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According to Newton's law of universal gravitation, a point mass, m1, exerts a gravitational force, , on another point mass, m2. In modern language, the law states: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Newton's law has since been superseded by Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity in most applications.

A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1660s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons". In 1686, when the first book of Newton's Principia was presented to the Royal Society, Robert Hooke accused Newton of plagiarism by claiming that he had taken from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions.

Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, he would still have some rights to it in view of his demonstrations of its accuracy. This remark refers among other things to Newton's finding, supported by mathematical demonstration, that if the inverse square law applies to tiny particles, then even a large spherically symmetrical mass also attracts masses external to its surface, even close up, exactly as if all its own mass were concentrated at its center. In regard to evidence that still survives of the earlier history, manuscripts written by Newton in the 1660s show that Newton himself had, by 1669, arrived at proofs that in a circular case of planetary motion, "endeavour to recede" (what was later called centrifugal force) had an inverse-square relation with distance from the center.[20] After his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. On the other hand, Newton did accept and acknowledge, in all editions of the 'Principia', that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Since the time of Newton and Hooke, scholarly discussion has also touched on the question of whether Hooke's 1679 mention of 'compounding the motions' provided Newton with something new and valuable, even though that was not a claim actually voiced by Hooke at the time. Nevertheless, a number of authors have had more to say about what Newton gained from Hooke and some aspects remain controversial.[24] The fact that most of Hooke's private papers had been destroyed or have disappeared does not help to establish the truth. Newton's role in relation to the inverse square law was not as it has sometimes been represented. If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force. The portion of the mass that is located at radii r < r0 causes the same force at r0 as if all of the mass enclosed within a sphere of radius r0 was concentrated at the center of the mass distribution (as noted above).

The portion of the mass that is located at radii r > r0 exerts no net gravitational force at the distance r0 from the center. As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere. Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. The gravitational field is a vector field that describes the gravitational force which would be applied on an object in any given point in space, per unit mass.

It is a generalisation of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent.

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used.

In situations where either dimensionless parameter is large, then general relativity must be used to describe the system.

Newton's theory of gravitation requires that the gravitational force be transmitted instantaneously.

Newton's Theory does not fully explain the precession of the perihelion of the orbits of the planets, especially of planet Mercury, which was detected long after the life of Newton.[32] There is a 43 arcsecond per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th Century.

The predicted angular deflection of light rays by gravity that is calculated by using Newton's Theory is only one-half of the deflection that is actually observed by astronomers. In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey Newton's law of universal gravitation. The observed fact that the gravitational mass and the inertial mass is not the same for all objects is unexplained within Newton's Theories. Substituting the inertial mass dilation expression from Special relativity yields the following expression. While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" that his equations implied. These objections were explained by Einstein's theory of general relativity, in which gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. The n-body problem is an ancient, classical problem[36] of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. The gravitational force is attractive, proportional to the product of the masses, and inversely proportional to the square of the distance, r, between the masses. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies.

Relativity is required only when there is a need for extreme precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the sun). The same author does credit Hooke with a significant and even seminal contribution, but he treats Hooke's claim of priority on the inverse square point as uninteresting since several individuals besides Newton and Hooke had at least suggested it, and he points instead to the idea of "compounding the celestial motions" and the conversion of Newton's thinking away from "centrifugal" and towards "centripetal" force as Hooke's significant contributions.

This is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy. Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, together with a principle of linear inertia.

Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.[7] He also did not provide accompanying evidence or mathematical demonstration. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hooke's 1679 letter.[12] Newton also pointed out and acknowledged prior work of others,[13] including Bullialdus,[14] (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and Borelli[15] (who suggested, also without demonstration, that there was a centrifugal tendency in counterbalance with a gravitational attraction towards the Sun so as to make the planets move in ellipses).

Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center.

Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles.[18] In addition, Newton had formulated in Propositions 43-45 of Book 1,[19] and associated sections of Book 3, a sensitive test of the accuracy of the inverse square law, in which he showed that only where the law of force is accurately as the inverse square of the distance will the directions of orientation of the planets' orbital ellipses stay constant as they are observed to do apart from small effects attributable to inter-planetary perturbations. As described above, Newton's manuscripts of the 1660s do show him actually combining tangential motion with the effects of radially directed force or endeavour, for example in his derivation of the inverse square relation for the circular case. It took place 111 years after the publication of Newton's Principia and 71 years after Newton's death, so none of Newton's calculations could use the value of G; instead he could only calculate a force relative to another force.

In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at r0, cancel each other out. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity. Attempts by physicists to identify the relationship between the gravitational force and other known fundamental forces are not yet resolved, although considerable headway has been made over the last 50 years (See: Theory of everything and Standard Model).

Given the classical assumptions of the nature of space and time before the development of General Relativity, a significant propagation delay in gravity leads to unstable planetary and stellar orbits. Calculations using General Relativity are in much closer agreement with the astronomical observations.

