BRITISH SPECIES OF THE GENUS MNIUM. 23 
specific. As the character under consideration is variable in each species, 
it is impossible to say to which of the two belongs a stem, the longest 
leaf of which has a length intermediate between the two figures ; and if the 
individual figure of the unknown stem is, for instance, 4 mm. or 8 mm., we 
can give no more than a guess at its identity. 
$15. MaxiMAL AND MINIMAL VALUES.—In a series of measurements 
representing the individual variation of a given property of a species, the 
MAXIMAL value and (with certain restrictions) the minimal value have a 
more definite BIOLOGICAL significance than the mean value. Moreover, 
it is possible to find these two limits, once for all, by a sufficient number of 
Observations. "THEREFORE THE MAXIMAL AND MINIMAL VALUES ARE THE 
CHARACTERISTIO CONSTANTS OF THE SPECIES. 
Adopting this view, I deviate from the classical principle of the mean 
value introduced in biometrical science by Quetelet (1846). This principle, 
borrowed from the theory of errors, is quite justified from a mathematical 
standpoint, and may do good service in certain cases (see $14); but from a 
biological point of view a mean value is very often deceptive. 
There are serious physiological reasons for considering the minimal and 
maximal limits as constants and for ascribing to them a definite biological 
and mathematical significance. I hope to expound this subject more 
completely in another publication *. 
$16. PRACTICAL USE OF THE LIMIT VALUES.—In the present paper I want 
to content myself by pointing out the practical importance of the minimal 
and maximal values for the description of species and the identification of 
specimens. 
We take, as first example, the length of the leaves of the tenth interval 
of the fertile stem of two species, Mnium cuspidatum and M. hornum. The 
measurement of a certain number of specimens, belonging to patches collected 
in several localities, has given the following limits :— 
cuspidatum. iTia MESE 373 mm. 7'44 mm. 
hornum :.......- ERU. DIT 80817 
In this case the variation is transgressive ; in other words, the two series 
(variation eurves) overlap each other. 
Let us suppose that we want to identify a specimen a, and that we are 
hesitating between the two mentioned species. The length of the longest 
leaf of w is 5:33 mm. ; this figure being between the minimum of hornum 
and the maximum of cuspidatum, it is impossible to make a choice. As the 
examples of transgressive variability are very numerous, a similar difficulty 
* I beg the reader of the present paper to accept as a postulate that the maximal valne 
and (with certain restrictions) the minimal value are specific constants. 
