CO > UJ
(< OU_1 62958 >m
OSMAN1A KNIVEKSITY XJBBARY
OaU No, &"7& - &/&^A'&%([2l Accession No,
Author
Title , _ ^
This book should be t^ttMf7ie^(on or before the date last marked! b^low
BY THE SAME AUTHOR
STATISTICAL METHODS
FOR RESEARCH WORKERS
Third Edition, 1930. Oliver and
Boyd, Edinburgh
THE
GENETICAL THEORY OF
NATURAL SELECTION
OXFORD UNIVERSITY PRESS
AMEN HOUSE, E.G. 4
LONDON EDINBURGH GLASGOW
LEIPZIG NEW YORK TORONTO
MELBOURNE CAPETOWN BOMBAY
CALCUTTA MADRAS SHANGHAI
HUMPHREY MILFORD
PUBLISHER TO THE
UNIVERSITY
PLATE I. MODELS AND MIMICS IN AUSTRALIAN (Figs. 1-3) AND
TROPICAL AMERICAN (Figs. 4-7) INSECTS
For description of the frontispiece see p. xiii
THE
GENETICAL THEORY OF
NATURAL SELECTION
BY
R. A. FISHER, Sc.D., F.R.S.
OXFORD
AT THE CLARENDON PRESS
193O
Printed in Great Britain
TO
MAJOR LEONARD DARWIN
In gratitude for the encouragement,
given to the author, during the last
fifteen years, by discussing many
of the problems dealt with
in this book
PEEFACE
NATURAL Selection is not Evolution. Yet, ever since the two
words have been in common use, the theory of Natural Selection
has been employed as a convenient abbreviation for the theory of
Evolution by means of Natural Selection, put forward by Darwin
and Wallace. This has had the unfortunate consequence that the
theory of Natural Selection itself has scarcely ever, if ever, received
separate consideration. To draw a physical analogy, the laws of con-
duction of heat in solids might be deduced from the principles of
statistical mechanics, yet it would have been an unfortunate limita-
tion, involving probably a great deal of confusion, if statistical
mechanics had only received consideration in connexion with the
conduction of heat. In this case it is clear that the particular physical
phenomena examined are of little theoretical interest compared to
the principle by which they can be elucidated. The overwhelming
importance of evolution to the biological sciences partly explains
why the theory of Natural Selection should have been so fully identi-
fied with its role as an evolutionary agency, as to have suffered neglect
as an independent principle worthy of scientific study.
The other biological theories which have been put forward, either
as auxiliaries, or as the sole means of organic evolution, are not quite
in the same position. For advocates of Natural Selection have not
failed to point out, what was evidently the chief attraction of the
theory to Darwin and Wallace, that it proposes to give an account
of the means of modification in the organic world by reference only
to * known ', or independently demonstrable, causes. The alternative
theories of modification rely, avowedly, on hypothetical properties
of living matter which are inferred from the facts of evolution them-
selves. Yet, although this distinction has often been made clear, its
logical cogency could never be fully developed in the absence of a
separate investigation of the independently demonstrable modes of
causation which are claimed as its basis. The present book, with all
the limitations of a first attempt, is at least an attempt to consider
the theory of Natural Selection on its own merits.
When the theory was first put forward, by far the vaguest element
in its composition was the principle of inheritance. No man of learn-
ing or experience could deny this principle, yet, at the time, no
approach could be given to an exact account of its working. That an
3653
viii PREFACE
independent study of Natural Selection is now possible is principally
due to the great advance which our generation has seen in the science
of genetics. It deserves notice that the first decisive experiments,
which opened out in biology this field of exact study, were due to
a young mathematician, Gregor Mendel, whose statistical interests
extended to the physical and biological sciences. It is well known
that his experiments were ignored, to his intense disappointment,
and it is to be presumed that they were never brought under the
notice of any man whose training qualified him to appreciate their
importance. It is no less remarkable that when, in 1900, the genetic
facts had been rediscovered by De Vries, Tschermak, and Correns, and
the importance of Mendel's work was at last recognized, the principal
opposition should have been encountered from the small group of
mathematical statisticians then engaged in the study of heredity.
The types of mind which result from training in mathematics and
in biology certainly differ profoundly; but the difference does not
seem to lie in the intellectual faculty. It would certainly be a mistake
to say that the manipulation of mathematical symbols requires more
intellect than original thought in biology ; on the contrary, it seems
much more comparable to the manipulation of the microscope and
its appurtenances of stains and fixatives ; whilst original thought in
both spheres represents very similar activities of an identical faculty.
This accords with the view that the intelligence, properly speaking,
is little influenced by the effects of training. What is profoundly
susceptible of training is the imagination, and mathematicians and
biologists seem to differ enormously in the manner in which their
imaginations are employed. Most biologists will probably feel that
this advantage is all on their side. They are introduced early to the
immense variety of living things ; their first dissections, even if only
of the frog or dog fish, open up vistas of amazing complexity and
interest, at the time when the mathematician seems to be dealing
only with the barest abstractions, with lines and points, infinitely
thin laminae, and masses concentrated at ideal centres of gravity.
Perhaps I can best make clear that the mathematician's imagination
also has been trained to some advantage, by quoting a remark
dropped casually by Eddington in a recent book
* We need scarcely add that the contemplation in natural science of a
wider domain than the actual leads to a far better understanding of the
actual.* (p. 267, The Nature of the Physical World.)
PREFACE ix
For a mathematician the statement is almost a truism. From a
biologist, speaking of his own subject, it would suggest an extra-
ordinarily wide outlook. No practical biologist interested in
sexual reproduction would be led to work out the detailed con-
sequences experienced by organisms having three or more sexes;
yet what else should he do if he wishes to understand why the sexes
are, in fact, always two ? The ordinary mathematical procedure in
dealing with any actual problem is, after abstracting what are
believed to be the essential elements of the problem, to consider it
as one of a system of possibilities infinitely wider than the actual,
the essential relations of which may be apprehended by generalized
reasoning, and subsumed in general formulae, which may be applied
at will to any particular case considered. Even the word possibilities
in this statement unduly limits the scope of the practical procedures
in which he is trained ; for he is early made familiar with the advan-
tages of imaginary solutions, and can most readily think of a wave, or
an alternating current, in terms of the square root of minus one.
The most serious difficulty to intellectual co-operation would seem
to be removed if it were clearly and universally recognized that the
essential difference lies, not in intellectual methods, and still less
in intellectual ability, but in an enormous and specialized extension
of the imaginative faculty, which each has experienced in relation
to the needs of his special subject. I can imagine no more beneficial
change in scientific education than that which would allow each to
appreciate something of the imaginative grandeur of the realms of
thought explored by the other.
In the future, the revolutionary effect of Mendelisin will be seen
to flow from the particulate character of the hereditary elements.
On this fact a rational theory of Natural Selection can be based, and
it is, therefore, of enormous importance. The merit for this discovery
must mainly rest with Mendel, whilst among our countrymen, Bateson
played the leading part in its early advocacy. Unfortunately he was
unprepared to recognize the mathematical or statistical aspects of
biology, and from this and other causes he was not only incapable
of framing an evolutionary theory himself, but entirely failed to see
how Mendelism supplied the missing parts of the structure first
erected by Darwin. His interpretation of Mendelian facts was from
the first too exclusively coloured by his earlier belief in the dis-
continuous origin of specific forms. Though his influence upon
x PREFACE
evolutionary theory was thus chiefly retrogressive, the mighty body
of Mendelian researches throughout the world has evidently out-
grown the fallacies with which it was at first fostered. As a pioneer of
genetics he has done more than enough to expiate the rash polemics
of his early writings.
To treat Natural Selection as an agency based independently on
its own foundations is not to mimimize its importance in the theory
of evolution. On the contrary, as soon as we require to form opinions
by other means than by comparison and analogy, such an indepen-
dent deductive basis becomes a necessity. This necessity is particu-
larly to be noted for mankind ; since we have some knowledge of the
structure of society, of human motives, and of the vital statistics of
this species, the use of the deductive method can supply a more
intimate knowledge of the evolutionary processes than is elsewhere
possible. In addition it will be of importance for our subject to call
attention to several consequences of the principle of Natural Selection
which, since they do not consist in the adaptive modification of specific
forms, have necessarily escaped attention. The genetic phenomena of
dominance and linkage seem to offer examples of this class, the future
investigation of which may add greatly to the scope of our subject.
No efforts of mine could avail to make the book easy reading.
I have endeavoured to assist the reader by giving short summaries
at the ends of all chapters, except Chapter IV, which is summarized
conjointly with Chapter V. Those who prefer to do so may regard
Chapter IV as a mathematical appendix to the corresponding part
of the summary. The deductions respecting Man are strictly in-
separable from the more general chapters, but have been placed
together in a group commencing with Chapter VIII. I believe no
one will be surprised that a large number of the points considered
demand a far fuller, more rigorous, and more comprehensive treat-
ment. It seems impossible that full justice should be done to the
subject in this way, until there is built up a tradition of mathematical
work devoted to biological problems, comparable to the researches
upon which a mathematical physicist can draw in the resolution of
special difficulties.
R. A. F.
BOTHAMSTED, June 1929.
CONTENTS
List of Illustrations ...... xiii
I. The Nature of Inheritance 1
The consequences of the blending theory, as drawn by Darwin.
Difficulties felt by Darwin. Parfciculate inheritance. Conservation of
the variance. Theories of evolution worked by mutations. Is all
inheritance particulate ? Nature and frequency of observed mutations.
[IJThe Fundamental Theorem of Natural Selection . . 22
The life table and the table of reproduction. The Malthusian para-
meter of population increase. Reproductive value. The genetic
element in variance. Natural Selection. The nature of adaptation.
Deterioration of the environment. Changes in population. Summary.
III. The Evolution of Dominance ..... 48
The dominance of wild genes. Modification of the effects of Mendelian
factors. Modifications of the heterozygote. Special applications of
the theory. The process of modification. Inferences from the theory
of the evolution of dominance. Summary.
