144 THE LEAVES. 



make a horizontal plan, we have the parts disposed as in the dia- 

 gram, Fig. 173, the lower leaves being of course the exterior. 



240. The eight-ranked arrangement, the next in order, is like- 

 wise not uncommon. 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. In 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 angular divergence of 

 f of the circumference. 



241. 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 re- 

 lations of the various methods brought to light, by sagacious recent 

 researches, principally those of Schimper and Braun. If we write 

 down in order the series of fractions which represent the simpler 

 forms of phyllotaxis already noticed, as determined by observation, 

 viz. , , f , f , 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 de- 

 nominator of the sum of the two preceding denominators. (Also 

 the numerator of each fraction is the denominator of the next but 

 one preceding.) We may carry out the series by applying this 

 simple law, when we obtain the further terms, y 5 ^, F 8 T , j, f |, &c. 

 Now these numbers are those which are actually verified by obser- 

 vation, and, with some abnormal exceptions, this series comprises 

 all the cases that occur. These higher forms are the most common 

 where the leaves are crowded on the stem, as in the rosettes of the 



Houseleek (Fig. 174), and the scales of Pine- 

 cones (for the arrangement extends to all parts 

 that are modifications of leaves), or where they 

 are numerous and small in proportion to the cir- 

 cumference of the stem, as the leaves of Firs, 

 &c. In fact, when the internodes are long and 

 the base of the leaves large in proportion to the 

 size of the stem, it is difficult, and often impossi- 

 ble to tell whether the 8th, 13lh, or 21st leaf 

 stands exactly over the first. When, on the other hand, the inter- 



FIG. 174. An offset of the Houseleek, with the rosette of leaves unexpanded, exhibiting the 

 5-13 arrangement; the fourteenth leaf being directly over the first. 



