In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector p = mv; this definition can be applied to each point in physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and the orbital angular momentum; the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin; the total angular momentum of an object is the sum of orbital angular momenta.
The orbital angular momentum vector of a particle is always parallel and directly proportional to the orbital angular velocity vector ω of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. However, the spin angular momentum of the object is proportional but not always parallel to the spin angular velocity Ω, making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is additive. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density over the entire body. Torque can be defined as the rate of change of angular momentum, analogous to force; the net external torque on any system is always equal to the total torque on the system. Therefore, for a closed system, the total torque on the system must be 0, which means that the total angular momentum of the system is constant; the conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, the precession of gyroscopes.
In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is. In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning that at any time, only one component can be measured with definite precision; because of this, it turns out that the notion of an elementary particle "spinning" about an axis does not exist. For technical reasons, elementary particles still possess a spin angular momentum, but this angular momentum does not correspond to spinning motion in the ordinary sense. Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum.
Thus, where linear momentum p is proportional to mass m and linear speed v, p = m v, angular momentum L is proportional to moment of inertia I and angular speed ω, L = I ω. Unlike mass, which depends only on amount of matter, moment of inertia is dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which does not depend upon the choice of origin, angular velocity is always measured with respect to a fixed origin; therefore speaking, L should be referred to as the angular momentum relative to that center. Because I = r 2 m for a single particle and ω = v r for circular motion, angular momentum can be expanded, L = r 2 m ⋅ v r, reduced to, L = r m v, the product of the radius of rotation r and the linear momentum of the particle p = m v, where v in this case is the equivalent linear speed at the radius; this simple analysis can apply to non-circular motion if only the component of the motion, perpendicular to the radius vector is considered. In that case, L
A gravimeter is an instrument used to measure gravitational acceleration. Every mass has an associated gravitational potential; the gradient of this potential is an acceleration. A gravimeter measures this gravitational acceleration; the first gravimeters were vertical accelerometers, specialized for measuring the constant downward acceleration of gravity on the earth's surface. The earth's vertical gravity varies from place to place over the surface of the Earth by about +/- 0.5%. It varies by about +/- 1000 nm/s^2 at any location because of the changing positions of the sun and moon relative to the earth; the change from calling a device an "accelerometer" to calling it a "gravimeter" occurs at the point where it has to make corrections for earth tides. Though similar in design to other accelerometers, gravimeters are designed to be much more sensitive, their first uses were to measure the changes in gravity from the varying densities and distribution of masses inside the earth, from temporal "tidal" variations in the shape and distribution of mass in the oceans and earth.
Gravimeters can detect vibrations and gravity changes from human activities. Depending on the interests of the researcher or operator, this might be counteracted by integral vibration isolation and signal processing; the resolution of the gravimeters can be increased by averaging samples over longer periods. Fundamental characteristic of gravimeters are the accuracy of a single measurement, the sampling rate. Resolution = SingleMeasurementResolution NumberOfSamples for example: Resolution per minute = Resolution per second 60 Gravimeters display their measurements in units of gals, nanometers per second squared, parts per million, parts per billion, or parts per trillion of the average vertical acceleration with respect to the earth; some newer units are pm/s2, fm/s2, am/s2 for sensitive instruments. Gravimeters are used for petroleum and mineral prospecting, geodesy, geophysical surveys and other geophysical research, for metrology, their fundamental purpose is to map the gravity field in time.
Most current work is earth-based, with a few satellites around earth, but gravimeters are applicable to the moon, planets, stars and other bodies. Gravitational wave experiments monitor the changes with time in the gravitational potential itself, rather than the gradient of the potential which the gravimeter is tracking; this distinction is somewhat arbitrary. The subsystems of the gravitational radiation experiments are sensitive to changes in the gradient of the potential; the local gravity signals on earth that interfere with gravitational wave experiments are disparagingly referred to as "Newtonian noise", since Newtonian gravity calculations are sufficient to characterize many of the local signals. The term "absolute gravimeter" has most been used to label gravimeters which report the local vertical acceleration due to the earth. "Relative gravimeter" refer to differential comparisons of gravity from one place to another. They are designed to subtract the average vertical gravity automatically.
