CHAPTER III

THE RATE OF GROWTH

When we study magnitude by itself, apart, that is to say, from the

gradual changes to which it may be subject, we are dealing with a

something which may be adequately represented by a number, or by means

of a line of definite length; it is what mathematicians call a _scalar_

phenomenon. When we introduce the conception of change of magnitude,

of magnitude which varies as we pass from one direction to another in

space, or from one instant to another in time, our phenomenon becomes

capable of representation by means of a line of which we define both

the length and the direction; it is (in this particular aspect) what is

called a _vector_ phenomenon.

When we deal with magnitude in relation to the dimensions of space, the

vector diagram which we draw plots magnitude in one direction against

magnitude in another,—length against height, for instance, or against

breadth; and the result is simply what we call a picture or drawing of

an object, or (more correctly) a “plane projection” of the object. In

other words, what we call Form is a _ratio of magnitudes_, referred to

direction in space.

When in dealing with magnitude we refer its variations to successive

intervals of time (or when, as it is said, we _equate_ it with time),

we are then dealing with the phenomenon of _growth_; and it is evident,

therefore, that this term growth has wide meanings. For growth may

obviously be positive or negative; that is to say, a thing may grow

larger or smaller, greater or less; and by extension of the primitive

concrete signification of the word, we easily and legitimately apply

it to non-material things, such as temperature, and say, for instance,

that a body “grows” hot or cold. When in a two-dimensional diagram, we

represent a magnitude (for instance length) in relation to time (or

“plot” {51} length against time, as the phrase is), we get that kind

of vector diagram which is commonly known as a “curve of growth.” We

perceive, accordingly, that the phenomenon which we are now studying is

a _velocity_ (whose “dimensions” are Space/Time or _L_/_T_); and this

phenomenon we shall speak of, simply, as a rate of growth.

In various conventional ways we can convert a two-dimensional into

a three-dimensional diagram. We do so, for example, by means of

the geometrical method of “perspective” when we represent upon

a sheet of paper the length, breadth and depth of an object in

three-dimensional space; but we do it more simply, as a rule, by means

of “contour-lines,” and always when time is one of the dimensions

to be represented. If we superimpose upon one another (or even set

side by side) pictures, or plane projections, of an organism, drawn

at successive intervals of time, we have such a three-dimensional

diagram, which is a partial representation (limited to two dimensions

of _space_) of the organism’s gradual change of form, or course of

development; and in such a case our contour-lines may, for the purposes

of the embryologist, be separated by intervals representing a few hours

or days, or, for the purposes of the palaeontologist, by interspaces of

unnumbered and innumerable years[79].

Such a diagram represents in two of its three dimensions form, and in

two, or three, of its dimensions growth; and so we see how intimately

the two conceptions are correlated or inter-related to one another.

In short, it is obvious that the form of an animal is determined by

its specific rate of growth in various directions; accordingly, the

phenomenon of rate of growth deserves to be studied as a necessary

preliminary to the theoretical study of form, and, mathematically

speaking, organic form itself appears to us as a _function of

time_[80]. {52}

At the same time, we need only consider this part of our subject

somewhat briefly. Though it has an essential bearing on the problems

of morphology, it is in greater degree involved with physiological

problems; and furthermore, the statistical or numerical aspect of the

question is peculiarly adapted for the mathematical study of variation

and correlation. On these important subjects we shall scarcely touch;

for our main purpose will be sufficiently served if we consider the

characteristics of a rate of growth in a few illustrative cases,

and recognise that this rate of growth is a very important specific

property, with its own characteristic value in this organism or that,

in this or that part of each organism, and in this or that phase of its

existence.

The statement which we have just made that “the form of an organism is

determined by its rate of growth in various directions,” is one which

calls (as we have partly seen in the foregoing chapter) for further

explanation and for some measure of qualification. Among organic forms

we shall have frequent occasion to see that form is in many cases due

to the immediate or direct action of certain molecular forces, of

which surface-tension is that which plays the greatest part. Now when

surface-tension (for instance) causes a minute semi-fluid organism to

assume a spherical form, or gives the form of a catenary or an elastic

curve to a film of protoplasm in contact with some solid skeletal rod,

or when it acts in various other ways which are productive of definite

contours, this is a process of conformation that, both in appearance

and reality, is very different from the process by which an ordinary

plant or animal _grows_ into its specific form. In both cases, change

of form is brought about by the movement of portions of matter, and in

both cases it is _ultimately_ due to the action of molecular forces;

but in the one case the movements of the particles of matter lie for

the most part _within molecular range_, while in the other we have

to deal chiefly with the transference of portions of matter into the

system from without, and from one widely distant part of the organism

to another. It is to this latter class of phenomena that we usually

restrict the term growth; and it is in regard to them that we are in

a position to study the _rate of action_ in different directions,

and to see that it is merely on a difference of velocities that the

modification of form essentially depends. {53} The difference between

the two classes of phenomena is somewhat akin to the difference between

the forces which determine the form of a rain-drop and those which, by

the flowing of the waters and the sculpturing of the solid earth, have

brought about the complex configuration of a river; _molecular_ forces

are paramount in the conformation of the one, and _molar_ forces are

dominant in the other.

At the same time it is perfectly true that _all_ changes of form,

inasmuch as they necessarily involve changes of actual and relative

magnitude, may, in a sense, be properly looked upon as phenomena of

growth; and it is also true, since the movement of matter must always

involve an element of time[81], that in all cases the rate of growth

is a phenomenon to be considered. Even though the molecular forces

which play their part in modifying the form of an organism exert an

action which is, theoretically, all but instantaneous, that action is

apt to be dragged out to an appreciable interval of time by reason of

viscosity or some other form of resistance in the material. From the

physical or physiological point of view the rate of action even in such

cases may be well worth studying; for example, a study of the rate of

cell-division in a segmenting egg may teach us something about the work

done, and about the various energies concerned. But in such cases the

action is, as a rule, so homogeneous, and the form finally attained is

so definite and so little dependent on the time taken to effect it,

that the specific rate of change, or rate of growth, does not enter

into the _morphological_ problem.

To sum up, we may lay down the following general statements. The form

of organisms is a phenomenon to be referred in part to the direct

action of molecular forces, in part to a more complex and slower

process, indirectly resulting from chemical, osmotic and other forces,

by which material is introduced into the organism and transferred from

one part of it to another. It is this latter complex phenomenon which

we usually speak of as “growth.” {54}

Every growing organism, and every part of such a growing organism, has

its own specific rate of growth, referred to a particular direction.

It is the ratio between the rates of growth in various directions by

which we must account for the external forms of all, save certain

very minute, organisms. This ratio between rates of growth in various

directions may sometimes be of a _simple_ kind, as when it results in

the mathematically definable outline of a shell, or in the smooth curve

of the margin of a leaf. It may sometimes be a very _constant_ one, in

which case the organism, while growing in bulk, suffers little or no

perceptible change in form; but such equilibrium seldom endures for

more than a season, and when the _ratio_ tends to alter, then we have

the phenomenon of morphological “development,” or steady and persistent

change of form.

This elementary concept of Form, as determined by varying rates of

Growth, was clearly apprehended by the mathematical mind of Haller,—who

had learned his mathematics of the great John Bernoulli, as the latter

in turn had learned his physiology from the writings of Borelli. Indeed

it was this very point, the apparently unlimited extent to which, in

the development of the chick, inequalities of growth could and did

produce changes of form and changes of anatomical “structure,” that

led Haller to surmise that the process was actually without limits,

and that all development was but an unfolding, or “_evolutio_,” in

which no part came into being which had not essentially existed

before[82]. In short the celebrated doctrine of “preformation” implied

on the one hand a clear recognition of what, throughout the later

stages of development, growth can do, by hastening the increase in

size of one part, hindering that of another, changing their relative

magnitudes and positions, and altering their forms; while on the other

hand it betrayed a failure (inevitable in those days) to recognise

the essential difference between these movements of masses and the

molecular processes which precede and accompany {55} them, and which

are characteristic of another order of magnitude.

By other writers besides Haller the very general, though not strictly

universal connection between form and rate of growth has been clearly

recognised. Such a connection is implicit in those “proportional

diagrams” by which Dürer and some of his brother artists were wont to

illustrate the successive changes of form, or of relative dimensions,

which attend the growth of the child, to boyhood and to manhood. The

same connection was recognised, more explicitly, by some of the older

embryologists, for instance by Pander[83], and appears, as a survival

of the doctrine of preformation, in his study of the development of

the chick. And long afterwards, the embryological aspect of the case

was emphasised by His, who pointed out, for instance, that the various

foldings of the blastoderm, by which the neural and amniotic folds

were brought into being, were essentially and obviously the resultant

of unequal rates of growth,—of local accelerations or retardations

of growth,—in what to begin with was an even and uniform layer of

embryonic tissue. If we imagine a flat sheet of paper, parts of which

are caused (as by moisture or evaporation) to expand or to contract,

the plane surface is at once dimpled, or “buckled,” or folded, by

the resultant forces of expansion or contraction: and the various

distortions to which the plane surface of the “germinal disc” is

subject, as His shewed once and for all, are precisely analogous.

An experimental demonstration still more closely comparable to the

actual case of the blastoderm, is obtained by making an “artificial

blastoderm,” of little pills or pellets of dough, which are caused to

grow, with varying velocities, by the addition of varying quantities of

yeast. Here, as Roux is careful to point out[84], we observe that it

is not only the _growth_ of the individual cells, but the _traction_

exercised through their mutual interconnections, which brings about the

foldings and other distortions of the entire structure. {56}

But this again was clearly present to Haller’s mind, and formed an

essential part of his embryological doctrine. For he has no sooner

treated of _incrementum_, or _celeritas incrementi_, than he proceeds

to deal with the contributory and complementary phenomena of expansion,

traction (_adtractio_)[85], and pressure, and the more subtle

influences which he denominates _vis derivationis et revulsionis_[86]:

these latter being the secondary and correlated effects on growth in

one part, brought about, through such changes as are produced (for

instance) in the circulation, by the growth of another.