IV J Variation as determined by Mutation and Selection . 70
The measurement of gene frequency. The chance of survival of an
individual gene; relation to Poisson series. Low mutation rates of
beneficial mutations. Single origins not improbable. Distribution of
gene ratios in factors contributing to the variance. Slight effects
of random survival. The number of the factors contributing to the
variance.
^
V.y Variation &c. (continued) ..... 97
The observed connexion between variability and abundance. Stable
gene ratios. Equilibrium involving two factors. Simple metrical
characters. Meristic characters. Biometrical effects of recent
selection. Summary.
VI. Sexual Reproduction and Sexual Selection . .121
The contrast between sexual and asexual reproduction. The nature
of species. Fission of species. Sexual preference. Sexual selection.
Sex limitation of modifications. Natural Selection and the sex ratio.
Summary.
. Mimicry 146
The relation of mimicry theory to the parent theory of Natural
Selection. Theories of Bates and Muller. Supposed statistical limita-
tion of Miillerian theory. Observational basis of mimicry theory.
The evolution of distastefulness. The theory of saltations. Stability
of the gene-ratio. Summary.
xii CONTENTS
VIII. Man and Society 170
On Man, prominence of preliminary studies. The decay of civiliza-
tions. Sociological views. Insect communities. Summary.
IX.) The Inheritance of Human Fertility . . . .188
^ y The great variability of human reproduction. The mental and moral
qualities determining reproduction. Direct evidence of the inheritance
of fertility. The evolution of the conscience respecting voluntary re-
production. Analogies of animal instinct and immunity to disease.
Summary.
X. Reproduction in relation to Social Class . . . 210
Economic and biological aspects of class distinctions. Defects of
current data. Early investigations. British data. Position in the
U.S.A. Effects of differential fertility. Summary.
XI. Social Selection of Fertility 228
History of the theory. Infertility in all classes, irrespective of its
cause, gains social promotion. Selection the predominant cause of the
inverted birth-rate. The decay of ruling classes. Contrast with
barbarian societies. Heroism and the higher human faculties. The
place of social class in human evolution. Analogy of parasitism
among ants. Summary.
XII. Conditions of Permanent Civilization . . . 256
Apology. A permanent civilization not necessarily unprogressive.
Redistribution of births. Social promotion of fertility. Inadequacy of
French system. Problem of existing populations. Summary.
Works Cited 266
Index 269
DESCRIPTION OF COLOURED PLATES
PLATE I
Frontispiece
All the figures are of the natural size.
FIG. 1. Abispa (Monerebia) ephippium, Fab., a common member of the
predominant group of Australian wasps, characterized by a dark
brownish- orange ground-colour and the great size and reduced number
of the black markings. They are mimicked by many other insects of
different groups including bees, flies, moths, and numerous beetles. That
the resemblance may be produced by quite different methods is illus-
trated in Figs. 2 and 3.
FIG. 2. Tragocerus formosus, Pascoe, a Longicorn beetle. Almost the
whole of the mimetic pattern is developed 'on the elytra or wing-covers
which hide the unwasplike abdomen, shown from above in Fig. 2A.
Free movement of the wings is permitted by an arched excavation in the
side of each wing-cover.
FIG. 3. Esthesis ferrugineus, Macleay. Another Longicorn beetle in which
the mimetic pattern is developed on the abdomen itself ; the wing-covers
are reduced to small rounded scales, thus freeing the wings, but at the
same time exposing the abdomen.
FIG. 4. Heliconius erato erato. Linn., a distasteful tropical American butter-
fly with a conspicuous pattern beautifully mimicked by the day-flying
Hypsid moth represented below.
FIG. 5. Pericopis pJiyleis, Druce. This moth and its butterfly model were
taken in Peru. The striking mimetic resemblance does not extend to the
antennae, which are threadlike and inconspicuous in the model.
FIG. 6. Meihona confusa, Butler, another tropical American butterfly
captured with one of its moth mimics (Fig. 7) in Paraguay. In this
butterfly and its allies, the antennae are rendered conspicuous by
terminal orange knobs, resembled by many of the mimics in different
groups of butterflies and moths.
FIG. 7. Castnia linus, Cramer. The antennal knobs of this day-flying moth,
in spite of the marked resemblance, possess a form quite different from
those of the model, but one characteristic of the Castniidae. This
example of mimetic likeness to a normally inconspicuous feature, here
exceptionally emphasized, may be compared with Figs. 1, lA-3, SA on
Plate II.
This model and mimic also illustrate, as do Figs. 1-3, the different
methods by which the resemblance may be obtained. The pale trans-
parent areas of the model are produced by the great reduction in the
size of the scales ; in the mimic, without reduction, by their transparency
xiv DESCRIPTION OF COLOURED PLATES
and by their being set at a different angle so that the light passes between
them. The important mimetic association illustrated by Figs. 6 and 7
includes numerous other species belonging to several distantly related
groups of butterflies and moths, and among these transparency is
attained by various different methods.
PLATE II
Facing p. 156
FIGS. 1-3. The head of the abundant East African Acraeine butterfly
Acraea zetes acara, Hew., as seen from the front (1), from above (2) and
the side (3), showing that the palpi, which are inconspicuous in most
butterflies, are a prominent feature with their orange colour displayed
against the black background.
FIGS. 1A-3A. Similar aspects of the head of the Nymphaline butterfly
Pseudacraea boisduvali trimenii, Butler, a mimic of A. z. acara and found
in the same part of Africa. It is evident that the resemblance here ex-
tends to the exceptionally emphasized feature, as was observed in the
American examples shown in Figs. 6 and 7 of Plate I.
FIG. 4. Danaida tytia, Gray, a conspicuous Oriental Danaine butterfly
taken with its mimic (Fig. 5) in the Darjiling district.
FIG. 5. Papilio agestor, Gray, a swallowtail butterfly mimicking the pattern
of tytia.
FIG, 6. Neptis imitans, Oberth., a Nymphaline butterfly from S. W. China,
mimicking the geographical form of D. tytia which is found in the same
area.
Thus these two butterflies of widely separated groups both mimic this
peculiar Danaine pattern.
The butterflies and moths here represented illustrate by single examples
the widespread mimicry of the chief distasteful families in the tropics on
Plate I the Ithomiinae (Fig. 6) and Heliconinae (Fig. 4) of the New World ;
on Plate II, the Daiiainae (Fig. 5), and Acraeinae (Figs. 1-3) of the Old.
I
THE NATURE OF INHERITANCE
The consequences of the blending theory, as drawn by Darwin. Difficulties felt by
Darwin. Particulate inheritance. Conservation of the variance. Theories of evolution
worked by mutations. Is all inheritance particulate ? Nature and frequency of observed
mutations.
But at present, after drawing up a rough copy on this subject, my conclusion
is that external conditions do extremely little, except in causing mere variability.
This mere variability (causing the child not closely to resemble its parent)
I look at as very different from the formation of a marked variety or new
species. DARWIN, 1856. (Life and Letters, ii, 87.)
As Samuel Butler so truly said: 'To me it seems that the "Origin of
Variation ", whatever it is, is the only true " Origin of Species 'V w. BATESON,
1909.
The consequences of the blending theory
THAT Charles Darwin accepted the fusion or blending theory of
inheritance, just as all men accept many of the undisputed beliefs
of their time, is universally admitted. That his acceptance of this
theory had an important influence on his views respecting variation,
and consequently on the views developed by himself and others on
the possible causes of organic evolution, was not, I think, apparent
to himself, nor is it sufficiently appreciated in our own times. In the
course of the present chapter I hope to make clear the logical con-
sequences of the blending theory, and to show their influence, not
only on the development of Darwin's views, but on the change of
attitude towards these, and other suppositions, necessitated by the
acceptance of the opposite theory of particulate inheritance.
It is of interest that the need for an alternative to blending in-
heritance was certainly felt by Darwin, though probably he never
worked out a distinct idea of a particulate theory. In a letter to
Huxley probably dated in 1857 occur the sentences (More Letters,
vol. i, Letter 57).
Approaching the subject from the side which attracts me most, viz.,
inheritance, I have lately been inclined to speculate, very crudely and
indistinctly, that propagation by true fertilization will turn out to be
a sort of mixture, and not true fusion, of two distinct individuals, or
rather of innumerable individuals, as each parent has its parents and
2 THE NATURE OF INHERITANCE
ancestors. I can understand on no other view the way in which crossed
forms go back to so large an extent to ancestral forms. But all this, of
course, is infinitely crude.
The idea apparently was never developed, perhaps owing to the
rush of work which preceded and followed the publication of the
Origin. Certainly he did not perceive that the arguments on varia-
tion in his rough essays of 1842 and 1844, which a year later (1858)
he would be rewriting in the form of the first chapter of the Origin,
would on a particulate theory have required him entirely to recast
them. The same views indeed are but little changed when 'The causes
of variability' came to be discussed in Chapter XXII of Variation of
Animals and Plants published in 1868.
The argument which can be reconstructed from these four sources
may be summarized as follows :
(a) with blending inheritance bisexual reproduction will tend
rapidly to produce uniformity ;
(6) if variability persists, causes of new variation must be con-
tinually at work ;
(c) the causes of the great variability of domesticated species, of
all kinds arid in all countries, must be sought for in the condi-
tions of domestication ;
(d) the only characteristics of domestication sufficiently general
to cover all cases are changed conditions and increase of food ;
(e) some changes of conditions seem to produce definite and
regular effects, e. g. increased food causes (hereditary) increase
in size, but the important effect is an indefinite variability in
all directions, ascribable to a disturbance, by change of condi-
tions, of the regularity of action of the reproductive system ;
(/) wild species also will occasionally, by geological changes, suffer
changed conditions, and occasionally also a temporary increase
in the supply of food; they will therefore, though perhaps
rarely, be caused to vary. If on these occasions no selection is
exerted the variations will neutralize one another by bisexual
reproduction and die away, but if selection is acting, the
variations in the right direction will be accumulated and a per-
manent evolutionary change effected.