They can be calibrated at a location where the gravity is known and transported to the location where the gravity is to be measured. Or they can calibrated in absolute units at their operating location. There are many methods for displaying acceleration fields called "gravity fields"; this includes traditional 2D maps, but 3D video. Since gravity and acceleration are the same, "acceleration field" might be preferable, since "gravity" is an oft misused prefix. Gravimeters for measuring the earth's gravity as as possible, are getting smaller and more portable. A common type measures the acceleration of small masses free falling in a vacuum, when the accelerometer is attached to the ground; the mass terminates one arm of a Michelson interferometer. By counting and timing the interference fringes, the acceleration of the mass can be measured. A more recent development is a "rise and fall" version that tosses the mass upward and measures both upward and downward motion; this allows cancellation of some measurement errors, however "rise and fall" gravimeters are not yet in common use.
Absolute gravimeters are used in the calibration of relative gravimeters, surveying for gravity anomalies, for establishing the vertical control network. Atom interferometric and atomic fountain methods are used for precise measurement of the earth's gravity, atomic clocks and purpose-built instruments can use time dilation measurements to track changes in the gravitational potential and gravitational acceleration on the earth; the term "absolute" does not convey the instrument's stability, accuracy, ease of use, bandwidth. So it and "relative" should not be used; the most common gravimeters are spring-based. They are used in gravity surveys over large areas for establishing the figure of the geoid over those areas, they are a weight on a spring, by measuring the amount by which the weight stretches the spring, local gravity can be measured. However, the strength of the spring must be calibrated by placing the instrument in a locati
Charles Marie de La Condamine
Charles Marie de La Condamine was a French explorer and mathematician. He spent ten years in present-day Ecuador measuring the length of a degree latitude at the equator and preparing the first map of the Amazon region based on astronomical observations. Furthermore he was a contributor to the Encyclopédie ou Dictionnaire raisonné des sciences, des arts et des métiers. Charles Marie de La Condamine was born in Paris as a son of well-to-do parents, Charles de La Condamine and Louise Marguerite Chourses, he studied at the Collège Louis-le-Grand where he was trained in humanities as well as in mathematics. After finishing his studies, he fought in the war against Spain. After returning from the war, he became acquainted with scientific circles in Paris. On 12 December 1730 he became a member of the Académie des Sciences and was appointed Assistant Chemist at the Academy; the next year he sailed with the Levant Company to Constantinople. After returning to Paris, La Condamine submitted in November 1732 a paper to the Academy entitled Mathematical and Physical Observations made during a Visit of the Levant in 1731 and 1732.
Three years he joined the French Geodesic Mission to present-day Ecuador which had the aim of testing a hypothesis of Isaac Newton. Newton had posited that the Earth is not a perfect sphere, but bulges around the equator and is flattened at the poles. Newton's opinion had raised a huge controversy among French scientists. Pierre Louis Maupertuis, Alexis Claude Clairaut, Pierre Charles Le Monnier traveled to Lapland, where they were to measure the length of several degrees of latitude orthogonal to the arctic circle, while Louis Godin, Pierre Bouguer, La Condamine were sent to South America to perform similar measurements around the equator. On 16 May 1735, La Condamine sailed from La Rochelle accompanied by Godin, a botanist, Joseph de Jussieu. After stopovers in Martinique, Saint-Domingue, Cartagena, they came to Panama where they crossed the continent; the expedition arrived at the Pacific port of Manta. La Condamine's associations with his colleagues were unhappy; the expedition was beset by many difficulties, La Condamine split from the rest and made his way to Quito, Ecuador separately following the Esmeraldas River, becoming the first European to encounter rubber in the process.
