490 
Church.-*— 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 1 . 
Botanic Gardens, Oxford. 
May, 1901. 
1 Cf. Church, On the Relation of Phyllotaxis to Mechanical Laws. Part I, 
Construction by Orthogonal Trajectories. 1901. 
