106 PROBLEMS OF RELATIVE GROWTH 



along the axis of the body, then clearly the proportions of the 

 animal must be continually changing so long as it is increas- 

 ing in absolute size, or at least over a long space of time. 

 But the fish's outline and the system of co-ordinates drawn 

 to fit it, represent the state of affairs only at one particular 

 moment of its life-history. If the fish had grown to twice 

 the bulk, its proportions would have changed, and the co- 

 ordinate grid would have to be altered ; yet the underlying 

 growth-gradient might have remained wholly unaltered. 



An improvement would be effected if the absolute sizes of 

 the two outlines which mark the onset and stoppage of orderly 

 differential growth could be given : but even so, the real 

 invariable, namely the growth-gradient expressing the values 

 of the growth-coefficients along the body, could only be 

 obtained from this by calculation, and is not deducible by 

 inspection. 



For this reason, the co-ordinate method, while of the utmost 

 importance as affording a graphic and immediate proof of 

 the need for postulating regularities in the distribution of 

 growth throughout the body, is of little use for detailed analysis, 

 because by its nature it neglects the fundamental attribute 

 of differential growth, namely the change of relative propor- 

 tions with absolute size : it is static instead of dynamic, and 

 substitutes the short cut of a geometrical solution for the more 

 complex realities actually underlying biological transformation. 



None the less, it is invaluable as demonstrating the need 

 for thinking in terms of growth-gradients and, in general, of 

 an orderly system in the distribution of growth-activity 

 throughout the body : and this whether we are considering 

 individual or evolutionary change of form. We have only to 

 glance at D'Arcy Thompson's figures of brachyuran carapaces, 

 ungulate limbs, and so forth, to realize immediately its utility 

 in this respect (Figs. 48, 58). 



Another interesting use has been to deduce the course of 

 evolution over gaps where actual fossil data are missing. It 

 is easy to say that we can do this by common-sense, simply 

 inserting hypothetical intermediate stages between known 

 end-points. But this is insufficient. There are, for instance, 

 hundreds of possible ways of bridging the gap between the 

 pelves of Archaeopteryx and the cretaceous bird Apatornis, 

 to take an example worked out by Thompson ; but if the 

 evolutionary modification of such a structure be due to growth- 

 changes, and if orderly and regular growth-gradients be the 



