CH. x] A. NUMERICAL 97 



shown by the universality of the hollow curve — we shall expect 

 to find that there will be a gap between the numbers of species in 

 the two largest genera of the family. This gap will obviously be 

 due to the fact that the two first genera of a family will usually 

 have been formed the one a good while before the other. During 

 that interval of time the first one will probably be able upon the 

 average to throw one or more species before the second genus 

 appears. It will thus get the start of the latter, and will con- 

 tinually gain upon it. It follows from this that the larger the 

 original genus now is, the greater, on the average, should be the 

 gap between it and the second genus. As we have just seen, 

 the third genus of a family will, upon the average, be separated 

 from the second by less time than is the second from the first, and 

 the time-separations will become less and less as we go downward 

 to the smaller genera. We shall, therefore, expect the gaps in 

 numbers of species also to lessen. 



Turning to the facts, this is exactly what we do find. Out of 

 all the families given in my Dictionary — about 240 with two or 

 more genera — only eleven, mostly very small, show no difference 

 between first and second genus. The two larger families are 

 Bignoniaceae and Sapotaceae, where the four top genera are all 

 given as having 100 species. But on the average of all the families, 

 the first difference is ninetj-nine, while the second gap is only 

 thirty-two, the third eleven, the fourth eleven, the fifth six. 



The result of this test case, therefore, is in favour of differentia- 

 tion. In fact, one cannot, with progressive arithmetical links 

 like these between the genera, consider natural selection or gradual 

 adaptation as having had much to do with the evolution. 



TEST-CASE IV. PROPORTIONS OF SMALL 

 GENERA IN FAMILIES 



If natural selection of gradual adaptation be the moving power 

 of evolution, and the small genera and local species be the relics 

 of what must be regarded as the failures, then one would certainly 

 expect that these ought to be more numerous in proportion in the 

 small and local families, which are also regarded as relics. If, on 

 the other hand, differentiation has been the mechanism, one will 

 expect that the larger a family grows, the more rapid will be its 

 proportionate production of small genera, for each genus, whether 

 small or large, may be able to throw new ones, so that the small 

 genera will be increasing (in number, not in size) more rapidly 

 than the large, the family following the hollow curve. . pV A m 



WED .J\5^ ll J^ / 