He joined the group again on 4 June 1736 in the city of Quito. La Condamine is credited with introducing samples of rubber to the Académie Royale des Sciences of France in 1736. In 1751, he presented a paper by François Fresneau to the Académie which described many of the properties of rubber; this has been referred to as the first scientific paper on rubber. The meridian arc which La Condamine and his colleagues chose to measure the length of passed through a high valley perpendicular to the equator, stretching from Quito in the north to Cuenca in the south; the scientists spent a month performing triangulation measurements in the Yaruqui plains — from 3 October to 3 November 1736 — and returned to Quito. After they had come back to Quito, they found. La Condamine, who had taken precautions and had made a deposit on a bank in Lima, traveled in early 1737 to Lima to collect money, he prolonged this journey somewhat to study the cinchona tree with its medicinally active bark, the tree being hardly known in Europe.
After returning to Quito on 20 June 1737, he found that Godin refused to disclose his results, whereupon La Condamine joined forces with Bouguer. The two men continued with their length measurements in the mountainous and inaccessible region close to Quito; when in December 1741 Bouguer detected an error in a calculation of La Condamine's, the two explorers got into a quarrel and stopped speaking to each other. However, working separately, the two completed their project in May 1743. Insufficient funds prevented La Condamine from returning to France directly, thus La Condamine chose to return by way of the Amazon River, a route, longer and more dangerous. His was the first scientific exploration of the Amazon, he reached the Atlantic Ocean on 19 September 1743, having made observations of astronomic and topographic interest on the way. He made some botanical studies, notably of cinchona and rubber trees. In February 1744 he arrived in the capital of French Guiana, he did not dare to travel back to France on a French merchant ship because France was at war, he had to wait for five months for a Dutch ship, but made good use of his waiting time by observing and recording physical and ethnological phenomena.
Leaving Cayenne in August 1744, he arrived in Amsterdam on 30 November 1744, where he stayed for a while, arrived in Paris in February 1745. He brought with him many notes, natural history specimens, art objects that he donated to the naturalist Buffon. La Condamine published the results of his measurements and travels with a map of the Amazon in Mém. de l'Académie des Sciences, 1745. This included the first descriptions by a European of the Casiquiare canal and the curare arrow poison prepared by the Amerindians, he noted the correct use of quinine to fight malaria. The journal of his ten-year-long voyage to South America was published in Paris in 1751; the scientific results of the expedition were unambiguous: the Earth is indeed a spheroid flattened at the poles as was believed by Newton. Not La Condamine and Bouguer failed to
Parallax is a displacement or difference in the apparent position of an object viewed along two different lines of sight, is measured by the angle or semi-angle of inclination between those two lines. Due to foreshortening, nearby objects show a larger parallax than farther objects when observed from different positions, so parallax can be used to determine distances. To measure large distances, such as the distance of a planet or a star from Earth, astronomers use the principle of parallax. Here, the term parallax is the semi-angle of inclination between two sight-lines to the star, as observed when Earth is on opposite sides of the Sun in its orbit; these distances form the lowest rung of what is called "the cosmic distance ladder", the first in a succession of methods by which astronomers determine the distances to celestial objects, serving as a basis for other distance measurements in astronomy forming the higher rungs of the ladder. Parallax affects optical instruments such as rifle scopes, binoculars and twin-lens reflex cameras that view objects from different angles.
Many animals, including humans, have two eyes with overlapping visual fields that use parallax to gain depth perception. In computer vision the effect is used for computer stereo vision, there is a device called a parallax rangefinder that uses it to find range, in some variations altitude to a target. A simple everyday example of parallax can be seen in the dashboard of motor vehicles that use a needle-style speedometer gauge; when viewed from directly in front, the speed may show 60. As the eyes of humans and other animals are in different positions on the head, they present different views simultaneously; this is the basis of stereopsis, the process by which the brain exploits the parallax due to the different views from the eye to gain depth perception and estimate distances to objects. Animals use motion parallax, in which the animals move to gain different viewpoints. For example, pigeons down to see depth; the motion parallax is exploited in wiggle stereoscopy, computer graphics which provide depth cues through viewpoint-shifting animation rather than through binocular vision.
