41 



ess may be followed in detail in sections II and IV of the Chart. 

 If the primary spiral were a left-hand one, all of these features 

 would of course be reversed. 



4. The actual amount of change in the interfoliar arc required 

 in order to accomplish these transformations is astonishingly 

 small. The dimensions of the arcs of the three regular patterns 

 with which we are here concerned are given — as fractions of the 

 circumference — in the very formulas b} which we distinguish 

 them, namely, %, ^.3, and ^%4. The change required to accom- 

 plish the first transition is therefore the diiTerence between the 

 first and second of these fractions; and to accomplish the second,. 

 the difference between the second and third. Reduced to decimal 

 form these fractions become 0.400, 0.3846, and 0.3823 ; and the 

 dift'erences are 0.0154 and 0.0023 — the latter amounting only to- 

 three fourth of a degree of arc, or one seventh of the minute-in- 

 terval on the face of a watch. Yet this infinitesimal quantity must 

 be subdivided and distributed over perhaps half a hundred leaf- 

 intervals ! De minimis curat Natitra. 



It may seem difficult to account for decrease in the interfoliar 

 arc while the girth of the tree is rapidly increasing. But greater 

 girth simply means larger surface for the insertion of more 

 leaves ; and in the case of an endogenous and branchless tree like 

 our palm, it is imperative that no space be wasted — that the grow— 

 ing leaves be crowded together as closely as they can be made to- 

 stand. Such being the case, so long as increase in girth keeps 

 ahead of the demand for foot-space for larger leaves, the record 

 will indicate that fact in a changing pattern ; because even a con- 

 stant space on an increasing circumference subtends a diminish- 

 ing angle at the center. If demand for foot-space catches up with 

 increase of girth, that fact will appear as a pause in the shifting 

 ranks. If increase again gets ahead, change will begin again. If: 

 finally equilibrium is established, the pattern reached at that point 

 becomes permanent, and with it the interfoliar arc. Such in brief 

 seems to be the explanation of the strange metamorphoses we 

 have been watching.* 



*Among later theories concerning changes in leaf-patterns to which the 

 writer's attention has been kindly directed by the editor of Torreya, the one 



