74 ARRANGEMENT OF LEAVES ON THE STEM. [LESSON 10. 



ranks, namely 5. In the same way the fraction \ stands for the 

 two-ranked mode, and for the three-ranked : and so these different 

 sorts are expressed by the series of fractions J, 

 , . And the other cases known follow in the 

 same numerical progression. 



189. The next is the eight-ranked arrange- 

 ment, where the ninth leaf stands over the first, 

 and three turns are made around the stem to 

 reach it ; so it is expressed by the fraction f . 

 This is seen in the Holly, and in the common 

 Plantain. Then comes the thirteen-ranked ar- 

 rangement, in which the fourteenth leaf is over 

 the first, after five turns around the stem. Of 

 this we have a good example in the common 

 Houseleek (Fig. 146). 



190. The series so far, 

 then, is , , f , f , T 5 -j ; the 

 numerator and the denomi- 

 nator of each fraction being 

 those of the two next pre- 

 ceding ones added together. 

 At this rate the next higher 

 should be ^ 8 T , then ^f, and 

 so on ; and in fact just such 



cases are met with, and (commonly) no others. 

 These higher sorts are found in the Pine Fam- 

 ily, both in the leaves and the cones (Fig. 324), 

 and in many other plants with small and crowd- 

 ed leaves. But the number of the ranks, or of 

 leaves in each cycle, can here rarely be made 

 out by direct inspection: they may be ascer- 

 tained, however, by certain simple mathematical 

 computations, which are rather too technical for 

 these Lessons. 



FIG. 143. Shoot with its leaves 5-ranked, the sixth leaf over the first ; as in the Apple-tree. 



FIG. 144. Diagram of this arrangement, with a spiral line drawn from the attachment of. 

 one leaf to the next, and so on ; the parts on the side turned from the eye are fainter. 



FIG. 145. A ground-plan of the same ; the section of the leaves similarly numbered-, a 

 dotted line drawn from the edge of one leaf to that of the next completes the spiral. 



FIG. 14(1. A young plant of the Houseleek, with the leaves (not yet expanded"* numbered, 

 and exhibiting the Kt-raukud arraugemei.it 



