to the Method of Least Squares. 173 



Contains discussions on the probable errors of the observations. 

 See also AI>handl. Akad. Berlin for 1825, pp. 23-35. 



[1820] Laplace. ' Troisihve Supplement d la TMorie analytique 

 des ProhahilWes.'' Paris, 4to, pp. 36. — Fourth edition of the Theorie 

 . . . . , pp. 624-660. 



See 1819. At the end of the Supplement is an investigation of the 

 general case of " observations assujetiesa plusieurs sources d'erreurs," 

 See ToDHUNTER, Mist, of Prohability, p, 612. 



[1820] Legendre, *■ Nouvelles methodes pour la determination de 

 Vorbite des comhtes. Second supplemetit.'' Paris, 4to, pp. 80. 



In pages 3 and 4 Legendre says : " Jai donne le premier deux 

 methodes stires pour obtenir la solution a la fois la plus simple et la 



plus exacte, savoir : la methode des corrections ind'etermin'ees , 



et la methode des nioindres carres qui paraisait alors pour la premiere 

 fois." See K89 and 1805. 



Pages 79-80 contain a " Note par M * * * " in which the honor of 

 the discovery of the Method of Least Squares is claimed for Legen- 

 dre on the ground of priority of publication, and in which Gauss 

 although not mentioned by name receives several sharp hits. 



1821 SvANBERG. Om roterande systemers princijial-axlar och san- 

 nolikaste medel-resultatet af gifna observationer. Vetensl: Akad. 

 Handl. Stockholm for 1821, p]x 388-408. 



1821 . 'Dissertation sur la rercherche du milieu le plus 



probable, entre les resultats de plusieurs observations ou experiences.' 

 Gergomie's Annates de Math., Vol. XII, pp. 181-204, 



This paper discusses at some length the different methods which 

 may be imagined for finding a mean value, and concludes that the 

 problem is indeterminate because it is impossible to render it inde- 

 pendent of the law of facility of error, concerning which there may 

 be " une infinite d'hypotheses." It tries to determine a mean, first 

 supposing that the probability of each given measurement is in- 

 versely proportional to the error committed and secondly supposing 

 that that probability is inversely proportional to the square of the 

 error, and concludes that the arithmetical mean can only be used 

 when the observations differ but slightly among themselves. 



The paper ends by offering a method for the correction of the 

 arithmetical mean, which amounts to this : First find the average of 

 the measured quantities and compute the residuals. Then take the 

 reciprocal of each residual as the weight of its corresponding obser- 

 vation and find the mean of these weighted observations. Or as 

 weights the reciprocals of the squares of the residuals may be taken. 

 The new mean gives new residuals from which a second approxima- 

 tion may be made, and so on. In a note at the end, the editor 

 {Ger(;onne) suggests that this approximation will always tend to 

 one of the given measurements as the mean. 