Let us admit that, on the physiological side, Haller’s or His’s methods

of explanation carry us back but a little way; yet even this little

way is something gained. Nevertheless, I can well remember the harsh

criticism, and even contempt, which His’s doctrine met with, not merely

on the ground that it was inadequate, but because such an explanation

was deemed wholly inappropriate, and was utterly disavowed[87].

Hertwig, for instance, asserted that, in embryology, when we found one

embryonic stage preceding another, the existence of the former was,

for the embryologist, an all-sufficient “causal explanation” of the

latter. “We consider (he says), that we are studying and explaining a

causal relation when we have demonstrated that the gastrula arises by

invagination of a blastosphere, or the neural canal by the infolding

of a cell plate so as to constitute a tube[88].” For Hertwig,

therefore, as {57} Roux remarks, the task of investigating a physical

mechanism in embryology,—“der Ziel das Wirken zu erforschen,”—has no

existence at all. For Balfour also, as for Hertwig, the mechanical or

physical aspect of organic development had little or no attraction.

In one notable instance, Balfour himself adduced a physical, or

quasi-physical, explanation of an organic process, when he referred the

various modes of segmentation of an ovum, complete or partial, equal or

unequal and so forth, to the varying amount or the varying distribution

of food yolk in association with the germinal protoplasm of the

egg[89]. But in the main, Balfour, like all the other embryologists of

his day, was engrossed by the problems of phylogeny, and he expressly

defined the aims of comparative embryology (as exemplified in his own

textbook) as being “twofold: (1) to form a basis for Phylogeny. and

(2) to form a basis for Organogeny or the origin and evolution of

organs[90].”

It has been the great service of Roux and his fellow-workers of the

school of “Entwickelungsmechanik,” and of many other students to

whose work we shall refer, to try, as His tried[91] to import into

embryology, wherever possible, the simpler concepts of physics, to

introduce along with them the method of experiment, and to refuse to be

bound by the narrow limitations which such teaching as that of Hertwig

would of necessity impose on the work and the thought and on the whole

philosophy of the biologist.

――――――――――

Before we pass from this general discussion to study some of the

particular phenomena of growth, let me give a single illustration, from

Darwin, of a point of view which is in marked contrast to Haller’s

simple but essentially mathematical conception of Form.

There is a curious passage in the _Origin of Species_[92], where Darwin

is discussing the leading facts of embryology, and in particular Von

Baer’s “law of embryonic resemblance.” Here Darwin says “We are so

much accustomed to see a difference in {58} structure between the

embryo and the adult, that we are tempted to look at this difference

as in some necessary manner contingent on growth. _But there is no

reason why, for instance, the wing of a bat, or the fin of a porpoise,

should not have been sketched out with all their parts in proper

proportion, as soon as any part became visible._” After pointing out

with his habitual care various exceptions, Darwin proceeds to lay down

two general principles, viz. “that slight variations generally appear

at a not very early period of life,” and secondly, that “at whatever

age a variation first appears in the parent, it tends to reappear

at a corresponding age in the offspring.” He then argues that it is

with nature as with the fancier, who does not care what his pigeons

look like in the embryo, so long as the full-grown bird possesses

the desired qualities; and that the process of selection takes place

when the birds or other animals are nearly grown up,—at least on the

part of the breeder, and presumably in nature as a general rule. The

illustration of these principles is set forth as follows; “Let us take

a group of birds, descended from some ancient form and modified through

natural selection for different habits. Then, from the many successive

variations having supervened in the several species at a not very early

age, and having been inherited at a corresponding age, the young will

still resemble each other much more closely than do the adults,—just

as we have seen with the breeds of the pigeon .... Whatever influence

long-continued use or disuse may have had in modifying the limbs or

other parts of any species, this will chiefly or solely have affected

it when nearly mature, when it was compelled to use its full powers

to gain its own living; and the effects thus produced will have been

transmitted to the offspring at a corresponding nearly mature age.

Thus the young will not be modified, or will be modified only in a

slight degree, through the effects of the increased use or disuse of

parts.” This whole argument is remarkable, in more ways than we need

try to deal with here; but it is especially remarkable that Darwin

should begin by casting doubt upon the broad fact that a “difference

in structure between the embryo and the adult” is “in some necessary

manner contingent on growth”; and that he should see no reason why

complicated structures of the adult “should not have been sketched out

{59} with all their parts in proper proportion, as soon as any part

became visible.” It would seem to me that even the most elementary

attention to form in its relation to growth would have removed most of

Darwin’s difficulties in regard to the particular phenomena which he

is here considering. For these phenomena are phenomena of form, and

therefore of relative magnitude; and the magnitudes in question are

attained by growth, proceeding with certain specific velocities, and

lasting for certain long periods of time. And it is accordingly obvious

that in any two related individuals (whether specifically identical or

not) the differences between them must manifest themselves gradually,

and be but little apparent in the young. It is for the same simple

reason that animals which are of very different sizes when adult,

differ less and less in size (as well as in form) as we trace them

backwards through the foetal stages.

――――――――――

Though we study the visible effects of varying rates of growth

throughout wellnigh all the problems of morphology, it is not very

often that we can directly measure the velocities concerned. But owing

to the obvious underlying importance which the phenomenon has to the

morphologist we must make shift to study it where we can, even though

our illustrative cases may seem to have little immediate bearing on the

morphological problem[93].

In a very simple organism, of spherical symmetry, such as the single

spherical cell of Protococcus or of Orbulina, growth is reduced to

its simplest terms, and indeed it becomes so simple in its outward

manifestations that it is no longer of special interest to the

morphologist. The rate of growth is measured by the rate of change in

length of a radius, i.e. _V_ = (_R′_ − _R_)/_T_, and from this we may

calculate, as already indicated, the rate of growth in terms of surface

and of volume. The growing body remains of constant form, owing to the

symmetry of the system; because, that is to say, on the one hand the

pressure exerted by the growing protoplasm is exerted equally in all

directions, after the manner of a hydrostatic pressure, which indeed it

actually is: while on the other hand, the “skin” or surface layer of

the cell is sufficiently {60} homogeneous to exert at every point an

approximately uniform resistance. Under these conditions then, the rate

of growth is uniform in all directions, and does not affect the form of

the organism.

But in a larger or a more complex organism the study of growth, and of

the rate of growth, presents us with a variety of problems, and the

whole phenomenon becomes a factor of great morphological importance. We

no longer find that it tends to be uniform in all directions, nor have

we any right to expect that it should. The resistances which it meets

with will no longer be uniform. In one direction but not in others it

will be opposed by the important resistance of gravity; and within the

growing system itself all manner of structural differences will come

into play, setting up unequal resistances to growth by the varying

rigidity or viscosity of the material substance in one direction or

another. At the same time, the actual sources of growth, the chemical

and osmotic forces which lead to the intussusception of new matter, are

not uniformly distributed; one tissue or one organ may well manifest a

tendency to increase while another does not; a series of bones, their

intervening cartilages, and their surrounding muscles, may all be

capable of very different rates of increment. The differences of form

which are the resultants of these differences in rate of growth are

especially manifested during that part of life when growth itself is

rapid: when the organism, as we say, is undergoing its _development_.

When growth in general has become slow, the relative differences in

rate between different parts of the organism may still exist, and

may be made manifest by careful observation, but in many, or perhaps

in most cases, the resultant change of form does not strike the eye.

Great as are the differences between the rates of growth in different

parts of an organism, the marvel is that the ratios between them are

so nicely balanced as they actually are, and so capable, accordingly,

of keeping for long periods of time the form of the growing organism

all but unchanged. There is the nicest possible balance of forces and

resistances in every part of the complex body; and when this normal

equilibrium is disturbed, then we get abnormal growth, in the shape of

tumours, exostoses, and malformations of every kind. {61}

_The rate of growth in Man._

Man will serve us as well as another organism for our first

illustrations of rate of growth; and we cannot do better than go for

our first data concerning him to Quetelet’s _Anthropométrie_[94], an

epoch-making book for the biologist. For not only is it packed with

information, some of it still unsurpassed, in regard to human growth

and form, but it also merits our highest admiration as the first great

essay in scientific statistics, and the first work in which organic

variation was discussed from the point of view of the mathematical

theory of probabilities.

[Illustration: Fig. 3. Curve of Growth in Man, from birth to 20 yrs

(♂); from Quetelet’s Belgian data. The upper curve of stature from

Bowditch’s Boston data.]

If the child be some 20 inches, or say 50 cm. tall at birth, and

the man some six feet high, or say 180 cm., at twenty, we may say

that his _average_ rate of growth has been (180 − 50)/20 cm., or 6·5

centimetres per annum. But we know very well that this is {62} but

a very rough preliminary statement, and that the boy grew quickly

during some, and slowly during other, of his twenty years. It becomes

necessary therefore to study the phenomenon of growth in successive

small portions; to study, that is to say, the successive lengths, or

the successive small differences, or increments, of length (or of

weight, etc.), attained in successive short increments of time. This

we do in the first instance in the usual way, by the “graphic method”

of plotting length against time, and so constructing our “curve of

growth.” Our curve of growth, whether of weight or length (Fig. 3), has

always a certain characteristic form, or characteristic _curvature_.

This is our immediate proof of the fact that the _rate of growth_

changes as time goes on; for had it not been so, had an equal increment

of length been added in each equal interval of time, our “curve” would

have appeared as a straight line. Such as it is, it tells us not

only that the rate of growth tends to alter, but that it alters in a

definite and orderly way; for, subject to various minor interruptions,

due to secondary causes, our curves of growth are, on the whole,

“smooth” curves.

The curve of growth for length or stature in man indicates a rapid

increase at the outset, that is to say during the quick growth of

babyhood; a long period of slower, but still rapid and almost steady

growth in early boyhood; as a rule a marked quickening soon after the

boy is in his teens, when he comes to “the growing age”; and finally

a gradual arrest of growth as the boy “comes to his full height,” and

reaches manhood.

