from its axis of motion, as its proper and adequate effect; but relativemotions, in one and the same body, are innumerable, according to the variousrelations it bears to external bodies, and like other relations, arc altogetherdestitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system whosuppose that our heavens, revolving below the sphere of the fixed stars,carry the planets along with them ; the several parts of those heavens, andthe planets, which are indeed relatively at rest in their heavens, do yetreally move. For they change their position one to another (which neverhappens to bodies truly at rest), and being carried together with theirheavens, partake of their motions, and as parts of revolving wholes,endeavour to recede from the axis of their motions.Wherefore relative quantities are not the quantities themselves, whosenames they bear, but those sensible measures of them (either accurate crinaccurate), which arc commonly used instead of the measured quantitiesthemselves. And if the meaning of words is to he determined bv their82 THE MATHEMATICAL PRINCIPLESuse, then by the names time, space, place and motion, their measures arvproperly to be understood ; and the expression will be unusual, and purelymathematical, if the measured quantities themselves are meant. Uponwhich account, they do strain the sacred writings, who there interpretthose words for the measured quantities. Nor do those less defile thepurity of mathematical and philosophical truths, who confound real quantities themselves with their relations and vulgar measures.It is indeed a matter of great difficulty to discover, and effectually todistinguish, the true motions of particular bodies from the apparent ; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yetthe thing is not altogether desperate : for we have some arguments toguide us, partly from the apparent motions, which are the differences ofthe true motions ; partly from the forces, which are the causes and effectsof the true motions. For instance, if tAvo globes, kept at a given distanceone from the other by means of a cord that connects them, were revolvedabout their common centre of gravity, we might, from the tension of thecord, discover the endeavour of the globes to recede from the axis of theirmotion, and from thence we might compute the quantity of their circularmotions. And then if any equal forces should be impressed at once on thealternate faces of the globes to augment or diminish their circular motions,from the increase or decr ase of the tensicn of 1 le cord, we might inferthe increment or decrement of their motions : and thence would be foundon what faces those forces ought to be impressed, that the motions of theglobes might be most augmented ; that is, we might discover their hindermostfaces, or those which, in the circular motion, do follow. But thefaces which follow being known, and consequently the opposite ones thatprecede, we should likewise know the determination of their motions. Andthus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing externalor sensible with which the globes could be compared. But now, if in thatspace some remote bodies were placed that kept always a given positionone to another, as the fixed stars do in our regions, we could not indeeddetermine from the relative translation of the globes among those bodies,whether the motion did belong to the globes or to the bodies. But if weobserved the cord, and found that its tension was that very tension whichthe motions of the globes required, we might conclude the motion to be inthe globes, and the bodies to be at rest ; and then, lastly, from the translation of the globes among the bodies, we should find the determination oitheir motions. But how we are to collect the true motions from theircauses, effects, and apparent differences ; and, vice versa, how from the motions, either true or apparent, we may come to the knowledge of theiicauses and effects, shall be explained more at large in the following tra<;tFor to this end it was that I composed it.OF NATURAL PHILOSOPHY.AXIOMS, OR LAWS OF MOTION.LAW I.Hvery body perseveres in its state of rest, or of uniform motion in a ri^htline, unless it is compelled to change that state by forces impressedthereon.PROJECTILES persevere in their motions, so far as they are not retardedby the resistance of the air, or impelled downwards by the force of gravityA top, whose parts by their cohesion are perpetually drawn aside fromrectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meetingwith less resistance in more free spaces, preserve then jDotions both progressive and circular for a much longer time.LAW II.The alteration of motion is ever proportional to the motive force impreused ; and is made in the direction of the right line in. which that forceis impressed.If any force generates a motion, a double force will generate double themotion, a triple force triple the motion, whether that force be impressedaltogether and at once, or gradually and successively. And this motion(being always directed the same way with the generating force), if