Ill] CONCERNING GRADIENTS 193 



A like problem occurs when we deal with rates of growth in 

 successive natural internodes; and we may then pass from the 

 actual growth of the internodes to the varying number of leaves 

 which they successively produce. Where we have whorls of leaves 

 at each node, as in Equisetum or in many water-weeds, then the 

 problem is simphfied; and one such case has been studied by 

 Ra3rmond Pearl*. In Ceratophyllum the mean number of leaves 

 increases with each successive whorl, but the rate of increase 

 diminishes from whorl to whorl as we ascend. On the main stem 

 the rate of change is very slow; but in the small twigs, or tertiary 

 branches, it becomes rapid, as we see from the following abbreviated 

 table : 



Number of leaves per whorl on the tertiary branches of 

 Ceratophyllum 



Raymond Pearl gives a logarithmic formula to fit the case; but 

 the main point is that the numbers form a graded series, and can 

 be plotted as a simple curve. 



In short, a large part of the morphology of the organism depends 

 on the fact that there is not only an average, or aggregate, rate of 

 growth common to the whole, but also a gradation of rate from one 

 part to another, tending towards a specific rate characteristic of each 

 part or organ. The least change in the ratio, one to another, of 

 these partial or locahsed rates of growth will soon be manifested 

 in more and more striking differences of form; and this is as much 

 as to say that the time-element, which is imphcit in the idea oigrowthy 

 can never (or very seldom) be wholly neglected in our consideration 

 of form I . 



A flowering spray of Montbretia or lily-of-the-valley exemplifies 

 a growth-gradient, after a simple fashion of its own. Along the 



* On variation and differentiation in Ceratophylly,m, Carnegie Inst. Publications, 

 No. 58, 1907; see p. 87. 



t Herein lies the easy answer to a contention raised by Bergson, an(J to which 

 he ascribes much importance, that "a mere variation of size is one thing, and 

 a change of form is another." Thus he considers "a change in the form of leaves" 

 to constitute "a profound morphological difference" {Creative Evolution, p. 71). 



