FLOEAL SYMMETRY, ETC. 127 



sion, eight leaves being required to complete tlie 

 cycle. Here you have eight perpendicular rows of 

 leaves, with an angular divergence of f the circum- 

 ference of the stem ; it is, therefore, called the f ar- 

 rangement. 



In some plants the leaf-cycle includes five turns 

 of the spiral and thirteen leaves, so that the four- 

 teenth is placed over the first. This is the -% ar- 

 rangement. There are also the / T , the ^-f- arrange- 

 ments, and so on. But these more complex modes 

 are only found where leaves grow in rosettes, as the 

 houseleek, or in the case of crowded radical 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 

 simple observation. 



There is a curious feature of the fractions express- 

 ing the angular divergence of leaves. Observe that 

 any one of the fractions of the series is the sum of 

 the two preceding simpler ones. For example, the 

 angles of divergence in Figs. 268 and 269 are % and 

 J-. Adding these numerators and these denomina- 

 tors, we have f , the pentastichous, or next more com- 

 plex arrangement. By adding, in the same way, -J 

 and f , we get f, while f and f give -f^, and so on. 



The , , and -f- modes of arrangement are so defi- 

 nite 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 length- 

 ening of internodes, or their absence altogether, ren- 

 ders observation difficult, and the decision uncertain. 

 So, when commencing the study of leaf-arrangement, 



