644 ON LEAF-ARRANGEMENT [ch. 



as to be considered "normal" and characteristic features of the 

 general phenomenon of phyllotaxis. The following account is 

 based on a short paper by Professor P. G. Tait*. 



Of the two following diagrams, Fig. 325 represents the general 



case, and Fig. 326 a particular one, 

 for the sake of possibly greater 

 simplicity. Both diagrams re- 

 present a portion of a branch, or 

 fir-cone, regarded as cylindrical, 

 and unwrapped to form a plane 

 surface. A, a, at the two ends 

 of the base-line, represent the 

 ^^ ggg same initial leaf or scale : is a 



leaf which can be reached from 

 A by m steps in a right-hand spiral (developed into the straight 

 line AO), and by n steps from a in a left-handed spiral aO. Now 

 it is obvious in our fir-cone, that we can include all the scales 

 upon the cone by taking so many spirals in the one direction, 

 and again include them all by so many in the other. Accordingly, 

 in our diagrammatic construction, the spirals AO and aO must, 

 and always can, be so taken that m spirals parallel to aO, and n 

 spirals parallel to AO, shall separately include all the leaves upon 

 the stem or cone. 



If m and n have a common factor, I, it can easily be shewn that 

 the arrangement is composite, and that there are I fundamental, 

 or genetic spirals, and I leaves (including A) which are situated 

 exactly on the line Aa. That is to say, we have here a whorled 

 arrangement, which we have agreed to leave unconsidered in 

 favour of the simpler case. We restrict ourselves, accordingly, 

 to the cases where there is but one genetic spiral, and when 

 therefore m and n are prime to one another. 



Our fundamental, or genetic, spiral, as we have seen, is that 

 which passes from A (or a) to the leaf which is situated nearest to 

 the base-line Aa. The fundamental spiral will thus be right- 

 handed {A, P, etc.) if P, which is nearer to A than to a, be this 

 leaf — left-handed if it be p. That is to say, we make it a con- 

 vention that we shall always, for our fundamental spiral, run 

 * Proc. Roy. Soc. Edin. vn, p. 391, 1872. 



