CHOICE AND CHANCE. 41 
‘‘errors” are met less frequently than the smaller ones. The results are repre- 
sented in Fig. 2 of the plate where the values (d) are laid off on the horizontal 
line and estimated from the point marked zero (0). 
The line o 2 represents a certain positive ‘¢error,” and the distance from 2 
to the curve represents the number of times it is met in 100 observations. 0 2 in 
in the opposite direction represents an equal negative ‘‘error,” while the vertical 
line at the point 2 represents its frequency. It can here be observed how the 
frequency diminishes, as the magnitude of the error increases, until cells the 
curve intersects the base line, indicating that larger ‘‘ errors” do not occur.* 
The law expressing the divergence of these values from the mean is the same 
as that which expresses the probability in drawing balls from an urn containing an 
equal number of white and black balls. 
In the case supposed, the differences are due to imperfections in the instru- 
ments and in the observer, to unknown variations in temperature, etc. 
Let us consider another case, which will perhaps aid us in gaining a clear idea 
of the subject. Let us suppose that we have a rifle mounted like a swivel-gun, so 
that it can turn only in a horizontal plane. In the same plane let a thin, hard board 
be placed with its edge turned toward the gun __If we fire at a certain point in the 
edge of the board, a sufficient number of times, we should cut into the board a 
gap which would be bounded by a curve precisely like the one before given. In 
case of a good marksman, the gap would be narrow and deep. (Fig. 3.) With 
an equal number of shots a very bad marksman would cut out a wide and shallow 
curve, while if all kinds of people were allowed to try their skill, we should get the 
curve of average human marksmanship. The errors of the former case are here re- 
presented by deviations from the mark, small deviations being most common. It 
might be said that these deviations are caused by chance. ‘They are in fact the 
result of unknown but purely mechanical causes, such as gusts of wind, irregularities 
of the balls, fouling of the gun, or deviations caused by pulling the trigger. Chance 
is neither a thing nor a cause; it is simply a name to cover over ignorance of the 
real causes; it is a matter of experience, that a great number of simultaneously 
acting and constantly varying causes, affect the result in such a regular way that 
we can predict in a statistical way, the frequency of different errors or deviations. 
When these causes are unknown, we say that it is a matter of chance. 
A curve of this kind would at once exhibit the success of any person in 
shooting at a mark, and it seems probable that the success with which we accom- 
plish any other thing involving a great number of acts, would be represented by 
a similar curve. 
We have a large class of people in this world who aim to be as good as other 
people. Some of them resemble the good marksman. ‘Their deeds all lie very 
close to the mark. They are never very good and never very bad; but always 
‘‘indifferent honest.” Others are like wild shooters. If we could grade their 
acts, we should find some decidedly bad, others far above the average of good_ 
+As the mean value is not the true value, it is not strictly correct to call these differences “ errors.” 
