The method of least squares
A rule: from discordant observations one takes those values of the parameters for which the sum of the squares of the residuals is minimal. An excess of measurements turns from a nuisance into a resource.
The rule was printed in 1805 by the Frenchman Adrien-Marie Legendre. The German Carl Friedrich Gauss in 1809 gave it a probabilistic justification, tying it to the distribution of errors, and claimed to have been using the method since 1795; the priority dispute went on for decades.
The method became the standard computational technique of nineteenth-century astronomy and geodesy: the more measurements there are, the more precise the estimate.
Why this date
The date is that of the first publication: the appendix Sur la méthode des moindres quarrés to Legendre's Nouvelles méthodes pour la détermination des orbites des comètes, Paris, 1805. Gauss printed the method in 1809 in Theoria motus and claimed to have been using it since 1795; there is no printed trace of it by him before 1805, but Stigler, examining the documents, leans towards Gauss having had the method earlier, while stating that this is not conclusive. The priority remains disputed.
What it grew from
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The laws of planetary motion
influenced · Confidence: medium
Both founding publications of the method are works on determining orbits from redundant and discordant observations, and in the preface to Theoria motus Gauss derives his problem directly from Kepler's extraction of orbital elements from the observations of Tycho Brahe.
Dispute: The descent from the problem of orbit determination is known from Gauss's own preface, that is, from the testimony of a participant rather than from an independent examination by a historian. The priority of the method is disputed as well. In Legendre's favour stands print: he published the method first, in 1805, and wrote to Gauss 'je n'ai jamais appelé principium nostrum un principe qu'un autre avait publié avant moi'. In Gauss's favour stand his own statement in Theoria motus of 1809 ('my principle, which I have made use of since 1795'), the diary entry of 17 June 1798, and Stigler's examination, which leans towards Gauss having had the method well before Legendre while stating outright that this cannot be regarded as conclusive.
Link sources
- Carl Friedrich Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, Charles Henry Davis, Boston (1857) Verified 20 Sep 2026
After the laws of planetary motion were discovered, the genius of Kepler was not without resources for deriving from observations the elements of motion of individual planets.
- A. Abdulle, G. Wanner, 200 years of least squares method, Elemente der Mathematik 57, 45-60 (2002) Verified 20 Sep 2026
In 1805 appeared the work Nouvelles méthodes pour la determination des orbites des comètes by A.-M. Legendre, containing in an appendix an extremely beautiful presentation of the méthode des moindres quarrés.
- Stephen M. Stigler, Gauss and the Invention of Least Squares, Annals of Statistics 9 (3), 465-474 (1981) Verified 20 Sep 2026
It is argued (though not conclusively) that Gauss probably possessed the method well before Legendre, but that he was unsuccessful in communicating it to his contemporaries.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Carl Friedrich Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, Charles Henry Davis, Boston (1857) Verified 20 Sep 2026
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Mathematical Principles of Natural Philosophy
influenced · Confidence: low
In the preface to Theoria motus Gauss names Newton's demonstration of motion in conic sections as what generalized the problem of orbit determination, and it is such an orbit that he fitted to observations; no independent source for the connection was found, the testimony comes from Gauss himself.
Dispute: The Newtonian formulation can be traced only in Gauss. Legendre's book of 1805 is titled as a work on the orbits of comets, but the method itself is set out in the appendix Sur la méthode des moindres quarrés and is shown there on a numerical example drawn from the French meridian measurements, that is, on the problem of the figure of the earth rather than on a cometary orbit, so this descent does not hold through Legendre.
Link sources
- Carl Friedrich Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, Charles Henry Davis, Boston 1857 (1857) Verified 25 Sep 2026
On the other hand, Newton, starting from the principle of universal gravitation, demonstrated à priori that all bodies controlled by the attractive force of the sun must move in conic sections, of which the planets present one form to us, namely, ellipses, while the remaining forms, parabolas and hyperbolas, must be regarded as being equally possible, provided there may be bodies encountering the force of the sun with the requisite velocity;
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Carl Friedrich Gauss, Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, Charles Henry Davis, Boston 1857 (1857) Verified 25 Sep 2026
What grew out of it
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'Sur l'homme': the astronomers' law of error applied to people
influenced · Confidence: medium
The error curve with which astronomers and surveyors described the scatter of measurements (Gauss, Legendre) was in Quetelet applied for the first time not to errors of observation but to the scatter of traits in a human population. Brush states outright that the law discovered by Adrain and Gauss was 'extensively applied by Quetelet'. The link is drawn as "influenced" rather than as a prerequisite: Quetelet did not use the method of least squares itself, he took the error distribution from the same tradition.
