Editing Newton's laws of motion (section)

==History==
{{Gallery
|File:Portrait of Sir Isaac Newton, 1689.jpg
|Isaac Newton (1643–1727), in a 1689 portrait by [[Godfrey Kneller]]
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|File:NewtonsPrincipia.jpg
|Newton's own copy of his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'', with hand-written corrections for the second edition, in the [[Wren Library]] at [[Trinity College, Cambridge]]
|alt2=
|File:Newtons laws in latin.jpg
|Newton's first and second laws, in Latin, from the original 1687 ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]''
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|align=center}}

The concepts invoked in Newton's laws of motion — mass, velocity, momentum, force — have predecessors in earlier work, and the content of Newtonian physics was further developed after Newton's time. Newton combined knowledge of celestial motions with the study of events on Earth and showed that one theory of mechanics could encompass both.{{refn|group=note|As one physicist writes, "Physical theory is possible because we ''are'' immersed and included in the whole process – because we can act on objects around us. Our ability to intervene in nature clarifies even the motion of the planets around the sun – masses so great and distances so vast that our roles as participants seem insignificant. Newton was able to transform Kepler's kinematical description of the solar system into a far more powerful dynamical theory because he added concepts from Galileo's experimental methods – force, mass, momentum, and gravitation. The truly external observer will only get as far as Kepler. Dynamical concepts are formulated on the basis of what we can set up, control, and measure."<ref>D. Bilodeau, quoted in {{cite book |first=Christopher A. |last=Fuchs |title=Coming of Age with Quantum Information |date=6 January 2011 |publisher=Cambridge University Press |pages=310–311 |isbn=978-0-521-19926-1 |oclc=759812415}}</ref> See, for example, Caspar and Hellman.<ref>{{cite book|first=Max |last=Caspar |translator-first=C. Doris |translator-last=Hellman |translator-link=C. Doris Hellman |title=Kepler |page=178 |publisher=Dover |year=2012 |orig-year=1959 |isbn=978-0-486-15175-5 |oclc=874097920}}</ref>}}

As noted by scholar [[I. Bernard Cohen]], Newton's work was more than a mere synthesis of previous results, as he selected certain ideas and further transformed them, with each in a new form that was useful to him, while at the same time proving false of certain basic or fundamental principles of scientists such as [[Galileo Galilei]], [[Johannes Kepler]], [[René Descartes]], and [[Nicolaus Copernicus]].<ref>{{Cite book |last=Cohen |first=I. Bernard |title=The Newtonian Revolution: With Illustrations of the Transformation of Scientific Ideas |date=1980 |publisher=Cambridge University Press |isbn=978-0-521-22964-7 |location= |pages=157–162}}</ref> He approached natural philosophy with mathematics in a completely novel way, in that instead of a preconceived natural philosophy, his style was to begin with a mathematical construct, and build on from there, comparing it to the real world to show that his system accurately accounted for it.<ref>{{Cite book |url=https://books.google.com/books?id=1VhC63yV-WgC&pg=PA178 |title=The Scientific Revolution: The Essential Readings |date=2003 |publisher=Blackwell Pub |isbn=978-0-631-23629-0 |editor-last=Hellyer |editor-first=Marcus |edition=Elektronische Ressource |series= |location=Malden, MA |pages=178–193 |language=en}}</ref>

