334 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



Note II. — Mathematical Orthostichies in Log. Spiral Systems. 



To obtain a colUnear intersection of m and n spirals, i.e, a 

 second point of intersection in the same straight line with the 

 origin, it is clear that in the Fibonacci series, for example, it is 

 only when m : n :: Jb — 1 -.2 that such an intersection will take 

 place at infinity. 



But with m and n finite integers, radially superposed inter- 

 sections will take place ; and taking the case of m spirals crossing 

 n, the nearest point collinear with the origin and any given point 

 on the same side of it will be the {m?+n^)th. term : that is to say, 

 in the general case in which m and n have no common factor. 



Thus, in the system (3 + 5) a true orthostichy will exist between 

 any member taken as and the (9 + 25) = 34th. In the case of 

 Sempervivum tectorum, for example (figs. 1, 2, 83), the contact- 

 parastichies at the apex being (3 + 5) (fig. 83), the line drawn 

 through No. 1 and No. 35 (fig. 2) should be mathematically a 

 radius, and a true orthostichy line to produce which the onto- 



genetic spiral would wind 13 times {i.e. — ). 



It will be noticed that such points are beyond the range of the 

 construction diagrams, which only include a portion of one 

 revolution of the pair of generating log. spirals ; and also beyond 

 observation on the plant, owing to the fact that minute differences 

 in the growth of older leaves would suf&ce to produce slight 

 displacements which would destroy the effect of these mathemati- 

 cally straight lines. For practical purposes these true orthostichies 

 pass beyond the limit of consideration, but the fact that such are 

 possible is still of botanical interest ; while the curious relation of 

 such an empirical formula of the Schimper-Braun series to the 

 actual construction in the case of Sempervivum is noteworthy. 

 The phyllotaxis may here be thus accurately written in the 



13 



Schimper-Braun terms — , but such a formula can only be deduced 



from the consideration of the properties of a (3 + 5) system, and 

 not from any inspection of the external characters of the leaf- 

 arrangement on the plant itself, in which superposition effects 



