290 PRINCIPLES OF EMBRYOLOGY 



programme. We can express the growth rates of entity x in animals A 

 and B as two time-functions: 



Ai ±Ji 



Now we could choose another unit for measuring time, f^, such that 



X 



fa{t^) = fhif)- This would amount to measuring the growth rate of— ^ 



in time-units which made it identical with that of — . The important point 



is that this same transformation of the time-unit would automatically 



convert the growth rate of — into that of— , and that of -j- into that of ^ 



A2 -B2 A2 -B3 



etc. Thus one change in time-scale would convert the whole chemical 

 growth system of one animal into that of another (apart from the compli- 

 cation due to the constants Ai, A^, etc. which express the initial state of 

 the system when growth starts). We have here an approach to a concept 

 of 'biological time', by which is meant the notion that events in, say, a 

 mouse or an elephant, are similar but are all uniformly speeded up in the 

 former as compared with the latter. It is not yet clear to what sort of 

 entities such a notion can be applied : for instance it seems most improb- 

 able that any such relation can hold for molecular enzymatic processes. 

 Further discussions of it will be found in Brody (1937) and du Noiiy 

 (1936) and some highly critical remarks in Medawar (1945). 



3. Growth gradients and transformations of shape 



In a complex organ, it is often found that the growth rate, relative 

 to some standard part, varies in a graded manner from place to place. The 

 simplest expression of this can be seen when there is a series of more or 

 less comparable parts, for each of which an allometric growth constant 

 {a) can be ascertained. Huxley (1932, Reeve and Huxley 1945) has de- 

 scribed many examples, relating for instance to the joints within a crus- 

 tacean limb, or the series of limbs attached to the different segments of the 

 body. Fig. 13.3!^ shows how the a\ for the different segments fall into an 

 orderly sequence, which can be taken as defining a growth gradient. 



If one measures the growth constants for a series of distinct sub-units 

 within an organ such as a limb, there are of course defmite jumps in its 

 value between adjacent segments, and what should, perhaps, be a continu- 

 ous gradient, is described in terms of a discontinuous series of steps. 

 Examination of other cases indicates that in fact the gradients within a 

 single mass of tissue are usually, if not always, continuous in gradation. 



