# The Lawes of Motion

## The Lawes of Motion

How solitary bodyes are moved.

Sect: 1^{[1]}. There is an uniform extension, space, or expansion continued every way w^{th}out bounds: in w^{ch} all bodyes are, each in severall ꝑ^{ts} of space possessed {illeg} adequately felled by their ꝑ^{ts} of it: w^{ch} ꝑ^{ts} of space possesse {illeg} adequately filled by y^{m} are their places. And their passing out of one place or ꝑ^{t} of space into another, through all y^{e} intermediate space is their {p}|m|otion. Which motion is done w^{th} more or lesse velocity acordingly {sic} as tis done through more or lesse space in equal times or through equall spaces in more or lesse time. But y^{e} motion it selfe & y^{e} force to p^{ꝫ}severe in y^{t} motion is more or lesse accordingly as y^{e} factus of y^{e} bodys bulk into its velocity is more or lesse. And y^{t} force is equivalent to that motion w^{ch} it is able to beget or destroy.

2^{[2]}. The motion of a body tends one way directly & severall other ways obliqly. As if y^{e} body A move directly towards y^{e} point B it also moves obliquely towards all y^{e} lines BC, BD, BE & w^{ch} passe through y^{t} point B: & shall arrive {illeg}to y^{t}|y^{m}| all i|a|t y^{e} same time. Whence its velocity towards y^{m} is in such proportion as its distance from them y^{t} is, as AB, C|A|C, AD, AE &c.

3^{[3]}. If a body A move towards B w^{th} the velocity R, & by y^{e} way hath some new force done to it w^{ch} had y^{e} body rested would have propeld|{e}|d it towards C w^{th} y^{e} velocity S. Then making AB∶AC∷R∶S, & Completing y^{e} Parallelogram BC y^{e} body shall move in y^{e} Diagonall AD & arive {sic} at y^{e} point D in y^{e} same time w^{th} this compound motion in y^{e} same time it would have arrived at y^{e} point B w^{th} its single motion.

4^{[4]} In every body there is a certaine point, called its center of motion about w^{ch} if y^{e} body bee any way circulated y^{e} endeavours of its ꝑ^{ts} every way from y^{e} center are exactly counterpoised by opposite endeavours. And y^{e} progressive motion of y^{e} body is y^{e} same w^{th} y^{e} motion of this its center w^{ch} always moves in a streight line & uniformly w^{n} y^{e} body is free from occursions w^{th} other bodys{illeg}|.| And so doth y^{e} common center of two bodys; w^{ch} is found by dividing y^{e} distance twixt their propper centers in reciprocall proportion to their bulk. And so y^{e} common center of 3 or more bodys &c. And all y^{e} lines passing through these centers of motion are axes of motion.

5^{[5]}. The angular quantity of a bodys circular motion & velocity is more or lesse accordingly as y^{e} body makes one revolution in more {or} lesse time but y^{e} reall quantity of its circular motion is more or lesse accordingly as y^{e} body hath more or lesse power & force to p^{ꝫ}severe in y^{t} motion; w^{ch} motion divided by y^{e} bodys bulke is more the reall quantity of its circular velocity. Now to know y^{e} reall quantity of a bodys circular {illeg}|m|otion & velocity about any given axis EF; Suppose it hung upon y^{e} two end E & F of y^{t} axis as upon two poles: And y^{t} another globular body of y^{e} same bignesse, whose center is A, is so placed y^{t} y^{e} circulating body shall hit it in y^{e} {illeg}|p|oint F & strike it away in y^{e} line BAG (w^{ch} lyeth in y^{e} same plane w^{th} one of y^{e} circles described about y^{e} axis EF) & thereby just loose all its owne motion. Then hath y^{e} {illeg} Globe gotten y^{e} same quantity of progressive motion & velocity w^{ch} y^{e} other had of circular, {w^{ch}} /its\ velocity being y^{e} same w^{th} y^{t} of y^{e} point C w^{ch} describes a circle touching y^{e} line BG. The Radius DC of w^{ch} circle I may therefore call y^{e} radius of Circular motion or velocity about y^{t} axis EF. And y^{e} circle described w^{th} y^{e} said Radius of Circulation in that plane w^{ch} cuts y^{e} axis EF perpendicularly in y^{e} center of motion I call y^{e} Equator of circulation about that axis, and those circles w^{ch} passe through y^{e} poles, me{d}ridians &c.

