160 



which to some extent can be formulated in a way comparable with 

 the second problem, by defining it as the question about the arrange- 

 ment of the different parts within these limited organs themselves. 



But as a plant or an animal increases its volume continuously 

 by growth, and only secondary influences like the exhaust of life- 

 energy, sexual functions, etc., will help to put a limit to this growth 

 within a finite time, while without these hindering causes it 

 properly would go on infinitely, - - the living organism can also 

 be looked upon as being endless and an unlimited system like those 

 we have discussed in the above. Exactly as the growth of a crystalline 

 medium is only determined by secondary circumstances existing in 

 its mother-liquid or its immediate environment, while from a 

 theoretical point of view it is also an endlessly extended system 

 of regularly arranged units. 



.In this connection some considerations may be inserted here 

 concerning the remarkable views about phyllotaxis, i. e. about the 

 way of arrangement of leaves in plants. As we shall see, these phe- 

 nomena are in many. points very analogous to those dealt with in 

 the preceding paragraphs. Closely related to them are the pecu- 

 liarities observed in the arrangement of buds, of scales, and of the 

 different parts of muscles etc., as observed in oceanic conchifers. 



The fact that the leaves of plants are arranged in spiral series 

 about their axis, has long been observed and recognised by botanists. 

 The spiral- theory of phyllotaxis has since the days of Goethe and 

 Bonnet 1 ) often been a subject of investigation and speculation, 

 and for a considerable time it has been an object of botanical interest, 

 since its development by Schimper and Braun' 2 ) and by A. and 

 L. Bravais 3 ). 



Its fundamental conception was originally, that the arrangement of 

 such leaves occurs in series which form alternating rows when viewed 

 in a horizontal or vertical direction. Thus proceeding along such a 

 spiral line, we shall meet a definite number of leaves ("members" of 

 the series), until after one or more revolutions a leaf is reached, which 

 stands exactly vertically above the first one. The members included 

 in such a series form together a cycle; the row of vertically superposed 



!) Ch. Bonnet, Recherches sur I' Usage des Feuilles dans les Plantes, Goettinge 

 et Leyde, (1754), p. 159. 



2 ) K. F. Schimper and A. Braun, Flora, 2, (1835); A. Braun, Nova Acta 

 Acad. Carol. Leopold. Nat. Curios., Halle, 15, 1, p. 195, (1831). 



3 ) A. and L. Bravais, Ann. des Sciences naturelles, (2), 7, p. 42, 67, (1837). 



