1893.] On the Mathematical Theory of Evolution. 329 



observed discrepancy between two corresponding values are nob of a 

 very serious kind. So that in any discussion of the variation of the 

 twenty-three pairs of organs discussed in the present paper, or of the 

 pairs of shrimp organs discussed in my previous communication, it 

 may be assumed as at least an empirical working rule that Galton's 

 function has the same value in all local races. The question whether 

 this empirical rule is rigidly true will have to be determined by fuller 

 investigation, based on larger samples : but the value of a merely em- 

 pirical expression for the relation between abnormality of one organ and 

 that of another is very great. It cannot be too strongly urged that the 

 problem of animal evolution is essentially a statistical problem : that 

 before we can properly estimate the changes at present going on in a 

 race or species we must know accurately (a) the percentage of 

 animals which exhibit a given amount of abnormality with regard to 

 a particular character ; (&) the degree of abnormality of other organs 

 which accompanies a given abnormality of one ; (c) the difference 

 between the death rate per cent, in animals of different degrees of 

 abnormality with respect to any organ ; (d) the abnormality of off- 

 spring in terras of the abnormality of parents, and vice versa. These 

 are all questions of arithmetic; and when we know the numerical 

 answers to these questions for a number of species we shall know 

 the direction and the rate of change in these species at the present 

 day a knowledge which is the only legitimate basis for speculations 

 as to their past history and f nture fate. 



III. "Contributions to the Mathematical Theory of Evolution." 

 By KARL PEARSON, M.A., Professor of Applied Mathematics, 

 University College. Communicated by Professor HENRIOI, 

 F.R.S. Received October 18, 1893. 



(Abstract.) 



1. If a series of measurements, physical, biological, anthropo- 

 logical, or economical, not of the same object, but of a group 

 of objects of the same type or family, be made, and a curve be 

 constructed by plotting up the number of times the measurements 

 fall within a given small unit of range to the range, this curve may 

 be termed a frequency curve. As a rule this frequency curve takes 

 the well known form of the curve of errors, and such a curve may be 

 termed a normal frequency curve. The latter curve is symmetrical 

 about its maximum ordinate. Occasionally, however, frequency 

 curves do not take the normal form, and are then generally, but not 

 necessarily, asymmetrical. Such abnormal curves arise particularly 

 in biological measurements ; they have been found by Professor 