Astrophysicists, however, explain this spectacular phenomenon in the framework of Newton's laws, with the presence of large amounts of Dark matter. The equations for Newtonian gravitation and Newtonian inertia use the same "mass", while in those of General Relativity they diverge. When it is near to the speed of light, the denominator approaches zero and the momentum explodes because effective inertial mass explodes.

In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity (although he invented two mechanical hypotheses in 1675 and 1717). In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. Solving this problem a€” from the time of the Greeks and on a€” has been motivated by the desire to understand the motions of the Sun, planets and the visible stars. Coulomb's law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.

Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way.

They also show Newton clearly expressing the concept of linear inertiaa€”for which he was indebted to Descartes' work, published in 1644 (as Hooke probably was).[23] These matters do not appear to have been learned by Newton from Hooke. Newton himself felt that the concept of an inexplicable action at a distance was unsatisfactory (see "Newton's reservations" below), but that there was nothing more that he could do at the time. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science.

This allowed a description of the motions of light and mass that was consistent with all available observations. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.[37] The n-body problem in general relativity is considerably more difficult to solve. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s.

Examples of this include: the experiments of Galileo Galilei that, decades before Newton, established that objects that have the same air or fluid resistance are accelerated by the force of the Earth's gravity equally, regardless of their different inertial masses. He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature".

In general relativity, the gravitational force is a fictitious force due to the curvature of spacetime, because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime. The lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis.[21] This background shows there was basis for Newton to deny deriving the inverse square law from Hooke. In these cases, the forces and energies that are required to accelerate various masses is completely dependent upon their different inertial masses, as can be seen from Newton's Second Law of Motion, F = ma. These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Newton's law has since been superseded by Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity in most applications.

A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1660s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons". In 1686, when the first book of Newton's Principia was presented to the Royal Society, Robert Hooke accused Newton of plagiarism by claiming that he had taken from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions.

Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea. Newton further defended his work by saying that had he first heard of the inverse square proportion from Hooke, he would still have some rights to it in view of his demonstrations of its accuracy. This remark refers among other things to Newton's finding, supported by mathematical demonstration, that if the inverse square law applies to tiny particles, then even a large spherically symmetrical mass also attracts masses external to its surface, even close up, exactly as if all its own mass were concentrated at its center. In regard to evidence that still survives of the earlier history, manuscripts written by Newton in the 1660s show that Newton himself had, by 1669, arrived at proofs that in a circular case of planetary motion, "endeavour to recede" (what was later called centrifugal force) had an inverse-square relation with distance from the center.[20] After his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. On the other hand, Newton did accept and acknowledge, in all editions of the 'Principia', that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Since the time of Newton and Hooke, scholarly discussion has also touched on the question of whether Hooke's 1679 mention of 'compounding the motions' provided Newton with something new and valuable, even though that was not a claim actually voiced by Hooke at the time. Nevertheless, a number of authors have had more to say about what Newton gained from Hooke and some aspects remain controversial.[24] The fact that most of Hooke's private papers had been destroyed or have disappeared does not help to establish the truth. Newton's role in relation to the inverse square law was not as it has sometimes been represented. If the bodies in question have spatial extent (rather than being theoretical point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. For points inside a spherically-symmetric distribution of matter, Newton's Shell theorem can be used to find the gravitational force. The portion of the mass that is located at radii r < r0 causes the same force at r0 as if all of the mass enclosed within a sphere of radius r0 was concentrated at the center of the mass distribution (as noted above).

The portion of the mass that is located at radii r > r0 exerts no net gravitational force at the distance r0 from the center. As a consequence, for example, within a shell of uniform thickness and density there is no net gravitational acceleration anywhere within the hollow sphere. Newton's law of universal gravitation can be written as a vector equation to account for the direction of the gravitational force as well as its magnitude. It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. The gravitational field is a vector field that describes the gravitational force which would be applied on an object in any given point in space, per unit mass.

It is a generalisation of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent.

Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used.

In situations where either dimensionless parameter is large, then general relativity must be used to describe the system.

Newton's theory of gravitation requires that the gravitational force be transmitted instantaneously.

Newton's Theory does not fully explain the precession of the perihelion of the orbits of the planets, especially of planet Mercury, which was detected long after the life of Newton.[32] There is a 43 arcsecond per century discrepancy between the Newtonian calculation, which arises only from the gravitational attractions from the other planets, and the observed precession, made with advanced telescopes during the 19th Century.

The predicted angular deflection of light rays by gravity that is calculated by using Newton's Theory is only one-half of the deflection that is actually observed by astronomers. In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey Newton's law of universal gravitation. The observed fact that the gravitational mass and the inertial mass is not the same for all objects is unexplained within Newton's Theories. Substituting the inertial mass dilation expression from Special relativity yields the following expression. While Newton was able to formulate his law of gravity in his monumental work, he was deeply uncomfortable with the notion of "action at a distance" that his equations implied. These objections were explained by Einstein's theory of general relativity, in which gravitation is an attribute of curved spacetime instead of being due to a force propagated between bodies. The n-body problem is an ancient, classical problem[36] of predicting the individual motions of a group of celestial objects interacting with each other gravitationally. The gravitational force is attractive, proportional to the product of the masses, and inversely proportional to the square of the distance, r, between the masses. Both are inverse-square laws, where force is inversely proportional to the square of the distance between the bodies.