To modern readers this will seem a very strange argument with
which to introduce the case for Natural Selection ; all that is gained
THE NATURE OF INHERITANCE 3
by it is the inference that wild as well as domesticated species will at
least occasionally present heritable variability. Yet it is used to
introduce the subject in the two essays and in the Origin. It should
be remembered that, at the time of the essays, Darwin had little
direct evidence on this point ; even in the Origin the second chapter
on 'Variation under Nature' deals chiefly with natural varieties
sufficiently distinct to be listed by botanists, and these were certainly
regarded by Darwin not as the materials but as the products of
evolution. During the twenty-six years between 1842 and 1868 evi-
dence must have flowed in sufficiently at least to convince him that
heritable variability was as widespread, though not nearly as extensive,
in wild as in domesticated species. The line of reasoning in question
seems to have lost its importance sufficiently for him to introduce the
subject in 1868 (Variation, Chapter XXII) with the words 'The sub-
ject is an obscure one ; but it maybe useful to probe our ignorance.'
It is the great charm of the essays that they show the reasons
which led Darwin to his conclusions, whereas the later works often
only give the evidence upon which the reader is to judge of their
truth. The antithesis is not so heterodox as it sounds, for every
active mind will form opinions without direct evidence, else the
evidence too often would never be collected. Impartiality and
scientific discipline come in in submitting the opinions formed to as
much relevant evidence as can be made available. The earlier steps
in the argument set out above appear only in the two essays, while
the conclusions continue almost unchanged up to the Variation of
Animals and Plants. Indeed the first step (a), logically the most
important of all, appears explicitly only in 1842. In 1844 it is clearly
implied by its necessary consequences. I believe its significance for
the argument of the Origin, would scarcely ever be detected from
a study only of that book. The passage in the 1842 MS. is (Founda-
tions, p. 2):
Each parent transmits its peculiarities, therefore if varieties allowed
freely to cross, except by the chance of two characterized by same
peculiarity happening to marry, such varieties will be constantly de-
molished. All bisexual animals must cross, hermaphrodite plants do
cross, it seems very possible that hermaphrodite animals do cross
conclusion strengthened :
together with a partly illegible passage of uncertain position,
If individuals of two widely different varieties be allowed to cross,
4 THE NATURE OF INHERITANCE
a third race will be formed a most fertile source of the variation in
domesticated animals. If freely allowed, the characters of pure parents
will be lost, number of races thus [illegible] but differences [ ?] besides
the [illegible]. But if varieties differing in very slight respects be allowed
to cross, such small variation will be destroyed, at least to our senses
a variation just to be distinguished by long legs will have offspring not
to be so distinguished. Free crossing great agent in producing uni-
formity in any breed.
The proposition is an important one, marking as it docs the great
contrast between the blending and the particulate theories of in-
heritance. The following proof establishes it in biometrical terms.
Let x and y represent the deviations in any measurement of the
two parents from the specific mean ; if the measurement is affected
not only by inheritance, but by non-heritable (environmental)
factors also, x and y stand for the heritable part of these deviations.
The amount of variability present in any generation of individuals
will be measured by the variance, defined as the mean value of the
square of x, or of y. In purely blending inheritance the heritable
portions of the deviations of the offspring will be, apart from muta-
tions, equal to %(x + y) ; in the absence of such mutations, therefore,
the variance of the progeny generation will be the mean value of
The mean values of x and y are both zero, since they are both
defined as deviations from the mean of the species ; consequently, in
the absence of selective mating, the mean value of xy is also zero, and
the variance of the progeny generation is found to be exactly half the
variance of the parental generation. More generally the ratio is not |
but \(l +r), where r is the correlation between x and y. r cannot
exceed unity, else the average value of the positive quantities (x-y) 2
would have to be negative, and can only be unity, if they are all zero,
that is, if the size of each individual prescribes exactly the size of its
possible mates. Darwin's 'except by the chance of two individuals
characterized by same peculiarities happening to marry' is his way
of rejecting high correlations as improbable.
The effect of correlation between mates is to hasten, if the correla-
tion is negative, or to retard if positive, the tendency of blending
inheritance to reduce the variance ; such effects are not of importance,
for even if the correlation were as high as 0-5, and mates had to be
as much alike as parent and child usually are, the rate of decay would
THE NATURE OF INHERITANCE 6
be little more than halved. The important consequence of the blend-
ing is that, if not safeguarded by intense marital correlation, the
heritable variance is approximately halved in every generation. To
maintain a stationary variance fresh mutations must be available in
each generation to supply the half of the variance so lost. If vari-
ability persists, as Darwin rightly inferred, causes of new variability
must continually be at work. Almost every individual of each genera-
tion must be a mutant, i. e. must be influenced by such causes, and
moreover must be a mutant in many different characters.
An inevitable inference of the blending theory is that the bulk of
the heritable variance present at any moment is of extremely recent
origin. One half is new in each generation, and of the remainder one
half is only one generation older, and so on. Less than one-thousandth
of the variance can be ten generations old; even if by reason of
selective mating we ought to say twenty generations, the general
conclusion is the same ; the variability of domesticated species must
be ascribed by any adherent of the blending theory to the conditions
of domestication as they now exist. If variation is to be used by the
human breeder, or by natural selection, it must be snapped up at
once, soon after the mutation has appeared, and before it has had
time to die away. The following passage from the 1844 essay shows
that Darwin was perfectly clear on this point (pp. 84-6).
Let us then suppose that an organism by some chance (which might
be hardly repeated in 1,000 years) arrives at a modern volcanic island
in process of formation and not fully stocked with the most appropriate
organisms ; the new organism might readily gain a footing, although the
external conditions were considerably different from its native ones.
The effect of this we might expect would influence in some small degree
the size, colour, nature of covering, &c., and from inexplicable influences
even special parts and organs of the body. But we might further (and
this is far more important) expect that the reproductive system would
be affected, as under domesticity, and the structure of the offspring
rendered in some degree plastic. Hence almost every part of the body
would tend to vary from the typical form in slight degrees, and in no
determinate way, and therefore without selection the free crossing of
these small variations (together with the tendency to reversion to the
original form) would constantly be counteracting this unsettling effect
of the extraneous conditions on the reproductive system. Such, I con-
ceive, would be the unimportant result without selection. And here
I must observe that the foregoing remarks are equally applicable to
6 THE NATURE OF INHERITANCE
that small and admitted amount of variation which has been observed
in some organisms in a state of nature ; as well as to the above hypo-
thetical variation consequent on changes of condition.
Let us now suppose a Being with penetration sufficient to perceive
differences in the outer and innermost organization quite imperceptible
to man, and with forethought extending over future centuries to watch
with unerring care and select for any object the offspring of an organism
produced under the foregoing circumstances ; I can see no conceivable
reason why he could not form a new race (or several were he to separate
the stock of the original organism and work on several islands) adapted
to new ends. As we assume his discrimination, and his forethought, and
his steadiness of object, to be incomparably greater than those qualities
in man, so we may suppose the beauty and complications of the adapta-
tions of the new races and their differences from the original stock to be
greater than in the domestic races produced by man's agency: the
ground-work of his labours we may aid by supposing that the external
conditions of the volcanic island, from its continued emergence, and the
occasional introduction of new immigrants, vary ; and thus to act on the
reproductive system of the organism, on which he is at work, and so keep
its organization somewhat plastic. With time enough, such a Being
might rationally (without some unknown law opposed him) aim at
almost any result.
Difficulties felt by Darwin
The argument based on blending inheritance and its logical con-
sequences, though it certainly represents the general trend of
Darwin's thought upon inheritance and variation, for some years
after he commenced pondering on the theory of Natural Selection,
did not satisfy him completely. Reversion he recognized as a fact
which stood outside his scheme of inheritance, and that he was not
altogether satisfied to regard it as an independent principle is shown
by his letter to Huxley already quoted. By 1857 he was in fact on
the verge of devising a scheme of inheritance which should include
reversion as one of its consequences. The variability of domesticated
races, too, presented a difficulty which, characteristically, did not
escape him. He notes (pp. 77, 78, Foundations) in 1844 that the most
anciently domesticated animals and plants are not less variable, but,
if anything more so, than those more recently domesticated; and
argues that since the supply of food could not have been becoming
much more abundant progressively at all stages of a long history of
THE NATURE OF INHERITANCE 7
domestication, this factor cannot alone account for the great varia-
bility which still persists. The passage runs as follows:
If it be an excess of food, compared with that which the being obtained
in its natural state, the effects continue for an improbably long time ;
during how many ages has wheat been cultivated, and cattle and sheep
reclaimed, and we cannot suppose their amount of food has gone on
increasing, nevertheless these are amongst the most variable of our
domestic productions.
This difficulty offers itself also to the second supposed cause of
variability, namely changed conditions, though here it may be
argued that the conditions of cultivation or nurture of domesticated
species have always been changing more or less rapidly. From a
passage in the Variation of Animals and Plants (p. 301), which runs:
Moreover, it does not appear that a change of climate, whether more
or less genial, is one of the most potent causes of variability; for in
regard to plants Alph. De Candolle, in his Geographic Botanique, re-
peatedly shows that the native country of a plant, where in most cases
it has been longest cultivated, is that where it has yielded the greatest
number of varieties.
it appears that Darwin satisfied himself that the countries in which
animals or plants were first domesticated, were at least as prolific
of new varieties as the countries into which they had been imported,
and it is natural to presume that his inquiries under this head were
in search of evidence bearing upon the effects of changed conditions.
It is not clear that this difficulty was ever completely resolved in
Darwin's mind, but it is clear from many passages that he saw the
necessity of supplementing the original argument by postulating
that the causes of variation which act upon the reproductive system
must be capable of acting in a delayed and cumulative manner so
that variation might still be continued for many subsequent genera-
tions.
