LAPLACE. 469 



entre plusieurs observations donnees d'un meme phenomene. II j a 

 deux ans que j'en donnai uue a 1' Academic, a la suite du Memoire sur 

 les Series recurrorecurrentes, imi^rime dans ce volume ; mais le peu 

 d'usage dont elle pouvoit etre, me la fit supprimer lors de Timpression. 

 J'ai appris depuis par le Journal astronomique de M, Jean Bernoulli, 

 que M". Daniel Bernoulli et la Grange se sont occupes du meme pro- 

 bleme dans deux Memoires manuscrits qui ne sont point venus a ma 

 connoissance. Cette annonce jointe a I'utilite de la matiere, a reveille 

 mes idees sur cet objet ; et quoique je ne doute point que ces deux 

 illustres Geometres ne Taient traite beaucoup plus heureusement que 

 raoi, je vais cependant exposer ici les reflexions qu'il m'a fait naitre, 

 persuade que les differentes manieres dont on pent I'envisager j^roduiront 

 une methode moins hypothetiqiie et plus sure pour determiner le milieu 

 que Ton doit prendre entre plusieurs observations. 



875. Laplace then enunciates his problem thus : 



Determiner le milieu que I'on doit prendre entre trois observations 

 donnees d'un meme phenomene. * 



Laplace supposes positive and negative errors to be equally 

 likely, and he takes for the probability that an error lies between 



X and x+ dx the expression — e~^^^ dx\ for this he offers some rea- 



sons, which however are very slight. He restricts himself as his 

 enunciation states, to three observations. Thus the investigation 

 cannot be said to have any practical value. 



876. Laplace says that by the mean which ought to be taken 

 of several observations, two things may be understood. We may 

 understand such a value that it is equally likely that the true 

 value is above or below it ; this he says we may call the milieu 

 de probdbilite. Or we may understand such a value that the sum 

 of the errors, each multiplied by its probability, is a minimum ; 

 this he says we may call the milieu derrein\ or the milieu astro- 

 nomique, as being that which astronomers ought to adopt. The 

 errors are here supposed to be all taken positively. 



It might have been expected from Laplace's words that these 

 two notions of a mean value w^ould lead to different results ; he 

 shews however that they lead to the same result. In both cases 

 the mean value corresponds to the point at which the ordinate to 



