74 



ARUANGEMKNT OF LEAVES ON THE STEM. [lESSON 10. 



rank?, namely o. In the same Avay the fraction J stands for the 



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



113 sorts are expressed by the series of fractions i, 



•J, |. And tlie otlier cases known follow in the 



same numerical progression. 



18y. 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 g. 

 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 

 Ilouseleek (Fig. 14G). 

 190. The series so far, 

 ^ then, is ^, ^, 1,1,1^-3 ; 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 ^\, then ^§, and 

 so on ; and in fact just such 

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



n\ 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. 



FIO. 143. glinot with its leaves .^-ranked, the sixth leaf ov^r the first ; as in the Apple-tree. 



riG. 111. rtiagram of this arraiigcmont, willi a spiral lino drawn froiri the attachment of 

 one leaf to tlie next, and so on ; the parts on the side tiiriie<l fmni the eye are fainter. 



FIG. 1-15. A frronnil-plan of the same ; the section of tlie leaves similarly nunihered ; a 

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



FIO. lU). A yoiinfr plant of tlie llonscleek, with the leaves (not yet expanded) numbered, 

 and cxhibiling tlio i:i-ranked arrangement 