Parallax arises due to change in viewpoint occurring due to motion of the observer, of the observed, or of both. What is essential is relative motion. By observing parallax, measuring angles, using geometry, one can determine distance. Astronomers use the word "parallax" as a synonym for "distance measurement" by other methods: see parallax #Astronomy. Stellar parallax created by the relative motion between the Earth and a star can be seen, in the Copernican model, as arising from the orbit of the Earth around the Sun: the star only appears to move relative to more distant objects in the sky. In a geostatic model, the movement of the star would have to be taken as real with the star oscillating across the sky with respect to the background stars. Stellar parallax is most measured using annual parallax, defined as the difference in position of a star as seen from the Earth and Sun, i. e. the angle subtended at a star by the mean radius of the Earth's orbit around the Sun. The parsec is defined as the distance.
Annual parallax is measured by observing the position of a star at different times of the year as the Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars; the first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from the Earth to the Sun, now based on radar reflection off the surfaces of planets; the angles involved in these calculations are small and thus difficult to measure. The nearest star to the Sun, Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec. This angle is that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. The fact that stellar parallax was so small that it was unobservable at the time was used as the main scientific argument against heliocentrism during the early modern age.
It is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed implausible: it was one of Tycho's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere. In 1989, the satellite Hipparcos was launched for obtaining improved parallaxes and proper motions for over 100,000 nearby stars, increasing the reach of the method tenfold. So, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy; the European Space Agency's Gaia mission, launched in December 2013, will be able to measure parallax angles to an accuracy of 10 microarcseconds, thus mapping nearby stars up to a distance of tens of thousands of ligh
A seconds pendulum is a pendulum whose period is two seconds. A pendulum is a weight suspended from a pivot; 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 combined with 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, to a slight degree on its weight distribution and the amplitude of the pendulum's swing. Time in physics is defined by its measurement: time is what a clock reads. In classical, non-relativistic physics it is a scalar quantity and, like length and charge, is described as a fundamental quantity. Time can be combined mathematically with other physical quantities to derive other concepts such as motion, kinetic energy and time-dependent fields.
Timekeeping is a complex of technological and scientific issues, part of the foundation of recordkeeping. The pendulum clock was invented in 1656 by Dutch scientist and inventor Christiaan Huygens, patented the following year. Huygens contracted the construction of his clock designs to clockmaker Salomon Coster, who built the clock. Huygens was inspired by investigations of pendulums by Galileo Galilei beginning around 1602. Galileo discovered the key property that makes pendulums useful timekeepers: isochronism, which means that the period of swing of a pendulum is the same for different sized swings. Galileo had the idea for a pendulum clock in 1637, constructed by his son in 1649, but neither lived to finish it; the introduction of the pendulum, the first harmonic oscillator used in timekeeping, increased the accuracy of clocks enormously, from about 15 minutes per day to 15 seconds per day leading to their rapid spread as existing'verge and foliot' clocks were retrofitted with pendulums.
These early clocks, due to their verge escapements, had wide pendulum swings of 80–100°. In his 1673 analysis of pendulums, Horologium Oscillatorium, Huygens showed that wide swings made the pendulum inaccurate, causing its period, thus the rate of the clock, to vary with unavoidable variations in the driving force provided by the movement. Clockmakers' realisation that only pendulums with small swings of a few degrees are isochronous motivated the invention of the anchor escapement around 1670, which reduced the pendulum's swing to 4–6°; the anchor became the standard escapement used in pendulum clocks. In addition to increased accuracy, the anchor's narrow pendulum swing allowed the clock's case to accommodate longer, slower pendulums, which needed less power and caused less wear on the movement; the seconds pendulum, 0.994 m long, in which each swing takes one second, became used in quality clocks. The long narrow clocks built around these pendulums, first made by William Clement around 1680, became known as grandfather clocks.
