8 PROF. JULIUS MACLEOD ON TEN 
a certain number of comparable individuals. The form of such a curve is 
governed by the laws of chance (Quetelet, Galton). 
The Gradation (variation along an axis) does not depend on chance, but 
on a certain cause (force or factor) which varies regularly all along the 
axis according to a certain law which is quite different from the laws of 
chance. In other words, in our example (fig. 1), the length y' of a given 
leaf 7 depends on (is a function of) the distance «' between the base of the 
leaf | and the origin o of the axis*. Chance has only an influence in 
producing slight irregularities of the curve, which are easily discerned when 
the curve is drawn. In fig. 1 such irregularities are less conspicuous than 
in the majority of the gradation curves (fig. 2). 
$3. ImpoRTANCE OF THE LONGEST LEAF.—In the fertile stem of all the 
species of Mnium the gradation curve of the length of the leaves is in a 
certain degree variable, but on the whole similar to the example represented 
in fig. l. The longest leaf of a given fertile stem (corresponding to the 
summit of the curve) is therefore comparable with the longest leaf of any 
fertile stem of any species of Mnium whatever (see $ 12, Remark II.). 
$4. SIMULTANEOUS GRADATION OF SEVERAL CHARACTERS ALONG THE SAME 
AXIS.— When we measure a second charaeter (for instance, the breadth of 
the successive leaves of a fertile stem) we obtain a second gradation curve.— 
Example: The gradation of the character length and the character breadth of 
the leaves of a fertile stem of Mnium serratum ; see Table II. and fig. 2. 
Tipun Lb 
Mnium serratum, Schrad. : a fertile stem with 27 leaves ; length and 
breadth of the suecessive leaves in mm. 
Leaves: 15 2. 9. 4. 5. 6. pa 8. 9. 
ength 5. ET 2:00.86. 2°06 2:15 2:28 3242 2:44 9:44 
Breadth .... a m f TOG CIO sol. DPI. 118. 115 
Leaves: 10. BE 12: 13. 14. 15. 16. 17. 18. 
264 2836 302 295 3819 309 3824 3:40 
Length ...... OYA 2 
Breadth... 1:36 1:27 1:35 1:34 1:20 i 140. 1:52 1: Or 
Leaves: 189. 90. 2]. 22. 93. 24. 25. 26. 27. 
Length ...... 29:99. 350. 905. | 9:84, 468. 497. AV 402 .- 3°22 
ibrendihie dece «1244. AD. 11:40. 1272, 1-24. 1:07, 080. 0:55 
Here the two characters are independent of each other with reference to 
their gradation: the maximal length corresponds to leaf 25, the maximal 
breadth to leaf 17. 
* The relation between the value y and the distance x may be expressed by y=fv. The 
nature of f is hitherto unknown. 
