96 M. Poisson's Memoir on the 



astronomy, the quantity which it is proposed to determine by 

 observations, is a given function of several elements which are 

 already approximately known, and in which it is only necessary 

 to make very small corrections, the products of which and all 

 the powers higher than the first are neglected. The given 

 function then becomes a linear function of these unknown cor- 

 rections. It is made equal successively to all the values 

 resulting from experiment, which affords as many equations of 

 condition as there are observations. The employment of 

 these linear equations for determining the corrections of the 

 elements, by adopting for the purpose a great number of k ob- 

 servations, has contributed much to the improvement of the 

 astronomical tables. It appears that Euler and Mayer are the 

 first who employed them ; one in his Memoir on the Perturba- 

 tions of Saturn and Jupiter, which received the prize in 1750 

 from our academy, and the other in his Memoir on the Libration 

 of the Moon. But their number being always superior to that 

 of the unknown quantities, the solving of them occasioned some 

 embarrassment, and this serious inconvenience ensued, that 

 the calculators could deduce, from the same system of equa- 

 tions, different results according to the method of calculation 

 they employed. When there was only one unknown quantity 

 to be determined, it was agreed to render its coefficient positive 

 in all the equations that were subsequently added, to form the 

 final equation, from which the value of the unknown quantity 

 was to be deduced. When these unknown quantities were two 

 or more in number, the combination of the equations of con- 

 dition that was made to reduce them to an equal number of 

 final equations was absolutely arbitrary. This embarrassment, 

 and the inconvenience which resulted from it, remained to the 

 period when M. Legendre proposed a direct and uniform me- 

 thod of forming the final equations, which was generally 

 adopted under the name of Method of least squares of the 

 errors, which was assigned to it by its author. It consists, as 

 is well known, in deducting from the result of each observation 

 the linear function of which it furnishes an approximate value : 

 the difference is the error of observation ; the sum of the 

 squares of all these differences is taken : its differentials taken 

 successively, with respect to the corrections of all the elements, 



