page_title: Pendulum
section_title: Early observations
hierarchical_section_title: Pendulum / Gravity measurement / Early observations
caption_attribution_description: English: Drawing of pendulum experiment to determine the length of the seconds pendulum at Paris, conducted in 1792 by Jean-Charles de Borda and Jean-Dominique Cassini. From their original paper. They used a pendulum that consisted of a 1 1/2 inch (3.8 cm) platinum ball suspended by a 12 foot (3.97 m) iron wire (F,Q). It was suspended in front of the pendulum (B) of a precision clock (A). Alterations to image: moved observer (O) closer to wall to reduce unwieldy size of image, converted to PNG. Image shows aliasing artifacts (crosshatching) from scanning of original halftone image.
context_page_description: A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum's swing.
  From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s. The pendulum clock invented by Christian Huygens in 1658 became the world's standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s.
context_section_description: 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.
  1644: Even before the pendulum clock, French priest Marin Mersenne first determined the length of the seconds pendulum was 39.1 inches (990 mm), by comparing the swing of a pendulum to the time it took a weight to fall a measured distance.
  1669: Jean Picard determined the length of the seconds pendulum at Paris, using a 1-inch (25 mm) copper ball suspended by an aloe fiber, obtaining 39.09 inches (993 mm).
  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 ​2 ¹⁄₂ minutes per day; its seconds pendulum had to be shortened by ​1 ¹⁄₄ lignes (2.6 mm) shorter than at Paris, to keep correct time.  In 1687 Isaac Newton in Principia Mathematica showed this was because the Earth had a slightly  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.  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, 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).
  1737: French mathematician Pierre Bouguer made a sophisticated series of pendulum observations in the Andes mountains, Peru.  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 density of the Earth.
  1747: Daniel Bernoulli showed how to correct for the lengthening of the period due to a finite angle of swing  θ₀ by using the first order correction  θ₀²/16, giving the period of a pendulum with an extremely small swing.
  1792: To define a pendulum standard of length for use with the new metric system, in 1792 Jean-Charles de Borda and Jean-Dominique Cassini made a precise measurement of the seconds pendulum at Paris. They used a ​1 ¹⁄₂-inch (14 mm) platinum ball suspended by a 12-foot (3.7 m) 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 Δt between the recurring instants when the two pendulums swung in synchronism was timed. From this the difference between the periods of the pendulums, T₁ and T₂, could be calculated:
  1821: Francesco Carlini made pendulum observations on top of Mount Cenis, Italy, from which, using methods similar to Bouguer's, he calculated the den
caption: Borda & Cassini's 1792 measurement of the length of the seconds pendulum
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