180 Mansfield Merriman — List of Writings relating 



The measurements are discussed by Gauss's method of correlatives 

 (1S2S) whose algorithm is given in full, and also by a new method of 

 Hansen. Rosenberger had in 1827 examined the same measure- 

 ments. 



1831 LiTTROw. Bemerkungen zum practischen Gebi-auche der 

 Wahrscheinlichkeitsrechnung. Baunigartner''s Zeitschr. f. Phys., 

 Vol. IX, pp. 43.3-449. 



1831 Puissant. 'Application du calcul des ])robal)ilites a la 

 mesure de la precision d'un grand nivellement trigonometrique.' 

 Mem. Acad. JParis, Vol. X, pp. 533-547. — Gonnaiss. des Temps for 

 1834, pp. 3-17. 



A modification of the method of Laplace's Tliir<l Supplement 

 (1820), illustrated by a practical example. 



1832 Encke. 'Ueberdie Begriindung der Methode der kleinsten 

 Quadrate.' Ahhandl. Akad. Berlin for 1831, pp. 73-78 of the mathe- 

 matical part. 



After brief notices of five proofs of the Method of Least Squares, 

 Enckf. gives the preference to Gauss's of 1809. To establish this 

 more rigidly he ofl:ers a demonstration to show that for direct obser- 

 vations the rule of the arithmetical mean gives the most probable 

 result. This demonstration (in my opinion not a rigorous one) has 

 been followed by many subsequent writers. It is repeated by Encke 

 in the article quoted next below, and is particularly stated wnth confi- 

 dence by Chauvenet in 1864. For criticisms see 1843 Reuschle and 

 1872 Glaisher. See also Encke's later opinion, below under 1850. 



1832 Encke. 'Ueber die Methode der kleinsten Quadrate.' Ber- 

 lin. Astron. Jahrhuch for 1834, pp. 249-312; for 1835, pp. 253-320; 

 for 1836, pp. 253-308. — Republished in Encke^s astronomische 

 Ahhandhmgen (Berlin, 1866), Vol. I, Nos. xii, xiii, xiv. 



These memoirs form a treatise on the Method of Least Squares, 

 from which many text-books have been compiled. 



The first memoir contains the proof of 1809 Gauss, reinforced by 

 Encke's attempted demonstration of the validity of the arithmetical 

 mean, the discussion of weight aiul probable errors, and two tables 



of the probability integral — ^- I e~^ dt, the first between the limits 



V 77-*/ 



X 



and ?, and the second between the limits and 0.476986- (;*• being any 



erroi- and r the probable error). These were computed from Kramp's 

 tables of 1799 as quoted by Bessel in 1818. See Land. Phil. Mag.., 

 1871, Vol. XLII, p. 431, et sq. A translation of this first memoir and 

 a reprint of the tables is given in Taylor'' s Scientific Memoirs, 1841, 

 Vol. II, pp. 317-369. 



