THE USES AND ORIGIN OF THE ARRANGEMENT OF LEAVES 01 PLANTS. 880 



frequent in nature, a = 1, and if we denote by k the ultimate value to win h the 

 fractions of this series more and more approximate, or what is supposed to be 

 the type form of them, then, since k = 1 



1 -f- id. a<l inf.. we have k z=z . , 



Hence k-j r k 2 = 1, or A? = l — &. In the form of a proportion this is. I k = k: 1 — & ; 

 or k is the ratio of the extreme and mean proportion. Its value found by solving this 



equation is k = | (y 7 5 — 1) = 0.6180, approximately. From the above equation 

 we obtain by multiplying by k successix iy the following : k 2 = 1 — / ifc* = A 



& 2 ; & 4 = /; 2 — & 3 , and in general k n z=^~ 2 — k n ~ y ; that in, any power of this 

 quantity is equal to the difference between the two next low< r powers. Its 

 square is equal to its complement; the cube to the difference between it and 

 its square or complement, and so on. Or ft sss 0.( • 1 - s ; k 2 = 0.382; F = 0.236; 

 k*= 0.146; k b = 0.090, etc. On this peculiar arithmeti il property of k d< nd 

 the geometrical one of the spiral arrangement, which it repres sits; namely, that 

 such an arrangement would effect the most thorough and rapid distribution of 

 the leaves around the stem, each new or higher leaf falling over the ir 



space between the two older ones which are nearest in direction so e> to subdivide 

 it in the same ratio, k, in which the first two, or any two succ ssive one 

 divide the circumference. But according to such an arrangement thei eould 

 be no limited systems or cycles, or no leaf would ever fall exactly ovr am 

 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 the leaves 

 to lidit and air around the stem, and the most ample elbow-room or .pace for 



expansion in the bud, is to realize a property that exist separately 

 abstraction, like a line without breadth. Nevertheless practically, and so far as 

 observation can go, we find that the last two fractions, | and ft, and all far- 

 ther ones of the first series, like ft ete > which are dl *****?** " 

 measured values in the plant, do actually realize this property with all needful 



Thus 4 = 0.625; A=0.615; and H= 0.010, and differ from k by 



acy. ±nus £=0.0^0; j$ 



0.007, 0.003, and 0.001, respectively; or they all differ by inappreciable v.due. 

 from the quantity which might therefore be made to stand for all of them 

 But in putting k for all the values of the 6rst series after the first three, 

 it should be with the understanding that it is not so employed >n ,u c,,ac„y 

 as the grand type, or the source of the distributive character winch they ha- , ; 



VOL. IX. 



53 



