THE PHYLLOTAXIS OF HELIANTHUS TUBEROSUS. 



055 



V. Transitions from Divergences of the Primary to those of the Secondary series. 

 Transitions from one series to another upon the same stem do not appear to be readily 



effected. It rarely if 



occurs amongst the spirals of the ILelia»(hus without 



intermediate verticillate (either actual or approximate) arrangement. 



If it be remembered, however, that for the primary series all divergences lie betwec 

 120° and 180° inclusively — for the secondary series, between 90° and 120°— and for tli 



tertiary between 72° and 90 



it 



ill b 



seen that to pass from the primary to 



secondary the angle must be as near 120° as possible, or that represented by -J-. Now I 



common hoth to the primary and secondary series, ice con conceive of 



3 



this fraction is commo: 



passage from one series to another through that divergence. Similarly tliroi 



( .)0°, a passage could be effected from the secondary to the tertiary. Hence the 



\, j, &c, would be the fractions indicating what might be called transitional divergences. 



Such, however, was not the method obtaining amongst the leaves of HelianthuB 

 tuberostis. 



The diagram (fig. 8) illustrates an instance where the leaves at the base of the stem 

 were arranged for the first cycle 

 revolving to the right. 



according to the f divergence (primary series) and 

 In consequence, however, of some misplacement in the second 

 ycle, the 9th leaf fell over the 6th, i. e. the 1st leaf in that cycle, which must therefore 



be represented by 



3 



14th leaf being over the 9th 



In the third cycle there was a return to the |, as shown by the 



In this cycle the 11th and 12th, as also the 14th and 15th 



in the next, became confluent, indicating therefore an attempt at a verticillate grouping. 

 The 16th, 17th and 18th leaves were free, and, though distant about 120° from each other, 

 were not at the same level. Prom this point a sudden change to two perfect whorls of three 

 each occurred. Then a new spiral (of the secondary series) abruptly followed, revolving 

 to the left, and containing four leaves in each coil, starting from No. 1 (bis). Of thi 

 spiral the first two leaves most nearly in the same vertical line were the 13th and 24th ; 

 and as there were three coils between them, the angular divergence now represented by 

 the leaves was -j^. No further change occurred up the stem. 



I may here mention that MM. Bravais and Martin observe that when transitions occur 

 from one series to another, it is by nature selecting, as it were, for the change a new 



denominator as 



ly the same as possible to the precedin 



thus A (in the tertiary 



series) may be followed by ■& (in the secondary series). They, however, do not cite an 

 example; and 1 would add that I met with no such instance; for in the Kel'anithus the 

 change was almost invariably effected by an intermediate (either actual or approximate) 



verticillate arrangement. 

 The following examples will illustrate the principal cases of change from one kind of 



angular divergence to another on the same stem : 



notice the occurrence of changes from verticils to spirals and vice versa. They state that if two different whorls have 



arithmetic 



mean between the divergences of the whorls. Thus if a whorl represented by (£) be connected with a whorl of (1 ), 



tliA miM^.J.'.j _ ™ , . .,• _ • j:^^«r»l />ocna /'Ann. dfiS Sc. Nat. 2 lnf Her. 



toe intermediate 



• t • 



known bv A 



They do not mention individual cases. (Ann. des Sc. Nat. 2 



No such arrangement appeared amongst the leaves of the Hehanthus. 



