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 circ.u.mstances 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 s.e.x; 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 a.n.a.lysis 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 h.o.m.ogamy must always be a.s.sociated with some other form of h.o.m.ogamy 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 h.o.m.ogamy. 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 a.s.sociation with selective fertility is concerned, the mathematical proof that such an a.s.sociation 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.

APPENDIX C.

SOME EXTRACTS FROM THE AUTHOR'S NOTE-BOOKS.

_Bearing of Weismannism on Physiological Selection._--If in view of other considerations I could fully accept Professor Weismann's theory of heredity, it would appear to me in no small measure to strengthen my own theory of physiological selection. For Weismann's theory supposes that all changes of specific type must have their origin in variations of a continuous germ-plasm. But _the more the origin of species is referred directly to variations arising in the s.e.xual elements, the greater is the play given to the principles of physiological selection_[67]; while, on the other hand, the less standing-ground is furnished to the theory that cross-infertility between allied species is due to "external conditions of life," "prolonged exposure to uniform change of conditions," "structural modifications re-acting on the s.e.xual functions"; or, in short, that "somatogenetic" changes of any kind can of themselves induce the "blastogenetic" change of cross-infertility between progeny of the same parental stock.

[67] _Doctrine of Descent and Darwinism_, Eng. trans. p. 139.

_Cross-infertility and Diversity of Life._--Observe that one great consequence of duly recognizing the importance of intercrossing is indefinitely to raise our estimate of the part played by the principle of cross-infertility in diversifying organic nature. For whenever in any line of descent the bar of sterility arises, there the condition is given for a new crop of departures (species of a genus); and when genera are formed by the occurrence of this bar, there natural selection and all other equilibrating causes are supplied with new material for carrying on adaptational changes in new directions. Thus, owing to cross-infertility, all these causes are enabled to work out numberless adaptations in many directions (i. e. lines of descent) simultaneously.

_Cross-infertility and Stability._--The importance of sterility as a diagnostic feature is obvious if we consider that more than any other feature it serves to give _stability_ to the type; and unless a type is stable or constant, it cannot be ranked as a species. That Darwin himself attributes the highest importance to this feature as diagnostic, see _Forms of Flowers_, pp. 58, 64.

_Cross-infertility and Specific Differentiation._--In their elaborate work on the many species of the genus Hieracium, Nageli and Peter are led to the general conclusion that the best defined species are always those which display absolute sterility _inter se_; while the species which present most difficulty to the systematist are always those which most easily hybridize. Moreover, they find, as another general rule applicable to the whole genus, that there is a constant correlation between inability to hybridize and absence of intermediate varieties, and, conversely, between ability to hybridize and the presence of such varieties.

_Cross-infertility in Domesticated Cattle._--Mr. J. W. Crompton, who has had a large experience as a professional cattle-breeder, writes to me (March 2, 1887)--

"That form of barrenness, very common in some districts, which makes heifers become what are called 'bullers'--that is, irregularly in 'season,' wild, and failing to conceive--is certainly produced by excess of iron in their drinking-water, and I suspect also by a deficiency of potash in the soil."

He also informs me that pure white beasts of either s.e.x are so well known by experienced breeders to be comparatively infertile together, that they are never used for breeding purposes, so that "in some parts of the country, where a tendency to sterility had become so confirmed in the white race that they utterly died out," only the coloured breeds are now to be found. He goes on to say that if "a lot of white heifers were put to a lot of white bulls, I think you would probably get a fertile breed of pure white cattle.... I think, in short, that domestication has produced just what your theory suggests, a new variety inclined to prove sterile with its parent stock."

