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MORPHOLOGY OF MEMBERS, 
inexplicable curiosity. This is not however exactly the case. There are many phyl- 
iotaxes 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^ or &c.; 
I + I I + I 
I. . . . I. . . . 
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 
those divergences only 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 particular 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 development, to the classification of plants, or to the mechanics of growth, has been 
established, 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 ^. 
Sect. 27. Directions of Growth^. — (i) In every thallus, branch, stem, leaf, 
hair, and root, it is easy to distinguish between two opposite ends, the Base and the 
Apex. The base is the place where the member originated and began to grow ; 
the apex is the extremity in the direction of the growth. The direction from the 
^ [Chauncey Wright (Memoirs of Amer. Acad, vol, ix. p. 389) has painted out an interesting 
property of the series |, ^, f , f 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 ^, f , f , f the terms of which are successive convergents of the con- 
tinued fraction i + i 
1 + I 
1+ &c. 
If we put this = K then K = -I— 
^ i + K 
or K^=i-K 
.-. i'.K = K:\-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 K.^ Practically, therefore, all terms of the series above the third fulfil the 
condition of that leaf-distribution which is theoretically the most efficient by distributing the leaves 
most completely to the action of the surrounding atmosphere.] 
2 H. von Mohl, Ueber die Symmetrie der Pflanzen, in his Vermischte Schriften. 1846. — 
Wichura, Flora, 1844, pp. 161 et s^y. —Hofmeister, Allgemeine Morphologie, §§ i, 23, 24. — Pfeffer, 
Arbeiten des botan. Instituts in Würzburg, 1871, p. 77. 
