40 KANSA>s CITY REVIEW OF SCIENCE. 
In order to represent these chances to the eye, divide the horizontal line 
of Fig. 1 into six equal parts, laying off at right angles to it, and at the points of 
division, the vertical lines whose lengths are proportional to the number of draws 
for the various cases. The fact that the continuous curve passed through the 
extremities of these vertical lines, is symmetric on each side of the middle, means 
that white balls and black balls stand an equal chance of being drawn. If the 
black balls were less in number or were smaller, and had a tendency to sink to 
the bottom, then the curve would be unsymmetric, the probability being in favor 
of the white balls.* 
We come now to the application of what we have here learned. 
If any dimension—for instance the length of an iron rod—be measured with 
the greatest attainable accuracy, the successive measurements will, in general, all 
disagree. Taking a great many such measurements, all differing from each other, 
which shall we adopt as the true value? We may assume that all the observations 
are made with equal care; one can have no preference over another. ‘The fact 
that they all disagree, and that the tendency to disagree increases with the deli- 
cacy of the determination, is sufficient to show that no one observation can be 
adopted as the true value; and further, that it is impossible for us ever to deter- 
mine the true value. The best that we can do is to take the mean of all determi- 
nations. ‘To this mean we give greater or less weight, according as the separate 
determination more or less nearly approximates to the mean. 
In order to illustrate this point we may take the 470 determinations of the 
right ascension of the star Sirius, as made by the astronomer Bradley. In the 
following table d represents the difference between the observation and the mean 
of all observations (in tenths of a second); n represents the number of times the 
respective differences were found to occur. 
The positive and negative distances are thrown together without regard to 
sign : . 
es 
5 
Oo wn AM BPW DN 
on 
Lt 
DOW ta aie ee iy 
Tit Wal Oye a 4 
Zz. €., Out of 470 observations, 94 showed differences of less than 1-10 second, 88 
fell between 1-10 and 2-10, etc. As was to have been expected, the greater 
*Jn this case the terms of the binomial would be no longer equal. 
