to the Method of Least Squares. 175 



ritlim for the solution of normal equations by the method of substi- 

 tution (1816), the determination of weights and of formulas for mean 

 error occupy the second part of the memoir. The value of the mean 



error of an observation of the weight nnity being m = \ ~"^~ , Gauss 



"^ n 



takes 2x^ as referring to the true errors, and determines 7n =z L_^ 



as a practical foi-mula, 2v'^ referring to the computed residuals, n be- 

 ing the number of observations and q that of the unknown quanti- 

 ties : the investigation however is not very clear. See 1816 Bessel 

 and Gauss, and 1856 Bienaymb. 



For Gauss's own account of the contents of these memoirs see the 

 Gotthigische gelehrte Anzeif/en, Feb. 26, 1821 and Feb. 24, 1823. 

 These reviews are reprinted in Vol. IV. of Gauss Werke, pp. 95-104. 

 Gauss here states that in the year 1797 he found that the determina- 

 tion of the most prohahle values of observed quantities was impossi- 

 ble, unless the law of facility of error was knoAvn ; and that since 

 1801 he had used the Method of Least Squares almost daily. See 



1830 RiESE. 



1824 Berlin. Explanatio tnethodi quadratorum minbnoricin. 

 Lundae, 4to. 



1824 Fourier. 'Regie usuelle pour la recherche des resultata 

 moyens d'un grand nombre d'observations.' Ferussac's Bull. Sci. 

 Math., Vol. II, pp. 88-90. 



The rule given is expressed by the formula 



1^^ - 1^\ 

 ...0-4769^'^A'^^ 



^ In 

 aj, ttg) ^*3? • • • being the results of the n observations. 



1824 PoissoN. ' Sur la probabilite des resultats moyens des obser- 

 vations.' Comiaiss. des Terns for 1827, pp. 273-302; for 1832, 

 pp. 3-22. 



These memoirs are a commentary on Laplace's fourth Chapter 

 (1812) and seem to form a kind of translation which Poissox made 

 of Laplace's investigations for his own satisfaction. A large part 

 of the memoirs are reproeluced in his Reeherches . . . , see 1837. See 

 also 1830 Hauher, 1847 Galloway and Todhunter's Historij of 

 Prohahility, pp. 560-588. See Jahrh. Chem. ic. Fhys., Vol. IV," pp. 

 38-42. 



" PoissoN confines himself to the case in which one element is to 

 be determined from a large number of observations, but he treats 

 this case in a more general manner than Laplace had done. La- 

 place had assumed that positive and negative errors were equally 



