principle, Przibram accounts for certain
curious phenomena which accompany the process of regeneration. As his
experiments and those of Morgan shew, if the large or knobby claw (_A_)
be removed, there are certain cases, e.g. the common lobster, where it
is directly regenerated. In other cases, e.g. Alpheus[195], the other
claw (_B_) assumes the size and form of that which was amputated, while
the latter regenerates itself in the form of the other and weaker one;
_A_ and _B_ have apparently changed places. In a third case, as in the
crabs, the _A_-claw regenerates itself as a small or _B_-claw, but the
_B_-claw remains for a time unaltered, though slowly and in the course
of repeated moults it later on assumes the large and heavily toothed
_A_-form.
Much has been written on this phenomenon, but in essence it is
very simple. It depends upon the respective rates of growth, upon a
ratio between the rate of regeneration and the rate of growth of the
uninjured limb: complicated a little, however, by {151} the possibility
of the uninjured limb growing all the faster for a time after the
animal has been relieved of the other. From the time of amputation,
say of _A_, _A_ begins to grow from zero, with a high “regenerative”
velocity; while _B_, starting from a definite magnitude, continues to
increase, with its normal or perhaps somewhat accelerated velocity. The
ratio between the two velocities of growth will determine whether, by a
given time, _A_ has equalled, outstripped, or still fallen short of the
magnitude of _B_.
That this is the gist of the whole problem is confirmed (if
confirmation be necessary) by certain experiments of Wilson’s. It
is known that by section of the nerve to a crab’s claw, its growth
is retarded, and as the general growth of the animal proceeds the
claw comes to appear stunted or dwarfed. Now in such a case as that
of Alpheus, we have seen that the rate of regenerative growth in an
amputated large claw fails to let it reach or overtake the magnitude of
the growing little claw: which latter, in short, now appears as the big
one. But if at the same time as we amputate the big claw we also sever
the nerve to the lesser one, we so far slow down the latter’s growth
that the other is able to make up to it, and in this case the two
claws continue to grow at approximately equal rates, or in other words
continue of coequal size.
――――――――――
The phenomenon of regeneration goes some way towards helping us to
comprehend the phenomenon of “multiplication by fission,” as it is
exemplified at least in its simpler cases in many worms and worm-like
animals. For physical reasons which we shall have to study in another
chapter, there is a natural tendency for any tube, if it have the
properties of a fluid or semi-fluid substance, to break up into
segments after it comes to a certain length; and nothing can prevent
its doing so, except the presence of some controlling force, such for
instance as may be due to the pressure of some external support, or
some superficial thickening or other intrinsic rigidity of its own
substance. If we add to this natural tendency towards fission of a
cylindrical or tubular worm, the ordinary phenomenon of regeneration,
we have all that is essentially implied in “reproduction by fission.”
And in so far {152} as the process rests upon a physical principle, or
natural tendency, we may account for its occurrence in a great variety
of animals, zoologically dissimilar; and also for its presence here
and absence there, in forms which, though materially different in a
physical sense, are zoologically speaking very closely allied.
CONCLUSION AND SUMMARY.
But the phenomena of regeneration, like all the other phenomena
of growth, soon carry us far afield, and we must draw this brief
discussion to a close.
For the main features which appear to be common to all curves of growth
we may hope to have, some day, a physical explanation. In particular we
should like to know the meaning of that point of inflection, or abrupt
change from an increasing to a decreasing velocity of growth which all
our curves, and especially our acceleration curves, demonstrate the
existence of, provided only that they include the initial stages of the
whole phenomenon: just as we should also like to have a full physical
or physiological explanation of the gradually diminishing velocity
of growth which follows, and which (though subject to temporary
interruption or abeyance) is on the whole characteristic of growth in
all cases whatsoever. In short, the characteristic form of the curve
of growth in length (or any other linear dimension) is a phenomenon
which we are at present unable to explain, but which presents us with
a definite and attractive problem for future solution. It would seem
evident that the abrupt change in velocity must be due, either to a
change in that pressure outwards from within, by which the “forces of
growth” make themselves manifest, or to a change in the resistances
against which they act, that is to say the _tension_ of the surface;
and this latter force we do not by any means limit to “surface-tension”
proper, but may extend to the development of a more or less resistant
membrane or “skin,” or even to the resistance of fibres or other
histological elements, binding the boundary layers to the parts within.
I take it that the sudden arrest of velocity is much more likely to be
due to a sudden increase of resistance than to a sudden diminution of
internal energies: in other words, I suspect that it is coincident with
some notable event of histological differentiation, such as {153} the
rapid formation of a comparatively firm skin; and that the dwindling
of velocities, or the negative acceleration, which follows, is the
resultant or composite effect of waning forces of growth on the one
hand, and increasing superficial resistance on the other. This is as
much as to say that growth, while its own energy tends to increase,
leads also, after a while, to the establishment of resistances which
check its own further increase.
