638 ON LEAF-ARRANGEMENT [ch, 



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. 



The first of these appears to me to present no diflEiculty. It 

 is a mere matter of strictly mathematical "deformation." The 

 stem which we have begun to speak of as a cylinder is not strictly 

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

 which Avinds 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 logarithmic spiral, and which is in 

 fact a logarithmic 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 necessarily becomes in the disc a logarithmic spiral 

 described about a focus 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, of course, equally true, and equally obvious, namely that any 

 logarithmic 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 cone 

 approximating to a cylinder. This mathematical conception is 

 translated, in botany, into actual fact. The fir-cone may be 

 looked upon as a cylindrical axis contracted at both ends, until 



