58^ LAPLACE. 



at every observation, but he does not assume that positive and 

 negative errors are equally likely, or have equal ranges. 



1013. Laplace says, on his page 833, that hitherto he has 

 been considering observations not yet made ; but he "will now 

 consider observations that have been already made. 



Suppose that observations assign values a^, a^, ^g, ... to an 

 unknown element ; let (z) be the function of the facility of an 

 error z, the function being supposed the same at every observa- 

 tion. Let us now determine the probability that the true value 

 of the element is x, so that the errors are a^ — x, a^ — x, a^ — x,... 

 at the various observations. 



Let P = (j) {a^ — x) . (j> (a^ — x) . <f) {a^ — x). ... 



Then, by the ordinary principles of inverse probability, the pro- 

 ba^bility that the true value lies between x and x + dx is 



Pdx 



I 



Pdx 



the integral in the denominator being supposed to extend over all 

 the values of which x is susceptible. 



Let H be such that, with the proper limits of integration, 



nlpdx^i. 



'I 



and let y = Hcj) {a^ — x) . ^ [a^ — x) . cf) {a^ — x) . . . . 



Laplace conceives that we draw the curve of which the ordi- 

 nate is 7/ corresponding to the abscissa x. He says that the value 

 which we ought to take as the mean result of the observations is 

 that which renders the mean error a minimum, every error being 

 considered positive. He shews that this corresponds to the point 

 the ordinate of which bisects the area of the curve just drawn ; 

 that is the mean result which he considers the best is such that 

 the true result is equally likely to exceed it or to fall short of it. 

 See Arts. 876, 918. 



Laplace says, on his page 335, 



Des geometres celebres ont pris ponr le milieu qu'il faut choisir, 

 celui qui rend le resiiltat observe, le plus probable, et par consequent 



