XIV] OR PHYLLOTAXIS 927 



Less tnan two there can obviously never be. When there are 

 exactly two, we have the simplest of all possible arrangements, 

 namely that in which the leaves are placed alternately on opposite 

 sides of the stem. When there are more than two, but less than 

 three, we have the elementary condition for the production of the 

 series which we have been considering, namely 1, 2; 2, 3; 3, 5, etc. 

 To put the latter part of this argument in more precise language, 

 let us say that: If, in our descending series, we come to steps 1 

 and t, where t is determined by the condition that 1 and t + 1 would 

 give spirals both right-handed, or both left-handed; it follows that 

 there are less than t + 1 leaves in a single turn of the fundamental 

 spiral. And, determined in this manner, it is found in the great 

 majority of cases, in fir-cones, and a host of other examples of 

 phyllotaxis, that t= 2. In other words, in the great majority of 

 cases, we have what corresponds to an arrangement next in order 

 of simphcity 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 completely 

 the so-called mysterious appearance of terms of the recurring series 

 1, 2, 3, 5, 8, 13, etc.* The other natural series, usually but mis- 

 leadingly 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 

 recorded, as more or less rare abnormalities, by Dickson f and other 

 writers. 



We have now learned, among other elementary facts, that wherever 

 any one system of spiral steps is present, certain others invariably 

 and of necessity accompany it, and are definitely related to it. In 

 any diagram, such as Fig. 451, 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 



* 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, 1863, pp. 358-372. Leslie Ellis, Whewell's brother-in-law, was 

 a man of great originahty. He is best remembered, perhaps, for his views on the 

 Theory of Probabilities (cf. J. M. Keynes, Treatise an Probabilities, 1921, p. 92), 

 and for his association with Stebbing as editor of Bacon. 



t Proc. R.S.E. vn, p. 397, 1872; Trans. E.S.E. xxvi, pp. 505-520, 1872. 