If we carried the curve further, we should see a very curious thing.

We should see that a man’s full stature endures but for a spell;

long before fifty[95] it has begun to abate, by sixty it is notably

lessened, in extreme old age the old man’s frame is shrunken and

it is but a memory that “he once was tall.” We have already seen,

and here we see again, that growth may have a “negative value.” The

phenomenon of negative growth in old age extends to weight also, and is

evidently largely chemical in origin: the organism can no longer add

new material to its fabric fast enough to keep pace with the wastage of

time. Our curve {63} of growth is in fact a diagram of activity, or

“time-energy” diagram[96]. As the organism grows it is absorbing energy

beyond its daily needs, and accumulating it at a rate depicted in our

_Stature, weight, and span of outstretched arms._

(_After Quetelet_, _pp._ 193, 346.)

Stature in metres Weight in kgm. Span of % ratio

arms, of stature

Age Male Female % F/M Male Female % F/M male to span

0 0·500 0·494 98·8 3·2 2·9 90·7 0·496 100·8

1 0·698 0·690 98·8 9·4 8·8 93·6 0·695 100·4

2 0·791 0·781 98·7 11·3 10·7 94·7 0·789 100·3

3 0·864 0·854 98·8 12·4 11·8 95·2 0·863 100·1

4 0·927 0·915 98·7 14·2 13·0 91·5 0·927 100·0

5 0·987 0·974 98·7 15·8 14·4 91·1 0·988 99·9

6 1·046 1·031 98·5 17·2 16·0 93·0 1·048 99·8

7 1·104 1·087 98·4 19·1 17·5 91·6 1·107 99·7

8 1·162 1·142 98·2 20·8 19·1 91·8 1·166 99·6

9 1·218 1·196 98·2 22·6 21·4 94·7 1·224 99·5

10 1·273 1·249 98·1 24·5 23·5 95·9 1·281 99·4

11 1·325 1·301 98·2 27·1 25·6 94·5 1·335 99·2

12 1·375 1·352 98·3 29·8 29·8 100·0 1·388 99·1

13 1·423 1·400 98·4 34·4 32·9 95·6 1·438 98·9

14 1·469 1·446 98·4 38·8 36·7 94·6 1·489 98·7

15 1·513 1·488 98·3 43·6 40·4 92·7 1·538 99·4

16 1·554 1·521 97·8 49·7 43·6 87·7 1·584 98·1

17 1·594 1·546 97·0 52·8 47·3 89·6 1·630 97·9

18 1·630 1·563 95·9 57·8 49·0 84·8 1·670 97·6

19 1·655 1·570 94·9 58·0 51·6 89·0 1·705 97·1

20 1·669 1·574 94·3 60·1 52·3 87·0 1·728 96·6

25 1·682 1·578 93·8 62·9 53·3 84·7 1·731 97·2

30 1·686 1·580 93·7 63·7 54·3 85·3 1·766 95·5

40 1·686 1·580 93·7 63·7 55·2 86·7 1·766 95·5

50 1·686 1·580 93·7 63·5 56·2 88·4 — —

60 1·676 1·571 93·7 61·9 54·3 87·7 — —

70 1·660 1·556 93·7 59·5 51·5 86·5 — —

80 1·636 1·534 93·8 57·8 49·4 85·5 — —

90 1·610 1·510 93·8 57·8 49·3 85·3 — —

curve; but the time comes when it accumulates no longer, and at last

it is constrained to draw upon its dwindling store. But in part, the

slow decline in stature is an expression of an unequal contest between

our bodily powers and the unchanging force of gravity, {64} which

draws us down when we would fain rise up[97]. For against gravity we

fight all our days, in every movement of our limbs, in every beat of

our hearts; it is the indomitable force that defeats us in the end,

that lays us on our deathbed, that lowers us to the grave[98].

Side by side with the curve which represents growth in length, or

stature, our diagram shows the curve of weight[99]. That this curve

is of a very different shape from the former one, is accounted for in

the main (though not wholly) by the fact which we have already dealt

with, that, whatever be the law of increment in a linear dimension,

the law of increase in volume, and therefore in weight, will be that

these latter magnitudes tend to vary as the cubes of the linear

dimensions. This however does not account for the change of direction,

or “point of inflection” which we observe in the curve of weight at

about one or two years old, nor for certain other differences between

our two curves which the scale of our diagram does not yet make clear.

These differences are due to the fact that the form of the child is

altering with growth, that other linear dimensions are varying somewhat

differently from length or stature, and that consequently the growth in

bulk or weight is following a more complicated law.

Our curve of growth, whether for weight or length, is a direct picture

of velocity, for it represents, as a connected series, the successive

epochs of time at which successive weights or lengths are attained.

But, as we have already in part seen, a great part of the interest

of our curve lies in the fact that we can see from it, not only that

length (or some other magnitude) is changing, but that the _rate of

change_ of magnitude, or rate of growth, is itself changing. We have,

in short, to study the phenomenon of _acceleration_: we have begun by

studying a velocity, or rate of {65} change of magnitude; we must

now study an acceleration, or rate of change of velocity. The rate,

or velocity, of growth is measured by the _slope_ of the curve; where

the curve is steep, it means that growth is rapid, and when growth

ceases the curve appears as a horizontal line. If we can find a means,

then, of representing at successive epochs the corresponding slope,

or steepness, of the curve, we shall have obtained a picture of the

rate of change of velocity, or the acceleration of growth. The measure

of the steepness of a curve is given by the tangent to the curve, or

we may estimate it by taking for equal intervals of time (strictly

speaking, for each infinitesimal interval of time) the actual increment

added during that interval of time: and in practice this simply amounts

to taking the successive _differences_ between the values of length (or

of weight) for the successive ages which we have begun by studying. If

we then plot these successive _differences_ against time, we obtain

a curve each point upon which represents a velocity, and the whole

curve indicates the rate of change of velocity, and we call it an

acceleration-curve. It contains, in truth, nothing whatsoever that was

not implicit in our former curve; but it makes clear to our eye, and

brings within the reach of further investigation, phenomena that were

hard to see in the other mode of representation.

The acceleration-curve of height, which we here illustrate, in Fig. 4,

is very different in form from the curve of growth which we have just

been looking at; and it happens that, in this case, there is a very

marked difference between the curve which we obtain from Quetelet’s

data of growth in height and that which we may draw from any other

series of observations known to me from British, French, American or

German writers. It begins (as will be seen from our next table) at

a very high level, such as it never afterwards attains; and still

stands too high, during the first three or four years of life, to be

represented on the scale of the accompanying diagram. From these high

velocities it falls away, on the whole, until the age when growth

itself ceases, and when the rate of growth, accordingly, has, for

some years together, the constant value of _nil_; but the rate of

fall, or rate of change of velocity, is subject to several changes or

interruptions. During the first three or four years of life the fall is

continuous and rapid, {66} but it is somewhat arrested for a while in

childhood, from about five years old to eight. According to Quetelet’s

data, there is another slight interruption in the falling rate between

the ages of about fourteen and sixteen; but in place of this almost

insignificant interruption, the English and other statistics indicate a

sudden

[Illustration: Fig. 4. Mean annual increments of stature (♂), Belgian

and American.]

and very marked acceleration of growth beginning at about twelve

years of age, and lasting for three or four years; when this period

of acceleration is over, the rate begins to fall again, and does

so with great rapidity. We do not know how far the absence of this

striking feature in the Belgian curve is due to the imperfections of

Quetelet’s data, or whether it is a real and significant feature in the

small-statured race which he investigated.

_Annual Increment of Stature (in cm.) from Belgian and American

Statistics._

D: Belgian (Quetelet, p. 344)

E: Paris* (Variot et Chaumet, p. 55)

F: Toronto† (Boas, p. 1547)

G: Worcester‡, Mass. (Boas, p. 1548)

H: Ann. increment

I: Increment

J: Boys

K: Girls

V: Variability of do.

────[D]─── ───────[E]─────── ──────[F]────── ────────[G]───────

Height Height [I] Height [V] [H] [H]

Age (Boys) [H] [J] [K] [J] [K] (Boys) (6) [H] ([J]) [V] ([K]) [V]

0 50·0 — — — — — — — — — — — —

1 69·8 19·8 74·2 73·6 — — — — — — — — —

2 79·1 9·3 82·7 81·8 8·5 8·2 — — — — — — —

3 86·4 7·3 89·1 88·4 6·4 6·6 — — — — — — —

4 92·7 6·3 96·8 95·8 7·7 7·4 — — — — — — —

5 98·7 6·0 103·3 101·9 6·5 6·1 105·90 4·40 — — — — —

6 104·0 5·9 109·9 108·9 6·6 7·0 111·58 4·62 5·68 6·55 1·57 5·75 0·88

7 110·4 5·8 114·4 113·8 4·5 4·9 116·83 4·93 5·25 5·70 0·68 5·90 0·98

8 116·2 5·8 119·7 119·5 5·3 5·7 122·04 5·34 5·21 5·37 0·86 5·70 1·10

9 121·8 5·6 125·0 124·7 5·3 4·8 126·91 5·49 4·87 4·89 0·96 5·50 0·97

10 127·3 5·5 130·3 129·5 5·3 5·2 131·78 5·75 4·87 5·10 1·03 5·97 1·23

11 132·5 5·2 133·6 134·4 3·3 4·9 136·20 6·19 4·42 5·02 0·88 6·17 1·85

12 137·5 5·0 137·6 141·5 4·0 7·1 140·74 6·66 4·54 4·99 1·26 6·98 1·89

13 142·3 4·8 145·1 148·6 7·5 7·1 146·00 7·54 5·26 5·91 1·86 6·71 2·06

14 146·9 4·6 153·8 152·9 8·7 4·3 152·39 8·49 6·39 7·88 2·39 5·44 2·89

15 151·3 4·4 159·6 154·2 5·8 1·3 159·72 8·78 7·33 6·23 2·91 5·34 2·71

16 155·4 4·1 — — — — 164·90 7·73 5·18 5·64 3·46 — —

17 159·4 4·0 — — — — 168·91 7·22 4·01 — — — —

18 163·0 3·6 — — — — 171·07 6·74 2·16 — — — —

19 165·5 2·5 — — — — — — — — — — —

20 167·0 1·5 — — — — — — — — — — —

* Ages from 1–2, 2–3, etc.

