RELATIVE POSITIONS OF LATERAL MEMBERS. i8l 



successive convergents of the continued fraction 



2 -I- I 

 I + I 

 I. . . . 



Were it possible to combine all kinds of phyllotaxis without exception in this manner into 

 one single continued fraction, we should actually have a kind of natural law, in which there 

 would be no relation of cause and effect, and which would hence stand out as an inex- 

 plicable curiosity. It is not however so bad. There are many such kinds which cannot 

 be expressed by this continued fraction ; and in order to carry out the method, new 

 continued fractions have to be constructed, e, g. 



I I 



3+1 4+1 



I. . , . I. . 



&c. ; 



of which indeed only one or two convergents are for the most part met with as actual 

 angles of divergence. And since it is possible immediately to construct a new continued 

 fraction for every phyllotaxis which cannot be arranged under those already in existence, 

 it is of course possible to represent by this method all varieties of phyllotaxis ; but it 

 follows at the same time that the method itself thus loses all deeper significance. If only 

 those divergences occurred on one and the same axis or on one system of axes which 

 can be represented by convergents of one and the same continued fraction, or if the 

 different values of one and the same continued fraction occurred exclusively in a genus, 

 family, or order, the method would even in that case be of some value. But this is not 

 the case. Since moreover no actual relationship of the method to the history of deve- 

 lopment, to the classification of plants, or to the mechanics of growth, has been esta- 

 blished, in spite of numberless observations, it seems to me absolutely impossible to 

 imagine what value the method can have for a deeper insight into the laws of phyllotaxis. 

 But even as a mnemonic assistance it appears to me not only superfluous, but even dis- 

 advantageous, since the use of it diverts the attention from relationships which are of 

 real importance \ 



* [Chauncey Wright (Memoirs of Amer. Acad. ix. p. 389) has pointed out an interesting 

 property of the series ^, ^, |, ^ which includes all the more common arrangements of phyllo- 

 taxis. If the spiral line passing through successive leaves be traced the long way round, we obtain 

 the complementary series j, |, 3, | the terms of which are successive convergents of the con- 

 tinued fraction 1 + i 



1 + I 



7T &c. 



If we put this = K then K = 



i+A' 

 or K''=i-K 



.'.I .K=K: i-K 



or K is the ratio of the extreme and mean proportion : and generally 



K is therefore the angular divergence which would effect ' the most thorough and rapid distribution 

 of the leaves round the stem, each new or higher leaf falling over the angular space between the two 

 older ones which are nearest in direction, so as to divide it in the same ratio, K, in which the first 

 two or any two successive ones divide the circumference. Now f and all successive fractions differ 

 inappreciably from A'.' Practically, therefore, all terms of the series above the third fulfil the 

 condition of that leaf-distribution which is theoretically the most efficient. — Ed.] 



