Who is the father(or mother) of the linear least squares analysis as we know it? 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"?
 A: 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
