The Inheritance of Complex Growth Forms, such as Stature, on Mendel’s Theory*

That is, stature tends to be graded according to the ordinary point binomial, it being granted probable that each group as above placed has essentially the same stature. With regard to the second, there are several things to note. For convenience we may write the point binomial as (1 + n + 1) p. If we expand by the multinominal theorem, arrange the terms, and then determine the moments, we have: the second moment the third moment as the multinominal is symmetrical, the fourth moment The relationship of these moments is such that for all values of or the resulting curve partakes of the qualities of Type IV rather than of the normal curve. But = 4 is not the dividing point. When the point binomial obviously is the square of (1 + 1) and therefore is of the type which gives rise to the normal curve. In this case We may therefore take it that this curve represents the dividing line, and that when is greater than 2, a curve with moment relationships more nearly those of the symmetrical form of Type IV should describe the result and not the normal curve. In the same way if <2 we have a curve with the moment relationships of Type II. In the case considered = 3·3, or the point binomial is represented by (3 + 10 + 3)p. To illustrate the manner in which this curve approximates to the normal we may take p = 4, i.e. there are eight pairs of allomorphs originally present in each parent. The distribution of this and the corresponding normal distribution are then:- Point binomial – 81, 1080, 5724, 15 240, 21 286, 15 240, 5724, 1080, 81 Normal curve – 115.2, 1136, 5764, 15 226, 21 042, 15 226, 5764, 1136, 115.2 It is to be noticed that the point binomial gives a ‘leptokurtic’ distribution, that is, one in which the radius of curvature at the apex is less than that of the normal curve; in other words, one corresponding to that of Type IV, but not identical with it. On the numbers given by the total, namely, 66 536, the difference between the two is most marked; but on one-tenth of the numbers, namely, 6654, a number much in excess of that obtained in ordinary observations, the normal curve is an exceedingly good fit, giving by the test χ2 = 3·1 or P = ·91. It is interesting to observe that with only eight pairs of allomorphs a normal distribution of stature within the limits of error of observation is at once derived. With free mating and equal fertility the proportion of the mixed population is (3 + 10 + 3)p. The meaning of the value of in the point binomial (1 + + 1)p is therefore as follows: If is equal to 3·3 all matings are equally fertile, if is >3·3 then the matings of hybrid with hybrid are most fertile and if is <3·3, the matings of the purer races. The manner in which asymmetry arises may be seen by going back to the simple case. Let If the hypotheses above discussed are granted, two points emerge demanding consideration: How far does this grouping accord with the observed statistics? How far does the grouping permit of the correlation coefficients found by observation? Every large series of statistics of a population give groups to which Type IV corresponds better than the normal curve. Those of Pearson,† Powys,‡ etc., may be taken as types. In the case of the diagrams given by the former, it is very noticeable that at the apex the statistics give a value in excess of that shown by the normal curve fitted to the statistics, and at the limits there is a defect on the part of the statistics resembling that just found theoretically. With the curves of the Scottish insane given by Tocher§ the same is also found. With regard to the values of the correlation coefficients we are on firm ground. As we have seen, the hypothesis of blended inheritance leads to correlation coefficients as follows: Parent and offspring ·5 “ “ grand-offspring ·25 “ “ great-grand-offspring ·125 “ “ great-great-grand-offspring ·0625 As before observed, these values arise on a Mendelian basis with dominance if a coefficient of assortive mating of be assumed. In the case of stature in man this is even exceeded. Professor Pearson finds that for this quality ⁠, so that although for purposes of calculation it is assumed that ⁠, this value is in defect. With the grouping assumed in this paper, the coefficient of correlation between parent and child without assortive mating is found to be ⁠, exactly as in the case considered by Professor Pearson.†† In this case, however, the increase produced by the degree of assortive mating considered is not so large as in that where dominance is assumed to come exclusively from one side. The coefficient of correlation between parent and offspring is raised from to ⁠, somewhat in defect of the value found from observation. It is however, in the highest degree improbable that pure dominance regulates inheritance. Blending, as already remarked, must also occur, and as a like coefficient of assortive mating must be held to apply in this case, the increase in the value of the coefficient to the neighbourhood of is almost a necessity, since for pure blended inheritance the correlation of parent and offspring is in the neighbourhood of when the coefficient of assortive mating is given by ⁠. To settle this question, inquiries into the constitution of races will require to be made, but I think that I have shown that there is nothing necessarily antagonistic between the evidence advanced by the biometricians and the Mendelian theory. If the inheritance of stature depends upon a Mendelian mechanism, then the distribution of the population as regards height will be that which is actually found, namely, a distribution closely represented by the normal curve. There is nothing in the values of the coefficients of inheritance found by Sir Francis Galton and Professor Pearson which cannot be explained on the basis of Mendelian inheritance. This curve then tends to approach the normal curve of error in its moment relationships, but it is to be remembered that must be small compared with ⁠, and also that if dominance exist perfect normality is impossible. In case of misunderstanding, it may be well to state that I have used the word blend in the sense that the quality resulting from the combination of two different elements lies between that of the separate elements, and not in the sense that either of the elements is modified by the combination, as is sometimes done. (Issued separately January 13, 1911).