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<TITLE>Newton's law of universal gravitation - Wikipedia</TITLE>

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			<H1 id="firstHeading" class="firstHeading" lang="en">Newton's law of universal gravitation</H1>
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<P><B>Newton's law of universal gravitation</B> states that a <A href="/wiki/Particle" title="Particle">particle</A> attracts every other particle in the universe using a <A href="/wiki/Force" title="Force">force</A> that is <A href="/wiki/Proportionality_(mathematics)#Direct_proportionality" title="Proportionality (mathematics)">directly proportional</A> to the product of their masses and <A href="/wiki/Proportionality_(mathematics)#Inverse_proportionality" title="Proportionality (mathematics)">inversely proportional</A> to the square of the distance between their centers.<SUP id="cite_ref-1" class="reference"><A href="#cite_note-1">[note 1]</A></SUP> This is a general <A href="/wiki/Physical_law" title="Physical law">physical law</A> derived from <A href="/wiki/Empirical_observation" class="mw-redirect" title="Empirical observation">empirical observations</A> by what <A href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</A> called <A href="/wiki/Inductive_reasoning" title="Inductive reasoning">inductive reasoning</A>.<SUP id="cite_ref-2" class="reference"><A href="#cite_note-2">[1]</A></SUP> It is a part of <A href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</A> and was formulated in Newton's work <I><A href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</A></I> ("the <I>Principia</I>"), first published on 5 July 1687. When Newton's book was presented in 1686 to the <A href="/wiki/Royal_Society" title="Royal Society">Royal Society</A>, <A href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</A> made a claim that Newton had obtained the inverse square law from him.</P>
<P>In today's language, the law states: Every <A href="/wiki/Point_particle" title="Point particle">point</A> <A href="/wiki/Mass" title="Mass">mass</A> attracts every single other point mass by a <A href="/wiki/Force" title="Force">force</A> pointing along the <A href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</A> intersecting both points. The force is <A href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</A> to the <A href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</A> of the two masses and <A href="/wiki/Proportionality_(mathematics)#Inverse_proportionality" title="Proportionality (mathematics)">inversely proportional</A> to the <A href="/wiki/Square_(algebra)" title="Square (algebra)">square</A> of the distance between them.<SUP id="cite_ref-Newton1_3-0" class="reference"><A href="#cite_note-Newton1-3">[2]</A></SUP> The first test of Newton's theory of gravitation between masses in the laboratory was the <A href="/wiki/Cavendish_experiment" title="Cavendish experiment">Cavendish experiment</A> conducted by the <A href="/wiki/United_Kingdom" title="United Kingdom">British</A> scientist <A href="/wiki/Henry_Cavendish" title="Henry Cavendish">Henry Cavendish</A> in 1798.<SUP id="cite_ref-The_Michell-Cavendish_Experiment_4-0" class="reference"><A href="#cite_note-The_Michell-Cavendish_Experiment-4">[3]</A></SUP> It took place 111 years after the publication of Newton's <I>Principia</I> and approximately 71 years after his death.</P>
<P>Newton's law of gravitation resembles <A href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</A> of electrical forces, which is used to calculate the magnitude of the electrical force arising between two charged bodies. Both are <A href="/wiki/Inverse-square_law" title="Inverse-square law">inverse-square laws</A>, where force is inversely proportional to the square of the distance between the bodies. Coulomb's law has the product of two charges in place of the product of the masses, and the <A href="/wiki/Electrostatic_constant" class="mw-redirect" title="Electrostatic constant">electrostatic constant</A> in place of the <A href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</A>.</P>
<P>Newton's law has since been superseded by <A href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</A>'s theory of <A href="/wiki/General_relativity" title="General relativity">general relativity</A>, but it continues to be used as an excellent approximation of the effects of <A href="/wiki/Gravity" title="Gravity">gravity</A> in most applications. 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 <A href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</A>'s orbit around the Sun).</P>
<P></P>
<DIV id="toc" class="toc">
<DIV class="toctitle">
<H2>Contents</H2>
<SPAN class="toctoggle"> [<A role="button" tabindex="0" class="togglelink">hide</A>] </SPAN></DIV>
<UL>
<LI class="toclevel-1 tocsection-1"><A href="#History"><SPAN class="tocnumber">1</SPAN> <SPAN class="toctext">History</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-2"><A href="#Early_history"><SPAN class="tocnumber">1.1</SPAN> <SPAN class="toctext">Early history</SPAN></A></LI>
<LI class="toclevel-2 tocsection-3"><A href="#Plagiarism_dispute"><SPAN class="tocnumber">1.2</SPAN> <SPAN class="toctext">Plagiarism dispute</SPAN></A></LI>
<LI class="toclevel-2 tocsection-4"><A href="#Hooke.27s_work_and_claims"><SPAN class="tocnumber">1.3</SPAN> <SPAN class="toctext">Hooke's work and claims</SPAN></A></LI>
<LI class="toclevel-2 tocsection-5"><A href="#Newton.27s_work_and_claims"><SPAN class="tocnumber">1.4</SPAN> <SPAN class="toctext">Newton's work and claims</SPAN></A></LI>
<LI class="toclevel-2 tocsection-6"><A href="#Newton.27s_acknowledgment"><SPAN class="tocnumber">1.5</SPAN> <SPAN class="toctext">Newton's acknowledgment</SPAN></A></LI>
<LI class="toclevel-2 tocsection-7"><A href="#Modern_priority_controversy"><SPAN class="tocnumber">1.6</SPAN> <SPAN class="toctext">Modern priority controversy</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-8"><A href="#Modern_form"><SPAN class="tocnumber">2</SPAN> <SPAN class="toctext">Modern form</SPAN></A></LI>
<LI class="toclevel-1 tocsection-9"><A href="#Bodies_with_spatial_extent"><SPAN class="tocnumber">3</SPAN> <SPAN class="toctext">Bodies with spatial extent</SPAN></A></LI>
<LI class="toclevel-1 tocsection-10"><A href="#Vector_form"><SPAN class="tocnumber">4</SPAN> <SPAN class="toctext">Vector form</SPAN></A></LI>
<LI class="toclevel-1 tocsection-11"><A href="#Gravitational_field"><SPAN class="tocnumber">5</SPAN> <SPAN class="toctext">Gravitational field</SPAN></A></LI>
<LI class="toclevel-1 tocsection-12"><A href="#Problematic_aspects"><SPAN class="tocnumber">6</SPAN> <SPAN class="toctext">Problematic aspects</SPAN></A>
<UL>
<LI class="toclevel-2 tocsection-13"><A href="#Theoretical_concerns_with_Newton.27s_expression"><SPAN class="tocnumber">6.1</SPAN> <SPAN class="toctext">Theoretical concerns with Newton's expression</SPAN></A></LI>
<LI class="toclevel-2 tocsection-14"><A href="#Observations_conflicting_with_Newton.27s_formula"><SPAN class="tocnumber">6.2</SPAN> <SPAN class="toctext">Observations conflicting with Newton's formula</SPAN></A></LI>
<LI class="toclevel-2 tocsection-15"><A href="#Newton.27s_reservations"><SPAN class="tocnumber">6.3</SPAN> <SPAN class="toctext">Newton's reservations</SPAN></A></LI>
<LI class="toclevel-2 tocsection-16"><A href="#Einstein.27s_solution"><SPAN class="tocnumber">6.4</SPAN> <SPAN class="toctext">Einstein's solution</SPAN></A></LI>
</UL>
</LI>
<LI class="toclevel-1 tocsection-17"><A href="#Extensions"><SPAN class="tocnumber">7</SPAN> <SPAN class="toctext">Extensions</SPAN></A></LI>
<LI class="toclevel-1 tocsection-18"><A href="#Solutions_of_Newton.27s_law_of_universal_gravitation"><SPAN class="tocnumber">8</SPAN> <SPAN class="toctext">Solutions of Newton's law of universal gravitation</SPAN></A></LI>
<LI class="toclevel-1 tocsection-19"><A href="#See_also"><SPAN class="tocnumber">9</SPAN> <SPAN class="toctext">See also</SPAN></A></LI>
<LI class="toclevel-1 tocsection-20"><A href="#Notes"><SPAN class="tocnumber">10</SPAN> <SPAN class="toctext">Notes</SPAN></A></LI>
<LI class="toclevel-1 tocsection-21"><A href="#References"><SPAN class="tocnumber">11</SPAN> <SPAN class="toctext">References</SPAN></A></LI>
<LI class="toclevel-1 tocsection-22"><A href="#External_links"><SPAN class="tocnumber">12</SPAN> <SPAN class="toctext">External links</SPAN></A></LI>
</UL>
</DIV>
<P></P>
<H2><SPAN class="mw-headline" id="History">History</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=1" title="Edit section: History">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<H3><SPAN class="mw-headline" id="Early_history">Early history</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=2" title="Edit section: Early history">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>A recent assessment (by Ofer Gal) about the early history of the inverse square law is "by the late 1670s", 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".</P>
<P>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 "<A href="/wiki/Centrifugal_force" title="Centrifugal force">centrifugal</A>" and towards "<A href="/wiki/Centripetal_force" title="Centripetal force">centripetal</A>" force as Hooke's significant contributions.</P>
<P>Newton himself gave credit in his <I>Principia</I> to two persons: <A href="/wiki/Isma%C3%ABl_Bullialdus" title="Ismaël Bullialdus">Bullialdus</A><SUP id="cite_ref-Bullialdus_1645_5-0" class="reference"><A href="#cite_note-Bullialdus_1645-5">[4]</A></SUP> (he wrote without proof that there was a force on the earth towards the sun), and <A href="/wiki/Giovanni_Alfonso_Borelli" title="Giovanni Alfonso Borelli">Borelli</A><SUP id="cite_ref-Borelli.2C_G._A._1666_6-0" class="reference"><A href="#cite_note-Borelli.2C_G._A._1666-6">[5]</A></SUP> (wrote that all planets were attracted towards the sun). Whiteside wrote that the main influence was Borelli, because Newton had a copy of his book.<SUP id="cite_ref-T_Whiteside_1970_pages_5-19_7-0" class="reference"><A href="#cite_note-T_Whiteside_1970_pages_5-19-7">[6]</A></SUP></P>
<H3><SPAN class="mw-headline" id="Plagiarism_dispute">Plagiarism dispute</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=3" title="Edit section: Plagiarism dispute">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>In 1686, when the first book of <A href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</A>'s <I><A href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Principia</A></I> was presented to the <A href="/wiki/Royal_Society" title="Royal Society">Royal Society</A>, <A href="/wiki/Robert_Hooke" title="Robert Hooke">Robert Hooke</A> accused Newton of <A href="/wiki/Plagiarism" title="Plagiarism">plagiarism</A> 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". At the same time (according to <A href="/wiki/Edmond_Halley" title="Edmond Halley">Edmond Halley</A>'s contemporary report) Hooke agreed that "the Demonstration of the Curves generated thereby" was wholly Newton's.<SUP id="cite_ref-1686letters_8-0" class="reference"><A href="#cite_note-1686letters-8">[7]</A></SUP></P>
<P>In this way, the question arose as to what, if anything, Newton owed to Hooke. This is a subject extensively discussed since that time and on which some points, outlined below, continue to excite controversy.</P>
<H3><SPAN id="Hooke's_work_and_claims"></SPAN><SPAN class="mw-headline" id="Hooke.27s_work_and_claims">Hooke's work and claims</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=4" title="Edit section: Hooke's work and claims">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>Robert Hooke published his ideas about the "System of the World" in the 1660s, when he read to the Royal Society on March 21, 1666, a paper "On gravity", "concerning the inflection of a direct motion into a curve by a supervening attractive principle", and he published them again in somewhat developed form in 1674, as an addition to "An Attempt to Prove the Motion of the Earth from Observations".<SUP id="cite_ref-attempt_9-0" class="reference"><A href="#cite_note-attempt-9">[8]</A></SUP> Hooke announced in 1674 that he planned to "explain a System of the World differing in many particulars from any yet known", based on three "Suppositions": that "all Celestial Bodies whatsoever, have an attraction or gravitating power towards their own Centers" [and] "they do also attract all the other Celestial Bodies that are within the sphere of their activity";<SUP id="cite_ref-10" class="reference"><A href="#cite_note-10">[9]</A></SUP> that "all bodies whatsoever that are put into a direct and simple motion, will so continue to move forward in a straight line, till they are by some other effectual powers deflected and bent..."; and that "these attractive powers are so much the more powerful in operating, by how much the nearer the body wrought upon is to their own Centers". 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.</P>
<P>Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses.<SUP id="cite_ref-11" class="reference"><A href="#cite_note-11">[10]</A></SUP> He also did not provide accompanying evidence or mathematical demonstration. On the latter two aspects, Hooke himself stated in 1674: "Now what these several degrees [of attraction] are I have not yet experimentally verified"; and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e. "prosecuting this Inquiry").<SUP id="cite_ref-attempt_9-1" class="reference"><A href="#cite_note-attempt-9">[8]</A></SUP> It was later on, in writing on 6 January 1679|80<SUP id="cite_ref-12" class="reference"><A href="#cite_note-12">[11]</A></SUP> to Newton, that Hooke communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance."<SUP id="cite_ref-13" class="reference"><A href="#cite_note-13">[12]</A></SUP> (The inference about the velocity was incorrect.<SUP id="cite_ref-14" class="reference"><A href="#cite_note-14">[13]</A></SUP>)</P>
<P>Hooke's correspondence with Newton during 1679–1680 not only mentioned this inverse square supposition for the decline of attraction with increasing distance, but also, in Hooke's opening letter to Newton, of 24 November 1679, an approach of "compounding the celestial motions of the planets of a direct motion by the tangent & an attractive motion towards the central body".<SUP id="cite_ref-15" class="reference"><A href="#cite_note-15">[14]</A></SUP></P>
<H3><SPAN id="Newton's_work_and_claims"></SPAN><SPAN class="mw-headline" id="Newton.27s_work_and_claims">Newton's work and claims</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=5" title="Edit section: Newton's work and claims">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>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. Among the reasons, Newton recalled that the idea had been discussed with Sir Christopher Wren previous to Hooke's 1679 letter.<SUP id="cite_ref-16" class="reference"><A href="#cite_note-16">[15]</A></SUP> Newton also pointed out and acknowledged prior work of others,<SUP id="cite_ref-june1686_17-0" class="reference"><A href="#cite_note-june1686-17">[16]</A></SUP> including <A href="/wiki/Isma%C3%ABl_Bullialdus" title="Ismaël Bullialdus">Bullialdus</A>,<SUP id="cite_ref-Bullialdus_1645_5-1" class="reference"><A href="#cite_note-Bullialdus_1645-5">[4]</A></SUP> (who suggested, but without demonstration, that there was an attractive force from the Sun in the inverse square proportion to the distance), and <A href="/wiki/Giovanni_Alfonso_Borelli" title="Giovanni Alfonso Borelli">Borelli</A><SUP id="cite_ref-Borelli.2C_G._A._1666_6-1" class="reference"><A href="#cite_note-Borelli.2C_G._A._1666-6">[5]</A></SUP> (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). D T Whiteside has described the contribution to Newton's thinking that came from Borelli's book, a copy of which was in Newton's library at his death.<SUP id="cite_ref-T_Whiteside_1970_pages_5-19_7-1" class="reference"><A href="#cite_note-T_Whiteside_1970_pages_5-19-7">[6]</A></SUP></P>
