264 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



by different observers. The fact remains that in many cases the 

 primordia not only do not appear to touch, but are certainly 

 not in any relation of mutual pressure {Aspidium). While this 

 standpoint of origin in contact creates a prejudice in favour of 

 regarding the primordia as of a bulky nature from their first 

 inception, the growth-theory here proposed, by postulating an 

 initial growth-centre of the nature of a mathematical point, leads 

 directly to the view that contact-relations are secondary, and that 

 theoretically growth must proceed around each centre until it 

 reaches the field of adjacent growth-centres, although the actual 

 boundary may be beyond observation. What is really established 

 by the observations of Schwendener is the fact that these contact- 

 relations are characteristically of a special type, to which the 

 conventional terminology of a "concentrated system" has been 

 applied. It is clear, however, that the growth-centres of the 

 primordia are always quite distinct from each other, and that the 

 contact-relations must be ultimately produced if growth only 

 continues long enough, and in the great majority of leafy shoots 

 contacts are soon established. But such contacts will not 

 necessarily imply any displacements, and the phenomena 

 of displacement of the edges of the members according to 

 a definite plan is perhaps one of the most remarkable features 

 of ordinary spirally constructed leafy shoots; and there can be 

 no doubt that the existence of such slipping or sliding-growth 

 effect, which apparently implies a forcible displacement of tlie 

 members, was the fundamental fact which led Schwendener to 

 postulate a forcible displacement under mutual pressure. The 

 actual significance of the regular displacement of leaf edges in 

 asymmetrical systems, which it must also be noted does not take 

 place in symmetrical constructions, may be left for the present, 

 since it only becomes apparent from the mathematical considera- 

 tion of the subject. It is only necessary to point out that the 

 necessity for such displacement naturally follows from the general 

 conception of a phyllotaxis system as built up of primary and 

 secondary growth-centres, and the fact that the required dis- 

 placement effect does occur is one of the strongest proofs of the 

 practicability of the log. spiral theory (c/. Mathematical Notes). 



