PHYLLOTAXY. 131 



over tiie first. This is the -^ arrangement. There are 

 also the -/ f , the f arrangements, and so on. But these 

 more complex modes are only found where leaves grow in 

 rosettes, as the house-leek, or in the case of crowded radi- 

 cal leaves, or in the scales of cones. In these cases the 

 vertical rows are not distinguishable, and the order has to 

 be made out by processes of reasoning rather than by sim- 

 ple observation. 



There is a curious feature of the fractions expressing 

 the angular divergence of leaves. Observe that any one 

 of the fractions of the series is the sum of the two pre- 

 ceding simpler ones. For example, the angles of diver- 

 gence in Figs. 393 and 394 are ^ and -J. Adding these 

 numerators and these denominators, we have -f, the pen- 

 tastichous, or next more complex arrangement. By add- 

 ing in the same way J and f, we get f, while f and f give 

 y 5 ^-, and so on. 



The J, -J, and f modes of arrangement are so definite 

 and simple as to be easily discovered ; but it is not worth 

 while, ordinarily, to continue the study of a specimen if it 

 does not belong to one of these modes. A slight twisting 

 of the stem, a considerable lengthening of internodes, or 

 theii absence altogether, renders observation difficult, and 

 the decision uncertain. So, when commencing the study 

 of leaf-arrangement, take care to select the straightest and 

 thriftiest stems for the purpose. 



Examine the arrangement of bracts, and see if they 

 follow the same order as leaves. 



Observe whether the spirals take the same direction in 

 branches as in the parent stem. When they do, they are 

 called homodromous ; but when they turn in opposite di- 

 rections, they are said to be heterodromous. 



Give the numbers of the leaves in each perpendicular 

 series in your specimen showing the \ arrangement (Fig. 



393). 



In the arrangement, what leaf stands over the first ? 



