THE LAW OF ERRORS OF OBSERVATIONS. 
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should be so, it being quite easy to conceive a different economy of nature in which no 
such accordance would subsist *. 
It is possible ci priori to conceive that the law of single errors of observation might be 
of any form whatever, varying with each kind of observation : how far it is true that in 
practice one general law will be found to prevail, is essentially a question of facts — an 
inquiry, not into what might he , but what is. Now the hypothesis above mentioned, — 
namely, that errors in rerum naturd result from the superposition of a large number of 
minuter errors arising from a number of independent sources, — when submitted to 
mathematical analysis, leads to the law which is generally received ; as far therefore as 
this hypothesis is in accordance with fact, so far is" the law practically true. Fully to 
decide how far this hypothesis does agree with facts is an extremely subtle question in 
philosophy, which would embrace not only an extended inquiry into the laws of the 
material universe, but an examination of the senses and faculties of man, which form an 
important element in the generation of error. Still, without pretending to enter on a 
demonstration of the truth of this hypothesis, a few reflections upon the facts, especially 
in the case of Astronomy (which is par excellence the science of observation, and where 
accordingly the lessons of experience are the clearest and most complete), will, I think, 
at least convince us of its reasonableness in certain large classes of errors of observations. 
Now if we attend to what has taken place in the history of astronomical observation, we 
find that the gross errors of the earlier observers proceeded mainly from three or four 
principal causes — for instance, refraction, imperfect measurement of time, and the use 
of the naked eye in pointing to objects. When these few capital occasions of error were 
removed (at least approximately), refraction being discovered and allowed for, and the 
pendulum and telescopic sights introduced, it was found that observations at once 
attained a high order of accuracy, showing that the principal sources of error had been 
eliminated. It would seem, in fact, that in coarse and rude observations the errors 
proceed from a very few principal causes; and in this case, consequently, our hypothesis 
will probably represent the facts only imperfectly, and the frequency of the errors will 
only approximate roughly and vaguely to the law which follows from itf. But when 
* The extreme simplicity of the exponential relation itself, whether considered as expressing the law of single 
errors, or that of the mean results of a large number of observations, as contrasted with the long and difficult 
methods by which it was established, has naturally led to several attempts to dispense with or simplify the 
latter ; in some the hypothesis we here adopt is taken as a basis ; but, so far as the present writer is aware, 
every process given, except Poisson’s, fails in generality. In a recent Memoir on the Law of Frequency of 
Error by Professor Tait (Edin. Trans, vol. xxiv.) (where, it should be stated, the learned author speaks with 
some hesitation, and only gives his method as an attempt), it is assumed that each of the elementary errors 
which are combined can be assimilated to the deviation from its most probable value of the number of white 
balls among a given large number of balls drawn from an urn, which contains white and black in a given pro- 
portion. It is then shown (as indeed is done in Laplace’s 3rd chapter) that this error follows the exponential 
law. Thus the proof only applies to the combination of a number of elementary errors, each of which follows 
that law. But it is quite certain that many simple errors do not follow that law ; hence the method is altogether 
deficient in generality. 
t We cannot, however, assert this positively, if there is reason to believe that the error which arises from 
each principal cause is itself a composite error, which certainly is often the case. The “ error in time,” for 
