A DISCOURSE OF LOCAL MOTION.  
Undertaking to Demonstrate The LAWS of MOTION, And withal to prove, That of the SEVEN RULES Delivered By M. Des-Cartes on this SUBjECT, He hath Mistaken SIX.  
BY A. M.  
Englished out of French.  
London, Printed by W. G. and are to be sold by Moses Pitt at the White-Hart in Little-Britain, 1670.   

The PREFACE.  
I Pretend not to celebrate in this place the Mechanics, and to set forth the advantages, which the Knowledge of Motion affords us. It is sufficiently known, that all the Productions, which come either from the Industry of Men, or from the Causes of Nature, are made no otherwise but by Motion. So that it is not possible to penetrate into the Secrets of Nature, nor to succeed in the Invention and Practice of Arts, without the assistance of the Mechanics, that is, without the Knowledge of the Laws of Motion.  
Neither do I undertake here to go through this whole Subject. It is too vast to be comprised in so brief a Discourse as this is intended. I have confined myself to what may be called the Elements of this Knowledge; and I insist particularly on considering the Communication which is made of Motion in Percussions.  
'Tis true, that this Subject hath been handled by very Eminent Men; but I take it quite otherwise in hand, me thinks, than they have done. For, without making any particular Hypothesis, I make it my business to search in the very sources of Nature the Causes of all the Effects we find in Motions, and I undertake to give the Demonstrations of them, which, without supposing any Experiment at all, are only founded upon incontrolable Principles of pure Metaphysics.  
The Design will doubtless appear bold to those, who know the difficulty, there is in thus preventing Experiments, and in prescribing to Nature such Laws as she is afterwards to observe.  
It may also come to pass, that the Difference, to be found between the Rules, which I endeavour to establish here, and those which M. Des-Cartes hath laid down in his Principles of Physics, will furnish matter to exercise the curiosity of those, who love the Philosophy of that Author, and engage them, to make search, wherein my Paralogisms may consist, in regard that the ratiocinations, which I employ, are so opposite to those, which many have taken hitherto for true Demonstrations. For I advow, that of the Seven Rules of Motion, which M. Des-Cartes delivers, there is but One only, which agreeth with mine: So that either this Philosopher hath not hit aright in this point, or I myself am fallen into considerable Errors.  
Mean time I cannot be ignorant of what hath been published through all France touching the Rules of percussion, which have been proposed by some famous Mathematicians of the Royal Academies of London and Paris. If there be honour in inventing any thing new in Sciences, I do not contest with these persons about that, which they pretend to, of having found the Secret of the Laws of Motion. I willingly and fully yield it to them, and I claim nothing therein. Yet this I can say, that 'tis now three years, that I gave abroad all what I deliver here in this Discourse; and that, if my Rules be compared with theirs, there may possibly be found conformity enough to make Men believe that I have lighted together with them upon the truth: But there will also be found difference enough to make Men judge, that I have not learned it from them. Besides, that they have done no more than merely to propose their Rules without proving them; whereas I undertake to demonstrate all those, I advance. And although M. Hugens hath given us hopes of publishing shortly a Book, wherein he would prove all his Rules; yet not withstanding, without comparing myself to so excellent a person, I dare affirm, that his Method will be quite different from mine; forasmuch as he hath already explained himself sufficiently, to give us to understand, that his Demonstrations are grounded upon particular Hypotheses.  
However, I have already declared myself about the little pretention, I have to the glory of passing for the Inventor of these things: I leave it altogether to those Gentlemen; and if they will allow me a share therein, I shall receive it as a favour, and take it kindly, that they will acknowledge I have hit upon their thoughts, or at least not considerably shot besides the Mark.   

A TABLE of the HEADS.  

I. A Body is in itself indifferent to Rest or Motion. pag. 1.  
II. If a Body be once at Rest, it will ever remain therein. pag. 2.  
III. And if it be once in Motion, it continues also to move always. pag. 3.  
iv That Rest is not a mere Negation. pag. 4.  
V That there is as much positive Action in Rest, as in Motion. pag. 5.  
VI Objections. pag. 8.  
VII. A Finite Cause may have an Effect that lasts always. pag. 9  
VIII. This Quality, which is called Impetuosity, lasts always. pag. 11.  
IX. The Bodies which we move, do cease to move, because they are impeded. pag. 13.  
X. A Demand for the safety of the following Demonstrations. pag. 14.  
XI. A Body receiving successively many Determinations, remains only affected with the last. pag. 16.  
XII. A free Body cannot be determined to move in a Curve Line, nor with unequal celerity. pag. 17.  
XIII. Every Body that moveth about a Centre, endeavours to recede from it. pag. 19  
XIV. The Stars cannot move of themselves. pag. 20.  
XV. How a Body may be moved circularly. pag. 21.  
XVI. One Body moving against another Body giveth it its whole Motion. pag. 23.  
XVII. In the meeting of two Bodies there is made a percussion, which is mutual, and equally received in both. pag. 24.  
XVIII. A moving Body, meeting with another Body that is quiescent, gives it all its Motion, and remains itself moveless. pag. 26.  
XIX. What is meant by absolute and respective velocity. pag. 28.  
XX. The Percussions are as the respective Velocities. pag. 29.  
XXI. Two Bodies meeting one another, turn back, making an exchange of their velocity. pag. 31.  
XXII. Two Bodies moving towards the same places, continue after their encounter by exchanging their velocities. pag. 33.  
XXIII. An hard Body coming to hit another Body that cannot be shaken, is reflected with its whole Motion. pag. 35.  
XXIV. The Angle of Reflection is equal to the Angle of Incidence. pag 38.  
XXV. It may be imagined, that the obliqne Motion is composed of two Motions. pag. 40.  
XXVI. A Remark upon the Argument of P. Riccioli. pag. 42.  
XXVII. A Remark upon some Citadels. pag. 45.  
XXVIII. A general Rule of all Percussions. pag. 46.  
XXIX. There is always equal quantity of respective Motion. p. 48.  
XXX. The midst of two Bodies is always uniformly moved in a direct Line. pag. 49.  
XXXI. All these Rules are true, whether the Bodies be equal, or not. pag. 50.  
XXXII. A Body moveth in pleno as freely as in vacuo. pag. 52.  
XXXIII. Motions diminish little by little in the Air. pag. 54.  
XXXIV. The Percussions of equal Bodies are made in pleno as in vacuo. pag. 56.  
XXXV. When the Bodies are unequal, the percussions are made in pleno otherwise than in vacuo. pag. 57  
XXXVI. The Percussions of unequal Bodies cannot be reduced to one General Rule. pag. 59  
XXXVII. Of Refraction. pag. 61.  
XXXVIII. The Conclusion. pag. 62.  
An Appendix containing a Review of this Discourse, made by the Author himself. pag. 67.    

A DISCOURSE OF LOCAL MOTION.  

I. A Body is in itself indifferent to Rest or Motion.  
IF we should imagine, that in the World there were nothing corporeal but one or two Balls, and sever from the same whatever might cause any kind of secret Commerce, whereby the one might attract or repel the other: Or, if we should consider such Balls free from all kind of particular Determination; without Levity, without Gravity; in Vacuo, or at least in a Space altogether uniform, where nothing were that might carry them rather this than that way, or hinder them to move freely, if they should happen to be propelled towards a place: Then should we conceive these Balls to be absolutely indifferent to touch one another, or to be severed; to be here, or there; forasmuch as they would find nothing more in one place than in another, and consequently be equally indifferent for Rest or Motion.   

