The principle of unit-characters and of elementary species leads at once
to the recognition of two kinds of variability. The changes of wider
amplitude consist of the acquisition of new units, or the loss of
already existing ones. The lesser variations are due to the degree of
activity of the units themselves.
Facts illustrative of these distinctions were almost wholly lacking at
the time of the first publication of Darwin's theories. It was a bold
conception to point out the necessity for such distinction on purely
theoretical grounds. Of course some sports were well known and
fluctuations were evident, but no exact analysis of the details was
possible, a fact that was of great importance in the demonstration of
the theory of descent. The lack of more definite knowledge upon this
matter was keenly felt by Darwin, [716] and exercised much influence
upon his views at various times.
Quetelet's famous discovery of the law of fluctuating variability
changed the entire situation and cleared up many difficulties. While a
clear conception of fluctuations was thus gained, mutations were
excluded from consideration, being considered as very rare, or
non-existent. They seemed wholly superfluous for the theory of descent,
and very little importance was attached to their study. Current
scientific belief in the matter has changed only in recent years.
Mendel's law of varietal hybrids is based upon the principle of
unit-characters, and the validity of this conception has thus been
brought home to many investigators.
A study of fluctuating or individual variability, as it was formerly
called, is now carried on chiefly by mathematical methods. It is not my
purpose to go into details, as it would require a separate course of
lectures. I shall consider the limits between fluctuation and mutation
only, and attempt to set forth an adequate idea of the principles of the
first as far as they touch these limits. The mathematical treatment of
the facts is no doubt of very great value, but the violent discussions
now going on between mathematicians such as Pearson, Kapteyn and others
should warn biologists to abstain [717] from the use of methods which
are not necessary for the furtherance of experimental work.
Fortunately, Quetelet's law is a very clear and simple one, and quite
sufficient for our considerations. It claims that for biologic phenomena
the deviations from the average comply with the same laws as the
deviations from the average in any other case, if ruled by chance only.
The meaning of this assertion will become clear by a further discussion
of the facts. First of all, fluctuating variability is an almost
universal phenomenon. Every organ and every quality may exhibit it. Some
are very variable, while others seem quite constant. Shape and size vary
almost indefinitely, and the chemical composition is subject to the same
law, as is well known for the amount of sugar in sugar-beets. Numbers
are of course less liable to changes, but the numbers of the rays of
umbels, or ray-florets in the composites, of pairs of blades in pinnate
leaves, and even of stamens and carpels are known to be often
exceedingly variable. The smaller numbers however, are more constant,
and deviations from the quinate structure of flowers are rare.
Complicated structures are generally capable of only slight deviations.
From a broad point of view, fluctuating variability [718] falls under
two heads. They obey quite the same laws and are therefore easily
confused, but with respect to questions of heredity they should be
carefully separated. They are designated by the terms individual and
partial fluctuation. Individual variability indicates the differences
between individuals, while partial variability is limited to the
deviations shown by the parts of one organism from the average
structure. The same qualities in some cases vary individually and in
others partially. Even stature, which is as markedly individual for
annual and biennial plants as it is for man, becomes partially variant
in the case of perennial herbs with numbers of stems. Often a character
is only developed once in the whole course of evolution, as for
instance, the degree of connation of the seed-leaves in tricotyls and in
numerous cases it is impossible to tell whether a character is
individual or partial. Consequently such minute details are generally
considered to have no real importance for the hereditary transmission of
the character under discussion.
Fluctuations are observed to take place only in two directions. The
quality may increase or decrease, but is not seen to vary in any other
way. This rule is now widely established by numerous investigations, and
is fundamental to [719] the whole method of statistical investigation.
It is equally important for the discussion of the contrast between
fluctuations and mutations, and for the appreciation of their part in
the general progress of organization. Mutations are going on in all
directions, producing, if they are progressive, something quite new
every time. Fluctuations are limited to increase and decrease of what is
already available. They may produce plants with higher stems, more
petals in the flowers, larger and more palatable fruits, but obviously
the first petal and the first berry, cannot have originated by the
simple increase of some older quality. Intermediates may be found, and
they may mark the limit, but the demonstration of the absence of a limit
is quite another question. It would require the two extremes to be shown
to belong to one unit, complying with the simple law of Quetelet.
