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UN ie darm E ing es oy ag 
18 PROF. JULIUS MACLEOD ON TEN 
Individual variation depends on chance, but the variation of the twenty-seven 
leaves in fig. 2 is independent of it *. 
The two forms of variability are often confounded, and the fruit of many 
laborious researches has been spoiled by that confusion. 
Let us suppose that we want to study the variation of the length of the 
leaves of the fertile stem of M. hornum (or any other species). If we 
collect and measure a certain number of leaves taken at random from a 
certain number of stems, and if we construct a variation curve by means of 
the collected figures, we bring together material which is not homogeneous. 
The variation depends here on two causes—chance and gradation,—the 
effects of which are not governed by the same laws. If we start from the 
idea that variation, in this and all similar cases, is merely governed by the 
law of chance, the calculation of a mean value and other more complicated 
computations will give artificial results. 
As this question is important, I may be allowed to give one more example. 
In Table XII. one finds the mean interval curves of four characters of the 
leaves of Mnium orthorrhynehum. 
‘TABLE X Ll 
Mnium orthorrhynchum, Bruch, Schimp. & Guemb.— Mean interval curves of 
four characters of the leaves of the fertile stem. 
Intervals: 1. 2. 3. 4, 5. 6. i 8. 9. 10, 
Length (mm............. 076 0:99 193 141 159 2:06 234 291 319 396 
Breadth (mm) .......... 0:36 042 050 056 0-65 084 093 0:93 083 061 
PON. B et i3 jj HH 14 M 14 1 
Marginal teeth (number) .. 1 2 5 7 dO 145 . 21:960 96 920 
Although only four characters are taken into account in this example, the 
enormous differences existing between the leaves of the ten intervals are 
obvious (compare, for instance, the combination of figures in the vertical 
column 3 with the combination in column 8, etc.). The knowledge of the 
gradation curves is our leading clue among this disconcerting variation, 
which does not depend on chance. 
Remark II.: The principle of gradation enables us to answer the question, 
which leaf of a given specimen is comparable with a given leaf of another 
specimen : the leaves which are situafed at the same degree of the 
gradation axist are comparable.— Ezample: In a stem with 15 leaves 
(from the lowest leaf to the longest one) the 8th leaf (numbered from 
below) is at 500° (see the method of computation in $6) ; in a stem with 
33 leaves the 17th leaf is also at 500°, and is thus comparable with the 
8th leaf in the first stem. In the same way, the longest leaves of all 
the stems are comparable, as they are all at 1000? (see $ 3). 
* We leave out of account the fortuitous irregularities of the curves (see $ 21. 
T In other words, the leaves which have the same value « (see § 2). 
