Newton's law of universal gravitation
Newton's law of universal gravitation states that a particle attracts
every other particle in the universe with a force which is directly
proportional to the product of their masses and inversely proportional
to the square of the distance between their centers.[note 1] This is a
general physical law derived from empirical observations by what Isaac
Newton called inductive reasoning. It is a part of classical
mechanics and was formulated in Newton's work Philosophiæ Naturalis
Principia Mathematica ("the Principia"), first published on 5 July
1686. When Newton's book was presented in 1686 to the Royal Society,
Robert Hooke made a claim that Newton had obtained the inverse square
law from him.
In today's language, the law states: Every point mass attracts every
single other point mass by a force pointing along the line
intersecting both points. The force is proportional to the product of
the two masses and inversely proportional to the square of the
distance between them. The first test of Newton's theory of
gravitation between masses in the laboratory was the Cavendish
experiment conducted by the British scientist
Henry Cavendish in
1798. It took place 111 years after the publication of Newton's
Principia and approximately 71 years after his death.
Newton's law of gravitation resembles
Coulomb's law of electrical
forces, which is used to calculate the magnitude of the electrical
force arising between two charged bodies. Both are inverse-square
laws, 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 electrostatic
constant in place of the gravitational constant.
Newton's law has since been superseded by Albert Einstein's theory of
general relativity, but it continues to be used as an excellent
approximation of the effects of gravity 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 Mercury's orbit around the Sun).
1.1 Early history
1.3 Hooke's work and claims
1.4 Newton's work and claims
1.5 Newton's acknowledgment
1.6 Modern priority controversy
2 Modern form
3 Bodies with spatial extent
4 Vector form
5 Gravitational field
6 Problematic aspects
6.1 Theoretical concerns with Newton's expression
6.2 Observations conflicting with Newton's formula
6.3 Newton's reservations
6.4 Einstein's solution
8 Solutions of Newton's law of universal gravitation
9 See also
12 External links
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".
The same author does credit Hooke with a significant and even seminal
contribution, but he treats Hooke's claim of priority on the inverse
square point as uninteresting since several individuals besides Newton
and Hooke had at least suggested it, and he points instead to the idea
of "compounding the celestial motions" and the conversion of Newton's
thinking away from "centrifugal" and towards "centripetal" force as
Hooke's significant contributions.
Newton himself gave credit in his Principia to two persons:
Bullialdus (he wrote without proof that there was a force on the
earth towards the sun), and Borelli (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.
In 1686, when the first book of Newton's Principia was presented to
the Royal Society,
Robert Hooke accused Newton of plagiarism by
claiming that he had taken from him the "notion" of "the rule of the
decrease of Gravity, being reciprocally as the squares of the
distances from the Center". At the same time (according to Edmond
Halley's contemporary report) Hooke agreed that "the Demonstration of
the Curves generated thereby" was wholly Newton's.
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.
Hooke's work and claims
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". 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"; 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
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. 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"). It was later on, in writing on 6
January 167980 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." (The inference about the velocity was incorrect.)
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".
Newton's work and claims
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. Newton also
pointed out and acknowledged prior work of others, including
Bullialdus, (who suggested, but without demonstration, that there
was an attractive force from the
Sun in the inverse square proportion
to the distance), and Borelli (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.
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."
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. In addition, Newton had formulated in Propositions
Book 1, 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.
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. 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. This background shows there was basis for Newton to deny
deriving the inverse square law from Hooke.
On the other hand, Newton did accept and acknowledge, in all editions
of the Principia, that Hooke (but not exclusively Hooke) had
separately appreciated the inverse square law in the solar system.
Newton acknowledged Wren, Hooke and Halley in this connection in the
Scholium to Proposition 4 in
Book 1. 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
Modern priority controversy
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). These matters do not appear to have been learned by Newton
Nevertheless, a number of authors have had more to say about what
Newton gained from Hooke and some aspects remain controversial.
The fact that most of Hooke's private papers had been destroyed or
have disappeared does not help to establish the truth.
Newton's role in relation to the inverse square law was not as it has
sometimes been represented. 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
About thirty years after Newton's death in 1727, Alexis Clairaut, 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
In modern language, the law states the following:
Every point mass attracts every single other point mass by a force
pointing along the line intersecting both points. The force is
proportional to the product of the two masses and inversely
proportional to the square of the distance between them:
displaystyle F=G frac m_ 1 m_ 2 r^ 2
F is the force between the masses;
G is the gravitational constant
(6.674×10−11 N · (m/kg)2);
m1 is the first mass;
m2 is the second mass;
r is the distance between the centers of the masses.
