Chapter X: Appendix: B

AN EXAMINATION BY MR. FLETCHER MOULTON OF MR. WALLACE'S CALCULATION TOUCHING THE POSSIBILITY OF PHYSIOLOGICAL SELECTION EVER ACTING ALONE.

We have seen that the only important point of difference between Mr. Wallace's more recent views and my own on the problem of inter-specific sterility, has reference to the question whether variations in the way of cross-infertility can _ever_ arise and act "alone, in an otherwise undifferentiated species," or whether they can _never_ so arise and act. It is Mr. Wallace's opinion that, even if they ever do arise alone, at all events they can never act in differentiating a specific type, seeing that the chances against their suitable mating must be so great: only if they be from the first associated with some other form of homogamy, which will have the effect of determining their suitable mating, does he think that they can act in the way supposed by our theory of "selective fertility"[62]. On the other hand, as previously and frequently stated, I have so strong a belief in the segregating power of physiological selection, or selective fertility, that I do not think it is necessary for this principle to be _always___ associated with some other form of homogamy. From the first, indeed, I have laid great stress (as, also, has Mr. Gulick) on the re-enforcing influence which association with any other form of homogamy must exercise upon the physiological form, and vice versa; but I have also said that, in my opinion, the physiological form may in many cases be able to act entirely alone, or without assistance derived from any other source. The question here is, as we have already so fully seen, a question of but secondary importance; since, whether or not the physiological form of homogamy ever acts alone, even Mr. Wallace now allows, or rather argues, that it acts in combination--and this so habitually, as well as with so much effect, that it constitutes a usual condition to the origination of species. Nevertheless, although the only relevancy of his numerical computation of chances--whereby he thinks that he overturns my theory _in toto_--is such relevancy as it bears to this question of secondary importance, I have thought it desirable to refer the question, together with Mr. Wallace's views upon it, to the consideration of a trained mathematician.

[62] His sentence, "all fertility not correlated with some _useful_
variation has a constant tendency to effect its own elimination,"
still further restricts the possible action of physiological
selection to cases where at least one of the other forms of homogamy
with which it is associated is natural selection. Or, in other
words, it is represented that physiological selection must always be
associated with natural selection, even if it be likewise associated
with any other form of exclusive breeding. But as this further
limitation appears to me self-evidently unjustifiable (seeing that
utility is not the only possible means of securing effective
isolation) I here neglect it, and take the wider ground marked out
above. It is needless to say that this is giving Mr. Wallace every
possible advantage, by not holding him to his still narrower ground.

As this "subordinate question" depends entirely on numerical computations involving the doctrine of chances, I should first of all like to remark, that in reference to biological problems of the kind now before us, I do not myself attach much importance to a merely mathematical analysis. The conditions which such problems involve are so varied and complex, that it is impossible to be sure about the validity of the _data_ upon which a mathematical analysis is founded. Nevertheless, for the sake of meeting these criticisms upon their own ground, I will endeavour to show that, even as mathematical calculations, they are quite untrustworthy. And, in order to do this effectually, I will quote the results of a much more competent, as well as a much more thorough, inquiry. I applied to Mr. Moulton for this purpose, not only because he is one of the ablest mathematicians of my acquaintance; but also because his interest in biology, and his knowledge of Darwinian literature, render him well fitted to appreciate exactly, and in all their bearings, the questions which were submitted to his consideration. I need only add that his examination was completely independent, and in no way influenced by me. Having previously read my paper on _Physiological Selection_, Mr. Gulick's paper on _Divergent Evolution_, and Mr. Wallace's book on _Darwinism_, he was in possession of all the materials; and I merely requested the favour of his opinion upon the whole case from a mathematical point of view. The following is his reply; and I give it _in extenso_, because it serves to place in another light some of the general considerations which it has already been my endeavour to present[63].

