384 



MEMOIRS OF THE AMERICAN ACADEMY 



f leaves in the spiral 



gements 



as to those of whorls, and refer rather 



please, than to actually defi- 



of 



in these arrangements is 



to abstract numbers, counted from any point we 



nite groups. The actual system, cycle, or group 



indefinite extent, or comprises the whole stem, so far as it is developed, and 



even extends into the undeveloped leaves of the terminal bud. In speaking of 



a 

 b 



ycl 



of 



these 



arrangements no definitely situated group is meant 

 definite number counted from any one we may choose for an origin. 



gements of this typ 



we find that, after thus counting some 



In almost all arrai 



definite number of leaves from some one assumed as the first, we 



at a leaf which stands directly over the first. Such a group, so determined, 



next 



makes what is called a cycle ; or, 



as 



we may sometimes prefer to call it, a 



system. Within it leaves succeed each other at successively greater and greater 

 heights, and are so placed around the stem that the same angular interval or 



angle of divergence is contained between 

 of divergence 



any two successive ones. 



This angle 



is commensurate with the circumference, but is not always an 



aliquot part of 



in 



the 



gular interval of the le 



of whorls. It 



many plants some multiple of an aliquot part, and in counting the leaves suc- 

 cessively through the cycle, we have to turn several times around the stem. 

 This number of revolutions, divided by the number of leaves in the cycle, is the 

 ratio of the angle of divergence to the whole circumference ; and the fraction 



expressing 



this ratio is used to denote the particular arrangement 



of 



su 



ch a 



system. Thus the fraction i denotes the alternate arrangement, in which there 



the name 



are two leaves in one turn, the third leaf falling over the first 



} is 



of the three-leaved system, in which there are three leaves in one turn, the 



the name of the system in which five leaves 



fourth falling over the first. 



I is 



occur in two turns, and the sixth falls over the first. In order that such defi- 



nite 



rical systems, or cycles, should exist in the leaves of any pi 



only necessary that the ratio of the angle of divergence to the circumference 

 should be some proper fraction, and this fraction would be in the same way the 



name of the system. But any proper fraction whatever would 



the prop- 



erty I have pointed out; namely, that after the number of leaves denoted by 



$ number of turns denoted by its numerator, the next 



its denominat 



d th 



succeeding leaf would fall over the first. Whatever may be the purpose or 



advantage of the spiral an 

 that some other purpose is 

 actual 



gement, and of this feature in 



ht, or some other adv 



'© 



it is obvious 

 gained, by the 



arrangements 



of this sort in nature ; or else it would appear on 



the 



