386 ' MEMOIRS OF THE AMERICAN ACADEMY. 



arrangements, and the more frequent occurrence of some of them than of others. 

 It is a well-known property of the fractions of these series, that after the first 

 two in each, the others can be deduced from the preceding ones, and continued 

 indefinitely, by a very simple process. The numerator of each after the first two 

 is equal to the sum of the numerators of the two preceding, and its denominator 

 to the sum of their denominators. This law, as a matter of observation, was 

 actually discovered only in the first four fractions of the first or second series, 

 Avhich are by far the commonest of actually observed arrangements in nature. 

 Other less frequently occurring fractions were arranged on the same principle, 

 and extended so as to give the last two series. The four series, or the three 

 lower ones, contain, therefore, more than all the fractions that are known to 

 belong to natural arrangements. This will be sufficiently evident when we observe 

 that the fractions f and T 8 3 in the first series, or their complements, f and T \, in 

 the second series, would be indistinguishable in actual measurement; since they 

 differ from each other by T ^j, or by less than a hundredth, which is much less 

 than can be observed, or than stems are often twisted by irregular growth. 

 For the same reason we must reject all but the first three terms of the third 



and fourth series as being distinguishable only in theory. We are thus left 

 with a very slight basis of facts on which to erect the superstructure of theory. 

 We shall see further on a still more cogent reason for calling in question the 

 validity of this induction ; namely, that limiting the evidence as we are thus 

 obliged to do, we have still left so large a number of actually observed ar- 

 rangements, that they include almost all that are possible among equally simple 

 and distinguishable fractions within the observed limits of natural arrangements; 



all, in fact, but two ; namely, the fractions -J and f . The range is not a nar- 



row one, but extends from \ to |, or from \ to ^, since the fractions above J 

 are complements of those below, and express the same arrangements, but in an 

 opposite direction around the circumference. The problem of Phyllotaxy, there- 

 fore, seems at first sight to be reduced to this ; not why the other fractions 

 do occur in nature, but why these two do not? But to answer the latter 

 question is really also to answer the former, though it will go but very little 

 way towards justifying the theory of the typical or unique angle. It will go 

 much further if we exclude from this list of fractions those which are of very 

 infrequent occurrence, namely, those peculiar to the third and fourth series; or, 

 in other words, take account of the relative frequency in nature of the several 



o 



ements. This, indeed, entirely changes the aspects of the question, for 



