GEOMETRICAL REPRESENTATION OF GROWTH. 35 



Since the longitudinal section affords no clue, it is therefore 

 necessary to fall back on the transverse components of the growing 

 system. 



A transverse section shows a simple concentric circular structure 

 in which cell- walls follow the paths of circles and radii, intersecting 

 therefore orthogonally. That is to say, the circles and radii re- 

 present reciprocal paths of equal action, and since the protoplasm 

 is a semi-fluid mass, such paths may be compared to the lines of equal 

 pressure and flow in a plane circular system. 



Thus, if fluid films are laid down in connection with radial lines 

 of equal pressure, the periclinal walls will be established, and may 

 be subsequently fixed by a deposit of cellulose. In the same 

 nianner, because the anticlinal walls follow the paths of radii, it 

 follows that their position results from another uniform action 

 along the circular paths. These orthogonal paths are interchange- 

 able, and what can be said of one can be inferred of the other. 

 The formation of anticlinal and periclinal walls in such a theoretical 

 apex may be considered therefore as resulting from two motions in 

 the fluid protoplasm, one a radiating current, the other a free circular 

 vortex. Main current movements of protoplasm in the whole 

 growing apex, apart from subsidiary currents in individual cells, 

 may thus be regarded as following along the general lines readily 

 observable in single cells, and known respectively as movements 

 of Circulation and Botaiion. The diagram for the paths of equal 

 action in a transverse section of an apex would be the same as 

 that for the circulation and rotation of protoplasm in an isolated 

 spherical cell, and the mechanical law underlying the geometrical 

 construction of Sachs for the orthogonal formation and intersec- 

 tion of cell-walls would be that such orthogonal paths represent 

 the geometrical consequence of the fact that lines of equal 

 pressure and flow in a fluid medium are mutually at right angles. 



bodies might be due to crystallization formed the keystone of the MiceUar 

 Theory. With such a standpoint it is the more remarkable that Sachs did not 

 explain the layering of the tree trunk along the lines of an ovoid starch-grain, 

 and did not note that the small end of a typical starch-grain is equally indis- 

 tinguishable by the eye from a parabola, and presents an equally good imitation 

 of the construction lines of a growing apex. 



