136 



THE LEAVES. 



a stem. If we shorten the axis, as it Avas in the bud, or make a 



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 Holly, the Callistemon 

 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. All these modes, or nearly all of them, 

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

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

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

 Schiniper, Braun, and others. If we Avrite down in order the series 

 of fractions which represent the simpler forms of phyUotaxis already 

 noticed, as determined by observation, viz. ^, \, f , |, we can hai-dly 

 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 denominatoi's. Also the numerator of each fraction is 

 the denominator of the next but one preceding. Extending this 

 series, we obtain the further terms, y^r, j\, -^f , f -g^, &c. Now these 

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

 tion^, this series g-, ^, f , f , y^^, Jy, ^|, f ^, comprises all the varia- 



FI6. 204. A branch exhibiting the five-ranked arrangement of leaves. 



FIG. 205. Diagram of the same : a spiral line is tlrawn 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 directly before, or within, the first. 