<P>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. 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. According to Newton, while the 'Principia' was still at pre-publication stage, there were so many a-priori reasons to doubt the accuracy of the inverse-square law (especially close to an attracting sphere) that "without my (Newton's) Demonstrations, to which Mr Hooke is yet a stranger, it cannot believed by a judicious Philosopher to be any where accurate."<SUP id="cite_ref-18" class="reference"><A href="#cite_note-18">[17]</A></SUP></P>
<P>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. Thus Newton gave a justification, otherwise lacking, for applying the inverse square law to large spherical planetary masses as if they were tiny particles.<SUP id="cite_ref-19" class="reference"><A href="#cite_note-19">[18]</A></SUP> In addition, Newton had formulated in Propositions 43-45 of Book 1,<SUP id="cite_ref-20" class="reference"><A href="#cite_note-20">[19]</A></SUP> 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.</P>
<P>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.<SUP id="cite_ref-21" class="reference"><A href="#cite_note-21">[20]</A></SUP> After his 1679-1680 correspondence with Hooke, Newton adopted the language of inward or centripetal force. According to Newton scholar J. 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 involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The lesson offered by Hooke to Newton here, although significant, was one of perspective and did not change the analysis.<SUP id="cite_ref-22" class="reference"><A href="#cite_note-22">[21]</A></SUP> This background shows there was basis for Newton to deny deriving the inverse square law from Hooke.</P>
<H3><SPAN id="Newton's_acknowledgment"></SPAN><SPAN class="mw-headline" id="Newton.27s_acknowledgment">Newton's acknowledgment</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=6" title="Edit section: Newton's acknowledgment">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>On the other hand, Newton did accept and acknowledge, in all editions of the <I>Principia</I>, that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the solar system. Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1.<SUP id="cite_ref-23" class="reference"><A href="#cite_note-23">[22]</A></SUP> Newton also acknowledged to Halley that his correspondence with Hooke in 1679-80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ..."<SUP id="cite_ref-june1686_17-1" class="reference"><A href="#cite_note-june1686-17">[16]</A></SUP></P>
<H3><SPAN class="mw-headline" id="Modern_priority_controversy">Modern priority controversy</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=7" title="Edit section: Modern priority controversy">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>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. 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. They also show Newton clearly expressing the concept of linear inertia—for which he was indebted to Descartes' work, published in 1644 (as Hooke probably was).<SUP id="cite_ref-dtw1970_24-0" class="reference"><A href="#cite_note-dtw1970-24">[23]</A></SUP> These matters do not appear to have been learned by Newton from Hooke.</P>
<P>Nevertheless, a number of authors have had more to say about what Newton gained from Hooke and some aspects remain controversial.<SUP id="cite_ref-25" class="reference"><A href="#cite_note-25">[24]</A></SUP> The fact that most of Hooke's private papers had been destroyed or have disappeared does not help to establish the truth.</P>
<P>Newton's role in relation to the inverse square law was not as it has sometimes been represented. He did not claim to think it up as a bare idea. What Newton did was to show how the inverse-square law of attraction had many necessary mathematical connections with observable features of the motions of bodies in the solar system; and that they were related in such a way that the observational evidence and the mathematical demonstrations, taken together, gave reason to believe that the inverse square law was not just approximately true but exactly true (to the accuracy achievable in Newton's time and for about two centuries afterwards – and with some loose ends of points that could not yet be certainly examined, where the implications of the theory had not yet been adequately identified or calculated).<SUP id="cite_ref-26" class="reference"><A href="#cite_note-26">[25]</A></SUP><SUP id="cite_ref-27" class="reference"><A href="#cite_note-27">[26]</A></SUP></P>
<P>About thirty years after Newton's death in 1727, <A href="/wiki/Alexis_Clairaut" title="Alexis Clairaut">Alexis Clairaut</A>, a mathematical astronomer eminent in his own right in the field of gravitational studies, wrote after reviewing what Hooke published, that "One must not think that this idea ... of Hooke diminishes Newton's glory"; and that "the example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".<SUP id="cite_ref-28" class="reference"><A href="#cite_note-28">[27]</A></SUP><SUP id="cite_ref-29" class="reference"><A href="#cite_note-29">[28]</A></SUP></P>
<H2><SPAN class="mw-headline" id="Modern_form">Modern form</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=8" title="Edit section: Modern form">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>In modern language, the law states the following:</P>
<TABLE style="border-top:1px solid #aaa; border-bottom:1px solid #aaa; margin-bottom:1em">
<TBODY><TR>
<TD style="padding:0.1em 2em 0 0.5em"><SPAN style="white-space:nowrap">Every <A href="/wiki/Point_mass" class="mw-redirect" title="Point mass">point</A> <A href="/wiki/Mass" title="Mass">mass</A> attracts every single other point mass by a <A href="/wiki/Force" title="Force">force</A> pointing along</SPAN> the <A href="/wiki/Line_(mathematics)" class="mw-redirect" title="Line (mathematics)">line</A> intersecting both points. The force is <A href="/wiki/Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</A> to the <A href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</A> of the two masses and <A href="/wiki/Proportionality_(mathematics)#Inverse_proportionality" title="Proportionality (mathematics)">inversely proportional</A> to the <A href="/wiki/Square_(algebra)" title="Square (algebra)">square</A> of the distance between them:<SUP id="cite_ref-Newton1_3-1" class="reference"><A href="#cite_note-Newton1-3">[2]</A></SUP></TD>
<TD rowspan="2" style="width:99%; vertical-align:top">
<DIV class="floatleft"><A href="/wiki/File:NewtonsLawOfUniversalGravitation.svg" class="image" title="Diagram of two masses attracting one another"><IMG alt="Diagram of two masses attracting one another" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/200px-NewtonsLawOfUniversalGravitation.svg.png" width="200" height="140" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/300px-NewtonsLawOfUniversalGravitation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0e/NewtonsLawOfUniversalGravitation.svg/400px-NewtonsLawOfUniversalGravitation.svg.png 2x" data-file-width="400" data-file-height="280" /></A></DIV>
</TD>
</TR>
<TR>
<TD style="padding:0 2em 0.1em 0.5em; white-space:nowrap">
<P><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>F</mi>
        <mo>=</mo>
        <mi>G</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msub>
                <mi>m</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1</mn>
                </mrow>
              </msub>
              <msub>
                <mi>m</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msub>
            </mrow>
            <msup>
              <mi>r</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mtext> </mtext>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}\ }</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6ee5510ba3c7d6664775c0e76b53e72468303a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:14.339ex; height:5.009ex;" alt="F=G{\frac {m_{1}m_{2}}{r^{2}}}\ " /></SPAN><BR />
where:</P>
<UL>
<LI><I>F</I> is the force between the masses;</LI>
<LI><I>G</I> is the <A href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</A> (6.674×10<SUP>−11</SUP> N <B>·</B> (m/kg)<SUP>2</SUP>);</LI>
<LI><I>m</I><SUB>1</SUB> is the first mass;</LI>
<LI><I>m</I><SUB>2</SUB> is the second mass;</LI>
<LI><I>r</I> is the distance between the centers of the masses.</LI>
</UL>
</TD>
</TR>
</TBODY></TABLE>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:442px;"><A href="/wiki/File:Gravity_Big_G_Measurements_NIST.png" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Gravity_Big_G_Measurements_NIST.png/440px-Gravity_Big_G_Measurements_NIST.png" width="440" height="308" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Gravity_Big_G_Measurements_NIST.png/660px-Gravity_Big_G_Measurements_NIST.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Gravity_Big_G_Measurements_NIST.png/880px-Gravity_Big_G_Measurements_NIST.png 2x" data-file-width="3000" data-file-height="2100" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Gravity_Big_G_Measurements_NIST.png" class="internal" title="Enlarge"></A></DIV>
Error plot showing experimental values for big G.</DIV>
</DIV>
</DIV>
<P>Assuming <A href="/wiki/International_System_of_Units" title="International System of Units">SI units</A>, <I>F</I> is measured in <A href="/wiki/Newton_(units)" class="mw-redirect" title="Newton (units)">newtons</A> (N), <I>m</I><SUB>1</SUB> and <I>m</I><SUB>2</SUB> in <A href="/wiki/Kilogram" title="Kilogram">kilograms</A> (kg), <I>r</I> in meters (m), and the constant <I>G</I> is approximately equal to <SPAN class="nowrap"><SPAN style="display:none" class="sortkey">6989667400000000000♠</SPAN>6.674<SPAN style="margin-left:0.25em;margin-right:0.15em;">×</SPAN>10<SUP>−11</SUP> N m<SUP>2</SUP> kg<SUP>−2</SUP></SPAN>.<SUP id="cite_ref-30" class="reference"><A href="#cite_note-30">[29]</A></SUP> The value of the constant <I>G</I> was first accurately determined from the results of the <A href="/wiki/Cavendish_experiment" title="Cavendish experiment">Cavendish experiment</A> conducted by the <A href="/wiki/United_Kingdom" title="United Kingdom">British</A> scientist <A href="/wiki/Henry_Cavendish" title="Henry Cavendish">Henry Cavendish</A> in 1798, although Cavendish did not himself calculate a numerical value for <I>G</I>.<SUP id="cite_ref-The_Michell-Cavendish_Experiment_4-1" class="reference"><A href="#cite_note-The_Michell-Cavendish_Experiment-4">[3]</A></SUP> This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's <I>Principia</I> and 71 years after Newton's death, so none of Newton's calculations could use the value of <I>G</I>; instead he could only calculate a force relative to another force.</P>
<H2><SPAN class="mw-headline" id="Bodies_with_spatial_extent">Bodies with spatial extent</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=9" title="Edit section: Bodies with spatial extent">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:202px;"><A href="/wiki/File:Earth-G-force.png" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth-G-force.png/200px-Earth-G-force.png" width="200" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth-G-force.png/300px-Earth-G-force.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Earth-G-force.png/400px-Earth-G-force.png 2x" data-file-width="600" data-file-height="600" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Earth-G-force.png" class="internal" title="Enlarge"></A></DIV>
Gravitational field strength within the Earth</DIV>
</DIV>
</DIV>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:222px;"><A href="/wiki/File:Gravity_field_near_earth.gif" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/74/Gravity_field_near_earth.gif/220px-Gravity_field_near_earth.gif" width="220" height="218" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/7/74/Gravity_field_near_earth.gif 1.5x" data-file-width="301" data-file-height="298" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Gravity_field_near_earth.gif" class="internal" title="Enlarge"></A></DIV>
Gravity field near the surface of the Earth – an object is shown accelerating toward the surface</DIV>
</DIV>
</DIV>
<P>If the bodies in question have spatial extent (as opposed to being point masses), then the gravitational force between them is calculated by summing the contributions of the notional point masses which constitute the bodies. In the limit, as the component point masses become "infinitely small", this entails <A href="/wiki/Integral" title="Integral">integrating</A> the force (in vector form, see below) over the extents of the two <A href="/wiki/Physical_body" title="Physical body">bodies</A>.</P>
<P>In this way, it can be shown that an object with a spherically-symmetric distribution of mass exerts the same gravitational attraction on external bodies as if all the object's mass were concentrated at a point at its centre.<SUP id="cite_ref-Newton1_3-2" class="reference"><A href="#cite_note-Newton1-3">[2]</A></SUP> (This is not generally true for non-spherically-symmetrical bodies.)</P>
<P>For points <I>inside</I> a spherically-symmetric distribution of matter, Newton's <A href="/wiki/Shell_theorem" title="Shell theorem">Shell theorem</A> can be used to find the gravitational force. The theorem tells us how different parts of the mass distribution affect the gravitational force measured at a point located a distance <B>r</B><SUB>0</SUB> from the center of the mass distribution:<SUP id="cite_ref-31" class="reference"><A href="#cite_note-31">[30]</A></SUP></P>
<UL>
<LI>The portion of the mass that is located at radii r < <B>r</B><SUB>0</SUB> causes the same force at <B>r</B><SUB>0</SUB> as if all of the mass enclosed within a sphere of radius <B>r</B><SUB>0</SUB> was concentrated at the center of the mass distribution (as noted above).</LI>
<LI>The portion of the mass that is located at radii r > <B>r</B><SUB>0</SUB> exerts <I>no net</I> gravitational force at the distance <B>r</B><SUB>0</SUB> from the center. That is, the individual gravitational forces exerted by the elements of the sphere out there, on the point at <B>r</B><SUB>0</SUB>, cancel each other out.</LI>
</UL>
<P>As a consequence, for example, within a shell of uniform thickness and density there is <I>no net</I> gravitational acceleration anywhere within the hollow sphere.</P>
