150 THE AMERICAN NATURALIST. [VoL. XXXVI. 
ausdehnt, und zur Bestimmung der constanten Mischcurve der 
Art ist eine sehr grosse Anzahl von Zahlungen nóthig." 
I agree with Ludwig that a large number of counts is 
necessary for the determination of the constants in material of 
this kind; but how great must it be? It is to be noted that, 
in his tables representing the variation in number of achenes 
in the heads of Ranunculus acris L., he gives the results of two 
sets of observations, each consisting of counts of 1000 heads. 
The maxima of the two multimodal curves formed, alternate 
with each other, and their summation gives but few maxima 
where they ought (?) to be, — the material is insufficient ; yet he 
makes virtue of the fact that the summation of Lucas’s data, 
only 831 counts, presents maxima upon the series of Fibonacci. 
This series of Fibonacci, which is of recognized importance 
in the phyllotaxy of flowering plants, should be accepted with 
caution as the key to all variation among plants. The members 
of the series, along with Ludwig's “ Unterzahlen," which are 
made up from the numbers of the Fibonacci series by multipli- 
cation or addition, —¢.g., 10 = (2 X 5), 29 = (8 + 21), etc., — 
include so large a proportion of all the smaller numbers that 
many modes must fall on or near one of them, even if there be 
no fundamental relation existing between this complex series 
and the number of floral parts or other organs under considera- 
tion. To account for modes which do not fall on any of these, 
Ludwig creates the ‘‘Scheingipfel,” which is formed by the 
overlapping of curves having their modes on adjacent numbers 
of the Fibonacci-Ludwig complex. Thus, if the maximum 
falls upon 9, it is a “ Scheingipfel" formed by the union of 
curves having maxima upon 8 and 10; if it fall upon 11, it is 
made up of curves having maxima upon 10 and 13, etc. It is 
evident that such a scheme will furnish an Pep Apamon f of 
almost any condition which might arise. 
7 In conclusion, it needs to be said that there remains much 
s to be done in determining the many causes of variation. The 
quantitative study of variation shows only the existing condi- 
tion of the material studied. It deals only with results, and if 
there has been no intentional selection of material, it indicates 
nothing as to the causes which have brought about those results. 
