6 VARIATION AND DIFFERENTIATION IN CERATOPHYLLUM. 



have been obtained in the field of ' ' Entwickelungsmechanik, * ' and to for- 

 get that there is still a very great deal to be done in this field. Indeed, 

 hardly more than a good start has been made towards the analysis of the 

 factors concerned in form production. 



It is the belief of the writer that in the methods of biometry we 

 have an analytical tool capable of rendering great aid in the investiga- 

 tion of the problems of morphogenesis, and in just the direction where 

 aid is most needed. Observation and experiment yield results which, 

 whether they are quantitative or qualitative, certainly demand quanti- 

 tative analysis if we are to get at their full meaning. In the physical 

 sciences not only has the necessity for quantitative (i. e., mathematical) 

 analysis of observational data long been recognized; but further, the 

 greatest generalizations of those sciences have come as the result of 

 such analysis. Unless one is prepared to maintain that the phenomena 

 of the inorganic world are fundamentally different in kind from those in 

 the organic, I can see no reason why a method of investigation which 

 has proven so valuable in the physical sciences should not, with proper 

 development, prove equally fruitful in the biological. 



We may now consider briefly the precise nature of the present study. 

 As has been stated above, our problem was to determine and formulate 

 so far as possible the laws according to which differentiation with growth 

 occurs in Ceratophyllum. If we take an individual Ceratophyllum plant, 

 we find it to include a number of whorls of leaves all generally like each 

 other but differing in detail. For example, some whorls have a larger, 

 some a smaller number of leaves. A very little study suffices to show 

 that whorls with different numbers of leaves distribute themselves 

 about a typical condition in a characteristic way. Whorls with one par- 

 ticular number of leaves occur in a different proportion than do those 

 with either more or fewer leaves. In this way, if we take into account 

 all the whorls on the plant, we get a characteristic frequency distribu- 

 tion for the different numbers of leaves, such as is shown, for example, 

 in fig. 5 (p. 25, infra). Further, as we shall see, the character of the 

 frequency distribution is fundamentally the same, whatever may be the 

 absolute size or source of the plant. Our problem is to determine so 

 far as possible the biological factors that result in the production of this 

 characteristic distribution. A whorl of leaves is the product of a definite 

 morphogenetic process in the growing bud, and it seems not unreason- 

 able to suppose that there is a definite set of factors (internal and 

 external) which determine the number of leaves which each particular 

 whorl shall bear. Moreover, we can be reasonably certain that since the 

 nature of the distribution is the same for Ceratophyllum plants gener- 

 ally, some of these factors at least are constant in their action. Through 



