394 MEMOIRS OF THE AMERICAN ACADEMY. 



occur in nature, or are deduced from theory. The theoretical source of many of 

 them, of more than half, is more than probable; for the limit of the de- 

 nominations assumed for the table is undoubtedly beyond the limits of distin- 

 guishable forms in actual measurements. If, however, this assumed limit had been 

 less, the number of the other fractions would have been reduced in greater pro- 

 portion than the phyllotactic ones. If it had been greater, they would have been 

 increased in a greater proportion. In other words, the ratio of the number of 

 fractions given by the theory of Phyllotaxy to all other possible ones within 

 the same range of denominations decreases from a very large value (nearly the 

 whole) to smaller and smaller values. This ratio, therefore, is a fact of no im- 



portance as a fact of observ 



almost wholly a formal fact 



material one, as the logicians say ; or is involved in the form of expression, 

 or of representation, or in the nature of the method of investigation. The 

 important fact is that there are fractions, however few, which would be dis- 



tinguishable if they existed in nature, but are not found, though their magni 

 tudes are within the range of those that do exist. Such are the fractions 

 and f. If the fractions of our table were arranged, not only in the order of 

 their magnitudes, but at corresponding distances, and if we disregarded altogether 

 the character of simplicity or complexity in these fractions, and the numbers of 

 them within any limit of denomination, and considered only the ranges of geo- 

 metrical values between them, we should find between £ and f a difference of 

 T \ = 0.100 of the circumference, which is greater than the whole range of the 

 other fractions of Phyllotaxy in the first group, namely, T ^=rr|_-| — 0.067 of 

 the circumference; and between the last of these and the first fraction of the 

 second group we have the difference f — f = ^ T = 0.047 of the circumference, 



\ 



which is not much less than the range T V In these two spaces, therefore, of 

 A and 2 1> tner e would be room for fractional intervals as distinguishable from 

 each other as those of the first group are; though, in the space A, or 0.047, 



betw 



sen the first and second groups, no simple fractions, or of less denomination 

 than T V, could occur, and no interval has been observed which belongs to 

 either of these spaces. It is therefore sufficiently obvious that the fractions f and 

 * (which would, perhaps, be with difficulty distinguished from each other, since they 

 differ by only &) are real omissions from natural arrangements. T % and -^ 

 would be really indistinguishable from each other if they existed ; but the for 

 mer could be as readily distinguished from any real arrangements as these are 

 from one another. It ought, therefore, to be regarded as also a real exception, 



