1874.] Dr. H. Airy on Leaf -Arrangement. 303 



of the shoot as retaining the ancient order, and the more exposed terminal 

 portion as having undergone protective modification. 



The various degrees of obliquity of spiral ranks in the alternate orders 

 of leaf -arrangement, and the complicated numerical relations existing 

 between those various ranks, are all fully accounted for by the conden- 

 sation theory. 



Analyzing the spiral arrangement seen in a sunflower-head, a dandelion- 

 head, a house-leek rosette, and an apple-twig, the result is found to be 

 that any leaf (or fruit, in the first two instances), taken as zero, has for next 

 neighbours successively, in rising steps of complexity of order, the 1st, 2nd, 

 3rd, 5th, 8th, 13th, 21st, 34th, 55th, 89th, 144th, &c. (in order of growth) 

 alternately on the right side and on the left, producing alternately right- 

 and left-handed spirals in sets of 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 

 &c. ; and these numbers are identical with those which would result 

 from condensation of one of the lower orders of series A. Similar 

 considerations apply to series B and C. 



It is a significant relation that, in the sunflower and similar examples, 

 the arrangement of the fruits in the composite head is such as would 

 result from condensation of the arrangement of the leaves on the stem. 



Among the ivliorled orders also there is equally strong evidence of the 

 working of the same force of condensation. 



First there is a series (a) derivable from the crucial arrangement. 

 (This is shown by diagrams.) In the orders thus formed it is seen that 

 conspicuous sets of parallel spirals will form the most striking feature, 

 and that these spirals will be found in sets of 2, 4, 6, 10, 16, 26, 42, &c. 

 (series a). 



Instances are seen in the genera Mercurialis and Sagina, and the order 

 Bipsacacece, in which last the whole series a finds exemplification. 



Here also it is a significant relation that the fruit-order in the com- 

 posite heads of Dipsacacece is such as would result from condensation of 

 the crucial order of their stem-leaves. Some of these plants exhibit in 

 their radical leaves a minor degree of the same condensation. 



In like manner it is shown that condensation of whorls of three 

 would produce orders with spirals in sets of 3, 6, 9, 15, 24, 39, 63, &c. 

 (series /3). For examples see Hofrneister, op. cit. p. 460. 



Condensation (if any) of whorls of four would give spirals in sets of 

 4, 8, 12, 20, 32, &c. (series y). 



It is contended that the preceding evidence, drawn from both divisions 

 of leaf-arrangement (alternate and whorled), is sufficient to establish the 

 principle of condensation as having played an important part in. the 

 history of leaf-arrangement. 