II. If a Body be once at Rest, it will ever remain therein.  
ANd so if we further conceive, that one of these Balls is at Rest, having been put in that state by some Cause, that hath power to stir or stop Bodies; we at the same time conceive, it will eternally remain at Rest, if there be not some new Cause displacing it, by putting it into Motion; because this Ball, being of itself indifferent to Rest or Motion, and being once determined to Rest, it is impossible it should of itself quit that Rest, and fall to Motion: Wherefore it must needs continue forever in that Rest, if nothing happen to make it change that state.   

III. And if it be once in Motion, it continues also to move always.  
BY the same Reason we must conceive, that if one of these Balls be put in Motion, by some Cause or other, it will continue to move forever, if no new Cause come to stop it: Because this Ball being of itself indifferent to Motion and Rest, and being once determined to Motion, it is impossible it should determine itself to cease from that Motion, to take Rest: And so it must ever remain in this Motion, if nothing else come to stop it.   

iv That Rest is not a mere Negation.  
I Find that we are generally inclined to consider Rest as a Cessation of Action, and to take Motion for a positive Action, which we experiment in ourselves, when we move ourselves, or will move another Body: Whereas we conceive a Body to be at Rest from the time that no Body touches it, or that there is no other Cause which actually imprints in it this Quality or this Action necessary to Motion. And so it seems, that although a Body, being once at Rest, remains therein forever, yet it should not follow, that if it be once in Motion, it should ever persist therein; since that for to be moved there is required a positive Action, but that Rest is nothing else but a Negation or a Ceasing from Action or Motion.   

V That there is as much positive Action in Rest, as in Motion.  
BUt if the Weight of our Bodies, which we must bear; the rigidness of our Limbs, which we must bend; the agitation of the Spirits, which we must employ; and many other things make us feel some resistance, and oblige us to use some force to overcome these impediments: We cannot draw from thence any Sequel against our Hypothesis, in which we suppose, there is no impediment neither of Gravity, nor of particular Inclination, nor of any Body resisting from without. In this Case 'tis manifest, that there needs no more Action for Motion than for Rest; and that for the Rest of a Body it is not less requisite, it should be put at Rest, than it is necessary for its Motion, that it should be put into it. And indeed if we consider well the nature of Rest and Motion, we shall find, that Motion may as well be called a Cessation of Rest, as Rest a Cessation of Motion; or rather we shall find, that both are something positive, in regard that Motion is a state, by which a Body corresponds successively ●o many places; or, a passing Presence; or, a sequel of divers Presences in divers places: As Rest is a state, by which a Body always corresponds to one and the same place; or, one and the same Presence in one and the same place: So that Rest as well as Motion is a State, or a Presence; but differing in this, that Rest is a State of Consistency, and a Constant Presence, always kept to be the same; whereas Motion is a Changing State, and a Transitory Presence. Now in what manner soever, these constant or passing Presences be considered, if there be any Action, or any Power, or any kind of Cause in the Body, which is to produce this Consecution of divers Presences in Motion, there is no less Action or Force necessary in Rest, to preserve the same Presence, in regard that to preserve a thing, is to produce it continually. It is therefore evident, that after the Presence hath been produced by a Body in the first instant (I speak in the sense of those, who hold, that there is a true production of these Presences) it must needs be also produced a new in the instant following by the same Body, to make it remain Quiescent: But, methinks, there is in that as much Action and as much Power, as there is for the producing in the second instant a new Presence, instead of reproducing the first.  
Nec minor est virtus, quàm quaerere, parta tueri.  
So that, whether there be to be produced every instant a new Presence for Motion, or reproduced the same Presence for Rest; it will always amount to the same, and a Body will have no less work to preserve to itself this same Presence, and to remain Quiescent, than to produce new Presences, and conserve itself in Motion. Whence it is to be concluded, that as a Body from the very time, it hath been once determined to Rest, is sufficiently determined always to keep itself in the same Presence; so also from the very moment, it hath been once determined to Motion, it is sufficiently determined always to produce new Presences, and so to move itself without ceasing.   

VI Objections.  
I Shall not stay to answer all the cavilling Scruples that may be cast in upon this Subject, seeing they are easy enough to resolve: For instance, 'tis said, That a Finite Cause cannot produce an Infinite Effect, and that this Motion would be Infinite, since it would last forever. 'Tis further alleged, That whoever moveth a Body, impresseth therein a certain Quality, called Impetuosity, and that as long as this Quality lasts, the Motion lasts also; but when that ceaseth, the Motion ceaseth likewise. And 'tis added, That this Quality cannot last always, being in its nature so imperfect, that it cannot last long. Besides, it is Objected, That Experience shows, that all Motions do cease little by little, as appeareth in a Wheel that hath been violently agitated, in a Ball that hath been rolled on a Billiard-Table, in a Ball suspended and vibrated, and in other innumerable Bodies; the Motions of which, do by little and little diminish, and at last are quite extinguished.   

VII. A Finite Cause may have an Effect that lasts always.  
I Say, it is very easy to Answer all these Objections, and many such others. If any one will maintain, that Motion is an Infinite Effect, because it lasts forever; he must also say, that Rest will be an Infinite Effect, if it thus last eternally; and, that consequently, a Finite Cause not being able to have an Infinite Effect, it must be said, that after a Man hath put a Body at Rest, this Body cannot remain in that Rest forever, but that Rest must at last cease, and the Body begin to move: which is not consonant to reason. There is a great difference between an Infinite and an Ever-during Effect. And if it be true, that a Finite Cause cannot produce an Infinite Effect; it is as true, that a Cause, how bounded soever it be, may produce an Ever-subsisting Effect, if it be not destroyed by some new Cause. For if I make a square Figure upon Wax, this Figure will last always, if nothing survene to spoil it, or to destroy the Wax itself. So that 'tis not incongruous at all, to say, that if Rest or Motion be once produced in a Body, this Rest or Motion shall last without end, if nothing come to destroy it.   

VIII. This Quality, which is called Impetuosity, lasts always.  
AS to that Quality, which is pretended to be produced in the Body by him that striketh it; 'tis all one to me, whether it be believed to be so or not: But this I say, that if that Quality be necessary, it will last forever, after it hath been once produced, and that it will never cease to be, till some new Cause destroy it. And herein the Sentiment of Vasquez 1. 2. d. 81. c. 2 & 3. is very remarkable, when he teacheth generally of all Forms, substantial and accidental, and particularly of Motion and Impetuosity, That, if they can subsist one moment without needing the influence of their first Efficient Cause, they will last always, until they be destroyed by the production of a new contrary Form. If Men will still persist in this Opinion, and say, That this Quality is so weak in its own nature, that it destroys itself; I do maintain, that, after this Quality shall have been destroyed, the Motion notwithstanding must continue for the reasons already delivered, in regard that Motion cannot cease, unless Rest be produced a new: But there must always be a positive Cause to produce a new, what Effect soever it be; whereas there needs none such to make that subsist, which is already in being. And this is the true reason, why a square Figure, made in Wax, would last eternally, if God should keep all external Agents from destroying any thing in that Wax, because this square Body of Wax could not lose this Figure, unless another Figure were produced: And as a Figure cannot begin to be a new, unless there be some positive Cause to produce it, and we also suppose, that there is none such in this Case; it must needs follow, that this first Figure, which is already produced, keeps forever the possession of its existence, 'Tis the same thing with Motion: And although this pretended Impetuosity ceaseth to be, yet the Motion, which is already produced, is not therefore to cease also, because there is no new Cause, producing Rest, and Motion cannot cease, but Rest must be produced instead thereof.   

