THE PHYLLOTAXIS OF HELIANTHUS TUBEROSUS. ('49 



On the Transitions from one kind of dicer gence to another of the same or i 



different series. 



On referring to the diagram (fig. 1.) of the divergence / f of the primary series, it will 



be seen that in consequence of no projected coil, as described above, containing more 

 than three leaves, the angular divergences of all the spirals of these Beries will neces- 

 sarily lie between 120° and 180° inclusive, or those represented by \ or -1, and particular 

 numbers will arrange themselves right and left, and nearest to the assumed vertical line 

 corresponding to any generating spiral. Thus, for the fraction - 2 S ,, two numbers in close 

 proximity to this line, and situated below the 22nd leaf (which commences the second 

 evele), are the 9th and 14th. These, together with the initial leaf selected .is No. 1 of 

 the generating spiral, form the commencement of two secondary spirals through the 



numbers 1, 9, 17, 25, 83, 41, &c, and 1, 14, 27, 40, &c., the angular divergences of 



which, as generating spirals, would be represented by the fractions j! and , r, ;J - respectively. 

 Hence it can be seen that by shifting, as it were, the 22nd leaf to the one side or the 

 other, some other leaf will fall exactly or approximately over the first, and the generatin 

 spiral will no longer be represented by - a \, but by some other fraction. And as the 

 leaves nearest to the vertical line passing through the 1st and 22nd leaf are those which 

 commence the second cycles of spirals, represented by fractions ^ on the one hand, and 

 H on the other, these are found to be the divergencies into which r would most 





r> 



d 



readily pass. 



Similarly, if a vertical line be drawn corresponding to the -^ divergence, by a si 

 movement to the left (suppose), the 22nd leaf comes most nearly over the first, 

 the spiral arrangement of / f is obtained; but if a greater displacement to the righl had 

 taken place, the 9th leaf would have fallen over it ; or, again, by a still greater displace- 

 ment to the left, the sixth will be vertically over the first ; and we thus pass into the 

 arrangements for the generating spirals represented by the fractions f and f respectively. 



Exactly analogous results can be obtained from the secondary and other series of 



fractions ; for if we select the line passing through the 1st and 19th leaves, or the vertical 

 line for the divergence A, we can, by supposing a slight deviation to the right, bring the 

 29th leaf over the 1st, so that the generating spiral would now become &• Likewis. 

 by a deviation to the left the 12th leaf would arrive over the first, and a transition 1). 



thus obtained into the divergence A 5 or > if we had first chosen the vcrtical for tllis 



arrangement, *. e. a line through the 1st and 12th leaves, then it is easy to see how a 



change can be effected, either into the lower members of the series 7 or i, or to the 

 higher one ^. 



Generalizing these remarks, it becomes clear that similar transitions can be presumed 

 possible in all other series, and, further, that any one series can pass into another, pro- 

 vided it be represented by a generating spiral, the angular divergence of which is a low 

 one in that series, i. e. either itself being one of the divergences J, h h h &C or capabh 

 of passing into one of them, as, from these, passages may be presumed possible from the 

 primary into the secondary, secondary into tertiary, and rice versd respectively, mas- 

 much as each of these fractions is common to two series. 





vol. xxvi 



4x 



