The Lawes of MotionIsaac Newtonc.2,280 wordsThe Newton ProjectFalmer2010Newton Project, University of Sussex
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c. 1665 - c. 1672, c. 2,291 words, 6 pp.6 pp.MS Add. 3958.5, ff. 81r-83v, Cambridge University Library, Cambridge, UKUKCambridgeCambridge University LibraryPortsmouth CollectionMS Add. 3958.5, ff. 81r-83vc. 1665 - c. 1672EnglandEnglishUnknown HandIsaac NewtonScienceMathematicsCatalogue information compiled by Yvonne SantacreuYvonne Santacreu tagged transcriptionCatalogue information updated by Michael HawkinsProofed by Robert IliffeTranscription audited by Michael Hawkins
81 The Lawes of Motion How solitary bodyes are moved.
Sect: 1Of Place motion velocity & force.. There is an uniform extension, space, or expansion continued every way wthwithout bounds: in wchwhich all bodyes are, each in severall ꝑtsparts of space possessed adequately felled by theirꝑtsparts of it: wchwhichꝑtsparts of space possesse adequately filled by ymthem are their places. And their passing out of one place or ꝑtpart of space into another, through all yethe intermediate space is their pmotion. Which motion is done wthwith more or lesse velocity acordinglyaccordingly as tis done through more or lesse space in equal times or through equall spaces in more or lesse time. But yethe motion it selfe & yethe force to pꝫpersevere in ytthat motion is more or lesse accordingly as yethe factus of yethe bodys bulk into its velocity is more or lesse. And ytthat force is equivalent to that motion wchwhich it is able to beget or destroy.
2wthwithwtwhat velocity a body moves severall ways at once.. The motion of a body tends one way directly & severall other ways obliqly. As if yethe body A move directly towards yethe point B it also moves obliquely towards all yethelines BC, BD, BE & wchwhich passe through ytthat point B: & shall arrive to ytthatymthem all iat yethe same time. Whence its velocity towards ymthem is in such proportion as its distance from them ytthat is, as AB, CAC, AD, AE &c.
3How two progressive motions are joyned into one.. If a body A move towards B wthwiththe velocity R, & by yethe way hath some new force done to it wchwhich had yethe body rested would have propelded it towards C wthwithyethe velocity S. Then making AB∶AC∷R∶S, & Completing yethe Parallelogram BC yethebody shall move in yethe Diagonall AD & arivearrive at yethe point D in yethe same timewthwith this compound motion in yethe same time it would have arrived at yethe point B wthwith its single motion.
4Of centers & axes of motion & yethe motion of those centers. In every body there is a certaine point, called its center of motion about wchwhich if yethe body bee any way circulated yethe endeavours of its ꝑtsparts every way from yethe center are exactly counterpoised by opposite endeavours. And yetheprogressive motion of yethe body is yethe same wthwithyethe motion of this its center wchwhich always moves in a streight line & uniformly wnwhenyethe body is free from occursions wthwith other bodys. And so doth yethe common center of two bodys; wchwhich is found by dividing yethe distance twixt their propper centers in reciprocall proportion to their bulk. And so yethe common center of 3 or more bodys &c. And all yethe lines passing through these centers of motion are axes of motion.
5Of circular motion and velocity about those axes.. The angular quantity of a bodys circular motion & velocity is more or lesse accordingly as yethebody makes one revolution in more or lesse time but yethe reall quantity of its circular motion is more or lesse accordingly as yethe body hath more or lesse power & force to pꝫpersevere in ytthat motion; wchwhich motion divided by yethe bodys bulke is morethe reall quantity of its circular velocity. Now to know yethereall quantity of a bodys circular motion & velocity about any given axis EF; Suppose it hung upon yethe two end E & F of ytthat axis as upon two poles: And ytthat another globular body of yethe same bignesse, whose center is A, is so placed ytthatyethe circulating body shall hit it in yethepoint F & strike it away in yethe line BAG (wchwhich lyeth in yethe same plane wthwithone of yethe circles described about yethe axis EF) & thereby just loose all its owne motion. Then hath yethe Globe gotten yethe same quantity of progressive motion & velocity wchwhichyethe other had of circular, wchwhichits velocity being yethe same wthwithytthat of yethe point C wchwhich describes a circle touching yethe line BG. The Radius DC of wchwhich circle I may therefore call yethe radius of Circular motion or velocity about ytthat axis EF. And yethecircle described wthwithyethe said Radius of Circulation in that plane wchwhich cuts yethe axis EF perpendicularly in yethe center of motion I call yethe Equator of circulation about that axis, and those circles wchwhich passe through yethe poles, medridians &c.