IX. The Bodies which we move, do cease to move, because they are impeded.  
LAstly, when we see, that Bodies moved by us do in a little time cease to move, that proveth nothing against us; it being certain, that those Bodies meet with impediments to their Motion: Whence we see, that the more or the less we remove of those impediments, the more or less do those Motions continue. Thus a Ball rolleth much longer over a very smooth Alley, than in a rugged way: A Wheel turns much better, if its Axletree be slender and well turned, than when 'tis big and irregular. A Stone is cast much farther in the Air, than in Water. But I shall endeavour in the Sequel of this Discourse to explain, how all these Impediments do by little and little make the Motion of Bodies to cease.   

X. A Demand for the safety of the following Demonstrations.  
ALl I have been just now deducing about the Nature and Perpetuity of Motion, is in a manner necessary for the understanding what I pretend to demonstrate in this Discourse. But as this Question can never be handled so clearly, but that it will always be obnoxious to the Cavils of Disputants; I foresee well enough, that after all my reasonings it will doubtless so fall out, that all will not be convinced of what I shall have undertaken to prove. And besides, not being willing to clash with any, nor to leave ground to believe, that I build my Discourse upon a doubtful Principle; I declare, that for the firmness of my Demonstrations I need not it should be thought, that Motion would in effect be perpetual; so it be but allowed me (which no Man can deny) that Motion, once begun, lasts at least for some time, and continues the more uniformly, the less impediments there are to stop or diminish it. Let this Continuance of Motion be explained by the production of an impressed Quality, or by a simple Determination, or by whatever you please, 'tis indifferent to me: I only demand, it may be allowed me to take this as a Postulatum of Geometry, That, after a Body is once moved, it continues to move for some time, and that this time is considerable, when there is nothing without, able to stop or lessen the Motion. By the means of which Demand, I hope, that all the following Demonstrations will be found of full force.   

XI. A Body receiving successively many Determinations, remains only affected with the last.  
A Body not only persevereth in Rest or Motion, according as it hath once begun to be in either; but it persists also in the same kind of Motion, and with the same degree of Celerity in which it hath been put.  

For Example:  
If it have begun to move in a strait Line Eastward with one degree of Celerity, it continues to move with the same degree without ever receding a jot from the same Line: Which is evident from the same reasons, I alleged to prove, the Motion to last always.  
But it is to be Noted, that, when a Body hath successively received many different Determinations, it remains affected with the last of them, the precedent making no impression at all upon it.   

For Example:  
If a Ball be propelled with the hand, or otherwise from a to b, and that afterwards the same Ball be carried from b to d, and there abandoned; I say, that 
Fig. 1.  the Ball will continue to move towards e, in the same Line b d e, and with that celerity it moved from b to d; and that first determination, it had received from a to b, and which would have carried it to c, serveth nothing now, no more than if it had never been, because it is destroyed by this second determination.    

XII. A free Body cannot be determined to move in a Curve Line, nor with unequal celerity.  
THence it follows, that a Body cannot be determined to move in a Curve Line, or with unequal velocity; but that every Body that's free, continues to move in a strait Line, and with an uniform velocity.  

For Example:  
Let a Body be moved in a Curve Line from a through b c d e unto f (as a Stone in a Sling) and let this Body be abandoned in f, to see what will become 
Fig. 2.  of it. I say, that it will not continue to move in a Curve Line towards h, but that it will pass towards g in a strait Line, which will touch the Curve in the point f. For, however the Body were first moved from a to b, that is nothing to this last determination; and it would now move the same way, though it had only begun to move from the point b, or from c, or from d or e, or yet nearer; provided it had still in f the same degree of celerity: Because that these first Motions are so many different determinations, the latter of which destroy the former; and so the Body remains affected with the last of all: But this last did carry it towards g, that is to say, you are to take the inclination, which the Curve Line hath at the point f, and this inclination is measured by the Tangent, as Geometers know: And so it is according to this Tangent that the Body hath been determined last of all; and consequently 'tis according to this Line that it continues to move.    

XIII. Every Body that moveth about a Centre, endeavours to recede from it.  
THence it appears, that the Axiom is very true, which saith, That every Body moving round endeavours to recede from the Centre of its Motion: As a Stone in a Sling, which maketh the hand sensible of its endeavour to move in a strait Line, and consequently to go from the hand, which is the Centre of its Motion. So also do the drops of water, or the grains of sand, which fly out into a strait Line as soon as they can get free from the Wheel of the Cutler; and the like.   

XIV. The Stars cannot move of themselves.  
IT appears also, that those are deceived, who supposing the Celestial Matter liquid and , do believe, that the Sun and the other Stars may have received a first impetuosity, which lasts still, and maketh them move circularly about the Centre of the World. For it is manifest, that if an Angel, or some other Cause whatsoever, had thus moved a Star in a Circle about the Centre of the World; assoon as that Angel, or that other Cause, should abandon that Star, it would cease at the same instant to move in a Circle, and fly out into a strait Line towards the extremities of the World.   

XV. How a Body may be moveed circularly.  
BUt if a Body fastened, as might be a Ball suspended by a Thread, or a Wheel fixed upon its Axis, or, if it be liquid and enclosed in a Vessel, as Water in a Basin; then this Ball, or this Wheel, being once agitated with sufficient violence, or this Liquor being also stirred; all these Bodies will continue to move in a Circle; the Ball about the Nail, by which it is suspended; the Wheel about its Axis, where 'tis fastened; and the Liquor about the Centre of the Vessel, in which 'tis enclosed. So also if two Bodies being tied together, are equally agitated towards different places, it must needs happen, that these two opposite Bodies do move circularly about the point which is in the midst of them: And thus it is, that a Fusee, or a Whirligig, continue to move in a Circle; because the opposite parts being fastened and united among themselves, and besides moved by ones fingers two different ways, one, one way, and the other, another way; this Fusee must needs move about itself. And then, if these opposite parts are moved unequally, so that the one be carried a little faster one way; then this Body, besides its circular Motion about itself, will have another Motion, which will carry it altogether in some different Lines, according to the diversity and combination of these Determinations. And thus it is, that a Whirligig describeth by its Axis upon a Table divers figures interlaced, whilst it moveth with an incredible swiftness about its own Centre.   

XVI. One Body moving against another Body giveth it its whole Motion.  
NOw let us take a Body moving in a strait Line, and encountering another, and see, what will become of these two Bodies.  
First, In regard that Bodies are impenetrable, 'tis impossible the Body A should move, but that the Body B 
Fig. 3.  hitting against it will move also; because that otherwise these two Bodies would penetrate one another: And as I elsewhere suppose, that the Body B is there altogether indifferent, either to remain quiescent, or to take that Motion that may be given it; assoon as the Body A shall come to hit against it, it will determine it also to a like Motion: And so, there being no impediment, this Body B will take full as much Motion as the Body A had, and pass towards the same place, in the same Line, with the same celerity; and all this for the same reason, to wit, because the Bodies being impenetrable, and the Body a tending to move towards b, and then the Body B meeting there with an absolute indifferency, and free from all impediment; 'tis evident, that the Body B must move towards b with the same celerity, that the Body a did move towards the same place. And thus it seems, that there is no more difficulty to understand, that naturally a Body can move another Body, than there is to conceive, that two Bodies are impenetrable, and that one Body in its Motion may meet another.   