Nourishment is the potent factor of fluctuating variability. Of course
in thousands of cases our knowledge is not sufficient to allow us to
analyze this relation, and a number of phases of the phenomenon have
been discovered only quite recently. But the fact itself is thoroughly
manifest, and its appreciation is as old as horticultural science.
Knight, who lived at the beginning of the last century, has laid great
stress upon it, and it has since influenced practice in a [720] large
measure. Moreover, Knight pointed out more than once that it is the
amount of nourishment, not the quality of the various factors, that
exercises the determinative influence. Nourishment is to be taken in the
widest sense of the word, including all favorable and injurious
elements. Light and temperature, soil and space, water and salts are
equally active, and it is the harmonious cooperation of them all that
rules growth.
We treated this important question at some length, when dealing with the
anomalies of the opium-poppies, consisting of the conversion of stamens
into supernumerary pistils. The dependency upon external influences
which this change exhibited is quite the same as that shown by
fluctuating variability at large. We inquired into the influence of good
and bad soil, of sunlight and moisture and of other concurrent factors.
Especial emphasis was laid upon the great differences to which the
various individuals of the same lot may be exposed, if moisture and
manure differ on different portions of the same bed in a way unavoidable
even by the most careful preparation. Some seeds germinate on moist and
rich spots, while their neighbors are impeded by local dryness, or by
distance from manure. Some come to light on a sunny day, and increase
their first leaves rapidly, while on [721] the following day the weather
may be unfavorable and greatly retard growth. The individual differences
seem to be due, at least in a very great measure, to such apparent
trifles.
On the other hand partial differences are often manifestly due to
similar causes. Considering the various stems of plants, which multiply
themselves by runners or by buds on the roots, the assertion is in no
need of further proof. The same holds good for all cases of artificial
multiplication by cuttings, or by other vegetative methods. But even if
we limit ourselves to the leaves of a single tree, or the branches of a
shrub, or the flowers on a plant, the same rule prevails. The
development of the leaves is dependent on their position, whether
inserted on strong or weak branches, exposed to more or less light, or
nourished by strong or weak roots. The vigor of the axillary buds and of
the branches which they may produce is dependent upon the growth and
activity of the leaves to which the buds are axillary.
This dependency on local nutrition leads to the general law of
periodicity, which, broadly speaking, governs the occurrence of the
fluctuating deviations of the organs. This law of periodicity involves
the general principle that every axis, as a rule, increases in strength
when [722] growing, but sooner or later reaches a maximum and may
afterwards decrease.
This periodic augmentation and declination is often boldly manifest,
though in other cases it may be hidden by the effect of alternate
influences. Pinnate leaves generally have their lower blades smaller
than the upper ones, the longest being seen sometimes near the apex and
sometimes at a distance from it. Branches bearing their leaves in two
rows often afford quite as obvious examples, and shoots in general
comply with the same rule. Germinating plants are very easy of
observation on this point. When they are very weak they produce only
small leaves. But their strength gradually increases and the subsequent
organs reach fuller dimensions until the maximum is attained. The
phenomenon is so common that its importance is usually overlooked. It
should be considered as only one instance of a rule, which holds good
for all stems and all branches, and which is everywhere dependent on the
relation of growth to nutrition.
The rule of periodicity not only affects the size of the organs, but
also their number, whenever these are largely variable. Umbellate plants
have numerous rays on the umbels of strong stems, but the number is seen
to decrease and to become very small on the weakest lateral [723]
branches. The same holds good for the number of ray-florets in the
flower-heads of the composites, even for the number of stigmas on the
ovaries of the poppies, which on weak branches may be reduced to as few
as three or four. Many other instances could be given.
One of the best authenticated cases is the dependency of partial
fluctuation on the season and on the weather. Flowers decline when the
season comes to an end, become smaller and less brightly colored. The
number of ray-florets in the flower-heads is seen to decrease towards
the fall. Extremes become rarer, and often the deviations from the
average seem nearly to disappear. Double flowers comply with this rule
very closely, and many other cases will easily occur to any student of
nature.
Of course, the relation to nourishment is different for individual and
partial fluctuations. Concerning the first, the period of development of
the germ within the seed is decisive. Even the sexual cells may be in
widely different conditions at the moment of fusion, and perhaps this
state of the sexual cells includes the whole matter of the decision for
the average characters of the new individual. Partial fluctuation
commences as soon as the leaves and buds begin to form, and all later
changes in nutrition can only cause partial differences. All leaves,
[724] buds, branches, and flowers must come under the influence of
external conditions during the juvenile period, and so are liable to
attain a development determined in part by the action of these factors.
