ON ERRORS OF OBSERVATION, 145 



measured, BK is P, and the chance of any error 

 falling between and B N is such a fraction of unity as 

 the area BPN is of the whole area of the curve. The 

 case before us is precisely that of an observer with a 

 personal error equal to P., in addition to casualties. If 

 we imagine a very large number of observations, made 

 under circumstances equally favourable to positive and 

 negative error, and if the error P be added to or taken 

 away from each of the results, according as P is a 

 positive or a negative error, we shall then have such a 

 succession of results as might be looked for on the 

 present hypothesis. For instance, let the quantity P 

 be 10, then whatever may be the prospect of having 

 the error 1, when positive and negative errors are 

 equally likely, we have the same chance for the error 

 11, in the present case. 



For simplification, I shall adopt the algebraical 

 method of signifying positive and negative errors. 

 Thus -f 3 means 3 too much, or 3 added to the truth ; 

 4 means 4 too little. Thus + 4 5 is 1 ; mean- 

 ing that four too much from one cause, and 5 too little 

 from another, gives on the whole 1 too little. Again, 

 we consider 2 less than 1, since the effect of the 

 former is to lessen the result more than would be done 

 by the latter. To show that the supposition now before 

 us is equivalent to that of an observer with a personal 

 error, or an instrument with an individual error, imagine 

 an instrument wrongly graduated, so that for every 

 reading we ought to read 10 less : thus for 125 we 

 ought to read 115. In other respects let the instru- 

 ment be equally likely to give positive and negative 

 departures from truth. If then an observation give 

 176, we know immediately that the truth is 176" 10 



