The Distribution of Senescence 



Three types of growth-pattern are theoretically possible in 

 vertebrates — growth to a maximum size, ceasing when this is 

 reached: growth toward a limiting size which is approached 

 asymptotically: and growth without a limiting size. In the third 

 of these cases, the specific growth-acceleration can be negative 

 — i.e. the growth rate continually declines — but it could 

 theoretically do so in such a way that, given a sufficiently 

 long life, any final size could be reached. These last two modes 

 of growth correspond to convergent and divergent series. Thus 

 in the series 



(1) 1 +£+i+J...lim2, 

 and the series 



(2) 1 +*+*+*+*..., 



the increment at each term decreases (the specific growth rate 

 falls), but whereas in (1) the series tends to a limiting size 

 (specific size), in (2) it does not, and can be indefinitely con- 

 tinued so that any sum is ultimately attained. The terms 'in- 

 determinate growth' and 'indeterminate size' have been differ- 

 ently used by different writers. D'Arcy Thompson wrote, 'It is 

 the rule in fishes and other cold-blooded vertebrates that 

 growth is asymptotic and size indeterminate' (1942). If the 

 growth of an animal is in fact asymptotic, its size is limited by 

 the sum of the asymptotic series. 'Indeterminate' growth with- 

 out limit, but with a decline in the specific growth-rate, strictly 

 follows the pattern of the divergent series. For this reason it 

 would be desirable, but it is not empirically possible, given real 

 biological material, to distinguish between 'asymptotic' and 

 'indeterminate' growth. In both cases the rate of growth 

 declines with advancing age; but in the second case the potential 

 size is unlimited. 



Distinctions of this kind, however, are based upon the fitting 

 of equations to points derived by averaging observations upon 

 populations of animals, and in spite of the real value of such 

 biometric applications, in the study of growth-curves they have 

 tended to lose contact with the real behaviour of observable 

 material. It is possible in practice to distinguish only between 

 species, or particular populations of a given species, which 



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