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THE USES AND ORIGIN OF THE ARRANGEMENTS OP LEAVES IN I'LWTR. $87 



find that, instead of two, there are six fractions of the simpler denomination* 

 or within the limits of distinguishable vnlues), which either do not occur in 

 nature at all, or occur very rarely; while those that are common are t UT in 

 number, or less than half of all. But we shall find that those of the six which 

 occur rarely differ from the two really unique ones among then and agrc* 

 with the common ones in respect to the law on which the ans\ftf to our 

 question really depends. This answer will be found to depend on the law 

 which was observed in the first four fractions of the first or second series, and 



was extended in the continuation of these and the formation of the oth rs. 



This law, or the dependence of these fractions on each other, wa en to b< 



» 



a simple case of the relations of dependence in the Furnish appr imati ru 

 of continued fractions, and thus lead to the induction of tin—- fractions; namely, 



the continued fraction 1 



1+1 



1 + 1 



1 -|- &c. for the first series, or 1 



2 + 1 



1 + 



1+&C 



for the second. The ultimate values of these continued fractions exi-nded in- 

 finitely are complements of each other, as their successive appi ximations lie, 

 and are in effect the same fraction; namely, the irrational or incommemurat* 

 interval which is supposed to be the perfect form of the spiral arrangement. 

 This does, in fact, possess in a higher degree than any rational fr tion tie 

 property common to those which have been observed in nature; though prac- 

 tically, or so far as observation can go, this higher degree is a mere refinement 

 of theory. For, as we shall find, the typical irrational interval diners from that 

 of the fraction f (and its complement differs from f) by almost exactly i5Vo , 

 a quantity much less than can be observed in the actu.l angles of leaf- 

 arrangements. The conception of such a typical angle as an actual value ,n 



d as a 



point of departure for more specialized ones, 



ting eith 



among the normal patterns, or formative principles of vegetable life, as lb. 

 theory of types supposes, or in some nnknown law of development or phys-o- 

 • ieal necessity, - sueh a eoneeption is a very attractive one. And as esh.b, mg 

 the abstract and in its most perfect form a property pecuhar, as we ,W1 



see, to natnral arrangements, but belonging to them in inferior and in anons 

 degrees, -as exhibiting this separated from the property wh.ch such arrangement, 



