926 ON LEAF-ARRANGEMENT [ch. 



shall always, for our fundamental spiral, run round the system, from 

 one leaf to the next, hy the shortest way. 



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

 shewn in Fig. 451),- that, besides the spirals running along AO and 

 aO, we have a series running /rom the steps on aO to the steps on ^0. 

 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 spiral fine 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. 



If our original lines along AO and aO contain, for instance, 13 

 and 8 steps respectively (i.e. m = 13, n= 8), then our next series, 

 observable in the same cone, will be 8 and (13 — 8) or 5; the next 

 5 and (8 — 5) or 3; the next 3, 2; and the next 2, 1 ; leading to the 

 ultimate condition of 1, 1. These are the very series which we have 

 found to be common, or normal; and so far as our investigation 

 has yet gone, it has proved to us that, if one of these exists, it entails, 

 ipso facto, the presence of the rest. 



In following down our series, according to the above construction, 

 we have seen that at every step we have changed direction, the 

 longer and the shorter sides of our triangle changing places every 

 time. Let us stop for a moment, when we come to the 1, 2 series, 

 or AT, aT of Fig. 451. It is obvious that there is nothing to prevent 

 us making a new 1, 3 series if we please, by continuing the generating 

 spiral through three leaves, and connecting the leaf so reached 

 directly with our initial one. But in the case represented in Fig. 451, 

 it is obvious that these two series {A, 1, 2, 3, etc., and a, 3, 6, etc.) 

 will be running in the same direction; i.e. they will both be right- 

 handed, or both left-handed spirals. The simple meaning of this 

 is that the third leaf of the generating spiral was distant from our 

 initial leaf by more than the circumference of the cyhndrical stem; 

 in other words, that there were more than two, but less than three 

 leaves in a single- turn of the fundamental spiral. 