=== Antiquity and medieval background ===

==== Aristotle and "violent" motion ====
[[File:Statue at the Aristotle University of Thessaloniki (cropped).jpg|alt=Statue of Aristotle|thumb|205x205px|Aristotle <br/>(384–322 [[BCE]])]]
The subject of physics is often traced back to [[Aristotle]], but the history of the concepts involved is obscured by multiple factors. An exact correspondence between Aristotelian and modern concepts is not simple to establish: Aristotle did not clearly distinguish what we would call speed and force, used the same term for [[density]] and [[viscosity]], and conceived of motion as always through a medium, rather than through space. In addition, some concepts often termed "Aristotelian" might better be attributed to his followers and commentators upon him.<ref>{{Cite journal |last=Ugaglia |first=Monica |date=2015 |title=Aristotle's Hydrostatical Physics |url=https://www.jstor.org/stable/43915795 |journal=Annali della Scuola Normale Superiore di Pisa. Classe di Lettere e Filosofia |volume=7 |issue=1 |pages=169–199 |jstor=43915795 |issn=0392-095X}}</ref> These commentators found that Aristotelian physics had difficulty explaining projectile motion.{{refn|group=note|Aristotelian physics also had difficulty explaining buoyancy, a point that Galileo tried to resolve without complete success.<ref>{{Cite journal |last1=Straulino |first1=S. |last2=Gambi |first2=C. M. C. |last3=Righini |first3=A. |date=January 2011 |title=Experiments on buoyancy and surface tension following Galileo Galilei |url=http://aapt.scitation.org/doi/10.1119/1.3492721 |journal=[[American Journal of Physics]] |language=en |volume=79 |issue=1 |pages=32–36 |doi=10.1119/1.3492721 |bibcode=2011AmJPh..79...32S |hdl=2158/530056 |issn=0002-9505 |quote=Aristotle in his ''Physics'' affirmed that solid water should have a greater weight than liquid water for the same volume. We know that this statement is incorrect because the density of ice is lower than that of water (hydrogen bonds create an open crystal structure in the solid phase), and for this reason ice can float. [...] The Aristotelian theory of buoyancy affirms that bodies in a fluid are supported by the resistance of the fluid to being divided by the penetrating object, just as a large piece of wood supports an axe striking it or honey supports a spoon. According to this theory, a boat should sink in shallow water more than in high seas, just as an axe can easily penetrate and even break a small piece of wood, but cannot penetrate a large piece.|hdl-access=free }}</ref>}} Aristotle divided motion into two types: "natural" and "violent". The "natural" motion of terrestrial solid matter was to fall downwards, whereas a "violent" motion could push a body sideways. Moreover, in Aristotelian physics, a "violent" motion requires an immediate cause; separated from the cause of its "violent" motion, a body would revert to its "natural" behavior. Yet, a javelin continues moving after it leaves the thrower's hand. Aristotle concluded that the air around the javelin must be imparted with the ability to move the javelin forward. 

==== Philoponus and impetus ====
[[John Philoponus]], a [[Byzantine Greek]] thinker active during the sixth century, found this absurd: the same medium, air, was somehow responsible both for sustaining motion and for impeding it. If Aristotle's idea were true, Philoponus said, armies would launch weapons by blowing upon them with bellows. Philoponus argued that setting a body into motion imparted a quality, [[Theory of impetus|impetus]], that would be contained within the body itself. As long as its impetus was sustained, the body would continue to move.<ref>{{cite book|first=Richard |last=Sorabji |chapter=John Philoponus |title=Philoponus and the Rejection of Aristotelian Science |jstor=44216227 |year=2010 |edition=2nd |publisher=Institute of Classical Studies, University of London |isbn=978-1-905-67018-5 |oclc=878730683}}</ref>{{Rp|47}} In the following centuries, versions of impetus theory were advanced by individuals including [[Nur ad-Din al-Bitruji]], [[Avicenna]], [[Abu'l-Barakāt al-Baghdādī]], [[John Buridan]], and [[Albert of Saxony (philosopher)|Albert of Saxony]]. In retrospect, the idea of impetus can be seen as a forerunner of the modern concept of momentum.{{refn|group=note|[[Anneliese Maier]] cautions, "Impetus is neither a force, nor a form of energy, nor momentum in the modern sense; it shares something with all these other
concepts, but it is identical with none of them."<ref>{{cite book|first=Anneliese |last=Maier |author-link=Anneliese Maier |title=On the Threshold of Exact Science |publisher=University of Pennsylvania Press |year=1982 |editor-first=Steven D. |editor-last=Sargent |isbn=978-0-812-27831-6 |oclc=495305340}}</ref>{{Rp|79}}}} The intuition that objects move according to some kind of impetus persists in many students of introductory physics.<ref>See, for example:
*{{Cite journal|last1=Eaton|first1=Philip|last2=Vavruska|first2=Kinsey|last3=Willoughby|first3=Shannon|date=2019-04-25|title=Exploring the preinstruction and postinstruction non-Newtonian world views as measured by the Force Concept Inventory|journal=[[Physical Review Physics Education Research]]|language=en|volume=15|issue=1|pages=010123|doi=10.1103/PhysRevPhysEducRes.15.010123|bibcode=2019PRPER..15a0123E |s2cid=149482566 |issn=2469-9896|doi-access=free}}
*{{Cite journal |last1=Robertson |first1=Amy D. |last2=Goodhew |first2=Lisa M. |last3=Scherr |first3=Rachel E. |author3-link=Rachel Scherr|last4=Heron |first4=Paula R. L. |date=March 2021 |title=Impetus-Like Reasoning as Continuous with Newtonian Physics |url=https://aapt.scitation.org/doi/10.1119/10.0003660 |journal=[[The Physics Teacher]] |language=en |volume=59 |issue=3 |pages=185–188 |doi=10.1119/10.0003660 |s2cid=233803836 |issn=0031-921X}}
*{{Cite journal|last1=Robertson|first1=Amy D.|last2=Goodhew|first2=Lisa M.|last3=Scherr|first3=Rachel E.|author3-link=Rachel Scherr|last4=Heron|first4=Paula R. L.|date=2021-03-30|title=University student conceptual resources for understanding forces|journal=[[Physical Review Physics Education Research]]|language=en|volume=17|issue=1|pages=010121|doi=10.1103/PhysRevPhysEducRes.17.010121|bibcode=2021PRPER..17a0121R |s2cid=243143427 |issn=2469-9896|doi-access=free}}</ref>

