I. A Body is in it self indifferent to Rest or Motion.

IF we should imagin, 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, [Page 2] 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 sever'd; 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 it self indifferent to Rest [Page 3] or Motion, and being once deter­mined to Rest, it is impossible it should of it self 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 it self indifferent to Motion and Rest, and be­ing once determined to Motion, it is impossible it should determine it self 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 meer 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 our selves, when we move our selves, 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 mov'd 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 [Page 6] 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 consider'd, 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, [Page 7] 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.

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 it self this same Presence, and to remain Qui­escent, than to produce new Presences, and conserve it self in Motion. Whence it is to be concluded, that as a Body [Page 8] from the very time, it hath been once determin'd to Rest, is sufficiently de­termin'd always to keep it self in the same Presence; so also from the very moment, it hath been once determin'd to Motion, it is sufficiently determin'd always to produce new Presences, and so to move it self 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 easie 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 al­ledged, That whoever moveth a Body, impresseth therein a certain Quality, call'd Impetuosity, and that as long as this Quality lasts, the Motion lasts also; but when that ceaseth, the Mo­tion ceaseth likewise. And 'tis added, [Page 9] That this Quality cannot last always, being in its nature so imperfect, that it cannot last long. Besides, it is Ob­jected, That Experience shews, 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 extinguish'd.

VII. A Finite Cause may have an Effect that lasts always.

I Say, it is very easie 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 [Page 10] 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 it self. 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, [Page 12] and say, That this Quality is so weak in its own nature, that it destroys it self; I do maintain, that, after this Quality shall have been destroy'd, the Motion notwithstanding must continue for the reasons already deliver'd, 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: [Page 13] 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 Axle-tree be slender and well turned, than when 'tis big [Page 14] and irregular. A Stone is cast much farther in the Air, than in Water. But I shall endevour 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 [Page 15] 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 allow'd 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 explain'd 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 allow'd me to take this as a Postulatum of Geometry, That, after a Body is once mov'd, 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.

XII. A free Body cannot be deter­min'd to move in a Curve Line, nor with unequal celerity.

THence it follows, that a Body can­not be determin'd to move in a Curve Line, or with unequal velocity; but that every Body that's free, con­tinues to move in a straight Line, and with an uniform velocity.

XIII. Every Body that moveth about a Center, endevours to recede from it.

THence it appears, that the Axiom is very true, which saith, That every Body moving round indevours to recede from the Center of its Motion: As a Stone in a Sling, which maketh the hand sensible of its endevour to move in a straight Line, and consequently to go from the hand, which is the Cen­ter of its Motion. So also do the drops of water, or the grains of sand, which fly out into a straight 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 immoveable, 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 Center 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 Center 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 straight Line towards the extremities of the World.

XV. How a Body may be move'd circularly.

BUt if a Body fastened, as might be a Ball suspended by a Thread, or a Wheel fix'd upon its Axis, or, if it be liquid and inclosed in a Vessel, as Water in a Bason; 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 Center of the Vessel, in which 'tis in­closed. So also if two Bodies being tyed 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 [Page 22] 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 it self. And then, if these opposite parts are mov'd unequally, so that the one be carried a little faster one way; then this Body, besides its circular Motion about it self, 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 enterlaced, whilst it moveth with an incredible swiftness about its own Center.

XVI. One Body moving against another Body giveth it its whole Motion.

NOw let us take a Body moving in a straight 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 determin 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, [Page 24] 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 consider'd, that in this encounter of two Bodies [Page 25] 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 it self. 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 imagin them to have feeling, and capable to resent pain when they are struck: For then it is manifest, that the Body a coming to hit against B, will it self feel as much pain as the Body B; as we see, that a hand striking another hand, does as much hurt to it self 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 [Page 26] 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 then 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 it self moveless.

LEt us resume our Example. The Body A is mov'd with one degree [Page 27] of velocity towards a; and there it meets in a straight 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 [Page 28] 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 it self 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 absol [...]te and resp [...]ctive velocity of a Body. I call that absolute velocity, which is consider'd in a Body compared with the Space wherein it moveth; and res [...]ctive, that which is considered in two Bodies compared together, by [Page 29] 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 call'd one degree of absolute celerity; but if we compare it with the Body A, which is mov'd 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 measur'd, not by the abso­lute, but the respective Velocity; be­cause [Page 30] 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 mov'd with two degrees of absolute celerity, and [Page 31] 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 [Page 32] of necessity be, that the Body a turn only back with one degree of celerity towards A, because it is carryed 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 swi [...]tness 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 [Page 33] 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 straight 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 straight 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 [Page 34] 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 shew'd, 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 [Page 35] 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 it self 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 [Page 36] 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 prov'd § 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 [Page 37] 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 it self 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 [Page 38] 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 easie 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 straight Line be drawn e d, then 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 [Page 39] 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 hindring their Motion. But the Motion which car­rieth the Ball towards c a, is not at all hindred by the Body B. There is but the Motion, which carried the Body A towards e B, that is hindred 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 [Page 40] 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 tbe oblique Motion is composed of two Motions.

BEcause this is important, it will be worth while to explain it yet after another manner. Let us imagin 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 imagin, 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 [Page 41] 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 oblique 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 [Page 42] 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 ob­lique 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 [Page 43] 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 hight 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 [Page 44] greatest; and the force of these ob­lique percussions is measured, as I have shew'd, 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 easie 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 Remarque upon some Citadels.

THe other Remarque is upon what I have seen in some of our Citadels, where those that have raised them have preferr'd 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 joyning 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 oblique Bat­teries of Guns the same advantage and the same force, which direct Batteries have: So that a Bullet, which coming [Page 46] 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 easie, thus slopingly to carry away a stone, which giving hold-fast 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 oblique Mo­tion, 'tis easie 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 oblique Percussions, and even of the direct [Page 47] 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. Joyn the Centers 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, [Page 48] 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 easie 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 [Page 49] the Motions made in pleno, I believe it would be easie 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 [Page 50] straight 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 wondred, 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 verifie 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 sheweth, 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 [Page 51] 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, alledged 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 [Page 52] 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 shews 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 subtil 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 [Page 53] rarifi'd, 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 doe 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 it self 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 [Page 54] 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 [Page 55] 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 it self and to open before, and to follow and close it self 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 it self, 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 [Page 57] 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 [Page 58] 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 [Page 59] 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 [Page 60] of other impediments, the com­binations whereof may infinitely diver­sifie all the effects of the percussions. Only I may say, that making a cer­tain Hypothesis, which appears natu­ral enough, it may be shewed 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 it self 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 [Page 62] 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 thred do vibrate to and fro. Somewhat also should be spoken of the Motion of Liquors, as well of [Page 63] 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.