OP ARTS AND SCIENCES. 193 



XII. 



ON THE METHOD OF LEAST SQUARES. 



Bt Truman Henry Safford. 

 Presented, Oct. 12, 1875. 



The method of least squares with the theory of errors of observa- 

 tion upon which it depends forms the foundation of modern practical 

 astronomy, and is largely used in subjects akin to astronomy, as 

 geodesy and physics : the object of the present note is to assist in 

 their practical application, both in planning and reducing series of 

 observations. 



Legendre first seems to have published the method of least squares, 

 mainly as a convenient and pretty method of computation. Gauss 

 soon after showed that it was the same in principle as the ordinary 

 method of taking the arithmetical means, in the simpler cases to 

 which that is applicable ; and that either method presupposes a distri- 

 bution of the errors of observation', such that the probability of larger 

 errors is less than that of smaller ; and proportional to the function 



where A is an error of observation, and h a factor which reduces all 

 systems of such errors to be comparable with each other. 



Of course the probability of an error of observation exactly equal to 

 a given amount, no hair more nor less, is infinitesimal ; and the definite 

 integral 



denotes the probability that there will be errors between the limits A 

 and Aj; while the total probability of all the errors of observation 

 denoted by unity will be equal to 



VOL. xr. (n. 8. III.) 13 



