MATHEMATICAL CONSIDERATIONS 81 



would be seen to gather together again in a comparatively 

 few individuals if the pedigree of the species could be 

 traced in its entirety. 



Such a reductio ad absurdum is not altogether value- 

 less. It shows how essential it is for one to recognize the 

 unavoidable limitations, the desirability of definite analy- 

 sis, the necessity of precise methods of attack, in any 

 consideration of the proposition he may undertake. 



There are three distinct phases of the inbreeding prob- 

 lem, as Pearl has pointed out: 



1. The system of mating with regard to the relation 

 of the actual number of ancestors making up the pedigree 

 of an individual to the total possible number. 



2. The constitution of each individual with respect to 

 Mendelian unit factors which results from the continued 

 operation of a given system of mating which is inbreeding. 



3. The physiological effect produced upon the indi- 

 vidual by the constitution derived from this system 

 of mating. 



The first two phases of the problem are capable of 

 abstract mathematical treatment. The third can be 

 solved only by experimental investigation. 



Precise methods of measuring and comparing systems 

 of mating have been devised by Pearl by the use of a 

 Coefficient of Inbreeding and a Coefficient of Relation- 

 ship. The first is a measure of the actual number of 



a Pearl has made a somewhat more precise analysis of the Inbreeding 

 and Relationship Coefficients in later papers, 175, 176, and has suggested 

 a Partial Inbreeding Index, in a percentage which one-half of the Relation- 

 ship Coefficient is of the Inbreeding Coefficient. This constant is a measure 

 of the amount of inbreeding due to relationship between the sire and dam. 

 Further, he has described a single numerical measure of inbreeding for 

 bisexual organisms, in the ratio of the area of the inbreeding curve in any 

 pedigree to the area of the maximum) (brother X sister) curve. 



For our purposes, it is unnecessary to consider these extensions of 

 Pearl's studies in detail, though technically they are very valuable. 