=== Inertia and the first law ===
{{See also|Galileo Galilei#Inertia}}
The French philosopher [[René Descartes]] introduced the concept of inertia by way of his "laws of nature" in ''[[The World (book)|The World]]'' (''Traité du monde et de la lumière'') written 1629–33. However, ''The World'' purported a [[Heliocentrism|heliocentric]] worldview, and in 1633 this view had given rise a great conflict between [[Galileo Galilei]] and the [[Roman Inquisition|Roman Catholic Inquisition]]. Descartes knew about this controversy and did not wish to get involved. ''The World'' was not published until 1664, ten years after his death.<ref name=":8">{{Cite journal |last=Blackwell |first=Richard J. |date=1966 |title=Descartes' Laws of Motion |url=https://www.jstor.org/stable/227961 |journal=Isis |volume=57 |issue=2 |pages=220–234|doi=10.1086/350115 |jstor=227961 |s2cid=144278075 }}</ref>
[[File:Galileo.arp.300pix.jpg|alt=Justus Sustermans - Portrait of Galileo Galilei|thumb|184x184px|Galileo Galilei <br/>(1564–1642)]]
 
The modern concept of inertia is credited to Galileo. Based on his experiments, Galileo concluded that the "natural" behavior of a moving body was to keep moving, until something else interfered with it. In ''Two New Sciences'' (1638) Galileo wrote:<ref>{{Cite book |last=Galilei |first=G. |url=http://galileoandeinstein.physics.virginia.edu/tns_draft/tns_244to279.html |title=Dialogues Concerning Two New Sciences |publisher=Dover Publications Inc |year=1954 |editor1=Crew, H. |editor2=De Salvio, A. |pages=268 |orig-date=1638, 1914}}</ref><ref>{{Cite book |last=Galilei |first=G. |url=http://archive.org/details/twonewsciencesin0000gali |title=Two new sciences, including centers of gravity & force of percussion |date=1974 |publisher=University of Wisconsin Press |pages=217 [268] |translator-last=Drake |translator-first=S. |orig-date=1638}}</ref>{{Blockquote|text=Imagine any particle projected along a horizontal plane without friction; then we know, from what has been more fully explained in the preceding pages, that this particle will move along this same plane with a motion which is uniform and perpetual, provided the plane has no limits.}}[[File:Frans_Hals_-_Portret_van_René_Descartes_(cropped)2.jpg|alt=Portrait of René Descartes|thumb|153x153px|René Descartes <br/>(1596–1650)]]Galileo recognized that in projectile motion, the Earth's gravity affects vertical but not horizontal motion.<ref>{{Cite journal |last=Hellman |first=C. Doris |author-link=C. Doris Hellman |date=1955 |title=Science in the Renaissance: A Survey |url=https://www.cambridge.org/core/product/identifier/S0277903X00013281/type/journal_article |journal=[[The Renaissance Society of America|Renaissance News]] |language=en |volume=8 |issue=4 |pages=186–200 |doi=10.2307/2858681 |issn=0277-903X |jstor=2858681}}</ref> However, Galileo's idea of inertia was not exactly the one that would be codified into Newton's first law. Galileo thought that a body moving a long distance inertially would follow the curve of the Earth. This idea was corrected by [[Isaac Beeckman]], Descartes, and [[Pierre Gassendi]], who recognized that inertial motion should be motion in a straight line.<ref>{{Cite book|last=LoLordo|first=Antonia|url=https://www.worldcat.org/oclc/182818133|title=Pierre Gassendi and the Birth of Early Modern Philosophy|date=2007|publisher=Cambridge University Press|isbn=978-0-511-34982-9|location=New York|pages=175–180|oclc=182818133}}</ref> Descartes published his laws of nature (laws of motion) with this correction in ''[[Principles of Philosophy]]'' (''Principia Philosophiae'') in 1644, with the heliocentric part toned down.<ref name=":02">{{Cite book |last=Descartes |first=R. |url=https://www.earlymoderntexts.com/assets/pdfs/descartes1644part2.pdf |title=Principles of philosophy |year=2008 |editor-last=Bennett |editor-first=J. |at=Part II, § 37, 39. |orig-date=1644}}</ref><ref name=":8" />
[[File:Breaking_String.PNG|thumb|Ball in circular motion has string cut and flies off tangentially.]]
{{Blockquote|text=First Law of Nature: Each thing when left to itself continues in the same state; so any moving body goes on moving until something stops it.}}{{Blockquote|text=Second Law of Nature: Each moving thing if left to itself moves in a straight line; so any body moving in a circle always tends to move away from the centre of the circle.}}