The increased accuracy resulting from these developments caused the minute hand rare, to be added to clock faces beginning around 1690. The 18th- and 19th-century wave of horological innovation that followed the invention of the pendulum brought many improvements to pendulum clocks; the deadbeat escapement invented in 1675 by Richard Towneley and popularised by George Graham around 1715 in his precision "regulator" clocks replaced the anchor escapement and is now used in most modern pendulum clocks. The observation that pendulum clocks slowed down in summer brought the realisation that thermal expansion and contraction of the pendulum rod with changes in temperature was a source of error; this was solved by the invention of temperature-compensated pendulums. With these improvements, by the mid-18th century precision pendulum clocks achieved accuracies of a few seconds per week. At the time the second was defined as a fraction of the Earth's rotation time or mean solar day and determined by clocks whose precision was checked by astronomical observations.
Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day. Two types of solar time are mean solar time. Mean solar time is the hour angle of the mean Sun plus 12 hours; this 12 hour offset comes from the decision to make each day start at midnight for civil purposes whereas the hour angle or the mean sun is measured from the zenith. The duration of daylight varies during the year but the length of a mean solar day is nearly constant, unlike that of an apparent solar day. An apparent solar day can be 30 seconds longer than a mean solar day. Long or short days occur in succession, so the difference builds up until mean time is ahead of apparent time by about 14 minutes near February 6 and behind apparent time by about 16 minutes near November 3; the equation of time is this difference, cyclical and does not accumulate from year to year. Mean time follows the mean sun. Jean Meeus describes the mean sun as follows: Consider a first fictitious Sun travelling along the ecliptic with a constant speed and coinciding with the true sun at the perigee and apogee.
Consider a second fictitious Sun travelling along the celestial equator
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 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, 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, achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s.
Pendulums are used in scientific instruments such as accelerometers and seismometers. They were used as gravimeters to measure the acceleration of gravity in geophysical surveys, as a standard of length; the word "pendulum" is new Latin, from the Latin pendulus, meaning'hanging'. The simple gravity pendulum is an idealized mathematical model of a pendulum; this is a weight on the end of a massless cord suspended without friction. When given an initial push, it will swing forth at a constant amplitude. Real pendulums are subject to friction and air drag, so the amplitude of their swings declines; the period of swing of a simple gravity pendulum depends on its length, the local strength of gravity, to a small extent on the maximum angle that the pendulum swings away from vertical, θ0, called the amplitude. It is independent of the mass of the bob. If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is: T ≈ 2 π L g θ 0 ≪ 1 r a d i a n where L is the length of the pendulum and g is the local acceleration of gravity.
For small swings the period of swing is the same for different size swings: that is, the period is independent of amplitude. This property, called isochronism, is the reason. Successive swings of the pendulum if changing in amplitude, take the same amount of time. For larger amplitudes, the period increases with amplitude so it is longer than given by equation. For example, at an amplitude of θ0 = 23 ° it is 1 % larger; the period increases asymptotically as θ0 approaches 180°, because the value θ0 = 180° is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms, one example being the infinite series: T = 2 π L g where θ 0 is in radians; the difference between this true period and the period for small swings above is called the circular error. In the case of a typical grandfather clock whose pendulum has a swing of 6° and thus an amplitude of 3°, the difference between the true period and the small angle approximation amounts to about 15 seconds per day.
For small swings the pendulum approximates a harmonic oscillator, its motion as a function of time, t, is simple harmonic motion: θ = θ 0 cos where φ is a constant value, dependent on initial conditions. For real pendulums, the period varies with factors such as the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string. In precision applications, corrections for these factors may need to be applied to eq. to give the period accurately. Any swinging rigid body free to rotate about a fixed horizontal axis is called a compound pendulum or physical pendulum; the appropriate equivalent length L for calculating the period of any such pendulum is the distance from the pivot to the center of oscillation. This point is located under the center of mass at a distance from the pivot traditionally called the radius of oscillation, which depends on the mass distribution of the pendulum.
