840 SEMI-CENTENNIAL OF TORREY BOTANICAL CLUB 
per umbel in the Umbelliferae all follow the series of Fibonacci 
(1, 2, 3, 5, 8, 13, 21, etc.), the maxima differing for different species, 
or for different races of the same species. He sought to relate 
deviations from this series to the ‘‘Nebenzahlen”’ of the series, 
39, 42, 55, 63, which may be considered as “duplica” or 
triplica” of certain of the main series (7о7а). 
However, in discussing various views regarding phyllotaxy, 
Ludwig (’97b) suggests the development of other series than the 
Fibonacci (1/2, 1/3, 2/5, 3/8, 5/13, etc.). The Trientalis, for 
example, differs in giving the series 1/3, 1/4, 2/7, 3/11, 5/18, etc. 
Evolution in regard to flower number was thus held by Ludwig 
to be discontinuous, so that the various species in a phylogeny 
represent a series of discontinuous variations with values for 
flower number which depend on that of the original species. 
The number of flowers realized in ontogeny was considered to 
be determined first by the divisions initiated in the mother organ 
("Mutterorgan", and, second, by the processes that determine 
phyllotaxy. The suggestion is made that the development of a 
flower head or of the number of rays involves one complete turn 
of the spiral. Ludwig, however, does not attempt to correlate 
flower number with the phyllotaxy of the species, and in the Com- 
positae he does not find maxima that correspond to any other than 
the Fibonacci series or duplica or triplica of its various numbers. 
Ludwig's later ('95). theoretical conceptions of the morpho- 
genetic processes involved in the development of the different 
numbers of ray-flowers are based chiefly on the observations of 
Otto Müller ('83) on Melosira. In this diatom the individual 
cells remain attached, forming filaments. The development of 
the filamentous colony Müller claims to be as follows. Cell 
division, as always in diatoms, occurs in such a way that of the 
two daughter cells one is larger. The larger then divides while 
the smaller one rests. Then the latter divides simultaneously 
with the larger of the newer pair. Thus one cell divides, giving 
two; of these, one divides making three cells in the filament; two 
of these next divide, making five in all; three of these divide next, 
making eightinall,etc. It is thus claimed that there are rhythmic 
and periodic divisions in which all the older cells divide together 
with one half of the newer cells. Asa result, the number of cells 