XVII. In the meeting of two Bodies there is made a percussion, which is mutual, and equally received in both.  
NExt, It is to be considered, that in this encounter of two Bodies there is made a certain percussion, which is nothing else but a shock or hitting of two Bodies, which meeting do hinder one another by their impenetrability. But although very often there be but one Body moving and striking, whilst the other remains moveless and receives the stroke; yet notwithstanding, the percussion is always mutual and equally received by both Bodies: So that as much as the Body a striketh the Body B, so much is it struck itself. Which we may easily conceive, if we 
Fig. 3.  suppose, that these two Bodies are altogether like in bulk, shape, and hardness, and if besides we imagine them to have feeling, and capable to resent pain when they are struck: For than it is manifest, that the Body a coming to hit against B, will itself feel as much pain as the Body B; as we see, that a hand striking another hand, does as much hurt to itself as it doth to the other, if that be as tender. The same is also to be understood, if you suppose, that there are two Nails, altogether equal, half fixed, the one to the Body a, and the other to the Body B, and that in the Motion of the Body a against B the two heads of the Nails do meet; for than we conceive, that in this percussion these two Nails are struck deeper in; and that there is no reason to make us believe, the Nail B should be sunk deeper than a: On the contrary, since both the Nails are equal, and equally sharp, and the Bodies equally hard, without any other difference; the two Nails must needs equally be struck in, and the one as much fixed as the other. Thus we may make it a general Maxim; That when two Bodies are struck, the percussion is mutual and equal on both sides.   

XVIII. A moving Body, meeting with another Body that is quiescent, gives it all its Motion, and remains itself moveless.  
LEt us resume our Example. The Body A is moved with one degree of velocity towards a; and there it meets in a strait Line the Body B, and by the percussion communicates to it its Motion, which will carry the Body B with one degree of velocity towards b, according to what hath been demonstrated in § 16. Since therefore the percussion, which the Body B receiveth, is of one degree, that is, 
Fig. 3.  capable to carry the Body B with one degree of celerity towards b; it must needs be, that the percussion, which the Body a receiveth at the same time, be also of one degree, that is, be able to carry the Body a with one degree of celerity toward the opposite parts, namely, towards A. (For these percussions strike and drive the two Bodies toward the opposite places, the one, towards b, the other, towards A.) And as the Body a had already one degree of impetuosity or swiftness to go towards b; and that now it receiveth such another to return towards A; this Body must needs remain moveless at the Point a, without going forwards or backwards, forasmuch as it is equally driven toward the opposite places. Thus in this percussion the Body a gives its motion and celerity to the Body B, and mean while remains itself moveless.   

XIX. What is meant by absolute and respective velocity.  
NOw let us suppose, that the two Bodies move towards one another in the same Line; the one, from b with one degree of celerity towards B; the other, from A with the same degree of celerity towards a, where they meet; and let us see, what will follow. The percussion will here not only be of one degree, but of two; and to understand this, we are to distinguish between the absolute and respective velocity of a Body. I call that absolute velocity, which is considered in a Body compared with the Space wherein it moveth; and respective, that which is considered in two Bodies compared together, by which velocity these two Bodies mutually approach to, or recede from, one another. As in our Example: If we consider the Body b, comparing it to 
Fig. 3.  the Space, for Example, of one Foot, which it moveth in one Minute; that shall be called one degree of absolute celerity; but if we compare it with the Body A, which is moved on its part towards a with the same degree of absolute celerity, passing also one foot through, in one minute; then the respective celerity of both will be of two degrees; because they mutually approach one another with this celerity, and make in one minute two feet, by which they were before distant from one another.   

XX. The Percussions are as the respective Velocities.  
NOw the force of the Percussion is to be measured, not by the absolute, but the respective Velocity; because the percussion proceeds only, as we have said, from the impenetrability of two Bodies, which mutually approaching one another do hinder their first Motion, and receive also new impressions. Whence it appears also, that the percussion will be so much the greater, by how much swifter that mutual approach shall be made. So that the Percussions are always as the respective Velocities, caeteris omnibus paribus. Thus two Bodies approaching, each with one degree of absolute celerity, and making each a foot on its part in one minute; 'tis manifest, that the percussion, which each Body will receive in a B, will be the same, that it would be, if one had remained moveless 
Fig. 3.  in A, until the other were come forth from b to A with two degrees of absolute celerity, making in one minute both the two feet, that are from b unto A: Forasmuch as the respective celerities are still the same, whether we suppose, that whilst the one remains moveless in A, the other is moved with two degrees of absolute celerity, and maketh both the feet in one minute; or, that both Bodies move, by approaching to one another, each with one only degree of velocity; so that in one minute they shall have made by their approach both the feet, that were betwixt them at the beginning of the minute.   

XXI. Two Bodies meeting one another, turn back, making an exchange of their velocity.  
IT being therefore certain, that the percussion, which is made in this encounter, is of two degrees; and that each of these Bodies receiveth in this shock an impression, that would carry them with two degrees of velocity towards the opposite places; that is to say, that the Body a receiveth a stroke, which would carry it towards A with two degrees of velocity; and that the 
Fig. 3.  Body B receiveth likewise one, which would carry it with the same two degrees of velocity towards b: It must of necessity be, that the Body a turn only back with one degree of celerity towards A, because it is carried by two impressions unequal and altogether contrary; by one, of two degrees, towards A, which it receiveth in the percussion; by the other, of one degree, towards b, which it had before; and so there remaineth to it only one free degree of impression and celerity, which carrieth it towards A. And likewise B will be carried towards b with one degree also of celerity; so that both turn back in the same Line with the same swiftness they came. If we suppose, that the one advanceth with more celerity than the other; for Example, that A moveth with one degree and an half of celerity, running one foot and an half in one minute, and that b moveth with half a degree of celerity, running half a foot only; then, the percussion being of two degrees as well as in the precedent case, since the respective celerity is the same, although the absolute ones are different, each Body must receive two degrees of impression and celerity to turn back; and by consequence, the Body B, which had half a degree only of celerity, will return towards A with one degree and an half; whereas a, which formerly had one degree and an half, will return towards b with half a degree only. And after this manner it may be proved, That two Bodies, moving towards one another in a strait Line, turn both backward, after their encounter, by making an exchange of their velocities.   

XXII. Two Bodies moving towards the same places, continue after their encounter by exchanging their velocities.  
IF the two Bodies move towards the same places in a strait Line, so that the slowest, moving first, be at last overtaken by that which moveth faster after it; then both will continue to move in the same Line towards the same place, but they will exchange their velocities. Let the Body A be moved with two degrees of velocity towards b, making in one minute two feet as far as to a. At the same time let the Body 
Fig. 4.  B be moved in the same Line with one degree of celerity, making only one foot as far as to b, and that there it be overtaken by the Body a. The force of the percussion being measured, as I have showed, by the respective celerity; this percussion must here be but of one degree, because the respective celerity is but of one degree, seeing that these two Bodies do not approach one another but with this degree of celerity, and that in one minute they make, the one in respect of the other, but one foot of space, which was betwixt both at the beginning. Now, since the Body b had, before, one degree of celerity, which carried it towards a, and that now in the percussion it receiveth another towards the same places; it must move with two degrees and make two feet as far as to b; whereas the Body a, which before had two degrees of velocity towards b, and receiveth now one, to turn back towards B, is constrained to go towards a with one degree of velocity.   

