388 Boyal Society : — 



It also appears that the necessary sequence of these successive 

 steps of coudensation, thus detemiined by the geometry of the 

 case, does necessarily exclude the non-existent orders, |, f, f , |, 



Numbering the spheres from upwards, it appears that, under 

 contraction, 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 g, f, §' tt ^^' • ^^^'^ ^^ ^^^^ very failure the results 

 of the experiment are more closely in agreement 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 moi-e, 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 " ^ 

 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 seA^eral sets of spirals, 

 presents a striking agreement with the above synthetical process. 

 In the sunflow^er, 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 conden- 

 sation, 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 which w^e have already seen possessing that relation in 

 the experimental condensation of the order |. 



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 g in marine Alr/w, and in Gmvnnece, 

 a low-developed gregarious group, and its singular freedom from 

 individual variation in that group and in elm, beech, &c., support 

 the view that this order is the original of the spiral orders. 



In many plants we find actual transition from the order ^ to an 

 order more complex, as, for instance, in SjxinisJi chestnut, laurels, 

 nut, ivy ; and these instances agree in presenting the complex 

 order in the buds that occupy the most exposed situations, while 

 they retain the simple | in the less-exposed lateral buds. Several 

 kinds of aloe have the order | in their basal leaves, and a higher 

 order in the remainder. A species of cactus often contains a 



