16 
METHODS. 
The exactness of these measurements is indeed so great, that I detected, in 
working with this apparatus, certain slight errors in the micrometers employed, 
which had been obtained from a firm of excellent standing. These errors this 
firm itself found after having, at my request, reexamined the micrometers. 
6. Biometry. 
To utilize the measurements taken biometrically, all those of the same 
dimension in different individuals must be arranged in groups of suitable extent. 
In each group all the measurements lying between certain limits are placed, 
and the number of dimensions lying between these limits ascertained. These 
numbers are then plotted on equidistant ordinates in a graph and the points 
thus obtained connected by a line. This line is the biometrical frequency-curve 
of the dimension studied. 
In the method generally employed the groups of dimensions represented 
by the ordinates of the graph are equal in range. That is to say, the mean 
dimensions of the groups to which the ordinates correspond form an arithmetrical 
progression like 1, 2, 3, 4, .... n. This method involves a systematic error which 
makes the resulting biometrical curve wrong and misleading. When a dimen- 
sion examined varies between limits small in comparison to itself, that error 
is slight and generally overlooked. When, however, as is the case in the amphi- 
discs of the Hexactinellida for instance, the dimensions examined vary so much 
that the largest may be twenty-five times as great as the smallest, the error 
leads to results so glaringly wrong that it is noticed at once. This error is caused 
by the equality of the extent of the successive groups and by their mean dimen- 
sions, which are represented by the ordinates of the graph, forming an arith- 
metrical progression. For it is obvious that a difference, say of 10 m, in the length 
of objects only 10-20 n long must be of far greater biological importance than 
the same difference of 10 n in the length of objects 500-510 /j. long. To avoid 
this error I divide the measurements taken into groups .of uniformly increasing 
range. The increment selected is such that the extent of each successive group 
is 10 % greater than the extent of the preceding group, so that the ordinates, 
which are also placed by me at equal distances in the graph, represent a series 
of mean-dimensions of groups which form the geometrical progression 1.1, l.l 2 , 
l.l 3 , l.l 4 l.l n . 
The biometrical curves obtained in this manner express identically the 
character of dimensional variation of all the individuals compared, however 