Error plot showing experimental values for big G.
Assuming SI units, F is measured in newtons (N), m1 and m2 in
kilograms (kg), r in meters (m), and the constant G is approximately
equal to 6989667400000000000♠6.674×10−11 N m2 kg−2.
The value of the constant G was first accurately determined from the
results of the
Cavendish experiment conducted by the British scientist
Henry Cavendish in 1798, although Cavendish did not himself calculate
a numerical value for G. 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 Principia and
71 years after Newton's death, so none of Newton's calculations could
use the value of G; instead he could only calculate a force relative
to another force.
Bodies with spatial extent
Gravitational field strength within the Earth
Gravity field near the surface of the Earth – an object is shown
accelerating toward the surface
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 integrating the force (in
vector form, see below) over the extents of the two bodies.
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. (This is not
generally true for non-spherically-symmetrical bodies.)
For points inside a spherically-symmetric distribution of matter,
Shell theorem 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
r0 from the center of the mass distribution:
The portion of the mass that is located at radii r < r0 causes the
same force at r0 as if all of the mass enclosed within a sphere of
radius r0 was concentrated at the center of the mass distribution (as
The portion of the mass that is located at radii r > r0 exerts no
net gravitational force at the distance r0 from the center. That is,
the individual gravitational forces exerted by the elements of the
sphere out there, on the point at r0, cancel each other out.
As a consequence, for example, within a shell of uniform thickness and
density there is no net gravitational acceleration anywhere within the
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
Gravity field surrounding Earth from a macroscopic perspective.
Newton's law of universal gravitation
Newton's law of universal gravitation can be written as a vector
equation to account for the direction of the gravitational force as
well as its magnitude. In this formula, quantities in bold represent
displaystyle mathbf F _ 21 =-G m_ 1 m_ 2 over vert mathbf r
_ 12 vert ^ 2 ,mathbf hat r _ 12
F21 is the force applied on object 2 exerted by object 1,
G is the gravitational constant,
m1 and m2 are respectively the masses of objects 1 and 2,
r12 = r2 − r1 is the distance between objects 1 and 2, and
displaystyle mathbf hat r _ 12 stackrel mathrm def =
frac mathbf r _ 2 -mathbf r _ 1 vert mathbf r _ 2 -mathbf r
_ 1 vert
is the unit vector from object 1 to 2.
It can be seen that the vector form of the equation is the same as the
scalar form given earlier, except that F is now a vector quantity, and
the right hand side is multiplied by the appropriate unit vector.
Also, it can be seen that F12 = −F21.
Main article: Gravitational field
The gravitational field is a vector field that describes the
gravitational force which would be applied on an object in any given
point in space, per unit mass. It is actually equal to the
gravitational acceleration at that point.
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 r instead of r12 and m instead of
m2 and define the gravitational field g(r) as:
displaystyle mathbf g (mathbf r )=-G m_ 1 over vert mathbf
r vert ^ 2 ,mathbf hat r
so that we can write:
displaystyle mathbf F (mathbf r )=mmathbf g (mathbf r ).
This formulation is dependent on the objects causing the field. The
field has units of acceleration; in SI, this is m/s2.
Gravitational fields are also conservative; 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 V(r)
displaystyle mathbf g (mathbf r )=-nabla V(mathbf r ).
If m1 is a point mass or the mass of a sphere with homogeneous mass
distribution, the force field g(r) outside the sphere is isotropic,
i.e., depends only on the distance r from the center of the sphere. In
displaystyle V(r)=-G frac m_ 1 r .
the gravitational field is on, inside and outside of symmetric masses.
As per Gauss Law, field in a symmetric body can be found by the
displaystyle partial V
displaystyle mathbf g(r) cdot dmathbf A =-4pi GM_ text enc ,
displaystyle partial V
is a closed surface and
displaystyle M_ text enc
is the mass enclosed by the surface.