Particulate inheritance
It is a remarkable fact that had any thinker in the middle of the
nineteenth century undertaken, as a piece of abstract and theoretical
analysis, the task of constructing a particulate theory of inheritance,
he would have been led, on the basis of a few very simple assump-
tions, to produce a system identical with the modern scheme of
Mendelian or factorial inheritance. The admitted non-inheritance of
8 THE NATURE OF INHERITANCE
scars and mutilations would have prepared him to conceive of the
hereditary nature of an organism as something none the less definite
because possibly represented inexactly by its visible appearance.
Had he assumed that this hereditary nature was completely deter-
mined by the aggregate of the hereditary particles (genes), which
enter into its composition, and at the same time assumed that
organisms of certain possible types of hereditary composition were
capable of breeding true, he would certainly have inferred that each
organism must receive a definite portion of its genes from each parent,
and that consequently it must transmit only a corresponding portion
to each of its offspring. The simplification that, apart from sex and
possibly other characters related in their inheritance to sex, the
contributions of the two parents were equal, would not have been
confidently assumed without the evidence of reciprocal crosses ; but
our imaginary theorist, having won so far, would scarcely have
failed to imagine a conceptual framework in which each gene had its
proper place or locus, which could be occupied alternatively, had the
parentage been different, by a gene of a different kind. Those
organisms (homozygotes) which received like genes, in any pair of
corresponding loci, from their two parents, would necessarily hand on
genes of this kind to all of their offspring alike ; whereas those (hetero-
zygotes) which received from their two parents genes of different
kinds, and would be, in respect of the locus in question, crossbred,
would have, in respect of any particular offspring, an equal chance of
transmitting either kind. The heterozygote when mated to either
kind of homozygote would produce both heterozygotes and homo-
zygotes in a ratio which, with increasing numbers of offspring, must
tend to equality, while if two heterozygotes were mated, each
homozygous form would bo expected to appear in a quarter of the
offspring, the remaining half being heterozygous. It thus appears
that, apart from dominance and linkage, including sex linkage, all
the main characteristics of the Mendelian system flow from assump-
tions of particulate inheritance of the simplest character, and could
have been deduced a priori had any one conceived it possible that
the laws of inheritance could really be simple and definite.
The segregation of single pairs of genes, that is of single factors,
was demonstrated by Mendel in his paper of 1865. In addition Mendel
demonstrated hi his material the fact of dominance, namely that the
heterozygote was not intermediate in appearance, but was almost or
THE NATURE OF INHERITANCE 9
quite indistinguishable from one of the homozygous forms. The fact
of dominance, though of the greatest theoretical interest, is not an
essential feature of the factorial system, and in several important
cases is lacking altogether. Mendel also demonstrated what a theorist
could scarcely have ventured to postulate, that the different factors
examined by him in combination, segregated in the simplest possible
manner, namely independently. It was not till after the rediscovery
of Mendel's laws at the end of the century that cases of linkage were
discovered, in which, for factors in the same linkage group, the pair
of genes received from the same parent are more often than not
handed on together to the same child. The conceptual framework
of loci must therefore be conceived as made of several parts, and
these are now identified, on evidence which appears to be singularly
complete, with the dark-staining bodies or chromosomes which are
to be seen in the nuclei of cells at certain stages of cell division.
The mechanism of particulate inheritance is evidently suitable for
reproducing the phenomenon of reversion, in which an individual
resembles a grandparent or more remote ancestor, in some respect
in which it differs from its parents ; for the ancestral gene combina-
tion may by chance be reproduced. This takes its simplest form when
dominance occurs, for every union of two heterozygotes will then
produce among the offspring some recessivcs, differing in appearance
from their parents, but probably resembling some grandparent or
ancestor.
Conservation of the variance
It has not been so clearly recognized that particulate inheritance
differs from the blending theory in an even more important fact.
There is no inherent tendency for the variability to diminish. In
a population breeding at random in which two alternative genes of
any factor, exist in the ratio p to g, the three genotypes will occur in
the ratio p 2 : 2pq : g 2 , and thus ensure that their characteristics will
be represented in fixed proportions of the population, however they
may be combined with characteristics determined by other factors,
provided that the ratio p : q remains unchanged. This ratio will
indeed be liable to slight changes ; first by the chance survival and
reproduction of individuals of the different kinds ; and secondly by
selective survival, by reason of the fact that the genotypes are pro-
bably unequally fitted, at least to a slight extent, to their task of
10 THE NATURE OF INHERITANCE
survival and reproduction. The effect of chance survival is easily
susceptible of calculation, and it appears, as will be demonstrated
more fully (Chapter IV), that in a population of n individuals breed-
ing at random the variance will be halved by this cause acting alone
in 1-4 n generations. Since the number of individuals surviving to
reproduce in each generation must in most species exceed a million,
and in many is at least a million-fold greater, it will be seen that this
cause of the diminution of hereditary variance is exceedingly minute,
when compared to the rate of halving in one or two generations by
blending inheritance.
It will be seen in Chapter IV that selection is a much more impor-
tant agency in keeping the variability of species within limits. But
even relatively intense selection will change the ratio p : q of the
gene frequencies relatively slowly, and no reasonable assumptions
could be made by which the diminution of variance due to selection,
in the total absence of mutations, would be much more than a ten-
thousandth of that ascribable to blending inheritance. The immediate
consequence of this enormous contrast is that the mutation rate
needed to maintain a given amount of variability is, on the particulate
theory, many thousand times smaller than that which is required on
the blending theory. Theories, therefore, which ascribe to agencies
believed to be capable of producing mutations, as was 'use and
disuse' by Darwin, a power of governing the direction in which
evolution is taking place, appear in very different lights, according
as one theory of inheritance, or the other, is accepted. For any
evolutionary tendency which is supposed to act by favouring muta-
tions in one direction rather than another, and a number of such
mechanisms have from time to time been imagined, will lose its force
many thousand-fold, when the particulate theory of inheritance, in
any form, is accepted; whereas the directing power of Natural
Selection, depending as it does on the amount of heritable variance
maintained, is totally uninfluenced by any such change. This con-
sideration, which applies to all such theories alike, is independent of
the fact that a great part of the reason, at least to Darwin, for
ascribing to the environment any considerable influence in the pro-
duction of mutations, is swept away when we are no longer forced to
consider the great variability of domestic species as due to the
comparatively recent influence of their artificial environment.
The striking fact, of which Darwin was well aware, that whole
THE NATURE OF INHERIATNCE 11
brothers and sisters, whose parentage, and consequently whose
entire ancestry is identical, may differ greatly in their hereditary
composition, bears under the two theories two very different inter-
pretations. Under the blending theory it is clear evidence of new
and frequent mutations, governed, as the greater resemblance of
twins suggests, by temporary conditions acting during conception
and gestation. On the particulate theory it is a necessary consequence
of the fact that for every factor a considerable fraction, not often
much less than one half, of the population will be heterozygotes, any
two offspring of which will be equally likely to receive unlike as like
genes from their parents. In view of the close analogy between the
statistical concept of variance and the physical concept of energy,
we may usefully think of the heterozygote as possessing variance in
a potential or latent form, so that instead of being lost when the
homozygous genotypes are mated it is merely stored in a form from
which it will later reappear. A population mated at random immedi-
ately establishes the condition of statistical equilibrium between the
latent and the apparent form of variance. The particulate theory of
inheritance resembles the kinetic theory of gases with its perfectly
elastic collisions, whereas the blending theory resembles a theory of
gases with inelastic collisions, and in which some outside agency is
required to be continually at work to keep the particles astir.
The property of the particulate theory of conserving the variance
for an indefinite period explains at once the delayed or cumulative
effect of domestication in increasing the variance of domesticated
species, to which Darwin calls attention. Many of our domesticated
varieties are evidently ill-fitted to survive in the wild condition. The
mutations by which they arose may have been occurring for an
indefinite period prior to domestication without establishing them-
selves, or appreciably affecting the variance, of the wild species. In
domestication, however, not only is the rigour of Natural Selection
relaxed so that mutant types can survive, and each such survival add
something to the store of heritable variance, but novelties of form or
colour, even if semi-monstrous, do undoubtedly attract human
attention and interest, and are valued by man for their peculiarity.
The rapidity with which new variance is accumulated will thus be
enhanced. Without postulating any change hi the mutation rates
due to domestication, we should necessarily infer from what is known
of the conditions of domestication that the variation of domesticated
12 THE NATURE OF INHERITANCE
species should be greater than that of similar wild species, and that
this contrast should be greatest with those species most anciently
domesticated. Thus one of the main difficulties felt by Darwin is
resolved by the particulate theory.
Theories of evolution worked by mutations
The theories of evolution which rely upon hypothetical agencies,
capable of modifying the frequency or direction in which mutations
are taking place, fall into four classes. In stating these it will be
convenient to use the term 'mutation', to which many meanings have
at different times been assigned, to denote simply the initiation of any
heritable novelty.
(A) It may be supposed, as by Lamarck in the case of animals,
that the mental state, and especially the desires of the organism,
possess the power of producing mutations of such a kind, that these
desires may be more readily gratified in the descendants. This view
postulates (i) that there exists a mechanism by which mutations are
caused, and even designed, in accordance with the condition of the
nervous system, and (ii) that the desires of animals in general are
such that their realization will improve the aptitude of the species
for life in its natural surroundings, and also will maintain or improve
the aptitude of its parts to co-operate with one another, both in
maintaining the vital activity of the adult animal, and in ensuring
its normal embryological development. The desires of animals must,
in fact, be very wisely directed, as well as being effective in provoking
suitable mutations.
