CH. x] A. NUMERICAL 99 



TEST-CASE V. THE HOLLOW CURVE 



Many years ago it was shown that this curve, which is described 

 in Age and Area, p. 195, and in Chap, iv above, is a universal 

 feature of distribution in plants and in animals, both in regard to 

 the areas occupied, and to the sizes of the genera in families by 

 number of species contained. When plotted logarithmically, in 

 the latter case, they give close approximations to straight lines, 

 showing that they have the same mathematical form, and must 

 be due to the operation of the same law. The production of such 

 curves seems to the writer to place an almost insuperable ob- 

 stacle in the path of those who would explain evolution and 

 distribution in terms of gradual adaptation by means of natural 

 selection. Yule has shown (75) that the curve would result from 

 the continual doubling of the species and genera concerned, 

 when one supposes the parent to survive as well as the offspring, 

 as is the case according to the theory of differentiation. The curve 

 then becomes a normal and necessary feature of the evolution 

 that is going on, whereas under the theory of natural selection it 

 is totally inexplicable. Opponents have tried to belittle it by 

 showing that one can get similar curves from the names in the 

 telephone book, and such like conglomerations of inanimate 

 things. I have lately shown that the distribution of family 

 surnames of farmers in Canton Vaud (69) is just like the distribu- 

 tion of species, and therefore must follow the same laws, as it 

 gives the same curve. Natural selection could not determine it, 

 therefore it cannot be the determinant in the general distribution 

 of plants. 



Nothing but a uniform pressure would ensure that results 

 could be expressed in hollow curves. Family by family, and 

 genus by genus, whether in numbers or in areas, all alike obey 

 the same law. Natural selection could not produce results like 

 this, and the only cause yet suggested is age, which represents 

 the resultant of all the forces acting. If they produce an average 

 result of <r in a long time 1, they will produce 2x in time 2. Age 

 thus forms a measure of distribution, but one cannot compare 

 unrelated forms, and must always work in tens of allied species, 

 to average out the differences that there may be between them. 



It is clear that this test gives an unqualified verdict in favour 

 of differentiation. 



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