THE 



PHILOSOPHICAL MAGAZINE 

 AND JOURNAL. 



30 th SEPTEMBER 1826. 



XXII. On the Method of the Least Squares. By J. Ivory, 

 Esq. M.A. F.R.S.* 

 HAVE already treated of the method of the least squares 

 ■^ in this Journal for January 1825, and the two following 

 months. By altering a little an idea first suggested by Cotes, 

 we may represent the errors of observation by the parts of a 

 rigid line, the positive errors being on one side, and the nega- 

 tive errors on the other side, of a common fulcrum ; and if 

 we append to the extremities of the errors, weights proportional 

 to the coefficients of the correction in the equations of condi- 

 tion, the equilibrium of the weights is the rule of the least 

 squares, and determines the most advantageous value of the 

 correction sought. When there are several elements to be 

 corrected simultaneously, the coefficients of the different cor- 

 rections in the equations of condition will form so many se- 

 parate sets of weights, and the levers must be in equilibria 

 whichever set is appended; by which means we obtain as 

 many equations as there are unknown quantities to be found. 

 This method is at least precise and free from every thing con- 

 jectural or tentative. It is likewise founded on just princi- 

 ples ; for every coefficient in the equations of condition has its 

 due influence in the quantity of the result. But it must be 

 allowed that the introducing of the properties of the lever and 

 of equilibrium in a demonstration of this kind is not altogether 

 unexceptionable. Such considerations would not be necessary 

 if the truth to be proved were entirely disengaged from what 

 is foreign to it. It is however extremely desirable that a me- 

 thod, which is of great practical utility, should be clearly and 

 simply deduced from the real principles alone concerned; 

 because it is only when this is done that the method can be 

 fully understood, and that we can apply it with confidence 

 and without danger of mistake. My intention in returning 

 to this subject is to attempt an explanation of the ground of 

 the method of the least squares that may in some degree an- 

 swer the description here given. 



* Communicated by the Author. 

 Vol. 68. No, 341. Sept. 1826. X In 



