GENERAL DISCUSSION OF RESULTS. 127 



It seems somewhat remarkable that so important a matter for the 

 understanding of many problems of morphology as is the study of post- 

 embryonic growth from both of the points of view noted should have 

 been so much neglected. On the botanical side there is a good deal 

 of literature dealing with special phases of the subject, but for our 

 present point of view most of this material has little direct bearing. 

 On the zoological side the principal work is due to anthropologists who 

 have studied post-embryonic growth in man. In this field the available 

 evidence regarding individual growth, so far as it goes, appears to be in 

 good accord with what we have found in Ceratophyllum for organal 

 growth. Thus, for example, Pearson ( :04) has shown that the growth 

 in auricular height of the head in children follows a logarithmic 

 curve, and in a recent memoir by Lewenz and Pearson ( :04) it is stated 

 that such a curve has been found to represent the growth changes in 

 other characters. Probably the most thorough and in all respects the 

 best study of growth in any other animal than man which has been 

 published is the classical investigation of Minot ('91) on growth in the 

 guinea-pig. Speaking of his statistics Minot (p. 148) says: "They 

 demonstrate two fundamental facts: First, the rate of growth dimin- 

 ishes almost uninterruptedly from the time onwards when the animal 

 recovers from the post-natal loss of weight; second, the diminution is 

 rapid at first, but slower afterwards." It will be seen that these 

 statements exactly agree with those we have made above for growth 

 in Ceratophyllum. That is, it would appear that the "individual" 

 growth of the guinea-pig follows a logarithmic law. A careful study 

 of Minot's data indicates that this is in fact the case. There is a great 

 need for special investigations of growth directed towards determining 

 exactly the laws which the changes follow. From such investigations 

 we may hope to get some idea of the extent to which a logarithmic law 

 is general. In any event, it is clear that such a growth law is not entirely 

 unique in Ceratophyllum. On the contrary one has been convinced by 

 going over the older material available in the literature, which it would 

 take too much space to cite in detail here, that a logarithmic law is 

 probably very general for growth in both plants and animals,* and for 

 "individual" as well as "organal" growth. 



There can be no doubt that what has been found in Ceratophyllum 

 with reference to the variation of repeated parts is simply an example 



*It has doubtless occurred to the reader that this logarithmic law of growth super- 

 ficially resembles in form the well-known Weber-Fechner law regarding the quantitative 

 relation of stimulation and sensation, as it was formulated by Fechner. Unfortunately 

 later research in physiological psychology has shown that Fechner' s statement of the 

 law either does not hold at all, or at most only in a. very limited range of cases. 