According to American philosopher [[Richard J. Blackwell]], Dutch scientist [[Christiaan Huygens]] had worked out his own, concise version of the law in 1656.<ref name=":9">{{Cite journal |last1=Blackwell |first1=Richard J. |last2=Huygens |first2=Christiaan |date=1977 |title=Christiaan Huygens' The Motion of Colliding Bodies |url=https://www.jstor.org/stable/230011 |journal=Isis |volume=68 |issue=4 |pages=574–597|doi=10.1086/351876 |jstor=230011 |s2cid=144406041 }}</ref> It was not published until 1703, eight years after his death, in the opening paragraph of ''De Motu Corporum ex Percussione''.

{{Blockquote|text=Hypothesis I: Any body already in motion will continue to move perpetually with the same speed and in a straight line unless it is impeded.}}

According to Huygens, this law was already known by Galileo and Descartes among others.<ref name=":9" />

=== Force and the second law ===
[[File:Christiaan Huygens-painting (cropped).jpeg|thumb|Christiaan Huygens <br />(1629–1695)|155x155px]]
Christiaan Huygens, in his ''[[Horologium Oscillatorium]]'' (1673), put forth the hypothesis that "By the action of gravity, whatever its sources, it happens that bodies are moved by a motion composed both of a uniform motion in one direction or another and of a motion downward due to gravity." Newton's second law generalized this hypothesis from gravity to all forces.<ref>{{Cite journal |last=Pourciau |first=Bruce |date=October 2011 |title=Is Newton's second law really Newton's? |url=http://aapt.scitation.org/doi/10.1119/1.3607433 |journal=[[American Journal of Physics]] |language=en |volume=79 |issue=10 |pages=1015–1022 |doi=10.1119/1.3607433 |bibcode=2011AmJPh..79.1015P |issn=0002-9505}}</ref>

