64 THE PALM-STEM: 



shorter, looking at its absolute size, the nearer it is to the 

 inner end, but the angular divergence of the leaves will 

 be the same. If this be the case, and I believe that all 

 investigations hitherto made testify that such is the con- 

 dition, it is evident that the movement of every leaf takes 

 place in the radial direction, and is simply a consequence 

 of the elongation and expansion of the axis ; that the 

 motion of any leaf in a spiral direction is only apparent, 

 and results from the fact that the spiral line has no defi- 

 nite place upon the upper surface of the bud, but its 

 point of origin continually advances in the direction in 

 which the leaf-spiral runs, into the circle which is formed 

 by the connexion of the cylindrical surface of the stem 

 with the depressed surface of the bud. If such be the 

 condition, the leaves must appear to traverse the outer 

 turns of the spiral more rapidly than the inner, since the 

 same angles of divergence correspond to larger segments 

 of turns in the outer parts of the spiral, while, according 

 to Meneghini's theory, the contrary occurs : the motion of 

 the leaves diminishes in regard to the angle of divergence 

 in the outer turns of the spiral, but in regard to the spaces 

 passed through remains the same, and thus must appear 

 to become slower. Since, although I had investigated 

 the terminal buds of some large Monocotyledonous plants 

 in reference to this point, and had found no notable 

 deviation with regard to the divergence of their leaves, 

 my judgment on this matter could not be nearly so va- 

 luable as that of my honoured friend, Professor Alexander 

 Braun, I wrote to ask him whether he had ever met with 

 cases of such a deviation in the divergence in the terminal 

 bud as is required by Meneghini's theory, and obtained 

 the answer, that he is indeed of opinion that it is very 

 difficult to settle this point with certainty by direct obser- 

 vation, but from his own researches, he believes the diver- 

 gence in the terminal bud to be the same as in the stem. 

 From this important confirmation of my views, it appears 

 to me that the entire doctrine of an actual spiral motion 

 of the leaves, and the deduction from it of the oblique 



