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Background:
The least-squared error fit has been around for some time.

Laplace, P. S. "Des méthodes analytiques du Calcul des Probabilités." Ch. 4 in Théorie analytique des probabilités, Livre 2, 3rd ed. Paris: Courcier, 1820.

Gauss, C. F. "Theoria combinationis obsevationum erroribus minimis obnoxiae." Werke, Vol. 4. Göttingen, Germany: p. 1, 1823.

Wikipedia attributes Gauss and Legendre for it. (link)

Many software tools perform basic linear fits with analysis of quality of fit. (JMP, R 'lm', ...)

There is a 200 year span between 2020 and 1820. Somewhere in there the details were added.

Question:
Who is the effective "father" (or mother) of the analysis as we know it?

There has to be someone "back in day" who made the "first" that is ~80% (or more) like what this foundational analysis method?

Can you give a reference to this "first work"?

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    $\begingroup$ I'm not sure this question has a better answer than Laplace and Gauss. Are you sure that the refinements on the method since them haven't been gradual? $\endgroup$ – Kodiologist Jun 15 '17 at 19:03
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    $\begingroup$ Why do you say "Wikipedia blames" rather than attributes? The statement implies that there is something wrong with it. $\endgroup$ – Michael R. Chernick Jun 15 '17 at 19:27
  • $\begingroup$ @MichaelChernick - something is wrong with Wikipedia. It is not considered an authoritative source. Even a broken clock is accurate twice a day. Even if Wikipedia is accurate now, there is no assurance it will retain that tomorrow. I will edit the text. $\endgroup$ – EngrStudent - Reinstate Monica Jun 15 '17 at 19:32
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    $\begingroup$ I think people like Galton definitely fall within any reasonable "ish". I am not sure how a less precise reading of the question would improve the problem I am getting at. $\endgroup$ – Glen_b -Reinstate Monica Jun 17 '17 at 0:28
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    $\begingroup$ @Glen_b Along the lines of "as we [currently] know it," a strong case could also be made for R.A. Fisher and the influential, 1925 book Statistical Methods for Research Workers. $\endgroup$ – Matthew Gunn Jun 18 '17 at 20:07
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I highly recommend Prof. Stephen Stigler's A History of Statistics: Measurement of Uncertainty before 1900. Chapter 1 discusses your question in depth. (A link is available here.)

Legendre first developed the method of least squares and derived the normal equations in 1805 as a way to solve an overdetermined system of linear equations.

Quoting Stigler, "For stark clarity of exposition the presentation [of Legendre] is unsurpassed; it must be counted as one of the clearest and most elegant introductions of a new statistical method in the history of statistics."

The context of ordinary least squares

A fascinating point is that development of least squares came before the discovery of the normal distribution and modern justifications for the use of least squares.

Least squares was developed as a way of combining multiple, imperfect astronomical observations to recover underlying parameters governing the movements of heavenly bodies.

Each astronomical observation defines a linear equation, and with more observations than parameters, the astronomers of the time were faced with an inconsistent system. What to do? Mayer developed a method where observations were separated into $k$ groups, the equations in each group were averaged together, and then the underlying system could be solved (Stigler 1990). Legendre instead proposed introducing an error term and minimizing the sum of squared error.

References

Stigler, Stephen, A History of Statistics: Measurement of Uncertainty Before 1900, 1990. Belknap Press

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