No. 422.] QUANTITATIVE STUDY OF VARIATION 115 
were carried out to the tenth decimal place, and wherever 
practicable they were checked. It is probable, therefore, that 
all calculations are correct to the fifth decimal place. 
In plotting the variations I have used the “method of rec- 
tangles" as best representing the actual conditions found. 
In those cases in which material was insufficient compared 
with the range of variation, I have doubled the classes, but 
in no case have I grouped more than two classes together. 
In this way the essential features of the curves have remained 
unchanged, while the lesser irregularities have been eliminated. 
It must be remarked, however, in regard to the grouping of 
classes, that the method should be used with much caution. 
By combining a sufficient number of classes every multimodal 
curve may be made monomodal. 
It is also essential, in doubling, that a definite plan be 
adopted in order that the results may be comparable. The 
writer followed the plan of throwing together the two classes 
nearest the mean. As the mean is a constant, this makes the 
resulting double classes of the various curves strictly compar- 
able. If I had begun at the lower limit of range instead of at 
the mean, the count of one more variate might have lowered 
the range by one, and thus have changed the combinations, 
with the result that the character of the curve might be much 
modified. 
IV. RESULTS. 
Aster shortii Hook. — Fig. 1 represents the frequency poly- 
gon of the bracts. The range, 28 to 49, was so great that the 
226 counts were insufficient, and it was deemed best to double 
the classes. In this way the curve becomes monomodal, with 
the mode on 36-37, and the mean on 36.800884. The coeffi- 
cient of variability was 10.727157. 
The frequency curve of the rays (Fig. 2) shows a remarkably 
strong mode on 13, 38 per cent of the variates falling into that 
class, with the mean on 14. This strong mode on 13 suggests 
at once Ludwig's (95,'96,'98) interesting results on Chrysan- 
themum leucanthemum L., etc.; but, as nowhere else in the 
Asters studied has there been any apparent tendency of modes 
