to the Method of Least Squares. 157 



consists in comparing tlie mean with a new mean found after reject- 

 ing that observation deviating most from the first mean. See Nova 



Acta Li2)slae, 1700, p. 5(30; also Lani/f erf s J3eytrage Vol.1, 



p. 426. 



1765 Lambekt. 'Theorieder Zuverlassigkeit der Beobachtnngen 

 und Versnche.' Beytrage zum Gehrauche der Mathematik . . . . (Ber- 

 lin, 1765-70, 3 vols., 8vo; second ed., 1792), Vol.1, pp. 424-488. 



Contains a method of adjusting simple observations founded on 

 the principle that the algebraic sum of the errors shall be zero. The 

 method is illustrated by the determination of empirical formulse for 

 the length of the seconds pendulum, the declination of the magnetic 

 needle, etc. 



1774 Laplace. 'Determiner le milieu que I'on doit prendre entre 

 trois observations donnees d'un meme phenomene.' Mem. Acad. 

 Paris, par divers savans [etrangers], Vol. IV, pp. 684-644. 



This is Part V of the ' Memoire sur la probabilite des causes par 

 les evenemens,' which occupies pages 621-656 of the volume. It 

 contains the first attempt to deduce a rule for the combination of 

 observations from the principles of Probability. 



Laplace begins by saying that the law of probability of errors of 

 observation may be represented by a curve whose equation is 

 y:=qj(x), X being any eri-or and y its probability; and this curve 

 must have three properties : 1st, it must be symmetrical with refer- 

 ence to the axis of y, since positive and negative errors are equally 

 probable; 2nd, the axis of ,r must be an asymptote, since tlie proba- 

 bility of the error co is ; 3rd, the area of the curve must be unity, 

 since it is certain that an error will be committed. 



Laplace takes qj{x) as (p(x)=.l'/ne^'"''' (x being regarded as always 

 positive), but his reasons for doing so are slight. With this law he 

 finds the mean of three observations, regarding it as corresponding 

 to an ordinate which divides a curve u-=.cp(^x^)(/j(x2)(p{xQ) into two 

 equal parts. His result is as follows: Let J/j, 31^ and M^ be the 

 three measurements, of which Mj is the least ; let M^ -\-x be the 

 mean ; it is required to find x. Put M^ — M ^z=zp and M^ — M^zzzq: 

 then X is given by 



X =p-\ log, (1-^4- e-'"^' - 4- e-""'), 



9)1. ^ ^ ' ' 



in which ra is a constant depending upon the precision of the obser- 

 vation. Laplace then shows that this value cannot agree Math the 

 rule of the arithmetical mean, and he computes a table "for finding x 

 for certain given ratios of q to />. For instance 



if (/ = 0.0;> x=z 0.860 j», 



q=Q.\ p _. a;= 0.894 jt), 



q = 0.2p xz=^.9)\<dp, 



q = Q.^p x= 0.932 p. 



