THK 



PHILOSOPHICAL MAGAZINE 

 AND JOURNAL. 



28"' FE B E UA R Y 1825. 



XIII. On the Method of the Least Squares. By J. Ivory, Esq. 

 M.A. F.R.S. 



[Continued from p. 10.] 



I 



N the last Number of this Journal I have attempted to de- 

 monstrate the method of the least squares without havino- 

 recourse to the doctrine of probabilities. It must be remem- 

 bered that the demonstration, in whatever way we attempt it, 

 must inevitably contain something vague. But the arguments 

 that have been adduced for preferring the solution by the me- 

 thod of the least squares to every other, appear to be pei'fectly 

 sufficient and conclusive as far as the nature of the problem 

 will admit. If the proof by the doctrine of probabilities be 

 more formal, it is not on that account more satisfactory ; be- 

 cause it involves precarious suppositions which cannot possi- 

 bly be rigorously exact. But in order to derive the greatest 

 practical utility from a system of equations of condition, it 

 becomes necessary to connect the solution of them with the 

 rules of cliance : and we now proceed to treat the problem in 

 this view of it. 



1. It is perfectly impossible to ascertain with mathemati- 

 cal precision the probability of the error of an observation. 

 There is even no proof that the probability will in all cases 

 depend precisely in the same manner upon the magnitude of 

 the error. All that can be accomplished in this theory is to 

 adopt the suppositions that are least likely to deviate much 

 from the truth. 



Experience shows that great errors occur less frequently 

 than small or middling ones. If we push this principle to 

 the utmost length, the function <p{e), which denotes tlie pro- 

 bability of the error c, will be greatest when e = 0, and it 

 will decrease as e increases. Again, abstracting from errors 

 that are affected by a constant cause, and confining our at- 

 tention to such only as are irregular and fortuitous, the 

 errors + e will be e(jually probable, which requires that 

 Vol. 65. No. 322. Feb. 1825. L f {e) 



