RHYTHM. 225 



regularity in the initiation of the impulses which produce new 

 centres of lateral growth. 



Thus it follows from observation of plant systems that such 

 lateral growth-centres producing a sequence of similar foliage 

 members are always similar figures at similar ages, and that these 

 are so arranged that they make similar contacts with adjacent 

 members. In other words, if the lateral growth-centres are repre- 

 sented as circles, they must be arranged in some manner after 

 such schemes as shown in figs. 19, 20, 22, 23, so long as the 

 simple case of uniform growth is postulated. That is to say, in 

 that the lateral members are similar figures they will fall along 

 lines plotted by equiangular spirals, intermediate between the 

 limiting cases of the straight line and the circle ; and in that 

 they may be represented by "circles" in lateral contact which 

 would be contained in the quasi-squares, the contact lines of such 

 series must necessarily be orthogonally intersecting equiangular 

 spirals. The log. spiral theory of phyllotaxis is thus the necessary 

 outcome of : — 



I. The theory of the geometrical representation of a uniform 



growth-movement. 

 II. The hypothesis that a lateral growth-centre is essentially of 

 the same nature as the symmetrical growth-centre origin- 

 ally postulated.* 



* It liaa been stated above that the lateral growth-centres would be ex- 

 pressed as true circles or as the homologues of circles inscribed in the meshes 

 of the square meshwork. That the latter is probably the case in the formation 

 of leaf -members is very clear from the fact that circles cannot be placed in 

 the accurate contact relations required ; this being especially noticed in low 

 systems which in the plant are apparently as regular in formation as higher 

 ones. 



A simple and fundamental conception of a leaf as opposed to a branch is 

 thus brought out, which constitutes, in fact, a true mathematical distinction 

 between an axis and an wpTpendage. A leaf is a, primary appendage belonging 

 to a system controlled by a central growth-centre, a subsidiary development 

 of it, differing from it in its increased rate of growth, and is thus represented 

 by the quasi-circle homologue, the controlling growth-centre remaining at 

 the apex of the shoot. A branch or lateral axis, on the other hand, is repre- 

 sented by a true circle, that is to say, as a new growth-centre wholly uncon- 

 trolled by the growth-centre of the parent shoot, and maintaining its own 



