646 ON LEAF-AREANGEMENT [ch. 



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 con- 

 struction, 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. 326. 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 con- 

 necting the leaf so reached directly with our initial one. But in 

 the case represented in Fig. 326, 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 cylindrical stem ; in other words, that there 

 were more than two, but less than three leaves in a single turn of 

 the fundamental spiral. 



Less than two there can obviously never be. When there are 

 exactly two, we have the simplest of all possible arrangements, 

 namely that in which the leaves are placed alternately on opposite 

 sides of the stem. When there are more than two, but less than 

 three, we have the elementary condition for the production of the 

 series which we have been considering, namely 1, 2; 2, 3; 3, 5, 

 etc. To put the latter part of this argument in more precise 

 language, let us say that : If, in our descending series, we come to 

 steps 1 and t, where t is determined by the condition that 1 and 

 t + 1 would give spirals both right-handed, or both left-handed ; 

 it follows that there are less than t + 1 leaves in a single turn of 

 the fundamental spiral. And, determined in this manner, it is 

 found in the great majority of cases, in fir-cones and a host of 

 other examples of phyllotaxis, that t = 2. In other words, in the 



