IDEAL ANGLES. 71 



biological conditions, acquires a greatly enhanced value, in that 

 the ratios naturally adopted by the plant for its parastichies are 

 those which, being the successive terms of the simplest continuous 

 fraction 1 



1 + 1 



1, etc., give the optimum approach to symmetry in 

 an asymmetrical system. 



Just as it has been previously shown that the plant in normal 

 asymmetrical phyllotaxis makes use of — 



(1) The optimum concentration system. 



(2) Those ratios of a set series which more nearly approximate 

 the symmetrical position of equality ; 



so also, 



(3) It utilizes that summation series of ratios which allows the 

 optimum approach to radially symmetrical construction. 



All these three factors appear then inherent in the protoplasm, 

 and wholly independent of extraneous forces. From the fact that all 

 of them are illustrated in Helianlhvs, for which a normal structure 

 was postulated, they may be regarded as constituting the funda- 

 mental principles of normal phyllotaxis ; while cases in which 

 any one of them happens to be omitted or impaired may be re- 

 garded as secondary and induced by subsequent speciahzation or 

 degeneration. 



Othee Seeibs. 



While the series (1 : 2), (2 : 3), (3 : 5), etc., thus becomes the 

 normal system for asymmetrical phyllotaxis, the fact that other 

 continued series have been proposed, and are generally accepted, 

 remains to be considered. At the same time, it must be pointed 

 out that their recognition m. virtue merely of the method of 

 " orthostichies " is wholly unreliable, and it is only in those eases 

 in which the fractions have been determined by the method of 

 parastichies that the ratios can be regarded as correct. Many 

 such cases were recognised by the latter method by the brothers 

 Bravais, though more recently the orthostichy method has been 

 considered the most important (Schwendener, Weisse). Further, 

 since such cases, though widely distributed, are relatively less 



