390 MEMOIRS OP THE AMERICAN ACADEMY. 



in its capacity as an irrational fraction, — but simply as being indistinguishable 

 practically from these rational ones, and as being entirely consistent practically 

 with the property that rational proper fractions also have of forming limited 

 systems or cycles. Much mystification has come from the irrational character of 

 this fraction; scepticism on the part of non-mathematical botanists, and mysticism 

 on the part of mathematicians. The simpler or the first three fractions of this series 

 have also in a less degree the same distributive quality, and so in a still less 

 degree have the fractions of the two lower series. But all the fractions left among 

 possible ones, within the limits considered, that are sufficiently simple to be readily 

 identified, are the fractions | and -|, or their complements f- and | ; and these 

 exceptions, as I have said, are all the grounds of fact which at first sight give any 

 plausibility to the theory of Phyllotaxy, or make its laws anything other apparently 

 than the necessary consequences of purely numerical properties in the simpler 

 fractions. Yet beside the fact that these two have not the distributive character 



of the others, the fact should be taken account of, that by confining ourselves 

 to the limits \ to £ we have neglected several other simple fractions, that are 

 even worse adapted for the purpose which the great majority appear to serve. 

 These fractions are f, f, |, and f, or their complements. Moreover, we should 

 consider that as the fractions peculiar to the two lower series are much less 

 fitted for this purpose than those of the first series, so they are much less 

 frequently found in nature. Taking account of all these facts, we find the 

 hypothesis that nature has chosen certain intervals in the spiral arrangements 

 of leaves, and for the purpose I have indicated, to be sufficiently probable to 

 justify a more careful consideration of it. Wide divergences from the most 

 perfect realization of this purpose, such as we have among the more frequent 

 forms in the fractions £ and f, or in the alternate and three-leaved systems, 

 and also among the less frequent forms, indicate the existence of other condi- 

 tions or purposes in these arrangements, which I propose to consider further 

 on. I may remark here, however, that these two classes of exceptions from the 

 most perfect realization of the distributive property, namely, those of the first 

 series which belong to the most advanced forms of life, and those peculiar to 

 the two other series, are probably due to widely different causes ; the one having, 

 m fact, a high degree of specialization, and the other falling short in respect to 

 this distributive property on account of a low degree of specialization. This 

 view which is one of the consequences of theoretical considerations on the origin 

 ot these arrangements, that will be presented when we come to consider the 



