352 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



Page 116. Fibonacci Ratios. — The relation of the sequence 2 ■ 1 • 2 • 1 • 2, etc., is 

 not quite so definite as stated, the summation being correct up to 34 ; since 34 

 members of the series do not add up to 55 but 54 ; similarly, 55 members add 

 up to 88. To keep the ratio correct in an expansion system an extra curve 

 must be put in : this does not affect the value of the convention, since the series 

 must be arranged around a circle, and the sequence must be broken somewhere. 

 Since a strict adherence to the 2 • 1 • 2 • 1 • 2 sequence would in these cases 

 result in the formation of multijugate systems, this property of the numbers 

 involved renders the accuracy with which the expansion ratios succeed one 

 another still more remarkable. 



Page 153, line 25, for 36, read 63. 



Note on Dichotomous Systems in Helianthus annuus. 



Dichotomy of the shoot-apex of the type described in Lycopodium Selago 

 (fig. 79, II., III.), and found characteristically in the Lycopodiaceae and allied 

 forms, is possibly to be regarded as the most primitive type of ramification of 

 the main axis of aerial plants. At any rate the causes which have directed the 

 evolution of the axillary branching of the strict type met with in higher 

 plants still remain far to seek, though there can be little doubt as to the 

 biological utility of the method so widely adopted. Dichotomy of such a 

 strict type is less frequent among higher plants, and its occurrence would as 

 a matter of fact be usually classed as a monstrosity. It is clear, however, that 

 such dichotomy, however anomalous it may be considered, represents the first 

 step in the production of the still more complicated growth-systems included 

 under the heading of Fasciation, and that the division of a growth-centre into 

 two equal centres is the simplest case of irregularity. 



Helianthus annuus, which has so frequently been taken as the most typical 

 representative of phyllotaxis phenomena, owing to the marvellous accuracy of 

 its inflorescence construction scheme, has already been shown to present in 

 addition all the typical phenomena of symmetrical and asymmetrical construc- 

 tions, the perfection of Fibonacci relationships, and also the peculiar relation- 

 ships of bijugate construction : it again becomes a plant of special interest 

 from the frequency of the occurrence of strict dichotomy, which in garden 

 specimens may affect the whole of the lateral branch system of the plant. 

 Similar phenomena, but in a less perfect manner, may be noticed in such 

 allied forms as H. rigidus and H. strumosus, in their garden varieties. In 

 such cases the dichotomy may occur (I.) in the foliage region, giving long- 

 stalked pairs of capitula ; (II.) close behind the involucral region, giving twin- 

 heads ; and (III.) within the involucre, resulting in the phenomenon of 

 "two-eyed" capitula with a more or less perfect ray series between the 

 two disks. 



Observation of such systems shows that the irregularities recorded for 

 Lycopodium Selago (p. 207) also hold good for Helianthus ; there being thus 

 no necessary connection between the distribution of the primordia of the 

 secondary centres, either between themselves or between these and that of the 