[63] In our _Nature_ correspondence of 1890-1891, Mr. Wallace
remarked: "If Dr. Romanes will carefully work out numerically (as I
have attempted to do) a few cases showing the preservative and
accumulative agency of pure physiological selection within an
otherwise undifferentiated species, he will do more for his theory
than volumes of general disquisition or any number of assertions
that it _does_ possess this power." Several months before this was
written I had already in my hands Mr. Moulton's letter, with its
accompanying calculations.

After some introductory remarks on Mr. Wallace's "adoption of the theory of physiological selection pure and simple," and "the pure caricature of it which he puts forward as" mine, the letter proceeds thus:--

The reason why it is so easy to attack your theory is that it is so
easy to confuse the survival of an _individual_ with the survival
of a _peculiarity_ of _type_. No one has ever said that an
_individual_ is _assisted_ by the possession of selective
fertility: that is a matter which cannot affect his chance of
_life_. Nor has any one said that the possession of selective
fertility in an _individual_ will _of itself_ increase the chance
of his having _progeny_ that will survive, and in turn become the
progenitors of others that will survive. Taken by itself, the fact
that an _individual_ is capable of fertility with some only of the
opposite sex lessens the chance of his having progeny. Whether or
not he is more or less favourably situated than his _confreres_ for
the battle of life must be decided by the _total sum_ of his
peculiarities; and the question whether or not this selective
fertility will be a hindrance must be decided by considerations
depending on the other peculiarities associated with it.

But when we come to consider the survival or permanence of a _type_
or _peculiarity_, the case is quite different. It then becomes not
only a favourable circumstance, but, in my opinion, almost a
necessary condition, that the peculiarity should be associated with
selective fertility[64].

[64] As, for example, in the case of sexuality in general. It is not
to the advantage of such individual male Arthropoda as perish after
the performance of the sexual act that they should perform it; but
its performance is necessary for the perpetuation of their
species.--G. J. R.

Take the case of the Jews. I don't think that intermarriage with
other nations would lessen their fertility, or diminish the number
of their progeny; nor is there any reason to think that this
progeny would be unequal to the struggle for existence. But no one
doubts that the abandonment of their voluntary isolation (which
operates so far as this is concerned as a selective fertility),
would lead to the disappearance of the familiar Jewish type. All
the world would get some of it; but as a whole it would be
"swamped."

Now although no doubt Wallace would admit all this, he fails to
give it the weight it ought to have. In discussing the question of
its operation he considers too exclusively the case of the
individual.

Of course, a type can only be perpetuated through the medium of
individuals, and all that his argument amounts to is, that
selective fertility would be so fatal to individuals that _no_ type
which presents it could be formed or perpetuated--a conclusion
which is not only absurd in itself, but contradicted by his own
subsequent adoption of your theory. Besides, apart from
calculations (with which I will deal when I write next), such
reasoning brings its own refutation. Selective fertility is not in
the same category as some of the other influences to which an
important share has been ascribed in the formation of the existing
types. _It exists as a recognized phenomenon._ Hence all these
numerical proofs that it would lead to extinction, because it is so
disadvantageous to the possessor, prove too much. They would show
that the degree of selective fertility which so frequently
characterizes species is a most onerous gift; and that, were it not
present, there would be a vastly increased chance of fertility,
which would render the races fitter and lead to their increased
survival. Why then has it not been got rid of?

The two answers which no doubt would be given seem to me to support
rather than to make against your theory. In the first place,
Wallace might say that this infertility is an advantage because it
keeps pure a type which is specially fitted to its surroundings, as
shown by its continued existence. But if this be so, and it is
necessary to protect the _developed_ type, how much more necessary
to protect the _incipient_ type! In the second place, he might say
that this selective fertility is not so disadvantageous when the
species has been formed, because the individual can choose his mate
from his like; whereas, when it is beginning to be formed, he must
mate blindly, or without what you call "psychological selection."
But this seems to me to be wholly inapplicable to at least half the
animal, and to all the vegetable kingdom. Moreover, with regard to
the other half of the animal kingdom, it merely raises the
question,--How soon will such an incipient type recognize itself?
Seeing it is probable that many families [broods] will belong to
the same [incipient] type, I should not be surprised if it were
found that this sexual recognition and preference sets in very
early.

