Note on Phyllotaxis. 
BY 
ARTHUR H. CHURCH, M.A., D.Sc., 
Lecturer in Natural Science , Jesus College , Oxford , 
With two Figures in the Text. 
RITERS on Phyllotaxis are generally agreed in 
V V accepting the series of formulae known as the 
Schimper-Braun series of divergences, f, |, T 5 ^, &c., as 
fundamental expressions of the primary phenomena of the 
arrangement of lateral members. This series of fractional 
expressions, which involves the utilization of the Fibonacci 
ratio series 2, 3, 5, 8, 13, &c., has thus proved for over sixty 
years the ground- work of all theories of phyllotaxis, and is 
usually described in the early pages of textbooks. Taking 
the ‘ f ’ as a type of these values, this expression implies that 
in placing five members on a spiral which makes two complete 
revolutions of an axis, the sixth member is mathematically 
superposed to the first, and that successive members differ by 
a divergence-angle of 144 0 . So simple are these relations and 
so thoroughly well known that it is not necessary to dwell 
further on the vast superstructure of morphological theory 
which has been built up on this foundation. However, as 
a matter of fact, taking the § divergence again as an example, 
it is beyond doubt that observation of the actual plant shows 
that these relations do not strictly hold, and various theories 
[Annals of Botany, Vol. XV. No. LIX. September, 1901.] 
