PHYLLOTAXY. 85 



than tlie number of the secondary spirals which follow an opposite direction. The 

 same result can be obtained from the succeeding fractions. 



On the contrary (fig. 459 e), with the fractions f, ^5_, if, and so on, we find that 

 the right-hand radius is occupied by a leaf sooner than the left-hand one, and that 

 in consequence the number of the first leaf on the right-hand radius is less than the 

 number of the first leaf on the left-hand radius. Therefore the number of secondary 

 spirals which can be followed from left to right is less than those from right to left, 

 or, in other words, the most numerous secondary spirals turn in the same direction 

 as the generating spiral, and knowing the direction of the one, we know the direc- 

 tion of the other. 



The direction of the generating spiral varies not merely in the individuals of a 

 species, but sometimes in the same individual. Thus, in cones from the same speci- 

 men of Maritime Pine, right-to-left secondary spirals will be more frequent in some, 

 and left-to-right in others ; but in all cases the relative direction of the generating 

 spiral follows the law just enunciated. 



The angle of divergence itself is constant only in the fractions i, |, f , and 

 when these cycles are more numerous, the one is often substituted for the other, 

 which is owing to the distance between them being extremely small, and to the fact 

 that the angles expressed by the fractions ^-^, f^, if, |J-, |^, &c., if reduced to 

 degrees and minutes, differ by a few minutes only ; so that the angles of divergence 

 actually oscillate between 137° and 138°. A slight twist of the stem or axis is 

 sufficient to account for so small a variation, and may well occur in rosettes of 

 leaves, in involucral bracts, and in cones, and cast a doubt on the value of the angle 

 of divergence. Thus, in Pines (fig. 459 (^, the rectilinear series indicating the suc- 

 cessive cycles may deviate more or less to right or left, so that the secondary spirals, 

 which were the most obvious at the base of the cone, become less so in ascending, 

 and render it difficult to determine such fractions as |-, ■^, -£-j-. A change in the 

 shape of the stem will also lead to the substitution of one cycle for another, as in 

 certain Gacti with ribbed or angular stems bearing tufts of prickles, and whose ribs 

 double as they ascend, and offer cycles of a higher number. 



Lastly, there are exceptional cases which perplex the student of Phyllotaxy ; 

 the above-named fractions are not the only ones which may be observed ; f , ^, f , -^^, 

 &c., do occur, though very rarely ; but when they do, they preserve among them- 

 selves the same relations as the preceding, i.e. that each successive fraction may be 

 obtained by the addition of the numerators and denominators of the two preceding. 

 We have seen that whorled leaves present a succession of circular groups ; but here 

 also, as in alternate leaves, the spiral arrangement is discernible. In a branch of 

 Oleander, for instance, where the leaves are whorled in threes, a relation exists 

 between any three vertically superimposed leaves of successive whorls ; and a line 

 successively passing through their insertions will describe a regular spiral ; and if 

 we examine the relations between the other leaves of these whorls, we shall perceive 

 that the number of whorls represents as many parallel spirals as there are leaves in 

 each of them. 



