482 Church. — Note on Phyllotaxis. 



have at different times been proposed to show why this should 

 be so ; these again agree in taking the fractional expressions 

 as representative of some mathematical law, all deviations 

 from which must be due to the action of secondary forces, 

 real or hypothetical. Such speculations include the original 

 prosenthesis theory of Schimper and Braun, various torsion 

 and displacement theories, culminating in the contact-pressure 

 theory of Schwendener. These various views have been 

 recently critically examined by Winkler (Pringsh. Jahrb., 1901, 

 Heft I). 



Since the general plan of these investigations consists, how- 

 ever, in superimposing some new hypothesis on the original 

 conception of Schimper and Braun, a strict analysis of the 

 subject demands a preliminary investigation of the views of 

 Schimper and Braun and the scientific evidence underlying 

 these fractional expressions, which become translated into 

 accurate divergence-angles of degrees, minutes, and seconds. 

 So long have these numbers been accepted that it appears 

 somewhat gratuitous to point out that these generalizations 

 rest on no scientific basis whatever, and that what passed for 

 evidence in 1830 does not necessarily hold at the present day. 

 Thus Schimper and Braun elaborated these expressions of 

 divergence on the plan of the original f or quincimcial system 

 proposed by Bonnet in 1754. The starting-point in dealing 

 with phyllotaxis is therefore the elucidation of the exact point 

 of view of Bonnet, which has determined the path along 

 which all subsequent investigation has proceeded. Now 

 Bonnet, who had the assistance of the mathematician Calan- 

 drini, studied adult axes only, and devised, as an expression 

 of the facts observed on elongated leafy shoots, a helix winding 

 round a cylinder and spacing out at equal angles five members 

 in two complete revolutions, the sixth member falling on the 

 same vertical line as the first ; a simple mathematical concep- 

 tion was thus utilized to express the observed phenomena. 

 The fact which Bonnet thoroughly understood, that on a plant- 

 shoot the sixth leaf did not fall exactly over the first, but that 

 the series formed by every fifth leaf itself wound along a spiral 