Commenting on the origin of domesticated cattle, Professor Oscar Schmidt remarks (_Doctrine of Descent_, p. 139)--

"Rutimeyer's minute researches on domestic cattle have shown that, in Europe at least, three well-defined species of the diluvial period have contributed to their formation--_Bos primigenius_, _longifrons_, and _frontosus_. These species once lived geographically separate, but contemporaneously; and they and their specific peculiarities have perished, to rise again in our domestic races. These races breed together with unqualified fertility. In the form of skull and horns they recall one or other of the extinct species; but collectively they const.i.tute a new main species. That from their various breeds, the three or any one of the aboriginal species would ever emerge in a state of pristine purity, would be an utterly ludicrous a.s.sertion."

Now, seeing that these "aboriginal species," although living "contemporaneously," were "geographically separate," we can well understand that their divergence of type from a common ancestor did not require, as a condition to their divergence, that any cross-sterility should have arisen between them. The geographical isolation was enough to secure immunity from mutual intercrossing, and therefore, as our present theory would have expected as probable, morphological divergence occurred without any corresponding physiological divergence, as must almost certainly have been the case if such polytypic evolution had occurred on a common area. Indeed, one of the two lines of experimental verification of our theory consists in selecting cases where nearly allied species are separated by geographical barriers, and proving that, in such cases, there is no cross-sterility.

_Fertility of Domesticated Varieties._--Some writers have sought to explain the contrast between domesticated varieties and natural species in respect of fertility when crossed, by the consideration that it is only those natural species which have proved themselves so far flexible as to continue fertile under changed conditions of life that can have ever allowed themselves to become domesticated. But although this condition may well serve to explain the unimpaired fertility under domestication of such species as for this very reason have ever become domesticated, I fail to see how it explains the further and altogether different fact, that this fertility continues unimpaired between all the newly differentiated morphological types which have been derived from the original specific type. It is one thing that this type should continue fertile after domestication: it is quite another thing that fertility should continue as between all its modified descendants, even although the amount of modification may extend much further than that which usually obtains between different natural species.

_Testing for Cross-infertility_ among varieties growing on the same area is a much more crucial line of verification than testing for unimpaired fertility between allied species which occupy different areas, because while in the former case we are dealing with "incipient species" with a view to ascertaining whether the divergence which they have already undergone is accompanied by physiological isolation, in the latter case we can never be sure that two allied species, which are now widely disconnected geographically, have always been so disconnected. They may both have originated on the same area; or one may have diverged from the other before it migrated from that area; or even if, when it migrated, it was unchanged, and if in its new home it afterwards split into two species by physiological selection, the newer species would probably prove infertile, not only with its parent type, but also with its grand-parent in any other part of the world.

_Seebohm on Isolation._--Seebohm is so strongly influenced by the difficulty from "the swamping effects of free intercrossing," that he is driven by it to adopt Asa Gray's hypothesis of variations as teleological. Indeed, he goes as far as Wagner, for he maintains that in no case can there be divergence or multiplication of species without isolation. He makes the important statement that "the more the geographical distribution of birds is studied, the more doubtful it seems to be that any species of bird has ever been differentiated without the aid of geographical isolation" (_Charadriidae_, p. 17). If this is true, it makes in favour of physiological selection by showing the paramount importance of the swamping effects of intercrossing, and consequent importance of isolation. But it makes against physiological selection by showing that the geographical form of isolation is sufficient to explain all the cases of specific differentiation in birds. But I must remember that the latter point rests largely on negative inference, and that birds, owing to their highly locomotive habits, are the cla.s.s of animals where physiological selection is likely to be most handicapped.

_Herbert on Hybridization._--Herbert tells us that when he first astonished the Horticultural Society by laying before them the results of his experiments on hybridization, his brother botanists took serious alarm. For it appeared to them that this "intermixture of species would confuse the labours of botanists, and force them to work their way through a wilderness of uncertainty." Therefore he was bluntly told by several of these gentlemen, "I do not thank you for your mules." Now, although naturalists have travelled far and learnt much since those days, it appears to me that a modern evolutionist might still turn to the horticulturist with the same words. For a.s.suredly he has no reason to thank the horticulturist for his mules, until he has found a satisfactory answer to the question why it is that natural species differ so profoundly as regards their capacity for hybridizing.

