DISTRIBUTION OF LEAVES ON THE STEM. 125 



cases actually occur, and ordinarily only these. 1 The -^ and 

 A are not uncommon in foliage. The rosettes of the House- 

 leek exhibit the ^ or thirteen-ranked arrangement, as also does 

 the cone of Pinus Strobus, the 14th leaf falling over the first. 

 (Fig. 246.) The & is perhaps little less common in foliage 

 upon very short internodes, as likewise are higher ranked 

 numbers ; and in many pine-cones and similar structures ^f 

 and f phyllotaxy may be readily made out. This actual series, 

 , , f , |, &c., answers to and may be expressed by the con- 

 tinued fraction, % + \ 



t + i + |, &c. 2 



1 When other instances are detected, they are found to belong to other 

 series, following the same law, such as the rare one of \, \, $ , T \. 



2 " The ultimate values of these continued fractions extended infinitely 

 are complements of each other, as their successive approximations are, and are 

 in effect the same fraction, namely, the irrational or incommeasurate inter- 

 val which is supposed to be the perfect form of the spiral arrangement. 

 This does, in fact, possess in a higher degree than any rational fraction the 

 property common to those which have been observed hi nature; though 

 practically, or so far as observation can go, this higher degree is a mere 

 refinement of theory. For, as we shall find, the typical irrational inter- 

 val differs from that of the fraction f by almost exactly \^6> a quantity 

 much less than can be observed in the actual angles of leaf -arrangements." 

 "On this peculiar arithmetical property .... depends the geometrical one, 

 of the spiral arrangement, which it represents ; namely, that such an arrange- 

 ment would effect the most thorough and rapid distribution of the leaves 

 around the stem, each new or higher leaf falling over the angular space be- 

 tween the two older ones which are nearest in direction, so as to subdivide it in 

 the same ratio in which the first two, or any two successive ones, divide the 

 circumference. But, according to such an arrangement, no leaf would ever 

 fall exactly over any other ; and, as I have said, we have no evidence, and 

 could have none, that this arrangement actually exists in nature. To realize 

 simply and purely the property of the most thorough distribution, the most 

 complete exposure of light and air around the stem, and the most ample 

 elbow-room, or space for expansion in the bud, is to realize a property that 

 exists separately only in abstraction, like a line without breadth. Neverthe- 

 less, practically, and so far as observation can go, we find that the fractions 

 | and -fa, ^ T , &c., which are all indistinguishable as measured values in the 

 plant, do actually realize this property with all needful accuracy. Thus, 

 f = 0.375, & = 0.385, and & == 0.381, and differ from k [the ultimate value 

 to which the fractions of this series approximate, or what is supposed to be 

 the type-form of them] by 0.007, +0.003, and 0.001 respectively ; or they 

 all differ by inappreciable values from the quantity which might therefore be 

 made to stand for all of them. But, in putting k for all the values of the 

 series after the first three, it should be with the understanding that it is not 

 so employed in its capacity as the grand type, or source of the distributive 

 character which they have, in its capacity as an irrational fraction, but 

 simply as being indistinguishable practically from those rational ones." 

 Chaucey Wright, in Mem. Amer. Acad. ix. 387-390. 