Hence, for a hollow sphere of radius
and total mass
displaystyle mathbf g(r) = begin cases 0,& mbox if
r<R\\ dfrac GM r^ 2 ,& mbox if rgeq Rend cases
For a uniform solid sphere of radius
and total mass
displaystyle mathbf g(r) = begin cases dfrac GMr R^ 3
,& mbox if r<R\\ dfrac GM r^ 2 ,& mbox if rgeq Rend
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 φ/c2 and (v/c)2 are both
much less than one, where φ is the gravitational potential, v is the
velocity of the objects being studied, and c is the speed of
light. For example, Newtonian gravity provides an accurate
description of the Earth/
Sun system, since
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 2pi r_ mathrm orbit (1 mathrm yr )c
right)^ 2 sim 10^ -8
where rorbit is the radius of the Earth's orbit around the Sun.
In situations where either dimensionless parameter is large, then
general relativity 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.
Theoretical concerns with Newton's expression
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
Theory of everything
Theory of everything and Standard Model). Newton himself
felt that the concept of an inexplicable action at a distance was
unsatisfactory (see "Newton's reservations" below), but that there was
nothing more that he could do at the time.
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.
Observations conflicting with Newton's formula
Newton's Theory does not fully explain the precession of the
perihelion of the orbits of the planets, especially of planet Mercury,
which was detected long after the life of Newton. There is a 43
arcsecond per century discrepancy between the Newtonian calculation,
which arises only from the gravitational attractions from the other
planets, and the observed precession, made with advanced telescopes
during the 19th century.
The predicted angular deflection of light rays by gravity that is
calculated by using Newton's Theory is only one-half of the deflection
that is actually observed by astronomers. Calculations using General
Relativity are in much closer agreement with the astronomical
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 dark
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: "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."
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 mechanical hypotheses 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
General Scholium in the second edition of Principia: "I
have not yet been able to discover the cause of these properties of
gravity from phenomena and I feign no hypotheses.... 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."
Part of a series on
Doubly special relativity
Introduction to general relativity
Mathematics of general relativity
Einstein field equations
Introduction to gravitation
Newton's law of universal gravitation
Derivations of relativity
Mathematics of general relativity
These objections were explained by Einstein's theory of general
relativity, in which gravitation is an attribute of curved spacetime
instead of being due to a force propagated between bodies. 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 fictitious force due
to the curvature of spacetime, because the gravitational acceleration
of a body in free fall is due to its world line being a geodesic of
Newton was the first to consider in his Principia an extended
expression of his law of gravity including an inverse-cube term of the
displaystyle F=G frac m_ 1 m_ 2 r^ 2 +B frac m_ 1 m_ 2 r^
attempting to explain the Moon's apsidal motion. Other extensions were
proposed by Laplace (around 1790) and Decombes (1913):
displaystyle F(r)=k frac m_ 1 m_ 2 r^ 2 exp(-alpha r)
displaystyle F(r)=k frac m_ 1 m_ 2 r^ 2 left(1+ alpha over
r^ 3 right)
In recent years, quests for non-inverse square terms in the law of
gravity have been carried out by neutron interferometry.
Solutions of Newton's law of universal gravitation
Main article: n-body problem
The n-body problem is an ancient, classical problem of predicting
the individual motions of a group of celestial objects interacting
with each other gravitationally. Solving this problem — from the
time of the Greeks and on — has been motivated by the desire to
understand the motions of the Sun, planets and the visible stars. In
the 20th century, understanding the dynamics of globular cluster star
systems became an important n-body problem too. The n-body problem
in general relativity is considerably more difficult to solve.
The classical physical problem can be informally stated as: given the
quasi-steady orbital properties (instantaneous position, velocity and
time) of a group of celestial bodies, predict their interactive
forces; and consequently, predict their true orbital motions for all
The two-body problem has been completely solved, as has the restricted
Book: Isaac Newton
Gauss's law for gravity
Jordan and Einstein frames
Newton's laws of motion
Static forces and virtual-particle exchange
^ It was shown separately that large, spherically symmetrical masses
attract and are attracted as if all their mass were concentrated at
^ Isaac Newton: "In [experimental] philosophy particular propositions
are inferred from the phenomena and afterwards rendered general by
Book 3, General Scholium, at p.392 in Volume
2 of Andrew Motte's English translation published 1729.
