THE THEORY OF NATURAL SELECTION FROM A 
MATHEMATICAL POINT OF VIEW* 
“T°HE fascinating hypothesis of Darwinism has, within the last 
few years, so completely taken hold of the scientific mind, 
both in this country and in Germany, that almost the whole of our 
rising men of science may be classed as belonging to this school 
of thought. Probably since the time of Newton no man has had 
so great an influence over the development of scientific thought 
as Mr. Darwin; and no one can over-estimate the debt which 
Science owes to his patient researches and his clear insight into 
some of the hidden ways of Nature. The advocates of Darwinism 
have, however, almost invariably failed to recognise that the 
theory consists of two essentially distinct portions, one of which 
may be admitted while the other is denied. The first portion is 
that with which the name of Darwin is popularly associated, 
although its origination is by no means due to him, namely, the 
probable ancestry of all forms of living organism from a single 
or a few original germs ; the other portion, and that which we 
especially owe to his genius, is the theory that the infinite modi- 
fications of existing forms owe their origin to a process of 
Natural Selection from spontaneous variations. These two per- 
fectly distinct hypotheses have generally been so confounded 
together that those who have attacked or defended the one have 
also attacked or defended the other. My object in the present 
paper is to show that, while the former hypothesis may be con- 
sidered as established, as nearly as it is possible to establish a 
theory which requires thousands or millions of years for its 
complete development, the arguments in support of the second 
hypothesis are far less satisfactory. 
The principle that new forms of organic life have been pro- 
duced by modifications of older nearly-allied forms is by no 
means a new one; its inherent reasonableness and probability 
commended it to Lamarck and the author of the ‘‘ Vestiges of 
Creation” long before it was elaborated in a more scientific form 
by Mr. Darwin and Mr. Wallace. It has been opposed, of course, 
by theologians ; but, were it not that the theological mind is 
inherently averse to the reception of new ideas, it would have 
been seen that the supposition that the Creative Power works by 
continuous modification and adaptation of contrivance to end, 
by a constant exercise of His prerogative, is a far higher tribute 
to His exalted attributes, than the popular dogma that all living 
things were created as we now see them by one single gigantic 
effort, after which the power collapsed, and has never since been 
exercised. Why should organic life be the one thing in the 
world not subject to change? The coup de grace may be con- 
sidered to have been given to the anciently received theory by 
the investigations so ably carried out by Mr. Darwin and Dr. 
Hooker on the characteristics of Insular Floras. The fact that 
no island which has been separated from the mainland during 
recent geological epochs has genera, and scarcely even species, 
of animals or plants peculiar to itself, while islands which have 
remained isolated during lengthened geological periods have 
faunze and florz almost entirely peculiar to themselves, is in- 
explicable on any other hypothesis than that of the gradual 
differentiation of species by long-continued separation. No 
more striking instance of this law has been given than that 
afforded by the East Indian Islands, as shown in Mr. Wallace’s 
“*Malay Archipelago.” Two great types of animals and plants 
are found in different regions of the archipelago, the Indo- 
Malayan and the Australian ; and these two types are separated, 
not by any diversity of climate and soil—not even by any of the 
wide but shallow channels which indicate recent separation, such 
as that between Borneo and Sumatra—but by the narrow 
but very deep channel separating Bali from Lombok, which 
indicates a lengthened geological separation of two continents at 
this point. 
The hypothesis that the prime agent in all these infinite modi- 
fications is the principle of Natural Selection from spontaneous 
variations, has been recently further illustrated by Mr. Wallace’s 
volume of Essays, ‘‘Contributions to the Theory of Natural 
Selection ;” and it is mainly from the illustrations furnished in 
this work that I propose to derive my arguments as to its 
inadequacy. In the first place I wish to call attention to the 
fact which the Duke of Argyll has already acutely pointed out 
in his *‘ Reign of Law,” that the theory does not even attempt 
to explain the most inexplicable phenomenon in the develipment 
of these organic changes, namely, the first commencement of a 
* Paper read before Section D of the British Assocjation, at Liverpool, 
September acth, 1870. 