But this leads me to the question of your letter. I understand you
to want me to examine and criticize the attempted numerical
arguments against or for your theory. Now it seems to me that it
will be best to take, in the first instance, the vegetable kingdom,
and with regard to it I cannot see how there can be any numerical
argument against the theory. For we often have species side by side
with others nearly allied, but much more numerous. The condition of
these is precisely analogous to that of your incipient species.
They are exposed to fertilization from, say, ten times as numerous
individuals of the allied species. They reject this in favour of
that from the relatively few individuals of their own. Yet the two
species are in competition. I could go through the numerical
arguments of your assailant word for word, applying them to such a
case as this, and they would triumphantly show that the specific
fertility of the rarer kind would lead to its certain extinction.
Yet we know that this is not so.

Indeed, the too triumphant character of the logic used against you
seems to me to be capable of being turned to your use. If
cross-infertility is so intensely disadvantageous to the
individuals presenting it, it cannot have been _that_ which made
these individuals and their progeny survive. It is therefore a
burden which they have carried. But we find that it is more or less
present in all the closely allied types that occur on common areas:
therefore it must be a necessary feature in the formation of such
types; for it cannot be an accident that it is present in so many.
In other words, it must be the price which the individual and his
progeny pay for their formation into a type. And this is your
theory pure and simple.

The more I consider the matter, the more I feel that it is
impossible to decide as to the sufficiency of selective fertility
to explain the formation of species, if we consider merely the
effect it would have on the number of individuals, as contrasted
with what it would be if no such peculiarity had developed itself.
Indeed, I may say that on pondering over the matter I have come to
the conclusion, that mere fertility is probably a comparatively
unimportant factor in the preservation of the species, after a
certain sufficient degree of fertility is attained. I do not wish
to be misunderstood. To a certain point fertility is not only
advantageous but necessary, in order to secure survival of the
type; but I feel that little reliance can be placed on calculations
based on the numerical co-efficient of fertility (i. e. the ratio
of the number of offspring to the number of parents) in determining
the relative chance of type-survival.

Take, for instance, the oak tree. It produces thousands of acorns,
almost the whole of which die without producing any progeny. Have
we any reason to believe that if the number of acorns borne by oak
trees were diminished, even so much as to one-tenth, the race of
oaks would perish? It may of course be said that, if all other
things are equal, the probabilities of survival must be increased
by increased fertility of this kind; but I feel convinced that when
numerical fertility has attained to a high point in circumstances
in which actual increase of the race cannot take place to any
substantial extent, the numerical value of this fertility sinks
down into a factor of the second or third order of importance--that
is to say, into the position of a factor whose effects are only to
be considered when we have duly allowed for the full effects of all
the main factors. Until we have done that, we gain little or
nothing in the way of accuracy of conclusion by taking into
consideration the minor factors. It may be very well to neglect the
effect of the attraction of Jupiter in our early researches on the
motion of the Moon; and our doing so will not prevent the results
being approximate and having considerable value, because we are
retaining the two main factors that establish the motion, viz. the
effects of the Earth and the Sun. But if we exclude the effect of
one of these main factors, our results would be worthless; and it
would not be rendered substantially less so by the fact that we had
taken Jupiter into account in arriving at them.

