168 Mansfield Merrhnan — lAat of Writings relating 



Laplace's proof has been greatly improved by subsequent writers. 

 Ellis in 1844 extended it to any number of unknown quantities, 

 ToBHUNTER in his History of Probability, pp. 560-588 supplied a 

 valuable commentary, and Glaisheh in 1872 presented it in a clear 

 and simple form. See also below, 1824 Poisson, 1847 DeMorgan, 

 1852 Biexayme, 1861 Airy, 1873 Laurent and 1875 Dienger. 



On pages 318-319 is given what is sometimes called Laplace's 

 second proof of the Method of Least Squares. This depends on the 

 definition that " la valeur moyenne de I'erreur a craindre en plus" 

 should be a minimum. The reasoning is similar to Gauss's proof of 

 1823 ; see Ellis's and Glaisher's papers quoted below under 1844 

 and 1872. 



[1813] Plana. Sur divers problemes de probabilite. Mem. Acad. 

 Turin for 1811-12, Vol. XX, pp. 355-408. 



1814 Cauchy. Memoire sur le syst^me de valeurs qu'il faut at- 

 tribuer a divers Siemens, determinees par un grand nombre d'obser- 

 vations. [Paris, Lith. MS.] 



Probably the same as his memoir of 1831. 



1814 Laplace. '• Essai philosophique sur les prohabilites.'' Paris, 

 4to. — Sixth edition, Paris, 1840, 8vo, pp. 274. — Introduction to 

 second edition of Theorie. . . .des Proh., pp. cvi; to third edition, j^p. 

 cxlii ; to fourth edition, pp. v-clxix. 



Near the end are some remarks concerning the history of the The- 

 ory of Errors of observation, and descriptive of Laplace's processes. 



1814 Legendre. ' Methode des moindres quarres, pour trouver 

 le milieu le plus probable entre les resultats de diiferentes observa- 

 tions.' Mem. Inst. France for 1810, Pt. II, pp. 149-154. 



Pages *11-^ih of the Nouvelles methodes. . . .\HOb, are here quoted 

 and reference made to the practical applications of the method given 

 in that work, in order to call attention to Legendre's priority of 

 publication. 



1815 Bessel. ' Ueber den Ort des Polarsterns.' Berlin. Astron. 



Jahrb. for 1818, pp. 233-240. 



We here find the first mention of probable error. After giving 48 

 observations on the right ascension of Polaris whose arithmetical 

 mean is 55"^ 48^5104, Bessel says: " Der loahrscheinliche Fehler 

 einer einzelnen Beobachtung ist, nach den wirklichen vorkommenden 

 Fehlern zu urtheilen :=1^067, und daher der wahrscheinliche Fehler 

 des Endresultats =::0M54. Die Grund dieser Schatzung des wahr- 

 scheinlichen Fehlers, beruhen auf der von Gauss gegebenen Ent- 

 wickelung der Wahrscheinlichheit, einen F'ehler von gegebener 



