918 ON LEAF-ARRANGEMENT [ch. 



precisely equivalent to one another; and we may call them A, B, C, 

 etc., and A', B', C\ and so on. These are the cases which we call 

 "whorled" leaves; or in the simplest case, where the whorl consists 

 of two opposite leaves only, we call them *' decussate." 



Among the phenomena of phyllotaxis, two points in particular 

 have been found difficult of explanation, and have aroused dis- 

 cussion. These are (1), the presence of the logarithmic spirals such 

 as we have already spoken of in the sunflower; and (2) the fact 

 that, as regards the number of the helical or spiral rows, certain 

 numerical coincidences are apt to recur again and again, to the 

 exclusion of others, and so to become characteristic features of the 

 phenomenon. 



As to the first of these, we have seen that the curves resemble, 

 and sometimes closely resemble, the logarithmic spiral; but that 

 they are, strictly speaking, logarithmic is neither proved nor capable 

 of proof. That they appear as spiral curves (whether equable or 

 logarithmic) is then a mere matter of mathematical "deformation." 

 The stem which we have begun to speak of as a cyhnder is not 

 strictly so, inasmuch as it tapers off towards its summit. The 

 curve which winds evenly around this stem is, accordingly, not a 

 true helix, for that term is confined to the curve which winds evenly 

 around the cylinder : it is a curve in space which (like the spiral curve 

 we have studied in our turbinate shells) partakes of the characters of 

 a helix and of a spiral, and which is in fact a spiral with its pole 

 drawn out of its original plane by a force acting in the direction of 

 the axis. If we imagine a tapering cylinder, or cone, projected by 

 vertical projection on a plane, it becomes a circular disc; and a 

 helix described about the cone becomes in the disc a spiral described 

 about a pole which corresponds to the apex of our cone. In like 

 manner we may project an identical spiral in space upon such surfaces 

 as (for instance) a portion of a sphere or of an ellipsoid ; and in all 

 these cases we preserve the spiral configuration, which is the more 

 clearly brought into view the more we reduce the vertical component 

 by which it was accompanied. The converse is equally true, and 

 equally obvious, namely that any spiral traced upon a circular disc 

 or spheroidal surface will be transformed into a corresponding spiral 

 helix when the plane or spheroidal disc is extended into an elongated 



