Historical Note on the Method of Least ISquares. 4:13 



fundamental principle of the doctrine of chance, the prohahi'.- 

 that both these errors happen together will be expre.^sed b.v t. 

 product XY. If now we were to determine the values ol 

 and y from the equations x+y=^ and XY = maxiiuuni, \\ 

 ought evidently to arrive at the equation -= ^ : ^^d since x 

 and y are rational functions of the simplest order possible of 

 «, h and E, we ought to arrive at the equation | = | witliout 

 tlie intervention of roots, in other words by simple equation? : 

 ■>r. which amounts to the same thing in effect, if there be several 

 forms of X and Y that will fulfill the required condition we 

 must choose the simplest possible, as having the greatest possi- 

 ble degree of probability. 



"Let X', Y' be the logarithms of X and Y, to^ any base or 

 modulus; and when XY = maa;. its logarithm X' + Y -max. 

 and therefore X' + Y' = 0, which fluxional equation we may 

 express by X"i 4 Y'V = ; for as X' involves only the vanable 

 quantity x, its fluxion 1' will evidently involve only the flux- 

 i<>n of X- in like manner the fluxion of Y' may be expressed 

 ^v Y'V; and from the equation ^"x ^Y"'j -0 viQ hd.-ve 

 X r = -Y"y : but since x + ^ = E we have also x + t/ = 0, and 

 '-": ~h}>J which dividing the equation tJ'x = -Y"y, we obtain 



"Now this equation ought to be equivalent to - = y ; and 

 this circumstance is effected in the simplest manner possible, by 

 assuming X" = ^, and Y" = ^; m being any fixed number 

 ^^hich the question may require. 



-Since, therefore, X'' == — we have X"i = X' = -^, and 

 taking the fluent, we have X' = a' + ^. The constant quan- 

 tity a' being either absolute, or some function of t^e distance o^ 



"We have discovered, therefore, that the loganthm of the 

 probability that the error x happens in the distance a is ex- 

 pressed by a' + '^= X' and consequently the probability it- 

 , 2a ' 



f is X = e^' = /"'^ ^). Such is the formula by which 

 he probabihties of different errors may be compared, when 

 ^he values of the detenninate quantities e, a ana m ^ 

 P'-operly adjusted. If this probability of the error x be denoted 