_Advance on Herbert's Position._--- If it be said that all my work amounts to showing what Herbert said long ago--viz. that the only true or natural distinction between organic types is the s.e.xual distinction--I answer that my work does much more than this. For it shows that the principle of sterility is the main condition to the differentiation, not merely of species and genera, but also to the evolution of adaptations everywhere, in higher as well as in lower taxonomic divisions. Moreover, even though naturalists were everywhere to consent to abandon specific designations, and, as Herbert advises, to "entrench themselves behind genera," there would still remain the facts of what are now called specific differences (of the secondary or morphological kind), and by whatever name these are called, they alike demand explanation at the hands of the evolutionist.

_Fritz Muller on Cross-infertility._--Fritz Muller writes, "Every plant requires, for the production of the strongest possible and most prolific progeny, a certain amount of difference between male and female elements which unite. Fertility is diminished as well when this degree is too low (in relatives too closely allied) as when it is too high (in those too little related)." Then he adds, as a general rule, "Species which are wholly sterile with pollen of the same stock, and even with pollen of nearly allied stocks, will generally be fertilized very readily by the pollen of another species. The self-sterile species of the genus Abutilon, which are, on the other hand, so much inclined to hybridization, afford a good example of this theory, which appears to be confirmed also by Lobelia, Pa.s.siflora, and Oncidium" (_American Naturalist_, vol. viii, pp. 223-4, 1874).

_Different groups of plants exhibit remarkable differences in the capability of their const.i.tuent species to hybridize._--In so far as these differences have reference only to first crosses, they have no bearing either for or against my theory. Only in so far as the differences extend to the production of fertile hybrids does any question arise for me. First of all, therefore, I must ascertain whether (or how far) there is any correlation between groups whose species manifest apt.i.tude to form first crosses, and groups where first crosses manifest apt.i.tude to produce fertile hybrids. Next, whatever the result of this inquiry should be, if I find that certain natural groups of plants exhibit comparatively well-marked tendencies to form fertile hybrids, the question will arise, Are these tendencies correlated with _paucity_ of species? If they are, the fact would make strongly in favour of physiological selection. For the fact would mean that in these natural groups, owing to "the nature of the organisms" included under them, less opportunity is given to physiological selection in its work of differentiating specific types than is given by other natural groups where the nature of the organism renders them more p.r.o.ne to mutual sterility. But in prosecuting this branch of verification, I must remember to allow for possibilities of differential degrees of geographical isolation in the different groups compared.

On this subject Focke writes me as follows:--"In a natural group (family, order, genus) showing considerable variability in the structure of the flower, we may expect to find [or do find] a greater number of mules than in a group whose species are only distinguished by differences in the shape of the leaves, or in growth, &c. I do not know, however, which in this connexion of things is the cause and which the effect. A useful ancestral structure of the flower may be conserved by an otherwise varying progeny, on condition that the progress of diversity be not disturbed by frequent intercrossings. [Therefore, if this condition be satisfied, the structure of the flower in different members of the group will continue constant: here the cause of _constancy_ in the flower (however much variability there may be in the leaves, &c.) is its original _inability_ to hybridize.] On the other hand, in species or groups ready to hybridize [or capable of hybridizing], the fixation of a new specific type will require some change in the structure of the flower, and a change considerable enough to alter the conditions of fertilization. [Here the reason of the _in_constancy of the flower in different members of the group is the original _apt.i.tude_ of their ancestral forms to hybridize.] Perhaps there is something in this suggestion, but certainly there are other efficient physiological relations, which are at present unknown. Your theory of physiological selection may serve to explain many difficult facts."

