530 



KNOWLEDGE & SCIENTIFIC NEWS. 



[September, 1906. 



the race is represented by the length of the base line 

 upon which the curve is erected. The frequency with 

 which any variation within that range occurs will be at 

 once seen to depend upon the shape of the curve, its 

 steepness, S;c. .Suppose IM to be the mean value of the 

 character as represented in our race; then the measure 

 of the character in any individual may be written M + D 

 where D is the deviation of the character from the 

 mean value for the race in that individual. The mean 

 value of the squares of all the individual deviations 

 represented in the curve is the square of a factor known 

 as the standard deviation, which gives a measure of the 

 variability of the character in the race under observa- 

 tion. Observation shows that, in general, related 

 individuals tend to resemble one another, in regard to 

 .some one character, more nearly than they resemble 

 all th(? other individuals of the same race. This re- 

 semblance between the characters of related individuals 

 is a ca.se of correlation. It is found, moreover, that if 

 one individual possesses the character in a certain 

 degree, there is a " most probable value " for the same 

 character in his relations. Inspection of the curve will 

 suggest to us that in general this " most probable 

 value " will lie between the value of the first chosen 

 individual and the mode. This is found to be the case, 

 and this phenomenon is known as regression. 



Regression and correlation may be represented 

 graphically by plotting the values of the first chosen 

 individuals with the mean values for some one group 

 of relations, say, for their brothers, or for their sons. 



If the corresponding values which are plotted to- 

 gether arc identical the curve will form a diagonal to 

 the square, correlation will be complete, and regression 

 will be ab.sent; but if, ^s is always the case, correlation 

 is only partial, the curve will form an angle with the 

 diagonal which represent.s complete correlation, the 

 angle being greater the less the correlation and the 

 greater the regression are. The slope of this line givers 

 a measure of the correlation known as the '" coefficent of 

 correlation," which is taken as unity in the hvpothetical 

 case where correlation is complete. In practice the 

 coefficient of correlaton is alwavs a positive fraction 

 less than unity. 



The numerical expression for correlation and re- 

 gression thus obtained may be applied to the .special 

 case in which the relation between the first-chosen 

 individual and the group of relatives is that of parent 

 and offspring. The character of the parent being 

 known, the coefficient of correlation and the standard 

 deviation for the race may be used to obtain a value for 

 the probable character of the offspring, that is, a value 

 to which the mean of the values for all the offspring 

 will approximate more and more nearly as the number 

 of offspring is larger and larger. 



The same method may be applied to the ca.se of 

 hiparental inheritance, when we require to find the 

 probable character of the offspring, taking into account 

 the known charact-ers of both parents. Taking as our 

 example the case of the stature of man, it will be noticed 

 nt once that the greater mean stature of the males, as 

 C()nipar:-d with that of the females, introduces a factor 

 for which allowance must be made. Men are, on the 

 average, taller than women, hence in calculating the 

 probable stature of sons from the known statures of 

 the father „and mother, a correction must be applied to 

 the observed stature of the mother. 



Galton showed how this might be done by replacing 

 the observed stature of the mother bv a value represenr- 

 ing the equivalent stature for the 'male. Prof. Karl 

 Pearson has shown that in general the probable charac- 



ter of the offspring may be calculated by the use of 

 what is known as a "mid-parent"; that is, a single 

 artificial parent whose characters are calculated so as 

 to combine those of the father and of the mother into 

 one. 



From the study of biparental inheritance by means 

 of the method indicated above we may pass directly to 

 the " Law of Ancestral Heredity," which, first formu- 

 lated by Francis Galton, has been somewhat modified 

 and extended by Karl Pearson. In this law, u.se is 

 made, not only of the correlation between offspring and 

 parent, but of that between offspring and grandparent, 

 and between offspring and more remote ancestors, in 

 order that the probable character of the offspring may 

 be calculated with the greatest possible approximation 

 to the truth. 



By means of the method already mentioned a " mid- 

 grandparent " is formed from the four grandparents; a 

 '■ mid-great-grandparent " from the eight great-grand- 

 parents, and so on for the more remote ancestors. 



Now it is obvious that as the generations are traced 

 backwards the ever-increasing number of ancestors 

 will, except in extreme cases of in-breeding, approxi- 

 mate more and more closely towards a fair sample of 

 the whole population at that period, so that the mid- 

 parent of, say, the loth generation, will show little or 

 no deviation from the racial type of that period. 



Francis (Jalton's " Law of Ancestral Heredity " was 

 the first attempt to give a numerical expression which 

 should indicate the proportion in which each generation 

 of the ancestry contributes to the characters of an 

 individual. 



The Law was enunciated thus : " Each parent con- 

 tributes, on an average, one quarter or (0.5)'*, each 

 grandparent one-sixteenth or (o.5)^ and so on, and 

 generally the occupier of each ancestral place in the 

 «th degree, whatever be the value of n contributes 

 (0.5)^" of the heritage." These numbers form a geo- 

 metrical series, of which the sum of an infinite number 

 of terms is unity. Professor Pearson h;LS shown that 

 some modification of this statement is necessary, and 

 has introduced a quite general mathematical expression 

 for the Law of .\ncestral Heredity, which avoids certain 

 assumptions made in Galton 's enunciation of the law. 

 For the particular case assumed by Galton, the original 

 series of factors would be replaced by the series 



03,0-15, 0075, 0-0375 • • • "6 X (*)" 



for unions whore the sexes are equipotent, blend their 

 characters and mate pangamously. 



Want of space compels us to turn now to the second 

 part of our subject. The previous pages have given 

 no more than the briefest outline of the methods 

 adopted in the statistical treatment of heredity. I 

 would suggest that any readers who may wish to make 

 themselves more familiar with the subject should refer 

 to Prof. Pearson's book, " The Grammar of Science," 

 in which the theory is simply explained, and to numer- 

 ous publications of the same author in the Transactions 

 of the Royal St/cicty. Further information may be 

 obtained in " Biometrika " (Camb. Univ. Press), a 

 journal in which the re,siilts obtained by the statistical 

 m-.'thod are published from time to time. 



The Mendelian Hypothesis. 



It was as long ago as 1865 that (uegor .Mendel, who 

 was then .\bbot of Briinn, communicated the results of 

 his experiments in plant hybridisation, which he had 

 pursued for eight years previously, to the Briinn Society 

 of Naturalists. 



