WILCOX: SPERMATOGENESIS. 21 
I have used the word “reduction” without indicating the particular 
sense in which I use it. The definition of reduction proposed by Weis- 
mann (92), and adopted by vom Rath, Hacker, and others, is that which 
I prefer, and according to which I have used the term. This is: * Unter 
Reductionstheilung verstehe ich eino jede Kerntheilung durch welche die 
Zahl der Ide welche im ruhenden Kern vorhanden war, für die Tochter- 
kerne auf die Hälfte herabgesetzt wird." It is not necessary to adopt 
Weismann's terms, “ids, idants,” etc., in order to use his definition, If 
the developmental possibilities are only one half as great in the daughter 
nucleus as in the mother nucleus, there has been a reduction in Weis- 
mann’s sense. If a nucleus contains four elements which happen to be 
two pairs of identical elements, the formula would be 
Now, if the division takes place along the line € y, there is a reduction 
in Weismann's sense ; but if the division be along the line m n, it is an 
equation division. Either division would reduce the chromatic mass, 
but only the first would reduce the number of different elements (ids) 
in the daughter as compared with the mother cell. 
Since the rings, or Vierergruppen, have already been found in the 
oógenesis and spermatogenesis of numerous species in different groups, 
this arrangement of the chromatin just before the maturation divisions 
is certainly very general, if not practically universal. In order, there- 
fore, to interpret properly these two divisions, and to come to any 
sound conclusions with regard to the reduction question, it is of funda- 
mental importance carefully to study the formation of the Vierergruppen. 
Häcker (93) and vom Rath (93) have already called attention to the 
fact that the double longitudinal splitting of the chromatic thread, main- 
tained by Boveri and Brauer, must bring about groups of four identical 
d 
elements. The formula for a Vierergruppe would then be isa]: 
There could not in this ease be any Weismannian reduction in either 
division, for there is only one kind of id in all the four elements of the 
group. If the Vierergruppen always arose as Brauer describes the 
process, — i. e. by two longitudinal splittings of the chromatic granules, 
which alone, he believes, possess an individuality, — then the four com- 
ponents of each group would be identical, and there could be no redue- 
