CHAPTER SIXTEEN 



Sorm Quafititative Aspects 

 of Evolutmi 



The realization of the fact that selection need not be an all-or-none 

 force, but can operate through gradual changes in the frequency of cer- 

 tain characters or character combinations in a species, required that an 

 attempt be made to analyze statistically the changes in the genetic com- 

 position of a species which might be expected under various conditions 

 of mutation, selection, and population structure. This problem has been 

 attacked mathematically by Fisher, Haldane, Wright, and others. The 

 results of their calculations are one of the major achievements of the neo- 

 Darwinian school. In the first edition of his brilliant book, Dobzhansky * 

 prefaces the discussion of the statistical analysis of variation in popula- 

 tions with the statement that ". . . Only in recent years, a number of in- 

 vestigators . . . have undertaken a mathematical analysis of these processes, 

 deducing their regularities from the known properties of the Mendelian 

 mechanism of inheritance. The experimental work that should test these 

 mathematical deductions is still in the future, and the data that are neces- 

 sary for the determination of even the most important constants in this 

 field are wholly lacking." In the intervening years, many investigators in 

 the new science of population genetics have ameliorated this picture, yet 

 it is still true that the theoretical mathematics of evolution is more highly 

 developed than is its experimental application to evolution in nature. It 

 is still possible that the mathematical analysis of evolutionary phenomena, 

 as it is now generally accepted, violates the dictum of Johannsen that 

 "Biology must be handled with mathematics, but not as mathematics." 

 Only some of the simplest and most important of the mathematical work 

 will be presented below. 



EVOLUTIONARY MATHEMATICS 



The Hardy-Weinberg Law. Evolutionary mathematics begins with the 

 Hardy-Weinberg Law, which states that, if alternative forms of a gene 



* Dobzhansky, Th., "Genetics and the Origin of Species," 1st Ed., Columbia Univer- 

 sity Press, 1937. 



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