(B) A power of adaptation may be widely observed, both among
plants and animals, by which particular organs, such as muscles or
glands, respond by increased activity and increased size, when addi-
tional physiological calls are made upon them. It may be suggested,
as it was by Darwin, that such responses of increased functional
activity induce, or are accompanied by, mutations of a kind tending
to increase the size or activity of the organ in question in future
generations, even if no additional calls were made upon this organ's
activity. This view implies (i) that the power which parts of organisms
possess, of responding adaptively to increased demands upon them,
is not itself a product of evolution, but must be postulated as a
primordial property of living matter: and requires (ii) that a mecha-
THE NATURE OF INHERITANCE 13
nism exists by which the adaptive response shall itself tend to cause,
or be accompanied by, an appropriate mutation.
Both these two suggested means of evolution expressly aim at
explaining, not merely the progressive change of organic beings, but
the aptitude of the organism to its place in nature, and of its parts to
their function in the organism.
(C) It may be supposed that the environment in which the or-
ganism is placed controls the nature of the mutations which occur
in it, and so directs its evolutionary course ; much as the course of
a projectile is controlled by the field of force in which it flies.
(D) It may be supposed that the mutations which an organism
undergoes are due to an 'inner urge' (not necessarily connected with
its mental state) implanted in its primordial ancestors, which thereby
directs its predestined evolution.
The two last suggestions give no particular assistance towards the
understanding of adaptation, but each contains at least this element
of truth; that however profound our ignorance of the causes of
mutation may be, we cannot but ascribe them, within the order of
Nature as we know it, either to the nature of the organism, or to
that of its surrounding environment, or, more generally, to the inter-
action of the two. What is common, however, to all four of these
suppositions, is that each one postulates that the direction of evo-
lutionary change is governed by the predominant direction in which
mutations are taking place. However reasonable such an assumption
might have seemed when, under the blending theory of inheritance,
every individual was regarded as a mutant, and probably a multiple
mutant, it is impossible to let it pass unquestioned, in face of the
much lower mutation rates appropriate to the participate theory.
A further hypothetical mechanism, guiding the evolution of the
species according to the direction in which mutations are occur-
ring, was suggested by Weismann. Weismann appreciated much more
thoroughly than many of his contemporaries the efficacy of Natural
Selection, in promoting the adaptation of organisms to the needs of
their lives in their actual habitats. He felt, however, that this action
would be aided in a subordinate degree if the process of mutation
could acquire a kind of momentum, so that a series of mutations
affecting the increase or decrease of a part should continue to occur,
as a consequence of an initial chance tendency towards such increase
or decrease. Such an assumed momentum in the process of mutation
14 THE NATURE OF INHERITANCE
he found useful in two respects : (i) it would enable an assumed minimal
mutation in an advantageous direction to be increased by further
mutations, until it ' attains selection value ' ; (ii) it explains the con-
tinuous decrease of a useless organ, without assuming that each step
of this decrease confers any advantage upon the organism mani-
festing it.
The concept of attaining selection value, which is fairly common
in biological literature, seems to cover two distinct cases. In the first
case we may imagine that, with increasing size, the utility of an
organ shows no increase up to a certain point, but that beyond this
point increasing size is associated with increasing utility. In such
a case, which, in view of the actual variability of every organism,
and of the parts of related organisms, must be regarded as somewhat
ideal, we are really only concerned with the question whether the
actual variability in different members of the species concerned, does
or does not reach as far as the critical point. If it does not do so the
species will not be able to take the advantage offered, simply because
it is not variable enough, and the postulate of an element of momen-
tum in the occurrence of mutations, was certainly not made in
order to allow organisms to be more variable than they would be
without it.
The second meaning, which is also common in the literature,
depends upon a curious assumption as to the manner in which
selective advantage increases with change of size of the organ upon
which this advantage is dependent ; for it is sometimes assumed that,
while at all sizes an increase of size may be advantageous, this
advantage increases, not continuously, but in a step-like manner; or
at least that increases below a certain limit produce an advantage
which may be called 'inappreciable', and therefore neglected, Both
the metaphor and the underlying idea appear to be drawn from
psychophysical experience. If we compare two physical sensations
such as those produced by the weights of two objects, then when the
weights are sufficiently nearly equal the subject will often be unable
to distinguish between them, and will judge them equal, whereas
with a greater disparity, a distinct or appreciable difference of weight
is discerned. If, however, the same test is applied to the subject
repeatedly with differences between the weights varying from what is
easily discernible to very much smaller quantities, it is found that
differences in the weights, which would be deemed totally inappreci-
THE NATURE OF INHERITANCE 16
able, yet make a significant and perfectly regular difference to the
frequency with which one is judged heavier than the other. The
discontinuity lies in our interpretation of the sensations, and not in
the sensations themselves. Now, survival value is measured by the
300
250
H
o
Osoo
150
D
O
1
5
LLJ
100
84
88 92 96 100 104
WEIGHT TESTED IN GRAMS
108
FIG. 1. The frequency with which test objects of different weights arc
judged heavier than a standard 100 gram weight. (Urban's data, for a
single subject.) Illustrating the fact that with a sufficient number of
trials, differences in weight, however 'inappreciable', will affect the
frequency of the judgement.
frequency with which certain events, such as death or reproduction,
occur, to different sorts of organisms exposed to the different chances
of the same environment, and, even if we should otherwise be in doubt,
the psychophysical experiments make it perfectly clear that the
selective advantage will increase or decrease continuously, even for
changes much smaller than those appreciable to our own senses, or
to those of the predator or other animal, which may enter into the
biological situation concerned. If a change of 1 mm. has selection
value, a change of 0-1 mm. will usually have a selection value
approximately one-tenth as great, and the change cannot be ignored
because we deem it inappreciable. The rate at which a mutation
increases in numbers at the expense of its allelomorph will indeed
depend on the selective advantage it confers, but the rate at which
a species responds to selection in favour of any increase or decrease
of parts depends on the total heritable variance available, and not
16 THE NATURE OF INHERITANCE
on whether this is supplied by large or small mutations. There is no
limen of appreciable selection value to be considered.
The remaining advantage which Weismann sought in postulating
his mechanism of germinal selection was to supply an explanation
of the progressive diminution of useless organs, even when these are
of so trifling a character that the selective advantage of their sup-
pression is questionable. The subject is an interesting one, and
deserves for its own sake a more extended discussion than would be
suitable in the present book. For our present purpose it will be
sufficient to notice (i) that to assert in any particular case that the
progressive suppression of an organ brings with it no progressive
selective advantage appears to be very far beyond the range of our
actual knowledge. To take a strong case from Weismann the
receptaculum seminis of an ant is assuredly minute ; but the ant her-
self is not very large, nor are we concerned only with the individual
ant, but with the whole worker population of the nest. As an economic
problem we certainly do not possess the data to decide whether the
suppression of this minute organ would or would not count as an
appreciable factor in the ant polity. Human parallels might be given
in which the elimination of very minute items of individual waste,
can lend an appreciable support to social institutions which are
certainly not negligible. I do not assert that the suppression of the
receptaculum has been useful to the ant, but that in this as in other
cases, if we pause to give the matter due consideration, it is at once
apparent that we have not the knowledge on which to base any
decided answer, (ii) In the second place Weismann's view that in
the absence of all selection a useless organ might diminish, degenerate,
and finally disappear, by the cumulative action of successive muta-
tions, and especially his view that this is the only type of progressive
change, which could take place by mutations only, without the
guidance of Natural Selection, is fully in accordance with modern
knowledge of the nature of mutations. The special mechanism,
however, by which he sought to explain the successive occurrence of
degenerative mutations must be judged to be superfluous. It is
moreover exposed to the logical objection that the driving force of
his mechanism of germinal selection is an assumed competition for
nutriment between the chromatin elements which represent the
degenerating organ, and those which represent the rest of the body.
The degenerating organ itself is assumed to be so unimportant that
THE NATURE OF INHERITANCE 17
its demands upon the general nutrition of the body are to be neglected ;
and it may well be asked if it is legitimate to bring in, in respect of
the well-nourished germ cell, the factor of nutritional competition
which is to be ignored in the occasionally ill-nourished body.
Is all inheritance participate ?
The logical case for rejecting the assumption that the direction of
evolutionary change is governed by the direction in which mutations
are taking place, and thereby rejecting the whole group of theories
in which this assumption is implicit, would be incomplete had not
modern researches supplied the answer to two further questions:
(i) May it not be that in addition to the mechanism of particulate
inheritance, which has been discovered and is being investigated,
there is also, in living organisms, an undiscovered mechanism of
blending inheritance ? (ii) Do the known facts within the particulate
system render a mechanism, which could control the predominant
direction of mutation, inoperative as a means of governing the
direction of evolutionary change ?
On the first point it should be noted briefly that, whereas at the
beginning of the century there were several outstanding facts of
inheritance which seemed to demand some sort of blending theory,
these have all in the course of research been shown, not only to be
compatible with particulate inheritance, but to reveal positive
indications that such is their nature. The apparent blending in colour
in crosses between white races of man and negroes is compatible
with the view that these races differ in several Mendelian factors,
affecting the pigmentation. Of these some may have intermediate
heterozygotes, and of the remainder in some the darker, and in some
the lighter tint may be dominant. The Mendelian theory is alone
competent to explain the increased variability of the offspring of the
mulattoes.
The biometrical facts as to the inheritance of stature and other
human measurements, though at first regarded as incompatible with
the Mendelian system, have since been shown to be in complete
accordance with it, and to reveal features not easily explicable on any
other view. The approximately normal distribution of the measure-
ments themselves may be deduced from the simple supposition that
the factors affecting human stature are approximately additive in
their effects. The correlations found between relatives of different
18 THE NATURE OF INHERITANCE
degrees of kinship are, within their sampling errors, of the magnitudes
which would be deduced from the assumption that the measurement
is principally determined by inheritance, and that the factors con-
trolling it show, like most Mendelian factors, complete or almost
complete dominance. The presence of dominance is a Mendelian
feature, which is shown in the biometrical data by the well-estab-
lished fact that children of the same parents are, on the average,
somewhat more alike than are parent and offspring.