the bodymoved before, is added to or subducted from the former motion, accordingas they directly conspire with or are directly contrary to each other ; orobliquely joined, when they are oblique, so as to produce a new motioncompounded from the determination of both.LAW III.To every action there is always opposed an equal reaction : or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.Whatever draws or presses another is as much drawn or pressed by thatother. If you press a stone with your finger, the finger is also pressed bythe stone. If a horse draws a stone tied to a rope, the horse (if I may sosay) will be equally drawn back towards the stone: for the distended rope,by the same endeavour to relax or unbend itself, will draw the horse asmuch towards the stone, as it does the stone towards the horse, and willobstruct the progress of the one as much as it advances that of the other.84 THE MATHEMATICAL PRINCIPLESIf a body impinge upon another, and by its force change the motion of (It*other, that body also (because of the equality of the mutual pressure) willundergo an equal change, in its own motion, towards the contrary part.The changes made by these actions are equal, not in the velocities but inthe motions of bodies ; that is to say, if the bodies are not hindered by anyother impediments. For, because the motions are equally changed, thechanges of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as willbe proved in the next scholium.COROLLARY I.A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it wovld describe the sides, by those forcesapart.If a body in a given time, by the force M impressedapart in the place A, should with an uniform motion /be carried from A to B ; and by the force N impressedapart in the same place, should be carried from A to c ~。)C ; complete the parallelogram ABCD, and, by both forces acting together,it will in the same time be carried in the diagonal from A to D. Forsince the force N acts in the direction of the line AC, parallel to BD, thisforce (by the second law) will not at all alter the velocity generated by theother force M, by which the body is carried towards the line BD. Thebody therefore will arrive at the line BD in the same time, whether therorce N be impressed or not ; and therefore at the end of that time it willhe found somewhere in the line BD. By the same argument, at the endof the same time it AY ill be found somewhere in the line CD. Therefore itwill be found in the point D, where both lines meet. But it will move in;i right line from A to D, by Law I.COROLLARY II.And hence is explained the composition of any one direct force AD, outof any two oblique forces AC and CD ; and, on the contrary, the resolution of any one direct force AD into two oblique forces AC andCD : which composition and resolution are abundantly confirmed from,mechanics.As if the unequal radii OM and ON drawn from the centre O of anywheel, should sustain the weights A and P by the cords MA and NP ; andthe forces of those weights to move the wheel were required. Through therentre O draw the right line KOL, meeting the cords perpendicularly inA and L; and from the centre O, with OL the greater of the distancesOF NATURAL PHILOSOPHY.OK arid OL, describe a circle, meeting the cordMA in D : and drawing OD, make AC paral- "^lei and DC perpendicular thereto. Now, itbeing indifferent whether the points K, L, D, ofthe cords be lixed to the plane of the wheel ornot, the weights will have the same effectwhether they are suspended from the points Kand L, or from D and L. Let the whole forceof the weight A be represented by the line AD,and let it be resolved into the forces AC andCD ; of which the force AC, drawing the radiusOD directly from the centre, will have no effect to move the wheel : butthe other force DC, drawing the radius DO perpendicularly, will have thesame effect as if it drew perpendicularly the radius OL equal to OD ; thatis, it w ill have the same effect as the weight P, if that weight is to theweight A as the force DC is to the force DA ;that is (because of the similar triangles ADC, DOK), as OK to OD or OL. Therefore the weights Aand P, which are reciprocally as the radii OK and OL that lie in the sameright line, will be equipollent, and so remain in equilibrio ; which is the wellknown property of the balance, the lever, and the wheel. If either weight isgreater than in this ratio, its force to move the wheel will be so much greater.If the weight p, equal to the weight P, is partly suspended by thecord NJO, partly sustained by the oblique plane pG ; draw p}i, NH, theformer perpendicular to the horizon, the latter to the plane pG ; and ifthe force of the weight p tending downwards is represented by the line/?H, it may be resolved into the forces joN, HN. If there was any plane/?Q, perpendicular to the cord y?N, cutting the other plane pG in a lineparallel to the horizon, and the weight p was supported only by thoseplanes pQ, pG, it would press those planes perpendicularly with the forcespN, HN; to wit, the plane joQ, with the force joN, and the plane pG withthe force