One important characteristic of Newtonian physics is that forces can [[action at a distance|act at a distance]] without requiring physical contact.{{refn|group=note|Newton himself was an enthusiastic [[alchemy|alchemist]]. [[John Maynard Keynes]] called him "the last of the magicians" to describe his place in the transition between [[protoscience]] and modern science.<ref>{{Cite journal|last=Fara|first=Patricia|author-link=Patricia Fara|date=2003-08-15|title=Was Newton a Newtonian?|url=https://www.science.org/doi/10.1126/science.1088786|journal=[[Science (journal)|Science]]|language=en|volume=301|issue=5635|pages=920|doi=10.1126/science.1088786|s2cid=170120455 |issn=0036-8075}}</ref><ref>{{Cite book|last=Higgitt|first=Rebekah|url=https://www.worldcat.org/oclc/934741893|title=Science and Culture in the Nineteenth Century: Recreating Newton |date=2015|publisher=Taylor & Francis|isbn=978-1-317-31495-0|location=New York|oclc=934741893 |page=147}}</ref> The suggestion has been made that alchemy inspired Newton's notion of "action at a distance", i.e., one body exerting a force upon another without being in direct contact.<ref>{{cite book|title=The Foundations of Newton's Alchemy: Or, "the Hunting of the Greene Lyon" |first=Betty Jo Teeter |last=Dobbs |author-link=Betty Jo Teeter Dobbs |year=1975 |publisher=Cambridge University Press |isbn=9780521273817 |oclc=1058581988 |pages=211–212}}</ref> This suggestion enjoyed considerable support among historians of science<ref>{{cite book|first=Richard |last=West |title=Never at Rest |year=1980 |publisher=Cambridge University Press |isbn=9780521231435 |oclc=5677169 |page=390}}</ref> until a more extensive study of Newton's papers became possible, after which it fell out of favor. However, it does appear that Newton's alchemy influenced his [[optics]], in particular, how he thought about the combination of colors.<ref name="Newman2016">{{cite book|first=William R. |last=Newman |author-link=William R. Newman |chapter=A preliminary reassessment of Newton's alchemy |title=The Cambridge Companion to Newton |edition=2nd |year=2016 |publisher=Cambridge University Press |isbn=978-1-107-01546-3 |pages=454–484 |oclc=953450997}}</ref><ref>{{Cite journal|last=Nummedal|first=Tara|author-link=Tara Nummedal|date=2020-06-01|title=William R. Newman. Newton the Alchemist: Science, Enigma, and the Quest for Nature's "Secret Fire"|url=https://www.journals.uchicago.edu/doi/10.1086/709344|journal=[[Isis (journal)|Isis]]|language=en|volume=111|issue=2|pages=395–396|doi=10.1086/709344|s2cid=243203703 |issn=0021-1753}}</ref>}} For example, the Sun and the Earth pull on each other gravitationally, despite being separated by millions of kilometres. This contrasts with the idea, championed by Descartes among others, that the Sun's gravity held planets in orbit by swirling them in a vortex of transparent matter, ''[[Aether theories|aether]]''.<ref>{{Cite book |last=Aldersey-Williams |first=Hugh |url=https://www.worldcat.org/oclc/1144105192 |title=Dutch Light: Christiaan Huygens and the Making of Science in Europe |date=2020 |publisher=Picador |isbn=978-1-5098-9333-1 |location=London |oclc=1144105192}}</ref> Newton considered aetherial explanations of force but ultimately rejected them.<ref name="Newman2016"/> The study of magnetism by [[William Gilbert (physician)|William Gilbert]] and others created a precedent for thinking of ''immaterial'' forces,<ref name="Newman2016" /> and unable to find a quantitatively satisfactory explanation of his law of gravity in terms of an aetherial model, Newton eventually declared, "[[Hypotheses non fingo|I feign no hypotheses]]": whether or not a model like Descartes's vortices could be found to underlie the ''Principia''<nowiki/>'s theories of motion and gravity, the first grounds for judging them must be the successful predictions they made.<ref>{{Cite journal |last=Cohen |first=I. Bernard |author-link=I. Bernard Cohen |date=1962 |title=The First English Version of Newton's Hypotheses non fingo |url=https://www.jstor.org/stable/227788 |journal=[[Isis (journal)|Isis]] |volume=53 |issue=3 |pages=379–388 |doi=10.1086/349598 |jstor=227788 |s2cid=144575106 |issn=0021-1753}}</ref> And indeed, since Newton's time [[Mechanical explanations of gravitation|every attempt at such a model has failed]].