† The epochs are, in this table, 5·5, 6·5, years, etc.

‡ Direct observations on actual, or individualised,

increase of stature from year to year: between the ages of

5–6, 6–7, etc.

Even apart from these data of Quetelet’s (which seem to constitute

an extreme case), it is evident that there are very {68} marked

differences between different races, as we shall presently see there

are between the two sexes, in regard to the epochs of acceleration of

growth, in other words, in the “phase” of the curve.

It is evident that, if we pleased, we might represent the _rate of

change of acceleration_ on yet another curve, by constructing a table

of “second differences”; this would bring out certain very interesting

phenomena, which here however we must not stay to discuss.

_Annual Increment of Weight in Man_ (_kgm._).

(After Quetelet, _Anthropométrie_, p. 346*.)

Increment

Age Male Female

0–1 5·9 5·6

1–2 2·0 2·4

2–3 1·5 1·4

3–4 1·5 1·5

4–5 1·9 1·4

5–6 1·9 1·4

6–7 1·9 1·1

7–8 1·9 1·2

8–9 1·9 2·0

9–10 1·7 2·1

10–11 1·8 2·4

11–12 2·0 3·5

12–13 4·1 3·5

13–14 4·0 3·8

14–15 4·1 3·7

15–16 4·2 3·5

16–17 4·3 3·3

17–18 4·2 3·0

18–19 3·7 2·3

19–20 1·9 1·1

20–21 1·7 1·1

21–22 1·7 0·5

22–23 1·6 0·4

23–24 0·9 −0·2

24–25 0·8 −0·2

* The values given in this table are not in precise accord

with those of the Table on p. 63. The latter represent

Quetelet’s results arrived at in 1835; the former are the

means of his determinations in 1835–40.

The acceleration-curve for man’s weight (Fig. 5), whether we draw

it from Quetelet’s data, or from the British, American and other

statistics of later writers, is on the whole similar to that which

we deduce from the statistics of these latter writers in regard to

height or stature; that is to say, it is not a curve which continually

descends, but it indicates a rate of growth which is subject to

important fluctuations at certain epochs of life. We see that it begins

at a high level, and falls continuously and rapidly[100] {69} during

the first two or three years of life. After a slight recovery, it runs

nearly level during boyhood from about five to twelve years old; it

then rapidly rises, in the “growing period” of the early teens, and

slowly and steadily falls from about the age of sixteen onwards. It

does not reach the base-line till the man is about seven or eight and

twenty, for normal increase of weight continues during the years when

the man is “filling out,” long after growth in height has ceased;

but at last, somewhere about thirty, the velocity reaches zero, and

even falls below it, for then the man usually begins to lose weight a

little. The subsequent slow changes in this acceleration-curve we need

not stop to deal with.

[Illustration: Fig. 5. Mean annual increments of weight, in man and

woman; from Quetelet’s data.]

In the same diagram (Fig. 5) I have set forth the acceleration-curves

in respect of increment of weight for both man and woman, according to

Quetelet. That growth in boyhood and growth in girlhood follow a very

different course is a matter of common knowledge; but if we simply

plot the ordinary curve of growth, or velocity-curve, the difference,

on the small scale of our diagrams, {70} is not very apparent. It is

admirably brought out, however, in the acceleration-curves. Here we see

that, after infancy, say from three years old to eight, the velocity in

the girl is steady, just as in the boy, but it stands on a lower level

in her case than in his: the little maid at this age is growing slower

than the boy. But very soon, and while his acceleration-curve is still

represented by a straight line, hers has begun to ascend, and until

the girl is about thirteen or fourteen it continues to ascend rapidly.

After that age, as after sixteen or seventeen in the boy’s case, it

begins to descend. In short, throughout all this period, it is a very

_similar_ curve in the two sexes; but it has its notable differences,

in amplitude and especially in _phase_. Last of all, we may notice that

while the acceleration-curve falls to a negative value in the male

about or even a little before the age of thirty years, this does not

happen among women. They continue to grow in weight, though slowly,

till very much later in life; until there comes a final period, in both

sexes alike, during which weight, and height and strength all alike

diminish.

From certain corrected, or “typical” values, given for American

children by Boas and Wissler (_l.c._ p. 42), we obtain the following

still clearer comparison of the annual increments of _stature_ in boys

and girls: the typical stature at the commencement of the period, i.e.

at the age of eleven, being 135·1 cm. and 136·9 cm. for the boys and

girls respectively, and the annual increments being as follows:

Age 12 13 14 15 16 17 18 19 20

Boys (cm.) 4·1 6·3 8·7 7·9 5·2 3·2 1·9 0·9 0·3

Girls (cm.) 7·5 7·0 4·6 2·1 0·9 0·4 0·1 0·0 0·0

Difference −3·4 −0·7 4·1 5·8 4·3 2·8 1·8 0·9 0·3

The result of these differences (which are essentially

_phase_-differences) between the two sexes in regard to the velocity

of growth and to the rate of change of that velocity, is to cause the

_ratio_ between the weights of the two sexes to fluctuate in a somewhat

complicated manner. At birth the baby-girl weighs on the average nearly

10 per cent. less than the boy. Till about two years old she tends to

gain upon him, but she then loses again until the age of about five;

from five she gains for a few years somewhat rapidly, and the girl of

ten to twelve is only some 3 per cent. less in weight than the boy. The

boy in his teens gains {71} steadily, and the young woman of twenty

is nearly 15 per cent. lighter than the man. This ratio of difference

again slowly diminishes, and between fifty and sixty stands at about

12 per cent., or not far from the mean for all ages; but once more as

old age advances, the difference tends, though very slowly, to increase

(Fig. 6).

[Illustration: Fig. 6. Percentage ratio, throughout life, of female

weight to male; from Quetelet’s data.]

While careful observations on the rate of growth in other animals are

somewhat scanty, they tend to show so far as they go that the general

features of the phenomenon are always much the same. Whether the animal

be long-lived, as man or the elephant, or short-lived, like horse or

dog, it passes through the same phases of growth[101]. In all cases

growth begins slowly; it attains a maximum velocity early in its

course, and afterwards slows down (subject to temporary accelerations)

towards a point where growth ceases altogether. But especially in the

cold-blooded animals, such as fishes, the slowing-down period is very

greatly protracted, and the size of the creature would seem never

actually to reach, but only to approach asymptotically, to a maximal

limit.

The size ultimately attained is a resultant of the rate, and of {72}

the duration, of growth. It is in the main true, as Minot has said,

that the rabbit is bigger than the guinea-pig because he grows the

faster; but that man is bigger than the rabbit because he goes on

growing for a longer time.

――――――――――

In ordinary physical investigations dealing with velocities, as for

instance with the course of a projectile, we pass at once from the

study of acceleration to that of momentum and so to that of force; for

change of momentum, which is proportional to force, is the product of

the mass of a body into its acceleration or change of velocity. But we

can take no such easy road of kinematical investigation in this case.

The “velocity” of growth is a very different thing from the “velocity”

of the projectile. The forces at work in our case are not susceptible

of direct and easy treatment; they are too varied in their nature and

too indirect in their action for us to be justified in equating them

directly with the mass of the growing structure.

It was apparently from a feeling that the velocity of growth ought

in some way to be equated with the mass of the growing structure

that Minot[102] introduced a curious, and (as it seems to me) an

unhappy method of representing growth, in the form of what he called

“percentage-curves”; a method which has been followed by a number of

other writers and experimenters. Minot’s method was to deal, not with

the actual increments added in successive periods, such as years or

days, but with these increments represented as _percentages_ of the

amount which had been reached at the end of the former period. For

instance, taking Quetelet’s values for the height in centimetres of a

male infant from birth to four years old, as follows:

Years 0 1 2 3 4

cm. 50·0 69·8 79·1 86·4 92·7

Minot would state the percentage growth in each of the four annual

periods at 39·6, 13·3, 9·6 and 7·3 per cent. respectively.

Now when we plot actual length against time, we have a perfectly

definite thing. When we differentiate this _L_/_T_, we have

_dL_/_dT_, which is (of course) velocity; and from this, by a second

differentiation, we obtain _d_^2 _L_/_dT_^2, that is to say, the

acceleration. {73}

But when you take percentages of _y_, you are determining _dy_/_y_,

and when you plot this against _dx_, you have

(_dy_/_y_)/_dx_, or _dy_/(_y_ ⋅ _dx_), or (1/_y_) ⋅ (_dy_/_dx_),

that is to say, you are multiplying the thing you wish to represent

by another quantity which is itself continually varying; and the

result is that you are dealing with something very much less easily

grasped by the mind than the original factors. Professor Minot is, of

course, dealing with a perfectly legitimate function of _x_ and _y_;

and his method is practically tantamount to plotting log _y_ against

_x_, that is to say, the logarithm of the increment against the time.

This could only be defended and justified if it led to some simple

result, for instance if it gave us a straight line, or some other

simpler curve than our usual curves of growth. As a matter of fact, it

is manifest that it does nothing of the kind.

_Pre-natal and post-natal growth._

In the acceleration-curves which we have shown above (Figs. 2, 3), it

will be seen that the curve starts at a considerable interval from the

actual date of birth; for the first two increments which we can as yet

compare with one another are those attained during the first and second

complete years of life. Now we can in many cases “interpolate” with

safety _between_ known points upon a curve, but it is very much less

safe, and is not very often justifiable (at least until we understand

the physical principle involved, and its mathematical expression), to

“extrapolate” beyond the limits of our observations. In short, we do

not yet know whether our curve continued to ascend as we go backwards

to the date of birth, or whether it may not have changed its direction,

and descended, perhaps, to zero-value. In regard to length, or stature,

however, we can obtain the requisite information from certain tables

of Rüssow’s[103], who gives the stature of the infant month by month

during the first year of its life, as follows:

Age in months 0 1 2 3 4 5 6 7 8 9 10 11 12

Length in cm. (50) 54 58 60 62 64 65 66 67·5 68 69 70·5 72

[Differences (in cm.) 4 4 2 2 2 1 1 1·5 ·5 1 1·5 1·5]

If we multiply these _monthly_ differences, or mean monthly velocities,

by 12, to bring them into a form comparable with the {74} _annual_

velocities already represented on our acceleration-curves, we shall see

that the one series of observations joins on very well with the other;

and in short we see at once that our acceleration-curve rises steadily

and rapidly as we pass back towards the date of birth.

[Illustration: Fig. 7. Curve of growth (in length or stature) of child,

before and after birth. (From His and Rüssow’s data.)]

But birth itself, in the case of a viviparous animal, is but an

unimportant epoch in the history of growth. It is an epoch whose

relative date varies according to the particular animal: the foal and

the lamb are born relatively later, that is to say when development

has advanced much farther, than in the case of man; the kitten and the

puppy are born earlier and therefore more helpless than we are; and the

mouse comes into the world still earlier and more inchoate, so much so

that even the little marsupial is scarcely more unformed and embryonic.

In all these cases alike, we must, in order to study the curve of

growth in its entirety, take full account of prenatal or intra-uterine

growth. {75}

According to His[104], the following are the mean lengths of the unborn

human embryo, from month to month.

Months 0 1 2 3 4 5 6 7 8 9 10

(Birth)

Length in mm. 0 7·5 40 84 162 275 352 402 443 472 490–500

Increment per

month in mm. — 7·5 32·5 44 78 113 77 50 41 29 18–28

[Illustration: Fig. 8. Mean monthly increments of length or stature of

child (in cms.).]

These data link on very well to those of Rüssow, which we have just

considered, and (though His’s measurements for the pre-natal months are

more detailed than are those of Rüssow for the first year of post-natal

life) we may draw a continuous curve of growth (Fig. 7) and curve of

acceleration of growth (Fig. 8) for the combined periods. It will at

once be seen that there is a “point of inflection” somewhere about

the fifth month of intra-uterine life[105]: up to that date growth

proceeds with a continually increasing {76} velocity; but after that

date, though growth is still rapid, its velocity tends to fall away.

There is a slight break between our two separate sets of statistics

at the date of birth, while this is the very epoch regarding which we

should particularly like to have precise and continuous information.

Undoubtedly there is a certain slight arrest of growth, or diminution

of the rate of growth, about the epoch of birth: the sudden change

in the {77} method of nutrition has its inevitable effect; but this

slight temporary set-back is immediately followed by a secondary, and

temporary, acceleration.

[Illustration: Fig. 9. Curve of pre-natal growth (length or stature) of

child; and corresponding curve of mean monthly increments (mm.).]

[Illustration: Fig. 10. Curve of growth of bamboo (from Ostwald, after

Kraus).]

It is worth our while to draw a separate curve to illustrate on a

larger scale His’s careful data for the ten months of pre-natal life

(Fig. 9). We see that this curve of growth is a beautifully regular

one, and is nearly symmetrical on either side of that point of

inflection of which we have already spoken; it is a curve for which

we might well hope to find a simple mathematical expression. The

acceleration-curve shown in Fig. 9 together with the pre-natal curve

of growth, is not taken directly from His’s recorded data, but is

derived from the tangents drawn to a smoothed curve, corresponding as

nearly as possible to the actual curve of growth: the rise to a maximal

velocity about the fifth month and the subsequent gradual fall are

now demonstrated even more clearly than before. In Fig. 10, which is

a curve of growth of the bamboo[106], we see (so far as it goes) the

same essential features, {78} the slow beginning, the rapid increase

of velocity, the point of inflection, and the subsequent slow negative

acceleration[107].

_Variability and Correlation of Growth._

The magnitudes and velocities which we are here dealing with are, of

course, mean values derived from a certain number, sometimes a large

number, of individual cases. But no statistical account of mean values

is complete unless we also take account of the _amount of variability_

among the individual cases from which the mean value is drawn. To do

this throughout would lead us into detailed investigations which lie

far beyond the scope of this elementary book; but we may very briefly

illustrate the nature of the process, in connection with the phenomena

of growth which we have just been studying.

It was in connection with these phenomena, in the case of man, that

Quetelet first conceived the statistical study of variation, on lines

which were afterwards expounded and developed by Galton, and which have

grown, in the hands of Karl Pearson and others, into the modern science

of Biometrics.

When Quetelet tells us, for instance, that the mean stature of the

ten-year old boy is 1·273 metres, this implies, according to the law of

error, or law of probabilities, that all the individual measurements

of ten-year-old boys group themselves _in an orderly way_, that is

to say according to a certain definite law, about this mean value of

1·273. When these individual measurements are grouped and plotted

as a curve, so as to show the number of individual cases at each

individual length, we obtain a characteristic curve of error or curve

of frequency; and the “spread” of this curve is a measure of the amount

of variability in this particular case. A certain mathematical measure

of this “spread,” as described in works upon statistics, is called the

Index of Variability, or Standard Deviation, and is usually denominated

by the letter σ. It is practically equivalent to a determination of

the point upon the frequency curve where it _changes its curvature_

on either side of the mean, and where, from being concave towards

the middle line, it spreads out to be convex thereto. When we divide

this {79} value by the mean, we get a figure which is independent

of any particular units, and which is called the Coefficient of

Variability. (It is usually multiplied by 100, to make it of a more

convenient amount; and we may then define this coefficient, _C_, as

= (σ/_M_) × 100.)

In regard to the growth of man, Pearson has determined this coefficient

of variability as follows: in male new-born infants, the coefficient

in regard to weight is 15·66, and in regard to stature, 6·50; in

male adults, for weight 10·83, and for stature, 3·66. The amount of

variability tends, therefore, to decrease with growth or age.

Similar determinations have been elaborated by Bowditch, by Boas and

Wissler, and by other writers for intermediate ages, especially from

about five years old to eighteen, so covering a great part of the whole

period of growth in man[108].

_Coefficient of Variability (σ/_M_ × 100) in Man, at various ages._

Age 5 6 7 8 9

Stature (Bowditch) 4·76 4·60 4·42 4·49 4·40

Stature (Boas and Wissler) 4·15 4·14 4·22 4·37 4·33

Weight (Bowditch) 11·56 10·28 11·08 9·92 11·04

Age 10 11 12 13 14

Stature (Bowditch) 4·55 4·70 4·90 5·47 5·79

Stature (Boas and Wissler) 4·36 4·54 4·73 5·16 5·57

Weight (Bowditch) 11·60 11·76 13·72 13·60 16·80

Age 15 16 17 18

Stature (Bowditch) 5·57 4·50 4·55 3·69

Stature (Boas and Wissler) 5·50 4·69 4·27 3·94

Weight (Bowditch) 15·32 13·28 12·96 10·40

The result is very curious indeed. We see, from Fig. 11, that the

curve of variability is very similar to what we have called the

acceleration-curve (Fig. 4): that is to say, it descends when the rate

of growth diminishes, and rises very markedly again when, in late

boyhood, the rate of growth is temporarily accelerated. We {80} see,

in short, that the amount of _variability_ in stature or in weight is a

function of the _rate of growth_ in these magnitudes, though we are not

yet in a position to equate the terms precisely, one with another.

[Illustration: Fig. 11. Coefficients of variability of stature in Man

(♂). from Boas and Wissler’s data.]

If we take not merely the variability of stature or weight at a given

age, but the variability of the actual successive increments in each

yearly period, we see that this latter coefficient of variability

tends to increase steadily, and more and more rapidly, within the

limits of age for which we have information; and this phenomenon is,

in the main, easy of explanation. For a great part of the difference,

in regard to rate of growth, between one individual and another is a

difference of _phase_,—a difference in the epochs of acceleration and

retardation, and finally in the epoch when growth comes to an end.

And it follows that the variability of rate will be more and more

marked, as we approach and reach the period when some individuals

still continue, and others have already ceased, to grow. In the

following epitomised table, {81} I have taken Boas’s determinations

of variability (σ) (_op. cit._ p. 1548), converted them into the

corresponding coefficients of variability ((σ/_M_) × 100), and then

smoothed the resulting numbers.

_Coefficients of Variability in Annual Increment of Stature._

Age 7 8 9 10 11 12 13 14 15

Boys 17·3 15·8 18·6 19·1 21·0 24·7 29·0 36·2 46·1

Girls 17·1 17·8 19·2 22·7 25·9 29·3 37·0 44·8 —

The greater variability of annual increment in the girls, as compared

with the boys, is very marked, and is easily explained by the more

rapid rate at which the girls run through the several phases of the

phenomenon.

Just as there is a marked difference in “phase” between the

growth-curves of the two sexes, that is to say a difference in the

periods when growth is rapid or the reverse, so also, within each sex,

will there be room for similar, but individual phase-differences.

Thus we may have children of accelerated development, who at a given

epoch after birth are both rapidly growing and already “big for their

age”; and others of retarded development who are comparatively small

and have not reached the period of acceleration which, in greater

or less degree, will come to them in turn. In other words, there

must under such circumstances be a strong positive “coefficient of

correlation” between stature and rate of growth, and also between

the rate of growth in one year and the next. But it does not by any

means follow that a child who is precociously big will continue to

grow rapidly, and become a man or woman of exceptional stature. On the

contrary, when in the case of the precocious or “accelerated” children

growth has begun to slow down, the backward ones may still be growing

rapidly, and so making up (more or less completely) to the others. In

other words, the period of high positive correlation between stature

and increment will tend to be followed by one of negative correlation.