<P>Furthermore, inside a uniform sphere the gravity increases linearly with the distance from the center; the increase due to the additional mass is 1.5 times the decrease due to the larger distance from the center. Thus, if a spherically symmetric body has a uniform core and a uniform mantle with a density that is less than 2/3 of that of the core, then the gravity initially decreases outwardly beyond the boundary, and if the sphere is large enough, further outward the gravity increases again, and eventually it exceeds the gravity at the core/mantle boundary. The gravity of the Earth may be highest at the core/mantle boundary.</P>
<DIV style="clear:both;"></DIV>
<H2><SPAN class="mw-headline" id="Vector_form">Vector form</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=10" title="Edit section: Vector form">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="thumb tright">
<DIV class="thumbinner" style="width:202px;"><A href="/wiki/File:Gravitymacroscopic.svg" class="image"><IMG alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitymacroscopic.svg/200px-Gravitymacroscopic.svg.png" width="200" height="200" class="thumbimage" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitymacroscopic.svg/300px-Gravitymacroscopic.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Gravitymacroscopic.svg/400px-Gravitymacroscopic.svg.png 2x" data-file-width="296" data-file-height="296" /></A>
<DIV class="thumbcaption">
<DIV class="magnify"><A href="/wiki/File:Gravitymacroscopic.svg" class="internal" title="Enlarge"></A></DIV>
Gravity field surrounding Earth from a macroscopic perspective.</DIV>
</DIV>
</DIV>
<P>Newton's law of universal gravitation can be written as a <A href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</A> <A href="/wiki/Equation" title="Equation">equation</A> to account for the direction of the gravitational force as well as its magnitude. In this formula, quantities in bold represent vectors.</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mrow class="MJX-TeXAtom-ORD">
            <mi mathvariant="bold">F</mi>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>21</mn>
          </mrow>
        </msub>
        <mo>=</mo>
        <mo>−<!-- − --></mo>
        <mi>G</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msub>
                <mi>m</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1</mn>
                </mrow>
              </msub>
              <msub>
                <mi>m</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msub>
            </mrow>
            <msup>
              <mrow class="MJX-TeXAtom-ORD">
                <mo fence="false" stretchy="false">|</mo>
                <msub>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="bold">r</mi>
                  </mrow>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mn>12</mn>
                  </mrow>
                </msub>
                <mo fence="false" stretchy="false">|</mo>
              </mrow>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mspace width="thinmathspace"></mspace>
        <msub>
          <mrow class="MJX-TeXAtom-ORD">
            <mrow class="MJX-TeXAtom-ORD">
              <mover>
                <mi mathvariant="bold">r</mi>
                <mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
              </mover>
            </mrow>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>12</mn>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {\vert \mathbf {r} _{12}\vert }^{2}}\,\mathbf {\hat {r}} _{12}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009790661ba6d872a59712cc884be14ca63b88c4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:21.035ex; height:6.009ex;" alt="{\displaystyle \mathbf {F} _{21}=-G{m_{1}m_{2} \over {\vert \mathbf {r} _{12}\vert }^{2}}\,\mathbf {\hat {r}} _{12}}" /></SPAN></DD>
</DL>
<P>where</P>
<DL>
<DD><B>F</B><SUB>21</SUB> is the force applied on object 2 exerted by object 1,</DD>
<DD><I>G</I> is the <A href="/wiki/Gravitational_constant" title="Gravitational constant">gravitational constant</A>,</DD>
<DD><I>m</I><SUB>1</SUB> and <I>m</I><SUB>2</SUB> are respectively the masses of objects 1 and 2,</DD>
<DD>|<B>r</B><SUB>12</SUB>| = |<B>r</B><SUB>2</SUB> − <B>r</B><SUB>1</SUB>| is the distance between objects 1 and 2, and</DD>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mrow class="MJX-TeXAtom-ORD">
            <mrow class="MJX-TeXAtom-ORD">
              <mover>
                <mi mathvariant="bold">r</mi>
                <mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
              </mover>
            </mrow>
          </mrow>
          <mrow class="MJX-TeXAtom-ORD">
            <mn>12</mn>
          </mrow>
        </msub>
        <mtext> </mtext>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-REL">
            <mover>
              <mrow class="MJX-TeXAtom-OP">
                <mo>=</mo>
              </mrow>
              <mrow class="MJX-TeXAtom-ORD">
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="normal">d</mi>
                  <mi mathvariant="normal">e</mi>
                  <mi mathvariant="normal">f</mi>
                </mrow>
              </mrow>
            </mover>
          </mrow>
        </mrow>
        <mtext> </mtext>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <msub>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">r</mi>
                </mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msub>
              <mo>−<!-- − --></mo>
              <msub>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">r</mi>
                </mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1</mn>
                </mrow>
              </msub>
            </mrow>
            <mrow>
              <mo fence="false" stretchy="false">|</mo>
              <msub>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">r</mi>
                </mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msub>
              <mo>−<!-- − --></mo>
              <msub>
                <mrow class="MJX-TeXAtom-ORD">
                  <mi mathvariant="bold">r</mi>
                </mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>1</mn>
                </mrow>
              </msub>
              <mo fence="false" stretchy="false">|</mo>
            </mrow>
          </mfrac>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {r}} _{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ce6372f21cf5211fa1fb706bd7ddcb467fab863" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:16.027ex; height:5.676ex;" alt="\mathbf {\hat {r}} _{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{2}-\mathbf {r} _{1}}{\vert \mathbf {r} _{2}-\mathbf {r} _{1}\vert }}" /></SPAN> is the <A href="/wiki/Unit_vector" title="Unit vector">unit vector</A> from object 1 to 2.</DD>
</DL>
<P>It can be seen that the vector form of the equation is the same as the <A href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</A> form given earlier, except that <B>F</B> is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that <B>F</B><SUB>12</SUB> = −<B>F</B><SUB>21</SUB>.</P>
<H2><SPAN class="mw-headline" id="Gravitational_field">Gravitational field</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=11" title="Edit section: Gravitational field">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV role="note" class="hatnote navigation-not-searchable">Main article: <A href="/wiki/Gravitational_field" title="Gravitational field">Gravitational field</A></DIV>
<P>The <B>gravitational field</B> is a <A href="/wiki/Vector_field" title="Vector field">vector field</A> that describes the gravitational force which would be applied on an object in any given point in space, per unit mass. It is actually equal to the <A href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</A> at that point.</P>
<P>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). For 2 objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write <B>r</B> instead of <B>r</B><SUB>12</SUB> and <I>m</I> instead of <I>m</I><SUB>2</SUB> and define the gravitational field <B>g</B>(<B>r</B>) as:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">g</mi>
        </mrow>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mo>−<!-- − --></mo>
        <mi>G</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>m</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>1</mn>
              </mrow>
            </msub>
            <mrow class="MJX-TeXAtom-ORD">
              <msup>
                <mrow class="MJX-TeXAtom-ORD">
                  <mo fence="false" stretchy="false">|</mo>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="bold">r</mi>
                  </mrow>
                  <mo fence="false" stretchy="false">|</mo>
                </mrow>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mspace width="thinmathspace"></mspace>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow class="MJX-TeXAtom-ORD">
            <mover>
              <mi mathvariant="bold">r</mi>
              <mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo>
            </mover>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} }</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf0689fbd05781a129e2df24ef5bd8b7edf2f93" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:17.114ex; height:6.009ex;" alt="\mathbf {g} (\mathbf {r} )=-G{m_{1} \over {{\vert \mathbf {r} \vert }^{2}}}\,\mathbf {\hat {r}} " /></SPAN></DD>
</DL>
<P>so that we can write:</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">F</mi>
        </mrow>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mi>m</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">g</mi>
        </mrow>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} ).}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef10f4f1d5c39458a54cfc8edcd1e5b9bdda13fc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.743ex; height:2.843ex;" alt="\mathbf {F} (\mathbf {r} )=m\mathbf {g} (\mathbf {r} )." /></SPAN></DD>
</DL>
<P>This formulation is dependent on the objects causing the field. The field has units of acceleration; in <A href="/wiki/SI" class="mw-redirect" title="SI">SI</A>, this is m/s<SUP>2</SUP>.</P>
<P>Gravitational fields are also <A href="/wiki/Conservative_field" class="mw-redirect" title="Conservative field">conservative</A>; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field <I>V</I>(<B>r</B>) such that</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">g</mi>
        </mrow>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mo>−<!-- − --></mo>
        <mi mathvariant="normal">∇<!-- ∇ --></mi>
        <mi>V</mi>
        <mo stretchy="false">(</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">r</mi>
        </mrow>
        <mo stretchy="false">)</mo>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} ).}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28ab8b74ac2595cca9eb982767171ceb79ebe90" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.561ex; height:2.843ex;" alt="\mathbf {g} (\mathbf {r} )=-\nabla V(\mathbf {r} )." /></SPAN></DD>
</DL>
<P>If <I>m</I><SUB>1</SUB> is a point mass or the mass of a sphere with homogeneous mass distribution, the force field <B>g</B>(<B>r</B>) outside the sphere is isotropic, i.e., depends only on the distance <I>r</I> from the center of the sphere. In that case</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>V</mi>
        <mo stretchy="false">(</mo>
        <mi>r</mi>
        <mo stretchy="false">)</mo>
        <mo>=</mo>
        <mo>−<!-- − --></mo>
        <mi>G</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <msub>
              <mi>m</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>1</mn>
              </mrow>
            </msub>
            <mi>r</mi>
          </mfrac>
        </mrow>
        <mo>.</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle V(r)=-G{\frac {m_{1}}{r}}.}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b8de5a197a1b13a6eb7844e692862b3da53ba2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:16.058ex; height:4.676ex;" alt="V(r)=-G{\frac {m_{1}}{r}}." /></SPAN></DD>
</DL>
<P>the gravitational field is on, inside and outside of symmetric masses.</P>
<P>As per <A href="/wiki/Gauss%27s_law_for_gravity" title="Gauss's law for gravity">Gauss Law</A>, field in a symmetric body can be found by the mathematical equation:</P>
<DL>
<DD><SPAN class="nowrap"><IMG alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/25px-OiintLaTeX.svg.png" width="25" height="44" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/38px-OiintLaTeX.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/50px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /><SPAN style="position:relative; right:8px; top:18px; margin-right:-8px;"><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">∂<!-- ∂ --></mi>
        <mi>V</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.126ex; height:2.343ex;" alt="\partial V" /></SPAN></SPAN> <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">g</mi>
          <mo mathvariant="bold" stretchy="false">(</mo>
          <mi mathvariant="bold">r</mi>
          <mo mathvariant="bold" stretchy="false">)</mo>
        </mrow>
        <mo>⋅<!-- ⋅ --></mo>
        <mi>d</mi>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">A</mi>
        </mrow>
        <mo>=</mo>
        <mo>−<!-- − --></mo>
        <mn>4</mn>
        <mi>π<!-- π --></mi>
        <mi>G</mi>
        <msub>
          <mi>M</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mtext>enc</mtext>
          </mrow>
        </msub>
        <mo>,</mo>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06858ef1880b87d71ba7d1e5f508944f81d75b2c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:24.335ex; height:2.843ex;" alt="{\displaystyle \mathbf {g(r)} \cdot d\mathbf {A} =-4\pi GM_{\text{enc}},}" /></SPAN></SPAN></DD>
</DL>
<P>where <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi mathvariant="normal">∂<!-- ∂ --></mi>
        <mi>V</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.126ex; height:2.343ex;" alt="\partial V" /></SPAN> is a closed surface and <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <msub>
          <mi>M</mi>
          <mrow class="MJX-TeXAtom-ORD">
            <mtext>enc</mtext>
          </mrow>
        </msub>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle M_{\text{enc}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36be47bb78936458374b87df625a0d9d28fd9136" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.893ex; height:2.509ex;" alt="{\displaystyle M_{\text{enc}}}" /></SPAN> is the mass enclosed by the surface.</P>
<P>Hence, for a hollow sphere of radius <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>R</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle R}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:2.176ex;" alt="R" /></SPAN> and total mass <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>M</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle M}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.453ex; height:2.176ex;" alt="M" /></SPAN>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mo stretchy="false">|</mo>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">g</mi>
          <mo mathvariant="bold" stretchy="false">(</mo>
          <mi mathvariant="bold">r</mi>
          <mo mathvariant="bold" stretchy="false">)</mo>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mo stretchy="false">|</mo>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow>
            <mo>{</mo>
            <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
