106 



THE AMERICAN NATURALIST [Vol. LIV 



quencies are out of exact proportion to the distances on a 

 straight linear map, that this is because the line is curved, 

 and the factors thus nearer together. If still more evi- 

 dence against the curved line idea be desired, it may be 

 added that when the curve is constructed so as to be in 

 good agreement (statistically) with the relations found 

 between the smaller frequencies, it is then not sufficiently 

 arched to permit the representation of the larger fre- 

 quencies by proportionate distances (see section 4). A 

 detailed compilation which I have made of all the data 

 has shown that experimental error will not well account 

 for the differences thus obtained between the two sets of 

 results. The curved line being abandoned, it becomes, 

 therefore, necessary to revert to ' ' double crossing over," 

 in explanation of the deviation of the large frequencies 

 from the straight map values. 2 



3. If we examine further into Castle 's argument, how- 

 ever, we find that he objects, not only because the larger 

 separation frequencies are not proportional to the dis- 

 tances in linear maps, but also because he believes that 

 the smaller frequencies are not proportional; in fact, 

 according to his solid models, none of the kinds of fre- 

 quencies, small or large, could even be a function of the 

 distances in a linear map. In his solid, or rather, three- 

 dimensional, models, which purport to have the factors 

 so spaced that all distances between them are exactly 

 proportional to the corresponding linkages, the factors 

 are scattered about at all angles to each other, in such a 



p The fact that the geometrical line which represents the linkages of the 

 factors should be taken as straight does not imply that the supposed physical 



kind of physical line at all, then, if their linkages are determined, in some 

 way, by their distances as measured along this line, these linkages should be 

 representable on the basis of a straight geometrical map, inasmuch as all 

 distan.-c- taken along a curved line must have the same interrelationships as 



