Carl Friedrich Gauss

b. 30 April 1777, Brunswick

d. 23 February 1855, Göttingen

The principle of least squares was first published in 1805 as an appendix to a work on determining the orbits of comets written by Adrien Legendre. Gauss, four years later, writes in the Theoria Motus:

"Our principle, which we have made use of since the year 1795, has lately been published by LEGENDRE in the work Nouvelles methodes pour la determination des orbites des cometes, Paris, 1806, where several other properties of this principle have been explained, which, for the sake of brevity, we here omit." (Davis translation.)

Thus a controvery over priority begins. The claim by Gauss puts his discovery at when he was 17 years of age and when he had become acquainted with Volume 1 of Lambert's Beyträge zum Gebrauche der Mathematik und deren Anwendung.

Gauss mentions Legendre in 1806 GAUSS. "II Comet vom .Jahr 1805," Zach's Monatl. Corres., Vol. XIV, pp. 181-186. For discussions of the controversy see:

R.L. Plackett, Studies in the History of Probability and Statistics XXIX. The Discovery of the Method of Least Squares, Biometrika, Vol. 59, No. 2 (Aug. 1972) 239-251.
S.M. Stigler, "An Attack on Gauss, published by Legendre in 1820," Historia Mathematica 4, (1977) 31-35.
S.M. Stigler, "Gauss and the Invention of Least Squares," Annals of Statistics, Vol. 9, No. 3, (1981) 465-474.
O.B. Sheynin, various publications for which see this web site

Gauss's writings on least squares and applications

The collected works of Gauss (Werke) have been published in 12 volumes. These are available through the GDZ: Göttinger Digitalisierungszentrum. Those relevant to this discussion are the following:

Volume 4 Wahrscheinlichkeitsrechnung und Geometrie

[1] ABHANDLUNGEN: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior. Commentationes societatis regiae scientarium Gottingensis recentiores, 5. pp. 33- 62. (1821 Feb. 15) Werke 4, 1-26.

[2] Theoria combinationis observationum erroribus minimis obnoxiae: Pars posterior. Commentationes societatis regiae scientarium Gottingensis recentiores, 5. pp. 63-90. (1823 Feb. 2) Werke 4, 27-53.

[3] Supplementum theoriae combinationis observationum erroribus minimis obnoxiae. Commentationes societatis regiae scientarium Gottingensis recentiores, 6. pp. 57-98. (1826 Sept. 16) Werke 4 55-94.

[4 ANZEIGEN EIGNER ABHANDLUNGEN: Theoria combinationis observationum erroribus minimis obnoxiae: Pars prior. Göttingische gelehrte Anzeigen, 33: 321-327. (1821 Feb. 26) Werke 4, pp. 95-100.

[5] Theoria combinationis observationum erroribus minimis obnoxiae: Pars posterior. Göttingische gelehrte Anzeigen, 32: 313-318. (1823 Feb. 24) Werke 4, pp. 100-104.

[6] Supplementum theoriae combinationis observationum erroribus minimis obnoxiae. Göttingische gelehrte Anzeigen, 153:1521-1527. (1826 Sept. 25) Werke 4, 104-108.

[7] AUFSATZ: Bestimmung der Genauigkeit der Beobachtungen Zeitschrift für Astronomie, 1. (1816 March, pp. 185-197) Werke 4, 109-117. (On the Determination of the Precision of Observations)

Volume 6 Astronomische Abhandlungen

[8] Disquisitio de elementis ellipticis Pallidis Commentationes societatis regiae scientarium Gottingensis recentiores, 1. pp. 1-26. (1810) Werke 6, 1-50. (Application of the Method of Least Squares to the Elements of the Planet Pallas) A partial translation into German with comments is given as "Über die elliptischen Elemente der Pallas," Monatliche Correspondenz, 1811, Vol. XXIV, pp. 449-465.

[9] Chronometrische Längenbestimmungen Astronomische Nachrichten Band 5, S. 227-240 (1826 Nov.). Werke 6, 455-458. (On the Chronometric Determination of Longitude)

Volume 7 Theoretische Astronomie

[10] Theoria Motus Corporum Coelestium in Sectionibus Conicis Solem Ambientium. Perthes and Besser, Hamburg. (1809) Reprinted in Werke 7, pp 1-261.

Volume 9 Geodäsie. Fortsetzung von Band 4.

[11] Bestimmung des Breitenunterschiedes zwischen den Sternwarten von Göttingen und Altona durch Beobachtungen am Ramsden'schen Zenithsector (1828) Werke 9, 1-64.

[12] Anwendungen der Wahrscheinlichkeitsrechnung auf eine Aufgabe der praktischen Geometrie Astronomische Nachrichten 1, S. 81-86. (1823) Werke 9, 231-237 (Application of Calculus of Probabilities to Practical Geometry)

Gauss in translation

1855 J. Bertrand, with the approval of Gauss himself, translated his writings into French as Méthode des moindres carrés, mémoires sur la combinaison des observations. Ballet-Bachelier, Paris.
1857 Charles Henry Davis translated the Theoria Motus as Theory of Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. Little, Brown and Co. Reprinted by Dover 1963.
1865 Carl Haase. Theorie der Bewegung der Himmelskörper... (Hannover). This is the Theoria Motus Corporum Coelestium.
1887 A large selection of writings of Gauss related to least squares was published in Germany as Abhandlungen zur Methode der kleinsten Quadrate. These were collected by A. Börsch and P. Simon.
1957 Hale F. Trotter made a translation directly from Bertand in 1957 as Gauss's Work (1803-1826) on the Theory of Least Squares, Technical Report No. 5, Statistical Techniques Research Group, Princeton University. I believe this to be in the public domain, but until verified I will refrain from making it available.
1995 G.W. Stewart Theory of the Combination of Observations Least Subject to Errors, SIAM, 1995.

In summary, Stewart contains translations from the Latin of [1] to [6]. Trotter's translation has [1] - [3], [7]-[9], [10] and [12] with several posted at the University of York. For [10], see Davis. Finally, [11] is only available in German through Börsch and Simon.

Further literature

The literature on Gauss and least squares is vast.

Seal, Hilary L. (1967) The historical development of the Gauss linear model. Biometrika 54: 1-24.
Sprott, David A. (1978) Gauss's contributions to statistics. Historia Mathematica 5: 183-203.
Sheynin, O.B. (1979), C. F. Gauss and the theory of errors. Arch. Hist. Exact Sci. , vol. 20, pp. 21 - 72.
Farebrother, R.W. (1985) The statistical estimation of the standard linear model 1756-1853. Proceedings of the First International Tampere Seminar on Linear Statistical Models and their Applications, pp. 77-99.
Sheynin, O. (1993) On the history of the principle of least squares. Arch. Hist. Exact Sci. 46, 39-54.
Celmins, Aivars (1998) The Method of Gauss in 1799. Statistical Science Vol. 13, No. 2, 123-135.

Finally, correspondence
1799 Vermischte Nachrichten no. 3. Allgemeine Geographische Ephemeridenz, 4, 378.

1839 Letter to Bessel, 28 February 1839. Briefwechsel zwischen Gauss und Bessel. pp. 523-525. Werke 8, 146-147.

1844 Letter to Schumacher 25 November 1844. Werke 8, 147-148.