Imperial Institute of France. 401 



which we may flatter ourselves we have obtained — adding, 

 as a second condition, that it' the sum of the errors, taken 

 with their natural sign, be reduced to zero, we cannot obtain 

 more accurate rcsuks than by a nuii)ber of equations of a 

 less unity than that of the unknown quantities. 



To conclude, M. Gauss states that the principle of the 

 small squares which hf. has made use of since the vear 1 793, 

 was published by M.Legendre in 1805 in his Memoir upon 

 Comets. 



From this declaration a new question arises. Tn speaking; 

 of the above method, both authors use the expression of 

 vinn principe des moindies carrcs. To whom belongs the 

 merit of this principle which M.Gauss made use of 16 

 years ago, and which M. Legendre seems to have become 

 acquainted with so recently? The answer is very simple. 

 It is impossible that M. Legendre can be under any obliga- 

 tions on this point to M. Gauss, who had not published 

 any thing : we are convinced that M. Gauss had discovered 

 the theorem, but it is equally clear that M. Legendre not 

 only discovered it for himself, but was the first to make it 

 public. 



Finally, we may remark that M. Laplace, although he 

 has in no shape laid claim to the honour of the discovery, 

 has at least directly demonstrated and clearly developed, bv 

 an analysis peculiar to himself, a truth which was scarcelv 

 suspected ; numcly, that the corrections furnished by the 

 methods of the small squares are the most precise which 

 can possibly be procured. W« shall add, for the benefit of 

 those wlu) are familiarised to astronomical calculation, and 

 who use the processes of transendent geometry, that it is 

 sufficient to follow attentively the course and mechanism of 

 the numerical calculation of M. Legendre, in order to be 

 thoroughly convinced that his method has all the advan- 

 tages which M. La Place's analysis ascribes to it. To con- 

 clude: as the results obtained are only the most probable, 

 the calculator ought not to dispense with ulterior proofs. 

 These cannot be obtained except by a rigorous calcula- 

 tion made upon the elements corrected and compared di- 

 rectly with all the observations. In fact, the equations 

 upon which he wrote, are onlv approximations, since thev 

 are linear ; and it is not impossible that this revision wil! 

 furnish him, for his elements, with slight modifications, 

 which, without carrying him far from the results of the 

 small squares, will give still more precision to his Tables. 



[To be continued.] 



IJX. J'K 



