16G Mr. Ivory on the Method of the Least Squares. 



lo the case of a great number of errors, in order to render the 

 calculations practicable. 



Thus, if we apply the doctrine of probabilities to find the 

 most advantageous way of combining a set of observations,^ 

 and likewise require that the final equation must be one of 

 the first degree, we are invariably led lo the method ot tlie 

 least squares, and to the law of probability expressed by 



the function kc~'^ ^\ in vvhatever way we pursue the investi- 

 gation. It must therefore be allowed that the evidence we 

 have for one of these two things is just the same as that which 

 can be obtained for the other. If it tan be well proved that 

 the particular law of probabilily will belong to every set of 

 observations, the rule of the least squares will be firmly esta- 

 blished ; but if hardly any good reason can be alleged in sup- 

 port of the first, the other will rest on foundations equally 

 feeble. When the investigation of Laplace is understood in 

 all the generality that, I apprehend, has hitherto been ascribed 

 to it, the proof of the method of the least squares is as strong 

 and convincing as the nature of the case will admit; because 

 among all the laws of probabihty that can be imagined, there 

 must be one that will nearly apply to the errors of any set of 

 observations in which only an ordinary degree of regularity 

 is supposed to prevail. But the complexicm of the proof is 

 entirely changed when it is shown that the author's reasoning 

 takes in only one particular law. 



What has now been said justifies the view taken of tliis 

 theory in the foregoing researches. The proof of the method 

 of the least squares by means of the doctrine of probabilities, 

 being entirely supposititious and mathematical, is insufficient 

 and unsatisfactory ; and we must therefore seek a better sup- 

 port for it in the nature of the equations of condition. I have 

 already given two different demonstrations independent of the 

 laws of chance. On re-considering these I do not find that 

 any thing material can be added to the second ; but some con- 

 siderations that have occurred since the first was written seem 

 to render it more complete. 



If, as in the first demonstration, we conceive the weights 

 a, a', a" &c. to be in equilibrio when suspended from the 

 levers e, e', e" &c., it is obvious that the equilibrium will not 

 be disturbed, if all the errors increase or decrease in the same 

 proportion. Now, when the errors vary in this manner, there 

 can be no manner of doubt that the most advantageous sys- 

 tem is that which makes the sum of their squares a minimum. 

 Suppose next that the errors vary, but not all in the same pro- 

 portion ; the equilibrium will be destroyed, and the force of 



prepondcrancy 



