340 RELATION OB" PHYLLOTAXIS TO MECHANICAL LAWS. 

 The angle is given by the formula : — 



^^^-^' of 360°. 

 m^ + w^ 



Thus for (1+ 2) the divergence angle = | of 360° = 216' or 144° 

 (2+ 3) „ „ T^ „ =138° 27' 42" 



(3+ 5) „ „ fi „ =137° 38' 50' 



(5+ 8) „ „ 11 „ =137° 31' 41" 



(8 + 13) „ „ m •. =137° 30' 38" 



while the limiting angle 



^^l^p^ =137° 30' 28" 



Li 



The " ideal angle " of the Fibonacci series remains the same as 

 in the Schimper-Braun series ; but the angles for lower members 

 of the series are not only very different from the conventional 

 series, but they are definitely very much more like the angles 

 obtained in measurements of plant specimens. 



The standpoint of the Bravais, that there might be quite reason- 

 ably only one angular divergence for normal Fibonacci phyllotaxis, 

 and that one the " ideal angle " of Schimper, is thus seen to 

 be well within the experimental facts. Since, it may be again 

 pointed out, these angles hold for growing systems, they do not 

 hold for systems which show progressive cessation of growth ; but 

 so long as growth proceeds uniformly in the system, that is to say, 

 the nearer the apex of a plant approximates uniform growth, so 

 will these angles be the true angles of phyllotaxis, and will be 

 found well within any error of observation on the plant. The 

 (1 + 2) system alone differs from the ideal angle by about 7°, and 

 as already noted, a (1 + 2) system which can be regarded as 

 approximately exhibiting uniform growth is not readily obtained, 

 owing to the effect of growth-retardation and secondary cessation. 

 It will be noticed that already at the (2 + 3) system the "ideal 

 angle" may be attained within an error of about one degree 

 (c/. fig. 94) in a primordium which subtends 100° ; while an error 

 of one per cent, is practically beyond consideration in dealing with 

 the plant. A slightly higher ratio (5 + 8) gives the "ideal angle" 

 within a theoretical error of owe minute. While the possible 



