Church. — Note on Phyllotaxis. 489 



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 ^^^^^^.^^ 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 Helianthus 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 



