146 ESSAY ON PROBABILITIES. 



+ a casual error, the effect of which disappears in a 

 large number of observations. The whole error then 

 is 10 + the casual error, 10 -f x and 10 # being equally 

 likely. This is precisely the hypothesis in question, 

 in the case in which P = 10. 



The average of observations, in the case before us, 

 does not necessarily give an approximation to the truth. 

 Calling the quantity P a fixed error, meaning by a fixed 

 error one which is as likely to be exceeded as not, we 

 see, in the following theorem, a justification of the term : 

 the most probable result of a large number of observations 

 is the truth, increased or diminished by the fixed error, 

 according as it is positive or negative. This error may 

 either be fixed in the instrument, fixed in the observer, 

 or in different degrees in both. 



Let the phenomenon to be observed be perfectly 

 unknown, except by what the instrument tells us. 

 Then it is totally impossible to discover the amount of 

 this error ; which, nevertheless, must be assumed to 

 exist, unless the contrary be shewn. For example, an 

 instrument wrongly graduated throughout could never 

 tell the truth, either in individual or average results. 

 But it is obvious that such an apparatus, incapable as 

 it is of telling absolute truths, might nevertheless detect 

 such results as are obtained by measuring the differences 

 of other results. Thus a clock which goes truly, but 

 is set too fast or too slow, will serve to find the time 

 elapsed between two events, though it will not show the 

 real time of either. Instruments which, on account of 

 some permanent error affecting all their results, can 

 only be used to determine differences, are called dif- 

 ferential instruments. 



All instruments, as well as observers, are subject 

 more or less to this species of error ; how then is it 

 possible to depend upon the results of any observations ? 

 The answer to this question will require some detail. 

 Since perfect exactness cannot be attained, either on 

 the part of the instrument, or of the observer, we can 

 only call either good, when positive errors are as likely 