So far we have merely established the negative fact that there are
no outstanding observations which require a blending system of
inheritance. There is, however, one group of modern researches
which, at least in the organisms investigated, seems to exclude it,
even as a possibility. In certain organisms which are habitually self-
fertilized, as Johannsen was the first to show with a species of bean,
it is possible to establish so-called pure lines, within which heritable
variability is, apart from exceptional mutations, completely absent.
Within these lines the selection of the largest or the smallest beans,
even where this selection was continued for ten or twenty generations,
constantly produced offspring of the same average size. This size
differed from one line to another, showing that heritable variability
existed abundantly in the species, and among the thousands of beans
examined two distinct mutants were reported. If, however, any
appreciable fraction of the variance in bean size were ascribable to
elements which blend, the mutations necessary to maintain such
heritable variability would, in ten generations, have had time to
supply it almost to its maximum extent, and must inevitably have
been revealed by selection. Experiments of this type seem capable
of excluding the possibility that blending inheritance can account
for any appreciable fraction of the variance observed.
Nature and frequency of observed mutations
The assumption that the direction of evolutionary change is actually
governed by the direction in which mutations are occurring is not
easily compatible with the nature of the numerous mutations which
have now been observed to occur. For the majority of these produce
strikingly disadvantageous deformities, and indeed much the largest
class are actually lethal. If we had to admit, as has been so often
assumed in theory, that these mutations point the direction of
evolution, the evolutionary prospects of the little fruit-fly Drosophila
THE NATURE OF INHERITANCE 19
would be deplorable indeed. Nor is the position apparently different
with man and his domesticated animals and plants ; as may be judged
from the frequency with which striking recessive defects, such as
albinism, deaf -mutism, and feebleness of mind in man, must have
occurred in the comparatively recent past, as mutations. Mutant
defects seem to attack the human eye as much as that of Drosophila,
and in general the mutants which occur in domesticated races are
often monstrous and predominantly defective, whereas we know in
many cases that the evolutionary changes which these creatures have
undergone under human selection have been in the direction of a
manifest improvement.
In addition to the defective mutations, which by their con-
spicuousness attract attention, we may reasonably suppose that
other less obvious mutations are occurring which, at least in certain
surroundings, or in certain genetic combinations, might prove them-
selves to be beneficial. It would be unreasonable, however, to assume
that such mutations appear individually with a frequency much
greater than that which is observed in the manifest defects. The
frequency of individual mutations in Drosophila is certainly seldom
greater than one in 100,000 individuals, and we may take this figure
to illustrate the inefficacy of any agency, which merely controls the
predominant direction of mutation, to determine the predominant
direction of evolutionary change. For even if selective survival were
totally absent, a lapse of time of the order of 100,000 generations
would be required to produce an important change with respect to
the factor concerned, in the heritable nature of the species. Moreover,
if the mutant gene were opposed, even by a very minute selective
disadvantage, the change would be brought to a standstill at a very
early stage. The ideas necessary for a precise examination of the
nature of selective advantage will be developed in Chapter II ; but
it will be readily understood that if we speak of a selective advantage
of one per cent., with the meaning that animals bearing one gene
have an expectation of offspring only one per cent, greater than those
bearing its allelomorph, the selective advantage in question will be
a very minute one; at least in the sense that it would require an
enormous number of experimental animals, and extremely precise
methods of experimentation, to demonstrate so small an effect
experimentally. Such a selective advantage would, however, greatly
modify the genetic constitution of the species, not in 100,000 but in
20 THE NATURE OF INHERITANCE
100 generations. If, moreover, we imagine these two agencies opposed
in their tendencies, so that a mutation which persistently occurs in
one in 100,000 individuals, is persistently opposed by a selective
advantage of only one per cent., it will easily be seen that an
equilibrium will be arrived at when only about one individual in
1,000 of the population will be affected by the mutation. This
equilibrium, moreover, will be stable ; for if we imagine that by some
chance the number of mutants is raised to a higher proportion than
this, the proportion will immediately commence to diminish under
the action of selection, and evolution will proceed in the direction
contrary to the mutation which is occurring, until the proportion of
mutant individuals again reaches its equilibrium value. For muta-
tions to dominate the trend of evolution it is thus necessary to postu-
late mutation rates immensely greater than those which are known
to occur, and of an order of magnitude which, in general, would be
incompatible with particulate inheritance.
Summary
The tacit assumption of the blending theory of inheritance led
Darwin, by a perfectly cogent argument, into a series of speculations,
respecting the causes of variations, and the possible evolutionary
effects of these causes. In particular the blending theory, by the
enormous mutation rates which it requires, led Darwin and others
to attach evolutionary importance to hypothetical agencies which
control the production of mutations. A mechanism (Mendelism) of
particulate inheritance has since been discovered, requiring mutations
to an extent less by many thousandfold. The 'pure line ' experiments
seem to exclude blending inheritance even as a subordinate possibility.
The nature of the mutations observed is not compatible with the
view that evolution is directed by their means, while their observed
frequency of occurrence shows that an agency controlling mutations
would be totally ineffectual in governing the direction of evolutionary
change.
The whole group of theories which ascribe to hypothetical physio-
logical mechanisms, controlling the occurrence of mutations, a power
of directing the course of evolution, must be set aside, once the
blending theory of inheritance is abandoned. The sole surviving
theory is that of Natural Selection, and it would appear impossible
to avoid the conclusion that if any evolutionary phenomenon
THE NATURE OF INHERITANCE 21
appears to be inexplicable on this theory, it must be accepted at
present merely as one of the facts which in the present state of
knowledge seems inexplicable. The investigator who faces this fact,
as an unavoidable inference from what is now known of the nature
of inheritance, will direct his inquiries confidently towards a study
of the selective agencies at work throughout the life history of the
group in their native habitats, rather than to speculations on the
possible causes which influence their mutations. The experimental
study of agencies capable of influencing mutation rates is of the
highest interest for the light which it may throw on the nature of
these changes. We should altogether misinterpret the value of such
researches were we to regard them as revealing the causes of evolu-
tionary modification.
II
THE FUNDAMENTAL THEOREM OF NATURAL
SELECTION
The life table and the table of reproduction. The Malthusian parameter of popu-
lation increase. Reproductive value. The genetic element in variance. Natural Selection.
The nature of adaptation. Deterioration of the environment. Changes in population.
Summary.
One has, however, no business to feel so much surprise at one's ignorance,
when one knows how impossible it is without statistics to conjecture the
duration of life and percentage of deaths to births in mankind. DARWIN,
1845. (Life and Letters, ii, 33.)
In the first place it is said and I take this point first, because the imputation
Is too frequently admitted by Physiologists themselves that Biology differs
f rom the Physico-chemical and Mathematical sciences in being 'inexact'.
HUXLEY, 1854.
The life table
IN order to obtain a distinct idea of the application of Natural
Selection to all stages in the life -history of an organism, use may be
made of the ideas developed in the actuarial study of human mor-
tality. These ideas are not in themselves very recondite, but being
associated with the laborious computations and the technical nota-
tion employed in the practical business of life insurance, are not so
familiar as they might be to the majority of biologists. The text-
books on the subject, moreover, are devoted to the chances of death,
and to monetary calculations dependent on these chances, whereas
in biological problems at least equal care and precision of ideas is
requisite with respect to reproduction, and especially to the combined
action of these two agencies in controlling the increase or decrease of
the population.
The object of the present chapter is to combine certain ideas
derivable from a consideration of the rates of death and reproduction
of a population of organisms, with the concepts of the factorial
scheme of inheritance, so as to state the principle of Natural Selection
in the form of a rigorous mathematical theorem, by which the rate
of improvement of any species of organisms in relation to its environ-
ment is determined by its present condition.
The fundamental apparatus of the actuary's craft is what is known
FUNDAMENTAL THEOREM OF NATURAL SELECTION 23
as a life table. This shows, for each year of age, of the population
considered, the proportion of persons born alive who live to attain
that age. For example, a life table may show that the proportion of
persons living to the age of 20 is 88 per cent., while only 80 per cent.
reach the age of 40. It will be easily inferred that 12 per cent, of
those born alive die in the first 20 years of life, and 8 per cent, in the
second 20 years. The life table is thus equivalent to a statement of
the frequency distribution of the age of death in the population con-
cerned. The amount by which each entry is less than the preceding
entry represents the number of deaths between these limits of age,
and this divided by the number living at the earlier age gives the
probability of death within a specified time of those living at that age.
Since the probability of death changes continuously throughout life,
the death rate at a given age can only be measured consistently by
taking the age interval to be infinitesimal. Consequently if l x is the
number living to age x, the death rate at age x is given by:
the logarithm being taken, as in most mathematical representations,
to be on the Natural or Naperian system. The life table thus contains
a statement of the death rates at all ages, and conversely can be
constructed from a knowledge of the course taken by the death rate
throughout life. This in fact is the ordinary means of constructing the
life tables in practical use.
It will not be necessary to discuss the technical procedure employed
in the construction of life tables, the various conventions employed
in this form of statement, nor the difficulties which arise in the inter-
pretation of the observational data available in practice for this
purpose. It will be sufficient to state only one point. As in all other
experimental determinations of theoretical values, the accuracy
attainable in practice is limited by the extent of the observations ;
the result derived from any finite number of observations will be
liable to an error of random sampling, but this fact does not, in any
degree, render such concepts as death rates or expectations of life
obscure or inexacjb. These are statements of probabilities, averages
&c., pertaining to the hypothetical population sampled, and depend
only upon its nature and circumstances. The inexactitude of our
methods of measurement has no more reason in statistics than it has
24 FUNDAMENTAL THEOREM OF NATURAL SELECTION
in physics to dim our conception of that which we measure. These
conceptions would be equally clear if we were stating the chances of
death of a single individual of unique genetic constitution, or of one
exposed to an altogether transient and exceptional environment.
