338 SEMI-CENTENNIAL OF TORREY BOTANICAL CLUB 
Ludwig (’95) published a summary of all his studies, which 
involve twenty-six genera and over sixty species of Compositae. 
The maxima determined for the number of ray-flowers per head 
are given for the different species. Often the counts were made 
from only a few flower heads so that the maximum is frequently 
not to be considered as established, as Ludwig himself fully recog- 
nized. 
The largest number of counts made of any composite was of 
Chrysanthemum Leucanthemum. Data for 17,000 heads were col- 
lected in Europe at different places and at different times of the 
year over a period of several years. Ludwig concludes that 21 
ray-flowers per head is the number characteristic of this species. 
In no instance does he give data showing that a race with a dif- 
ferent maximum was found growing in isolation, but he considered 
that such races were indicated by secondary maxima obtained 
from data of mixed populations. 
Data for ray-flower number of approximately 12,000 heads of 
Bellis perennis (Ludwig, '98) were presented to show that maxima 
may occur on the “ Nebenzahlen" of the Fibonacci series, i. e., 
39. 42, 55, 63, which were considered as “duplica” or “triplica” 
of various numbers of the main series. The only maxima that 
can be considered as at all established for Bellis perennis are 34 
(one of the main series) and 42 (twice 21). Study was also made 
of total flower-number per head in B. perennis: 860 heads were 
counted and for these the range was from 63 to 233, a very large 
range of variability. 
For Chrysanthemum inodorum (Ludwig, '95), data for 1,000 
heads reveal a range in ray-flower number from 10 to 32 and a 
pronounced maximum at 21. 
The data for Chrysanthemum segetum (Ludwig, 'o4) are of 
special interest as this is the species in which de Vries later made 
selection experiments. Ludwig reports 750 counts made in one 
locality and 250 made in another. Of the 750 from the one lo- 
cality, 150 were made on July 29 and боо on August 30. The 250 
counts in the other locality were made on August 16. This is 
the only instance where Ludwig reports the dates of collections. 
However, the two series show no significant differences either in 
maxima or in range of variability; for the 750 counts the maximum 
