THE USES AND ORIGIN 



iVES IN PLANTS. 399 



the first series, or for those most commonly occurring in nature, each leaf of the 

 cycle is so placed over the space between older leaves nearest in direction to it 

 as always to fall near the middle, and never beyond the middle third of the space. 



or by more than one sixth of the space from the middle, until the ryde is com- 

 pleted, when the new leaf is placed exactly over an older one. This property 

 depends mathematically on the character of the continued fractions, of which 

 these fractions are the approximations, according to the theory of Phyllotaxy. 

 The denominators in the characteristic part of the continued fractions, or 

 for the whole in the case of the fractions of the first group, are each a 

 unit plus a fraction, which, at the end, is also a unit, or the last denominator 



t 



, or 1 -|- -1. The first denominator is the ratio of the larger primary interval 

 to the whole circumference. These denominators are, in fact, the ratios of the 

 successive intervals and subintervals of our diagram. The other fractions, ex- 

 pressed in the form of continued ones, would have denominators expressing, in 

 the same way, the ratios of the successive subintervals, which the diagram rep- 

 resents ; and fractions in general may be classified according to their special 

 forms as continued fractions. Thus we have : 



1=1 8_1 4 _ 1 6 = 1 6_1 jt=i 



3 i+i, 6-y+i t-t+j, »~i+i, ii~T+i, iff i+i,, 



l + L T + l 1 + 1 1 + 1 l + l l+l 



12 3 



s 



Again we have : 



3 _ 1 61 7_1 9_1 11_1 



* T+l, T = T+l tV = T + 1, iS'l+i-.L, 16=T+1 



2 + 1 2~+l 2+1 2+1 2+1 



I $ f ¥ S 



The numerators and the denominators of the proper fractions of these series have 



constant successive differences. • 



The last denominators in these continued fractions represent the ratios of the 

 contiguous intervals of the diagram introduced in the second or third turns 

 by the third or fourth leaves. Only the first two fractions in each of these 



series conform to the above law. The others, like 4 and f, violate the law 



early in the cycle; and this explains the absence of them from natural 

 arrangements of the spiral type. The property common to the latter resembles 

 what we have observed in the arrangements of whorls, namely, that the leaves 

 of successive whorls are so placed that those of the upper one fall over the 

 middle positions of the spaces between those of the lower one; but those of 

 the next one above, or in the third whorl, are thus made to fall directly over 

 the leaves of the first. Two whorls thus constitute a cycle, in the sense in 