6WthWithwtwhat velocity a body circulates about severall axes at once.. A body circulates about one axis (as PC) directly & about severall other axes (as AC, BC, &c) obliquely. And yethe angular quantity of its circulations about those axes (PC, AC, BC &c) are as yethe sines (PC, AD, BE, &c) of yethe angles wchwhich those axes make wthwithyethe Equator (FG) of yethe principall & direct axis (PC).
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7How two circular motions are joyned into one.. If a body circulates about yethe axis AC wthwithyethe angular quantity of velocity R: & some new force is done to it, wchwhich, if yethe body had rested, would have made it circulate about another axis BC, wthwithyethe angular quantity of velocity S. Then in yethe plane of yethe two axes, & in one of those two opposite angles (made by yethe axes) in wchwhichyethe two circulations are contrary one to another, (as in yethe angle ACB). I find such a point P from wchwhichyetheperpendiclars (PK, PH) let fall to those axes are bee reciprocally proportional to yethe angular velocitys about those axes, (ytthat is PK∶PH∷R∶S). And drawing yethe line PC, it shall bee yethe new ab axis about wchwhichyethe compound motion is pꝫperformed. And yethe summe of & CD when yethe perpendiculars PH & PK fall on divers sides of yethe axis PC, otherwise their difference, is yethe angular quantity of circulation about ytthat axis: WchWhich in yethe angle tends contrary to yethe circulation about yethe axis
8In wtwhat cases a circulating body pꝫpersevere in yethe same state & in wtwhat it doth not.. Every body keepes yethe same reall quantity of circular motion & velocity so long as tis not opposed by other bodys. And it keeps yethe same axis too if yethe endeavour from yethe axis wchwhichyethe two opposite quarters twixt yethe Equator & every meridian of motion have, bee exactly counterpoised by onthe opposite endeavours of yethe 2 side quarters; & ynthen also its axis doth always keepe parallel to it selfe. But if yethe said endeavours from yethe axis bee not exactly counterpoised by such opposite endeavours: ynthen for want of such counterpoise yethepꝫprevalent ꝑtsparts shall by little & little get further from yethe axis & draw nearer & nearer to such a Counterpoise, but shall never bee exactly counterpoised. And as yethe axis is continually moved in yethe body, so it continually moves in yethe space too wthwith some kind or other of spirall motion; always drawing nearer & nearer to a center or parallelisme wthwith it selfe, but never attaining to it. Nay tis so far from ever keeping parallel to it sel selfe, ytthat it shall never bee twice in yethe same position.How Bodys are Reflected.9Some names & letters defined.. Suppose yethe bodys A & α did move in yethe lines DA & Eα till they met in the point B: ytthat BC is yethe plane wchwhich toucheth them in yethe point of contact B: ytthatyethe velocity of yethe Body A towards yethe said plane of contact is B, & yethe motion AB; & ytthatyethe change wchwhich is made by reflection in ytthat velocity & motion is X & AX. Suppose also ytthat from yethe body A its center of motion two lines are drawne yethe one AB to yethe point of contact yethe other AC to yethe plane of Contact: ytthatyethe intercepted line BC is F: that yetheaxis of motion wchwhich is perpendicular to yethe plane ABC & its Equator are called yethe axis & Equator of reflected circulation: ytthatyethe radius of ytthat Equator is G: ytthatyethe reall quantity of velocity about ytthat axis is D, & yethe AD: ytthatyethe change wchwhich reflection makes in ytthatvelocity & motion is y & AY: And ytthatyethe correspondent lines & motions of yethe other body α are β, αβ, ξ, αξ, φ, γ, δ, αδ, ν & αν. Lastly for brevity sake suppose ytthat. And . Observing ytthat at yethe time of reflection if in either body yethe center of motion doth move from yethe plane of contact, or those ꝑtsparts of it nearest yethe point of contact doe circulate from yethe plane of contact: ynthenyethe said motion is to bee esteemed negative & yethe signe of its velocity B, β, D or δ must bee made negative in yethe valor of Q.10The Rule for Reflection. The velocitys B, β, D & δ & they only are directly opposed & changed in Reflection; & ytthat according to these rules . . . & . Which mutations X ξ Y & ν tend all of them from yethe plane of Contact. And these four rules I gather thus: The whole velocity of yethetwo points of contact towards one another perpendicularly to yetheplane of contact is Q (arising ꝑtpartly from yethe bodys progressiove velocity B & β & ꝑtpartly from their circular D & δ): And yethe same points are reflected one from another wthwithyethe same quantity of such velocity. So ytthatyethe whole change of all ytthat their velocity wchwhich is perpendicular to yethe plane of contact is Q. Which change must bee distributed amongst yethe foure opposed velocitys B, βD & δ proportionably to yethe easinesse (or smallnesse of resistance) wthwithwchwhich those velocitys are changed, ytthat is, proportionably to , , , & . Soe ytthat. that is . . . & .11The conclusion. In wtwhat method yethe precedent rules must be used.. Now if any two reflecting bodys A & α, wthwithyethe quantity 83 of their progressive & angular motions; & their position at their meeting & consequently their point & plane of contact &: be given: to know how those bodys shall bee reflected, First find B & βby sec 2. Then yethe lines F & φby sec & yethe axis of reflected circulation by sec 9. ynthen & their Radij G & γ by sec 5. Then their angular quantity of velocity about yethe axes of reflected circulation by sec 6, & yethe reall quantity D & δ by sec 5. Then P & Q by sec 9 Then X, ξ, Y & ν by sec 10. Then yethebodys new progressive determinations & velocitys by sec 3. Then yethe angular quantity of ytthat circulation (Y & ν)wchwhich is generated by reflection by sec 5. And lastly yethe new axes & angular quantity of velocity about ymthem by sec 7.Some Observations about Motion.Only those bodyes which are absolutely hard are exactly reflected acording to those rules. Now the bodyes here amongst us (being an aggregate of smaller other bodyes) haue a relenting softnesse & springynesse, wchwhichmakes their contact be for some time & in more points then one. And yethe touching surfaces during yethe time of contact doe slide one upon another more or lesse or not at all acording to their roughnesse. And few or none of these bodyes haue a springynesse soe strong, as to force them one from another wthwithyethe same vigor that they came together. Besides ytthat their motions are continually impeded & slackened by yethe mediums in wchwhichthey move. Now hee ytthat would prescribe rules for yethereflections of these compound bodies, must consider in how many points yethe two bodies touch at their meeting, yethe position & pression of every point, wchwhich their planes of contact &c: & how all these are varyed every moment during yethe time of contact by yethe more or lesse relenting softnesse or springynesse of those bodies & their various slidings. And also what effect yethe air or other mediums compressed betwixt yethe bodies may haue.2 These are some cases of Reflections of bodies absolutely hard to which these rules extend not: As when two bodies meet wthwith their angular point, or in more points then one at once; Or with their superficies. But these cases are rare.3 In all reflections of any bodies wthwithwtwhat ever this rule is true that yethe commmon center of tiwo or more bodies changeth not its state of motion or rest by yethe reflection of those bodies one amongst another.4 Motion may be lost by reflection. As if two equall bodies Globes A & αwthwith equall motions from D & δ done in the perpendicular lines DA & δα, hit one another when the center of yethe body α is inyethe line DA. Then yethe body A shall loose all its motion & yet yethe motion of α is not doubled. For completing yethe square Bβ, yethe body α shall move in yethe Diagonall αC, & arrive at C but at yethe same time it would haue arrived at βwthwithout reflection. see yethe third section.5 Motion may be gained by reflection. for if yethe bodyFor if the body α return wthwithyethe same motion back again from C to α. The two bodyes A & α after reflection shall regain yethesame equall motions in yethe lines AD & αδ (though backwards) wchwhichthey had at first.