930 ON LEAF-ARRANGEMENT [ch. 



easy, then, to number the entire system. The generating spiral, 

 0, 1, 2, 3, ... , and the various secondary Fibonacci spirals, are then 

 easily recognised. 



A Fibonacci series, unwrapped from a cone or cylinder. m = 6x-\- Sy. 



1 6 11 16 21 26 31 36 



T 2 3' 8 13 18 23 28 



15 To 5" 5 10 15 20 



^r8l3'83 2 7 12 



■5T'2B2Tl6rT 6 T4 



The place of the first scale in each series is then found to be as 

 follows : 



Series 1 



y~ 2 



And this is the Fibonacci series over again. We also see how the 

 several spirals, of which these are the beginnings, alternate to the 

 right and left of an asymptotic line, where x/y = 0-618. . . . 



The Fibonacci numbers, so conspicuous in the fir-cone, make their 

 appearance also in the flower. The commonest of floral numbers 

 are 3 and 5; among the Composites we find 8 ray-florets in the 

 single dahlia, 13 in the ragwort, 21 in the ox-eye daisy or the mari- 

 gold. In the last two, heads with 34 ray-florets are apt to be 

 produced at certain times or in certain places*; and in C. segetum 

 these florets are said to vary in a bimo(fe,l curve of frequency, with a 

 high maximum at 13 and a lower at 21t. The simplest explanation 

 (though perhaps it does not go far) is to suppose that a ligulate 

 floret terminates, or tends to terminate, each of the principal spiral 

 series. But among the higher numbers these numerical relations 

 are only approximate, and the whole matter rests, so far, on some- 

 what scanty evidence. 



The determination of the precise angle of divergence of two con- 

 secutive leaves of the generating spiral does not enter into the above 

 general investigation (though Tait gives, in the same paper, a method 



* Cf. G. Henslow, On the origin of dimerous and trimerous whorls among the 

 flowers of Dicotyledons, Trans. Linn. Soc. (Bot.) (2), vii, p. 161, 1908. 

 f Cf. A. Gravis, Elements de Physiologic vegetale, 1921, p. 122. 



