Chronology of Combination of Observations and the Method of Least Squares

1632   Galileo Galilei, in analysis of Tycho Brahe's new star (1572) proposes
      that all observations are subject to errors that are (a) symmetrically
      distributed about 0; (b) small errors occur more frequently than
      large errors.  Proposes that best hypothesis is the one with the
      smallest sum of absolute deviations from the fitted value.
 
1714   British Parliament establishes the "Commission for the Establishment
      of Longitude at Sea," giving prizes for contributions to this topic
      over the next 100 years.
      
1722   Cotes' Rule for finding "the most probable place" from n observations,
        each with weights by a weighted mean.

1749   Leonhard Euler's work on Inequalities in the Movement of Saturn
      and Jupiter: attempts to solve for 8 unknowns describing the orbit
      of Saturn from 75 sets of observations made 1582--1745.

1750   Johann Tobias Mayer's 'Libration of the Moon' develops a method for
      solving overdetermined sets of (27) linear equations in 3 unknowns
      by grouping them into 3 sets of similar equations, and solving their
      summed equations.  Gives a bound on the error of estimation for one
      unknown.

1755-1770   Roger Boskovich proposes general principles to be used in solving
      observational equations related to the 'figure of the earth' from
      measurements of arc length at different latitudes.  These conditions
      amount to minimizing least absolute deviations.  Gives a geometrical
      solution.

1787   Pierre Simon Laplace extends Mayer's method by considering and
      solving sets of linear combinations of the 'equations of condition'
      for the orbit of Jupiter.

1789-1797   Laplace gives an algebraic formulation of Boskovich's method, 
      proves it minimizes the sum of absolute errors.  Then he extends
      this to minimize a *weighted* sum of absolute errors, with given
      weights.  [Laplace also proposed to minimize the largest absolute
      error -- a minimax procedure.]
       
1805   A. M. Legendre publishes the method of least squares (in a 9-page
      appendix to a work on determination of the orbits of comets),
      and applies this to measurements of lengths of the arc of the
      meridian through Paris

1809   C.F. Gauss gives a probabilistic justification of Legendre's least
      squares criterion, showing it is a maximum likelihood method when
      errors are normal.

1823   C.F. Gauss proves the Minimum Variance theorem, showing that, of
      all linear combinations of measurements estimating an unknown,
      least squares has minimum variance (smallest RMSE);  does not
      depend on distribution of errors.

1844   William Fishburn Donkin demonstrates that the method of least
      squares can be derived without recourse to probability, from
      a purely *physical* model of forces, so that the least squares
      estimate is the position of equilibrium, with all forces balanced.