This interesting and important point, due to Boas and Wissler[109], is

confirmed by the following table:―

_Correlation of Stature and Increment in Boys and Girls._

(_From Boas and Wissler._)

Age 6 7 8 9 10 11 12 13 14 15

Stature (B) 112·7 115·5 123·2 127·4 133·2 136·8 142·7 147·3 155·9 162·2

(G) 111·4 117·7 121·4 127·9 131·8 136·7 144·6 149·7 153·8 157·2

Increment (B) 5·7 5·3 4·9 5·1 5·0 4·7 5·9 7·5 6·2 5·2

(G) 5·9 5·5 5·5 5·9 6·2 7·2 6·5 5·4 3·3 1·7

Correlation (B) ·25 ·11 ·08 ·25 ·18 ·18 ·48 ·29 −·42 −·44

(G) ·44 ·14 ·24 ·47 ·18 −·18 −·42 −·39 −·63 ·11

{82}

A minor, but very curious point brought out by the same investigators

is that, if instead of stature we deal with height in the sitting

posture (or, practically speaking, with length of trunk or back), then

the correlations between this height and its annual increment are

throughout negative. In other words, there would seem to be a general

tendency for the long trunks to grow slowly throughout the whole period

under investigation. It is a well-known anatomical fact that tallness

is in the main due not to length of body but to length of limb.

The whole phenomenon of variability in regard to magnitude and to rate

of increment is in the highest degree suggestive: inasmuch as it helps

further to remind and to impress upon us that specific rate of growth

is the real physiological factor which we want to get at, of which

specific magnitude, dimensions and form, and all the variations of

these, are merely the concrete and visible resultant. But the problems

of variability, though they are intimately related to the general

problem of growth, carry us very soon beyond our present limitations.

_Rate of growth in other organisms[110]._

Just as the human curve of growth has its slight but well-marked

interruptions, or variations in rate, coinciding with such epochs as

birth and puberty, so is it with other animals, and this phenomenon is

particularly striking in the case of animals which undergo a regular

metamorphosis.

In the accompanying curve of growth in weight of the mouse (Fig. 12),

based on W. Ostwald’s observations[111], we see a distinct slackening

of the rate when the mouse is about a fortnight old, at which period it

opens its eyes and very soon afterwards is weaned. At about six weeks

old there is another well-marked retardation of growth, following on a

very rapid period, and coinciding with the epoch of puberty. {83}

Fig. 13 shews the curve of growth of the silkworm[112], during its

whole larval life, up to the time of its entering the chrysalis stage.

The silkworm moults four times, at intervals of about a week, the first

moult being on the sixth or seventh day after hatching. A distinct

retardation of growth is exhibited on our curve in the case of the

third and fourth moults; while a similar retardation accompanies the

first and second moults also, but the scale of our diagram does not

render it visible. When the worm is about seven weeks old, a remarkable

process of “purgation” takes place, as a preliminary to entering on the

pupal, or chrysalis, stage; and the great and sudden loss of weight

which accompanies this process is the most marked feature of our curve.

[Illustration: Fig. 12. Growth in weight of Mouse. (After W. Ostwald.)]

The rate of growth in the tadpole[113] (Fig. 14) is likewise marked

by epochs of retardation, and finally by a sudden and drastic change.

There is a slight diminution in weight immediately after {84} the

little larva frees itself from the egg; there is a retardation of

growth about ten days later, when the external gills disappear; and

finally, the complete metamorphosis, with the loss of the tail, the

growth of the legs and the cessation of branchial respiration, is

accompanied by a loss of weight amounting to wellnigh half the weight

of the full-grown larva. {85}

[Illustration: Fig. 13. Growth in weight of Silkworm. (From Ostwald,

after Luciani and Lo Monaco.)]

While as a general rule, the better the animals be fed the quicker

they grow and the sooner they metamorphose, Barfürth has pointed

out the curious fact that a short spell of starvation, just before

metamorphosis is due, appears to hasten the change.

[Illustration: Fig. 14. Growth in weight of Tadpole. (From Ostwald,

after Schaper.)]

The negative growth, or actual loss of bulk and weight which often,

and perhaps always, accompanies metamorphosis, is well shewn in the

case of the eel[114]. The contrast of size is great between {87} the

flattened, lancet-shaped Leptocephalus larva and the little black

cylindrical, almost thread-like elver, whose magnitude is less than

that of the Leptocephalus in every dimension, even, at first, in length

(Fig. 15).

[Illustration: Fig. 15. Development of Eel; from Leptocephalus larvae

to young Elver. (From Ostwald after Joh. Schmidt.)]

[Illustration: Fig. 16. Growth in length of Spirogyra. (From Ostwald,

after Hofmeister.)]

From the higher study of the physiology of growth we learn that such

fluctuations as we have described are but special interruptions in

a process which is never actually continuous, but is perpetually

interrupted in a rhythmic manner[115]. Hofmeister shewed, for instance,

that the growth of Spirogyra proceeds by fits and starts, by periods

of activity and rest, which alternate with one another at intervals

of so many minutes (Fig. 16). And Bose, by very refined methods of

experiment, has shewn that plant-growth really proceeds by tiny and

perfectly rhythmical pulsations recurring at regular intervals of a few

seconds of time. Fig. 17 shews, according to Bose’s observations[116],

the growth of a crocus, under a very high magnification. The stalk

grows by little jerks, each with an amplitude of about ·002 mm., every

{88} twenty seconds or so, and after each little increment there is a

partial recoil.

[Illustration: Fig. 17. Pulsations of growth in Crocus, in

micro-millimetres. (After Bose.)]

_The rate of growth of various parts or organs[117]._

The differences in regard to rate of growth between various parts or

organs of the body, internal and external, can be amply illustrated in

the case of man, and also, but chiefly in regard to external form, in

some few other creatures[118]. It is obvious that there lies herein

an endless field for the mathematical study of correlation and of

variability, but with this aspect of the case we cannot deal.

In the accompanying table, I shew, from some of Vierordt’s data, the

_relative_ weights, at various ages, compared with the weight at birth,

of the entire body, of the brain, heart and liver; {89} and also the

percentage relation which each of these organs bears, at the several

ages, to the weight of the whole body.

_Weight of Various Organs, compared with the Total Weight of the Human

Body (male)._ (_After Vierordt, Anatom. Tabellen, pp. 38, 39._)

Percentage weights compared

Weight Relative weights of with total body-weights

of body† ───────────────────────── ───────────────────────────

Age in kg. Body Brain Heart Liver Body Brain Heart Liver

0 3·1 1 1 1 1 100 12·29 0·76 4·57

1 9·0 2·90 2·48 1·75 2·35 100 10·50 0·46 3·70

2 11·0 3·55 2·69 2·20 3·02 100 9·32 0·47 3·89

3 12·5 4·03 2·91 2·75 3·42 100 8·86 0·52 3·88

4 14·0 4·52 3·49 3·14 4·15 100 9·50 0·53 4·20

5 15·9 5·13 3·32 3·43 3·80 100 7·94 0·51 3·39

6 17·8 5·74 3·57 3·60 4·34 100 7·63 0·48 3·45

7 19·7 6·35 3·54 3·95 4·86 100 6·84 0·47 3·49

8 21·6 6·97 3·62 4·02 4·59 100 6·38 0·44 3·01

9 23·5 7·58 3·74 4·59 4·95 100 6·06 0·46 2·99

10 25·2 8·13 3·70 5·41 5·90 100 5·59 0·51 3·32

11 27·0 8·71 3·57 5·97 6·14 100 5·04 0·52 3·22

12 29·0 9·35 3·78 (4·13) 6·21 100 4·88 (0·34) 3·03

13 33·1 10·68 3·90 6·95 7·31 100 4·49 0·50 3·13

14 37·1 11·97 3·38 9·16 8·39 100 3·47 0·58 3·20

15 41·2 13·29 3·91 8·45 9·22 100 3·62 0·48 3·17

16 45·9 14·81 3·77 9·76 9·45 100 3·16 0·51 2·95

17 49·7 16·03 3·70 10·63 10·46 100 2·84 0·51 2·98

18 53·9 17·39 3·73 10·33 10·65 100 2·64 0·46 2·80

19 57·6 18·58 3·67 11·42 11·61 100 2·43 0·51 2·86

20 59·5 19·19 3·79 12·94 11·01 100 2·43 0·51 2·62

21 61·2 19·74 3·71 12·59 11·48 100 2·31 0·49 2·66

22 62·9 20·29 3·54 13·24 11·82 100 2·14 0·50 2·66

23 64·5 20·81 3·66 12·42 10·79 100 2·16 0·46 2·37

24 — — 3·74 13·09 13·04 100 — — —

25 66·2 21·36 3·76 12·74 12·84 100 2·16 0·46 2·75

† From Quetelet.

From the first portion of the table, it will be seen that none of these

organs by any means keep pace with the body as a whole in regard to

growth in weight; in other words, there must be some other part of the

fabric, doubtless the muscles and the bones, which increase _more_

rapidly than the average increase of the body. Heart and liver both

grow nearly at the same rate, and by the {90} age of twenty-five they

have multiplied their weight at birth by about thirteen times, while

the weight of the entire body has been multiplied by about twenty-one;

but the weight of the brain has meanwhile been multiplied only

about three and a quarter times. In the next place, we see the very

remarkable phenomenon that the brain, growing rapidly till the child

is about four years old, then grows more much slowly till about eight

or nine years old, and after that time there is scarcely any further

perceptible increase. These phenomena are diagrammatically illustrated

in Fig. 18.

[Illustration: Fig. 18. Relative growth in weight (in Man) of Brain,

Heart, and whole Body.]

Many statistics indicate a decrease of brain-weight during adult life.

