274 PHYSIOLOGICAL BOTANY. 



that this divergence in most cases amounts to 137 30' 28", 

 an irrational angle ; in some other and more rare cases, 

 the angle, which is also irrational, is = 99 30' 6", or 

 77 57' 19", or 151 8' 8". None of these angles are 

 altered, at least their mean values, either by the dissimi- 

 larity of the members which follow each other or other 

 local circumstances. The increase is worthy of note, 

 at least according to its average value, especially in 

 Schimper's method of finding the angle of divergence, 

 since we cannot always meet with a foliar part occurring 

 exactly in a vertical line above it. It is remarked also, 

 that for this purpose, the outer rind often requires to be 

 removed in order to distinguish the false from the true 

 angles. The authors also extend their observations to 

 the false whorls, they show that the including generating 

 spiral extends to the underground stem, that the direction 

 of the spiral is the same in the stem and branches, but 

 exerts no influence upon the direction of twining stems. 

 The convergence of two spirals into one, which is some- 

 times noticed, may arise from the abortion of one spiral 

 or a confluence of two spirals into one, an entire series 

 may then be absent, whence the existence of several series 

 becomes doubtful. It appeared to me important to refer 

 to this memoir again, since it appears to be read less than 

 it deserves, for it not only contains a great many theo- 

 retical considerations, but also numerous investigations 

 made upon the plants themselves. Meyen's remarks upon 

 this subject, which have been given in previous Annual 

 Reports, do not appear to me altogether to the point. 



In my 'Elem. Philosophise Botanicse,' pp. 450-1, I 

 endeavoured to discover a general expression for the ex- 

 positions given by Schimper and Braun, by which they 

 might be more easily reviewed. I was unacquainted 

 with the memoir of Bravais ; it appeared in 1837, simul- 

 taneously with my ' Elemental I started from Schimper's 

 theory. Let m represent the number of leaves (including 

 in this term bracts also) between two leaves which come 

 next each other in a vertical line, but which are arranged 



