Editing Clairaut's theorem (gravity) (section)

== History ==
Although it had been known since antiquity that the Earth was spherical, by the 17th century evidence was accumulating that it was not a perfect sphere.   In 1672 [[Jean Richer]] found the first evidence that gravity was not constant over the Earth (as it would be if the Earth were a sphere); he took a [[pendulum clock]] to [[Cayenne]], [[French Guiana]] and found that it lost {{frac|2|1|2}} minutes per day compared to its rate at Paris.<ref>{{cite book
  | last = Poynting
  | first = John Henry
  |author2=Joseph John Thompson
  | title = A Textbook of Physics, 4th Ed.
  | publisher = Charles Griffin & Co.
  | year = 1907
  | location = London
  | page = [https://archive.org/details/atextbookphysic01thomgoog/page/n29 20]
  | url = https://archive.org/details/atextbookphysic01thomgoog
  }}</ref><ref name="Lenzen1964">{{cite conference
  | first = Lenzen
  | last = Victor F.
  |author2=Robert P. Multauf
  | title = Paper 44: Development of gravity pendulums in the 19th century
  | book-title = United States National Museum Bulletin 240: Contributions from the Museum of History and Technology reprinted in Bulletin of the Smithsonian Institution
  | pages = 307
  | publisher = [[Smithsonian Institution Press]]
  | year = 1964
  | location = Washington
  | url = https://books.google.com/books?id=A1IqAAAAMAAJ&pg=RA2-PA307
  | access-date = 2009-01-28}}</ref>  This indicated the [[Gravitational acceleration|acceleration of gravity]] was less at Cayenne than at Paris.  Pendulum gravimeters began to be taken on voyages to remote parts of the world, and it was slowly discovered that gravity increases smoothly with increasing latitude, gravitational acceleration being about 0.5% greater at the poles than at the equator.

British physicist [[Isaac Newton]] explained this in his ''[[Principia Mathematica Philosophiae Naturalis|Principia Mathematica]]'' (1687) in which he outlined his theory and calculations on the shape of the Earth.<ref>[https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookIII-Prop2 Propositions X-XXIV (Motions of celestial bodies and the sea)], Propositions XIX and XX. [https://la.wikisource.org/wiki/Philosophiae_Naturalis_Principia_Mathematica/Liber_III Original Latin].</ref> Newton theorized correctly that the Earth was not precisely a sphere but had an [[Oblate spheroidal coordinates|oblate]] [[ellipsoid]]al shape, slightly flattened at the poles due to the [[centrifugal force]] of its rotation. Using geometric calculations, he gave a concrete argument as to the hypothetical ellipsoid shape of the Earth.<ref>{{Cite book|title = Principia, Book III, Proposition XIX, Problem III|last = Newton|first = Isaac}}</ref>

The goal of ''[[Principia Mathematica Philosophiae Naturalis|Principia]]'' was not to provide exact answers for natural phenomena, but to theorize potential solutions to these unresolved factors in science.  Newton pushed for scientists to look further into the unexplained variables. Two prominent researchers that he inspired were [[Alexis Clairaut]] and [[Pierre Louis Maupertuis]]. They both sought to prove the validity of Newton's theory on the shape of the Earth. In order to do so, they went on an expedition to [[Lapland (Finland)|Lapland]] in an attempt to accurately measure a [[meridian arc]].  From such measurements they could calculate the [[eccentricity (mathematics)|eccentricity]] of the Earth, its degree of departure from a perfect sphere.

Clairaut confirmed that Newton's theory that the Earth was ellipsoidal was correct, but that his calculations were in error, and he wrote a letter to the [[Royal Society|Royal Society of London]] with his findings.<ref>{{Cite book|title = The Problem of the Earth's Shape from Newton to Clairaut|last = Greenburg|first = John|publisher = [[Cambridge University Press]]|year = 1995|isbn = 0-521-38541-5|location = New York|pages = [https://archive.org/details/problemofearthss1995gree/page/132 132]|url = https://archive.org/details/problemofearthss1995gree/page/132}}</ref> The society published an article in [[Philosophical Transactions of the Royal Society|Philosophical Transactions]] the following year, 1737.<ref>{{Cite journal|last1 = Clairaut|first1 = Alexis|last2 = Colson|first2 = John|date = 1737|title = An Inquiry concerning the Figure of Such Planets as Revolve about an Axis, Supposing the Density Continually to Vary, from the Centre towards the Surface|jstor = 103921|journal = Philosophical Transactions}}</ref> In it Clairaut pointed out (Section XVIII) that Newton's Proposition XX of Book 3 does not apply to the real earth. It stated that the weight of an object at some point in the earth depended only on the proportion of its distance from the centre of the earth to the distance from the centre to the surface at or above the object, so that the total weight of a column of water at the centre of the earth would be the same no matter in which direction the column went up to the surface. Newton had in fact said that this was on the assumption that the matter inside the earth was of a uniform density (in Proposition XIX). Newton realized that the density was probably not uniform, and proposed this as an explanation for why gravity measurements found a greater difference between polar regions and equatorial regions than what his theory predicted. However, he also thought this would mean the equator was further from the centre than what his theory predicted, and Clairaut points out that the opposite is true. Clairaut points out at the beginning of his article that Newton did not explain why he thought the earth was ellipsoid rather than like some other oval, but that Clairaut, and [[James Stirling (mathematician)|James Stirling]] almost simultaneously, had shown why the earth should be an ellipsoid in 1736.

Clairaut's article did not provide a valid equation to back up his argument as well. This created much controversy in the scientific community. It was not until Clairaut wrote ''Théorie de la figure de la terre'' in 1743 that a proper answer was provided. In it, he promulgated what is more formally known today as Clairaut's theorem.