392 



MEMOIRS OP THE AMERICAN ACADEMY 



2 



f 



3 

 "5 

 4 



4 



T 



5 



T 



5 



5. 



9 

 1 

 9 



10 



J5_ 

 1 1 



_7_ 



1 1 



Ji- 

 ll 



JL 



12 



7 

 13 



JL 

 13 



1 

 IS 



k 



.500 



.667 



.750 



.600 

 .800 



.571 

 .714 



.625 



.556 



.778 



.700 



.545 



.636 



.727 



.583 



.538 



.615 

 .692 



.769 

 .618 



.500 



.000 



CO 



.462 



.076 



6 



.455 



.091 



5 



.444 



.111 



4 



.429 



.143 



3 



.417 



.167 



2J 



.400 



.385 

 .382 



.375 

 .364 



.333 



.308 



.300 



.286 



.273 

 .250 

 .231 



.222 



.200 



.200 

 .231 

 .236 



.250 



.272 



.333 



.384 

 .400 

 .428 

 .454 



.500 

 .538 

 .556 

 .600 



2 



1^ 



i 3 



* 



1 



4 



3 



2 

 3 



1 



1 



2 

 8 



Y 



2 



The first point to be noticed in this table is the character of the ratios in the 



* 



last column. For all the fractions 

 whose complements are greater than 

 preceding, or in the smaller 

 twice. 



5 



(or 

 the first subinterval is contained in the 



here given of less magnitude than ■§, 



To the interval i or in 



i 



of the primary intervals, several times, or more than 



secondary 



the alternate arrangement there is no 



al 



* 



The cycle is completed at once, and no distribution is effected, except 



The next following fractions -J* and 



the simple opposition of successive leaves. 



6 



TT 



hav 



a similar character 



;pect to the property of distribution : that is, th 



subinterval introduced by the third leaf would 



* 



respectively 6 and 5 times smaller than the 

 have not the < 



sm 



these, be very 

 primary interva 



B 



being 

 they 



ycl 



simplicity of the alternate system, and thus lack whatever 



advantage belongs to it. The same is true in diminishing degrees of the follow 



ing fractions, until we arrive at f ; and none of these occur in nature, 

 first in order of magnitude afte 



is 



the 



lowing 

 fractioi 



r ^ which is found in nature, and immediately fol 



i 



2 



find all the 



phyllotactic fractions of the first 



or all the 



that are of common occurrence in 



eludes these, and also one intruder 

 range for these fractions from 2 to 



pt 



The bract 



namely 



i i- 



Th 



ratios in the 



1 ; indicating that for all these the sub 



terval introduced by the third leaf, though less th 



contained 



the smaller primary 



it more than twice. I 



the last fraction of this series, f> 