XXIII. An hard Body coming to hit another Body that cannot be shaken, is reflected with its whole Motion.  
IF the Body, which is struck, be altogether unshakeable, we must see, what force the percussion will have, and what will become of the percutient Body. Let us suppose, that the Body A do move with one degree of velocity towards a, and that there it meet the 
Fig. 5.  Body b, indifferent to move, yet so as that betwixt both there be found a plate or a surface indifferent in itself to Rest or Motion, but yet impenetrable. In this case, the Body a, striking this plate, striketh also thereby the Body b, which is met with close behind it: And as I elsewhere suppose, that this plate maketh no resistance at all, but only in being impenetrable; it is manifest (by what hath been proved in § 18.) that in this encounter the Body a remains moveless in a, and that as well the plate, as the Body b, doth move towards B with one degree of velocity. But if we suppose, that when A comes to strike the plate in a, at the same time B also striketh it in b; this plate will remain moveless, in regard it is struck equally from both the opposite sides, and each Body will turn back with its degree of celerity, wherewith it came. For, as I have said, these two Bodies strike one another, notwithstanding this plate, as if there were nothing betwixt them: But if there were nothing betwixt them, they would reflect with their same degree of velocity, as hath been proved § 21. And so, although this plate be there, they will not the less be reflected. Now let us consider, that this same plate, being impenetrable, be moreover quite firm, so as to be unshakeable and inflexible; and let us move as before the two Bodies A and B so as they may strike it at the same time in a and b: I say, that after this shock each Body must reflect with the same degree of velocity; because if the plate had been indifferent, and not firm, they would have reflected, and this plate been moveless: But the same effect must follow, though we suppose, that this plate be of itself moveless, firm, and unshakeable, in regard that either way it remains without any kind of Action or Motion. If lastly we suppose, that the sole Body A moveth towards a, and hits the plate fastened and beyond shaking, it must then also be said, that the Body a turns back towards A; because it would return, if at the same time the Body B had come to hit in b; therefore it reflecteth also, when the Body B does not come, because the plate being unshakeable, causeth still the same effect in respect of the Body a, whether b strike it or not. And thus you see, how it is demonstrated, That an hard Body coming to hit another Body that is hard, inflexible, and unshakeable, is reflected with all its Motion. Which I think no Man hath yet demonstrated.   

XXIV. The Angle of Reflection is equal to the Angle of Incidence.  
HItherto we have always supposed, that the percussions are made altogether direct: Let us now see, what will happen, when the Bodies strike one another obliquely. And to make this to be more clearly understood, I shall still employ Balls or flat Bodies; and it will afterwards be very easy to understand what shall happen in Bodies that have Figures less regular. Let the Ball A be moved toward a, striking 
Fig. 6.  obliquely the unshakeable Body B. Through the Point of contact let a strait Line be drawn e d, than a parallel A c a, the perpendiculars A e, a c, next c a or B d equal to c A, or to B c. I say, that the Ball will turn back by the Line a a, so as the Angle of Reflection a a d is always equal to the Angle of Incidence A a e. For proof, let us consider, that the Ball A receives at once two strokes or impressions; one, driving it towards e with 
Fig. 7.  one degree of velocity, and the other towards c with two degrees; it must then move in the diagonal A a, and there hit the Body B. But the force of percussion will be but of one degree, because the percussion is only made, as I have often said, by the impenetrability of the two Bodies hindering their Motion. But the Motion which carrieth the Ball towards c a, is not at all hindered by the Body B. There is but the Motion, which carried the Body A towards e B, that is hindered by the Body B, and consequently all the force of this percussion is measured by this respective velocity, which maketh the Body A approach towards the Line e B. In this case also, the percussion is the same, as if the Body A had only moved from c to a with this sole degree of celerity; and so in the percussion it must turn back with the same degree of swiftness, and be carried towards c a, as before it was carried towards e B, whilst the other Motion remains all entire towards a d. Whence it follows, that the Ball reflecteth in the Line a a.   

XXV. It may be imagined, that the obliqne Motion is composed of two Motions.  
BEcause this is important, it will be worth while to explain it yet after another manner. Let us imagine the Body B moveless; and another Body A a, moving parallel betwixt the Lines A c, a d, and hitting the moveless Body: 
Fig. 8.  Then (according to what hath been already proved in § 23.) this Body will be reflected wholly towards A a with its same velocity. Besides, let us imagine, that this Body is hollow Channel-wise, and that in this Channel there is a Ball rolling from A towards a, in such a manner that in the same time, wherein the whole Body moveth from A a unto the moveless Body B, the Ball maketh in its Channel the way A c. Thus, whilst the whole Body shall turn back after the percussion, the Ball shall continue to move in its Channel from c towards a with its same velocity. But the true way, which this Ball shall have made, will be A a a, so as the Angle of Reflection will be equal to the Angle of Incidence; in regard that as well the Lines A c, c a, as A e, d a, are equal. But it is manifest, that the same percussion, and consequently the same reflection would be made, if the Ball had hit immediately coming from A to a, than if it were the Channel A a that had hit, whilst the Ball had rolled in the Channel without any interruption. Whence we may conclude, that in all obliqne Motion when one Body hits another obliquely, we may distinguish as 'twere two Motions; one, which we shall call perpendicular, which carrieth it to hit the Body, and which receiveth a change in the percussion; the other, lateral, by which the Body only slideth against the other without hitting it, and which by consequence remains entire after the percussion. Here the perpendicular Motion is that, which carrieth the Ball towards e d, whose velocity is measured by the perpendicular A e; and the lateral Motion is measured by the parallel A c, which continueth after the percussion towards c a.   

XXVI. A Remark upon the Argument of P. Riccioli.  
I Cannot hold to make here two remarks on the occasion of the obliqne percussion. One is, touching the Argument, which one of the greatest Men of our Age maketh to decide the Question about the Motion of the Earth. He pretends, that if heavy Bodies did descend by a Curve Line, such as Galileo describeth, the percussions of heavy Bodies would not be made, as we see they are. For, according as a Body falls from a greater height, it striketh the more forcibly, so as the percussion will be ten or twenty times stronger of a fall of a hundred or four hundred times the height: Mean time in the Hypothesis, which this Author, of whom I speak, opposeth, the force of the percussion should be, thinks he, always the same, at least there would be no sensible difference, what difference soever there should be found in the heights of the descents; because the heavy Body would go in this Curve Line with an almost uniform velocity: And the force of the percussions being always proportionate to the velocity, he concludes, that the velocities being always equal in what height soever it be, the percussions would be so too. But this Argument is not concluding, because, the celerity remaining always the same, the percussions may diminish if they be made obliquely: And if we conceive, that the Bullets a, b, c, hit the Wall in d, all with the same celerity, but some more obliquely than others, the percussion certainly of that Bullet which hits more directly will be the greatest; and the force of these obliqne percussions is measured, as I have showed, by the perpendiculars c e, b f, a g. So that the Bullet c may hit so obliquely, that it shall only graze along the 
Fig. 9  Wall without having any considerable effect. Thus although the weights, which are supposed to fall in a Curve Line, should be moved in almost an uniform velocity, yet they would for all that hit more forcibly, falling from a greater height, because then the percussion would be more direct: And in effect if one shall go to make the calculation of it (which is very easy to do even upon that, which this Author hath made in Astronomia Reformata) it will be found, that the obliquity of these Motions is always fully that, which is requisite to make that diversity, which we see in the percussions of falling Bodies.   

