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JOURNAL OF THE ROYAL HORTICULTURAL SOCIETY. 



between its component fractions. It will be as follows : J> h !> f > &c. ; 

 such I have elsewhere* proposed to call the primary series. It cannot 

 fail to be noticed that the sum of any two successive numerators, or of 

 any two successive denominators, forms that of the next fraction 

 respectively, so that we might extend this series indefinitely ; thus : \, 

 h h |i Ai -iu H> th Hi &c. It will be also observed that the 

 numerator of any fraction is the same number as the denominator next 

 but one preceding it. There yet remains one more remarkable connection 

 between them, viz. that these fractions are the successive convergents 

 of the continued fraction 



1 



2 + 1 

 1 + 1 



1+ &e. 



That is to say, if we reduce, by the ordinary rules for simplifying fractions, 

 the portions 



i i i 



8*2+1' 2~+l' &C ' 

 1 1 + 1 



1 



and so on, the resulting fractions will be the same as those given above. 



I have said that the above series of fractions represent the arrange- 

 ments which exist in nature, and it is not usual to find any species 

 departing from the arrangement which may be characteristic of it ; in 

 other words, the phyllotaxis of any species is, as a rule, constant to that 

 species. The following are illustrations : — 



\. Iris, or Flag. The glumes (chaff) of all grasses. Some " orchids." 



^. Carcx, or Sedge. Leaves of several grasses. 



£. Oak, Hawthorn. This is one of the commonest arrangements. 



|. Yew, Greater Plantain. A common arrangement amongst mosses. 

 Fruits of Pineapple and many Fir-cones. 



ff 8 j . Scales of Spruce fir-cones. 



If, now, a semicircle be described, and one extremity of its diameter 

 represent the position of any leaf, assumed as the first (in the diagram), 

 and if a radius be drawn at the angular distance of 120° from this point, 

 then the point where the radius meets the circumference will be the 

 position of the second leaf of the tristichous arrangement. The opposite 

 extremity of the diameter will be that of the second leaf of the distichous 

 arrangement. And these points form the extreme positions for the second 

 leaves of spirals of the primary series, corresponding to the fractions J 

 and \ respectively. No second leaf ever lies nearer to the first than 120°, 

 nor further than 180°.t The positions of all the second leaves are upon 

 the arc included between those extreme points (viz. 120 and 180 degrees 

 from the extremity of the diameter corresponding to the position of the 



* " On the Variations of the Angular Divergences of the Leaves of Helianthus 

 HlberoBUi" by the Rev. George Henslow, Transactions of the Linnean Society, 

 vol. xxxi. p. 647. 



t If the second leaf be at a greater distance than 180, and not less than 240 

 degrees from the first, it will be seen that the conditions are simply reversed, and the 

 spiral will then run round in the opposite direction. 