_The Importance of Prepotency._--A. Kerner shows by means of his own observations on sundry species of plants which hybridize in the wild state, that they do so very much more frequently if both, or even if only one of the parent forms be rare in the neighbourhood. This fact can only be explained by supposing that, even in species most p.r.o.ne to hybridizing under Nature, there is some degree of prepotency of pollen of the same species over that of the other species; so that where both species are common, it is correspondingly rare that the foreign pollen gets a chance. But if there were no prepotency, the two species would blend; and this Kerner supposes must actually take place wherever two previously separated species, thus physiologically circ.u.mstanced, happen to be brought together. (Kerner's paper is published in _Oester. Bot.

Zeitschrift_, XXI, 1871, where he alludes to sundry other papers of his own advocating similar views.)

The relation of these observations to Jordan's _especes affines_ is obvious. We have only to suppose that some such slight and constant difference characterizes the s.e.xual elements of these allied varieties as demonstrably characterizes their morphology, and we can understand how pollen-prepotency would keep the forms distinct--such forms, therefore, being so many records of such prepotency.

Both from Kerner's work, and still more from that of Jordan and Nageli, I conclude that (at all events in plants) prepotency is the way in which physiological selection chiefly acts. That is to say, _sudden_ and _extreme_ variations in the way of s.e.xual incompatibility are probably rare, as compared with some degree of prepotency. According as this degree is small or great so will be the amount of the corresponding separation. This view would show that in plants the principle of physiological selection is one of immensely widespread influence, causing (on the same areas) more or less permanent varieties much below specific rank. And when we remember on how delicate a balance of physiological conditions complete correspondency of pollen to ovules depends, we may be prepared to expect that the phenomenon of prepotency is not of uncommon occurrence.

_Self-fertilization and Variability._--It occurred to Count Berg Sagnitz that, if physiological selection is a true principle in nature, vegetable species in which self-fertilization obtains ought to be more rich in constant varieties than are species in which cross-fertilization rules. For, although even in the latter case physiological isolation may occasionally arise, it cannot be of such habitual or constant occurrence as it must be in the former case. Acting on this idea, Count Berg Sagnitz applied himself to ascertain whether there is any general correlation between the habit of self-fertilization and the fact of high variability; and he says that in all the cases which he has. .h.i.therto investigated, the correlation in question is unmistakable.

_Additional Hypothesis concerning Physiological Selection._--In reciprocal crosses _A_ _B_ is often more fertile than _B_ _A_. If hybrid _AB_ is more fertile with _A_, and hybrid _BA_ with _B_, than vice versa, there would be given a good a.n.a.logy on which to found the following hypothesis.

Let _A_ and _B_ be two intergenerating groups in which segregate fecundity is first beginning. Of the hybrids, _AB_ will be more fertile with _A_, and _BA_ with _B_, than vice versa. The interbreeding of _AB_ with _A_ will eventually modify s.e.xual characters of _A_ by a.s.similating it to those of _AB_, while the interbreeding of _BA_ with _B_ will similarly modify s.e.xual characters of _B_ by a.s.similating it to those of _BA_. Consequently, _A_ will become more and more infertile with _B_, while _B_ becomes more and more infertile with _A_. Fewer and fewer hybrids will thus be produced till mutual sterility is complete.

To sustain this hypothesis it would be needful to prove experimentally, (1) that hybrid forms _AB_ are more fertile with _A_ than with _B_, while hybrid forms _BA_ are more fertile with _B_ than with _A_ [or, it may be possible that the opposite relations would be found to obtain, viz. that _AB_ would be more fertile with _B_, and _BA_ with _A_]; (2) that, if so, effect of intercrossing _AB_ with _A_ is to make progeny more fertile with _A_ than with _B_, while effect of intercrossing _BA_ with _B_ is to make progeny more fertile with _B_ than with _A_.

Such experiments had best be tried with species where there is already known to be a difference of fertility between reciprocal crosses (e.g.

Matthiola annua and M. glabra, see _Origin of Species_, p. 244).