Dispute: Brush's sentence asserts the Gauss-Quetelet link in passing, inside a statement about Maxwell; no separate historian's study analysing what exactly Quetelet took from the error theory of Gauss and Laplace, and what from Laplace's demographic work, was read here. The core node least_squares is the method of least squares, not the error distribution as such, so the link is called "influenced" and the confidence is medium.
Link sources
- Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026
The crucial step was his translation of the 'normal distribution law' or 'law of errors,' discovered by Adrain and Gauss and extensively applied by Quetelet, into a distribution law for molecular velocities.
- J. J. O'Connor, E. F. Robertson, Adolphe Quetelet, MacTutor History of Mathematics Archive, University of St Andrews (2006) Verified 20 Sep 2026
Influenced by Laplace and Fourier, Quetelet was the first to use the normal curve other than as an error law.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Stephen G. Brush, History of the Kinetic Theory of Gases (Storia della Scienza, vol. 7, ch. 44) (2004) Verified 20 Sep 2026
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Pearson's correlation coefficient and the chi-square test
influenced · Confidence: medium
Pearson fitted regression curves by least squares, but his problem was different from Legendre's and Gauss's. Eisenhart puts it as an extension beyond the original formulation: in the classical problem one variable was measured with negligible error and the other with error, whereas in the Galton-Pearson situation both traits are measurable with negligible error in a single individual but have a joint distribution in a population.
Dispute: Eisenhart stresses the difference of the two formulations rather than a borrowing: Pearson himself preferred the method of moments and, in Eisenhart's words, 'Pearson clearly favored his method of moments; but the method of least squares has prevailed'. The link is therefore drawn as "influenced" with medium confidence: least squares gave the language and the computational technique but was not a prerequisite for correlation.
Link sources
- Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026
However, use of the method of least squares to fit polynomial regression curves in a bivariate correlation situation involves an extension beyond the original formulation and development of the method of least squares by Legendre, Gauss, Laplace, and their followers in the nineteenth century.
- Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026
In this classical development of the method of least squares, one of the variables-x, for example—was a quantity that could be measured with negligible error, and the other, y, a quantity of interest functionally related to x, the observed values of which for particular values of x, Yx, were, however, subject to nonnegligible measurement errors.
- Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026
In the Galton-Pearson correlation situation, in contrast, the traits X and Y may both be measurable with negligible error with respect to any single individual but in some population of individuals have a joint frequency or probability distribution.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Churchill Eisenhart, «Pearson, Karl», Complete Dictionary of Scientific Biography (2008) Verified 20 Sep 2026
Node sources
- A. Abdulle, G. Wanner, 200 years of least squares method, Elemente der Mathematik 57, 45-60 (2002) Verified 20 Sep 2026
In 1805 appeared the work Nouvelles méthodes pour la determination des orbites des comètes by A.-M. Legendre, containing in an appendix an extremely beautiful presentation of the méthode des moindres quarrés.
- A. Abdulle, G. Wanner, 200 years of least squares method, Elemente der Mathematik 57, 45-60 (2002) Verified 20 Sep 2026
Despite of Legendre's work, Gauss called, in his famous treatise [9] Theoria motus corporum celestium, published 1809, obstinately the least squares idea 'my principle, which I have made use of since 1795'.
- Stephen M. Stigler, Gauss and the Invention of Least Squares, Annals of Statistics 9 (3), 465-474 (1981) Verified 20 Sep 2026
The most famous priority dispute in the history of statistics is that between Gauss and Legendre, over the discovery of the method of least squares.
- Stephen M. Stigler, Gauss and the Invention of Least Squares, Annals of Statistics 9 (3), 465-474 (1981) Verified 20 Sep 2026
It is argued (though not conclusively) that Gauss probably possessed the method well before Legendre, but that he was unsuccessful in communicating it to his contemporaries.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.