292 PRINCIPLES OF EMBRYOLOGY 



D' Arcy Thompson's suggestions opened up a large field for investigation, 

 but unfortunately this has not been as systematically studied as might have 

 been hoped. We can deal v^ith the recent work under three heads : firstly, 

 improvements in the method; secondly, the general physiology of growth 

 gradients and shape transformations, and thirdly, attempts to discover the 

 physiological mechanisms underlying them. 



Actually rather little has been done to make D'Arcy Thomson's method 

 of the distortion of a co-ordinate network into a means of exact analysis. 

 Medawar (1944, 1945) has made some steps in this direction (see also 

 Richards and Kavanaugh 1945). He considered the changing shape of 

 the human body during its development from the early foetus to the 

 adult. The body was first reduced to a two-dimensional shape by repre- 

 senting it as a series of outline drawings when seen from the front (Fig. 

 13.5). It becomes obvious then that in the early stages the legs grow faster 

 than the parts nearer the head, and it appears probable that there is a single 

 continuous growth gradient with its high point towards the feet, falling 

 off as one goes higher up the body. Medawar pointed out that this could 

 be represented by a transformation of co-ordinates and that this trans- 

 formation could theoretically be specified in algebraic terms. To illustrate 

 how this might be done he reduced the shape of the whole living body 

 still more drastically and considered only certain points on the vertical 

 midline; the foot, fork, navel, nipples, chin, etc. The original three- 

 dimensional shape was thus reduced to a line on which certain intervals 

 are marked. Suppose now that Pi, P2, P3, etc. are the heights from foot 

 to fork at successive points in time, and similarly Qi, Q2, Q3, etc. the 

 heights from fork to navel at the same times, and so on for the other 

 intervals. We can from the actual measurements work out empirical 

 equations connecting the successive Ps and again, another set of equations 

 connecting the successive Qs, and so on. Each equation will give the 

 changes of one part of the network as time proceeds. We can also fmd 

 algebraical relations between the equations relating to the Ps and those 

 relating to the Qs, the Rs and 5s, etc. at any given point in time. With the 

 aid of these two sets of equations, the whole series of transformations can 

 be expressed algebraically. It is clear, however, that quite a lot of arith- 

 metic is required to produce even a fairly clumsy algebraical description 

 of a series of shapes, notwithstanding that these have been reduced to 

 their very simplest form, the original three dimensions having been 

 whittled away to one. Such labour is only justified if it enables one to see 

 certain relations which would otherwise be missed. So far, such evidence 

 of a real usefulness of the method has not been forthcoming. 



There are contexts, however, in which it seems probable that Medawar s 