6^{[6]}. A body circulates about one axis (as PC) directly & about severall other axes (as AC, BC, &c) obliquely. And y^{e} angular quantity of its circulations about those axes (PC, AC, BC &c) are as y^{e} sines (PC, AD, BE, &c) of y^{e} angles w^{ch} those axes make w^{th} y^{e} Equator (FG) of y^{e} principall & direct axis (PC).

7^{[7]}. If a body circulates about y^{e} axis AC w^{th} y^{e} angular quantity of velocity R: & some new force is done to it, w^{ch}, if y^{e} body had rested, would have made it circulate about another axis BC, w^{th} y^{e} angular quantity of velocity S. Then in y^{e} plane of y^{e} two axes, & in one of those two opposite angles (made by y^{e} axes) in w^{ch} y^{e} two circulations are contrary one to another, (as in y^{e} angle ACB). I find such a point P from w^{ch} y^{e} perpendic{illeg}{la}rs \(PK, PH)/ let fall to those axes are bee reciprocally proportional to y^{e} angular velocitys about those axes, (y^{t} is PK∶PH∷R∶S). And drawing y^{e} line PC, it shall bee y^{e} new {ab} axis about w^{ch} y^{e} compound motion is p^{ꝫ}formed. And y^{e} summe of $\frac{\mathrm{CH}}{\mathrm{CP}}\times \mathrm{R}$ & {CD} $\frac{\mathrm{CR}}{\mathrm{CP}}\times \mathrm{S}$ when y^{e} perpendiculars PH & PK fall on divers sides of y^{e} axis PC, otherwise their difference, is y^{e} angular quantity of circulation about y^{t} axis: W^{ch} in y^{e} angle $\begin{array}{c}\mathrm{ACP}\\ \mathrm{BCP}\end{array}$ tends contrary to y^{e} circulation about y^{e} axis $\begin{array}{c}\mathrm{AC.}\\ \mathrm{BC.}\end{array}$

8^{[8]}. Every body keepes y^{e} same reall quantity of circular motion & velocity so long as tis not opposed by other bodys. And it keeps y^{e} same axis too if y^{e} endeavour from y^{e} axis w^{ch} y^{e} two opposite quarters twixt y^{e} Equator & every meridian of motion have, bee exactly counterpoised by {on}|th|e opposite endeavours of y^{e} 2 side quarters; & y^{n} also its axis doth always keepe parallel to it selfe. But if y^{e} said endeavours from y^{e} axis bee not exactly counterpoised by such opposite endeavours: y^{n} for want of such counterpoise y^{e} p^{ꝫ}valent ꝑ^{ts} shall by little & little get further from y^{e} axis & draw nearer & nearer to such a Counterpoise, but shall {illeg}|n|ever bee exactly counterpoised. And as y^{e} axis is continually moved in y^{e} body, so it continually moves in y^{e} space too w^{th} some kind or other of spirall motion; always drawing nearer & nearer to a center or parallelisme w^{th} it selfe, but never attaining to it. Nay tis so far from ever keeping parallel to it sel selfe, y^{t} it shall never bee twice in y^{e} same position.

