GENERAL DISCUSSION OP RESULTS. 125 



GENERAL DISCUSSION OF RESULTS. 



In bringing this paper to a close it appears desirable to discuss to 

 some extent certain general aspects of the work as a whole, and to 

 consider them on the theoretical side. Speaking broadly, the most 

 significant result of this work appears to the writer to lie in the fact 

 that, proceeding by quantitative analytical methods, it has been possible 

 to formulate two laws of growth, which serve to describe with a very 

 high degree of (a) precision, (6) completeness, and (c) generality the 

 observed results of those processes of morphogenesis which in the 

 growing Ceratophyllum plant lead to differentiation of parts. Further, 

 it has been shown by direct appeal to statistics that the characteristic 

 features of the variation of Ceratophyllum are obviously the result of 

 the fact that the organism grows in accordance with these laws. By 

 this, of course, is not meant that any kind of "explanation" of the 

 origin of variation in Ceratophyllum has been gained. What has been 

 gained, though, is the knowledge that, so far as our present material, 

 which includes a reasonably wide range of conditions as to habitat, time 

 of collection, etc., is concerned, the difference between two sets of 

 individuals in respect to their variation constants are capable of prac- 

 tically complete interpretation solely in terms of the two laws of growth. 

 The results actually observed are such as would be expected to arise in a 

 system differentiating in accordance with our two growth laws. 



The first of these laws of growth was stated on page 88 in the fol- 

 lowing way: "The mean number of leaves per whorl increases with 

 each successive whorl, and in such a way that not only does the absolute 

 increment diminish, but also the rate of increase diminishes, as the 

 ordinal number of the whorl measured from a fixed point increases." 

 If we let y stand for number of leaves in the whorl, and x denote the 

 position in a series of successively formed whorls, then we find that y is 

 a simple logarithmic function of x as follows: 



y = A + Clog {x — a) 



where A, C, and « are constants. The leaf whorls become differentiated 

 with growth according to a logarithmic law. Also, as was shown in the 

 last section (p. 125) , if we let y denote the number of lateral^branches 

 found in a given number of nodes, and x as before the position of these 

 particular nodes in the whole series, again we find y a simple logarithmic 

 function of x. So that, in general terms, we see that in Ceratophyllum 

 growth, whether expressed in the formation of leaf whorls or of lateral 

 branches, takes place in such a way that the product increases at the 

 same proportionate rate that the logarithm of the position in the whole 

 series of products increases. 



