112 THE AMERICAN NATURALIST [Vol. LIV 



tion— whereby each separation frequency is supposed to 

 be shown by an exactly proportionate distance on the 

 figure, no matter how many dimensions may be required 

 for this purpose. It will now be shown, however, that 

 such a system of representation is impossible, quite aside 

 from the fact that the models shown in Castle's papers 

 are based upon data which can not legitimately be com- 

 bined together. That is, no matter whether the data used 

 are all derived from one experiment, or whether the re- 

 sults of different experiments are combined according to 

 Castle's method, they could not be represented either in 

 a three-dimensional or in any other geometrical figure 

 in such a way that all the distances would be proportional 

 to the separation frequencies. 



This may be illustrated by the data reported in Table 

 II. It has been seen that the per cent, of separations be- 

 tween y and cl is exactly equal to the per cent, of separa- 

 tions between y and bi plus that between bi and cl. If 

 then we represent these frequencies by actual distances, 

 we must make the distance between points y and cl ex- 

 actly equal to the distance between y and bi plus that 

 between bi and cl. The only possible way to do this, on 

 any kind of geometry— one-dimensional, three-dimen- 

 sional or ^-dimensional— it to put these three points in 

 one straight line. In a similar manner we must place bi 

 cl v in a straight line, and also v s B. Cl v and s are in 

 almost a straight line, but there would have to be a slight 

 bend at v, owing to the fact that cl s is very slightly 

 shorter than cl v plus v s (on account of just one double 

 crossover having occurred between them) ; this is corre- 

 lated with the fact that cl s is a longer distance than the 

 others considered. The figure so constructed, on the 

 basis of Castle's own methods, is shown in Fig. 3; it is 

 quite evident that this is the only figure which will repre- 

 sent directly (proportionately) the frequencies above 

 considered. If, however, we now measure the distance 

 on this figure between the extreme points, y and B, we 

 find that it turns out to be 49.3, or very nearly the sum of 



