ARRANGEMENT OF LEA VES. 



151 



(1.) 21-ranked in the weak branches of Abies pectinata and Picea 

 excelsa, and in most cones of these species. 



(m.) 34-ranked on strong branches of Abies pectinata and Picea 

 excelsa, cones of Pinus laricio, and the interfloral 

 bracts of the inflorescence of Hudbeckia. 



(n.) 55-ranked in the uppermost shoots of many 

 pines and firs, in many Mamillarice, etc. 



(0.) 144-ranked in the interfloral bracts of 

 strong-grown flower-heads of Helianthus annuus. 



199. By an examination of various 

 leaf-arrangements, the following interest- 

 ing but not very important facts may be 

 noted (Fig. 129) : 



(1.) If we draw a line from the inser- 

 tion of one leaf to the one next above and 

 nearest to it, and continue this around the 

 stem to the next, and so on, a spiral will 

 be obtained agreeing with the order of 

 development of the young leaves on the 

 punctum vegetationis. To this line, so 

 drawn, the name of Generating Spiral 

 has been given. 



(2.) In most cases the spiral passes more 

 than once around the stem before inter- 

 secting leaves of all the ranks. 



(3.) The number of turns of the spiral 

 about the stem in intersecting leaves of 

 all the ranks equals the numerator of the rig. 139. Diagram of 



... i i j- J.I. i T eight - ranked arransje- 



fraction which indicates the angular di- ment. Theortiu 

 vergence of the leaves from each other. and "bottom fn 



(4.) Two sets of secondary spirals (Par- 

 astichies)* crossing each other at an acute 

 angle may be observed on the stem when toi ng u p w*ar? f - Aife 

 the leaves are close together, as in Fig. Pranti. ; 



129 ; the leaves numbered 1, 6, 11, and 16 form one of the 



* It is of great importance that the student should not regard these 

 spirals (genernting spirals and parastichies) as anything more than 

 convenient means for describing any particular leaf-arrangement. En- 

 tirely too much attention has been given to working out all kinds of curi- 

 ous mathematical laws, which are, to say the least, absolutely worthless 



