30G Dr. H. Airy on Leaf -Arrangement. [Apr. 30, 



series /3. If six ranks should fall into alternate order -, one obli- 

 quity would lead to a series i, i, -^ , j^ &c., the opposite to a series 



1 2 3 



6' IT If &C ' 



Seven vertical ranks would take alternate order -, facilitated by 



obliquity. Condensation would give series B. (It is needless to follow 

 other possible lines of condensation.) 



Eight vertical ranks would fall into whorls of four, with the same 

 general characters noted above in whorls of two and three. Condensation 

 would give series y. 



Nine would give -. Condensation would produce series C. 



Ten would give whorls of five. 



2 

 Eleven would give . 



Twelve would give whorls of six. 

 Thirteen would give -^ ; and so on. 



Thus it appears that the ivhorled orders would naturally arise from 

 economic arrangement of even numbers (except 2), and the alternate 

 orders from, economic arrangement of odd numbers (including also 2), of 

 vertical ranks. 



It also appears that, in the whorled division, the members of each 

 whorl will divide the circumference of the stem equally, and that succes- 

 sive whorls will alternate in angular position. 



It has already been shown that in the alternate division the spiral 

 arrangement of the leaves, with angular divergence limited to certain 

 series of fractional values (A, B, C, &c.), would follow on the hypothesis 

 of condensation. 



These are the " main facts of leaf -arrangement " set down on page 298 

 to be accounted for. 



It is possible that all the varieties of leaf -order at present existing may 

 have been derived from an original two-ranked arrangement, partly by 

 variation in the number of leaf-ranks, and partly by vertical condensa- 

 tion of the orders so formed. This view is supported by 



(1) the high probability that the simplest form has been the earliest ; 



(2) the prevalence of the two-ranked form among lower phanero- 



gamous plants (e. g. Graminece) ; 



(3) the numerous instances of transition from a two-ranked order 



at the base of a shoot to a more complex order in the higher 

 parts ; 

 (4.) the prevalence of the two-ranked arrangement of rootlets on 



