118 THE PHENOMENON OF 



existence of all the forms of a regular system bounded by 

 a definite number of surfaces, on the ground that they all 

 belonged to a single series terminating in the sphere, a 

 surface without any boundaries. I am far from intending 

 by this to take away all value from the labours of the 

 brothers Bravais ; the more complicated modes of ar- 

 rangement are often so difficult to distinguish that the 

 doubt as to their existence, as actually different, is often 

 readily explicable, but, on the other hand, in the simple 

 conditions, there cannot be the slightest doubt that they 

 exist in the greatest mathematical accuracy, which indeed 

 is admitted by the Bravais themselves. Hence arises 

 with the Bravais a diversity of explanation of phenomena 

 which evidently belong to the same series, a circumstance 

 which by no means harmonises with the formulary praise 

 which Schleiden gives to the theory of Bravais. Schleiden 

 would certainly not have arrived at such a judgment had 

 the subject of phyllotoxy been more experimentally 

 familiar to him. At the same time, the question here is 

 not so much of the preference as regards method of a 

 theoretical exposition, as the establishment of a series of 

 facts. I cite in support of my own convictions, gained by 

 many years' study of the modes of arrangement of leaves, 

 an observer whose trustworthiness is declared by most 

 important labours in an analogous field. Naumann has 

 not the least doubt of the actual and distinguishable 

 occurrence of most complicated conditions of arrangement 

 (^ || } in,) and refers the incredulous to the investigation 

 of the CactaceaB and Composite, remarking that in the 

 sun-flower, a blind person will be able to distinguish the 

 above-named allied plans of arrangement with the fingers, 

 although they depend upon immeasurably small differences 

 in the angle of divergence (Poggend. 'Annal.' 1843, p. 

 554 55). According to my own and Naumann's 

 (Ueber den Quincunx, pp. 70 71) observations, not only 

 | and occur in the Cacteas but also & & & in a manner 

 which even Bravais could not help acknowledging as not 

 merely apparently but actually rectilinear and rational. 