Boas[119] was inclined to attribute this apparent phenomenon to our

statistical methods, and to hold that it could “hardly be explained in

any other way than by assuming an increased death-rate among men with

very large brains, at an age of about twenty years.” But Raymond Pearl

has shewn that there is evidence of a steady and very gradual decline

in the weight of the brain with advancing age, beginning at or before

the twentieth year, and continuing throughout adult life[120]. {91}

The second part of the table shews the steadily decreasing weights of

the organs in question as compared with the body; the brain falling

from over 12 per cent. at birth to little over 2 per cent. at five and

twenty; the heart from ·75 to ·46 per cent.; and the liver from 4·57 to

2·75 per cent. of the whole bodily weight.

It is plain, then, that there is no simple and direct relation, holding

good _throughout life_, between the size of the body as a whole and

that of the organs we have just discussed; and the changing ratio

of magnitude is especially marked in the case of the brain, which,

as we have just seen, constitutes about one-eighth of the whole

bodily weight at birth, and but one-fiftieth at five and twenty.

The same change of ratio is observed in other animals, in equal or

even greater degree. For instance, Max Weber[121] tells us that in

the lion, at five weeks, four months, eleven months, and lastly when

full-grown, the brain-weight represents the following fractions of

the weight of the whole body, viz. 1/18, 1/80, 1/184, and 1/546. And

Kellicott has, in like manner, shewn that in the dogfish, while some

organs (e.g. rectal gland, pancreas, etc.) increase steadily and very

nearly proportionately to the body as a whole, the brain, and some

other organs also, grow in a diminishing ratio, which is capable of

representation, approximately, by a logarithmic curve[122].

But if we confine ourselves to the adult, then, as Raymond Pearl has

shewn in the case of man, the relation of brain-weight to age, to

stature, or to weight, becomes a comparatively simple one, and may be

sensibly expressed by a straight line, or simple equation.

Thus, if _W_ be the brain-weight (in grammes), and _A_ be the age,

or _S_ the stature, of the individual, then (in the case of Swedish

males) the following simple equations suffice to give the required

ratios:

_W_ = 1487·8 − 1·94  _A_ = 915·06 + 2·86  _S_.

{ 92}

These equations are applicable to ages between fifteen and eighty;

if we take narrower limits, say between fifteen and fifty, we can

get a closer agreement by using somewhat altered constants. In the

two sexes, and in different races, these empirical constants will be

greatly changed[123]. Donaldson has further shewn that the correlation

between brain-weight and body-weight is very much closer in the rat

than in man[124].

The falling ratio of weight of brain to body with increase of size or

age finds its parallel in comparative anatomy, in the general law that

the larger the animal the less is the relative weight of the brain.

Weight of Weight of

entire animal brain

gms. gms. Ratio

Marmoset 335 12·5 1 : 26

Spider monkey 1845 126 1 : 15

Felis minuta 1234 23·6 1 : 56

F. domestica 3300 31 1 : 107

Leopard 27,700 164 1 : 168

Lion 119,500 219 1 : 546

Elephant 3,048,000 5430 1 : 560

Whale (Globiocephalus) 1,000,000 2511 1 : 400

For much information on this subject, see Dubois, “Abhängigkeit des

Hirngewichtes von der Körpergrösse bei den Säugethieren,” _Arch. f.

Anthropol._ XXV, 1897. Dubois has attempted, but I think with very

doubtful success, to equate the weight of the brain with that of the

animal. We may do this, in a very simple way, by representing the

weight of the body as a _power_ of that of the brain; thus, in the

above table of the weights of brain and body in four species of cat,

if we call _W_ the weight of the body (in grammes), and _w_ the weight

of the brain, then if in all four cases we express the ratio by _W_

= _w_^{_n_}, we find that _n_ is almost constant, and differs little

from 2·24 in all four species: the values being respectively, in the

order of the table 2·36, 2·24, 2·18, and 2·17. But this evidently

amounts to no more than an empirical rule; for we can easily see

that it depends on the particular scale which we have used, and

that if the weights had been taken, for instance, in kilogrammes

or in milligrammes, the agreement or coincidence would not have

occurred[125]. {93}

_The Length of the Head in Man at various Ages._

(_After Quetelet, p. 207._)

Men Women

────────────────────────── ──────────────────────

Age Total height Head Ratio Height Head† Ratio

m. m. m. m.

Birth 0·500 0·111 4·50 0·494 0·111 4·45

1 year 0·698 0·154 4·53 0·690 0·154 4·48

2 years 0·791 0·173 4·57 0·781 0·172 4·54

3 years 0·864 0·182 4·74 0·854 0·180 4·74

5 years 0·987 0·192 5·14 0·974 0·188 5·18

10 years 1·273 0·205 6·21 1·249 0·201 6·21

15 years 1·513 0·215 7·04 1·488 0·213 6·99

20 years 1·669 0·227 7·35 1·574 0·220 7·15

30 years 1·686 0·228 7·39 1·580 0·221 7·15

40 years 1·686 0·228 7·39 1·580 0·221 7·15

† A smooth curve, very similar to this, for the growth in

“auricular height” of the girl’s head, is given by Pearson,

in _Biometrika_, III, p. 141. 1904.

As regards external form, very similar differences exist, which however

we must express in terms not of weight but of length. Thus the annexed

table shews the changing ratios of the vertical length of the head to

the entire stature; and while this ratio constantly diminishes, it will

be seen that the rate of change is greatest (or the coefficient of

acceleration highest) between the ages of about two and five years.

In one of Quetelet’s tables (_supra_, p. 63), he gives measurements

of the total span of the outstretched arms in man, from year to year,

compared with the vertical stature. The two measurements are so nearly

identical in actual magnitude that a direct comparison by means of

curves becomes unsatisfactory; but I have reduced Quetelet’s data to

percentages, and it will be seen from Fig. 19 that the percentage

proportion of span to height undergoes a remarkable and steady change

from birth to the age of twenty years; the man grows more rapidly in

stretch of arms than he does in height, and the span which was less

than {94} the stature at birth by about 1 per cent. exceeds it at the

age of twenty by about 4 per cent. After the age of twenty, Quetelet’s

data are few and irregular, but it is clear that the span goes on for

a long while increasing in proportion to the stature. How far the

phenomenon is due to actual growth of the arms and how far to the

increasing breadth of the chest is not yet ascertained.

[Illustration: Fig. 19. Ratio of stature in Man, to span of

outstretched arms.

(From Quetelet’s data.)]

The differences of rate of growth in different parts of the body

are very simply brought out by the following table, which shews the

relative growth of certain parts and organs of a young trout, at

intervals of a few days during the period of most rapid development. It

would not be difficult, from a picture of the little trout at any one

of these stages, to draw its approximate form at any other, by the help

of the numerical data here set forth[126]. {95}

_Trout (Salmo fario): proportionate growth of various organs._

(_From Jenkinson’s data._)

Days Total 1st Ventral 2nd Breadth

old length Eye Head dorsal fin dorsal Tail-fin of tail

49 100 100 100 100 100 100 100 100

63 129·9 129·4 148·3 148·6 148·5 108·4 173·8 155·9

77 154·9 147·3 189·2 (203·6) (193·6) 139·2 257·9 220·4

92 173·4 179·4 220·0 (193·2) (182·1) 154·5 307·6 272·2

106 194·6 192·5 242·5 173·2 165·3 173·4 337·3 287·7

While it is inequality of growth in _different_ directions that we can

most easily comprehend as a phenomenon leading to gradual change of

outward form, we shall see in another chapter[127] that differences of

rate at different parts of a longitudinal system, though always in the

same direction, also lead to very notable and regular transformations.

Of this phenomenon, the difference in rate of longitudinal growth

between head and body is a simple case, and the difference which

accompanies and results from it in the bodily form of the child and the

man is easy to see. A like phenomenon has been studied in much greater

detail in the case of plants, by Sachs and certain other botanists,

after a method in use by Stephen Hales a hundred and fifty years

before[128].

On the growing root of a bean, ten narrow zones were marked off,

starting from the apex, each zone a millimetre in breadth. After

twenty-four hours’ growth, at a certain constant temperature, the

whole marked portion had grown from 10 mm. to 33 mm. in length; but

the individual zones had grown at very unequal rates, as shewn in the

annexed table[129].

Zone Increment

mm.

Apex 1·5

2nd 5·8

3rd 8·2

4th 3·5

5th 1·6

6th 1·3

7th 0·5

8th 0·3

9th 0·2

10th 0·1

{96}

[Illustration: Fig. 20. Rate of growth in successive zones near the tip

of the bean-root.]

The several values in this table lie very nearly (as we see by Fig.

20) in a smooth curve; in other words a definite law, or principle of

continuity, connects the rates of growth at successive points along the

growing axis of the root. Moreover this curve, in its general features,

is singularly like those acceleration-curves which we have already

studied, in which we plotted the rate of growth against successive

intervals of time, as here we have plotted it against successive

spatial intervals of an actual growing structure. If we suppose for a

moment that the velocities of growth had been transverse to the axis,

instead of, as in this case, longitudinal and parallel with it, it is

obvious that these same velocities would have given us a leaf-shaped

structure, of which our curve in Fig. 20 (if drawn to a suitable scale)

would represent the actual outline on either side of the median axis;

or, again, if growth had been not confined to one plane but symmetrical

about the axis, we should have had a sort of turnip-shaped root, {97}

having the form of a surface of revolution generated by the same

curve. This then is a simple and not unimportant illustration of the

direct and easy passage from velocity to form.

A kindred problem occurs when, instead of “zones” artificially marked

out in a stem, we deal with the rates of growth in successive actual

“internodes”; and an interesting variation of this problem occurs when

we consider, not the actual growth of the internodes, but the varying

number of leaves which they successively produce. Where we have whorls

of leaves at each node, as in Equisetum and in many water-weeds, then

the problem presents itself in a simple form, and in one such case,

namely in Ceratophyllum, it has been carefully investigated by Mr

Raymond Pearl[130].

