GROWTH-GRADIENTS 81 



curve representing the steepness of the gradient, or in other 

 words the difference in absolute growth-potential between the 

 two ends of the gradient (as measured by growth-coefficients, 

 which for our present purpose afford the only comparable 

 standard for measuring intensity of growth-potential in a 

 number of different regions or forms) in relation to length of 

 the gradient — i.e. the relative length of the appendage. Such 

 a graph, however, is as I say only a crude representation of 

 the true growth-gradient. The most obvious reason for this 

 is the impossibility of assigning fixed points along the abscissa- 

 axis to the several joints, since the very fact of their differential 

 growth is causing their centres (or ends) to shift differentially 

 with increase in absolute size. 



And secondly, we have the difficulty that the values we 

 have obtained for the growth-coefficients are merely mean 

 values for large regions of the organ, whereas if the idea of 

 a growth-gradient be really justified, we should expect a pro- 

 gressive change of the growth-coefficient from point to point 

 along the axis, even within the limits of a single joint — a 

 theoretical consideration supported by certain actual evidence 

 in other forms (pp. 98, 261-2). 



The full solution of the problem, so as to obtain a quantita- 

 tively accurate picture of the graded change in growth-potential 

 along an organ, will be a matter of considerable difficulty, 

 partly owing to the formal difficulties arising from the con- 

 stant change of the relative size of the parts measured, partly 

 owing to the practical difficulty of finding sufficient distinctive 

 points within the limits of a region such as the segment of a 

 limb, on which to take measurements to determine the detailed 

 form of the gradient empirically and not by mere extrapolation 

 from a few mean values. 



Finally, there is still another difficulty. As pointed out 

 to me by Mr. J. B. S. Haldane, if y be the value of the 

 weight (or linear measurement) of the organ as a whole, and 

 if y 1} y 2 , . . . y n be the corresponding values of its constituent 

 joints or segments, then if y = bx k be a correct expression 

 for the limb as a whole, then y x = b^ 1 , y 2 = b 2 x kt , and so 

 forth cannot be accurate expressions for the separate parts 

 (or vice versa), since the sum of the several expressions for 

 the parts will not exactly fit the expression for the whole. 

 On the other hand, within certain limits of the value of k, 

 the discrepancy will only be slight, and we are justified, from 

 the actual figures obtained, in taking an expression of the 

 6 



