Volume Iii Part 10 (1/2)
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)--