XIV] OR PHYLLOTAXIS 647 



great majority of cases, we have what corresponds to an arrange- 

 ment next in order of simplicity to the simplest case of all : next, 

 that is to say, to the arrangement which consists of opposite and 

 alternate leaves. 



"These simple considerations," as Tait says, "explain com- 

 pletely the so-called mysterious appearance of terms of the 

 recurring series 1, 2, 3, 5, 8, 13, etc.* The other natural series, 

 usually but misleadingly represented by convergents to an infinitely 

 extended continuous fraction, are easily explained, as above, by 

 taking ^ = 3, 4, 5, etc., etc." Many examples of these latter series 

 have been given by Dickson f and other writers. 



We have now learned, among other elementary facts, that 

 wherever any one system of helical spirals is present, certain 

 others invariably and of necessity accompany it, and are definitely 

 related to it. In any diagram, such as Fig. 326, in which we 

 represent our leaf-arrangement by means of uniform and regularly 

 interspaced dots, we can draw one series of spirals after another, 

 and one as easily as another. But in our fir-cone, for instance, 

 one particular series, or rather two conjugate series, are always 

 conspicuous, while the others are sought and found with com- 

 parative difficulty. 



The phenomenon is illustrated by Fig. 327, a — d. The ground- 

 plan of all these diagrams is identically the same. The generating 

 spiral in each case represents a divergence of 3/8, or 135° of 

 azimuth ; and the points succeed one another at the same succes- 

 sional distances parallel to the axis. The rectangular outlines, 

 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 



* The necessary existence of these recurring spirals is also proved, in a 

 somewhat different way, by Leslie Ellis, On the Theory of Vegetable Spirals, in 

 Mathematical and other Writings, 1853, pp. 358-372. 



t Proc. Roy. Soc. Edin. vn, p. 397, 1872; Trans. Roy. Soc. Edin. xxvi, 

 p. 505, 1870-71. 



