Mr. Ivoiy on the Method of the Least Squares. 163 



contain a maxim which ought to have greater force in the 

 matliematics than in any other branch of learning. But it 

 might not be found altogether devoid of truth, if we were to 

 affirm that, in the present times, the voice of authority has a 

 more decisive influence in that science than in any other. It 

 is so much easier to approve or disapprove on the credit of a 

 few great names, than it is to find the skill and the patience 

 requisite to examine a knotty point of abstract science. 

 Having always exercised my own judgement in such s])ecula- 

 tions, I shall claim the same privilege in the present instance; 

 and I doubt not to be able to prove in a satisfactory manner, 

 that Laplace's demonstration is not general, as it is stated to 

 be, but is really confined to the particular law already men- 

 tioned. 



It will be sufficient for the purpose I have in view to consi- 

 der the simplest case of Laplace's investigation, that for finding 

 one element by means of a set of equations of condition: liv. 2. 

 cap. 4, No. 20. It is assumed that the errors are determined 

 by the ecjuation S.Ke = 0, that is, 



Xe + X'e + \"e" + &c. = 0, 

 where X, x', x"* &c. are integer numbers bearing any propor- 

 tions to one another ; and the scope of the investigation is to 

 determine these factors so as to obtain the most advantageous, 

 or the most probable, system. The author goes back to the 

 first principles of the doctrine of probabilities, or to the theory 

 of combinations ; and he investigates the chance that the func- 

 tion S . X e shall have a given value I. If we adopt the nota- 

 tion used here, and write h^ for „ , , the expression of the 

 probability sought, found in p. 317, 3d edition, is this, 



k -hK — 

 C S.A.-1 



Again, if we substitute the values of the errors given by the 

 equations of condition in the equation S.X<? = 0, we get the 



value of the element x equal to ^ ; and if we substitute 



'■ » . Xa 



^^-^ + u for X, in the equation S.he = I, we shall get 



I = u X S . Xa. 

 The foregoing expression therefore becomes 



/, -hi^ L.Wi 



::^:::^^ C S.X' 



v/frS.X' 



* As the letter m, which Laplace uses for the coefficients of the errors, 

 is prc-occiipied in the e(]uations of condition, it became necessary to intro- 

 duce another letter here. 



X 2 and 



