K. Pearson 
329 
source of change. But what permits of a change in the type ? Why, the existence 
of variation in the biometrical sense. If Mr Bateson terms the change in type 
variation, what name does he give to the distribution of deviations which alone 
render this change of type possible ? It is perhaps needless to add that, if discon- 
tinuous variation be summed up in the problem of whether positions of organic 
stability have ever existed in forms intermediate between recognised dimorphic 
types, Mr Bateson has not discussed discontinuous variation at all in his work, for 
we are certainly not given any data for dimorphic populations at different stages 
of their evolution, nor even statistics for several local races of one species of 
recognised dimorphic character. To those who have studied my memoir on 
Homotyposis there will be no need to say that discontinuity as described in either 
Mr Bateson's first or his third definition has no bearing whatever on that subject. 
If Mr Bateson, however, relies solely on his second definition, namely, that varia- 
tion is discontinuous when there is heterogeneity in the statistical distribution 
of frequency, then we may reasonably expect from him a study of frequency dis- 
tributions. Will he tell us what he understands by homogeneous and hetero 
geneous distributions ? Writing of a discontinuously varying population in 1897*, 
Mr Bateson says : 
" When such a population is seriated in respect of the varying character for 
statistical study in the manner with which naturalists have been familiarised by 
the writings of Galton and others the curve of variation has not one peak as in a 
monomorphic species, but has at least two peaks." 
Of course, from the statistical point of view this is an impossible definition of 
heterogeneity. Not only may two or many peaks occur in perfectly homogeneous 
material, but no peaks whatever in certainly heterogeneous material. It all 
depends on whether the peaks are significant or not, and on the distance between 
the modes of the mixed material. Indeed, if Mr Bateson's second definition be his 
final one, it can only be applied by a mathematical biologist, for the discrimination 
of modes is a most complex problem involving the theory of errors of random 
samplingf. Further the resolution of heterogeneous frequency distributions in 
biology will depend: (i) on an intimate and extensive knowledge of the distribution 
of frequency for organic characters for many types of life ; (ii) a selection on 
philosophical or on empirical grounds of theoretical distributions to represent these ; 
(iii) a test in any individual case of whether these theoretical distributions repre- 
sent the observed facts within the errors of random sampling, and (iv) supposing 
they do not, their resolution into component distributions. 
To sum up, Mr Bateson has given three distinct definitions of discontinuity : 
(a) A variation is treated as a deviation between parent and offspring, and 
variation is discontinuous if this be finite. 
* Science Progress, N. S. Vol. i. No. 5, October. 
t It would be interesting to know what degree of heterogeneity Mr Bateson supposes to exist in his 
"low" male group of Java beetles (p. 39). They have at least three apparent modes. Are the "low" 
males in themselves trimorphic ? 
Bioraetrika i 34 