It is found that the mean number of leaves per whorl increases with

each successive whorl; but that the rate of increment diminishes from

whorl to whorl, as we ascend the axis. In other words, the increase

in the number of leaves per whorl follows a logarithmic ratio; and if

_y_ be the mean number of leaves per whorl, and _x_ the successional

number of the whorl from the root or main stem upwards, then

_y_ = _A_ + _C_ log(_x_ − _a_),

where _A_, _C_, and _a_ are certain specific constants, varying with

the part of the plant which we happen to be considering. On the main

stem, the rate of change in the number of leaves per whorl is very

slow; when we come to the small twigs, or “tertiary branches,” it has

become rapid, as we see from the following abbreviated table:

_Number of leaves per whorl on the tertiary branches of Ceratophyllum._

Position of whorl 1 2 3 4 5 6

Mean number of leaves 6·55 8·07 9·00 9·20 9·75 10·00

Increment — 1·52 ·93 ·20 (·55) (·25)

We have seen that a slow but definite change of form is a common

accompaniment of increasing age, and is brought about as the simple

and natural result of an altered ratio between the rates of growth in

different dimensions: or rather by the progressive change necessarily

brought about by the difference in their accelerations. There are

many cases however in which the change is all but imperceptible to

ordinary measurement, and many others in which some one dimension is

easily measured, but others are hard to measure with corresponding

accuracy. {98} For instance, in any ordinary fish, such as a plaice or

a haddock, the length is not difficult to measure, but measurements of

breadth or depth are very much more uncertain. In cases such as these,

while it remains difficult to define the precise nature of the change

of form, it is easy to shew that such a change is taking place if we

make use of that ratio of length to weight which we have spoken of in

the preceding chapter. Assuming, as we may fairly do, that weight is

directly proportional to bulk or volume, we may express this relation

in the form _W_/_L_^3 = _k_, where _k_ is a constant, to be determined

for each particular case. (_W_ and _L_ are expressed in grammes and

centimetres, and it is usual to multiply the result by some figure,

such as 1000, so as to give the constant _k_ a value near to unity.)

_Plaice caught in a certain area, March, 1907. Variation of k (the

weight-length coefficient) with size. (Data taken from the Department

of Agriculture and Fisheries’ Plaice-Report, vol._ I, _p._ 107, 1908.)

Size in cm. Weight in gm. _W_/_L_^3 × 10,000 _W_/_L_^3 (smoothed)

23 113 92·8 —

24 128 92·6 94·3

25 152 97·3 96·1

26 173 98·4 97·9

27 193 98·1 99·0

28 221 100·6 100·4

29 250 102·5 101·2

30 271 100·4 101·2

31 300 100·7 100·4

32 328 100·1 99·8

33 354 98·5 98·8

34 384 97·7 98·0

35 419 97·7 97·6

36 454 97·3 96·7

37 492 95·2 96·3

38 529 96·4 95·6

39 564 95·1 95·0

40 614 95·9 95·0

41 647 93·9 93·8

42 679 91·6 92·5

43 732 92·1 92·5

44 800 93·9 94·0

45 875 96·0 —

{99}

Now while this _k_ may be spoken of as a “constant,” having a certain

mean value specific to each species of organism, and depending on the

form of the organism, any change to which it may be subject will be a

very delicate index of progressive changes of form; for we know that

our measurements of length are, on the average, very accurate, and

weighing is a still more delicate method of comparison than any linear

measurement.

[Illustration: Fig. 21. Changes in the weight-length ratio of Plaice,

with increasing size.]

Thus, in the case of plaice, when we deal with the mean values for a

large number of specimens, and when we are careful to deal only with

such as are caught in a particular locality and at a particular time,

we see that _k_ is by no means constant, but steadily increases to a

maximum, and afterwards slowly declines with the increasing size of the

fish (Fig. 21). To begin with, therefore, the weight is increasing more

rapidly than the cube of the length, and it follows that the length

itself is increasing less rapidly than some other linear dimension;

while in later life this condition is reversed. The maximum is reached

when the length of the fish is somewhere near to 30 cm., and it is

tempting to suppose that with this “point of inflection” there is

associated some well-marked epoch in the fish’s life. As a matter of

fact, the size of 30 cm. is approximately that at which sexual maturity

may be said to begin, or is at least near enough to suggest a close

connection between the two phenomena. The first step towards further

investigation of the {100} apparent coincidence would be to determine

the coefficient _k_ of the two sexes separately, and to discover

whether or not the point of inflection is reached (or sexual maturity

is reached) at a smaller size in the male than in the female plaice;

but the material for this investigation is at present scanty.

[Illustration: Fig. 22. Periodic annual change in the weight-length

ratio of Plaice.]

A still more curious and more unexpected result appears when we compare

the values of _k_ for the same fish at different seasons of the

year[131]. When for simplicity’s sake (as in the accompanying table and

Fig. 22) we restrict ourselves to fish of one particular size, it is

not necessary to determine the value of _k_, because a change in the

ratio of length to weight is obvious enough; but when we have small

numbers, and various sizes, to deal with, the determination of _k_ may

help us very much. It will be seen, then, that in the case of plaice

the ratio of weight to length exhibits a regular periodic variation

with the course of the seasons. {101}

_Relation of Weight to Length in Plaice of 55 cm. long, from Month to

Month. (Data taken from the Department of Agriculture and Fisheries

Plaice-Report, vol._ II, _p._ 92, 1909.)

Average weight

in grammes _W_/_L_^3 × 100 _W_/_L_^3 (smoothed)

Jan. 2039 1·226 1·157

Feb. 1735 1·043 1·080

March 1616 0·971 0·989

April 1585 0·953 0·967

May 1624 0·976 0·985

June 1707 1·026 1·005

July 1686 1·013 1·037

August 1783 1·072 1·042

Sept. 1733 1·042 1·111

Oct. 2029 1·220 1·160

Nov. 2026 1·218 1·213

Dec. 1998 1·201 1·215

With unchanging length, the weight and therefore the bulk of the fish

falls off from about November to March or April, and again between

May or June and November the bulk and weight are gradually restored.

The explanation is simple, and depends wholly on the process of

spawning, and on the subsequent building up again of the tissues and

the reproductive organs. It follows that, by this method, without ever

seeing a fish spawn, and without ever dissecting one to see the state

of its reproductive system, we can ascertain its spawning season, and

determine the beginning and end thereof, with great accuracy.

――――――――――

As a final illustration of the rate of growth, and of unequal growth in

various directions, I give the following table of data regarding the

ox, extending over the first three years, or nearly so, of the animal’s

life. The observed data are (1) the weight of the animal, month by

month, (2) the length of the back, from the occiput to the root of the

tail, and (3) the height to the withers. To these data I have added (1)

the ratio of length to height, (2) the coefficient (_k_) expressing

the ratio of weight to the cube of the length, and (3) a similar

coefficient (_k′_) for the height of the animal. It will be seen that,

while all these ratios tend to alter continuously, shewing that the

animal’s form is steadily altering as it approaches maturity, the

ratio between length and weight {102} changes comparatively little.

The simple ratio between length and height increases considerably, as

indeed we should expect; for we know that in all Ungulate animals the

legs are remarkably

_Relations between the Weight and certain Linear Dimensions of the Ox.

(Data from Przibram, after Cornevin†.)_

_k_ = _k′_ =

Age in _W_, wt. _L_, length _H_, _W_/_L_^3 _W_/_H_^3

months in kg. of back height _L_/_H_  × 10  × 10

0 37 ·78 ·70 1·114 ·779 1·079

1 55·3 ·94 ·77 1·221 ·665 1·210

2 86·3 1·09 ·85 1·282 ·666 1·406

3 121·3 1·207 ·94 1·284 ·690 1·460

4 150·3 1·314 ·95 1·383 ·662 1·754

5 179·3 1·404 1·040 1·350 ·649 1·600

6 210·3 1·484 1·087 1·365 ·644 1·638

7 247·3 1·524 1·122 1·358 ·699 1·751

8 267·3 1·581 1·147 1·378 ·677 1·791

9 282·8 1·621 1·162 1·395 ·664 1·802

10 303·7 1·651 1·192 1·385 ·675 1·793

11 327·7 1·694 1·215 1·394 ·674 1·794

12 350·7 1·740 1·238 1·405 ·666 1·849

13 374·7 1·765 1·254 1·407 ·682 1·900

14 391·3 1·785 1·264 1·412 ·688 1·938

15 405·9 1·804 1·270 1·420 ·692 1·982

16 417·9 1·814 1·280 1·417 ·700 2·092

17 423·9 1·832 1·290 1·420 ·689 1·974

18 423·9 1·859 1·297 1·433 ·660 1·943

19 427·9 1·875 1·307 1·435 ·649 1·916

20 437·9 1·884 1·311 1·437 ·655 1·944

21 447·9 1·893 1·321 1·433 ·661 1·943

22 464·4 1·901 1·333 1·426 ·676 1·960

23 480·9 1·909 1·345 1·419 ·691 1·977

24 500·9 1·914 1·352 1·416 ·714 2·027

25 520·9 1·919 1·359 1·412 ·737 2·075

26 534·1 1·924 1·361 1·414 ·750 2·119

27 547·3 1·929 1·363 1·415 ·762 2·162

28 554·5 1·929 1·363 1·415 ·772 2·190

29 561·7 1·929 1·363 1·415 ·782 2·218

30 586·2 1·949 1·383 1·409 ·792 2·216

31 610·7 1·969 1·403 1·403 ·800 2·211

32 625·7 1·983 1·420 1·396 ·803 2·186

33 640·7 1·997 1·437 1·390 ·805 2·159

34 655·7 2·011 1·454 1·383 ·806 2·133

† Cornevin, Ch., Études sur la croissance, _Arch. de

Physiol. norm. et pathol._ (5), IV, p. 477, 1892.

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