By cannal errors, the only ones to which the name of errors can properly bo given, are meant those which are absolutely inexplicable, or of which the cause and tendency aro equally uuknossai. They must be considered as equally likely to be positive or negative; so that in the long run the results which they give too great will be compensated by those which are too small. If this be not the case, that in, if there bo a greater tendency to too much than to too little, there must be a reason for this phenomenon, and a law of action, which must he sought for and detected. Let us suppose this done, so that any result of a single observation, corrected for all discoverable sources of error, is in itself as likely to be too small as too great.
If all the observations be equally good, the MEaN, or average, is more likely to be true than anything else. This is even true with reference to fixed or personal errors which may remain, but which aro totally unsuspected ; for there is an even chance of such errors acting in either way. In the article just cited is shown the way of finding. from the observations themselves, the probable error, as it is called, or that which there is an even chance of not exceeding; with references to further sources of information. This article (Mess], together with
the general considerations in PRORABILITIES, TFIEORY or, and WF.101IT or Onsenveemes, will contain all we shall find it necessary to say on the subject.
It might be supposed that the greater the number of obacrvatione, the less, in the same proportion, the probable error of the average ; but this is not true, since the probable error diminishes as the square root of the number of observations increases. Thus, suppose it to be well settled that twenty observations of a given observer will have an average of which it is an even chance that it does not err by (say) a unit : then the same observer must make four times as welly observa tions to get an average with an even chance of not more than half a unit of error ; nine times for one-third of a unit, and so mm.
Those who neglect sound principles of observation are apt to over rate the effect of multiplying observations ; which, though considerable, does not, as we see in the above rule, keep pace with the number of observations.