318 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



construction is not easy to find, owing to the fact that members 

 conventionally recognised as leaves always subserve special 

 functions, if developed to any extent, for which they become 

 secondarily adapted ; but the possibility of the conception of 

 such a theoretical and mathematical primary type of leaf-member 

 becomes increasingly clear, although now in a sense somewhat 

 different from that of the Ideal Leaf of Goethe. A remarkably 

 close approximation to such a hypothetical structure is found in the 

 protuberances associated with the primary leaf-points of a typical 

 Mamillaria. These conoidal masses, which physiologically replace 

 the primary leaves, are secondary outgrowths which certainly re- 

 present a leaf "idea" in the plant, worked out in a primitive 

 mechanical form : their special point of interest is that they 

 convey a very clear conception of the theoretical system, in that 

 they, unlike the primary leaf-appendages, have usually no special 

 tendency to become obviously bilateral. Thus Mamillaria, having 

 lost its primary " dorsiventral" assimilating members, in repeat- 

 ing its lateral outgrowth scheme a second time, expresses itself in 

 terms of the simplest possible type of appendage. 



Again, if growth in the whole system were uniform in the 

 transverse plane, all primordia, once they were formed, would 

 persist with their bases always in the same plane, and all would 

 appear in transverse section ; but owing to special phenomena of 

 retardation in the main axis, they fall back along it, and the tips 

 of the peripheral members gradually become lost in transverse 

 section of the apex : similar phenomena will also represent the 

 general result of an intercalated " second zone of elongation." 

 It is almost unnecessary to state the case in which growth is 

 uniform between axis and primordium {dormant centres), although 

 by assuming the proposition of uniform growth it becomes possible 

 to check the aberration from such a theoretical construction ; and 

 the rate of growth in the lateral members in the transverse 

 direction cannot be much greater than that of the axis at their 

 bases, since their insertions constitute the surface of the axis, 

 though at a higher level the growth may be unequally distributed 

 (bilateraliiy). The longitudinal rate may, however, be considerably 

 greater, and in such case the lateral members close over the apex 



