489 
Church.— Note on Phyllo taxis. 
which was deduced by Sachs from the analogy of the ortho- 
gonally intersecting planes of thickening observed in cell- 
walls and starch-grains. 
The readiness with which the several problems of phyllo- 
taxis may be solved from this standpoint, when once the key 
to the whole subject is grasped, is very remarkable, and these 
views have been elaborated to considerable length in a paper 
which awaits publication. The results are so varied and 
striking that it is difficult to give any summary of them in 
a small space : based as they are on the relative value of 
the spirals of Archimedes and logarithmic spirals as inter- 
preting the true developmental spiral of the plant-apex, it is 
evident that the discussion of such curves is beyond the 
province of the non-mathematical botanist. The object of* the 
present note is therefore merely to point out that the subject 
of phyllotaxis thus enters entirely new ground which promises 
results more fundamental than any yet obtained in the domain 
of plant morphology : for example, it follows in such con- 
structions that an equation may be given for the plane section 
of a lateral primordium which will serve as a true mathe- 
matical definition of a leaf, differentiating it from a stem : 
the true divergence-angles may be calculated, and a definite 
numerical value can be given to the ratio prin ^ r ai um which 
determines any given system ; while the geometrical con- 
structions, on the plan of Fig. 3, have the advantage that 
they do agree with the appearances observed in the plant ; 
they obey and amplify Hofmeister’s law, and from the stand- 
point of energy-distribution afford the clue to the subsequent 
building up of the elaborate ‘ expansion-systems ’ of which the 
capitulum of Helianthns may be taken as a type. 
It is not proposed at present to go into further detail as to 
these questions which are very fully discussed in the paper 
already prepared for publication ; until logarithmic spirals are 
more familiar to the botanist it will be sufficient to point out 
that the true key to phyllotaxis is undoubtedly to be found in 
the solution of the problems of symmetrical or asymmetrical 