Relativity is required only when there is a need for extreme precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the sun). The same author does credit Hooke with a significant and even seminal contribution, but he treats Hooke's claim of priority on the inverse square point as uninteresting since several individuals besides Newton and Hooke had at least suggested it, and he points instead to the idea of "compounding the celestial motions" and the conversion of Newton's thinking away from "centrifugal" and towards "centripetal" force as Hooke's significant contributions.

This is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy. Thus Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, together with a principle of linear inertia.

Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.[7] He also did not provide accompanying evidence or mathematical demonstration. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hooke's 1679 letter.[12] Newton also pointed out and acknowledged prior work of others,[13] including Bullialdus,[14] (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and Borelli[15] (who suggested, also without demonstration, that there was a centrifugal tendency in counterbalance with a gravitational attraction towards the Sun so as to make the planets move in ellipses).

Hooke, without evidence in favor of the supposition, could only guess that the inverse square law was approximately valid at great distances from the center.

Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles.[18] In addition, Newton had formulated in Propositions 43-45 of Book 1,[19] and associated sections of Book 3, a sensitive test of the accuracy of the inverse square law, in which he showed that only where the law of force is accurately as the inverse square of the distance will the directions of orientation of the planets' orbital ellipses stay constant as they are observed to do apart from small effects attributable to inter-planetary perturbations. As described above, Newton's manuscripts of the 1660s do show him actually combining tangential motion with the effects of radially directed force or endeavour, for example in his derivation of the inverse square relation for the circular case. It took place 111 years after the publication of Newton's Principia and 71 years after Newton's death, so none of Newton's calculations could use the value of G; instead he could only calculate a force relative to another force.

In the limit, as the component point masses become "infinitely small", this entails integrating the force (in vector form, see below) over the extents of the two bodies. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at r0, cancel each other out. General relativity reduces to Newtonian gravity in the limit of small potential and low velocities, so Newton's law of gravitation is often said to be the low-gravity limit of general relativity. Attempts by physicists to identify the relationship between the gravitational force and other known fundamental forces are not yet resolved, although considerable headway has been made over the last 50 years (See: Theory of everything and Standard Model).

Given the classical assumptions of the nature of space and time before the development of General Relativity, a significant propagation delay in gravity leads to unstable planetary and stellar orbits. Calculations using General Relativity are in much closer agreement with the astronomical observations.

Astrophysicists, however, explain this spectacular phenomenon in the framework of Newton's laws, with the presence of large amounts of Dark matter. The equations for Newtonian gravitation and Newtonian inertia use the same "mass", while in those of General Relativity they diverge. When it is near to the speed of light, the denominator approaches zero and the momentum explodes because effective inertial mass explodes.

In all other cases, he used the phenomenon of motion to explain the origin of various forces acting on bodies, but in the case of gravity, he was unable to experimentally identify the motion that produces the force of gravity (although he invented two mechanical hypotheses in 1675 and 1717). In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. Solving this problem a€” from the time of the Greeks and on a€” has been motivated by the desire to understand the motions of the Sun, planets and the visible stars. Coulomb's law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.

Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way.

They also show Newton clearly expressing the concept of linear inertiaa€”for which he was indebted to Descartes' work, published in 1644 (as Hooke probably was).[23] These matters do not appear to have been learned by Newton from Hooke. Newton himself felt that the concept of an inexplicable action at a distance was unsatisfactory (see "Newton's reservations" below), but that there was nothing more that he could do at the time. Moreover, he refused to even offer a hypothesis as to the cause of this force on grounds that to do so was contrary to sound science.

This allowed a description of the motions of light and mass that was consistent with all available observations. In the 20th century, understanding the dynamics of globular cluster star systems became an important n-body problem too.[37] The n-body problem in general relativity is considerably more difficult to solve. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s.

Examples of this include: the experiments of Galileo Galilei that, decades before Newton, established that objects that have the same air or fluid resistance are accelerated by the force of the Earth's gravity equally, regardless of their different inertial masses. He lamented that "philosophers have hitherto attempted the search of nature in vain" for the source of the gravitational force, as he was convinced "by many reasons" that there were "causes hitherto unknown" that were fundamental to all the "phenomena of nature".

In general relativity, the gravitational force is a fictitious force due to the curvature of spacetime, because the gravitational acceleration of a body in free fall is due to its world line being a geodesic of spacetime. The lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis.[21] This background shows there was basis for Newton to deny deriving the inverse square law from Hooke. In these cases, the forces and energies that are required to accelerate various masses is completely dependent upon their different inertial masses, as can be seen from Newton's Second Law of Motion, F = ma. These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found.

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