DIVERGENCES. 499 



parastichies ascending right and left : these however exhibit irregularities which are 

 inexplicable by the spiral theory, but otherwise easily intelhgible. The irregularities 

 simply depend upon the fact that the rhomboidal quadrangular fruits are every- 

 where nearly equal in size, but the receptacle on which they are situated becomes 

 smaller above, since it is conical where youngest. Hence the number of the fruits 

 occurring side by side at the same height decreases upwards ; also the number of 

 parastichies must be diminished by certain of them simply ceasing at different heights. 

 Thus the series b in the figure ceases between a a' and cc', and in the same way 

 several others besides m and n terminate on the right of the figure between / /' and (/ . 



The doctrine of phyllotaxis based on the spiral theory, which according to the 

 so-called principle of axillary branching was regarded as dominating the position of 

 the lateral shoots, laid, further, great stress on the fact that on shoot-axes richly 

 provided with leaves or ofif-shoots generally, certain ' divergences ' repeat themselves 

 more frequently than others. By divergence is understood that portion of the cir- 

 cumference of the shoot-axis which lies between two consecutive leaves ; thus, in 

 two-rowed phyllotaxis it amounts to \, in three-rowed to \, in five-rowed to f , in 

 eight-rowed to f of the circumference, and so on. In the case of some axes 

 abundantly provided with outgrowths (Pine cones, shoots with crowded leaves, &c.) 

 it is shown now that the divergences on the so-called genetic spiral remain constant 

 for more or fewer of the segments of the chain, and (a point on which particular 

 stress was laid) that the very divergences named — viz. \, \, f, |, ■^, &c. — recur 

 somewhat frequently. These fractions (parts of a continued fraction) appeared to 

 constitute the expression of a mysterious law which was assumed to dominate growth 

 in a supposed spiral manner. But it was nevertheless seen to be necessary, in addition 

 to the relations of phyllotaxis represented by that fraction, to add yet others, which 

 of course led to any continued fraction whatever, whereby however the point lost in 

 significance. The numerous cases of dorsi-ventral shoots however remained entirely 

 outside the system : in these, homologous off-shoots arise only on one side of the 

 axis or on two opposed flanks, and these could in no case be placed on a spiral 

 running round the axis. Further, it was fatal to the theory that the mysterious 

 divergences by no means remain constant even on one and the same shoot-axis, but 

 mostly begin with simple fractions, such as \ or \, or quite irregularly, and then pass 

 over into f , f , and so on, probably to be continually returning again to simpler ones, 

 or yet others. In addition to this it turned out that even in very closely aUied plants 

 the relations of position are often importantly different, and, which I would lay more 

 stress upon than on anything else, the whole spiral theory, with its divergences and 

 continued fractions, found no application whatever to the branching of roots -^the 

 roots had no existence, so to speak, for the spiral theory. 



Nevertheless the frequent occurrence of the divergences \, f, f, -j^^, &c. is 

 a fact of observation. The secret of this occurrence and the frequent absence of 

 other divergences is explained, according to Schwendener's investigations^, by 



' Hofmeister first attempted, in his ' Allgemeine Morphologic der Gewdchse' (186S), to shake 

 the foundations of the doctrine of phyllotaxis of Schimper and Braun, but he was himself in the 

 main still swayed by this doctrine. For my part, I have from the first regarded the theory of 

 phyllotaxis more as a sort of geometrical and arithmetical playing with ideas, and have especially 

 regarded the spiral theory as a mode of view gratuitously introduced into the plant, as may be 



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