136 



THE LEAVES. 



a Stem. 



If we shorten the axis, as it was in the bud, or make a 

 204 horizontal plan, we have 



the parts disposed as in the 

 diagram, Fig. 206, the low- 

 er leaves being of course 

 the exterior. 



241. The eight-ranked 

 arrangement, the next in 

 order, is likewise not un- 

 common. It is found in 

 the HoUy, the CalUstemon 

 of our conservatories, the 

 Aconite, the tuft of leaves 

 at the base of the common 

 Plantain, &c. Li this case the ninth leaf is 

 placed over the first, the tenth over the second, 

 and so on ; and the spiral line makes three turns 

 in laying down the cycle of eight leaves, each 

 separated from the preceding by an arc, or an- 

 gular divergence, of f of the circumference. 



242. AU these modes, or nearly all of them, 20s 



were pointed out by Bonnet as long ago as the middle of the last 

 century ; but they have recently been extended and generahzed, and 

 the mutual relations of the various methods brought to light, by 

 Schimper, Braun, and others. If we write down in order the series 

 of fractions wliich represent the simpler forms of phyllotaxis abeady 

 noticed, as determined by observation, viz. ^, ^, §, -|, we can hardly 

 fail to perceive the relation that they bear to each other. For the 

 numerator of each is composed of the sum of the numerators of the 

 two preceding fractions, and the denominator of the sum of the 

 two preceding denommators. Also the numerator of each fraction is 

 the denominator of the next but one preceding. Extending this 

 series, we obtain the further terms, -f^, ^j, ^J, §-^, &c. Now these 

 numbers are verified by observation, and, with some abnormal excep- 

 tions, this series J, J, f , f, J3-, /y, ^f , f i, comprises all the varia- 



riG. 204. A branch exbibiting the five-ranked arrangement of leaves. 



FIG. 205. Diagram of the same : a spiral line is drawn ascending the stem and passing 

 through the successive scars which mark the position of the leaves from 1 to 6. It is made a 

 dotted line where it passes on the opposite side of the stem, and the scars 2 and 5, which fall 

 on that side, are made fainter. 206. A plane, horizontal projection of the same ; the dotted 

 line passing from the edge of the first leaf to the second, and so on to the fifth leaf, which 

 completes the cycle ; as the sixth would come directlj before, or within, the first. 



