482 Church . — Note on Phy llot axis. 
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., 190 1, 
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 -§ or qumcuncial 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 
