490 Chtirck. — Note on Phyllotaxis. 



distribution of energy in orthogonally intersecting planes 

 around an initial ' growth-centre ' ; in the latter case the 

 whole of the spiral paths are log. spirals. The perfection of 

 such a construction involves uniform growth in the system ; 

 and owing to the obvious impairment of this uniform rate of 

 growth behind the plane portion of the apex, the true log- 

 spirals are possibly never to be observed on the plant, although 

 the approximation has been found in , certain cases to be 

 extremely close. Ultimately all these curves pass into spirals 

 of Archimedes as the members cease growth on the attain- 

 ment of constant volume, and these latter curves therefore 

 occur on adult axes and appeal to the eye in the macroscopic 

 view of the entire shoot. They were thus correctly isolated 

 by Bonnet, to whom the detailed construction of the growing 

 point was naturally unknown in 1754. The curves seen in 

 transverse section of an apical system of developing members 

 are thus probably curves transitional between log. spirals and 

 spirals of Archimedes. 



On the other hand it will be noted that the new con- 

 structions are equally incapable of absolute verification by 

 any angular measurements on the plant ; Schimper's ortho- 

 stichies have vanished, as pointed out by the Bravais, for 

 the more general examples of phyllotaxis, and the differ- 

 ence between the two spiral systems is very slight to the eye : 

 but, while the Schimper-Braun School only sought to imitate 

 the appearances seen on the plant, the log. spiral theory gives 

 at least an equally correct summary of the facts observed, and 

 is in addition founded on definite mechanical laws of con- 

 struction by orthogonal trajectories which have already been 

 accepted for plant anatomy ; it is so far then the logical 

 outcome of Sachs' theory of the orthogonal intersection of 

 cell-walls, and represents therefore another special case of the 

 distribution of energy along planes of equal action ^. 



Botanic Gardens, Oxford. 

 May, 1901. 



' Cf. Church, On the Relation of Phyllotaxis to Mechanical Laws. Part I, 

 Ctinstruction by Orthogonal Trajectories. 1901. 



