

THE USES AND ORIGIN OF THE ARRANGEMENT OF LEAVES IN FLANTS. 395 



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though it may have been omitted not only on account of its defective character 

 as a distributive fraction, but also for its lack of simplicity. Taking account of 

 the relative frequency of the fractions that are found in nature, we have 

 sufficient grounds to suppose that the distributive character of them is an 

 utility of actual importance to the welfare of plants. 



But I have not yet shown this distributive character throughout the cycle, 

 or what distinguishes them from the abnormal forms I have referred to. I 

 must first show, however, that these are only apparent anomalies, and that others 

 of the same character would occur if the table were extended. There is a 

 ready means of extending the table; the same law in fact which holds in the 

 series given above. Each value of the proper fractions, as arranged in the 

 second column, except the extreme ones, can, it will be seen, be derived from 

 the preceding and following ones by adding their numerators for a new numer- 

 ator, and their denominators for a new denominator, and, in some cases, reducing 

 the fraction thus obtained to lower terms. Moreover, in the table as it stands, 

 the difference of any two successive fractions is the reciprocal of the product 

 of their denominators, or when reduced to this as a common denominator, 

 thek numerators differ by a unit. From this property it follows in the theory 

 of numbers that intermediate values obtained in this way cannot be reduced to 

 lower terms, and are the fractions of the smallest denomination intermediate in 

 value between any two. It is obvious that by this process the table could be 

 extended indefinitely, without omitting any fraction of less than any assignable 

 denomination. Indeed, it could have been constructed from its limits by suc- 

 cessive interpolations of this sort. Thus taking the extreme limits | and 1, or, 

 as we may express the latter, {, which differ by J, or the reciprocal of the 

 product of their denominators, we obtain between them by this process §. Be- 

 tween J and | we find f . Between f and \ we find f . Continuing this process, 

 as in the subjoined example, we arrive at the results in the lower line, which 

 are, many of them, of higher denomination than those of the table. 



i 



2 



3 



* 4 



* iV iV IT TJ 



-i - 6 - ^ - 9 - 4 1 1 V 10 8 11 J IS 5 12 _7 9 ,| 9 « 

 2' 115 9? 1 6>y?195T2>TY? IJUHJ'2 l»tf»l 9» 1 1>1 4>3> 13>1 



7 JL .1 4 



6 



1 



By interpolating one more value between the first two of these fractions, 01 

 between } and T \, namely, T V, and by omitting those of higher denominations 

 we obtain all the proper fractions of our table. Among these higher denomi 



