RELATIVE POSITIONS OF LATERAL MEMBERS. 201 



always however met at the very outset by a consideration of great importance which 

 imposes itself upon us, I mean the /ou^ ensemble of properties which define the 

 character of the natural group, class, or order. By recognising a plant as a member 

 of a particular class, e.g. Muscineae, Filices, Equisetacese, Rhizocarpeae, Phanerogamia, 

 &c., an aggregate of properties is ascribed to it, which must be taken into account as 

 such. If we pay special regard to the point of view opened out by the Theory of 

 Descent, we must recognise in the law of heredity and the physiological adaptation 

 of organs, the difficulty or even impossibility of demonstrating the causes of any 

 morphological phenomenon in any other manner than genetically. Organic forms 

 are not the result of combinations of forces and materials given once for all, 

 and always again reproduced in exactly the same manner, as in the case of 

 a crystal which is first dissolved and then re-crystallised, but of combinations 

 which repeat themselves hereditarily and which at the same time undergo change. 

 To understand these it is necessary to refer to the past, and not merely to the im- 

 mediate present. 



I Abundant opportunity will be afforded in the description of the various classes in 

 3ook II. for a more exact observation of particular relations of position ; but what has 

 low been said is sufficient as a preliminary. Some additional remarks on the Spiral Theory 

 n the doctrine of phyllotaxis may however find a place here. It has already been 

 ihown that the construction required and employed in this theory is not in all cases 

 jossible, being sometimes arbitrary and without relation to development, and at other 

 imes simply meaningless; and that, finally, only those cases admit of the application 

 . of this theory without violence, where the shoot forms three or more rows of leaves 

 distributed singly and uniformly in all directions. The history of development often 

 points to quite different constructions, even in those cases in which the spiral is still 

 geometrically possible. But even where the connection of the leaves in order of suc- 

 cession in age by a spiral running round the stem in the same direction is possible and 

 even apparently useful, there is not in the phenomena connected with development 

 any sufficient reason for the hypothesis that the growth of the generating axis itself 

 actually follows a spiral ^. 



Closely connected with the spiral theory, which must be carefully distinguished from 

 the doctrine of phyllotaxis, is another very peculiar law connected with the angles of 

 divergence. It was thought, namely, that a kind of natural law was found when it was 

 discovered that some of the most commonly occurring constant divergences i, ^, f, f, 3^, 

 and some of the less common ones, as ^/i, ^|, |;j, xV\, &c.^, may be represented as succes- 

 sive convergents of the continued fraction 



I 



2 + I 



I + 1 

 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 



^ See on this point Hofmeister, Bot. Zeitg. 1867, nos. 5, 6, 7, and Allgemeine Morphologic, 

 p. 481. 



^ It m«st be observed in reference to this that it remains uncertain whether such complicate4 

 flivergences are ever so formed originally, or whether they are not always consequences of compli- 

 cated displacements, in consequence of which the direct observation of the growing point does 

 not give in these cases a certain conclusion. 



