Tmpei'ial histitute of Fravce. 241 



do better than conclude by giving this information, which 

 assuredly will excite the curiosity of all geometricians. If 

 amongst the applications which Count Laplace makes of 

 his formulae, we perceive a point which concerns two ce- 

 lebrated analytical philosophers, it certainly should find a 

 place in the present sketch. 



It is that passage in which Count Laplace mentions 

 small squares [petits carrees). He says that the method 

 proposed by Messrs. Legendre and Gauss corrects the ele- 

 ments in the most precise manner. The learned who have 

 not met with these works, it is probable may wish to 

 know this method of those eminent geometricians who 

 have already gained so much honour by their labours. 



M. Legendre, when he directed his attention to the pro- 

 blem of the comet in March 1805, first furnished astrono- 

 mers with a certain rule to guide them in their number of 

 approximative equations, much superior to those unknown 

 quantities the value of which it was left them to ascertain. 



The inevitable error of the observation on which the 

 equations are established, renders it impossible to explain 

 them all at once, and in taking the result of the system of 

 the observations it does not give more explicit satisfaction ; 

 all that can be gained is, that the errors are as trifling as 

 possible— that they are equally distributed — and that none 

 of them exceed the probable errors of the observations. To 

 approach nearest to the real value, M. Legendre proposes a 

 principle by which the sum of the squares of the errors 

 must be a minimum. 



This method, which he only mentions without giving the 

 analyses of it, he has made the subject of an Appendix at 

 the end of his Memoir, and in which he gives some further 

 developments. It is his opinion, that of all the principles 

 proposed on this subject, there are none more general, more 

 exact, or more easily to be applied. By this means, he con- 

 tinues, there is established amongst the errors a kind of 

 equilibrium which prevents the extremes from being pre- 

 valent. 



If by any singular event it were possible to obliterate all 

 the errors, he shows that it could only be done infallibly by 

 his method ; and this is an important remark. 



If after having determined the unknown quantities we 

 carry the value of them into each of the equations, instead 

 of seeing them reduced to zero ; in general we shall find a 

 value which will be for each of the observations, the errors 

 of the elements corrected, and it is impossible to diminish 

 their errors without augmenting the sum of their squares. 

 Vol. 39. No. 167. March 1812. Q M. Le- 



