124 PHYLLOTAXY, OR LEAF-ABE ANGEMENT. 



and the sixth is the first to come over any one below ; the 

 seventh comes over the second, the eighth over the third, &c. 

 The leaves are thus brought into five vertical ranks ; but these 

 five leaves are laid down on two turns of the 

 helix (the sixth beginning the second revolu- 

 tion) ; the angular divergence of the leaves in 

 order is f , or 144 ; the angular distance of 

 the vertical ranks, 72. This is a very advan- 

 tageous distribution for ordinary foliage on 

 erect or ascending branches. Its formula is f , expressing the 

 angular divergence, the denominator also indicating the number 

 of vertical ranks, the numer- 

 ator indicating the number 

 of revolutions made in add- 

 ing one leaf to each rank. 

 Fig. 244 illustrates this ar- 

 rangement on a cone of 

 American Larch, the scales 

 of which are homologous 

 with leaves, the numbers 

 in sight are affixed, and those of the whole 

 cone displayed on a plane at the side. 



Octostichous, or Eight-ranked, a less common 

 arrangement, occurs in the Holly, Aconite, 

 the radical leaves of Plantago. It has the 

 angular divergence of 135, or f of the cir- 

 cumference, and the leaves in eight ranks, 

 the ninth over the first and at the completion 

 of the third revolution : it is therefore repre- 

 sented by the fraction f . 



240. The obvious relations of the fractions 

 vi %> $> t> representing the primary forms of 

 spiral phyllotaxy, are that the sum of any two 

 numerators is the numerator of the next suc- 

 ceeding fraction, and the same is true of the 

 denominators ; also the numerator is the same 

 as the denominator of the next but one pro- 

 ceeding fraction. Following these indications, 

 the series may be extended to ^, ^, f , J, &c. Now these 



FIG. 244. A cone of the small-fruited American Larch (Larix Americana), with 

 the scales numbered, exhibiting the five-ranked arrangement. 



FIG. 245. An offset of the Houseleek, exhibiting the 5-13 arrangement; the leaves 

 in sight numbered, the 14th over the first, the 19th over the 6tb, &c. 



FIG. 246. Cone of White Pine (Pinus Strobus) with scales numbered from bottom, 

 and some secondary spirals marked. 



