Editing Pendulum (section)

=== Early observations ===
* '''1620''': British scientist [[Francis Bacon]] was one of the first to propose using a pendulum to measure gravity, suggesting taking one up a mountain to see if gravity varies with altitude.<ref>{{cite web
  | last = Baker
  | first = Lyman A.
  | title = Chancellor Bacon
  | website = English 233 – Introduction to Western Humanities
  | publisher = English Dept., Kansas State Univ.
  | date = Spring 2000
  | url = http://www-personal.ksu.edu/~lyman/english233/Voltaire-Bacon.htm
  | access-date = 2009-02-20}}</ref>
* '''1644''': Even before the pendulum clock, French priest [[Marin Mersenne]] first determined the length of the seconds pendulum was {{convert|39.1|in|mm}}, by comparing the swing of a pendulum to the time it took a weight to fall a measured distance.  He also was first to discover the dependence of the period on amplitude of swing. 
* '''1669''': [[Jean Picard]] determined the length of the seconds pendulum at Paris, using a {{convert|1|in|adj=on}} copper ball suspended by an aloe fiber, obtaining {{convert|39.09|in}}.<ref name="Poynting & Thompson 1907, p.9">[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA9 Poynting & Thompson 1907, p.9]</ref>  He also did the first experiments on thermal expansion and contraction of pendulum rods with temperature.
* '''1672''': The first observation that gravity varied at different points on Earth was made in 1672 by [[Jean Richer]], who took a [[pendulum clock]] to [[Cayenne]], [[French Guiana]] and found that it lost {{frac|2|1|2}} minutes per day; its seconds pendulum had to be shortened by {{frac|1|1|4}} ''[[ligne]]s'' (2.6&nbsp;mm) shorter than at Paris, to keep correct time.<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/bub_gb_TL4KAAAAIAAJ/page/n30 20]
  | url = https://archive.org/details/bub_gb_TL4KAAAAIAAJ
  }}</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>  In 1687 [[Isaac Newton]] in ''[[Principia Mathematica (Newton)|Principia Mathematica]]'' showed this was because the Earth had a slightly  [[Oblate spheroid|oblate]] shape (flattened at the poles) caused by the [[centrifugal force]] of its rotation.  At higher latitudes the surface was closer to the center of the Earth, so gravity increased with latitude.<ref name="Lenzen1964" />  From this time on, pendulums began to be taken to distant lands to measure gravity, and tables were compiled of the length of the seconds pendulum at different locations on Earth. In 1743 [[Alexis Claude Clairaut]] created the first hydrostatic model of the Earth, [[Clairaut's theorem]],<ref name="Poynting & Thompson 1907, p.9" /> which allowed the [[ellipticity]] of the Earth to be calculated from gravity measurements. Progressively more accurate models of the shape of the Earth followed.
* '''1687''': Newton experimented with pendulums (described in ''Principia'') and found that equal length pendulums with bobs made of different materials had the same period, proving that the gravitational force on different substances was exactly proportional to their [[mass]] (inertia).  This principle, called the [[equivalence principle]], confirmed to greater accuracy in later experiments, became the foundation on which [[Albert Einstein]] based his [[general theory of relativity]]. 
[[File:Borda and Cassini pendulum experiment.png|thumb|190px|Borda & Cassini's 1792 measurement of the length of the seconds pendulum]]
* '''1737''': French mathematician [[Pierre Bouguer]] made a sophisticated series of pendulum observations in the [[Andes]] mountains, Peru.<ref name="Poynting & Thompson, 1907, p.10">[https://books.google.com/books?id=TL4KAAAAIAAJ&pg=PA10 Poynting & Thompson, 1907, p.10]</ref>  He used a copper pendulum bob in the shape of a double pointed cone suspended by a thread; the bob could be reversed to eliminate the effects of nonuniform density. He calculated the length to the center of oscillation of thread and bob combined, instead of using the center of the bob. He corrected for thermal expansion of the measuring rod and barometric pressure, giving his results for a pendulum swinging in vacuum. Bouguer swung the same pendulum at three different elevations, from sea level to the top of the high Peruvian ''[[altiplano]]''. Gravity should fall with the inverse square of the distance from the center of the Earth. Bouguer found that it fell off slower, and correctly attributed the 'extra' gravity to the gravitational field of the huge Peruvian plateau. From the density of rock samples he calculated an estimate of the effect of the ''altiplano'' on the pendulum, and comparing this with the gravity of the Earth was able to make the first rough estimate of the [[Mass of the Earth|density of the Earth]].
* '''1747''': [[Daniel Bernoulli]] showed how to correct for the lengthening of the period due to a finite angle of swing ''θ''<sub>0</sub> by using the first order correction ''θ''<sub>0</sub><sup>2</sup>/16, giving the period of a pendulum with an extremely small swing.<ref name="Poynting & Thompson, 1907, p.10" />
* '''1792''': To define a pendulum standard of length for use with the new [[metric system]], in 1792 [[Jean-Charles de Borda]] and [[Dominique, comte de Cassini|Jean-Dominique Cassini]] made a precise measurement of the seconds pendulum at Paris. They used a {{frac|1|1|2}}-inch (14&nbsp;mm){{clarify|reason=1.5" is not 14mm|date=September 2019}} platinum ball suspended by a {{convert|12|ft|adj=on}} iron wire. Their main innovation was a technique called the "''method of coincidences''" which allowed the period of pendulums to be compared with great precision.  (Bouguer had also used this method). The time interval Δ''t'' between the recurring instants when the two pendulums swung in synchronism was timed. From this the difference between the periods of the pendulums, ''T''<sub>1</sub> and ''T''<sub>2</sub>, could be calculated: <math display="block">\frac {1}{\Delta t} = \frac {1}{T_1} - \frac {1}{T_2}</math>
* '''1821''': [[Francesco Carlini]] made pendulum observations on top of Mount Cenis, Italy, from which, using methods similar to Bouguer's, he calculated the density of the Earth.<ref>{{cite book
  | last = Poynting
  | first = John Henry
  | title = The Mean Density of the Earth
  | publisher = Charles Griffin
  | year = 1894
  | location = London
  | pages = [https://archive.org/details/meandensityeart00poyngoog/page/n44 22]–24
  | url = https://archive.org/details/meandensityeart00poyngoog
  }}</ref>  He compared his measurements to an estimate of the gravity at his location assuming the mountain wasn't there, calculated from previous nearby pendulum measurements at sea level. His measurements showed 'excess' gravity, which he allocated to the effect of the mountain. Modeling the mountain as a segment of a sphere {{convert|11|mi}} in diameter and {{convert|1|mi}} high, from rock samples he calculated its gravitational field, and estimated the density of the Earth at 4.39 times that of water. Later recalculations by others gave values of 4.77 and 4.95, illustrating the uncertainties in these geographical methods.