8 PROBLEMS OF RELATIVE GROWTH 



two rates of interest. (In our biological parallels, we know 

 nothing of the actual rates of growth, for since the organ and 

 the body have both existed for the same length of time when 

 we measure them, the time-factor cancels out, in point of fact.) 1 

 The actual rates, unlike those for the two sums of money, will 

 obviously be altering continuously; they will be high in youth, 

 low in age ; increased by high temperature, decreased by low ; 

 and so forth. What concerns us is that if our formula holds, 

 the ratio of the relative rates of growth remain constant. 



In such cases, therefore, we have a constant differential 

 growth-ratio , denoted by the value of k. If we prefer to con- 

 centrate upon the growth of the organ relative to the growth 

 of the body considered as a standard, then we may speak of 

 k as denoting the growth-coefficient of the organ. An organ 

 which is thus growing at a different rate from the body as a 

 whole may be called heterogonic, to use the convenient term 

 coined by Pezard (1918). If it is growing at the same rate 

 as the body it must be styled isogonic ; as will be apparent, 

 isogony is merely a special case of heterogony, as the circle 

 is a special case of the ellipse. 



It is clear that comparatively small variations in the value 

 of k will have large results provided that growth continues 

 over a considerable range of size. An attempt to show this 

 graphically has been made in Fig. 1. 



The best worked-out example of this law so far concerns 

 the large chela of male fiddler-crabs, Uca pugnax (Huxley, 

 1927A). This obeys the law of constant growth-ratio from crabs 

 of only about 60 milligrams total weight to the largest found, 

 weighing sixty times as much ; the value of k, however, changes 

 quite abruptly at about i-i g. total weight, a point which 

 probably denotes the onset of sexual maturity, decreasing here 

 to less than 80 per cent, of its value for the earlier growth- 

 phase. (It is a noteworthy and unexpected fact that the 

 growth-coefficient of this secondary sexual character should 

 be reduced instead of increased when the gonad begins to 

 function.) See Figs. 2 and 3. 



In our examples of the fiddler-crab, the weights of chela and 

 rest-of-body behave, over the earlier and longer growth-phase, 

 like £2 and £100 put out at 8 per cent, and 5 per cent, (continuous) 

 compound interest respectively 2 , and a calculation on this basis 



1 For cases where an organ is laid down considerably later than 

 the body as a whole, see Chapter IV. 



2 Strictly speaking, of course, 8-i and 5 per cent, (see p. 10). 