=== Momentum conservation and the third law ===
[[File:JKepler (cropped).jpg|alt=Portrait of Johannes Kepler|left|thumb|151x151px|Johannes Kepler <br/>(1571–1630)]]
[[Johannes Kepler]] suggested that gravitational attractions were reciprocal — that, for example, the Moon pulls on the Earth while the Earth pulls on the Moon — but he did not argue that such pairs are equal and opposite.<ref>{{Cite book |last=Jammer |first=Max |title=Concepts of Force: A Study in the Foundations of Dynamics |date=1999 |publisher=Dover Publications |isbn=978-0-486-40689-3 |location=Mineola, N.Y. |pages=91, 127 |oclc=40964671 |author-link=Max Jammer |orig-date=1962}}</ref> In his ''[[Principles of Philosophy]]'' (1644), Descartes introduced the idea that during a collision between bodies, a "quantity of motion" remains unchanged. Descartes defined this quantity somewhat imprecisely by adding up the products of the speed and "size" of each body, where "size" for him incorporated both volume and surface area.<ref>{{Cite web |last=Slowik |first=Edward |date=2021-10-15 |title=Descartes' Physics |url=https://plato.stanford.edu/entries/descartes-physics/ |access-date=2022-03-06 |website=[[Stanford Encyclopedia of Philosophy]]}}</ref> Moreover, Descartes thought of the universe as a [[Plenum (physics)|plenum]], that is, filled with matter, so all motion required a body to displace a medium as it moved. 

During the 1650s, Huygens studied collisions between hard spheres and deduced a principle that is now identified as the conservation of momentum.<ref>{{Cite journal |last=Erlichson |first=Herman |date=February 1997 |title=The young Huygens solves the problem of elastic collisions |url=http://aapt.scitation.org/doi/10.1119/1.18659 |journal=[[American Journal of Physics]] |language=en |volume=65 |issue=2 |pages=149–154 |doi=10.1119/1.18659 |bibcode=1997AmJPh..65..149E |issn=0002-9505}}</ref><ref>{{Cite journal |last=Smith |first=George E. |date=October 2006 |title=The vis viva dispute: A controversy at the dawn of dynamics |url=http://physicstoday.scitation.org/doi/10.1063/1.2387086 |journal=[[Physics Today]] |language=en |volume=59 |issue=10 |pages=31–36 |doi=10.1063/1.2387086 |bibcode=2006PhT....59j..31S |issn=0031-9228}}</ref> [[Christopher Wren]] would later deduce the same rules for [[Elastic collision|elastic collisions]] that Huygens had, and [[John Wallis]] would apply momentum conservation to study [[Inelastic collision|inelastic collisions]]. Newton cited the work of Huygens, Wren, and Wallis to support the validity of his third law.<ref>{{Cite journal |last=Davies |first=E. B. |date=2009 |title=Some Reflections on Newton's "Principia" |url=https://www.jstor.org/stable/25592244 |journal=[[The British Journal for the History of Science]] |volume=42 |issue=2 |pages=211–224 |doi=10.1017/S000708740800188X |jstor=25592244 |s2cid=145120248 |issn=0007-0874}}</ref>

Newton arrived at his set of three laws incrementally. In a [[De motu corporum in gyrum|1684 manuscript written to Huygens]], he listed four laws: the principle of inertia, the change of motion by force, a statement about relative motion that would today be called [[Galilean invariance]], and the rule that interactions between bodies do not change the motion of their center of mass. In a later manuscript, Newton added a law of action and reaction, while saying that this law and the law regarding the center of mass implied one another. Newton probably settled on the presentation in the ''Principia,'' with three primary laws and then other statements reduced to corollaries, during 1685.<ref>{{cite book|first=George E. |last=Smith |chapter=Newton's Laws of Motion |title=The Oxford Handbook of Newton |isbn=978-0-199-93041-8 |publisher=Oxford University Press |doi=10.1093/oxfordhb/9780199930418.013.35 |date=December 2020 |editor-first1=Eric |editor-last1=Schliesser |editor-first2=Chris |editor-last2=Smeenk |oclc=972369868 |no-pp=yes |at=Online before print}}</ref>