^ a b c - Proposition 75, Theorem 35: p.956 - I.Bernard Cohen and Anne
Whitman, translators: Isaac Newton, The Principia: Mathematical
Principles of Natural Philosophy. Preceded by A Guide to Newton's
Principia, by I.Bernard Cohen. University of California Press 1999
ISBN 0-520-08816-6 ISBN 0-520-08817-4
^ a b The Michell-Cavendish Experiment, Laurent Hodges
^ a b Bullialdus (Ismael Bouillau) (1645), "Astronomia philolaica",
^ a b Borelli, G. A., "Theoricae Mediceorum Planetarum ex causis
physicis deductae", Florence, 1666.
^ a b 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.
^ 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.
^ a b Hooke's 1674 statement in "An Attempt to Prove the Motion of the
Earth from Observations" is available in online facsimile here.
^ Purrington, Robert D. (2009). The First Professional Scientist:
Robert Hooke and the
Royal Society of London. Springer. p. 168.
ISBN 3-0346-0036-4. Extract of page 168
^ 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",
^ Calendar (New Style) Act 1750
^ Page 309 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol
2 (1676-1687), (Cambridge University Press, 1960), document #239.
^ See Curtis Wilson (1989) at page 244.
^ Page 297 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol
2 (1676-1687), (Cambridge University Press, 1960), document #235, 24
^ Page 433 in H W Turnbull (ed.), Correspondence of Isaac Newton, Vol
2 (1676-1687), (Cambridge University Press, 1960), document #286, 27
^ a b 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.
^ Page 436, Correspondence, Vol.2, already cited.
^ Propositions 70 to 75 in
Book 1, for example in the 1729 English
translation of the Principia, start at page 263.
^ Propositions 43 to 45 in
Book 1, in the 1729 English translation of
the Principia, start at page 177.
^ 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.
^ See J. Bruce Brackenridge, "The key to Newton's dynamics: the Kepler
problem and the Principia", (University of California Press, 1995),
especially at pages 20-21.
^ See for example the 1729 English translation of the Principia, at
^ 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.
^ 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),
^ See for example the results of Propositions 43-45 and 70-75 in Book
1, cited above.
^ See also G E Smith, in Stanford Encyclopedia of Philosophy,
"Newton's Philosophiae Naturalis Principia Mathematica".
^ 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.
^ 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".
^ Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA
Recommended Values of the Fundamental Physical Constants: 2006".
Reviews of Modern Physics. 80 (2): 633–730. arXiv:0801.0028 .
Direct link to value..
^ Equilibrium State
^ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973).
Gravitation. New York: W. H.Freeman and Company.
ISBN 0-7167-0344-0 Page 1049.
Max Born (1924), Einstein's Theory of Relativity (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.)
^ - The Construction of Modern Science: Mechanisms and Mechanics, by
Richard S. Westfall. Cambridge University Press. 1978
^ Leimanis and Minorsky: Our interest is with Leimanis, who first
discusses some history about the n-body problem, especially Ms.
Kovalevskaya's ~1868-1888, twenty-year complex-variables approach,
failure; Section 1: The Dynamics of Rigid Bodies and Mathematical
Exterior Ballistics (Chapter 1, the motion of a rigid body about a
fixed point (Euler and Poisson equations); Chapter 2, Mathematical
Exterior Ballistics), good precursor background to the n-body problem;
Section 2: Celestial Mechanics (Chapter 1, The Uniformization of the
Three-body Problem (Restricted Three-body Problem); Chapter 2, Capture
in the Three-Body Problem; Chapter 3, Generalized n-body Problem).
^ See References sited for Heggie and Hut. This page has
made their approach obsolete.
^ Quasi-steady 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 steady-state condition refers to
a system's state being invariant to time; otherwise, the first
derivatives and all higher derivatives are zero.
^ R. M. Rosenberg states the n-body problem similarly (see
References): 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? Rosenberg failed to
realize, like everyone else, that it is necessary to determine the
forces first before the motions can be determined.
^ A general, classical solution in terms of first integrals is known
to be impossible. An exact theoretical solution for arbitrary n can be
approximated via Taylor series, but in practice such an infinite
series must be truncated, giving at best only an approximate solution;
and an approach now obsolete. In addition, the n-body problem may be
solved using numerical integration, but these, too, are approximate
solutions; and again obsolete. See Sverre J. Aarseth's book
Gravitational N-body Simulations listed in the References.
Feather & Hammer Drop on Moon on YouTube
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