272 



SCIENCE. 



[Vol. YL, No. 138. 



was touched by a horizontal tangent, lay in a straight 

 line inclined to the vertical in the ratio off; those 

 where they were touched by a vertical tangent, lay 

 in a straight line inclined to the horizontal in the 

 ratio of -J. These ratios confirm the values of aver- 

 age regression already obtained by a different method, 

 of f from mid-parent to offspring, and of ^ from 

 offspring to mid-parent. These and other relations 

 were evidently a subject for mathematical analysis 

 and verification. They were all clearly dependent 

 on three elementary data, supposing the law of fre- 

 quency of error to be applicable throughout; these 

 data being 1°, the measure of racial variability; 2°, 

 that of co-family variability (counting the offspring . 

 of like mid-parentages as members of the same co- 

 family); and, 3°, the average ratio of regression. I 

 noted these values, and phrased the problem in ab- 

 stract terms such as a competent mathematician 

 could deal with, disentangled from all reference to 

 heredity, and in that shape submitted it to Mr. J. 

 Hamilton Dickson, of St. Peter's college, Cambridge. 

 I asked him kindly to investigate for me the surface 

 of frequency of error that would result from these 

 three data, and the various particulars of its sections, 

 one of which would form the ellipses to which I have 

 alluded. 



I may be permitted to say that I never felt such a 

 glow of loyalty and respect towards the sovereignty 

 and magnificent sway of mathematical analysis as 

 when his answer reached me, confirming, by purely 

 mathematical reasoning, my various and laborious 

 statistical conclusions with far more minuteness than 

 I had dared to hope; for the original data ran some- 

 what roughly, and I had to smooth them with tender 

 caution. His calculation corrected my observed 



value of mid-parental regression from - to : the 



^ ^ 3 17.6 



relation between the major and minor axis of the 

 ellipses was changed 3 per cent, their inclination 

 was changed less than 2°. It is obvious, then, that 

 the law of error holds throughout the investigation 

 with sufficient precision to be of real service, and 

 that the various results of my statistics are not cas- 

 ual determinations, but strictly interdependent. 



In the lecture at the Royal institution to which I 

 have referred, I pointed out the remarkable way in 

 which one generation was succeeded by another that 

 proved to be its statistical counterpart. I there had 

 to discuss the various agencies of the survival of the 

 fittest, of relative fertility, and so forth; but the 

 selection of human stature as the subject of investi- 

 gation now enables me to get rid of all these compli- 

 cations, and to discuss this very curious question 

 under its simplest form. How is it, I ask, that in 

 each successive generation, there proves to be the 

 same number of men per thousand who range 

 between any limits of stature we please to specify, 

 although the tall men are rarely descended from 

 equally tall parents, or the short men from equally 

 short ? How is the balance from other sources so 

 nicely made up ? The answer is, that the process 

 comprises two opposite sets of actions, one concen- 

 trative and the other dispersive, and of such a char- 



acter that they necessarily neutralize one another, 

 and fall into a state of stable equilibrium. By the 

 first set, a system of scattered elements is replaced 

 by another system which is less scattered; by the 

 second set, each of these new elements becomes a 

 centre, whence a third system of elements is dis- 

 persed. The details are as follows : In the first of 

 these two stages, the units of the population group 

 themselves, as it were by chance, into married cou- 

 ples, whence the mid-parentages are derived ; and 

 then by a regression of the values of the mid-parent- 

 ages the true generants are derived. In the second 

 stage, each generant is a centre whence the offspring 

 diverge. The stability of the balance between the 

 opposed tendencies is due to the regression being pro- 

 portionate to the deviation, — it acts like a spring 

 against a weight. 



A simple equation connects the three data of race 

 variability, of the ratio of regression, and of co- 

 family variability; whence, if any two are given, the 

 third may be found. My observations give separate 

 measures of all three, and their values fit well into 

 the equation, which is of the simple form, — 



+ /-^ 



where u = |, p = 1.7,/ = 1.5. 



It will therefore be understood that a complete 

 table of mid -parental and filial heights may be cal- 

 culated from two simple numbers. 



It will be gathered from what has been said, that 

 a mid-parental deviate of one unit implies a mid- 

 grandparental deviate of i, a mid-ancestral unit in 

 the next generation of ^, and so on. I reckon from 

 these and other data, by methods that I cannot stop 

 to explain, that the heritage derived on an average 

 from the mid-parental deviate, independently of what 

 it may imply, or of what may be known concerning 

 the previous ancestry, is only y- Consequently, that 

 similarly derived from a single parent is only \, and 

 that from a single grandparent is only rg-. 



The most elementary data upon which a complete 

 table of mid-parental and filial heights admits of 

 being constructed, are, 1°, the ratio between the mid- 

 parental and the rest of the ancestral influences; and, 

 2°, the measure of the co-family variability. 



I cannot now pursue the numerous branches that 

 spring from the data I have given, as from a root. I 

 will not speak of the continued domination of one 

 type over others, or of the persistency of unimpor- 

 tant characteristics, or of the inheritance of disease, 

 which is complicated in many cases by the requisite 

 concurrence of two separate heritages, the one of a 

 susceptible constitution, the other of the germs of 

 the disease. Still less can I enter upon the subject 

 of fraternal characteristics, which I have also worked 

 out. It will suffice for the present to have shown 

 some of the more important conditions associated 

 with the idea of race, and how the vague word ' type ' 

 may be defined by peculiarities in hereditary trans- 

 mission ; at all events, when that word is applied to 

 any single quality, such as stature. To include those 