              <mtr>
                <mtd>
                  <mn>0</mn>
                  <mo>,</mo>
                </mtd>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>if </mtext>
                  </mrow>
                  <mi>r</mi>
                  <mo><</mo>
                  <mi>R</mi>
                </mtd>
              </mtr>
              <mtr>
                <mtd></mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mstyle displaystyle="true" scriptlevel="0">
                      <mfrac>
                        <mrow>
                          <mi>G</mi>
                          <mi>M</mi>
                        </mrow>
                        <msup>
                          <mi>r</mi>
                          <mrow class="MJX-TeXAtom-ORD">
                            <mn>2</mn>
                          </mrow>
                        </msup>
                      </mfrac>
                    </mstyle>
                  </mrow>
                  <mo>,</mo>
                </mtd>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>if </mtext>
                  </mrow>
                  <mi>r</mi>
                  <mo>≥<!-- ≥ --></mo>
                  <mi>R</mi>
                </mtd>
              </mtr>
            </mtable>
            <mo fence="true" stretchy="true"></mo>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\mbox{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\mbox{if }}r\geq R\end{cases}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4076b9185d8ce136d722617125e8622a2a5fdd8d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.766ex; margin-bottom: -0.239ex; width:27.819ex; height:11.176ex;" alt="{\displaystyle |\mathbf {g(r)} |={\begin{cases}0,&{\mbox{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\mbox{if }}r\geq R\end{cases}}}" /></SPAN></DD>
</DL>
<P>For a uniform solid sphere of radius <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>R</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle R}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:2.176ex;" alt="R" /></SPAN> and total mass <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mi>M</mi>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle M}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.453ex; height:2.176ex;" alt="M" /></SPAN>,</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mo stretchy="false">|</mo>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mi mathvariant="bold">g</mi>
          <mo mathvariant="bold" stretchy="false">(</mo>
          <mi mathvariant="bold">r</mi>
          <mo mathvariant="bold" stretchy="false">)</mo>
        </mrow>
        <mrow class="MJX-TeXAtom-ORD">
          <mo stretchy="false">|</mo>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mrow>
            <mo>{</mo>
            <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false">
              <mtr>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mstyle displaystyle="true" scriptlevel="0">
                      <mfrac>
                        <mrow>
                          <mi>G</mi>
                          <mi>M</mi>
                          <mi>r</mi>
                        </mrow>
                        <msup>
                          <mi>R</mi>
                          <mrow class="MJX-TeXAtom-ORD">
                            <mn>3</mn>
                          </mrow>
                        </msup>
                      </mfrac>
                    </mstyle>
                  </mrow>
                  <mo>,</mo>
                </mtd>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>if </mtext>
                  </mrow>
                  <mi>r</mi>
                  <mo><</mo>
                  <mi>R</mi>
                </mtd>
              </mtr>
              <mtr>
                <mtd></mtd>
              </mtr>
              <mtr>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mstyle displaystyle="true" scriptlevel="0">
                      <mfrac>
                        <mrow>
                          <mi>G</mi>
                          <mi>M</mi>
                        </mrow>
                        <msup>
                          <mi>r</mi>
                          <mrow class="MJX-TeXAtom-ORD">
                            <mn>2</mn>
                          </mrow>
                        </msup>
                      </mfrac>
                    </mstyle>
                  </mrow>
                  <mo>,</mo>
                </mtd>
                <mtd>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mtext>if </mtext>
                  </mrow>
                  <mi>r</mi>
                  <mo>≥<!-- ≥ --></mo>
                  <mi>R</mi>
                </mtd>
              </mtr>
            </mtable>
            <mo fence="true" stretchy="true"></mo>
          </mrow>
        </mrow>
      </mstyle>
    </mrow>
    <annotation encoding="application/x-tex">{\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\mbox{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\mbox{if }}r\geq R\end{cases}}}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90896e2b9b3fefc2975e53c6e4e9c28aafc72587" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -6.163ex; margin-bottom: -0.175ex; width:28.878ex; height:13.843ex;" alt="{\displaystyle |\mathbf {g(r)} |={\begin{cases}{\dfrac {GMr}{R^{3}}},&{\mbox{if }}r<R\\\\{\dfrac {GM}{r^{2}}},&{\mbox{if }}r\geq R\end{cases}}}" /></SPAN></DD>
</DL>
<H2><SPAN class="mw-headline" id="Problematic_aspects">Problematic aspects</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=12" title="Edit section: Problematic aspects">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Newton's description of gravity is sufficiently accurate for many practical purposes and is therefore widely used. Deviations from it are small when the dimensionless quantities <I>φ</I>/<I>c</I><SUP>2</SUP> and <I>(v/c)<SUP>2</SUP></I> are both much less than one, where <I>φ</I> is the <A href="/wiki/Gravitational_potential" title="Gravitational potential">gravitational potential</A>, <I>v</I> is the velocity of the objects being studied, and <I>c</I> is the <A href="/wiki/Speed_of_light" title="Speed of light">speed of light</A>.<SUP id="cite_ref-32" class="reference"><A href="#cite_note-32">[31]</A></SUP> For example, Newtonian gravity provides an accurate description of the Earth/Sun system, since</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
    <mrow class="MJX-TeXAtom-ORD">
      <mstyle displaystyle="true" scriptlevel="0">
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mi mathvariant="normal">Φ<!-- Φ --></mi>
            <msup>
              <mi>c</mi>
              <mrow class="MJX-TeXAtom-ORD">
                <mn>2</mn>
              </mrow>
            </msup>
          </mfrac>
        </mrow>
        <mo>=</mo>
        <mrow class="MJX-TeXAtom-ORD">
          <mfrac>
            <mrow>
              <mi>G</mi>
              <msub>
                <mi>M</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="normal">s</mi>
                    <mi mathvariant="normal">u</mi>
                    <mi mathvariant="normal">n</mi>
                  </mrow>
                </mrow>
              </msub>
            </mrow>
            <mrow>
              <msub>
                <mi>r</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mrow class="MJX-TeXAtom-ORD">
                    <mi mathvariant="normal">o</mi>
                    <mi mathvariant="normal">r</mi>
                    <mi mathvariant="normal">b</mi>
                    <mi mathvariant="normal">i</mi>
                    <mi mathvariant="normal">t</mi>
                  </mrow>
                </mrow>
              </msub>
              <msup>
                <mi>c</mi>
                <mrow class="MJX-TeXAtom-ORD">
                  <mn>2</mn>
                </mrow>
              </msup>
            </mrow>
          </mfrac>
        </mrow>
        <mo>∼<!-- ∼ --></mo>
        <msup>
          <mn>10</mn>
          <mrow class="MJX-TeXAtom-ORD">
            <mo>−<!-- − --></mo>
            <mn>8</mn>
          </mrow>
        </msup>
        <mo>,</mo>
        <mspace width="1em"></mspace>
        <msup>
          <mrow>
            <mo>(</mo>
            <mrow class="MJX-TeXAtom-ORD">
              <mfrac>
                <msub>
                  <mi>v</mi>
                  <mrow class="MJX-TeXAtom-ORD">
                    <mrow class="MJX-TeXAtom-ORD">
                      <mi mathvariant="normal">E</mi>
                      <mi mathvariant="normal">a</mi>
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    <annotation encoding="application/x-tex">{\displaystyle {\frac {\Phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8}}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7466336e57a91d973d310299d1f94160b1bd818e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:58.727ex; height:6.843ex;" alt="{\frac {\Phi }{c^{2}}}={\frac {GM_{\mathrm {sun} }}{r_{\mathrm {orbit} }c^{2}}}\sim 10^{-8},\quad \left({\frac {v_{\mathrm {Earth} }}{c}}\right)^{2}=\left({\frac {2\pi r_{\mathrm {orbit} }}{(1\ \mathrm {yr} )c}}\right)^{2}\sim 10^{-8}" /></SPAN></DD>
</DL>
<P>where <I>r</I><SUB>orbit</SUB> is the radius of the Earth's orbit around the Sun.</P>
<P>In situations where either dimensionless parameter is large, then <A href="/wiki/General_relativity" title="General relativity">general relativity</A> must be used to describe the system. 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.</P>
<H3><SPAN id="Theoretical_concerns_with_Newton's_expression"></SPAN><SPAN class="mw-headline" id="Theoretical_concerns_with_Newton.27s_expression">Theoretical concerns with Newton's expression</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=13" title="Edit section: Theoretical concerns with Newton's expression">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<UL>
<LI>There is no immediate prospect of identifying the mediator of gravity. 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: <A href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</A> and <A href="/wiki/Standard_Model" title="Standard Model">Standard Model</A>). Newton himself felt that the concept of an inexplicable <I><A href="/wiki/Action_at_a_distance_(physics)" class="mw-redirect" title="Action at a distance (physics)">action at a distance</A></I> was unsatisfactory (see "<A href="#Newton.27s_reservations">Newton's reservations</A>" below), but that there was nothing more that he could do at the time.</LI>
<LI>Newton's theory of gravitation requires that the gravitational force be transmitted instantaneously. 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.</LI>
</UL>
<H3><SPAN id="Observations_conflicting_with_Newton's_formula"></SPAN><SPAN class="mw-headline" id="Observations_conflicting_with_Newton.27s_formula">Observations conflicting with Newton's formula</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=14" title="Edit section: Observations conflicting with Newton's formula">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<UL>
<LI>Newton's Theory does not fully explain the <A href="/wiki/Precession" title="Precession">precession</A> of the <A href="/wiki/Perihelion" class="mw-redirect" title="Perihelion">perihelion</A> of the <A href="/wiki/Orbit" title="Orbit">orbits</A> of the planets, especially of <A href="/wiki/Planet" title="Planet">planet</A> <A href="/wiki/Mercury_(planet)" title="Mercury (planet)">Mercury</A>, which was detected long after the life of Newton.<SUP id="cite_ref-33" class="reference"><A href="#cite_note-33">[32]</A></SUP> There is a 43 <A href="/wiki/Arcsecond" class="mw-redirect" title="Arcsecond">arcsecond</A> 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.</LI>
<LI>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. Calculations using <A href="/wiki/General_relativity#Bending_of_light" title="General relativity">General Relativity</A> are in much closer agreement with the astronomical observations.</LI>
<LI>In spiral galaxies, the orbiting of stars around their centers seems to strongly disobey Newton's law of universal gravitation. Astrophysicists, however, explain this spectacular phenomenon in the framework of Newton's laws, with the presence of large amounts of <A href="/wiki/Dark_matter" title="Dark matter">dark matter</A>.</LI>
</UL>
<H3><SPAN id="Newton's_reservations"></SPAN><SPAN class="mw-headline" id="Newton.27s_reservations">Newton's reservations</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=15" title="Edit section: Newton's reservations">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
<P>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. In 1692, in his third letter to Bentley, he wrote: <I>"That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it."</I></P>
<P>He never, in his words, "assigned the cause of this power". 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 <A href="/wiki/Mechanical_explanations_of_gravitation" title="Mechanical explanations of gravitation">mechanical hypotheses</A> in 1675 and 1717). 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. 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". These fundamental phenomena are still under investigation and, though hypotheses abound, the definitive answer has yet to be found. And in Newton's 1713 <I><A href="/wiki/General_Scholium" title="General Scholium">General Scholium</A></I> in the second edition of <I>Principia</I>: <I>"I have not yet been able to discover the cause of these properties of gravity from phenomena and I <A href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">feign no hypotheses</A>.... It is enough that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies."</I><SUP id="cite_ref-34" class="reference"><A href="#cite_note-34">[33]</A></SUP></P>
<H3><SPAN id="Einstein's_solution"></SPAN><SPAN class="mw-headline" id="Einstein.27s_solution">Einstein's solution</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=16" title="Edit section: Einstein's solution">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H3>
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<A href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</A><BR />
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<A href="/wiki/Introduction_to_the_mathematics_of_general_relativity" title="Introduction to the mathematics of general relativity">Mathematics of general relativity</A><BR />
<A href="/wiki/Einstein_field_equations" title="Einstein field equations">Einstein field equations</A></DIV>
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<P>These objections were explained by Einstein's theory of <A href="/wiki/General_relativity" title="General relativity">general relativity</A>, in which gravitation is an attribute of <A href="/wiki/Curved_spacetime" class="mw-redirect" title="Curved spacetime">curved spacetime</A> instead of being due to a force propagated between bodies. In Einstein's theory, energy and momentum distort spacetime in their vicinity, and other particles move in trajectories determined by the geometry of spacetime. This allowed a description of the motions of light and mass that was consistent with all available observations. In general relativity, the gravitational force is a <A href="/wiki/Fictitious_force" title="Fictitious force">fictitious force</A> due to the <A href="/wiki/Curvature_of_spacetime" class="mw-redirect" title="Curvature of spacetime">curvature of spacetime</A>, because the <A href="/wiki/Gravitational_acceleration" title="Gravitational acceleration">gravitational acceleration</A> of a body in <A href="/wiki/Free_fall" title="Free fall">free fall</A> is due to its <A href="/wiki/World_line" title="World line">world line</A> being a <A href="/wiki/Geodesic" title="Geodesic">geodesic</A> of <A href="/wiki/Spacetime" title="Spacetime">spacetime</A>.</P>
