:<l[ BOTANY OF THE LIVING PLANT 



folio* the arrangement of the leaves. This still further defines the 

 ial form of the mature organism. 



Very striking phenomena of spiral leaf-arrangement are afforded by many 

 p] tnt-shoota This- is the after-effect of rhythmic sequence in leaf-formation 

 and ia Been most clearly in transverse sections at the level of the bud-apex, as 

 in Fifl 136. Such patterns present systems of intersecting curves, which may 

 be expressed numerically ; the spiral effect being due to the fact that the 

 numbers as counted in different directions, left or right, are unequal, as, for 

 instance, 7 : ii, 5 : 8, 3 : 5. When the numbers are equal the system is whorled. 



Fig. 13;. 



Accurate drawing of a shoot of Rhododendron, seen from above. The successive 

 leaves are numbered, and it is seen that the ninth is covered by the first, while an 

 imaginary spiral including all their bases successively, will have encircled the stem 

 thriii-. That is, the angle of divergence between any two successive leaves is f. 

 (Reduced to J.) 



The fact that the numbers commonly observed at the shoot-apex, or residual 

 in older constructions, are in the great majority of cases successive numerals 

 of the Fibonacci ratio-series, 1 : 2 : 3 : 5 : 8 : 13, etc. (other ratios being 

 exceptional), has been taken to imply that the arrangement, as also working 

 out in terms of angular divergence of successive members as the Fibonacci 

 angle of approximately 137^, gives the optimum effect in distributing the 

 laminae to incident light (Wiesner) ; but though the mechanism may 

 be effective in such a direction it does not necessarily follow that this is the 

 1 sal f.ictor of the phenomenon (Church). 



The different types of spiral seen in the mature shoot may be designated 

 according to the angle between the median planes of the successive leaves. 

 This is called the angle of divergence, and it may be expressed as a fraction of 



