to the Method of Least Squares. 161 



In tlie preface Legendre gives an outline of his method for com- 

 ])utiiig orbits. On page viii he says : " II faxit ensiuite, lorsque 

 toutes les conditions du probleme sont exprimees convenablement, 

 dc'terminer les coefficiens de maniere a rendre les erreurs les plus 

 petites qu'il est possible. Pour cet effet, la methode qui me paroit la 

 ))lus simple et la plus generale, consiste a rendre mininmni la somme 

 des quarres des erreurs. On obtinent ainsi autant d'equations qu'il 

 y a de coefficiens inconnues; ce qui acheve determiner tons les elemens 



de I'orbite la methode dout je viens de parler, et que j'applle 



Methode des mohidres quarres, \)i^\\t etre d'une grande utilite " 



On page 64 is an application of the method to the solution of three 

 equations Avith two unknown quantities, in which the now well- 

 known rule for the solution of normal equations is followed. On 

 l)age 68 and 69 are references to its use. Pages 72-80 constitute an 

 Appendix " Sur la Methode des moindres quarres." In this, after 

 having mentioned that it is impossible that the sum of the errors 

 should be zero when the number of given equations exceeds that of 

 the unknown quantities, Legendre says : " De tons les principes 

 . . . . je pense qu'il n'en est pas de plus general, de plus exact, ni 

 d'une application plus facile que celui .... qui consiste a rendre 

 mhiimwiii la somme des quarres des erreurs. Par ce moyen, il 

 s'ctablit entre les erreurs une sorte d'equilibre qui empechant les ex- 

 tremes de prevaloir, est tres-propre a faire connoitre I'etat du systeme 

 le plus proche de la verite." 



Legendre then proceeds to deduce the rule for the formation of 

 "I'equation du minimum par rapport a I'une des inconnues," or as 

 we now say of a normal equation. His notation is the following : in 

 the n equations 



Q = a -{-hx -\- cy -\- . . . 

 = a! -(- b'x -\- c'y -\- . .. 

 = a" + h"x + c"y + . . . 



a, J, c...., a', b', c'....are known by observation or theory, and 

 £c, y, . . . . are to be determined. By forming the sum of the squares 

 of these equations, differentiating with I'eference to each unknown 

 sepai'ately and placing the derivatives equal to zero, he finds 



Q=fab + .>-fb^^yJhc-\-... 

 =fac + xj'bc + y/c2 -|- . . . 



which are the same in number as the unknown quantities x, y, 

 and in which 



fah = ab + a'b' + a"b" + ... 



f b^= b-^-{- Z>'2 + b"^- + ... 



Legendre next demonstrates that the rule of the arithmetical mean 

 is a particular case of his gei^.eral principle. He then supposes the 

 position of a point in space to be determined by three observations 

 and finds the values of its coordinates given by the method. Notic- 

 ing their identity with those for the centre of gravity of three points 

 in space, he announces that the sum of the squares of the distances of 



Trans. Conn. Acad., Vol. IV. 21 Oct., 1877. 