<H2><SPAN class="mw-headline" id="Extensions">Extensions</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=17" title="Edit section: Extensions">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<P>Newton was the first to consider in his Principia an extended expression of his law of gravity including an inverse-cube term of the form</P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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    <annotation encoding="application/x-tex">{\displaystyle F=G{\frac {m_{1}m_{2}}{r^{2}}}+B{\frac {m_{1}m_{2}}{r^{3}}}\ }</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34f01379b8d4f7cb6096a9e4311ee574f330087" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:26.025ex; height:5.009ex;" alt="F=G{\frac {m_{1}m_{2}}{r^{2}}}+B{\frac {m_{1}m_{2}}{r^{3}}}\ " /></SPAN>, <SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.774ex; height:2.176ex;" alt="B" /></SPAN> a constant</DD>
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<P>attempting to explain the Moon's apsidal motion. Other extensions were proposed by Laplace (around 1790) and Decombes (1913):<SUP id="cite_ref-35" class="reference"><A href="#cite_note-35">[34]</A></SUP></P>
<DL>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
  <semantics>
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    <annotation encoding="application/x-tex">{\displaystyle F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\exp(-\alpha r)}</annotation>
  </semantics>
</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a0717b81f146705e4d87c6c0e1e8470a4f38f49" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:26.209ex; height:5.009ex;" alt="F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\exp(-\alpha r)" /></SPAN> (Laplace)</DD>
<DD><SPAN class="mwe-math-element"><SPAN class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML">
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    <annotation encoding="application/x-tex">{\displaystyle F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\left(1+{\alpha  \over {r^{3}}}\right)}</annotation>
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</math></SPAN><IMG src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06cd8410781e1a0a4a4646529cf1a9dd601b4c84" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:26.842ex; height:6.176ex;" alt="F(r)=k{\frac {m_{1}m_{2}}{r^{2}}}\left(1+{\alpha  \over {r^{3}}}\right)" /></SPAN> (Decombes)</DD>
</DL>
<P>In recent years, quests for non-inverse square terms in the law of gravity have been carried out by <A href="/wiki/Neutron_interferometry" class="mw-redirect" title="Neutron interferometry">neutron interferometry</A>.<SUP id="cite_ref-36" class="reference"><A href="#cite_note-36">[35]</A></SUP></P>
<H2><SPAN id="Solutions_of_Newton's_law_of_universal_gravitation"></SPAN><SPAN class="mw-headline" id="Solutions_of_Newton.27s_law_of_universal_gravitation">Solutions of Newton's law of universal gravitation</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=18" title="Edit section: Solutions of Newton's law of universal gravitation">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV role="note" class="hatnote navigation-not-searchable">Main article: <A href="/wiki/N-body_problem" title="N-body problem">n-body problem</A></DIV>
<P>The <I>n</I>-body problem is an ancient, classical problem<SUP id="cite_ref-Leimanis_and_Minorsky_37-0" class="reference"><A href="#cite_note-Leimanis_and_Minorsky-37">[36]</A></SUP> of predicting the individual motions of a group of <A href="/wiki/Astronomical_object" title="Astronomical object">celestial objects</A> interacting with each other <A href="/wiki/Gravitation" class="mw-redirect" title="Gravitation">gravitationally</A>. Solving this problem — from the time of the Greeks and on — has been motivated by the desire to understand the motions of the <A href="/wiki/Sun" title="Sun">Sun</A>, <A href="/wiki/Planet" title="Planet">planets</A> and the visible <A href="/wiki/Star" title="Star">stars</A>. In the 20th century, understanding the dynamics of <A href="/wiki/Globular_cluster" title="Globular cluster">globular cluster</A> star systems became an important <I>n</I>-body problem too.<SUP id="cite_ref-Heggie_and_Hut_38-0" class="reference"><A href="#cite_note-Heggie_and_Hut-38">[37]</A></SUP> The <I>n</I>-body problem in <A href="/wiki/General_relativity" title="General relativity">general relativity</A> is considerably more difficult to solve.</P>
<P>The classical physical problem can be informally stated as: <I>given the quasi-steady orbital properties</I> (<I>instantaneous position, velocity and time</I>)<SUP id="cite_ref-39" class="reference"><A href="#cite_note-39">[38]</A></SUP> <I>of a group of celestial bodies, predict their interactive forces; and consequently, predict their true orbital motions for all future times</I>.<SUP id="cite_ref-40" class="reference"><A href="#cite_note-40">[39]</A></SUP></P>
<P>The <A href="/wiki/Two-body_problem" title="Two-body problem">two-body problem</A> has been completely solved, as has the restricted <A href="/wiki/Three-body_problem" title="Three-body problem">three-body problem</A>.<SUP id="cite_ref-41" class="reference"><A href="#cite_note-41">[40]</A></SUP></P>
<H2><SPAN class="mw-headline" id="See_also">See also</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=19" title="Edit section: See also">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
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<H2><SPAN class="mw-headline" id="Notes">Notes</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=20" title="Edit section: Notes">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
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<LI id="cite_note-1"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-1"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">It was shown separately that large, spherically symmetrical masses attract and are attracted <A href="/wiki/Shell_theorem" title="Shell theorem">as if all their mass were concentrated at their centers</A>.</SPAN></LI>
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<H2><SPAN class="mw-headline" id="References">References</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=21" title="Edit section: References">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<DIV class="reflist columns references-column-width" style="-moz-column-width: 30em; -webkit-column-width: 30em; column-width: 30em; list-style-type: decimal;">
<OL class="references">
<LI id="cite_note-2"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-2"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Isaac Newton: "In [experimental] philosophy particular propositions are inferred from the phenomena and afterwards rendered general by induction": "<A href="/wiki/Philosophiae_Naturalis_Principia_Mathematica" class="mw-redirect" title="Philosophiae Naturalis Principia Mathematica">Principia</A>", Book 3, General Scholium, at p.392 in Volume 2 of Andrew Motte's English translation published 1729.</SPAN></LI>
<LI id="cite_note-Newton1-3"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Newton1_3-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Newton1_3-1"><SUP><I><B>b</B></I></SUP></A> <A href="#cite_ref-Newton1_3-2"><SUP><I><B>c</B></I></SUP></A></SPAN> <SPAN class="reference-text">- Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne Whitman, translators: <A href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</A>, <I>The Principia</I>: <A href="/wiki/Mathematical_Principles_of_Natural_Philosophy" class="mw-redirect" title="Mathematical Principles of Natural Philosophy">Mathematical Principles of Natural Philosophy</A>. Preceded by <I>A Guide to Newton's Principia</I>, by I.Bernard Cohen. University of California Press 1999 <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-520-08816-6" title="Special:BookSources/0-520-08816-6">0-520-08816-6</A> <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-520-08817-4" title="Special:BookSources/0-520-08817-4">0-520-08817-4</A></SPAN></LI>
<LI id="cite_note-The_Michell-Cavendish_Experiment-4"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-The_Michell-Cavendish_Experiment_4-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-The_Michell-Cavendish_Experiment_4-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external text" href="http://www.public.iastate.edu/~lhodges/Michell.htm">The Michell-Cavendish Experiment</A>, Laurent Hodges</SPAN></LI>
<LI id="cite_note-Bullialdus_1645-5"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Bullialdus_1645_5-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Bullialdus_1645_5-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">Bullialdus (Ismael Bouillau) (1645), "Astronomia philolaica", Paris, 1645.</SPAN></LI>
<LI id="cite_note-Borelli.2C_G._A._1666-6"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-Borelli.2C_G._A._1666_6-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-Borelli.2C_G._A._1666_6-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">Borelli, G. A., "Theoricae Mediceorum Planetarum ex causis physicis deductae", Florence, 1666.</SPAN></LI>
<LI id="cite_note-T_Whiteside_1970_pages_5-19-7"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-T_Whiteside_1970_pages_5-19_7-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-T_Whiteside_1970_pages_5-19_7-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">D T Whiteside, "Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664-1684", Journal for the History of Astronomy, i (1970), pages 5-19; especially at page 13.</SPAN></LI>
<LI id="cite_note-1686letters-8"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-1686letters_8-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), giving the Halley-Newton correspondence of May to July 1686 about Hooke's claims at pp.431-448, see particularly page 431.</SPAN></LI>
<LI id="cite_note-attempt-9"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-attempt_9-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-attempt_9-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">Hooke's 1674 statement in "An Attempt to Prove the Motion of the Earth from Observations" is available in <A rel="nofollow" class="external text" href="http://echo.mpiwg-berlin.mpg.de/ECHOdocuView/ECHOzogiLib?mode=imagepath&url=/mpiwg/online/permanent/library/XXTBUC3U/pageimg">online facsimile here</A>.</SPAN></LI>
<LI id="cite_note-10"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-10"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book">Purrington, Robert D. (2009). <A rel="nofollow" class="external text" href="https://books.google.com/books?id=tJu97S3BtGIC"><I>The First Professional Scientist: Robert Hooke and the Royal Society of London</I></A>. Springer. p. 168. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/3-0346-0036-4" title="Special:BookSources/3-0346-0036-4">3-0346-0036-4</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+First+Professional+Scientist%3A+Robert+Hooke+and+the+Royal+Society+of+London&rft.pages=168&rft.pub=Springer&rft.date=2009&rft.isbn=3-0346-0036-4&rft.aulast=Purrington&rft.aufirst=Robert+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DtJu97S3BtGIC&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN> <A rel="nofollow" class="external text" href="https://books.google.com/books?id=tJu97S3BtGIC&pg=PA168">Extract of page 168</A></SPAN></LI>
<LI id="cite_note-11"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-11"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See page 239 in Curtis Wilson (1989), "The Newtonian achievement in astronomy", ch.13 (pages 233-274) in "Planetary astronomy from the Renaissance to the rise of astrophysics: 2A: Tycho Brahe to Newton", CUP 1989.</SPAN></LI>
<LI id="cite_note-12"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-12"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A href="/wiki/Calendar_(New_Style)_Act_1750" title="Calendar (New Style) Act 1750">Calendar (New Style) Act 1750</A></SPAN></LI>
<LI id="cite_note-13"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-13"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Page 309 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #239.</SPAN></LI>
<LI id="cite_note-14"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-14"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See Curtis Wilson (1989) at page 244.</SPAN></LI>
<LI id="cite_note-15"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-15"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Page 297 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #235, 24 November 1679.</SPAN></LI>
<LI id="cite_note-16"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-16"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Page 433 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #286, 27 May 1686.</SPAN></LI>
<LI id="cite_note-june1686-17"><SPAN class="mw-cite-backlink">^ <A href="#cite_ref-june1686_17-0"><SPAN class="cite-accessibility-label">Jump up to: </SPAN><SUP><I><B>a</B></I></SUP></A> <A href="#cite_ref-june1686_17-1"><SUP><I><B>b</B></I></SUP></A></SPAN> <SPAN class="reference-text">Pages 435-440 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol 2 (1676-1687), (Cambridge University Press, 1960), document #288, 20 June 1686.</SPAN></LI>
<LI id="cite_note-18"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-18"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Page 436, Correspondence, Vol.2, already cited.</SPAN></LI>
<LI id="cite_note-19"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-19"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Propositions 70 to 75 in Book 1, for example in the 1729 English translation of the <I>Principia</I>, <A rel="nofollow" class="external text" href="https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA263#v=onepage&q=&f=false">start at page 263</A>.</SPAN></LI>
<LI id="cite_note-20"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-20"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Propositions 43 to 45 in Book 1, in the 1729 English translation of the <I>Principia</I>, <A rel="nofollow" class="external text" href="https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA177#v=onepage&q=&f=false">start at page 177</A>.</SPAN></LI>
<LI id="cite_note-21"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-21"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">D T Whiteside, "The pre-history of the 'Principia' from 1664 to 1686", Notes and Records of the Royal Society of London, 45 (1991), pages 11-61; especially at 13-20.</SPAN></LI>