## How Bodys are Reflected.

9^{[9]}. Suppose y^{e} bodys A & α did move in y^{e} lines DA & Eα till they met in the point B: y^{t} BC is y^{e} plane w^{ch} toucheth them in y^{e} point of contact B: y^{t} y^{e} velocity of y^{e} Body A towards y^{e} said plane of contact is B, & y^{e} motion AB; & y^{t} y^{e} change w^{ch} is made \by reflection/ in y^{t} velocity & motion is X & AX. Suppose also y^{t} from y^{e} body A its center of motion two lines are drawne y^{e} one AB to y^{e} point of contact y^{e} other AC to y^{e} plane of Contact: y^{t} y^{e} intercepted line BC is F: that y^{e} axis of motion w^{ch} is perpendicular to y^{e} plane ABC & its Equator are called y^{e} axis & Equator of reflected circulation: y^{t} y^{e} radius of y^{t} Equator is G: y^{t} y^{e} reall quantity of velocity about y^{t} axis is D, & y^{e} AD: y^{t} y^{e} change w^{ch} reflection {m}akes in y^{t} velocity & motion is y & AY: And y^{t} y^{e} correspondent lines & motions of y^{e} other body α are β, αβ, ξ, αξ, φ, γ, δ, αδ, ν & αν. Lastly for brevity sake suppose y^{t} $\frac{1}{\mathrm{A}}+\frac{1}{\mathrm{\alpha}}+\frac{\mathrm{F}}{\mathrm{AG}}+\frac{\mathrm{\phi}}{\mathrm{\alpha \gamma}}=\mathrm{P}$. And $2\mathrm{B}+2\mathrm{\beta}+\frac{2\mathrm{DF}}{\mathrm{G}}+\frac{2\mathrm{\delta \phi}}{\mathrm{\gamma}}=\mathrm{Q}$. Observing y^{t} at y^{e} time of reflection if in either body y^{e} center of motion doth move from y^{e} plane of contact, or those ꝑ^{ts} of it nearest y^{e} point of contact doe circulate from y^{e} plane of contact: y^{n} y^{e} said motion is to bee esteemed negative & y^{e} signe of its velocity B, β, D or δ must bee made negative in y^{e} valor of Q.

10^{[10]}. The velocitys B, β, D & δ & they only are directly opposed & changed in Reflection; & y^{t} according to these rules {illeg} $\frac{\mathrm{Q}}{\mathrm{AP}}=\mathrm{X}$. $\frac{\mathrm{Q}}{\mathrm{\alpha P}}=\mathrm{\xi}$. $\frac{\mathrm{FQ}}{\mathrm{AGP}}=\mathrm{Y}$. & $\frac{\mathrm{\phi Q}}{\mathrm{\alpha \gamma P}}=\mathrm{\nu}$. Which mutations X ξ Y & ν tend all of them from y^{e} plane of Contact. And these four rules I gather thus: The whole velocity of y^{e} two points of contact towards one another perpendicularly to y^{e} plane of contact is $\frac{1}{2}$Q (arising ꝑ^{t}ly from y^{e} bodys progressio|v|e velocity B & β & ꝑ^{t}ly from their circular D & δ): And y^{e} same points are reflected one from another w^{th} y^{e} same quantity of such velocity. So y^{t} y^{e} whole change of all y^{t} their velocity w^{ch} is perpendicular to y^{e} plane of contact is Q. Which change must bee distributed amongst y^{e} foure opposed velocitys B, β D & δ proportionably to y^{e} easinesse (or smallnesse of resistance) w^{th} w^{ch} those velocitys are changed, y^{t} is, proportionably to $\frac{1}{\mathrm{A}}$, $\frac{1}{\mathrm{\alpha}}$, $\frac{\mathrm{F}}{\mathrm{AG}}$, & $\frac{\mathrm{\phi}}{\mathrm{\alpha \gamma}}$. Soe y^{t} $\frac{1}{\mathrm{A}}+\frac{1}{\mathrm{\alpha}}+\frac{\mathrm{F}}{\mathrm{AG}}+\frac{\mathrm{\phi}}{\mathrm{\alpha \gamma}}:\mathrm{Q}\colon\colon \frac{1}{\mathrm{A}}:\mathrm{X}\colon\colon \frac{1}{\mathrm{\alpha}}:\mathrm{\xi}\colon\colon \frac{\mathrm{F}}{\mathrm{AG}}:\mathrm{Y}\colon\colon \frac{\mathrm{\phi}}{\mathrm{\alpha \gamma}}:\mathrm{\nu}$. that is $\frac{\mathrm{Q}}{\mathrm{AP}}=\mathrm{X}$. $\frac{\mathrm{Q}}{\mathrm{\alpha P}}=\mathrm{\xi}$. $\frac{\mathrm{FQ}}{\mathrm{AGP}}=\mathrm{Y}$. & $\frac{\mathrm{\alpha Q}}{\mathrm{\alpha \gamma P}}$.