Before leaving these general considerations, we must direct our
attention to the question of utility. Obviously, fluctuating variability
is a very useful contrivance, in many cases at least. It appears all the
more so, as its relation to nutrition becomes manifest. Here two aspects
are intimately combined. More nutrient matter produces larger leaves and
these are in their turn more fit to profit by the abundance of
nourishment. So it is with the number of flowers and flower-groups, and
even with the numbers of their constituent organs. Better nourishment
produces more of them, and thereby makes the plant adequate to make a
fuller use of the available nutrient substances. Without fluctuation
such an adjustment would hardly be possible, and from all our notions of
usefulness in nature, we therefore must recognize the efficiency of this
form of variability.
In other respects the fluctuations often strike us as quite useless or
even as injurious. The numbers of stamens, or of carpels are dependent
on nutrition, but their fluctuation is not known to have any attraction
for the visiting insects.
[725] If the deviations become greater, they might even become
detrimental. The flowers of the St. Johnswort, or _Hypericum
perforatum_, usually have five petals, but the number varies from three
to eight or more. Bees could hardly be misled by such deviations. The
carpels of buttercups and columbines, the cells in the capsules of
cotton and many other plants are variable in number. The number of seeds
is thereby regulated in accordance with the available nourishment, but
whether any other useful purpose is served, remains an open question.
Variations in the honey-guides or in the pattern of color-designs might
easily become injurious by deceiving insects, and such instances as the
great variability of the spots on the corolla of some cultivated species
of monkey-flowers, for instance, the _Mimulus quinquevulnerus_, could
hardly be expected to occur in wild plants. For here the dark brown
spots vary between nearly complete deficiency up to such predominancy as
almost to hide the pale yellow ground-color.
After this hasty survey of the causes of fluctuating variability, we now
come to a discussion of Quetelet's law. It asserts that the deviations
from the average obey the law of probability. They behave as if they
were dependent on chance only.
Everyone knows that the law of Quetelet can [726] be demonstrated the
most readily by placing a sufficient number of adult men in a row,
arranging them according to their size. The line passing over their
heads proves to be identical with that given by the law of probability.
Quite in the same way, stems and branches, leaves and petals and even
fruits can be arranged, and they will in the main exhibit the same line
of variability. Such groups are very striking, and at the first glance
show that the large majority of the specimens deviate from the mean only
to a very small extent. Wider deviations are far more rare, and their
number lessens, the greater the deviation, as is shown by the curvature
of the line. It is almost straight and horizontal in the middle portion,
while at the ends it rapidly declines, going sharply downward at one
extreme and upward at the other.
It is obvious however, that in these groups the leaves and other organs
could conveniently be replaced by simple lines, indicating their size.
The result would be quite the same, and the lines could be placed at
arbitrary, but equal distances. Or the sizes could be expressed by
figures, the compliance of which with the general law could be
demonstrated by simple methods of calculation. In this manner the
variability of different organs can easily be compared. Another method
of demonstration consists in [727] grouping the deviations into
previously fixed divisions. For this purpose the variations are measured
by standard units, and all the instances that fall between two limits
are considered to constitute one group. Seeds and small fruits, berries
and many other organs may conveniently be dealt with in this way. As an
example we take ordinary beans and select them according to their size.
This can be done in different ways. On a small piece of board a long
wedge-shaped slit is made, into which seeds are pushed as far as
possible. The margin of the wedge is calibrated in such a manner that
the figures indicate the width of the wedge at the corresponding place.
By this device the figure up to which a bean is pushed at once shows its
length. Fractions of millimeters are neglected, and the beans, after
having been measured, are thrown into cylindrical glasses of the same
width, each glass receiving only beans of equal length. It is clear that
by this method the height to which beans fill the glasses is
approximately a measure of their number. If now the glasses are put in a
row in the proper sequence, they at once exhibit the shape of a line
which corresponds to the law of chance. In this case however, the line
is drawn in a different manner from the first. It is to be pointed out
that the glasses may be replaced by lines indicating [728] the height of
their contents, and that, in order to reach a more easy and correct
statement, the length of the lines may simply be made proportionate to
the number of the beans in each glass. If such lines are erected on a
common base and at equal distances, the line which unites their upper
ends will be the expression of the fluctuating variability of the
character under discussion.
