70 



l^SECTION 7. 



situated directly over any below (Fig. 18'J). Here the sixth leaf is over 

 the first; the leaves staud in five perpendicular ranks, with equal angular 

 distance from each other; and this distance between any two successive 

 leaves is just two fifths of the circumference of the stem. 



189. The five-ranked arrangement is expressed by the fraction |. Jhis 

 fraction denotes the divergence of 

 the successive leaves, i. e. the an- 

 gle they form with each other : the 

 'numerator also expresses the num- 

 ber of turns made round the stem 

 by the spiral line in completing 

 one cycle or set of leaves, namely, 

 two; and the denominator gives 

 the number of leaves in each cy- 

 cle, or the number of perpendic- 

 ular ranks, namely, five. In the 

 same way the fraction ^ stands for 

 the two-ranked mode, and J for 

 the three-ianked : and so these 



different sorts are expressed by 

 xue series of fractious •^, ^, |. Other cases follow in 

 the same numerical ])rogressiou, the next being the 



190. Eight-ranked arrangement. In this the ninth /' 

 leaf stands over the first, and three turns are made ill 

 around the stem to reach it ; so it is expressed by v 

 the fraction |. This is seen in the Holly, and in the 

 common Plantain. Then comes the 



191. Thirteen-ranked arrangement, in which the ^^ 

 fourteenth leaf is over the first, after five turns around the stem, 

 common Honseleek (Fig. 191) is a good example. 



192. The series so far, then, is ^, ^, |, |, ^^; the numerator and the 

 denominator of each fraction being those of tlie two next preceding ones 

 added together. At this rate the next higher should be -^j, then Jf , and 

 so on : and in fact just such cases are met with, and (commonly) no others. 

 These higher sorts are found in the Pine Family, both in the leaves and 

 the cones and in many other plants with small and crowded leaves. But 

 in those the number of the ranks, or of leaves in each cycle, can only rarely 



The 



Fig. 188. Shoot with its leave.s 5-rauked, the sixth leaf over the first ; as in the 

 Apple-tree. 



Fig. 189. Diagram of this arrangement, with a spiral line drawn from the attach, 

 ment of one leaf to the next, and so on ; the parts on the side turned from the eye 

 are fainter. 



Fig. 190. A ground-plan of the same; the section of the leaves similarly num- 

 bered ; a dotted line drawn from the edge of one leaf to that of the next marks out 

 the spiral. 



