xiv] 



OR PHYLLOTAXIS 



645 



round the system, from one leaf to the next, by the shortest 

 tvay. 



Now it is obvious, from the symmetry of the figure (as further 

 shewn in Fig. 326), that, besides the spirals running along AO and 

 aO, we have a series running /rom the steps on aO to the steps on 

 AO. In other words we can find a leaf (S) upon AO, which, like 

 the leaf 0, is reached directly by a spiral series from A and from 

 a, such that aS includes n steps, and AS (being part of the old 



Fig. 326. 



spiral line AO) now includes m — n steps. And, since m and n 

 are prime to one another (for otherwise the system would have 

 been a composite or whorled one), it is evident that we can 

 continue this process of convergence until we come down to a 

 1, 1 arrangement, that is to say to a leaf which is reached by a 

 single step, in opposite directions from A and from a, which leaf 

 is therefore the first leaf, next to A, of the fundamental or 

 generating spiral. 