XXVII. A Remark upon some Citadels.  
THe other Remark is upon what I have seen in some of our Citadels, where those that have raised them have preferred the pleasingness to the Eye before the strength of the Walls, when instead of making them all even and plain, they have diversified them with many Ornaments of Stones advancing above others; and besides, have cut each Stone Diamond-wise, or at least have made a Brim therein by notching them round about; so as the joining Stones leave betwixt them an hollowness after the manner of rural Archirecture: I say, that however all this variety may please the Eye, it is disadvantageous for Defence. For, these sinkings and sallies of Stones give to the obliqne Batteries of Guns the same advantage and the same force, which direct Batteries have: So that a Bullet, which coming side-ways would only graze along the Wall, if it were all flat and even, when it shall meet the sallies of those advancing stones, will have the same effect, and make as great a breach, as if it had hit direct perpendicularly; and even a greater, because it will be more easy, thus slopingly to carry away a stone, which giving holdfast to a Bullet is not supported by the others, than it would be, if it had been struck directly against the thickness of the Wall. But let us return to our subject.   

XXVIII. A general Rule of all Percussions.  
AFter we have made this distinction of two Motions in the obliqne Motion, 'tis easy to make a General Rule, explaining all the Effects of Percussions. You may here see the Proposition, 
Fig. 10, 11, 12, 13, 14, 15, 16.  together with the Figures, which express all the possible Cases of obliqne Percussions, and even of the direct ones, when the Bodies are not unshakeable. Let the Body A be moved towards a with the velocity of A a, and the Body B with the velocity of B b in the Line B b; or let one of the two be moveless, so as B b be but a point. Let the encounter be in a b. Join the Centres by the Line a b, continued both ways, if need be. Let there be drawn the perpendiculars A c, B d. We may here distinguish two Motions in each Ball; the one perpendicular, as if the Body A had moved from c unto a, and the Body B from d unto b: the other is the lateral, which carries the Body A towards c, and the Body B towards d, and this lateral remains entire after the percussion in both Bodies; whereas, the whole percussion being made by perpendicular Motions, these perpendicular Motions must be changed according to what hath been demonstrated, that is, the Body b will take the perpendicular Motion and velocity c a, and the Body a will take the velocity and Motion d b. Let therefore be drawn the Line a e equal and parallel to A c, and the Line e a equal and parallel to d b; I say, that the Body a shall move, after the percussion, in the Line a a with the velocity a a. Likewise let there be drawn b f equal and parallel to B d, and the Line f b equal and parallel to c a; I say, that the Body b shall move, in the Line b b with the velocity b b; and this needs no new proof.   

XXIX. There is always equal quantity of respective Motion.  
IT is to be observed, that 'tis not true, that there is always as much absolute Motion after the percussion, as there was before. But 'tis easy to demonstrate, that the respective Motion is always the same; so that the Bodies recede one from another after the percussion, as fast as they approached before it. Thus taking two equal times before and after percussion, the distance A B is always the same to the distance a b. And after I shall have also explained the Motions made in pleno, I believe it would be easy to me to prove, that having a respect generally to all the Bodies that are in the whole World, there is at present as much respective Motion, neither more nor less, than there was in the beginning of the Creation of the Universe.   

XXX. The midst of two Bodies is always uniformly moved in a direct Line.  
IT is also to be observed, that the point of the middle between two Bodies is always moved uniformly in a direct Line, drawing without any interruption towards the same places. Thus taking two equal times, before and after percussion, and supposing that o is the point of the middle between the two Bodies at the time of the percussion; and O being also the middle of the two Bodies before the percussion, as o is after; O o o will be in a strait Line, and O o will be equal to o o: Which I stay not to demonstrate, though that may be done Geometrically.   

XXXI. All these Rules are true, whether the Bodies be equal, or not.  
IT will perhaps be wondered, that in all the preceding Rules I have not made any mention of the Equality or Inequality of the Bodies, that strike one another. And it seems at first, that, to verify what I have been saying, I must suppose the Bodies to be perfectly Equal: For, if the one be bigger than the other, all those Rules must vary; and experience showeth, that a great Body striking a lesser that was before quiescent, the great Body ceaseth not from continuing to move after the shock, though it moveth more slowly, and quite contrarily, if it be the lesser Body that striketh, it reflecteth with a part of its velocity. But if I have here omitted to distinguish these Cases of Equality and Inequality of Bodies, I have done it with consideration: I have all along confounded the Velocity and the Motion, and I designed to give the Reader to understand, that all these Rules are true, whether the Bodies be Equal or not. And if notice be taken of the force of the reason, alleged by me in § 16. it is always the same, although the Bodies be of different Magnitudes. For, the Body struck being altogether indifferent to remain at Rest or to take Motion, and the whole effect of the percussion proceeding from the impenetrability of Bodies; if we suppose the Body struck to be greater, provided that all the parts thereof be well united together, it must move with the same velocity, with which the Body striking moveth, by the same reason, it doth so when they are equal, that is, because they are impenetrable, and the Body percutient cannot move forwards, unless the Body percussed, which is before, take all its velocity: And as otherwise the Greater is as indifferent, as the Equal Body, for Rest and Motion, certainly the Greater will make no more resistance than the Equal, forasmuch as neither of them will make not the least of all. If Experience shows the contrary, 'tis because the Motion of Bodies, which we see, are not made in vacuo, as we have hitherto supposed, but they are moved in a space filled with some fluid Body, such as the Air or some other yet more subtle substance. Now therefore we are to consider the Motion which is made of solid Bodies in a fluid substance.   

XXXII. A Body moveth in pleno as freely as in vacuo.  
IF this substance be perfectly fluid, that is, if all its parts, as well small as great, are flexible and liquid; if besides, this same substance be perfectly full, so as it cannot be condensed or rarified, as a Sponge is compressed or dilated by reason of its pores; if lastly it be enclosed in some place, whence it cannot at all issue: Then an Hard Body, that shall have begun to move in the midst of this Liquor, will continue to do it as freely as in vacuo, and will go to the extremities of the Liquor, where meeting with a firm unshakeable Body, it is reflected with the same velocity, and so it will move forever. The reason of it is, that when an hard Body moveth in a liquid substance, there is made a reflection of impetuosity, which communicateth itself in a moment to all the parts of the Liquor, in such a manner that the Body moving driveth all the parts of the Liquor, that are found before it, and so it should stop, if nothing else did survene (by § 18.) But these parts of the Liquor being thrust, do thrust others, and so on to the extreme, where is made a reflection, by which the parts that are found after the hard Body, are thrust with the same force to follow this same Body: Because, all the Liquor being shut in, and not capable to be condensed, and there being no vacuity; 'tis not possible, that all the parts which go before the Body should move, but the parts which follow the same Body must move also with the same force. Thus as much as the Hard Body is retarded by the parts preceding, so much it is driven back by those which follow; and by consequence, if the Motion have once begun, it must continue as if it were in vacuo. Whence it appears, that those who will prove the necessity of a vacuum from Motion, do not reason well.   