You must not imagine, however, that I think it wholly profitless to
see whether there would be any substantial effect on numerical
fertility were _selective_ fertility to manifest itself. But if we
want to derive any assistance from calculation, it must be by
applying it with a good deal more precision and definiteness than
anything that Wallace shows. And, in the first place, it is useless
to confuse the vegetable and animal kingdoms. In the former you
have union unaffected by choice; in the latter, so far at all
events as the higher animals are concerned, you have "psychological
selection." In order to give you a specimen of what can safely be
done by calculation if you take a problem of sufficient
definiteness, I have chosen the case of a flowering plant in which
a certain proportion of the race have developed the peculiarity of
being sterile with the remainder, while retaining the normal
fertility of the race in unions among themselves. In order to give
the greatest advantage to your critics, I have assumed that such
flowers as possess the peculiarity are not self-fertilizable; for
it is clear that if we suppose that they are self-fertilizable, the
fertility need be very slightly affected.

As I have excluded self-fertilization, it is necessary, if we are
to get any trustworthy results, that one should consider the mode
in which fertilization will be produced. I have taken the case of
fertilization by insects, and have assumed that each flower is
visited a certain number of times by insects during the period when
fertilization is possible; and, further, that the insects which
visit it have on the average visited a certain number of flowers of
the same species before they came there. Of course nothing but
observation can fix these latter numbers; but I should not be
surprised at finding that they are of considerable magnitude[65]. In
order to make the results a little more intelligible, I have
grouped them under the numbers which represent the average number
of flowers that an insect visits in a journey. This is a little
more than twice as great as the number which represents the number
of flowers he has on the average visited before coming to the
individual whose fertility we are considering.

[65] In this anticipation Mr. Moulton is right. The well-known
botanist, Mr. Bennett, read a most interesting paper on the subject
before the British Association in 1881. His results have since been
corroborated by other observers. In particular, Mr. R. M. Christy
has recorded the movements of 76 insects while visiting at least
2,400 flowers. (_Entomologist_, July 1883, and _Zool. Journal Lin.
Soc._, August 1883.) The following is an analysis of his results. In
the case of butterflies, in twelve observations on nearly as many
species, there are recorded altogether 99 visits to fifteen species
of flowers; and of these 99 visits 94 were constant to the same
species, leaving only 5 visits to any other, or second species. In
the case of the hive-bee, there were 8 individuals observed: these
visited altogether 258 flowers, and all the visits paid by the same
individual were paid to the same species in each of the eight cases.
Lastly, as regards bumble-bees, there were altogether observed 55
individuals belonging to four species. These paid altogether 1751
visits to 94 species of flowers. Of these 1751 visits, 1605 were
paid to one species, 131 to two species, 16 to three, 6 to four, and
1 to five. Adding all these results together, we find that 75
insects (butterflies and bees) visited 117 species of flowers: of
these visits, 1957 were constant to one species of flower; 136 were
paid also to a second species, 16 also to a third, 6 also to a
fourth, and 1 also to a fifth. Or, otherwise stated, while 1957 were
absolutely constant, from such absolute constancy there were only
159 deviations. Moreover, if we eliminate three individual humble
bees, which paid nearly an equal number of visits to two species
(and, therefore, would have ministered to the work of physiological
selection almost as well as the others), the 159 deviations become
reduced to 72, or about four per cent. of the whole.--G. J. R.

I send you the formula and the calculation on which it is based in
an Appendix; but as I know you have a holy horror of algebraical
formulae, I give you here a few numerical results.

The cases I have worked out are those in which the number of
insects visiting each flower is 5, or 10, or 15; and I have also
taken 5, 10, and 15, to represent the number of flowers which an
insect visits each journey. This makes nine cases in all; and I
have applied these to two instances--viz. one in which one-fifth of
the whole race have developed cross-infertility, and the other in
which one-tenth only have done so. Taking first the instance where
one-fifth have developed the peculiarity, I find that if on the
average five insects visit a flower, and each insect on the average
visits five flowers on a journey, the fertility is diminished by
about one-tenth. If, however, the average number of flowers the
insect visits is ten, the reduction of fertility is less than one
per cent. And it becomes inappreciable if the average number is
fifteen. If on the average ten insects visit each flower, then, if
each insect visits on the average five flowers on a journey, the
reduction of fertility is a little over one per cent.; but if it
visits ten or fifteen the reduction is inappreciable. If fifteen
insects visit the flower on an average, then, if these insects on
the average visit five or more flowers on a journey, the reduction
of fertility is inappreciable.

