RISING PHYLLOTAXIS. 119 



considerably iu excess of the theoretical minimum, and the conclu- 

 sion is warranted that the expansions may take place at intervals 

 at irregular distances from one another, although the actual tran- 

 sition may be limited to the minimum number of members ; the 

 new ratio being continued for an indefinite number of members 

 before a new change takes place. 



Thus the Daisy (fig. 47) shows a double transition, from (2 -I7 3) 

 foliage leaves, (5-1-8) involucre, to (13-)- 21) florets. The ends of 

 the system are not obtainable, but it is obvious that 13 involucral 

 members of the (5-t-8) system are retained when the complete 

 transition would have been completed in 8. 



Very remarkable relations occur in Helianthus in the number of 

 members constituting the involucre, and the tendency of Jlelianthios 

 to approximate a continuous expanding system is very marked. 

 Thus the involucre of a (55-1-89) capitulum should contain 34 

 sterile scales ; the range (Weisse) is 26 — 42, but it must be borne 

 in mind that the exact localisation of the involucral region is a 

 matter of difficulty, since the reason which determines which scales 

 shall be the first to produce florets is not known, and the range is 

 thus as equally open to minor variations as the number of the ray- 

 florets themselves. The variation curves tabulated by Weisse show 

 that similar ratios hold for capitula exhibiting the other parastichy 

 systems on the disk, and the number of sterile involucral scales is 

 thus one grade lower along the ratio series 3, 5, 8, etc., than the 

 theoretical number of ray-florets. The involucre of JELeUanthus 

 thus apparently includes the whole of the members involved in 

 one complete transition, while a . second transition takes place 

 immediately after, of which the ray-florets represent those transi- 

 tional members which have to negotiate the bulk change. Since 

 the head taken as a type in fig. 45 is only of a low order (21-1-34), 

 and the involucre therefore (8-1-13), it would follow that some of 

 the foliage leaves were laid down with a (3 -|- 5). 



But much larger heads are found, so that it becomes probable 

 that more than two such transitions may take place in such a rapidly 

 expanding axis. 



It has already been shown, for example, that Gyrmra shows two 

 transitions in the involucre, and another at the edge of the disk : 