The same inquiry may be made with other seeds, with fruits, or other
organs. It is quite superfluous to arrange the objects themselves, and
it is sufficient to arrange the figures indicating their value. In order
to do this a basal line is divided into equal parts, the demarcations
corresponding to the standard-units chosen for the test. The observed
values are then written above this line, each finding its place between
the two demarcations, which include its value. It is very interesting
and stimulating to construct such a group. The first figures may fall
here and there, but very soon the vertical rows on the middle part of
the basal line begin to increase. Sometimes ten or twenty measurements
will suffice to make the line of chance appear, but often indentations
will remain. With the increasing number of the observations the
irregularities gradually [729] disappear, and the line becomes smoother
and more uniformly curved.
This method of arranging the figures directly on a basal line is very
convenient, whenever observations are made in the field or garden. Very
few instances need be recorded to obtain an appreciation of the mean
value, and to show what may be expected from a continuance of the test.
The method is so simple and so striking, and so wholly independent of
any mathematical development that it should be applied in all cases in
which it is desired to ascertain the average value of any organ, and the
measure of the attendant deviations.
I cite an instance, secured by counting the ray-florets on the
flower-heads of the corn-marigold or _Chrysanthemum segetum_. It was
that, by which I was enabled to select the plant, which afterwards
showed the first signs of a double head. I noted them in this way;
47
47 52
41 54 68
44 50 62 75
36 45 58 65 72 __ 99
Of course the figures might be replaced in this work by equidistant dots
or by lines, but experience teaches that the chance of making mistakes
is noticeably lessened by writing down [730] the figures themselves.
Whenever decimals are made use of it is obviously the best plan to keep
the figures themselves. For afterwards it often becomes necessary to
arrange them according to a somewhat different standard.
Uniting the heads of the vertical rows of figures by a line, the form
corresponding to Quetelet's law is easily seen. In the main it is always
the same as the line shown by the measurements of beans and seeds. It
proves a dense crowding of the single instances around the average, and
on both sides of the mass of the observations, a few wide deviations.
These become more rare in proportion to the amount of their divergency.
On both sides of the average the line begins by falling very rapidly,
but then bends slowly so as to assume a nearly horizontal direction. It
reaches the basal line only beyond the extreme instances.
It is quite evident that all qualities, which can be expressed by
figures, may be treated in this way. First, of all the organs occurring
in varying numbers, as for instance the ray-florets of composites, the
rays of umbels, the blades of pinnate and palmate leaves, the numbers of
veins, etc., are easily shown to comply with the same general rule.
Likewise the amount of chemical substances can be expressed in
percentage numbers, as is done on a large [731] scale with sugar in
beets and sugar-cane, with starch in potatoes and in other instances.
These figures are also found to follow the same law.
All qualities which are seen to increase and to decrease may be dealt
with in the same manner, if a standard unit for their measurement can be
fixed. Even the colors of flowers may not escape our inquiry.
If we now compare the lines, compiled from the most divergent cases,
they will be found to exhibit the same features in the main. Ordinarily
the curve is symmetrical, the line sloping down on both sides after the
same manner. But it is not at all rare that the inclination is steep on
one side and gradual on the other. This is noticeably the case if the
observations relate to numbers, the average of which is near zero. Here
of course the allowance for variation is only small on one side, while
it may increase with out distinct limits on the alternate slope. So it
is for instance with the numbers of ray-florets in the example given on
p. 729. Such divergent cases, however, are to be considered as
exceptions to the rule, due to some unknown cause.
Heretofore we have discussed the empirical side of the problem only. For
the purpose of experimental study of questions of heredity this is
ordinarily quite sufficient. The inquiry [732] into the phenomenon of
regression, or of the relation of the degree of deviation of the progeny
to that of their parents, and the selection of extreme instances for
multiplication are obviously independent of mathematical considerations.
On the other hand an important inquiry lies in the statistical treatment
of these phenomena, and such treatment requires the use of mathematical
methods.
Statistics however, are not included in the object of these lectures,
and therefore I shall refrain from an explanation of the method of their
preparation and limit myself to a general comparison of the observed
lines with the law of chance. Before going into the details, it should
be repeated once more that the empirical result is quite the same for
individual and for partial fluctuations. As a rule, the latter occur in
far greater number, and are thus more easily investigated, but
individual or personal averages have also been studied.