=== After the ''Principia'' ===
[[File:Page_157_from_Mechanism_of_the_Heaven,_Mary_Somerville_1831.png|thumb|right|Page 157 from ''Mechanism of the Heavens'' (1831), [[Mary Somerville]]'s expanded version of the first two volumes of Laplace's ''Traité de mécanique céleste.''<ref>{{Cite journal |last=Patterson |first=Elizabeth C. |date=December 1969 |title=Mary Somerville |url=https://www.cambridge.org/core/product/identifier/S0007087400010232/type/journal_article |journal=[[The British Journal for the History of Science]] |language=en |volume=4 |issue=4 |pages=311–339 |doi=10.1017/S0007087400010232 |s2cid=246612625 |issn=0007-0874 |quote=In no sense was it a mere translation of Laplace's work. Instead it endeavoured to explain his method ". . . by which these results were deduced from one general equation of the motion of matter" and to bring the reader's mathematical skill to the point where the exposition of Laplace's mathematics and ideas would be meaningful—then to give a digest in English of his great work. Diagrams were added when necessary to the original text and proofs of various problems in physical mechanics and astronomy included. ... [F]or almost a hundred years after its appearance the book continued to serve as a textbook for higher mathematics and astronomy in English schools.}}</ref> Here, Somerville deduces the inverse-square law of gravity from [[Kepler's laws of planetary motion]].]]
Newton expressed his second law by saying that the force on a body is proportional to its change of motion, or momentum. By the time he wrote the ''Principia,'' he had already developed calculus (which he called "[[Fluxion|the science of fluxions]]"), but in the ''Principia'' he made no explicit use of it, perhaps because he believed geometrical arguments in the tradition of [[Euclid]] to be more rigorous.<ref>{{Cite book |last=Baron |first=Margaret E. |url=https://www.worldcat.org/oclc/892067655 |title=The Origins of Infinitesimal Calculus |publisher=Pergamon Press |date=1969 |isbn=978-1-483-28092-9 |edition=1st |location=Oxford |oclc=892067655 |author-link=Margaret Baron}}</ref>{{Rp|page=15}}<ref>{{Cite journal |last=Dunlop |first=Katherine |date=May 2012 |title=The mathematical form of measurement and the argument for Proposition I in Newton's Principia |url=http://link.springer.com/10.1007/s11229-011-9983-8 |journal=[[Synthese]] |language=en |volume=186 |issue=1 |pages=191–229 |doi=10.1007/s11229-011-9983-8 |s2cid=11794836 |issn=0039-7857}}</ref> Consequently, the ''Principia'' does not express acceleration as the second derivative of position, and so it does not give the second law as <math>F = ma</math>. This form of the second law was written (for the special case of constant force) at least as early as 1716, by [[Jakob Hermann]]; [[Leonhard Euler]] would employ it as a basic premise in the 1740s.<ref>{{cite web|url=https://plato.stanford.edu/entries/newton-principia/ |title=Newton's ''Philosophiae Naturalis Principia Mathematica'' |website=[[Stanford Encyclopedia of Philosophy]] |date=2007-12-20 |accessdate=2022-03-06 |first=George |last=Smith}}</ref> Euler pioneered the study of rigid bodies<ref>{{Cite journal |last1=Marquina |first1=J. E. |last2=Marquina |first2=M. L. |last3=Marquina |first3=V. |last4=Hernández-Gómez |first4=J. J. |date=2017-01-01 |title=Leonhard Euler and the mechanics of rigid bodies |url=https://iopscience.iop.org/article/10.1088/0143-0807/38/1/015001 |journal=[[European Journal of Physics]] |volume=38 |issue=1 |pages=015001 |doi=10.1088/0143-0807/38/1/015001 |bibcode=2017EJPh...38a5001M |s2cid=125948408 |issn=0143-0807}}</ref> and established the basic theory of fluid dynamics.<ref>{{Cite book |last=Hesse |first=Mary B. |url=https://www.worldcat.org/oclc/57579169 |title=Forces and Fields: The Concept of Action at a Distance in the History of Physics |date=2005 |publisher=Dover Publications |isbn=978-0-486-44240-2 |edition=Dover reprint |location=Mineola, N.Y. |pages=189 |oclc=57579169 |author-link=Mary Hesse |orig-date=1961}}</ref> [[Pierre-Simon Laplace]]'s five-volume ''[[Traité de mécanique céleste]]'' (1798–1825) forsook geometry and developed mechanics purely through algebraic expressions, while resolving questions that the ''Principia'' had left open, like a full theory of the [[Tide|tides]].<ref>{{cite web|url=https://plato.stanford.edu/entries/newton/ |title=Isaac Newton |website=[[Stanford Encyclopedia of Philosophy]] |first=George |last=Smith |date=2007-12-19 |access-date=2022-03-06 |quote=These advances in our understanding of planetary motion led Laplace to produce the four principal volumes of his ''Traité de mécanique céleste'' from 1799 to 1805, a work collecting in one place all the theoretical and empirical results of the research predicated on Newton's ''Principia''. From that time forward, Newtonian science sprang from Laplace's work, not Newton's.}}</ref>