If most of the mass is concentrated in a small bob compared to the pendulum length, the center of oscillation is close to the center of mass. The radiu
Jean-Charles de Borda
Jean-Charles, chevalier de Borda was a French mathematician and sailor. Borda was born in the city of Dax to Jeanne‐Marie Thérèse de Lacroix. In 1756, Borda wrote Mémoire sur le mouvement des projectiles, a product of his work as a military engineer. For that, he was elected to the French Academy of Sciences in 1764. Borda was a mariner and a scientist, spending time in the Caribbean testing out advances in chronometers. Between 1777 and 1778, he participated in the American Revolutionary War. In 1781, he was put in charge of several vessels in the French Navy. In 1782, he was captured by the English, was returned to France shortly after, he returned as an engineer in the French Navy, making improvements to pumps. He was appointed as France's Inspector of Naval Shipbuilding in 1784, with the assistance of the naval architect Jacques-Noël Sané in 1786 introduced a massive construction programme to revitalise the French navy based on the standard designs of Sané. In 1770, Borda formulated a ranked preferential voting system, referred to as the Borda count.
The French Academy of Sciences used Borda's method to elect its members for about two decades until it was quashed by Napoleon Bonaparte who insisted that his own method be used after he became president of the Académie in 1801. The Borda count is in use today in some academic institutions and several political jurisdictions; the Borda count has served as a basis for other methods such as the Quota Borda system and Nanson's method. In 1778, he published his method of reducing Lunar Distances for computing the longitude, still regarded as the best of several similar mathematical procedures for navigation and position-fixing in pre-chronometer days. Another of his contributions is his construction of the standard metre, basis of the metric system to correspond to the measurements of Delambre; as an instrument maker, he improved the reflecting circle and the repeating circle, the latter used to measure the meridian arc from Dunkirk to Barcelona by Delambre and Méchain. With the advent of the metric system after the French Revolution it was decided that the quarter circle should be divided into 100 degrees instead of 90 degrees, the degree into 100 seconds instead of 60 seconds.
This required the calculation of trigonometric tables and logarithms corresponding to the new size of the degree and instruments for measuring angles in the new system. Borda constructed instruments for measuring angles in the new units, used in the measurement of the meridian between Dunkirk and Barcelona by Delambre to determine the length of the metre; the tables of logarithms of sines and tangents were required for the purposes of navigation. Borda was an enthusiast for the metric system and constructed tables of these logarithms starting in 1792 but their publication was delayed until after his death and only published in the Year 9 as Tables of Logarithms of sines and tangents, co-secants, co-sines, co-tangents for the Quarter of the Circle divided into 100 degrees, the degree into 100 minutes, the minute into 100 seconds to ten decimals, including his tables of logarithms to 7 decimals from 10,000 to 100,000 with tables for obtaining results to 10 decimals; the division of the degree into hundredths was accompanied by the division of the day into 10 hours of 100 minutes and maps were required to show the new degrees of latitude and longitude.
The Republican Calendar was abolished by Napoleon in 1806, but the 400-degree circle lived on as the Gradian. Five French ships were named Borda in his honour; the crater Borda on the Moon is named after him. Asteroide 175726 has been called Borda in his honour, his name is one of the 72 names inscribed on the Eiffel Tower. Cape Borda on the northwest coast of Kangaroo Island in South Australia is named in his honour. Île Borda was the name given to Kangaroo Island in his honour by Nicholas Baudin. Borda count electoral method Borda–Carnot equation O'Connor, John J.. "Jean Charles de Borda", MacTutor History of Mathematics archive, University of St Andrews. Mascart, Jean, La vie et les travaux du chevalier Jean-Charles de Borda: Épisodes de la vie scientifique au XVIIIe siècle, Bibliothèque de la revue d'histoire maritime, Presses de l'Université de Paris-Sorbonne, ISBN 2-84050-173-2