PHYLLOTAXIS. 109 



159. Even if the leaves are placed singly and apparently 

 irregularly at intervals along the stem, it will be found on 

 examination that their arrangement is governed by definite 

 laws. Take, for instance, a branch of Poplar with a 

 number of leaves upon it. Fix upon any one leaf near 

 the lower end of the branch, and then from its point of 

 insertion draw a line, ly the nearest way, to the insertion 

 of the next higher leaf, and from this to the next, and so 

 on till you reach a leaf which is exactly over the first one- 

 If the branch itself has not been twisted out of its normal 

 shape, it will be found that the sixth leaf is always 

 precisely over the first, the seventh over the second, the 

 eighth over the third, and so on, and that the line joining 

 the points of insertion of successive leaves forms a spiral 

 round the stem. It will also be found that this spiral 

 goes twice round the stem before passing through the 

 sixth leaf. The sixth leaf, as standing exactly over the 

 first, begins a new set, which lasts in a similar manner till 

 we reach the eleventh. The leaves are therefore in sets or 

 cycles of five each, and the phyllotaxis in this case is 

 conveniently described by the fraction f , the denominator 

 of which gives the number of leaves in the cycle, and the 

 numerator the number of turns in the spiral. 



1 60. N ow,if through the insertions of the leaves which are 

 vertically over each other that is, through those numbered 

 1, 6, 11, 16, etc., and then through those numbered 2, 7, 

 12, 17, and so on lines be drawn, it is evident we shall 

 have five such vertical lines on the stem. These lines 

 mark the ranks of leaves, or orthostichies. The number 

 of orthostichies in any case always corresponds to the 

 number of leaves in the cycle. 



