Editing Least squares (section)

===Founding===
The method of least squares grew out of the fields of [[astronomy]] and [[geodesy]], as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the [[Age of Discovery]]. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on land sightings for navigation.

The method was the culmination of several advances that took place during the course of the eighteenth century:<ref name=stigler>{{cite book
  | last = Stigler
  | first = Stephen M.
  | title = The History of Statistics: The Measurement of Uncertainty Before 1900
  | publisher = Belknap Press of Harvard University Press
  | year = 1986
  | location = Cambridge, MA
  | isbn = 978-0-674-40340-6
  | url-access = registration
  | url = https://archive.org/details/historyofstatist00stig
  }}</ref>

*The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, first appeared in [[Isaac Newton]]'s work in 1671, though it went unpublished, and again in 1700.<ref>{{Cite book |last1=Buchwald |first1=Jed Z. |title=Newton and the Origin of Civilization |last2=Feingold |first2=Mordechai |date=2013 |publisher=Princeton University Press |isbn=978-0-691-15478-7 |location=Princeton Oxford |pages=90–93, 101–103}}</ref><ref>{{Cite web |last=Drum |first=Kevin |date=2013-05-10 |title=The Groundbreaking Isaac Newton Invention You've Never Heard Of |url=https://www.motherjones.com/kevin-drum/2013/05/groundbreaking-isaac-newton-invention-youve-never-heard/ |access-date=2024-12-21 |website=Mother Jones |language=en-US}}</ref> It was perhaps first expressed formally by [[Roger Cotes]] in 1722.
*The combination of different observations taken under the ''same'' conditions contrary to simply trying one's best to observe and record a single observation accurately. The approach was known as the method of averages. This approach was notably used by Newton while studying equinoxes in 1700, also writing down the first of the 'normal equations' known from [[ordinary least squares]],<ref>{{cite arXiv |eprint=0810.4948 |class=physics.hist-ph |first1=Ari |last1=Belenkiy |first2=Eduardo Vila |last2=Echague |title=Groping Toward Linear Regression Analysis: Newton's Analysis of Hipparchus' Equinox Observations |date=2008}}</ref> [[Tobias Mayer]] while studying the [[libration]]s of the Moon in 1750, and by [[Pierre-Simon Laplace]] in his work in explaining the differences in motion of [[Jupiter]] and [[Saturn]] in 1788.
*The combination of different observations taken under {{em|different}} conditions. The method came to be known as the method of ''[[least absolute deviation]]''. It was notably performed by [[Roger Joseph Boscovich]] in his work on the shape of the Earth in 1757 and by [[Pierre-Simon Laplace]] for the same problem in 1789 and 1799.
*The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved. Laplace tried to specify a mathematical form of the [[probability]] density for the errors and define a method of estimation that minimizes the error of estimation. For this purpose, Laplace used a symmetric two-sided exponential distribution we now call [[Laplace distribution]] to model the error distribution, and used the sum of absolute deviation as error of estimation. He felt these to be the simplest assumptions he could make, and he had hoped to obtain the arithmetic mean as the best estimate. Instead, his estimator was the posterior median.