CONSTANT DIFFERENTIAL GROWTH-RATIOS u 



total weight rather over 3-5 g.) : b = 0-083, k = I#2 55- As 

 purely graphic methods, especially with logarithmic plotting, 

 are not sufficient to establish the accuracy of an empirical 

 formula of this sort (see, e.g., Gray, 1929), I have calculated 

 the values to be expected from the formula. As will be seen 

 from Table I, the deviations from expectation are slight — 

 only in four cases over 5 per cent., and these all in the first 

 phase, where errors in weighing are liable to be relatively 

 greater ; the mean deviation for the second phase is only 

 + 0-19 per cent., for the second phase it is + 0-35 per cent., 

 and would be smaller but for the large deviation of the last 

 class, which consists of only a few individuals. Further, there 

 is no trend of the deviations from predominantly positive to 

 predominantly negative or vice versa. We may thus take 

 the formula as a rather surprisingly close approximation to 

 reality. It is possible that the delimitation of the beginning 

 of the second phase after the 14th instead of after the 

 15th class would have improved matters ; and also that 

 small alterations in the values of k would have given an 

 even better fit, 1 but I am only concerned to show that the 

 data conform to this type of mathematical expression, not 

 to obtain accuracy in an extra decimal place in the formula 

 itself. 



We are accordingly justified in saying that the large chela 

 of the male Uca grows in close approximation to the formula 

 of constant differential growth-ratio, namely : y — bx h . 



In passing, it is worth noting that the logarithmic method 

 of plotting brings into true relief an important point which 

 is entirely obscured by the usual method of plotting on the 

 absolute scale — namely the fact that growth is concerned 

 essentially with the multiplication of living substance. On 

 the logarithmic scale, equal spaces on the graph denote equal 

 amounts of multiplication, whereas on the ordinary absolute 

 scale they denote equal additions. From the point of view 

 of growth, the increase of weight of our fiddler-crabs from 

 5 mg. to 25 mg. is equivalent to that from 1 g. to 5 g. ; but 

 on the absolute scale the former interval cannot even be repre- 

 sented on the same graph as the latter. Thus when I speak 

 of a fraction of the growth-period, I shall invariably be think- 

 ing in terms of multiplicative growth, in which an «-fold 



1 The last two columns of the table give the expectation if 0-0074 

 be substituted for 0-0073 as the value of b, and show that this gives 

 a greater deviation, but one of opposite sign. 