NATURE 


i 
(Noo. 10, 1870 

tendency to variation. ‘The title of Mr. Darwin’s famous work, 
the text-book of the theory, seems to me, indeed, altogether a 
misnomer: ‘fThe Origin of Species by means of Natural Selec- 
tion.” Mr. Darwin admits the existence of what he terms a 
“ spontaneous”’ tendency to variation among the offspring from 
a common ancestor ; this ‘‘ spontaneous” tendency is the only 
natural law which can correctly be termed that of the origin of 
species ; all that Mr, Darwin arid his disciples attempt to ex- 
plain is the survival and propagation of certam among the 
diverse forms thus resulting in preference to others. Through- 
out the whole of Mr. Wallace’s volume he appears to have no 
consciousness that his theory does not go to the root of the 
matter. When once the tendency to change has set in, there 
can be no doubt that ‘‘ Natural Selection,” ‘*The Survival of 
the Fittest,” whatever you like to term the principle, is one 
among many causes which tend to the perpetuation of certain 
forms. When, however, Mr. Darwin asserts, ‘‘ I am convinced 
that Natural Selection has been the main, but not exclusive, 
means of modification,”* I am by no means prepared to go with 
him to that extent. Some of Mr. Darwin’s disciples go even 
further, and seem to consider it, in fact, as almost the only 
means. 
There is no phenomenon in Natural History which is more 
thoroughly relied on by the advocates of Natural Selection as 
furnishing a decisive argument in favour of their theory, than the 
one which forms the subject of the longest of the essays in Mr. 
Wallace’s volume, that of Mimicry or Mimetism. I propose, 
therefore, to occupy the greater part of this paper with an 
inquiry how far the facts which have been adduced support 
the conclusions first brought prominently forward by Mr. 
Bates in his ‘‘ Naturalist on the Amazons,” and more fully 
elaborated and illustrated by Mr. Wallace. There can be 
no doubt about the frequent occurrence of ‘‘ protective resem- 
blances” in the animal kingdom. Certain classes of animals en- 
joy, from various causes, exceptional immunity from the attacks 
of their natural enemies. In order to share in these immunities, 
it is found that other animals, belonging to an entirely different 
class or order, whilst retaining all the structural characters of 
their own’ class, so closely resemble in external features of 
colour and form particular species of the favoured races as to 
be readily mistaken for them. How do the advocates of the 
theory of Natural Selection attempt to account for this super- 
ficial resemblance? By the continuous preservation, through 
countless generations, of those particular individuals which 
spontaneously approach most nearly to the ultimate forms. 
Now, there are two principles admitted or insisted on by every 
advocate of Darwinism, which it is necessary to bear very clearly 
in mind in the following argument. The first is, that, in a 
state of nature those differences which ultimately become specific 
or generic are brought about by exceedingly slow gradations. And 
it is obvious that it must be so. For if by chance any strongly 
abnormal form is produced, even should it survive to generate. 
offspring, which is in itself doubtful, it must necessarily cross 
with other less abnormal individuals, and its descendants would 
thus have a tendency to revert towards the parental form. On 
this point Mr. Darwin himself says: ‘‘It may be doubted 
whether sudden and great deviations of structure, such as we 
occasionally see in’ our domestic productions, are ever perma- 
nently propagated in a state of nature.” + And again, ‘‘ Natural 
Selection always acts with extreme slowness.” The other 
point which I wish to be borne in mind is, that no change can 
possibly take place by the process of Natural Selection which 
1s not directly of advantage to the individual. On this point 
again all the supporters of the hypothesis are agreed. Mr. 
Darwin distinctly affirms that “only those variations which are 
in some way profitable, will be preserved or naturally se- 
lected ;” § and Mr. Wallace even more emphatically speaks of 
‘the principle which Mr. Darwin so earnestly impresses upon us, 
and which is, indeed, a necessary deduction from the theory of 
Natural Selection, namely, that none of the definite facts of 
organic nature, no special organ, zo characteristic form or 
marking, no peculiarities of instinct or of habit, no relations be- 
tweeu species or between groups of species—can exist, but 
which must now be or once have been wsefw/ to the individuals 
or the races which possess them.” || 
We have, therefore, established at the outset these two 
data: that the passage from the ordinary to the mimetic 
form is effected by a number of exceedingly small steps, and 
* Origin of Species,” 4th ed.,p.6. + Ibid, p. 47. t Ibid, p. 12x. 
§ Ibid, p. 131 
| “ Contributions tothe Theery of Natural Selection,” p. 47. 
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