INTKODUCTION. 5 



right or left, in individual cases; and the correlation of the | 

 arrangement with a 5-channelled stem. 



The fifth type of " Eedoubled Spirals " is of even greater interest, 

 in that it contains the germ not only of the parastichies of Braun, 

 but also of the multijugate systems of Bravais. 



Only two examples were noted : Pinus, in which three parallel 

 spirals of seven members each resulted in a cycle of 21 

 members, and Alies, in which five parallel spirals of eleven 

 members each gave a total of 55. These latter observations are 

 credited to Calandrini, who also drew the figures. 



The lack of higher divergences appears to be due to Bonnet's 

 preference for the longest leafy axes, and his special precautions to 

 avoid the terminal bud as much as possible, since this did not give 

 accurate results ! Notwithstanding this, he saw quite clearly in 

 the case of the Apricot (p. 180) that successive § cycles were 

 really not vertically superposed, and that, in fact, the first members 

 of each successive cycle also formed a spiral, and so in practice no 

 leaf was vertically superposed to another on the same axis. This 

 he regarded, not as the expression of any fault in the theory, but 

 as a confirmation of his law, since such a secondary displacement 

 would give room for the proper function of every leaf. 



Subsequently, arrangements in which eight and, thirteen parallel 

 spirals could be counted (the latter in the staminal cone of Gedrus) 

 were distinguished by De Candolle (A. P. de Candolle, OrgaTwgra'phie 

 Vigaale, 1827, vol. i. p. 329). . 



From such a medley of observations on vertical rows and parallel 

 spirals, the more modern theory of phyllotaxis was evolved by the 

 genius of Schimper and Braun. 



The vertical rows become " orthostichies," the parallel spirals 

 "parastichies," the number of leaves between two superposed 

 members becomes a " cycle," and these are tabulated in a series : — 



1 i 2 3 .a. _8- ptc * 

 2' t' 5' 8' 13> 21' ^^°-' 



* The properties of the Schimper-Braim series, 1, 2, 3, 5, 8, 13, etc., had long 

 been recognized by mathematicians (Qerhardt, Lam^), and appear to have been 

 first discussed by Leonardo da Pisa (Fibonacci) in the 13th century. 



Kepler, in 1611, speculated on the occurrence of these numbers in the vegetable 

 kingdom, and went so far as to suggest that the pentamerous flower owed its 