By the term inappreciable I mean that it is not substantially
greater than one-tenth of one per cent.--i.e. not more than
one-thousandth.

Of course, if the proportion of individuals acquiring the
peculiarity is less, the effect on the fertility under the above
hypothesis will be greater; and it will not be counteracted so
fully unless the number of insect visits is larger, or unless the
insects visit more flowers on a journey. Thus if only one-tenth of
the race have developed the peculiarity, then, if each flower is
visited on the average by five insects who visit five flowers on
each trip, the fertility will be reduced about one-third. If,
however, the insects visit on the average ten flowers per trip, it
will be only diminished about one-tenth; and if they visit fifteen
on each trip, it will be only diminished about one-fortieth. If in
the same case we suppose that each flower receives ten insect
visits, then, if the insects visit on an average five flowers per
trip, the fertility will be diminished about one-eighth. If they
visit ten on a trip, it will be diminished about one-hundredth, and
the diminution is inappreciable if they visit fifteen on a trip.
Similarly, if a flower receives fifteen insect visits, the
diminution is about one-twenty-fifth, if insects visit on the
average five flowers on a trip; and is inappreciable if they visit
ten or fifteen.

These figures will show you that it is exceedingly possible that a
peculiarity like this, the effect of which at first sight would
seem to be so prejudicial to fertility, may in fact have little or
no influence upon it; and if you set against this the overwhelming
importance of such a peculiarity in segregating the type so as to
give it a chance of becoming a fixed species, you will, I think,
feel that your hypothesis has nothing to fear from a numerical
examination.

I have not examined the case of fertilization by other means; nor
have I examined the case of fertilization in animals, where
psychological selection can come in. To obtain any useful results,
one would have to consider very carefully the circumstances of each
case; and at present, at all events, I do not think it would be
useful to do so. Nor have I attempted to show the converse of the
problem--viz. the effect of swamping where cross-fertilization is
possible. I shall be very glad to examine any one of these cases if
you want me to do so; but I should prefer to leave it until I hear
from you again.

If you contrast the results that I have given above with those
given on pages 181 to 183 of Wallace's book, you will see the
enormous difference. His calculations can only apply to the animal
kingdom in those cases in which there is only a union between one
individual of each sex; and before you can deal with the question
of such animals, you will have to take into consideration many
elements besides that of mere fertility, if you wish to get any
tolerably accurate result[66].

[66] Here follows the Appendix presenting the calculations on which
the above results are founded; but it seems unnecessary to reproduce
it on the present occasion.--G. J. R.

The above analysis leaves nothing to be added by me. But, in conclusion, I may once more repeat that the particular point with which it is concerned is a point of very subordinate importance. For even if Mr. Wallace's computation of chances had been found by Mr. Moulton to have been an adequate computation--and, therefore, even if it had been thus proved that physiological homogamy must always be associated with some other form of homogamy in order to produce specific divergence--still the importance of selective fertility as a factor of organic evolution would not have been at all diminished. For such a result would merely have shown that, not only "in many cases" (as I originally said), but actually in all cases, the selective fertility which I hold to have been so generally concerned in the differentiation of species has required for this purpose the co-operation of some among the numerous other forms of homogamy. But inasmuch as, by hypothesis, no one of these other or co-operating factors would of itself have been capable of effecting specific divergence in any of the cases where its association with selective fertility is concerned, the mathematical proof that such an association is _always_--and not merely _often_--necessary, would not have materially affected the theory of the origin of species by means of physiological selection. We have now seen, however, that a competent mathematical treatment proves the exact opposite; and, therefore, that Mr. Wallace's criticism fails even as regards the very subordinate point in question.