<LI id="cite_note-22"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-22"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See J. Bruce Brackenridge, "The key to Newton's dynamics: the Kepler problem and the Principia", (University of California Press, 1995), especially at <A rel="nofollow" class="external text" href="https://books.google.com/books?id=ovOTK7X_mMkC&pg=PA20#v=onepage&q=&f=false">pages 20-21</A>.</SPAN></LI>
<LI id="cite_note-23"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-23"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See for example the 1729 English translation of the <I>Principia</I>, <A rel="nofollow" class="external text" href="https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PA66#v=onepage&q=&f=false">at page 66</A>.</SPAN></LI>
<LI id="cite_note-dtw1970-24"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-dtw1970_24-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See page 10 in D T Whiteside, "Before the Principia: the maturing of Newton's thoughts on dynamical astronomy, 1664-1684", Journal for the History of Astronomy, i (1970), pages 5-19.</SPAN></LI>
<LI id="cite_note-25"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-25"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Discussion points can be seen for example in the following papers: N Guicciardini, "Reconsidering the Hooke-Newton debate on Gravitation: Recent Results", in Early Science and Medicine, 10 (2005), 511-517; Ofer Gal, "The Invention of Celestial Mechanics", in Early Science and Medicine, 10 (2005), 529-534; M Nauenberg, "Hooke's and Newton's Contributions to the Early Development of Orbital mechanics and Universal Gravitation", in Early Science and Medicine, 10 (2005), 518-528.</SPAN></LI>
<LI id="cite_note-26"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-26"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See for example the results of Propositions 43-45 and 70-75 in Book 1, cited above.</SPAN></LI>
<LI id="cite_note-27"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-27"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See also G E Smith, in Stanford Encyclopedia of Philosophy, <A rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/newton-principia/">"Newton's Philosophiae Naturalis Principia Mathematica"</A>.</SPAN></LI>
<LI id="cite_note-28"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-28"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The second extract is quoted and translated in W.W. Rouse Ball, "An Essay on Newton's 'Principia'" (London and New York: Macmillan, 1893), at page 69.</SPAN></LI>
<LI id="cite_note-29"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-29"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">The original statements by Clairaut (in French) are found (with orthography here as in the original) in "Explication abregée du systême du monde, et explication des principaux phénomenes astronomiques tirée des Principes de M. Newton" (1759), at Introduction (section IX), page 6: "Il ne faut pas croire que cette idée ... de Hook diminue la gloire de M. Newton", [and] "L'exemple de Hook" [serve] "à faire voir quelle distance il y a entre une vérité entrevue & une vérité démontrée".</SPAN></LI>
<LI id="cite_note-30"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-30"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation journal">Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). <A rel="nofollow" class="external text" href="http://physics.nist.gov/cuu/Constants/index.html">"CODATA Recommended Values of the Fundamental Physical Constants: 2006"</A>. <I><A href="/wiki/Reviews_of_Modern_Physics" title="Reviews of Modern Physics">Reviews of Modern Physics</A></I>. <B>80</B> (2): 633–730. <A href="/wiki/ArXiv" title="ArXiv">arXiv</A>:<SPAN class="plainlinks"><A rel="nofollow" class="external text" href="//arxiv.org/abs/0801.0028">0801.0028</A> <IMG alt="Freely accessible" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" title="Freely accessible" width="9" height="14" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></SPAN>. <A href="/wiki/Bibcode" title="Bibcode">Bibcode</A>:<A rel="nofollow" class="external text" href="http://adsabs.harvard.edu/abs/2008RvMP...80..633M">2008RvMP...80..633M</A>. <A href="/wiki/Digital_object_identifier" title="Digital object identifier">doi</A>:<A rel="nofollow" class="external text" href="//doi.org/10.1103%2FRevModPhys.80.633">10.1103/RevModPhys.80.633</A>.</CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Reviews+of+Modern+Physics&rft.atitle=CODATA+Recommended+Values+of+the+Fundamental+Physical+Constants%3A+2006&rft.volume=80&rft.issue=2&rft.pages=633-730&rft.date=2008&rft_id=info%3Aarxiv%2F0801.0028&rft_id=info%3Adoi%2F10.1103%2FRevModPhys.80.633&rft_id=info%3Abibcode%2F2008RvMP...80..633M&rft.aulast=Mohr&rft.aufirst=Peter+J.&rft.au=Taylor%2C+Barry+N.&rft.au=Newell%2C+David+B.&rft_id=http%3A%2F%2Fphysics.nist.gov%2Fcuu%2FConstants%2Findex.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN> <A rel="nofollow" class="external text" href="http://www.physics.nist.gov/cgi-bin/cuu/Value?bg">Direct link to value</A>..</SPAN></LI>
<LI id="cite_note-31"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-31"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external text" href="http://farside.ph.utexas.edu/teaching/336k/lectures/node109.html">Equilibrium State</A></SPAN></LI>
<LI id="cite_note-32"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-32"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><CITE class="citation book"><A href="/wiki/Charles_W._Misner" title="Charles W. Misner">Misner, Charles W.</A>; <A href="/wiki/Kip_Thorne" title="Kip Thorne">Thorne, Kip S.</A>; <A href="/wiki/John_Archibald_Wheeler" title="John Archibald Wheeler">Wheeler, John Archibald</A> (1973). <I>Gravitation</I>. New York: W. H.Freeman and Company. <A href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</A> <A href="/wiki/Special:BookSources/0-7167-0344-0" title="Special:BookSources/0-7167-0344-0">0-7167-0344-0</A></CITE><SPAN title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Gravitation&rft.place=New+York&rft.pub=W.+H.Freeman+and+Company&rft.date=1973&rft.isbn=0-7167-0344-0&rft.aulast=Misner&rft.aufirst=Charles+W.&rft.au=Thorne%2C+Kip+S.&rft.au=Wheeler%2C+John+Archibald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANewton%27s+law+of+universal+gravitation" class="Z3988"><SPAN style="display:none;"> </SPAN></SPAN> Page 1049.</SPAN></LI>
<LI id="cite_note-33"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-33"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">- <A href="/wiki/Max_Born" title="Max Born">Max Born</A> (1924), <I>Einstein's Theory of Relativity</I> (The 1962 Dover edition, page 348 lists a table documenting the observed and calculated values for the precession of the perihelion of Mercury, Venus, and the Earth.)</SPAN></LI>
<LI id="cite_note-34"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-34"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">- <I>The Construction of Modern Science: Mechanisms and Mechanics</I>, by Richard S. Westfall. Cambridge University Press. 1978</SPAN></LI>
<LI id="cite_note-35"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-35"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external free" href="http://physicsessays.org/doi/abs/10.4006/1.3038751?journalCode=phes">http://physicsessays.org/doi/abs/10.4006/1.3038751?journalCode=phes</A><SUP class="noprint Inline-Template"><SPAN style="white-space: nowrap;">[<I><A href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><SPAN title=" Dead link since August 2017">dead link</SPAN></A></I>]</SPAN></SUP></SPAN></LI>
<LI id="cite_note-36"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-36"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><A rel="nofollow" class="external free" href="http://journals.aps.org/prc/abstract/10.1103/PhysRevC.75.015501">http://journals.aps.org/prc/abstract/10.1103/PhysRevC.75.015501</A></SPAN></LI>
<LI id="cite_note-Leimanis_and_Minorsky-37"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Leimanis_and_Minorsky_37-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">Leimanis and Minorsky: Our interest is with Leimanis, who first discusses some history about the <I>n</I>-body problem, especially Ms. Kovalevskaya's ~1868-1888, twenty-year complex-variables approach, failure; <B>Section 1: The Dynamics of Rigid Bodies and Mathematical Exterior Ballistics</B> (Chapter 1, <I>the motion of a rigid body about a fixed point</I> (<B>Euler</B> and <B>Poisson</B> <I>equations</I>); Chapter 2, <I>Mathematical Exterior Ballistics</I>), good precursor background to the <I>n</I>-body problem; <B>Section 2: Celestial Mechanics</B> (Chapter 1, <I>The Uniformization of the Three-body Problem</I> (Restricted Three-body Problem); Chapter 2, <I>Capture in the Three-Body Problem</I>; Chapter 3, <I>Generalized n-body Problem</I>).</SPAN></LI>
<LI id="cite_note-Heggie_and_Hut-38"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-Heggie_and_Hut_38-0"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">See References sited for Heggie and Hut. This Wikipedia page has made their approach obsolete.</SPAN></LI>
<LI id="cite_note-39"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-39"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text"><I>Quasi-steady</I> loads refers to the instantaneous inertial loads generated by instantaneous angular velocities and accelerations, as well as translational accelerations (9 variables). It is as though one took a photograph, which also recorded the instantaneous position and properties of motion. In contrast, a <I>steady-state</I> condition refers to a system's state being invariant to time; otherwise, the first derivatives and all higher derivatives are zero.</SPAN></LI>
<LI id="cite_note-40"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-40"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">R. M. Rosenberg states the <I>n</I>-body problem similarly (see References): <I>Each particle in a system of a finite number of particles is subjected to a Newtonian gravitational attraction from all the other particles, and to no other forces. If the initial state of the system is given, how will the particles move?</I> Rosenberg failed to realize, like everyone else, that it is necessary to determine the forces <I>first</I> before the motions can be determined.</SPAN></LI>
<LI id="cite_note-41"><SPAN class="mw-cite-backlink"><B><A href="#cite_ref-41"><SPAN class="cite-accessibility-label">Jump up </SPAN>^</A></B></SPAN> <SPAN class="reference-text">A general, classical solution in terms of first integrals is known to be impossible. An exact theoretical solution for arbitrary <I>n</I> can be approximated via <A href="/wiki/Taylor_series" title="Taylor series">Taylor series</A>, but in practice such an <A href="/wiki/Infinite_series" class="mw-redirect" title="Infinite series">infinite series</A> must be truncated, giving at best only an approximate solution; and an approach now obsolete. In addition, the <I>n</I>-body problem may be solved using <A href="/wiki/Numerical_integration" title="Numerical integration">numerical integration</A>, but these, too, are approximate solutions; and again obsolete. See Sverre J. Aarseth's book <B>Gravitational N-body Simulations</B> listed in the References.</SPAN></LI>
</OL>
</DIV>
<H2><SPAN class="mw-headline" id="External_links">External links</SPAN><SPAN class="mw-editsection"><SPAN class="mw-editsection-bracket">[</SPAN><A href="/w/index.php?title=Newton%27s_law_of_universal_gravitation&action=edit&section=22" title="Edit section: External links">edit</A><SPAN class="mw-editsection-bracket">]</SPAN></SPAN></H2>
<UL>
<LI><A rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=5C5_dOEyAfk"><SPAN class="plainlinks">Feather & Hammer Drop on Moon</SPAN></A> on <A href="/wiki/YouTube" title="YouTube">YouTube</A></LI>
<LI><A rel="nofollow" class="external text" href="http://www.pythia.com.ar/?id=gravlaw">Newton‘s Law of Universal Gravitation Javascript calculator</A></LI>
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<DIV id="Isaac_Newton" style="font-size:114%;margin:0 4em"><A href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</A></DIV>
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<TH scope="row" class="navbox-group" style="width:1%">Publications</TH>
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<DIV style="padding:0em 0.25em">
<UL>
<LI><I><A href="/wiki/De_analysi_per_aequationes_numero_terminorum_infinitas" title="De analysi per aequationes numero terminorum infinitas">De analysi per aequationes numero terminorum infinitas</A></I> (1669, published 1711)</LI>
<LI><I><A href="/wiki/Method_of_Fluxions" title="Method of Fluxions">Method of Fluxions</A></I> (1671)</LI>
<LI><I><A href="/wiki/De_motu_corporum_in_gyrum" title="De motu corporum in gyrum">De motu corporum in gyrum</A></I> (1684)</LI>
<LI><I><A href="/wiki/Philosophi%C3%A6_Naturalis_Principia_Mathematica" title="Philosophiæ Naturalis Principia Mathematica">Philosophiæ Naturalis Principia Mathematica</A></I> (1687)
<UL>
<LI><A href="/wiki/General_Scholium" title="General Scholium">General Scholium</A> (1713)</LI>
</UL>
</LI>
<LI><I><A href="/wiki/Opticks" title="Opticks">Opticks</A></I> (1704)</LI>
<LI><I><A href="/wiki/The_Queries" title="The Queries">The Queries</A></I> (1704)</LI>
<LI><I><A href="/wiki/Arithmetica_Universalis" title="Arithmetica Universalis">Arithmetica Universalis</A></I> (1707)</LI>
</UL>
</DIV>
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</TR>
<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">Other writings</TH>
<TD class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">
<UL>
<LI><I><A href="/wiki/Notes_on_the_Jewish_Temple" title="Notes on the Jewish Temple">Notes on the Jewish Temple</A></I></LI>
<LI><A href="/wiki/Quaestiones_quaedam_philosophicae" title="Quaestiones quaedam philosophicae">Quaestiones quaedam philosophicae</A></LI>
<LI><I><A href="/wiki/The_Chronology_of_Ancient_Kingdoms_Amended" title="The Chronology of Ancient Kingdoms Amended">The Chronology of Ancient Kingdoms Amended</A></I> (1728)</LI>
<LI><I><A href="/wiki/An_Historical_Account_of_Two_Notable_Corruptions_of_Scripture" title="An Historical Account of Two Notable Corruptions of Scripture">An Historical Account of Two Notable Corruptions of Scripture</A></I> (1754)</LI>
</UL>
</DIV>
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<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">Discoveries    <BR />
and inventions</TH>
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<DIV style="padding:0em 0.25em">
<UL>
<LI><A href="/wiki/Calculus" title="Calculus">Calculus</A></LI>
<LI><A href="/wiki/Newton_disc" title="Newton disc">Newton disc</A></LI>
<LI><A href="/wiki/Newton_polygon" title="Newton polygon">Newton polygon</A>
<UL>
<LI><A href="/wiki/Newton%E2%80%93Okounkov_body" title="Newton–Okounkov body">Newton–Okounkov body</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Newton%27s_reflector" title="Newton's reflector">Newton's reflector</A></LI>
<LI><A href="/wiki/Newtonian_telescope" title="Newtonian telescope">Newtonian telescope</A></LI>
<LI><A href="/wiki/Newton_scale" title="Newton scale">Newton scale</A></LI>
<LI><A href="/wiki/Newton%27s_metal" title="Newton's metal">Newton's metal</A></LI>
