12 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



organs ; but, in the latter, they pointed out that the axis was often 

 conical or circular: in such case the straight orthostichies were 

 wanting and successive cycles were not accurately superposed. 

 More complete acquaintance with the structure of growing-points 

 would have shown them that the first case was wholly unnecessary, 

 and that the second hypothesis, based on a cone which might be 

 flattened to a circular disk, was alone required. Again, in common 

 with Schimper and Braun, they shared the view that the lateral 

 members were equal in bulk, or might be expressed by points, when 

 in point of fact they present in development a gradated series. 

 They, however, arrived safely at the conclusion that in such systems 

 the construction could not be expressed by a fractional divergence, 

 but only by the number of interesting parastichies (sinistrorsum 

 and dextrorsum), and the figure drawn for the theoretical structure 

 of a Composite inflorescence is very nearly correct, although its 

 method of construction (probably by modified Archimedean spirals) 

 is not described. Still more remarkable was the care with which 

 they worked out the multijugate types, in which the fractional 

 expression was divisible by a common factor (2-3), and thus clearly 

 pointed to the presence of two or more concurrent genetic spirals, 

 a case not contemplated by the spiral theory of Schimper and 

 Braun. 



Eestricted to the doubtful method of orthostichies, the Bravais 

 followed Schimper and Braun in the elaboration of other sets of 

 divergence fractions.* 



Thus if J, J, §, f, etc., pointed, as stages of a continuous frac- 

 tion, to an ideal angle of 137° 30' 28", why might not there be a 

 complementary system J, ^, f, f, ^ pointing to 151° 8' 8" ? As 

 also J, I, f, ^, etc., leading on to an ideal angle of 99° 30' 6", and 

 i if, A, etc., to 77° 57' 19"! 



It is clear that by such hypotheses any fraction that can be 



counted may be regarded as a member of some system; and, as 



Sachs has pointed out, this degenerates into mere "playing with 



figures"; while no progress along such lines is possible when a 



physiological reason is asked for. StUl, these formulse were founded 



* Bravaia, Ann. Sci. Nat., 1837, p. 87 ; Van Tieghem, TraM de Botanique 

 p. 55, 1891. ' 