The table of reproduction
The life table, although itself a very comprehensive statement, is
still inadequate to express fully the relation between an organism and
its environment ; it concerns itself only with the chances or frequency
of death, and not at all with reproduction. To repair this deficiency it
is necessary to introduce a second table giving rates of reproduction
in a manner analogous to the rates of death at each age. Just as a
person alive at the beginning of any infinitesimal age interval dx has
a/ chance of dying within that interval measured by n>xdx, so the
chance of reproducing within this interval will be represented by b x dx,
in which b x may be called the rate of reproduction at age x. Again, just
as the chance of a person chosen at birth dying within a specified
interval of age dx is l x p>xdx, so the chance of such a person living to
reproduce in that interval will be l x b x dx.
Owing to bisexual reproduction a convention must be introduced
into the measurement of b x , for each living offspring will be credited
bo both parents, and it will seem proper to credit each with one half
in respect of each offspring produced. This convention will evidently
be appropriate for those genes which are not sex -linked (autosomal
yenes) for with these the chance of entering into the composition of
3ach offspring is known to be one half. In the case of sex-linked genes
bhose of the heterogametic parent will be perpetuated or not accord-
ing as the offspring is male or female. These sexes, it is true, will not
be produced in exactly equal numbers, but since both must co-operate
in each act of sexual reproduction, it is clear that the different
frequencies at birth must ultimately be compensated by sexual
iifferences in the rates of death and reproduction, with the result that
bhe same convention appears in this case to be equally appropriate.
A similar convention, appropriate in the sense of bringing the
formal symbolism of the mathematics into harmony with the biologi-
3al facts, may be used with respect to the period of gestation. For it
Rdll happen occasionally that a child is born after the death of its
'ather. The children born to fathers aged x should in fact be credited
)o males aged three-quarters of a year younger. Such corrections are
FUNDAMENTAL THEOREM OF NATURAL SELECTION 25
not a necessity to an exact mathematical representation of the facts,
but are a manifest convenience in simplifying the form of expression ;
thus with mankind we naturally think of the stage in the life-history
as measured in years from birth. With other organisms the variable
x which with man represents this age, may in some cases be more
conveniently used to indicate rather the stage' in the life history
irrespective of chronological age, merely to give greater vividness to
the meaning of the symbolism, but without altering the content of
the symbolical statements.
The Malthusian parameter of population increase
If we combine the two tables giving the rates of death and repro-
duction, we may, still speaking in terms of human populations, at
once calculate the expectation of offspring of the newly-born child.
For the expectation of offspring in each element of age dx is l x b x dx,
and the sum of these elements over the whole of life will be the
total expectation of offspring. In mathematical terms this is
l x b x dx,
where the integral is extended from zero, at birth, to infinity, to cover
every possible age at which reproduction might conceivably take
place. If at any age reproduction ceases absolutely, b x will thereafter
be zero and so give automatically the effect of a terminating integral.
The expectation of offspring determines whether in the population
concerned the reproductive rates are more or less than sufficient to
balance the existing death rates. If its value is less than unity the
reproductive rates are insufficient to maintain a stationary popula-
tion, in the sense that any population which constantly maintained
the death and reproduction rates in question would, apart from
temporary fluctuations, certainly ultimately decline in numbers at a
calculable rate. Equally, if it is greater than unity, the population
biologically speaking is more than holding its own, although the
actual number of heads to be counted may be temporarily decreasing.
This consequence will appear most clearly in its quantitative aspect
if we note that corresponding to any system of rates of death and
reproduction, there is only one possible constitution of the population
in respect of age, which will remain unchanged under the action of
this system. For if the age distribution remains unchanged the
3653
26 FUNDAMENTAL THEOREM OF NATURAL SELECTION
relative rate of increase or decrease of numbers at all ages must be
the same ; let us represent the relative rate of increase by w ; which
will also represent a decrease if m is negative. Then, owing to the
constant rates of reproduction, the rate at which births are occurring
at any epoch will increase proportionately to e mt . At any particular
epoch, for which we may take 2 = 0, the rate at which births were
occurring x years ago will be proportional to eT mx , and this is the rate
at which births were occurring at the time persons now of age x were
being born. The number of persons in the infinitesimal age interval
Ax will therefore be e~ 7nx l x dx, for of those born only the fraction l x
survive to this age. The age distribution is therefore determinate if
the number m is uniquely determined. But knowing the numbers
living at each age, and the reproductive rates at each age, the rate at
which births are now occurring can be calculated, and this can be
equated to the known rate of births appropriate to 0. In fact, the
contribution to the total rate, of persons in the age interval dx, must
be e~ mx l x b x dx, and the aggregate for all ages must be
CO
I e~ mx l x b x dx,
o
which, when equated to unity, supplies an equation for w, of which
one and only one real solution exists. Since e~ mx is less than unity for
all values of x, if m is positive, and is greater than unity for all values
of #, if m is negative, it is evident that the value of m, which reduces
the integral above expressed to unity, must be positive if the expecta-
tion of offspring exceeds unity, and must be negative if it falls short
of unity.
The number m which satisfies this equation is thus implicit in any
given system of rates of death and reproduction, and measures the
relative rate of increase or decrease of a population when in the steady
state appropriate to any such system. In view of the emphasis laid
by Malthus upon the 'law of geometric increase ' m may appropriately
be termed the Malthusian parameter of population increase. It
evidently supplies in its negative values an equally good measure of
population decrease, and so covers cases to which, in respect of man-
kind, Malthus paid too little attention.
In view of the close analogy between the growth of a population
supposed to follow the law of geometric increase, and the growth of
capital invested at compound interest, it is worth noting that if we
FUNDAMENTAL THEOREM OF NATURAL SELECTION 27
regard the birth of a child as the loaning to him of a life, and the birth
of his offspring as a subsequent repayment of the debt, the method by
which m is calculated shows that it is equivalent to answering the
question At what rate of interest are the repayments the just
equivalent of the loan ? For the unit investment has an expectation
of a return l x bxdx in the time interval dx, and the present value of this
repayment, if m is the rate of interest, is e~ m *l x b x dx \ consequently the
Malthusian parameter of population increase is the rate of interest at
which the present value of the births of offspring to be expected is
equal to unity at the date of birth of their parent. The actual values
of the parameter of population increase, even in sparsely populated
dominions, do not, however, seem to approach in magnitude the rates
of interest earned by money, and negative rates of interest are, I
suppose, unknown to commerce.
Reproductive value
The analogy with money does, however, make clear the argument
for another simple application of the combined death and reproduction
rates. We may ask, not only about the newly born, but about persons
of any chosen age, what is the present value of their future offspring ;
and if present value is calculated at the rate determined as before, the
question has the definite meaning To what extent will persons of
this age, on the average, contribute to the ancestry of future genera-
tions ? The question is one of some interest, since the direct action of
Natural Selection must be proportional to this contribution. There
will also, no doubt, be indirect effects in cases in which an animal
favours or impedes the survival or reproduction of its relatives ; as a
suckling mother assists the survival of her child, as in mankind a
mother past bearing may greatly promote the reproduction of her
children, as a foetus and in less measure a sucking child inhibits
conception, and most strikingly of all as in the services of neuter
insects to their queen. Nevertheless such indirect effects will in very
many cases be unimportant compared to the effects of personal repro-
duction, and by the analogy of compound interest the present value
of the future offspring of persons aged x is easily seen to be
v x = ~j I e~ mt l t b t dt.
X
Each age group may in this way be assigned its appropriate
28 FUNDAMENTAL THEOREM OF NATURAL SELECTION
reproductive value. Fig. 2 shows the reproductive value of women
according to age as calculated from the rates of death and reproduc-
tion current in the Commonwealth of Australia about 1911. The
Malthusian parameter was at that time positive, and as judged from
20 30
AGE IN YEARS
FIG. 2. Reproductive value of Australian women.
The reproductive value for female persons calculated from the birth- and death-
rates current in the Commonwealth of Australia about 1911. The Malthusian
parameter is -f- 0-01231 per annum.
female rates was nearly equivalent to 1| per cent, compound interest ;
the rate would be lower for the men, and for both sexes taken together,
owing to the excess of men in immigration. The reproductive value,
which of course is not to be confused with the reproductive rate,
reaches its maximum at about 18f, in spite of the delay in repro-
duction caused by civilized marriage customs ; indeed it would have
been as early as 16, were it not that a positive rate of interest gives
higher value to the immediate prospect of progeny of an older woman,
compared to the more remote children of a young girl. If this is the
FUNDAMENTAL THEOREM OF NATURAL SELECTION 29
case among a people by no means precocious in reproduction, it would
be surprising if, in a state of society entailing marriage at or soon after
puberty, the age of maximum reproductive value should fall at any
later age than twelve. In the Australian data, the value at birth is
lower, partly by reason of the effect of an increasing population in
setting a lower value upon remote children and partly because of the
risk of death before the reproductive age is reached. The value shown
is probably correct, apart from changes in the rate since 1911, for
such a purpose as assessing how far it is worth while to give assistance
to immigrants in respect of infants (though of course, it takes no
account of the factor of eugenic quality), for such infants will usually
emigrate with their parents ; but it is overvalued from the point of
view of Natural Selection to a considerable extent, owing to the capa-
city of the parents to replace a baby lost during lactation. The
reproductive value of an older woman on the contrary is undervalued
in so far as her relations profit by her earnings or domestic assistance,
and this to a greater extent from the point of view of the Common-
wealth, than from that of Natural Selection. It is probably not without
significance in this connexion that the death rate in Man takes a
course generally inverse to the curve of reproductive value. The
minimum of the death rate curve is at twelve, certainly not far from
the primitive maximum of the reproductive value; it rises more
steeply for infants, and less steeply for the elderly than the curve of
reproductive value falls, points which qualitatively we should antici-
pate, if the incidence of natural death had been to a large extent
moulded by the effects of differential survival.