<LI><A href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</A></LI>
<LI><A href="/wiki/Sextant" title="Sextant">Sextant</A></LI>
</UL>
</DIV>
</TD>
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<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">Theory    <BR />
expansions</TH>
<TD class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">
<UL>
<LI><A href="/wiki/Kepler%27s_laws_of_planetary_motion" title="Kepler's laws of planetary motion">Kepler's laws of planetary motion</A></LI>
<LI><A href="/wiki/Problem_of_Apollonius" title="Problem of Apollonius">Problem of Apollonius</A></LI>
</UL>
</DIV>
</TD>
</TR>
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<TH scope="row" class="navbox-group" style="width:1%"><A href="/wiki/Newtonianism" title="Newtonianism">Newtonianism</A></TH>
<TD class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
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<UL>
<LI><A href="/wiki/Bucket_argument" title="Bucket argument">Bucket argument</A></LI>
<LI><A href="/wiki/Newton%27s_inequalities" title="Newton's inequalities">Newton's inequalities</A></LI>
<LI><A href="/wiki/Newton%27s_law_of_cooling" title="Newton's law of cooling">Newton's law of cooling</A></LI>
<LI><A class="mw-selflink selflink">Newton's law of universal gravitation</A>
<UL>
<LI><A href="/wiki/Post-Newtonian_expansion" title="Post-Newtonian expansion">Post-Newtonian expansion</A></LI>
<LI><A href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Parameterized post-Newtonian formalism</A></LI>
<LI><A href="/wiki/Newton%E2%80%93Cartan_theory" title="Newton–Cartan theory">Newton–Cartan theory</A></LI>
<LI><A href="/wiki/Schr%C3%B6dinger%E2%80%93Newton_equation" title="Schrödinger–Newton equation">Schrödinger–Newton equation</A></LI>
<LI><A href="/wiki/Gravitational_constant" title="Gravitational constant">Gravitational constant</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws of motion</A>
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<LI><A href="/wiki/Newtonian_dynamics" title="Newtonian dynamics">Newtonian dynamics</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Newton%27s_method_in_optimization" title="Newton's method in optimization">Newton's method in optimization</A>
<UL>
<LI><A href="/wiki/Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton algorithm</A></LI>
<LI><A href="/wiki/Truncated_Newton_method" title="Truncated Newton method">Truncated Newton method</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Newton%27s_rings" title="Newton's rings">Newton's rings</A></LI>
<LI><A href="/wiki/Newton%27s_theorem_about_ovals" title="Newton's theorem about ovals">Newton's theorem about ovals</A></LI>
<LI><A href="/wiki/Newton%E2%80%93Pepys_problem" title="Newton–Pepys problem">Newton–Pepys problem</A></LI>
<LI><A href="/wiki/Newtonian_potential" title="Newtonian potential">Newtonian potential</A></LI>
<LI><A href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</A></LI>
<LI><A href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</A></LI>
<LI><A href="/wiki/Newtonian_fluid" title="Newtonian fluid">Newtonian fluid</A></LI>
<LI><A href="/wiki/Corpuscular_theory_of_light" title="Corpuscular theory of light">Corpuscular theory of light</A></LI>
<LI><A href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton calculus controversy</A></LI>
<LI><A href="/wiki/Newton%27s_notation" class="mw-redirect" title="Newton's notation">Newton's notation</A></LI>
<LI><A href="/wiki/Rotating_spheres" title="Rotating spheres">Rotating spheres</A></LI>
<LI><A href="/wiki/Newton%27s_cannonball" title="Newton's cannonball">Newton's cannonball</A></LI>
<LI><A href="/wiki/Newton%E2%80%93Cotes_formulas" title="Newton–Cotes formulas">Newton–Cotes formulas</A></LI>
<LI><A href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</A>
<UL>
<LI><A href="/wiki/Newton_fractal" title="Newton fractal">Newton fractal</A></LI>
<LI><A href="/wiki/Generalized_Gauss%E2%80%93Newton_method" title="Generalized Gauss–Newton method">Generalized Gauss–Newton method</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Newton%27s_identities" title="Newton's identities">Newton's identities</A></LI>
<LI><A href="/wiki/Newton_polynomial" title="Newton polynomial">Newton polynomial</A></LI>
<LI><A href="/wiki/Newton%27s_theorem_of_revolving_orbits" title="Newton's theorem of revolving orbits">Newton's theorem of revolving orbits</A></LI>
<LI><A href="/wiki/Newton%E2%80%93Euler_equations" title="Newton–Euler equations">Newton–Euler equations</A></LI>
<LI>Newton number
<UL>
<LI><A href="/wiki/Kissing_number_problem" title="Kissing number problem">Kissing number problem</A></LI>
<LI><A href="/wiki/Power_number" title="Power number">Power number</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Difference_quotient" title="Difference quotient">Newton's quotient</A></LI>
<LI><A href="/wiki/Puiseux_series" title="Puiseux series">Newton–Puiseux theorem</A></LI>
<LI><A href="/wiki/Solar_mass" title="Solar mass">Solar mass</A></LI>
<LI><A href="/wiki/Dynamics_(mechanics)" title="Dynamics (mechanics)">Dynamics</A></LI>
<LI><A href="/wiki/Absolute_space_and_time" title="Absolute space and time">Absolute space and time</A></LI>
<LI><A href="/wiki/Finite_difference" title="Finite difference">Finite difference</A>
<UL>
<LI><A href="/wiki/Table_of_Newtonian_series" title="Table of Newtonian series">Table of Newtonian series</A></LI>
</UL>
</LI>
<LI><A href="/wiki/Impact_depth" title="Impact depth">Impact depth</A></LI>
<LI><A href="/wiki/Structural_coloration" title="Structural coloration">Structural coloration</A></LI>
<LI><A href="/wiki/Inertia" title="Inertia">Inertia</A></LI>
</UL>
</DIV>
</TD>
</TR>
<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">Phrases</TH>
<TD class="navbox-list navbox-even hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">
<UL>
<LI>"<A href="/wiki/Hypotheses_non_fingo" title="Hypotheses non fingo">Hypotheses non fingo</A>"</LI>
<LI>"<A href="/wiki/Standing_on_the_shoulders_of_giants" title="Standing on the shoulders of giants">Standing on the shoulders of giants</A>"</LI>
</UL>
</DIV>
</TD>
</TR>
<TR style="display: none;">
<TH scope="row" class="navbox-group" style="width:1%">Life</TH>
<TD class="navbox-list navbox-odd hlist" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px">
<DIV style="padding:0em 0.25em">
<UL>
<LI><A href="/wiki/Cranbury_Park" title="Cranbury Park">Cranbury Park</A></LI>
<LI><A href="/wiki/Woolsthorpe_Manor" title="Woolsthorpe Manor">Woolsthorpe Manor</A></LI>
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<LI><A href="/wiki/Isaac_Newton_in_popular_culture" title="Isaac Newton in popular culture">In popular culture</A></LI>
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<TH scope="row" class="navbox-group" style="width:1%">Related</TH>
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<LI><A href="/wiki/Writing_of_Principia_Mathematica" title="Writing of Principia Mathematica">Writing of <I>Principia Mathematica</I></A></LI>
<LI><A href="/wiki/List_of_things_named_after_Isaac_Newton" title="List of things named after Isaac Newton">List of things names after Newton</A>
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<LI><A href="/wiki/Isaac_Newton_Institute" title="Isaac Newton Institute">Isaac Newton Institute</A></LI>
<LI><A href="/wiki/Isaac_Newton_Medal" title="Isaac Newton Medal">Isaac Newton Medal</A></LI>
<LI><A href="/wiki/Isaac_Newton_Group_of_Telescopes" title="Isaac Newton Group of Telescopes">Isaac Newton Group of Telescopes</A></LI>
<LI><A href="/wiki/Isaac_Newton_Telescope" title="Isaac Newton Telescope">Isaac Newton Telescope</A></LI>
<LI><A href="/wiki/Newton_(unit)" title="Newton (unit)">Newton (unit)</A></LI>
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<LI><I><A href="/wiki/Elements_of_the_Philosophy_of_Newton" title="Elements of the Philosophy of Newton">Elements of the Philosophy of Newton</A></I></LI>
<LI><I><A href="/wiki/Isaac_Newton_S/O_Philipose" title="Isaac Newton S/O Philipose">Isaac Newton S/O Philipose</A></I></LI>
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<LI><A class="mw-selflink selflink">Newton's law of universal gravitation</A></LI>
<LI><A href="/wiki/History_of_gravitational_theory" title="History of gravitational theory">History of gravitational theory</A></LI>
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<TH scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><A href="/wiki/General_relativity" title="General relativity">General relativity (GR)</A></TH>
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<LI><A href="/wiki/Introduction_to_general_relativity" title="Introduction to general relativity">Introduction</A></LI>
<LI><A href="/wiki/History_of_general_relativity" title="History of general relativity">History</A></LI>
<LI><A href="/wiki/Mathematics_of_general_relativity" title="Mathematics of general relativity">Mathematics</A></LI>
<LI><A href="/wiki/General_relativity#Further_reading" title="General relativity">Resources</A></LI>
<LI><A href="/wiki/Tests_of_general_relativity" title="Tests of general relativity">Tests</A></LI>
<LI><A href="/wiki/Parameterized_post-Newtonian_formalism" title="Parameterized post-Newtonian formalism">Post-Newtonian formalism</A></LI>
<LI><A href="/wiki/Linearized_gravity" title="Linearized gravity">Linearized gravity</A></LI>
<LI><A href="/wiki/ADM_formalism" title="ADM formalism">ADM formalism</A></LI>
<LI><A href="/wiki/Gibbons%E2%80%93Hawking%E2%80%93York_boundary_term" title="Gibbons–Hawking–York boundary term">Gibbons–Hawking–York boundary term</A></LI>
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general relativity</A></TH>
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<TH scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Paradigms</TH>
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<UL>
<LI><A href="/wiki/Alternatives_to_general_relativity" title="Alternatives to general relativity">Classical theories of gravitation</A></LI>
<LI><A href="/wiki/Quantum_gravity" title="Quantum gravity">Quantum gravity</A></LI>
<LI><A href="/wiki/Theory_of_everything" title="Theory of everything">Theory of everything</A></LI>
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<TH scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Classical</TH>
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<UL>
<LI><A href="/wiki/Einstein%E2%80%93Cartan_theory" title="Einstein–Cartan theory">Einstein–Cartan</A></LI>
<LI><A href="/wiki/Bimetric_gravity" title="Bimetric gravity">Bimetric theories</A></LI>
<LI><A href="/wiki/Gauge_theory_gravity" title="Gauge theory gravity">Gauge theory gravity</A></LI>
<LI><A href="/wiki/Teleparallelism" title="Teleparallelism">Teleparallelism</A></LI>
<LI><A href="/wiki/Composite_gravity" title="Composite gravity">Composite gravity</A></LI>
<LI><A href="/wiki/F(R)_gravity" title="F(R) gravity"><I>f</I>(<I>R</I>) gravity</A></LI>
<LI><A href="/wiki/Infinite_Derivative_Gravity" class="mw-redirect" title="Infinite Derivative Gravity">Infinite Derivative Gravity</A></LI>
<LI><A href="/wiki/Massive_gravity" title="Massive gravity">Massive gravity</A></LI>
<LI><A href="/wiki/Modified_Newtonian_dynamics" title="Modified Newtonian dynamics">Modified Newtonian dynamics (MOND)</A></LI>
<LI><A href="/wiki/Nonsymmetric_gravitational_theory" title="Nonsymmetric gravitational theory">Nonsymmetric gravitation</A></LI>
<LI><A href="/wiki/Scalar%E2%80%93tensor_theory" title="Scalar–tensor theory">Scalar–tensor theories</A>
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<LI><SMALL><A href="/wiki/Brans%E2%80%93Dicke_theory" title="Brans–Dicke theory">Brans–Dicke</A></SMALL></LI>
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<LI><A href="/wiki/Scalar%E2%80%93tensor%E2%80%93vector_gravity" title="Scalar–tensor–vector gravity">Scalar–tensor–vector</A></LI>
<LI><A href="/wiki/Conformal_gravity" title="Conformal gravity">Conformal gravity</A></LI>
<LI><A href="/wiki/Scalar_theories_of_gravitation" title="Scalar theories of gravitation">Scalar theories</A>
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<LI><SMALL><A href="/wiki/Nordstr%C3%B6m%27s_theory_of_gravitation" title="Nordström's theory of gravitation">Nordström</A></SMALL></LI>
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<LI><A href="/wiki/Whitehead%27s_theory_of_gravitation" title="Whitehead's theory of gravitation">Whitehead</A></LI>
<LI><A href="/wiki/Geometrodynamics" title="Geometrodynamics">Geometrodynamics</A></LI>
<LI><A href="/wiki/Induced_gravity" title="Induced gravity">Induced gravity</A></LI>
<LI><A href="/wiki/Tensor%E2%80%93vector%E2%80%93scalar_gravity" title="Tensor–vector–scalar gravity">Tensor–vector–scalar</A></LI>
<LI><A href="/wiki/Chameleon_particle" title="Chameleon particle">Chameleon</A></LI>
<LI><A href="/wiki/Pressuron" title="Pressuron">Pressuron</A></LI>
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<LI><A href="/wiki/Euclidean_quantum_gravity" title="Euclidean quantum gravity">Euclidean quantum gravity</A></LI>
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<LI><SMALL><A href="/wiki/Wheeler%E2%80%93DeWitt_equation" title="Wheeler–DeWitt equation">Wheeler–DeWitt equation</A></SMALL></LI>
<LI><SMALL><A href="/wiki/Loop_quantum_gravity" title="Loop quantum gravity">Loop quantum gravity</A></SMALL></LI>
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<LI><A href="/wiki/Causal_dynamical_triangulation" title="Causal dynamical triangulation">Causal dynamical triangulation</A></LI>
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<LI><A href="/wiki/DGP_model" title="DGP model">DGP model</A></LI>
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<LI><A href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</A>
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<TH scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Unification and<BR />
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<LI><A href="/wiki/Noncommutative_quantum_field_theory" title="Noncommutative quantum field theory">Noncommutative geometry</A></LI>
<LI><A href="/wiki/Semiclassical_gravity" title="Semiclassical gravity">Semiclassical gravity</A></LI>
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<LI><A href="/wiki/String_theory" title="String theory">String theory</A>
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<LI><SMALL><A href="/wiki/M-theory" title="M-theory">M-theory</A></SMALL></LI>
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<LI><SMALL><A href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string theory</A></SMALL></LI>
<LI><SMALL><A href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string theory</A></SMALL></LI>
<LI><SMALL><A href="/wiki/Type_0_string_theory" title="Type 0 string theory">Type 0 string theory</A></SMALL></LI>
<LI><SMALL><A href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</A></SMALL></LI>
<LI><SMALL><A href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string theory</A></SMALL></LI>
<LI><SMALL><A href="/wiki/Little_string_theory" title="Little string theory">Little string theory</A></SMALL></LI>
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<LI><A href="/wiki/Twistor_theory" title="Twistor theory">Twistor theory</A>
