The necessary errors of observation arise from the imperfection of our perceptions and of the instruments which we use, and also from hasty or otherwise incorrect conclusions. The subject requires the separation of these errors into three claws, which may be mixed up with one another in results, and may be mistaken for one another. Wo may call them fired, peraoanl, and eaenal.
By ai.red error we mean ono which is inherent in the instrument or method employed, so that it must exist, and, all other things remaining the Patna, must have a given magnitude. Thus, if the axis of an equatorial (supposing such an instrument to be employed for absolbto measurements) do not absolutely coincide with that of the heavens, the right ascension and declination of a given star, measured when at a given distance from the meridian, must have a given error. It might be precisely the same in numerical effect, and would certainly produce an error of the same class if the observer used a wrong formula In the reduction of his observations. Thus, it would be perfectly possible to give to one observer an incorrect instrument and a correct formula, and to another a correct instrument and an incorrect formula, in such manner that their final results should coincide.
Errors of this kind cannot be detected by multiplying similar obser vations, since there can be no tendency to destroy error in the mere repetition of it. There aro many modes of detecting fixed errors, and of allowing for them; but the only mode of avoiding them is by taking advantage of the construction of the instrument to use it for the same purpose under different circumstances, in such manner that measure ments which are too large in one set of results must be as much too small in the other. If the same number of observations be contained in each set, this, an we shall presently see, is really a reduction of the fixed error to the clam of casual ones ; or rather, a destruction of the fixed error by the same process which gives the highest probability of destroying the casual errors.
All instruments must be more or leas erroneous in every particular. In the science of observation, as now understood, and in any matter in which the utmost attainable exactness is requisite, the assumption of perfection in an instrument, in any point whatsoever, is looked upon as nothing but the expression of the observer's unwillingness to take trouble. For even if ninety-nine anceessive days' trials have shown that any particular error does not exist to any sensible amount, it is not conclusive against the observations of the hundredth day being affected by some new circumstance, necessary or accidental, in which the instrument has' been placed in the intermediate time.
By a personal error is meant one of the same character as a fixed error, but arising from the temperament or habits of the observer, and not from the instrument. Thus if A should, in noting the time of a phenomenon by the clock, have a tendency to accelerate the moment of its happening, and B a similar tendency to retard it, the results of the two would differ by the Bum of their personal errors. It has been discovered that two individuals, observing the name phenomenon with the same species of instruments may differ sensibly (though but little) from each other; and this not once or twice, but nearly always, and in such a manner as to make the average of a set of observations of one observer differ from that of the other. For anything we can know to the contrary, this species of error may exist in every observer'; and its absolute quantity must be unknown until wo can compare the observations of men subject to it with those of some other beings who are not If indeed the personal error be purely casual, so that where one person measures too much, another measures as much too little, the average of tho results of a large number of observers would give the truth or very near It But should It be the case, which is not impossible, that all men are subject to an error of the same kind, some more and some less, namely, that all measure more or lea too much, or else, that all measure more or less too little, the average above mentioned would give, not the truth, but the truth affected by the average error of all the observers. Nor would the result's obtained over enable us to dis
tinguish whether personal errors have a fixed average or not ; for suppose the fact observed to be that A, one time with another, mess aura; more than B; this may mean either of several things :—either A measures truly, and II too little ; or B measures truly, and A too much ; or B measures too little, and A too much ; or both measure too little, but B more than A; or both measure too much, but A more than B. Now, if A and B were to observe together for a century, the mere comparison of their observations, though it would settle their average amount (of difference, would never enable us to give the least guess which of the preceding cases is the true one. If indeed we could convert the observer, as we have previously mentioned might generally be done with the instrument, into another observer with an error of the opposite kind, a true result, or one sensibly true, might be obtained. Suppose, for example, it is the observer's habit, in noting the transit of a star over a fixed wire In the field of a telescope, to take the transit too soon when the star conies in on the right side, and too late when on the left : consequently, by making a number of observations with an inverting telescope, and an equal number with one which does nut invert, the average of both sets world be as likely to give a true result RR if neither error had existed. cl:AirATION, PERSONAL) All the errors which precede, though called errors because they give a result which is not the one intended to be obtained, yet are in fact the consequences of an actually existing state of things, and their laws can be determined by tieing the right means, or at least must be supposed to arise from natural causes measurable by experiment in the same manner as other consequences of existing relations. They are then really measures of phenomena, called errors simply bCCAUFIC the effects of thei; causes are to be removed from the results. It is even possible that they might be made intentionally in a given form, with a view to prevent their occurrence in a more objectionable form. Thus, suppose an observer finds himself, in correcting discordant observations, apt to confound additions and subtractions, using one for the other : he will set his instrienent intentionally wrong to an amount which casual disconLances never reach, taking care, of course, to preserve means of correcting the intentional error with the rest ; so that the requisite correction shall always be of one kind, additive or subtractive. Nevertheless this arrangement, as it should be called, would go by the name of an error, simply as being to have its effect afterwards destroyed.