178 



Dr. H. Airy on Leaf- Arrangement. 



[Feb. 27, 



spheres are set a little away from the axis, the order becomes condensed 

 into (nearly) -J, with great precision and stability. And it appears that 

 further contraction, with increased distance of the spheres from the axis, 

 will necessarily produce the orders (nearly) §, -£ T , &c. in succession, 

 and that these successive orders represent successive maxima of stability 

 in the process of change from the simple to the complex. 



It also appears that the necessary sequence of these successive steps of 

 condensation, thus determined by the geometry of the case, does neces- 

 sarily exclude the non-existent orders, 5, f, f, J, 4 T , &c. 



Numbering the spheres from upwards, it appears that, under con- 

 traction, the following numbers are brought successively into contact 

 with 0, alternately to right and left:— 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 

 144, &c. None of them stands vertically above while in contact with 

 it, but a little to the right or a little to the left; and so far the results of 

 this experiment fall short of the perfect fractions 3, -f, §, &c. : but in 

 this very failure the results of the experiment are more closely in agree- 

 ment with nature than are those perfect fractions themselves ; for those 

 fractions give the angular divergence only in round numbers (so to 

 speak), and lose account of the little more, or the little less, which makes 

 all the difference between a vertical rank and a spiral. In the large 

 majority of spiral-leaved plants, one has to be content with " f- nearly " 

 or " § nearly," and it is difficult to find a specimen in which the fraction 

 represents the order exactly. 



The geometrical relations of the members of the above series 1, 2, 3, 

 5, 8, 13, &c. are as simple as their numerical relations. 



Analysis of the order seen in the head of the sunflower and other 

 examples, by consideration of their several sets of spirals, presents a 

 striking agreement with the above synthetical process. In the sunflower, 

 a marginal seed taken as is found to be in contact with the 34th, the 

 55th, and the 89th (counted in order of growth), and even with the 

 144th, if there is not contact with the 34th. The dandelion, with a lower 

 degree of condensation, has in contact with the 13th, the 21st, and the 

 34th in large specimens ; the house-leek in its leaf-order has in con- 

 tact with the 5th, 8th, and 13th ; the apple-bud has in contact with 

 the 2nd, 3rd, and 5th ; and thus we see that in nature the very same 

 series of numbers is found to have contact-relation with 0, which we 

 have already seen possessing that relation in the experimental conden- 

 sation of the order J. 



Difference of leaf -order in closely allied species (e. g. Plantago major 

 and P. coronopus) is found in close relation to their different habits and 

 needs. 



The prevalence of the order \ in marine Algw, and in Graminew, a low- 

 developed gregarious group, and its singular freedom from individual 