XXXIII. Motions diminish little by little in the Air.  
BUt if the hard Bodies are in a spongious Liquor, capable of compression, or if this Liquor be not so well bounded in, but that the extrimeties will yield a little; then the Motion will not be perpetual, but diminish by degrees, and be at last quite extinct. For the hard Body will find more resistance by the anteriour parts of the Liquor, than it will receive of impulse by the posteriour; because the Liquor, which is before, being compressed, or the extremities yielding, the communication of the impression cannot be made perfectly; and so the posteriour parts of the Liquor will not be so much thrust as the anteriour, and consequently will not so much thrust the hard Body, as the anteriour ones retard it. And 'tis for this reason that all the Motions cease in the Air and Water, or in other Liquors, because 'tis certain, that the Air is spongious and easily compressed: And that the Liquors are not bounded but by the Air when they are abroad, or at least by the sides of some Vessel that can yield and bend a little. For we know by certain experience, that Glass Vessels will stretch, and even those of Iron and Brass will bend to the strokes made upon them.   

XXXIV. The Percussions of equal Bodies are made in pleno as in vacuo.  
THe percussions, that are made of Bodies thus moving in Liquors, differ in something from those that are made in vacuo. To understand the Cause thereof, we are to note, that, when an hard Body is moved in a Liquor, it also communicateth its Motion to the same Liquor, in such a manner that it moveth also in following the hard Body, so as to divide itself and to open before, and to follow and close itself after the Body. And if the Body by any accident should come to lose its Motion; yet the Liquor being thus determined to move, would give again to that Body its Motion, and carry it away with itself, in some such manner as Rivers carry away with them the floating Wood If therefore a Body comes to hit another equal to it, the phoenomena will happen as in vacuo; because these two equal Bodies, being encompassed with the same quantity of Liquor, as much as the Liquor of the Body percussed hinders this same Body percussed from moving freely, so much an equal quantity of Liquor, which is about the Body percutient, driveth also anew the percutient as well as the percussed: Thus their Motion after the percussion will be made as in vacuo, forasmuch as the resistance of the Liquor from the Body percussed, is precisely recompensed by the impulsion of the Liquor from the Body percutient.   

XXXV. When the Bodies are unequal, the percussions are made in pleno otherwise than in vacuo.  
BUt if the Body percutient be greater, it must needs receive not so great an effect from the percussion, as the other, because 'tis carried away with more violence by the Liquor which environs it: For we see, that a Beam carried away by the Stream of a River hath much more effect, when it comes to hit against a Bridge or a Mill, than a stick would have being carried down by the same River; although the Beam should not move swifter than the Stick: And that, because the Beam coming to hit, is also carried by the great quantity of water surrounding it, whereas the Stick is but little so, by reason of the small space it taketh up, and of the little water, by which 'tis carried away. Thus therefore if the little Body be at Rest, and the greater come to hit it; this greater by communicating its motion to the smaller will not so be stopped as to become moveless, as it would do in vacuo; but it will continue to move, and to follow, though more slowly. On the contrary if the great one be quiescent, the smaller, after it shall have hit the other, and communicated to it a part of its Motion, will be reflected losing a part of its velocity. And from all this it appears, that Aristotle is not so much to be blamed as some pretend, when, to explicate the causes of the continuation of the Motions we see, he hath made use of the medium, that is, of the liquid substance wherein our Bodies are moved.   

XXXVI. The Percussions of unequal Bodies cannot be reduced to one General Rule.  
TO determine the excess, which there may be in the resistances or in the greatest impressions of these unequal Bodies, I esteem a thing not to be undertaken, at least if we consider the Bodies such as we have them among us; because that depends from the resistance made by the liquid Bodies, wherein the Hard Bodies, we see, are moved; from the facility, they have to be condensed or rarified; and from many other things, that cannot be known to us, no more than an infinity of other impediments, the combinations whereof may infinitely diversify all the effects of the percussions. Only I may say, that making a certain Hypothesis, which appears natural enough, it may be showed by the precedent Rules, that the percussions of Bodies unequal, shall be, after the manner delivered by Monsieur Hugens in the late 
* Published March 18. 1669. in French; and inserted also in the Philosophical Transactions, Numb. 46. pag. 927. Which Rules of Motion are conform to those of Dr. Christ. Wren, Printed in the Transactions, Numb. 43. pag. 867.  Journal Des Scavans. But I shall not stay longer upon that; I may possibly meet with another opportunity to discourse more amply of it.   

XXXVII. Of Refraction.  
THere appears also from what I have been explaining, the reason of the Refractions, that are made when an hard Body passeth out of one Liquor into another of different consistence. For if the Hard Body passeth out of a more free Liquor into one that is less so, it will lose somewhat of its velocity in the passage, finding more resistance in the Liquor which is before, than it feels itself thrust by that which follows; and so the Refraction will be made by receding from the Perpendicular. On the contrary, if the Body passeth out of a more impeding Liquor into another more free, the Refraction will be made by approaching to the Perpendicular, and the Body will increase its velocity in the passage, because it is thrust more by the Liquor which follows, than 'tis detained by that which is found before. And 'tis of this augmentation of velocity, which I think no body hath as yet given the reason of. I shall not note the measures of these Refractions, because that hath been done by others, and their Demonstrations may be very well accommodated to the things, here by me advanced. Nor do I speak in this place of the Refraction of Light, because I believe, that that is made quite otherwise, that is, by causes and means altogether different; as I could make out, if I should write other Discourses of Motion.   

XXXVIII. The Conclusion.  
THere would remain something to be said of the Motion of Heavy Bodies, as well of those that fall or are projected in the Air, as of those, which roll on inclined Plains, or which being suspended by a thread do vibrate to and fro. Somewhat also should be spoken of the Motion of Liquors, as well of their fall as their prosiliency, as also of their Undulations, and the like: But all those particulars deserve so many particular Discourses. And as I think, I have found something new concerning these things, I shall not scruple to publish my thoughts for examination, if I find, that this first Discourse hath not been judged altogether unworthy to be read by persons, who take a delight in such matters.    

AN ADVERTISEMENT TO THE READER.  
THe Author of this Tract about Local Motion, having been informed by a Friend, that some persons who had read these papers, as they were coming from the Press, gave out, that he followed altogether the Doctrine of Monsieur Des Cartes, and that, al●●ough in some places he seemed 〈◊〉 oppose him without naming him, yet he did establish all the Sentiments of that Philosopher concerning this Subject: He hath thought himself obliged to undeceive those, who should believe those persons upon their word, by the following Notes, which he thought fit to annex at the end of this Tract, before it should appear in public.   