11^{[11]}. Now if any two reflecting bodys A & α, w^{th} y^{e} quantity 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 {illeg} find B & β by sec 2. Then y^{e} lines F & φ by sec & y^{e} axis of reflected circulation by sec 9. y^{n} & their Radij G & γ by sec 5. Then their angular quantity of velocity about y^{e} axes of reflected circulation by sec 6, & y^{e} reall quantity D & δ by sec 5. Then P & Q by sec 9 Then X, ξ, Y & ν by sec 10. Then y^{e} bodys new progressive determinations & velocitys by sec 3. Then y^{e} angular quantity of y^{t} circulation \(Y & ν)/ w^{ch} is generated by reflection by sec 5. And lastly y^{e} new axes & angular quantity of velocity about y^{m} 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, w^{ch} makes their contact be for some time & in more points then one. And y^{e} touching surfaces during y^{e} 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 w^{th} y^{e} same vigor that they came together. Besides y^{t} their motions are continually impeded & slackened by y^{e} mediums in w^{ch} they move. Now hee y^{t} would prescribe rules for y^{e} reflections of these compound bodies, must consider in how many points y^{e} two bodies touch at their meeting, y^{e} position & pression of every point, w^{ch} their planes of contact &c: & how all these are varyed every moment during y^{e} time of contact by y^{e} more or lesse relenting softnesse or springynesse of those bodies & their various slidings. And also what effect y^{e} air or other mediums compressed betwixt y^{e} 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 w^{th} 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 w^{th}|w^{t}| ever this rule is true that y^{e} co^{m}on center of t{i}|w|o or more bodies changeth not its state of motion or rest by y^{e} reflection of those bodies one amongst another.

4 Motion may be lost by reflection. As if two equall {bodies} Globes A & α w^{th} equall motions from D & δ done in the perpendicular lines DA & δα, hit one another when the center of y^{e} body α is |in| y^{e} line DA. Then y^{e} body A shall loose all its motion & yet y^{e} motion of α is not doubled. For completing y^{e} square Bβ, y^{e} body α shall move in y^{e} Diagonall αC, & arrive at C but at y^{e} same time it would haue arrived at β w^{th}out reflection. see y^{e} third section.

5 Motion may be gained by reflection. for if y^{e} body For if the body α return w^{th} y^{e} same motion back again from C to α. The two bodyes A & α after reflection shall regain y^{e} same equall motions in y^{e} lines AD & αδ (though backwards) w^{ch} they had at first.

^{[1]} Of Place motion velocity & force.

^{[2]} w^{th} w^{t} velocity a body moves severall ways at once.

^{[3]} How two progressive motions are joyned into one.

^{[4]} Of centers & axes of motion & y^{e} motion of those centers.

^{[5]} Of circular motion and velocity about those axes.

^{[6]} W^{th} w^{t} velocity a body circulates about severall axes at once.

^{[7]} How two circular motions are joyned into one.

^{[8]} In w^{t} cases a circulating body p^{ꝫ}severe in y^{e} same state & in w^{t} it doth not.

^{[9]} Some names & letters defined.

^{[10]} The Rule for Reflection

^{[11]} The conclusion. In w^{t} method y^{e} precedent rules must be used.