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<LI><SMALL><A href="/wiki/Twistor_string_theory" title="Twistor string theory">Twistor string theory</A></SMALL></LI>
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<LI><A href="/wiki/Scale_relativity" title="Scale relativity">Scale relativity</A></LI>
<LI><A href="/wiki/Liouville_gravity" class="mw-redirect" title="Liouville gravity">Liouville gravity</A></LI>
<LI><A href="/wiki/Lovelock_theory_of_gravity" title="Lovelock theory of gravity">Lovelock theory</A></LI>
<LI><A href="/wiki/(2%2B1)-dimensional_topological_gravity" title="(2+1)-dimensional topological gravity">(2+1)-dimensional topological gravity</A></LI>
<LI><A href="/wiki/Gauss%E2%80%93Bonnet_gravity" title="Gauss–Bonnet gravity">Gauss–Bonnet gravity</A></LI>
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տեսություն – Armenian" lang="hy" hreflang="hy" class="interlanguage-link-target">Հայերեն</A></LI><LI class="interlanguage-link interwiki-hi"><A href="https://hi.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A4%BE%E0%A4%B0%E0%A5%8D%E0%A4%B5%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3_%E0%A4%95%E0%A4%BE_%E0%A4%B8%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%A7%E0%A4%BE%E0%A4%A8%E0%A5%8D%E0%A4%A4" title="न्यूटन का सार्वत्रिक गुरुत्वाकर्षण का सिद्धान्त – Hindi" lang="hi" hreflang="hi" class="interlanguage-link-target">हिन्दी</A></LI><LI class="interlanguage-link interwiki-hr"><A href="https://hr.wikipedia.org/wiki/Newtonov_zakon_gravitacije" title="Newtonov zakon gravitacije – Croatian" lang="hr" hreflang="hr" class="interlanguage-link-target">Hrvatski</A></LI><LI class="interlanguage-link interwiki-id"><A href="https://id.wikipedia.org/wiki/Hukum_gravitasi_universal_Newton" title="Hukum gravitasi universal Newton – Indonesian" lang="id" hreflang="id" class="interlanguage-link-target">Bahasa Indonesia</A></LI><LI class="interlanguage-link interwiki-it"><A href="https://it.wikipedia.org/wiki/Legge_di_gravitazione_universale" title="Legge di gravitazione universale – Italian" lang="it" hreflang="it" class="interlanguage-link-target">Italiano</A></LI><LI class="interlanguage-link interwiki-ka"><A href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A1%E1%83%9D%E1%83%A4%E1%83%9A%E1%83%98%E1%83%9D_%E1%83%9B%E1%83%98%E1%83%96%E1%83%98%E1%83%93%E1%83%A3%E1%83%9A%E1%83%9D%E1%83%91%E1%83%98%E1%83%A1_%E1%83%99%E1%83%90%E1%83%9C%E1%83%9D%E1%83%9C%E1%83%98" title="მსოფლიო მიზიდულობის კანონი – Georgian" lang="ka" hreflang="ka" class="interlanguage-link-target">ქართული</A></LI><LI class="interlanguage-link interwiki-kk"><A href="https://kk.wikipedia.org/wiki/%D0%91%D2%AF%D0%BA%D1%96%D0%BB_%D3%99%D0%BB%D0%B5%D0%BC%D0%B4%D1%96%D0%BA_%D1%82%D0%B0%D1%80%D1%82%D1%8B%D0%BB%D1%8B%D1%81_%D0%B7%D0%B0%D2%A3%D1%8B" title="Бүкіл әлемдік тартылыс заңы – Kazakh" lang="kk" hreflang="kk" class="interlanguage-link-target">Қазақша</A></LI><LI class="interlanguage-link interwiki-lv"><A href="https://lv.wikipedia.org/wiki/%C5%85%C5%ABtona_vispasaules_gravit%C4%81cijas_likums" title="Ņūtona vispasaules gravitācijas likums – Latvian" lang="lv" hreflang="lv" class="interlanguage-link-target">Latviešu</A></LI><LI class="interlanguage-link interwiki-lb"><A href="https://lb.wikipedia.org/wiki/Gravitatiounsgesetz" title="Gravitatiounsgesetz – Luxembourgish" lang="lb" hreflang="lb" class="interlanguage-link-target">Lëtzebuergesch</A></LI><LI class="interlanguage-link interwiki-lt"><A href="https://lt.wikipedia.org/wiki/Niutono_gravitacijos_d%C4%97snis" title="Niutono gravitacijos dėsnis – Lithuanian" lang="lt" hreflang="lt" class="interlanguage-link-target">Lietuvių</A></LI><LI class="interlanguage-link interwiki-hu"><A href="https://hu.wikipedia.org/wiki/Newton-f%C3%A9le_gravit%C3%A1ci%C3%B3s_t%C3%B6rv%C3%A9ny" title="Newton-féle gravitációs törvény – Hungarian" lang="hu" hreflang="hu" class="interlanguage-link-target">Magyar</A></LI><LI class="interlanguage-link interwiki-mk"><A href="https://mk.wikipedia.org/wiki/%D0%8A%D1%83%D1%82%D0%BD%D0%BE%D0%B2_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B7%D0%B0_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B0%D1%82%D0%B0" title="Њутнов закон за гравитацијата – Macedonian" lang="mk" hreflang="mk" class="interlanguage-link-target">Македонски</A></LI><LI class="interlanguage-link interwiki-ml"><A href="https://ml.wikipedia.org/wiki/%E0%B4%90%E0%B4%B8%E0%B4%95%E0%B5%8D_%E0%B4%A8%E0%B5%8D%E0%B4%AF%E0%B5%82%E0%B4%9F%E0%B5%8D%E0%B4%9F%E0%B4%A8%E0%B5%8D%E0%B4%B1%E0%B5%86_%E0%B4%97%E0%B5%81%E0%B4%B0%E0%B5%81%E0%B4%A4%E0%B5%8D%E0%B4%B5%E0%B4%BE%E0%B4%95%E0%B5%BC%E0%B4%B7%E0%B4%A3_%E0%B4%A8%E0%B4%BF%E0%B4%AF%E0%B4%AE%E0%B4%82" title="ഐസക് ന്യൂട്ടന്റെ ഗുരുത്വാകർഷണ നിയമം – Malayalam" lang="ml" hreflang="ml" class="interlanguage-link-target">മലയാളം</A></LI><LI class="interlanguage-link interwiki-mr"><A href="https://mr.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%9F%E0%A4%A8%E0%A4%9A%E0%A4%BE_%E0%A4%B5%E0%A5%88%E0%A4%B6%E0%A5%8D%E0%A4%B5%E0%A4%BF%E0%A4%95_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%BE%E0%A4%9A%E0%A4%BE_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्यूटनचा वैश्विक गुरुत्वाकर्षणाचा नियम – Marathi" lang="mr" hreflang="mr" class="interlanguage-link-target">मराठी</A></LI><LI class="interlanguage-link interwiki-ms"><A href="https://ms.wikipedia.org/wiki/Hukum_kegravitian_semesta_Newton" title="Hukum kegravitian semesta Newton – Malay" lang="ms" hreflang="ms" class="interlanguage-link-target">Bahasa Melayu</A></LI><LI class="interlanguage-link interwiki-cdo"><A href="https://cdo.wikipedia.org/wiki/U%C3%A2ng-i%C5%AB_%C4%ABng-l%C4%ADk_d%C3%AAng-l%C5%ADk" title="Uâng-iū īng-lĭk dêng-lŭk – Min Dong Chinese" lang="cdo" hreflang="cdo" class="interlanguage-link-target">Mìng-dĕ̤ng-ngṳ̄</A></LI><LI class="interlanguage-link interwiki-mn"><A href="https://mn.wikipedia.org/wiki/%D0%91%D2%AF%D1%85_%D0%B5%D1%80%D1%82%D3%A9%D0%BD%D1%86%D0%B8%D0%B9%D0%BD_%D1%82%D0%B0%D1%82%D0%B0%D0%BB%D1%86%D0%BB%D1%8B%D0%BD_%D1%85%D1%83%D1%83%D0%BB%D1%8C" title="Бүх ертөнцийн таталцлын хууль – Mongolian" lang="mn" hreflang="mn" class="interlanguage-link-target">Монгол</A></LI><LI class="interlanguage-link interwiki-nl"><A href="https://nl.wikipedia.org/wiki/Gravitatiewet_van_Newton" title="Gravitatiewet van Newton – Dutch" lang="nl" hreflang="nl" class="interlanguage-link-target">Nederlands</A></LI><LI class="interlanguage-link interwiki-ne"><A href="https://ne.wikipedia.org/wiki/%E0%A4%A8%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%9F%E0%A4%A8%E0%A4%95%E0%A5%8B_%E0%A4%97%E0%A5%81%E0%A4%B0%E0%A5%81%E0%A4%A4%E0%A5%8D%E0%A4%B5%E0%A4%BE%E0%A4%95%E0%A4%B0%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%B8%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%A8%E0%A5%8D%E0%A4%A7%E0%A5%80_%E0%A4%B5%E0%A4%BF%E0%A4%B6%E0%A5%8D%E0%A4%B5%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A4%BE%E0%A4%AA%E0%A5%80_%E0%A4%A8%E0%A4%BF%E0%A4%AF%E0%A4%AE" title="न्युटनको गुरुत्वाकर्षणसम्बन्धी विश्वव्यापी नियम – Nepali" lang="ne" hreflang="ne" class="interlanguage-link-target">नेपाली</A></LI><LI class="interlanguage-link interwiki-ja"><A href="https://ja.wikipedia.org/wiki/%E4%B8%87%E6%9C%89%E5%BC%95%E5%8A%9B" title="万有引力 – Japanese" lang="ja" hreflang="ja" class="interlanguage-link-target">日本語</A></LI><LI class="interlanguage-link interwiki-no"><A href="https://no.wikipedia.org/wiki/Newtons_gravitasjonslov" title="Newtons gravitasjonslov – Norwegian" lang="no" hreflang="no" class="interlanguage-link-target">Norsk</A></LI><LI class="interlanguage-link interwiki-oc"><A href="https://oc.wikipedia.org/wiki/Lei_de_la_gravitacion_universala" title="Lei de la gravitacion universala – Occitan" lang="oc" hreflang="oc" class="interlanguage-link-target">Occitan</A></LI><LI class="interlanguage-link interwiki-pa"><A href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%81%E0%A8%B0%E0%A9%82%E0%A8%A4%E0%A8%BE%E0%A8%95%E0%A8%B0%E0%A8%B8%E0%A8%BC%E0%A8%A3_%E0%A8%A6%E0%A8%BE_%E0%A8%B8%E0%A8%B0%E0%A8%B5-%E0%A8%B5%E0%A8%BF%E0%A8%85%E0%A8%BE%E0%A8%AA%E0%A9%80_%E0%A8%A8%E0%A8%BF%E0%A8%AF%E0%A8%AE" title="ਗੁਰੂਤਾਕਰਸ਼ਣ ਦਾ ਸਰਵ-ਵਿਅਾਪੀ ਨਿਯਮ – Punjabi" lang="pa" hreflang="pa" class="interlanguage-link-target">ਪੰਜਾਬੀ</A></LI><LI class="interlanguage-link interwiki-pnb"><A href="https://pnb.wikipedia.org/wiki/%D9%86%DB%8C%D9%88%D9%B9%D9%86_%D8%AF%D8%A7_%DA%A9%DA%BE%DA%86_%D8%AF%D8%A7_%D9%82%D9%86%D9%88%D9%86" title="نیوٹن دا کھچ دا قنون – Western Punjabi" lang="pnb" hreflang="pnb" class="interlanguage-link-target">پنجابی</A></LI><LI class="interlanguage-link interwiki-pms"><A href="https://pms.wikipedia.org/wiki/Lej_%C3%ABd_gravitassion_universal" title="Lej ëd gravitassion universal – Piedmontese" lang="pms" hreflang="pms" class="interlanguage-link-target">Piemontèis</A></LI><LI class="interlanguage-link interwiki-pl"><A href="https://pl.wikipedia.org/wiki/Prawo_powszechnego_ci%C4%85%C5%BCenia" title="Prawo powszechnego ciążenia – Polish" lang="pl" hreflang="pl" class="interlanguage-link-target">Polski</A></LI><LI class="interlanguage-link interwiki-pt"><A href="https://pt.wikipedia.org/wiki/Lei_da_gravita%C3%A7%C3%A3o_universal" title="Lei da gravitação universal – Portuguese" lang="pt" hreflang="pt" class="interlanguage-link-target">Português</A></LI><LI class="interlanguage-link interwiki-ro"><A href="https://ro.wikipedia.org/wiki/Legea_atrac%C8%9Biei_universale" title="Legea atracției universale – Romanian" lang="ro" hreflang="ro" class="interlanguage-link-target">Română</A></LI><LI class="interlanguage-link interwiki-ru"><A href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BB%D0%B0%D1%81%D1%81%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D1%82%D1%8F%D0%B3%D0%BE%D1%82%D0%B5%D0%BD%D0%B8%D1%8F_%D0%9D%D1%8C%D1%8E%D1%82%D0%BE%D0%BD%D0%B0" title="Классическая теория тяготения Ньютона – Russian" lang="ru" hreflang="ru" class="interlanguage-link-target">Русский</A></LI><LI class="interlanguage-link interwiki-sco"><A href="https://sco.wikipedia.org/wiki/Newton%27s_law_o_universal_gravitation" title="Newton's law o universal gravitation – Scots" lang="sco" hreflang="sco" class="interlanguage-link-target">Scots</A></LI><LI class="interlanguage-link interwiki-sq"><A href="https://sq.wikipedia.org/wiki/Ligji_gravitacional_universal_i_Njutonit" title="Ligji gravitacional universal i Njutonit – Albanian" lang="sq" hreflang="sq" class="interlanguage-link-target">Shqip</A></LI><LI class="interlanguage-link interwiki-si"><A href="https://si.wikipedia.org/wiki/%E0%B6%9C%E0%B7%94%E0%B6%BB%E0%B7%94%E0%B6%AD%E0%B7%8A%E0%B7%80%E0%B6%A2_%E0%B6%9A%E0%B7%8A%E0%B7%82%E0%B7%9A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%BA" title="ගුරුත්වජ ක්ෂේත්‍රය – Sinhala" lang="si" hreflang="si" class="interlanguage-link-target">සිංහල</A></LI><LI class="interlanguage-link interwiki-simple"><A href="https://simple.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation" title="Newton's law of universal gravitation – Simple English" lang="simple" hreflang="simple" class="interlanguage-link-target">Simple English</A></LI><LI class="interlanguage-link interwiki-sk"><A href="https://sk.wikipedia.org/wiki/Gravita%C4%8Dn%C3%BD_z%C3%A1kon" title="Gravitačný zákon – Slovak" lang="sk" hreflang="sk" class="interlanguage-link-target">Slovenčina</A></LI><LI class="interlanguage-link interwiki-sl"><A href="https://sl.wikipedia.org/wiki/Splo%C5%A1ni_gravitacijski_zakon" title="Splošni gravitacijski zakon – Slovenian" lang="sl" hreflang="sl" class="interlanguage-link-target">Slovenščina</A></LI><LI class="interlanguage-link interwiki-ckb"><A href="https://ckb.wikipedia.org/wiki/%DB%8C%D8%A7%D8%B3%D8%A7%DB%8C_%DA%95%D8%A7%DA%A9%DB%8E%D8%B4%D8%A7%D9%86%DB%8C_%DA%AF%DB%95%D8%B1%D8%AF%D9%88%D9%88%D9%86%DB%8C" title="یاسای ڕاکێشانی گەردوونی – Central Kurdish" lang="ckb" hreflang="ckb" class="interlanguage-link-target">کوردی</A></LI><LI class="interlanguage-link interwiki-sr"><A href="https://sr.wikipedia.org/wiki/%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D0%B7%D0%B0%D0%BB%D0%BD%D0%B8_%D0%B7%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%98%D0%B5" title="Универзални закон гравитације – Serbian" lang="sr" hreflang="sr" class="interlanguage-link-target">Српски / srpski</A></LI><LI class="interlanguage-link interwiki-sh"><A href="https://sh.wikipedia.org/wiki/Newtonov_zakon_gravitacije" title="Newtonov zakon gravitacije – Serbo-Croatian" lang="sh" hreflang="sh" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</A></LI><LI class="interlanguage-link interwiki-fi"><A href="https://fi.wikipedia.org/wiki/Painovoima#Newtonin_laki_vetovoimasta" title="Painovoima – Finnish" lang="fi" hreflang="fi" class="interlanguage-link-target">Suomi</A></LI><LI class="interlanguage-link interwiki-sv"><A href="https://sv.wikipedia.org/wiki/Newtons_gravitationslag" title="Newtons gravitationslag – Swedish" lang="sv" hreflang="sv" class="interlanguage-link-target">Svenska</A></LI><LI class="interlanguage-link interwiki-tl"><A href="https://tl.wikipedia.org/wiki/Batas_ng_unibersal_na_grabitasyon_ni_Newton" title="Batas ng unibersal na grabitasyon ni Newton – Tagalog" lang="tl" hreflang="tl" class="interlanguage-link-target">Tagalog</A></LI><LI class="interlanguage-link interwiki-ta"><A href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%AF%E0%AF%82%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%A9%E0%AE%BF%E0%AE%A9%E0%AF%8D_%E0%AE%88%E0%AE%B0%E0%AF%8D%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81_%E0%AE%B5%E0%AE%BF%E0%AE%A4%E0%AE%BF" title="நியூட்டனின் ஈர்ப்பு விதி – Tamil" lang="ta" hreflang="ta" class="interlanguage-link-target">தமிழ்</A></LI><LI class="interlanguage-link interwiki-te"><A href="https://te.wikipedia.org/wiki/%E0%B0%A8%E0%B1%8D%E0%B0%AF%E0%B1%82%E0%B0%9F%E0%B0%A8%E0%B1%8D_%E0%B0%B5%E0%B0%BF%E0%B0%B6%E0%B1%8D%E0%B0%B5%E0%B0%97%E0%B1%81%E0%B0%B0%E0%B1%81%E0%B0%A4%E0%B1%8D%E0%B0%B5%E0%B0%BE%E0%B0%95%E0%B0%B0%E0%B1%8D%E0%B0%B7%E0%B0%A3_%E0%B0%B8%E0%B0%BF%E0%B0%A6%E0%B1%8D%E0%B0%A7%E0%B0%BE%E0%B0%82%E0%B0%A4%E0%B0%82" title="న్యూటన్ విశ్వగురుత్వాకర్షణ సిద్ధాంతం – Telugu" lang="te" hreflang="te" class="interlanguage-link-target">తెలుగు</A></LI><LI class="interlanguage-link interwiki-th"><A href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%8E%E0%B8%84%E0%B8%A7%E0%B8%B2%E0%B8%A1%E0%B9%82%E0%B8%99%E0%B9%89%E0%B8%A1%E0%B8%96%E0%B9%88%E0%B8%A7%E0%B8%87%E0%B8%AA%E0%B8%B2%E0%B8%81%E0%B8%A5%E0%B8%82%E0%B8%AD%E0%B8%87%E0%B8%99%E0%B8%B4%E0%B8%A7%E0%B8%95%E0%B8%B1%E0%B8%99" title="กฎความโน้มถ่วงสากลของนิวตัน – Thai" lang="th" hreflang="th" class="interlanguage-link-target">ไทย</A></LI><LI class="interlanguage-link interwiki-tr"><A href="https://tr.wikipedia.org/wiki/Newton%27un_evrensel_k%C3%BCtle_%C3%A7ekim_yasas%C4%B1" title="Newton'un evrensel kütle çekim yasası – Turkish" lang="tr" hreflang="tr" class="interlanguage-link-target">Türkçe</A></LI><LI class="interlanguage-link interwiki-uk"><A href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B2%D1%81%D0%B5%D1%81%D0%B2%D1%96%D1%82%D0%BD%D1%8C%D0%BE%D0%B3%D0%BE_%D1%82%D1%8F%D0%B6%D1%96%D0%BD%D0%BD%D1%8F" title="Закон всесвітнього тяжіння – Ukrainian" lang="uk" hreflang="uk" 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document_title: Newton's law of universal gravitation

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question_text: when was the universal law of gravitation discovered