STOUT & Boas: STATISTICAL STUDIES IN CicHORIUM 341 
in the filament increases according to the Fibonacci series, I, 3, 5, 
5. 15, 21, etc. 
On the basis of these observations, Ludwig explained the occur- 
rence of maxima for ray-flowers, which he considered to correspond 
to the series of Fibonacci,on the hypothesis that in the develop- 
ment of such organs as ray-flowers, one part is like the mother 
organ and another is like the offspring. The mother organ forms 
new parts in rhythmic succession, the offspring goes through a 
ripening period and then divides. In Ludwig's own words, “ Das 
Mutterorgan grenzt fortgesetzt in rhythmischer Wiederholung 
neue Teile ab, der Sprossteil dagegen immer erst in der folgenden 
Teilungsperiode, nachdem derselbe herangewachsen ist." (Опо- 
tation from Vogler, 712) Тһе term “mother organ” is used 
vaguely. Whether it is considered as a fertilized egg or as the 
apical growing-point in the main stem or in lateral stems is not 
stated. What significance partial variability may have is not 
discussed. 
Ludwig later (’04) brings this conception forward in support 
of the mutation theory which had then recently been announced 
by de Vries. The emphasis was laid on the discontinuous increase 
in the number of cells and organs involved in such rhythmic 
divisions. In the Compositae, for example, the various species 
are considered to represent different steps in a series, the number 
of cell divisions are assumed to stop at certain points and become 
hereditary. However, in referring to de Vries's studies, later to 
be discussed here, Ludwig ('o4) considers that it is possible to 
change by cultivation the stage or step which has been reached by 
a species. 
Аз a theory of morphogenesis, the conception is interesting 
and suggestive. It is not indicated, however, how a series of 
rhythmic linear divisions, such as may occur discontinuously in 
flamentous forms, is to be applied to complex growing-points 
involving various histogens where various groups of cells are con- 
cerned in the production of an organ, as in the case of the formation 
of a composite flower head, and especially in the application to 
the number of differentiated ray-flowers constituting only a part 
of the head. Furthermore, the correlation of characteristic num- 
bers for a species with the phyllotaxy is not attempted. 
