THE PHYLLOTAXIS OF HELIANTHUS TUBEROSUS. <;.-; 



change came about, it can only be by mere speculation. AVe might, for butane* 



imagine that the crowded state of the leaves which must obtain where tl 



paratively large number in the same coil or circle might not be so advantageous to their 



re is a coin- 



development as a more scattered condition*. And hence we may be led to think that 

 natural selection may have had some influence in bringing about these changes. U 



•lit be induced to speculate still further, and say that the verticillate including 



5- "«' 



opp 



r> 





site leaves, was the earliest arrangement, and that the spiral condition a s a subsequent 

 evolution; while the various changes of divergence of the Helianthus might remind ua 

 that it appears much easier to resolve verticils into spirals than to convert spirals into 



verticils. So, too, if analogy might be brought to bear upon this point, we might thin] 

 that as "radial symmetry" (as we might call it) is characteristic of a lower organisation 

 amongst animals than the bilateral and integrated condition of organs belong-in" to those 

 of a higher organization, so the multiplication of leaves and their more or less verticillate 

 arrangement, might perhaps hint at an original truth, now well ni<?h obscured if not 

 obliterated by the evolution of ages. But putting all such speculations aside, no suc- 

 cessful attempt has hitherto been made to ascertain the cause of any definite arrai 

 ment existing at all. We are quite as unable to explain it as to give a reason why tin- 

 numbers 5 and 4 prevail among the parts of Dicotyledonous flowers, and 3 amon 

 those of Monocotyledons. 



So, again, if we limit our inquiries to the condition of the primary series alone, w< 

 find, as a fact in nature, that if we assume any leaf as No. 1, then No. 2 lies between 

 120° and 180° inclusively, and that, too, either to the right or to the left. If we 

 are asked why it is so, we can give no answer. Further, starting with this con- 

 dition, we find that the position of this 2nd leaf is not anywhere along that arc, 

 but that it has an inherent tendency to take up a definite point as its position; and, 

 again, when we compare the positions of all such 2nd leaves in a variety of plants, 

 or, as has been shown, in the Helianihus tuberosum alone, we find that the scries 

 of points between 120° and 180° affect an approach towards some limiting and inter- 

 mediate point, which point, however, no known example has ever reached. If we 

 ask, again, why the 2nd leaves endeavour to take up these definite positions and are 

 not anywhere over the arc, there is as yet no answer to this question, any more than 

 to the first. All that appears capable of exact statement is, that such are the condi- 

 tions found to exist in nature, and that when the angles between the first and second 

 leaves of all the different generating spirals are measured, and represented as frac- 

 tional parts of the circumference, they are found to bear such relations to one another, 

 when written down in succession, as obtain between the successive convergents of the 



continued fraction of the general form -L -L -L -I &c 



a+ 1+ 1+ 1 + 



not 



r aps the fact that in many cases where leaves _ 



"^frequently more or less inclined to be linear or acicular in form, may give some countenance 



occur either in a vertieiHate 



at least is the 



^yriophyllum. 



limb. 



y OL. XXVI. 



srenera 



Galiacea?, Equisetaceae ; Hippuns : 



imagine a more scattered condition favourable 



4 Y 



