MATHEMATICAL SECTION. 153 
two‘grand classes, and name them, from their consequences, instruc- 
tive errors and uninstructive errors. The latter class includes blun- 
ders in recording, pointing on wrong objects, &c. The former con- 
sists of errors that indicate error in other observations. 
I once tried the experiment of dropping a short straight piece of 
wire five hundred times upon a sheet of ruled paper and counting 
the number of intersections of the wire with a ruled line. When 
the end of the wire touched or nearly touched a line, and inter- 
section was doubtful, I counted it as half an intersection. I re- 
corded the number of intersections in groups of fifty trials, as fol- 
lows: 23, 26, 28.5, 24, 31.5, 28, 27, 14, 25, 28.5. These numbers 
may be regarded as observations from which may be deduced the 
probable ratio of the length of the wire to the distance between 
two consecutive lines; and it seems impossible to account for the 
remarkable smallness of the eighth number by any supposition of 
uninstructive error. It is almost certain that a ratio deduced from 
it alone is largely in error; but it indicates that the other nine 
observations are somewhat in error, and that its error is needed to 
counterbalance theirs. If we retain it, and regard the mean of all 
as the most probable truth, we infer that this observation is 11.55 
units in error. If we reject it, and take the mean of the other nine 
as the most probable truth, we infer that this observation is 12 5-6 
units in error. It should be remembered that the rejection of an 
observation does not sweep from existence the fact of its occurrence; 
but merely increases its already large estimate of error. Because 
an error of 11.55 units is so large as to be very improbable, shall 
we therefore infer that an error of 12 5-6 units is more probable? 
It seems very clear to me that the larger an instructive error is 
the more instructive it is, and the more important is it that the 
observation containing it should not be rejected. The mean of all 
the ten above-described observations being regarded as the most 
probable truth, any one of the other nine could be better spared 
than the eighth. On the other hand, the larger an uninstructive 
error is, the more important it is that the observation should be 
rejected. Whenever an observation is intelligently rejected, there 
is a comparison of two antecedent probabilities, viz.: that of the 
occurrence of an instructive error of the magnitude involved and 
that of the occurrence of an uninstructive error of the same mag- 
nitude. When an error is evidently so large that it cannot possibly 
belong to the instructive class, the antecedent probability of such 
14 
