344 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



construction; and when this condition becomes the rule, it is 

 possibly the result of unfavourable conditions of environnaent, and 

 may thus be correlated, as in the case of Cacti, etc., with other 

 xerophytic peculiarities. 



Note VII. — Continued FradioTis. 



Since the time of Schimper and Braun much importance has 

 been attached to the formulation and presentation of the ratios 

 commonly found in plants in the form of a summation-series 

 presenting certain mathematical properties; the ratios being 

 successive values of the stages of a continued fraction, the limiting 

 value of which became expressed as an " ideal angle." Hence 

 mathematical statements became read into the subject with which 

 Botany has nothing whatever to do. The formation of these 

 summation-series from observation of the plant kingdom, which 

 represents the great botanical discovery of Schimper on which all 

 his contributions to the theory of phyllotaxis were based, is a 

 mathematical conseqioence of the phenomena of intersecting spi/ral 

 curves radiating round a central point. The preceding geometrical 

 diagrams have rendered this sufficiently obvious (c/. figs. 25, 26). 

 Thus, if a certain number of curves cross another set, the same 

 points of intersection will be mapped out by two other sets re- 

 presenting the sum and difference of the first set (c/. p. 56) ; or, 

 if m curves cross n, {m + n) and (m — n) curves will also pass 

 through the same points and form diagonals of the original meshes : 

 four terms of a summation-series are thus involved, and other 

 terms may be obtained in the manner already described. Given 

 the intersecting curves, the mathematical manipulation and 

 description of continued fractions becomes a feature with which 

 Botany has nothing to do, nor is it at all helpful in any direction. 

 Such expressions may attract the mathematician, but they repel 

 the botanist, and it is hoped that the method of constructing 

 geometrical diagrams of the types indicated, on which the rela- 

 tions of the numbers can be more directly traced, will tend to 

 eliminate these expressions from botanical literature, together 

 with the curious prosenthesis formulae of older writers and 

 the Dachstuhl angles of a later school. 



