XIV] PR PHYLLOTAXIS 929 



in each case represents a divergence of 3/8, or 135° of azimuth; and 

 the points succeed one another at the same successional distances 

 parallel to the axis. The rectangular outhnes, which correspond to 

 the exposed surface of the leaves or cone-scales, are of equal area, 

 and of equal number. Nevertheless the appearances presented by 

 these diagrams are very different; for in one the eye catches a 5/8 

 arrangement, in another a 3/5 ; and so on, down to an arrangement 

 of 1/1. The mathematical side of this very curious phenomenon 

 I have not attempted to investigate. But it is quite obvious that, 

 in a system within which various spirals are imphcitly contained, 

 the conspicuousness of one set or another does not depend upon 

 angular divergence. It depends on the relative proportions in length 

 and breadth of the leaves themselves; or, more strictly speaking, 

 on the ratio of the diagonals of the rhomboidal figure by which 

 each leaf-area is circumscribed. When, as in the fir-cone, the scales 

 by mutual compression conform to these rhomboidal outlines, their 

 inclined edges at once guide the eye in the direction of some one 

 particular spiral; and we shall not fail to notice that in such cases 

 the usual result is to give us arrangements corresponding to the 

 middle diagrams in Fig. 452, which are the configurations in which 

 the quadrilateral outhnes approach most nearly to a rectangular 

 form, and give us accordingly the least possible ratio (under the 

 given conditions) of sectional boundary-wall to surface area. 



The manner in which one system of spirals may be caused to 

 slide, so to speak, into another, has been ingeniously demonstrated 

 by Schwendener on a mechanical model, consisting essentially of 

 a framework which can be opened or closed to correspond with one 

 another of the above series of diagrams*. 



The same curious fact, that one Fibonacci series leads to, or 

 involves the rest, is further shewn, in a very simple way, in the 

 following diagrammatic Table (p. 930). It shews, in the first 

 instance, the numerical order of the scales on a fir-cone, in so-called 

 5/8 phyllotaxis; that is to say, it represents the cone unwrapped, 

 with the two principal spirals lying along the axes of a rectangular 

 system. Starting from 0, the abscissae increase by 5, the ordinates 

 by 8; or, in other words, any given number m = bx + d>y\ it is 



* A common form of pail-shaped waste-paper basket, with wide rhomboidal 

 meshes of cane, is well-nigh as good a model as is required. 



