306 Sources of apparent Polymorphism in Plants, etc. 
misinterpretation of the criterion of significance, i.e. is not true multi-modalism at 
all, or, if such, is due to some heterogeneity of period, of season or of environment 
introduced by the gatherer. 
It is not contended that there is not a great deal of true polymorphism in 
plants — this is beyond dispute — only its true appreciation can scarcely be realised 
until the influences of environment and of stages in season on the modal values 
have been exhaustively studied. Take the case of the Fibonacci series 
8, 5, 8, 13, 21, 34, 55, etc., 
which has been so fully considered by Dr Ludwig*. There is no doubt that these 
numbers recur with somewhat remarkable persistency in the plant kingdom. 
Each number is the sum of the two immediately preceding it, and there may 
well be some mechanical explanation of the building up of flowers, by which type 
added to type is more probable than progression by units. But if we take 
Chrysanthemum leucanthemum itself, we can, not only pass at different periods of 
the season from one Fibonacci mode to a second, but other modes also come in 
which tend to destroy our faith in the absolute truth of the Fibonacci series. 
Mr Tower's second mode is quite definitely at 33 and not the Fibouacci 34. He 
has also in both his sub-groups an apparent mode between 22 and 25, which is 
only screened and not lost when he combines them. His series 284 and 168 
are somewhat small for this sort of work, but Mr Yule and one of the editors found 
in 1133 heads gathered in 1895, during some weeks in the Lake District (Keswick), 
a distribution which indicated modes between 14 and 16 an<l between 24 and 26, 
as well as the widespread typical mode of 21. 8, 13 and 34 as modes were 
unrepresented t- In Dr Ludwig's own classical series of 17,000 heads there is a 
significant mode between 24 and 26. Whatever be the value of the Fibonacci 
series, it seems impossible to look upon it as providing the only numbers which 
can arise as modes for the rays of the ox-eyed daisy. Its real significance can 
only be tested when all the flowers on a given small area are observed throughout 
the season, and the number of rays counted and the date noted as each flower 
opensj. Only thus shall we be able to test whether the mode changes continu- 
ously during the season, or springs from one number to a second, and, if the latter, 
whether these numbers are or are not really the Fibonacci series. 
* See especially Lis "Ueber Variation scurven uud Variationsflacheu der Pflanzen," Botaninches 
Centralhlatt, Bd. lxiv. 1895. 
t See p. 319 below. 
X Even in this case the flower should not be cut off, but effectively marked. The removal of flowers 
may tend to influence the characters of later flowers on the same plant. 
