ORIGIN OF THE ARRANGEMENT OF LEAVB IN PLA TS. 3i<3 



this subinterval is equal to the smaller primary one; the third leaf la 111 ex* \y 

 in the middle of the larger interval and completes the .vole. This fraction 

 thus next in cyclic simplicity to £, and has but Utile more of the distributive 

 character. Following £ we find two fractions, T » ff and T ^, which > semble it in 

 this respect as T ^ and T 6 T resemble J, and lacking like them tho cyclic -implicit 

 of their type they have still no superiority to it in the distributive property 

 The same is true in diminishing degrees of the following fractions. Hut these 



onward to the end are, with one intrusive 

 two lower series, or those of infrequent o 



pliyllotactic fractions of 



l nature. They includ 



the fractions peculiar to these series, that is, all but f , just as the group abo\ 

 includes all of the first series except £; and th se two fractions have tie let 

 of the distributive character and most of cyclic simplicity. They are the fiac* 



tions of the smallest denominations, and might properly be separated from tin 

 others as a special type; with more propriety, indeed, than the purely distribu- 

 tive fraction k could be separated from others of the first group. In all the 

 fractions of the lower group the disparity 'of the primary intervals, or tho gn it 

 difference between these fractions and their complements (the difl rences exceed- 

 ing these complements, and the ratios being proper fractions), unfit them for a 

 distributive arrangement, so far at least as the earlier st ps of the cycle or 

 the first three leaves are concerned. In other words, the primary intervals 

 being in greater ratios than two to one, the distribution i~ imperfect at the 

 outset. But it is better in subsequent leaves, for all those fractions that an 

 found in nature, or for all but the one intrusive exception I have referred to. 

 This exception is the fraction Jf, which, as well a< ,\ in the fn t -roup, - ems 

 at first sight a remarkable anomaly. They are not, as we shall see, anomalies at 



but, though differing from the other fractions of these groups in th 



of 



distribution higher up in the cycles than the steps we have yet con k red, 

 namely, the first three leaves, yet they occur thus isolated in these groups only 

 on account of the arbitrary limit I have assumed for the denomination, of the 

 fractions in the table, or the limit of 13ths. If fractions of higher denominations 

 had been included in the table, other exceptional fractions would have appeared 

 within the limits of these groups. But before proceeding to show this, I will 

 call attention to one other fact shown by this table, namely, how large within 

 the limits assumed for the table the number of pliyllotactic fractions is, compered 

 to the whole number, namely, more than half. Out of the nineteen pos.ble 

 proper fractions given in the table, ten are pliyllotactic; that is, either actually 