NOTES UPON THE DISCOURSE OF LOCAL MOTION.  
WHen the Author of this Discourse insisted to prove, that Motion is never destroyed but by a contrary determination, survening a new; he did sufficiently declare himself concerning the little addiction he had to this Sentiment. But as those, who have treated of this matter in Italy, England, Holland and France, agree almost all in that particular; he did not think, he was to recede from so common a Doctrine. Galileo, Gassendi, Hobbes, Regius, Magnan, Digby, Kircher, Fabri, and many others, do all maintain in some manner this Perpetuity of Motion; and they only differ in the way of proving it. Of all the proofs, alleged hitherto on its behalf, the weakest doubtless is that of Monsieur Des-Cartes. This Author pretends, that if Motion or Rest, once begun, should cease, God would be subject to change: Which is a ratiocination that maketh those smile, who have any tincture of Theology; there being none that knoweth not, that all these changes in the Creatures are made without any change in God. Apud Deum non est transmutatio, saith St Augustin; & ideo apud eum Cursus temporis, diei noctisque alternatione nequaquam variatur. And 'tis manifest, that the Cessation of Motion is no more repugnant to the Immutability of God, than the Creation of the World, or the actions of our Wills, or the vicissitude of Days and Nights. If this reason of Monsieur Des-Cartes were not so easy to answer, it would be a dangerous one; because it would prove, that God should have made from Eternity all the Motion, which is now found in the World.  
As many in the choice of Opinions have a regard to the Sentiment of the Ancients, and of the Scholastic Doctors, it may be added here, that besides what Vasquez hath said; who insists to prove at large this Perpetuity of Motion, affirming, that Motion, once begun, never ceaseth, unless there survene some new Cause, producing some positive form contrary to this Motion: Besides that, I say, three of those great Disputations of Lions, held at several times, affirm the same thing. Moreover, Aristotle is of the same mind. See, what he saith in his third Book of Meteors, Chap. 2. If a Body, that were without gravity or levity, be moved, it must needs be moved by some adventitious force, and being once so moved, it will move in infinitum: 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉. And in the fourth Book of his Physics, Text. 69. speaking of a Body, that had moved in vacuo, where 'tis supposed that there is no kind of impediment, he hath these words: No Man can say, why a Body, that were thus moved in vacuo, should stop any where. For, why should it rather stop here than there? And therefore it will not stir at all▪ or, if it begin to stir, it must move in infinitum, if something stronger doth not stop it. 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉.  
Monsieur Des-Cartes maketh very ill use of the Principle that hath been explained in § 13. That a Body which is moved about a Centre, endeavours to recede from it. It can be made out, that he hath mistaken in attempting thereby to explicate the Gravity of Bodies. Neither do we mean to allow to this Principle all that Latitude, which Monsieur Des-Cartes hath given it. And we approve much of that restriction, that hath been put by an Intelligent person, viz. That that is true in Artificial Motions, and that it cannot be so in the Natural.  
What hath been proved in § 16. and the following, sheweth, that Monsieur Des-Cartes hath been deceived in Six Rules of the 
* These Rules may be found in the second part of Des-Cartes 's Principles of Philosophy. Sect. 46, 47, 48, 49, 50, 51, 52, in my Edition, which is printed at Amsterdam, A. 1656.  Seven, which he hath delivered about Motion.  
In § 26. It is not at all pretended, to favour the Opinion of the Motion of the Earth. The Author of this Discourse is fully persuaded, that, although there were not the Holy Scriptures, the Hypothesis, which maketh the Earth moveless, is preferable to all others. He would only show, that that Argument of P. Ricciolo was not cogent. There are others that are better; especially that, which hath been prevalent on very good occasions, taken from the ●onique motion of the Loadstone.  
The 29. § is against Monsieur Des-Cartes, who hath not distinguished the Motion which is here called absolute from that, which is called respective. And when he saith, that there is always an equal quantity of Motion before or after the percussion, he means it of this absolute Motion; or it is very apparent, that he hath therein mistaken. For (in the figure 14.) before the percussion, the Motion of the two Balls A and B is A a and B b, and all the Motion after the percussion, reduced together in the sole Ball b, is only b b, the other Ball remaining moveless in a.  
When in § 21. mention is made of a substance more subtle than the Air, the Reader is not to imagine, that it is the subtle Matter of M. Des-Cartes. All Men acknowledge, that there are subtler Bodies, than the Air which we inspire. And as Aristotle in his Constitution of the Universe hath placed the Sphere of the Air above the Water, so hath he put the Fire above the Air, and the Aether above the Fire; which are all different substances, which the more subtle they are, the higher they are raised.  
It is pretended in § 37. that M. Des-Cartes hath not proved the Refractions of Bodies, and much less that of Light.  
FINIS.   

These Books are to be Sold by Moses Pitt at the White-Hart in Little-Britain.  

Folio.  
CAssandra, the famed Romance, 1667.  
Brigg's Logarithms.  
Francisci Suarez Metaphysica.   

Quarto.  
Dr John Bells Introduction to Algebra, Translated out of High-Duch into English by Thomas Branker, M. A. Also a Table of odd Numbers less than 100000, showing those that are Incomposits, and resolving the rest into their Factors or Coefficients, 1668.  
Nich. Mercatoris Logarithmo-Technia, sive Methodus construendi Logarithmos, 1668.  
Jacobi Gregorii Exercitationes Geometricae, 1668.  
Dr. Joh. Wallis Opera Mechanica, pars prima & secunda, 1670.  
Banisters Works of Chirurgery.  
Hugh Broughton's Consent of Scripture.  
Snellii 

Typhis Batavus, Lugd. Bat. 1624.  
Observat Hussiacae.    
Petrus Paaw, de ossibus Amstelreod. 1633.  
A Letter from a Gentleman of the Lord Howard's Retinue to his Friend in London. Dated at Fez, Nou. 1669. Wherein he gives a full Relation of the most remarkable passages in their Voyage thither, and of the present State of the Countries under the power of Taffaletta, Emperor of Morocco; with a brief account of the Merchandizing Commodities of Africa, as also the Manners and Customs of the People there.  
Lex ●alionis, sive Vindiciae Pharmacopoeorum: Or, A short Reply to Dr. Merret's Book, and others, written against the Apothecaries; wherein may be discovered the Frauds and Abuses committed by Doctors professing and practising Pharmacy.   

Octavo.  
Biblia Hebraea, Josephi Athias, 1661.  
Gualteri Needham, Disquisitio Anatomica de Formato Foetu, 1667.  
Buxtorfius' Fpitomy of his Hebrew Grammar, translated in English by John Davis, 1658.  
Crow, Scriptores in Scripturam: Now in the Press.  
The Fortunate Fool, or the Life of the Dr. Cenudo, 1670.  
The Adventures of Mr. T S. an English Merchant, taken Prisoner by the Turks of Algiers, and carried into the Inland Countries of Africa; with a Description of the Kingdom of Algiers, and of all the Towns and Places thereabouts; as also a Relation of the chief Commodities of the Country, and of the Actions and Manners of the People: Whereunto is annexed, an Observation of the Tide, and how to turn a Ship out of the straits Mouth the wind being westerly, 1670.  
Contemplations on Mortality, 1669.  
A Discourse written to a learned Friar by M. Des Fourneillis, showing, that the Systeme of M. Des Cartes, and particularly his Opinion concerning Brutes, does contain nothing dangerous; and that all he hath written of both, seems to have been taken out of the First Chapter of Genesis: To which is annexed the Systeme general of the Cartesian Philosophy.  
Basilius Valentinus of Natural and Supernatural Things; also of the first Tincture, Root and Spirit of Metals and Minerals, how the same are Conceived, Generated, Brought forth, Changed and Augmented: To which is added Alex. van Suchten of the Secrets of Antimony, 1670.  
Pharmacopoeia Lond. 24ᵒ. 1668.    

 geometric figures