THE USES AND ORIGIN OF THE ARRANGEMENTS OF LEAVES IN PLANTS. " 397 



This discussion will also show the resemblance of the distributive character of 

 the other arrangements of this group, as we ascend the cycle beyond the third 

 leaf, to the theoretical unique angle. In these, as in this angle k, the succes- 

 sive subintervals are simply the differences of the two preceding ones, continually 

 diminishing, but growing nearer and nearer in value until they become all the 



same aliquot parts of the circumference ; namely, that expressed by the denomi- 



nator of the fraction. This will be best seen in the accompanying diagram. 

 For the numerical illustration of it, let us take each fraction of this group, and 

 its complement; then the difference of these; then the difference of this from 

 the complement, and so on. We have in this way: 



I • 1 • i • i ■ 0- It wil1 be seen ky the (?) in the series for T 7 T , that 



V • A • tV • t% • tW • tV • 0. it violates the law which holds in all the other 



1 



1 



" 5 



3 

 13 



2 

 # 13 



fc 3 . 



k\ 



i 



1 



• 8 



k. & 2 . F. k*. & 5 . & 6 . &c. fractions. The fourth interval, or the second subin- 



¥ • I • t • i • i- 0- terval, is contained three times in the preceding one, 



IT • tt • tt • tt • tt • (•) instead of once, with a smaller remainder, or exactly 

 2 i i a ' J 



3 • 3 



i • o. 



twice without remainder as in the others. It com- 



pletes the cycle finally like the others, but introduces into it at the end of the 

 second and in subsequent turns great inequalities side by side. If these were of 

 sufficient absolute amount to be of importance in nature, we might be sure that 

 such an arrangement would never exist. But a twist of the stem by only one 

 eighth of the circumference in the length covered by eleven leaves, or a twist 

 of one eleventh in the range of eight leaves, would convert this arrangement into 

 the | system, or the f into this. We may see from this illustration how much 

 the mathematical theory of Phyllotaxy has refined upon the facts of observation. 



The property thus exhibited by the first group belongs also to the second 

 group, or the less frequently occurring fractions, but only after the first, revo- 

 lution. The complement of each of these fractions, or the smaller of the pri- 

 mary intervals, is contained more than twice in the larger, or in the fraction 

 itself. . We must, therefore, subtract from these fractions the largest multiple of 

 their complements contained in them for a first difference or subinterval, and 

 then proceed as in the above cases. This gives : 



. f X 2 . 1 . A . 0. Here, as before, the intrusive angle (-}- j) is found to 



tt • tt X 2 . T 2 T . J^ . J T . 0. violate the law which holds for the other fract 



! • i X 2 . i.O. 



15. _3_ v 3 1 2 



13 • 13 A 3 • i^ • T % _ 



* • I X 3. i . i 



beyond the first turn. The first difference or sub- 

 interval is contained three times in the second or 



X 3 . i o th e smaller primary interval. But here, also, as 



t 



ix. 54 



