No. 631] 



FACTORS OF HEREDITY 



99 



show that the factors must be bound together in a linear 

 series; the precise nature of the function (involving coin- 

 cidence, etc.) will then determine for us precisely the 

 mode of incidence of the crossing over— i. e., granted the 

 linear series, it is then possible to calculate from the data 

 the exact frequency of single crossing over, double cross- 

 ing over of the various possible types, and multiple cross- 

 ing over. Double crossing over thus becomes, not a 1 4 sub- 

 sidiary hypothesis," but a phenomenon directly demon- 

 strated. 



It may, however, be noted in passing that, even if there 

 had been no experimental evidence at all in regard to the 

 nature of the linkage it could not have been conceded 

 that Castle's alternative postulate— that no double cross- 

 ing over can ever occur at all— would have been any more 

 plausible a priori than that of the Drosophila workers 

 which admits the existence of double crossing over. For, 

 once the occurrence of single breaks in a chromosome is 

 admitted— a point agreed upon by both sides— it is just 

 as arbitrary to deny the possibility of double breaks as 

 to assert their existence. Although Castle nowhere does 

 explicitly admit that he has adopted this alternative 

 'Subsidiary hypothesis"— the denial of the possibility 

 of double crossing over— yet an inspection of the theory 

 of linkage which he himself has proposed shows that in 

 this it has been tacitly assumed throughout, being neces- 

 sary for the purposes of the solid models. Were double 

 crossing over once admitted to occur, it could no longer 

 be claimed that the distances between the factors in the 

 three-dimensional models are exactly proportional to 

 their per eents of separation— a condition which it is the 

 sole aim of the existence of the models to fulfill. 



We may now return to examine more carefully the 

 main argument upon which linear arrangement and its 

 corollaries (double crossing over, coincidence, etc.) is 

 based. The fact previously stated that the linkage rela- 

 tions between the genes are such that they are .all cal- 

 culable from the positions of points in a linear series 



