ON ERRORS OF OBSERVATION. 155 



possible. The considerations necessary foF this pur- 

 pose form a great part of the application of mathe- 

 matics to the sciences of observation : in which it 

 frequently happens that good methods of observing 

 are rendered useless by the multiplication of error which 

 the methods consequent upon them involve : and con- 

 versely, that formulae good in other respects, are in- 

 admissible from the tendency to error in the observations 

 which they require. And it has happened before now 

 that mistakes of serious amount have arisen from the 

 use of mathematical methods in which the errors of the 

 observations are much multiplied. 



I could hardly close such a chapter as the present 

 without some mention of the celebrated method of least 

 squares, on which the astronomy of the last thirty 

 years has depended for much of the increase of ac- 

 curacy which has been its characteristic. But as 

 any development of this very interesting subject is 

 impracticable without recourse to mathematical symbols 

 and reasoning, I content myself with a description 

 of one particular case, which is of very frequent 

 occurrence. 



Suppose a number of results to be obtained by observ- 

 ation, from which a consequence is to be drawn by mathe- 

 matical reasoning. If the observations were all correct, 

 the consequence deduced from any one would be the 

 same as that from any other ; but owing to the 

 errors of the observations, such agreement is of course 

 unattainable. It is, therefore, a question what method 

 of combining the several results should be adopted : 

 and mathematical analysis shows that the object is 

 attained by choosing such an intermediate result as 

 shall make the sum of the squares of the errors the 

 least possible. It might seem as if, positive and nega- 

 tive errors being equally probable, the average of results 

 is the most probable truth ; and this is the case when 

 the observation is itself made directly upon the result 

 which is required, or when there is only one datum 

 into which the uncertainty of observation is intro- 



