6 RELATION OF PHYLLOTAXIS TO MECHANICAL LAWS. 



from the central commonest type (§), the quincuncial system of 

 Bonnet. The essence of the Schimper-Braun theory, however, 

 consists in the fact that these ratios of the numbers of members 

 (denominator) to the turns of the spiral (numerator) being thus 

 expressed in fractional form, become reduced to angular measure- 

 ments expressed in degrees of arc (the divergence), and that a single 

 genetic spiral controls the whole system. 



When expressed in degrees, these divergences show an oscillation 

 between i and J, or 180° and 120°, towards a central station of 

 rest, an angle to which the term "ideal angle" was applied by 

 Schimper.* 



Thus, I = 144° 

 1 = 135° 



-/3 = 138°27'41"-o4 

 ^ = 137° 8'34"-28 

 if = 137°38'49"-41 



§1 = 137° 27' 16"-36 

 H = 137° 31' 41"-12 



1^ = 137° 30' 0" 

 "Ideal angle " = 137° 30' 27"-936 



It will be noticed that the differences become extremely minute 

 in the higher fractions, and that beyond ^ the difference is much 

 less than one degree of arc ; an angle quite impossible of observation 

 on most plants or of%,ccurate marking on a small diagram.-]- 



No satisfactory attempt could be made at measuring the angles ; 

 in fact, the brothers Bravais came to the conclusion that within the 

 error of observation all these higher divergences might be due to a 

 constant angle.+ 



structure to the fact that 5 was a member of the series. Cf. Ludwig, " Weiteres 

 iiber Fibonacci -curven," Bot. Centralh. Ixviii. p. 7, 1896. 



* It will be noted that Schimper's formulae are based on the type of the 

 quincuncial system (f ) of Bonnet. The construction proposed by the latter, with 

 the co-operation of the mathematician Calandrini, was that of a heUx drawn on a 

 cylinder. Such a system transferred to the plane representation of a floral 

 diagram, become a spiral of Archimedes, in which the sixth member falls on the 

 same radius vector as the first. The parastichies differing by two or three re- 

 spectively will similarly be Archimedean spirals. The truth of these systems 

 wiU therefore stand or fall acording as constructions by means of spirals of 

 Archimedes, derived from a consideration of adult cylindrical shoots, will explain 

 the facts observed' in the actual ontogeny of the members. 



t Cf. Bravais, Ann. Sci. Nat, 1837, pp. 67-71. 



f Cf. C. de Candolle, Throne de tangle unique en Phyllotaxie, 1865. 