Newton discovered that the law of chance can be expressed by very simple
mathematical calculations. Without going into details, we may at once
state that these calculations are based upon his binomium. If the form
(a + b) is calculated for some value of the exponent, and if the values
of the coefficients after development are alone considered, they yield
the basis [733] for the construction of what is called the line or curve
of probability. For this construction the coefficients are used as
ordinates, the length of which is to be made proportionate to their
value. If this is done, and the ordinates are arranged at equal
distances, the line which unites their summits is the desired curve. At
first glance it exhibits a form quite analogous to the curves of
fluctuating variability, obtained by the measurements of beans and in
other instances. Both lines are symmetrical and slope rapidly down in
the region of the average, while with increasing distance they gradually
lose their steep inclination, becoming nearly parallel to the base at
their termination.
This similarity between such empirical and theoretical lines is in
itself an empirical fact. The causes of chance are assumed to be
innumerable, and the whole calculation is based on this assumption. The
causes of the fluctuations of biological phenomena have not as yet been
critically examined to such an extent as to allow of definite
conceptions. The term nourishment manifestly includes quite a number of
separate factors, as light, space, temperature, moisture, the physical
and chemical conditions of the soil and the changes of the weather.
Without doubt the single factors are very numerous, but whether they are
numerous enough to be treated [734] as innumerable, and thereby to
explain the laws of fluctuations, remains uncertain. Of course the
easiest way is to assume that they combine in the same manner as the
causes of chance, and that this is the ground of the similarity of the
curves. On the other hand, it is manifestly of the highest importance to
inquire into the part the several factors play in the determination of
the curves. It is not at all improbable that some of them have a larger
influence on individual, and others on partial, fluctuations. If this
were the case, their importance with respect to questions of heredity
might be widely different. In the present state of our knowledge the
fluctuation-curves do not contribute in any large measure to an
elucidation of the causes. Where these are obvious, they are so without
statistics, exactly as they were, previous to Quetelet's discovery.
In behalf of a large number of questions concerning heredity and
selection, it is very desirable to have a somewhat closer knowledge of
these curves. Therefore I shall try to point out their more essential
features, as far as this can be done without mathematical calculations.
At a first glance three points strike us, the average or the summit of
the curve, and the extremes. If the general shape is once denoted by the
results of observations or by the coefficients [735] of the binomium,
all further details seem to depend upon them. In respect to the average
this is no doubt the case; it is an empirical value without need of any
further discussion. The more the number of the observations increases,
the more assured and the more correct is this mean value, but generally
it is the same for smaller and for larger groups of observations.
This however, is not the case with the extremes. It is quite evident
that small groups have a chance of containing neither of them. The more
the number of the observations increases, the larger is the chance of
extremes. As a rule, and excluding exceptional cases, the extreme
deviations will increase in proportion to the number of cases examined.
In a hundred thousand beans the smallest one and the largest one may be
expected to differ more widely from one another than in a few hundred
beans of the same sample. Hence the conclusion that extremes are not a
safe criterion for the discussion of the curves, and not at all adequate
for calculations, which must be based upon more definite values.
A real standard is afforded by the steepness of the slope. This may be
unequal on the two sides of one curve, and likewise it may differ for
different cases. This steepness is usually measured by means of a point
on the half curve and [736 ] for this purpose a point is chosen which
lies exactly half way between the average and the extreme. Not however
half way with respect to the amplitude of the extreme deviation, for on
this ground it would partake of the uncertainty of the extreme itself.
It is the point on the curve which is surpassed by half the number, and
not reached by the other half of the number of the observations included
in the half of the curve. This point corresponds to the important value
called the probable error, and was designated by Galton as the quartile.
For it is evident that the average and the two quartiles divide the
whole of the observations into four equal parts.
Choosing the quartiles as the basis for calculations we are independent
of all the secondary causes of error, which necessarily are inherent in
the extremes. At a casual examination, or for demonstrative purposes,
the extremes may be prominent, but for all further considerations the
quartiles are the real values upon which to rest calculations.
Moreover if the agreement with the law of probability is once conceded,
the whole curve is defined by the average and the quartiles, and the
result of hundreds of measurements or countings may be summed up in
three, or, in [737] the case of symmetrical curves, perhaps in two
figures.
