446 SEMI-CENTENNIAL OF TORREY BOTANICAL CLUB 
DISCUSSION AND CONCLUSION 
The aim of the earlier investigators who made statistical studies 
of number of flowers per head was to obtain exact descriptions of 
species assuch. By determining the number that is characteristic 
of various species, the degree of specific differentiation in respect 
to the character of flower number would appear. The results 
of the various statistical studies (of Ludwig especially) of flower 
number, although chiefly directed to ray-flowers, have been con- 
sidered as indicative of specific differentiation, as noted in the 
review of literature. That the number of rays is not closely 
or rigidly stable is very evident, however. A rather wide range 
of variability is nearly always in evidence, and this has led investi- 
gators to judge a species by some such value as the average or 
modal number. When thus judged, it seems clear that species 
differ and that a species may tend to maintain a particular maxi- 
mum. The studies with chicory, viewed from this standpoint, 
indicate that the total number of flowers per head may range in 
this species from 5 to 34 and that the number per head most fre- 
quently produced is somewhere from 17 to 19. 
The view held by Ludwig maintains that the numbers for 
ray-flowers characteristic for species as such conform to such a 
series as that of Fibonacci, hence there is proof of discontinuous 
variation or mutation in the evolution of species. However, 
the data are not fully in accord with this view. It appears that 
there are some maxima that fall on certain primary numbers of 
the Fibonacci series and such cases do suggest that fundamental 
rhythmic processes of development тау: have occurred giving spe- 
cific series of numbers for such organs as rays in flower heads. That 
all flower-number differences seen among and within species thus 
arose is not indicated, for there are evidently numerous cases of 
maxima that do not fall on either primary or well-recognized 
secondary numbers of such series. It will readily be recognized 
that the indiscriminate statistical study in populations of such a 
variable character as number of flowers per head centers the atten- 
tion on genéral or average performance and fails to recognize 
the effects of individual and partial variability of a character. 
Furthermore, a study of individual and partial variabilities 
