to the Method of Least Squares^. 167 



An account of the contents of this memoir, with liistorical remarks 

 on the subject of Least Squares, is given by Delaaibre, Bidl. iSoe. 

 Philom. Paris, 1811, Vol. II, pp. 262-266, an Enolisli transhition of 

 which is in Tilloeh's Phil. Mag., 1812, Vol. XXXIX, pp. 240-244. 



1811 Laplace. ' Du milieu qu'il faut choisir entre les resultats 

 (Pun grand nombre il'ol)servations.' Gonnaissance des Ihns for 

 1813, pp. 213-223. — German translation in ZaeNs Motiatl. ijorres., 

 1812, Vol. XXV, pp. 105-120. 



Contains matter Avhich is reproduced in the Theorie . . . des Prob.., 

 1812, pp. 322-329. 



1812 Laplace. '• Theorie analytiqye des Prohabilites.'' Paris, 4to. 

 pp. 464. — Third edition, 1820, 4to, pp. cxlii, 506, with Introduction 

 and three Supplements (see 1 8J 4, 1815, 1818, 1820). — Fourth edition 

 (Vol. VII of Oeuvres de Laplace) Paris, 1847, 4to, pp. cxcv, 691. 



" . . . . the gi-eater part of the Theorie des Probabilites is a reprint 

 of papers in the Memoirs of the Academy, which apj^ear to contain 

 the contents of the first papers on which he set down his processes. 

 These with preliminary chai)ters, descriptive not of what follows, but 

 of the general methods which he drew from the following j)arts, 

 make up the whole work." — De Morgan, Theory of Prohabilities in 

 Encyc.. 3Ietrop., \x 453. It "is by very much the most difficult 

 mathematical work w^e have met with." — Ibid, p. 418. Todhunter 

 in his History of Probability, p. 560, Ellis (1844) and other writers 

 have also testified to the abstruseness of Laplace's methods. 



The Method of Least Squares is developed in Chap. IV, (pages 

 304-348) of the second part of the work. The analysis only extends 

 to the case of two unknown quantities or elements, and the number 

 of observations is i-equired to be very large or infinite. LTnder these 

 restrictions the jNIethod is shown to give most advantageous results, 

 whatever be the law of facility of error provided only that positive 

 and negative errors ai-e equally probable. Laplace's definition of 

 t))ost advantayeoiis residts is the following : "... .si Ton multiplie les 

 erreurs possibles d'un element par leurs probabilites respectives, le 

 systeme le plus avantageux sera celui dans lequel la somme de ces 

 prodiiits tous pris positivement, est un niinimum.'''' The results thus 

 obtained are not necessarily the most probable. 



In the concluding paragraph of Chap. IV and in the opening pages 

 of the First Supplement (see 1815), Laplace has given a general ac- 

 count of his method of analysis. These remarks and the table of 

 contents at the end of the volume, give a much clearer idea of the 

 steps of the demonstration than does Chap. IV itself The principal 

 objection against the validity of the proof is that it requires an infi- 

 nite or very large number of obser\ations. With this requirement, 

 however, Gacss's proof of 1809 becomes perfectly logical and the 

 results are the most prolxMe, not merely most advantageous. 