Our knowledge of the whole complex phenomenon of growth is so scanty
that it may seem rash to advance even this tentative suggestion. But
yet there are one or two known facts which seem to bear upon the
question, and to indicate at least the manner in which a varying
resistance to expansion may affect the velocity of growth. For
instance, it has been shewn by Frazee[196] that electrical stimulation
of tadpoles, with small current density and low voltage, increases the
rate of regenerative growth. As just such an electrification would tend
to lower the surface-tension, and accordingly decrease the external
resistance, the experiment would seem to support, in some slight
degree, the suggestion which I have made.
Delage[197] has lately made use of the principle of specific rate of
growth, in considering the question of heredity itself. We know that
the chromatin of the fertilised egg comes from the male and female
parent alike, in equal or nearly equal shares; we know that the
initial chromatin, so contributed, multiplies many thousand-fold, to
supply the chromatin for every cell of the offspring’s body; and it
has, therefore, a high “coefficient of growth.” If we admit, with Van
Beneden and others, that the initial contributions of male and female
chromatin continue to be transmitted to the succeeding generations of
cells, we may then conceive these chromatins to retain each its own
coefficient of growth; and if these differed ever so little, a gradual
preponderance of one or other would make itself felt in time, and
might conceivably explain the preponderating influence of one parent
or the other upon the characters of the offspring. Indeed O. Hertwig
is said (according to Delage’s interpretation) to have actually shewn
that we can artificially modify the rate of growth of one or other
chromatin, and so increase or diminish the influence of the maternal
or paternal heredity. This theory of Delage’s has its fascination, but
it calls for somewhat large assumptions; and in particular, it seems
(like so many other theories relating to the chromosomes) to rest
far too much upon material elements, rather than on the imponderable
dynamic factors of the cell. {154}
We may summarise, as follows, the main results of the foregoing
discussion:
(1) Except in certain minute organisms and minute parts of organisms,
whose form is due to the direct action of molecular forces, we may look
upon the form of the organism as a “function of growth,” or a direct
expression of a rate of growth which varies according to its different
directions.
(2) Rate of growth is subject to definite laws, and the velocities in
different directions tend to maintain a _ratio_ which is more or less
constant for each specific organism; and to this regularity is due the
fact that the form of the organism is in general regular and constant.
(3) Nevertheless, the ratio of velocities in different directions is
not absolutely constant, but tends to alter or fluctuate in a regular
way; and to these progressive changes are due the changes of form which
accompany “development,” and the slower changes of form which continue
perceptibly in after life.
(4) The rate of growth is a function of the age of the organism, it
has a maximum somewhat early in life, after which epoch of maximum it
slowly declines.
(5) The rate of growth is directly affected by temperature, and by
other physical conditions.
(6) It is markedly affected, in the way of acceleration or retardation,
at certain physiological epochs of life, such as birth, puberty, or
metamorphosis.
(7) Under certain circumstances, growth may be _negative_, the organism
growing smaller: and such negative growth is a common accompaniment of
metamorphosis, and a frequent accompaniment of old age.
(8) The phenomenon of regeneration is associated with a large temporary
increase in the rate of growth (or “_acceleration_” of growth) of the
injured surface; in other respects, regenerative growth is similar to
ordinary growth in all its essential phenomena.
――――――――――
In this discussion of growth, we have left out of account a vast number
of processes, or phenomena, by which, in the physiological mechanism
of the body, growth is effected and controlled. We have dealt with
growth in its relation to magnitude, and to {155} that relativity
of magnitudes which constitutes form; and so we have studied it as a
phenomenon which stands at the beginning of a morphological, rather
than at the end of a physiological enquiry. Under these restrictions,
we have treated it as far as possible, or in such fashion as our
present knowledge permits, on strictly physical lines.
In all its aspects, and not least in its relation to form, the growth
of organisms has many analogies, some close and some perhaps more
remote, among inanimate things. As the waves grow when the winds strive
with the other forces which govern the movements of the surface of the
sea, as the heap grows when we pour corn out of a sack, as the crystal
grows when from the surrounding solution the proper molecules fall
into their appropriate places: so in all these cases, very much as in
the organism itself, is growth accompanied by change of form, and by
a development of definite shapes and contours. And in these cases (as
in all other mechanical phenomena), we are led to equate our various
magnitudes with time, and so to recognise that growth is essentially a
question of rate, or of velocity.
The differences of form, and changes of form, which are brought about
by varying rates (or “laws”) of growth, are essentially the same
phenomenon whether they be, so to speak, episodes in the life-history
of the individual, or manifest themselves as the normal and distinctive
characteristics of what we call separate species of the race. From one
form, or ratio of magnitude, to another there is but one straight and
direct road of transformation, be the journey taken fast or slow; and
if the transformation take place at all, it will in all likelihood
proceed in the self-same way, whether it occur within the life-time
of an individual or during the long ancestral history of a race. No
small part of what is known as Wolff’s or von Baer’s law, that the
individual organism tends to pass through the phases characteristic
of its ancestors, or that the life-history of the individual tends to
recapitulate the ancestral history of its race, lies wrapped up in this
simple account of the relation between rate of growth and form.
But enough of this discussion. Let us leave for a while the subject
of the growth of the organism, and attempt to study the conformation,
within and without, of the individual cell.
{156}