396 MEMOIRS OF THE AMERICAN ACADEMY. 



nations we find several new intruders in the phyllotactic groups, namely, f] 



18' 



J J, and j\ in the first, and j| in the second, as well as new fractions of the 

 series, namely, -J} of the first, ff o( the second, and }J of the third series, 

 which our limits had excluded from the table; and several others beyond these 

 groups. If, on the other hand, we make the denominations of our table smaller, 

 and exclude all above 9ths, as being either actually indistinguishable, or with 

 difficulty distinguished from the remaining ones, we have remaining J, |, ^ 3 



, f, f, £, |, all of which are included in observed arrangements, except the 

 two, ;"; and |. Our problem would therefore, as I have said, appear to be why 



these have been excluded from natural arrangements, rather than why the 

 others have been adopted or preserved. But the problem is more correctly as 

 follows : Account ought to be taken of the relative frequency of them, or 

 weights ought to be attached to them according to this consideration. The 

 weight of these two fractions would then be nothing. The weights of several 



. -.^~ "v & 



others would be very small. Now, what has determined these weights ? This 

 is our problem. The superior distributive character of the more frequently 

 occurring ones, is the only conceivable answer. We see from this, however, that 

 we ought not to attribute to Natural Selection the existence of the spiral 

 arrangements in general, at least not on account of the distributive property 

 we have considered, for, in fact, they include almost all possible intervals, as the 

 arrangements of the whorl do, and little selection is shown in them indepen- 

 dently of their relative frequency. This relative frequency, or infrequency, in 

 nature, amounting to total exclusion in the case of these two fractions, is. then. 



the only way in which Natural Selection could have been concerned in pro- 

 ducing or modifying the spiral arrangements, so far as that property of distri- 

 bution is concerned which is exhibited most perfectly by the typical or unique 

 angle of the theory of Phyllotaxy. But, supposing these arrangements to have 



come into existence through some other agency, or by Natural Selection acting 

 on some other ground of utility, or under some other phase of this one, we 

 then see sufficient reasons why, on this principle, they should be what they are. 

 We might exclude altogether from our consideration the intrusive angle in 

 our table, T ' T , as being a purely theoretical product, and indistinguishable in 

 nature from the simpler fraction above it, f; but I will include it in th 



further discussion of this group, for the sake of showing that it would, probably 

 have been excluded from nature, if plants had been more accurately constructed 

 and would not, therefore, be found, even if we had the power to distinguish it 



