ARRANGEMENT OF LEA VES. 



151 



IV 



(I.) 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 larico, and the interfloral 

 bracts of the inflorescence of Jttidbeckia. 



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

 piues and firs, in many Mdinillaricv, etc. 



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

 strong-grown flower-heads of HeMu-nthus 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 

 fraction which indicates the angular di- mem. 

 vergence of the leaves from each other. and bottom in Roman 



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

 astichies}* crossing each other at an acute 

 angle may be observed on the stem when 

 the leaves are close together, as in Fig. rranti. 

 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 (generating 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 