Also in comparing different curves with one another, the quartiles are
of great importance. Whenever an empirical fluctuation-curve is to be
compared with the theoretical form, or when two or more cases of
variability are to be considered under one head, the lines are to be
drawn on the same base. It is manifest that the averages must be brought
upon the same ordinate, but as to the steepness of the line, much
depends on the manner of plotting. Here we must remember that the mutual
distance of the ordinates has been a wholly arbitrary one in all our
previous considerations. And so it is, as long as only one curve is
considered at a time. But as soon as two are to be compared, it is
obvious that free choice is no longer allowed. The comparison must be
made on a common basis, and to this effect the quartiles must be brought
together. They are to lie on the same ordinates. If this is done, each
division of the base corresponds to the same proportionate number of
individuals, and a complete comparison is made possible.
On the ground of such a comparison we may thus assert that,
fluctuations, however different the organs or qualities observed, are
the same whenever their curves are seen to overlap one [738] another.
Furthermore, whenever an empirical curve agrees in this manner with the
theoretical one, the fluctuation complies with Quetelet's law, and may
be ascribed to quite ordinary and universal causes. But if it seems to
diverge from this line, the cause of this divergence should be inquired
into.
Such abnormal curves occur from time to time, but are rare.
Unsymmetrical instances have already been alluded to, and seem to be
quite frequent. Another deviation from the rule is the presence of more
than one summit. This case falls under two headings. If the ray florets
of a composite are counted, and the figures brought into a curve, a
prominent summit usually corresponds to the average. But next to this,
and on both sides, smaller summits are to be seen. On a close inspection
these summits are observed to fall on the same ordinates, on which, in
the case of allied species, the main apex lies. The specific character
of one form is thus repeated as a secondary character on an allied
species. Ludwig discovered that these secondary summits comply with the
rule discovered by Braun and Schimper, stating the relation of the
subsequent figures of the series. This series gives the terms of the
disposition of leaves in general, and of the bracts and flowers on the
composite flower [739] heads in our particular case. It is the series to
which we have already alluded when dealing with the arrangement of the
leaves on the twisted teasels. It commences with 1 and 2 and each
following figure is equal to the sum of its two precedents. The most
common figures are 3, 5, 8, 13, 18, 21, higher cases seldom coming under
observation. Now the secondary summits of the ray-curves of the
composites are seen to agree, as a rule, with these figures. Other
instances could readily be given.
Our second heading includes those cases which exhibit two summits of
equal or nearly equal height. Such cases occur when different races are
mixed, each retaining its own average and its own curve-summit. We have
already demonstrated such a case when dealing with the origin of our
double corn-chrysanthemum. The wild species culminates with 13 rays, and
the grandiflorum variety with 21. Often the latter is found to be
impure, being mixed with the typical species to a varying extent. This
is not easily ascertained by a casual inspection of the cultures, but
the true condition will promptly betray itself, if curves are
constructed. In this way curves may in many instances be made use of to
discover mixed races. Double curves may also result from the
investigation [740] of true double races, or ever-sporting varieties.
The striped snapdragon shows a curve of its stripes with two summits,
one corresponding to the average striped flowers, and the other to the
pure red ones. Such cases may be discovered by means of curves, but the
constituents cannot be separated by culture-experiments.
A curious peculiarity is afforded by half curves. The number of petals
is often seen to vary only in one direction from what should be expected
to be the mean condition. With buttercups and brambles and many others
there is only an increase above the typical five; quaternate flowers are
wanting or at least are very rare. With weigelias and many others the
number of the tips of the corolla varies downwards, going from five to
four and three. Hundreds of flowers show the typical five, and determine
the summit of the curve. This drops down on one side only, indicating
unilateral variability, which in many cases is due to a very intimate
connection of a concealed secondary summit and the main one. In the case
of the bulbous buttercup, _Ranunculus bulbosus_, I have succeeded in
isolating this secondary summit, although not in a separate variety, but
only in a form corresponding to the type of ever-sporting varieties.
[741] Recapitulating the results of this too condensed discussion, we
may state that fluctuations are linear, being limited to an increase and
to a decrease of the characters. These changes are mainly due to
differences in nourishment, either of the whole organism or of its
parts. In the first case, the deviations from the mean are called
individual; they are of great importance for the hereditary characters
of the offspring. In the second case the deviations are far more
universal and far more striking, but of lesser importance. They are
called partial fluctuations.
All these fluctuations comply, in the main, with the law of probability,
and behave as if their causes were influenced only by chance.
[742]