A property that well illustrates the significance of the method of
valuation, by which, instead of counting all individuals as of equal
value in respect of future population, persons of each age are assigned
an appropriate value v x , is that, whatever may be the age constitution
of a population, its total reproductive value will increase or decrease
according to the correct Malthusian rate m, whereas counting all
heads as equal this is only true in the theoretical case in which the
population is in its steady state. For suppose the number of persons
in the age interval dx is n x dx ; the value of each element of the popula-
tion will be n x v x dx ; in respect of each such group there will be a gain
in value by reproduction at the rate of n x b x v dx, a loss by death of
n x p, x v x dx, and a loss by depreciation of -nxdv x , or in all
30 FUNDAMENTAL THEOREM OF NATURAL SELECTION
but by differentiating the equation by which v x is defined, it appears
that
Idv I dl x -l x b x e~ mx _ Mo
7 ' 7 7 I*"
V x dx l x dx *,,_. v *
v a l * e
or that
dv x - n x v x dx -f b x v dx mv x dx.
Consequently the rate of increase in the total value of the population
is m times its actual total value, irrespective of its constitution in
respect of age. A comparison of the total values of the population at
two census epochs thus shows, after allowance for migration, the
genuine biological increase or decrease of the population, which may
be entirely obscured or reversed by the crude comparison of the
number of heads. The population of Great Britain, for example, must
have commenced to decrease biologically at some date obscured by
the war, between 1911 and 1921, but the census of 1921 showed a
nominal increase of some millions, and that of 1931 will, doubtless in
less degree, certainly indicate a further spurious period of increase,
due to the accumulation of persons at ages at which their reproduc-
tive value is negligible.
The genetic element in variance
Let us now consider the manner in which any quantitative individual
measurement, such as human stature, may depend upon the indi-
vidual genetic constitution. We may imagine, in respect of any pair
of alternative genes, the population divided into two portions, each
comprising one homozygous type together with half of the hetero-
zygotes, which must be divided equally between the two portions . The
difference in average stature between these two groups may then be
termed the average excess (in stature) associated with the gene sub-
stitution in question. This difference need not be wholly due to the
single gene, by which the groups are distinguished, but possibly also
to other genes statistically associated with it, and having similar
or opposite effects. This definition will appear the more appropriate
if, as is necessary for precision, the population used to determine its
value comprises, not merely the whole of a species in any one genera-
tion attaining maturity, but is conceived to contain all the genetic
combinations possible, with frequencies appropriate to their actual
FUNDAMENTAL THEOREM OF NATURAL SELECTION 31
probabilities of occurrence and survival, whatever these may be, and
if the average is based upon the statures attained by all these geno-
types in all possible environmental circumstances, with frequencies
appropriate to the actual probabilities of encountering these circum-
stances. The statistical concept of the excess in stature of a given
gene substitution will then be an exact one, not dependent upon
chance as must be any practical estimate of it, but only upon the
genetic nature and environmental circumstances of the species. The
excess in a factor will usually be influenced by the actual frequency
ratio p : q of the alternative genes, and may also be influenced, by
way of departures from random mating, by the varying reactions of
the factor in question with other factors ; it is for this reason that its
value for the purpose of our argument is defined in the precise
statistical manner chosen, rather than in terms of the average sizes
of pure genotypes, as would be appropriate in specifying such a value
in an experimental population, in which mating is under control, and
in which the numbers of the different genotypes examined is at the
choice of the experimenter.
For the same reasons it is also necessary to give a statistical
definition of a second quantity, which may be easily confused with
that just defined, and may often have a nearly equal value, yet which
must be distinguished from it in an accurate argument ; namely the
average effect produced in the population as genetically constituted.
by the substitution of the one type of gene for the other. By what-
ever rules mating, and consequently the frequency of different gene
combinations, may be governed, the substitution of a small propor-
tion of the genes of one kind by the genes of another will produce a
definite proportional effect upon the average stature. The amount oi
the difference produced, on the average, in the total stature of the popu-
lation, for each such gene substitution, may be termed the average
effect jof such substitution, in contra-distinction to the average excess
as defined above. In human stature, for example, the correlation
found between married persons is sufficient to ensure that each gene
tending to increase the stature must bo associated with other genes
having a like effect, to an extent sufficient to make the average
excess associated with each gene substitution exceed its average
effect by about a quarter.
If a is the magnitude of the average excess of any factor, and a the
magnitude of the average effect on the chosen measurement, we shall
32 FUNDAMENTAL THEOREM OF NATURAL SELECTION
now show that the contribution of that factor to the genetic variance
is represented by the expression pqaa.
The variable measurement will be represented byre, and the relation
of the quantities a to it may be made more clear by supposing that
for any specific gene constitution we build up an * expected' value, X,
by adding together appropriate increments, positive or negative,
according to the natures of the genes present. This expected value
will not necessarily represent the real stature, though it may be a
good approximation to it, but its statistical properties will be more
intimately involved in the inheritance of real stature than the
properties of that variate itself. Since we are only concerned with
variation we may take as a primary ingredient of the value of X, the
mean value of x in the population, and adjust our positive and nega-
tive increments for each factor so that these balance each other when
the whole population is considered. Since the increment for any one
gene will appear p times to that for its alternative gene q times in the
whole population, the two increments must be of opposite sign and in
the ratio q : ( #). Moreover, since their difference must be a, the
actual values cannot but be qa and (pa) respectively.
The value of the average excess a of any gene substitution was
obtained by comparing the average values of the measurement x in
two moieties into which the population can be divided. It is evident
that the values of a will only be properly determined if the same
average difference is maintained in these moieties between the values
of X, or in other words if in each such moiety the sum of the devia-
tions, x-X, is zero. This supplies a criterion mathematically
sufficient to determine the values of a, which represent in the popula-
tion concerned the average effects of the gene substitutions. It
follows that the sum for the whole population of the product X (x - X)
derived from each individual must be zero, for each entry qa or ( - pa)
in the first term will in the total be multiplied by a zero, and this will
be true of the items contributed by every factor severally. It follows
from this that if X and x are now each measured from the mean of
the population, the variance of X, which is the mean value of X 2 , is
equal to the mean value of Xx. Now the mean value of Xx will
involve a for each Mendelian factor ; for X will contain the item qa
in the p individuals of one moiety and ( -pa) in the q individuals of
the other, and since the average values of x in these two moieties
differ by a, the mean value of Xx must be the sum for all factors of the
FUNDAMENTAL THEOREM OF NATURAL SELECTION 33
quantities pqaa. Thus the variance of X is shown to be W=(pqaa)
the summation being taken over all factors, and this quantity we may
distinguish as the genetic variance in the chosen measurement x.
That it is essentially positive, unless the effect of every gene severally
is zero, is shown by its equality with the variance of X. An extension
of this analysis, involving no difference of principle, leads to a
similar expression for cases in which one or more factors have more
than two different genes or allelomorphs present.
The appropriateness of the term genetic variance lies in the fact
that the quantity X is determined solely by the genes present in the
individual, and is built up of the average effects of these genes. It
therefore represents the genetic potentiality of the individual con-
cerned, in the aggregate of the mating possibilities actually open to
him, in the sense that the progeny averages (of x, as well as of X) of
two males mated with an identical series of representative females
will differ by exactly half as much as the genetic potentialities of
their sires differ. Relative genetic values may therefore be determined
experimentally by the diallel method, in which each animal tested is
mated to the same series of animals of the opposite sex, provided
that a large number of offspring can be obtained from each such
mating. Without obtaining individual values, the genetic variance
of the population may be derived from the correlations between
relatives, provided these correlations are accurately obtained. For
this purpose the square of the parental correlation divided by the
grandparental correlation supplies a good estimate of the fraction, of
the total observable variance of the measurement, which may be
regarded as genetic variance.
It is clear that the actual measurements, x, obtained in individuals
may differ from their genetic expectations by reason of fluctuations
due to purely environmental circumstances. It should be noted that
this is not the only cause of difference, for even if environmental
fluctuations were entirely absent, and the actual measurements
therefore determined exactly by the genetic composition, these
measurements, which may be distinguished as genotypic, might still
differ from the genetic values, X. A good example of this is afforded
by dominance, for if dominance is complete the genotypic value of
the heterozygote will be exactly the same as that of the correspond-
ing dominant homozygote, and yet these genotypes differ by a gene
substitution which may materially affect the genetic potentiality
34 FUNDAMENTAL THEOREM OF NATURAL SELECTION
represented by X, and be reflected in the average measurement of
the offspring. A similar cause of discrepancy occurs when gene
substitutions in different factors are not exactly additive in their
average effects. The genetic variance as here defined is only a portion
of the variance determined genotypically, and this will differ from,
and usually be somewhat less than, the total variance to be observed.
It is consequently not a superfluous refinement to define the purely
genetic element in the variance as it exists objectively, as a statistical
character of the population, different from the variance derived from
the direct measurement of individuals.
Natural Selection
The definitions given above may be applied to any characteristic
whatever; it is of special interest to apply them to the special
characteristic m which measures the relative rate of increase or
decrease. The two groups of individuals bearing alternative genes,
and consequently the genes themselves, will necessarily either have
equal or unequal rates of increase, and the difference between the
appropriate values of m will be represented by a, similarly the
average effect upon m of the gene substitution will be represented
by a. Since m measures fitness to survive by the objective fact of
representation in future generations, the quantity pqaa will represent
the contribution of each factor to the genetic variance in fitness;
the total genetic variance in fitness being the sum of these contribu-
tions, which is necessarily positive, or, in the limiting case, zero.
Moreover, any increase dp in the proportion of one type of gene at the
expense of the other will be accompanied by an increase adp in the
average fitness of the species, where a may of course be negative;
but the definition of a requires that the ratio p : q must be increasing
in geometrical progression at a rate measured by a, or in mathe-
matical notation that
which may be written
/I IV
{ + ]dp a at,
\P 
