GROWTH 283 



producing 4, then 8, 16, 32, 64, etc. Designating by the letter A 

 the final size of the section, and by a the initial one; by t, the time 

 interval; and by r, the rate of percentage increase, the formula 

 A = aert is obtained, where e is the base of natural logarithms. 

 Changing this formula to a logarithmic form and converting the 

 natural logarithms into decimal ones, we obtain. 



, A 

 2.3026 log - = rt. 

 a 



whence it is easy to compute the percentage of growth per day, the 



initial and final size, and the number of days being known. 



Blackman's formula is applicable only in a limited way to the 



initial phase of the curve, for the exact moment of retardation is 



very difficult to determine. Since growth is based on the chemical 



transformation of assimilated or reserve substances into living 



molecules of protoplasm and the latter also begin to participate 



as soon as they are formed, in the process of growth, Robertson has 



made an attempt to apply to growth the formula of the monomolec- 



ular autocatalytic reaction which should embrace the entire process 



from beginning to end. If one considers the gradual retardation of 



the reaction as being due to the accumulation of its end products, 



dx 

 then the formula may be represented by — = K- (A — x), where K 



at 



is an empirical constant, x the size reached in t days, or other time 

 units, from the beginning of growth, and A the final size of the 

 growing organ. By integrating and other transformations 

 Robertson deduces from this formula another, which is more con- 

 venient for calculation: 



log = k(t - h), 



x 



in which h represents the time necessary for the growing organ to 

 reach one-half its final size. For many calculations it is convenient 

 to take relative values instead of absolute ones, and to assume the 

 final value of A to be 100. 



Besides Blackman's and Robertson's formulae, others have been 

 proposed. But all of them fail to express adequately the progress 

 of growth, since, in addition to internal conditions, the increments 

 per time unit depend also on environmental factors, such as tern- 



