2O6 KELLOGG AND BELL 



able by the existence of the large personal equation and prob- 

 ability of error elements in the methods necessarily employed 

 in the collection of the foundation data. However this may be 

 and the authors are not mathematicians enough fairly, per- 

 haps, to venture such an opinion it is certain that these 

 extremer biometrical papers not only fail to attract or interest 

 biologists, but repel them. And if mathematics is to revolu- 

 tionize biological inquiry or even to help it effectively it must 

 keep within hailing distance of the working biologists. 



In this paper mathematical expressions are used wherever 

 they are necessary or seem helpfully usable, but they are not 

 used where they seem unnecessary or their appearance of accu- 

 racy and quantitative exactness seems delusive because of the 

 character of the data on which they have to be based. 



Further, the writers believe that certain variations are more 

 advantageously and valuably expressed qualitatively than quan- 

 titatively. For these qualitative variations the mathematical 

 treatment may well stop with the erection of the frequency poly- 

 gons, which are always of unquestioned value both as graphic 

 and as quantitative expressions of the range and frequency 

 found in any statistical study of variation. Such variations as 

 difference in color, in pattern, in character of venation, etc., 

 are considered by the authors as qualitative variations (substan- 

 tive of Bateson) as contrasted with dimensions, and the number 

 of spines, spurs, segments, etc. (Bateson's meristic variations), 

 which lend themselves especially to thorough-going quantitative 

 treatment. 



Insects Specially Advantageous Subjects for Variation 

 Studies. That the insects offer special opportunities for the 

 study of variation is readily apparent when a few facts concern- 

 ing them are recalled. First to be noted is the overwhelming 

 dominance in number of species of this class of animals as com- 

 pared with the species total in other classes ; quite two-thirds of 

 all the known species of living animals are insects. These 

 animals, too, are unusually prolific ; multiplication proceeds by a 

 high ratio, and the life cycle is usually short. This results in 

 many species being represented by uncountable hosts of indi- 

 viduals ; recall the Rocky Mountain locust, the chinch bug, 