The concept of energy became a key part of Newtonian mechanics in the post-Newton period. Huygens' solution of the collision of hard spheres showed that in that case, not only is momentum conserved, but kinetic energy is as well (or, rather, a quantity that in retrospect we can identify as one-half the total kinetic energy). The question of what is conserved during all other processes, like inelastic collisions and motion slowed by friction, was not resolved until the 19th century. Debates on this topic overlapped with philosophical disputes between the metaphysical views of Newton and Leibniz, and variants of the term "force" were sometimes used to denote what we would call types of energy. For example, in 1742, [[Émilie du Châtelet]] wrote, "Dead force consists of a simple tendency to motion: such is that of a spring ready to relax; [[Vis viva|living force]] is that which a body has when it is in actual motion." In modern terminology, "dead force" and "living force" correspond to potential energy and kinetic energy respectively.<ref>{{Cite journal |last=Reichenberger |first=Andrea |date=June 2018 |title=Émilie Du Châtelet's interpretation of the laws of motion in the light of 18th century mechanics |url=https://linkinghub.elsevier.com/retrieve/pii/S0039368118300177 |journal=[[Studies in History and Philosophy of Science Part A]] |language=en |volume=69 |pages=1–11 |doi=10.1016/j.shpsa.2018.01.006|pmid=29857796 |bibcode=2018SHPSA..69....1R |s2cid=46923474 }}</ref> Conservation of energy was not established as a universal principle until it was understood that the energy of mechanical work can be dissipated into heat.<ref>{{Cite journal |last=Frontali |first=Clara |date=September 2014 |title=History of physical terms: "energy" |url=https://iopscience.iop.org/article/10.1088/0031-9120/49/5/564 |journal=[[Physics Education]] |volume=49 |issue=5 |pages=564–573 |doi=10.1088/0031-9120/49/5/564 |bibcode=2014PhyEd..49..564F |s2cid=122097990 |issn=0031-9120}}</ref><ref>{{Cite web |last=Gbur |first=Greg |author-link=Greg Gbur |date=2018-12-10 |title=History of the Conservation of Energy: Booms, Blood, and Beer (Part 1) |url=https://skullsinthestars.com/2018/12/10/history-of-the-conservation-of-energy-booms-blood-and-beer-part-1/ |access-date=2022-03-07 |website=Skulls in the Stars |language=en}} {{Cite web |date=2018-12-29 |title=History of the Conservation of Energy: Booms, Blood, and Beer (Part 2) |url=https://skullsinthestars.com/2018/12/28/history-of-the-conservation-of-energy-booms-blood-and-beer-part-2/ |access-date=2022-03-07  |language=en}} {{Cite web |date=2019-08-25 |title=History of the Conservation of Energy: Booms, Blood, and Beer (Part 3) |url=https://skullsinthestars.com/2019/08/24/history-of-the-conservation-of-energy-booms-blood-and-beer-part-3/ |access-date=2022-03-07 |language=en}}</ref> With the concept of energy given a solid grounding, Newton's laws could then be derived within formulations of classical mechanics that put energy first, as in the Lagrangian and Hamiltonian formulations described above.

Modern presentations of Newton's laws use the mathematics of vectors, a topic that was not developed until the late 19th and early 20th centuries. Vector algebra, pioneered by [[Josiah Willard Gibbs]] and [[Oliver Heaviside]], stemmed from and largely supplanted the earlier system of [[Quaternion|quaternions]] invented by [[William Rowan Hamilton]].<ref>{{Cite journal |last1=Silva |first1=Cibelle Celestino |last2=de Andrade Martins |first2=Roberto |date=September 2002 |title=Polar and axial vectors versus quaternions |url=http://aapt.scitation.org/doi/10.1119/1.1475326 |journal=[[American Journal of Physics]] |language=en |volume=70 |issue=9 |pages=958–963 |doi=10.1119/1.1475326 |bibcode=2002AmJPh..70..958S |issn=0002-9505}}</ref><ref>{{cite book|first=Karin |last=Reich |author-link=Karin Reich |chapter=The Emergence of Vector Calculus in Physics: The Early Decades |pages=197–210 |title=Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar |editor-first=Gert |editor-last=Schubring |series=Boston Studies in the Philosophy of Science |volume=187 |publisher=Kluwer |isbn=978-9-048-14758-8 |oclc=799299609 |year=1996}}</ref>