CONTACT-PRESSURES. 241 



determine the initiation of new growth-centres are perfectly 

 distinct from those which come into operation once they are 

 formed and have produced members of a definite visible 

 bulk. 



The weakness of Schwendener's argument is sufficiently clear — 

 the mere assumption of a cylindrical surface which may be 

 unrolled at once puts all developmental phenomena out of court : 

 the apex of a plant can never be regarded as a cylinder, although 

 on the other hand it may never be quite flat; the unrolled 

 cylinder representing, in fact, the longitudinal component of the 

 growth-system which is solely due to a retardation in the rate of 

 growth in a system which would remain always plane so long as 

 uniform growth persisted. Similarly, the primordia can never be 

 represented during development as equal spheres, nor possibly as 

 truly circular in section. The assumption of circular figures, 

 which will also be similar, and the orthogonal arrangement is 

 alone all that is required to mathematically deduce the log. spiral 

 theory ; since, when transferred to a plane projection of a growing- 

 point, no other spirals except log. spirals drawn in the manner 

 previously postulated will contimie to give either similar figures or 

 orthogonal intersections* 



As already pointed out, the attempt to eliminate inconvenient 

 spiral curves by unrolling the helix of Bonnet on to a plane is 

 the point at which the initial error crept in. The helix represents 

 the secondary stage of phyllotaxis, in which the members have 

 attained constant volume by a progressive cessation of growth. 

 A growing system is necessarily a log. spiral system or a derivative 

 of one, and the helix drawn on a cylinder is mathematically related 

 to both the spiral of Archimedes and the log. spiral of a plane 

 projection, and may, therefore, be derived from either. Two of 

 the five hypotheses of Schwendener, therefore, when applied to 

 the transverse component of a phyllotaxis system, are sufficient to 

 give the log. spiral theory, which agrees so closely with observed 

 facts that no external agency, whether of contact-pressures, con- 

 tact-stimulation, or anything else, is required to make the system 



* I am indebted to Mr H. Hilton for the mathematical proof of these 

 statements. 



