1899] VARIATION- STATISTICS IN ZOOLOGY 329 



but also in those of species belonging to different genera, even to 

 different families. This fact does not seem to me to have been 

 sufficiently regarded hitherto ; the explanation of it is, I suppose, the 

 constancy of the physiological capacity of a given organ for reacting to 

 the individual causes of variation (to be considered afterwards) with 

 respect to a given character. Some authors, however, seem to assume 

 a more or less constant relation between the height of the average 

 and that of the index of variability of a character. 



Average value and index of variability of a numerical character 

 are the first data necessary to the description of its variation. Both 

 ought always to be determined ; but they only give an approximate 

 idea of the variation of the character. Its complete description 

 requires the determination of the curve which rules the slope of its 

 polygon of variation, or in other words, on which the corner points of 

 the polygon are situated. To find this curve, we must find the mathe- 

 matical relations between the variants or their deviations from the 

 average value on one hand and their frequencies on the other. 



There is a striking likeness, even at the first glance, between the 

 polygons of variation and binomial polygons. We get the latter by 

 graphically representing the series of summation which arise by 

 developing binomial terms, as {p + qf. As a matter of fact, both are 

 closely related. In numerical characters we find variants deviating 

 from the average value in positive and in negative directions. Since 

 all processes in nature depend upon causes, we are obliged to assume 

 causes of variation with either positive or negative effects, of which 

 causes neither the number nor the intensity of effect is known. These 

 causes must be different from those which determine the average 

 character of the form-unit, and at the same time must be weaker in 

 effect than the latter. Now each individual has its own fate, which 

 word includes the total sum of enormously numerous and minute 

 factors acting on it in the most diverse combinations, which naturally 

 cannot be identical either for all individuals of the form-unit or in 

 every moment of the existence of the single individual. Thus we get 

 the conception of an enormous number of elementary causes of variation, 

 which may be regarded as equally effective so far as their small power 

 goes. Of these one set can effect positive deviations, the other negative 

 deviations from the average values of the different characters, all being 

 able to act on each individual of the form-unit, though as a matter of 

 fact they do not all act on it. The active set of causes is in each case 

 any random combination of positively and negatively acting causes, 

 and each of these combinations has a higher or lower degree of 

 probability, according to which its effect is more or less frequent in the 

 total population of individuals. The sum total of positively acting 

 causes may be equal in number to that of negatively acting ones, or be 

 different from it. 



Starting from such assumptions, mathematicians have investigated 



