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.

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].

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].

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.

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].

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.