HN. And therefore if the plane pQ was taken away, so thntthe weight might stretch the cord, because the cord, now sustaining theweight, supplies the place of the plane that was removed, it will be strainedby the same force joN which pressed upon the plane before. Therefore,the tension of this oblique cord joN will be to that of the other perpendicular cord PN as jt?N to joH. And therefore if the weight p is to theweight A in a ratio compounded of the reciprocal ratio of the least distancesof the cords PN, AM, from the centre of the wheel, and of the direct ratio ofpH tojoN, the weights will have the same effect towards moving the wheel,and will therefore sustain each other : as any one may find by experiment.But the weight p pressing upon those two oblique planes, may be considered as a wedge between the two internal surfaces of a body split by it;and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi8G THE MATHEMATICAL PRINCIPLESbecause the force with which the weight p presses the plane pQi is to theforce with which the same, whether by its own gravity, or by the blow ofa mallet, is impelled in the direction of the line joH towards both theplanes, as joN to pH ; and to the force with which it presses the otherplane pG, as joN to NH. And thus the force of the screw may be deducedfrom a like resolution of forces;it being no other than a wedge impelledwith the force of a lever. Therefore the use of this Corollary spreads farand wide, and by that diffusive extent the truth thereof is farther confirmed. For on what has been said depends the whole doctrine of mechanics variously demonstrated by different authors. For from hence are easilydeduced the forces of machines, which are compounded of wheels, pullics,levers, cords, and weights, ascending directly or obliquely, and other mechanical powers ; as also the force of the tendons to move the bones of animals.COROLLARY III.The (/uaittity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those thatare directed to contrary parts, suffers no change from the action ojbodies among themselves.For action and its opposite re-action are equal, by Law III, and therefore, by Law II, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts.whatever is added to the motion of the preceding body will be subductedfrom the motion of that which follows ; so that the sum will be the sameas before. If the bodies meet, with contrary motions, there will be anequal deduction from the motions of both ; and therefore the difference ofthe motions directed towards opposite parts will remain the same.Thus if a spherical body A with two parts of velocity is triple of aspherical body B which follows in the same right line with ten parts ofvelocity, the motion of A will be to that of B as 6 to 10. Suppose,then, their motions to be of 6 parts and of 10 parts, and the sum will be16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4,or 5 parts of motion, B will lose as many ; and therefore after reflexionA will proceed With 9, 10, or 11 parts, and B with 7, 6, or 5 parts; thesum remaining always of 16 parts as before. If the body A acquire 9,10, 11, or 12 parts of motion, and therefore after meeting proceed with15, 16, 17, or 18 parts, the body B, losing so many parts as A has got,will either proceed with 1 part, having lost 9, or stop and remain at rest,as having lost its whole progressive motion of 10 parts ; or it will go backwith 1 part, having not only lost its whole motion, but (if 1 may so say)one part more; or it will go back with 2 parts, because a progressive motion of 12 parts is taken off. And so the sums of the Conspiring motions15 ,1, or 16-1-0, and the differences of the contrary i otions 17 1 andOF NATURAL PHILOSOPHY.[S 2, will always be equal to 16 parts, as they were before tie meetingand reflexion of the bodies. But, the motions being known with whiclithe bodies proceed after reflexion, the velocity of either will be also known,by taking the velocity after to the velocity before reflexion, as the motionafter is to the motion before. As in the last case, where the motion of thobody A was of parts before reflexion and of IS parts after, and thevelocity was of 2 parts before reflexion, the velocity thereof after reflexionwill be found to be of 6 parts ; by saying, as the parts of motion beforeto 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after.But if the bodies are cither not spherical, or, moving in different rightlines, impinge obliquely one upon the other, and their mot ons after reflexion are required, in those cases we are first to determine the positionof the plane that touches the concurring bodies in the point of concourse ,then the motion of each body (by Corol. II) is to be resolved into two, oneperpendicular to that plane, and the other parallel to it. This done, because the bodies act upon each other in the direction of a line perpendicular to this plane, the parallel motions are to be retained the same afterreflexion as before ; and to the perpendicular motions we are to assignequal changes towards the contrary parts ;in such manner that the sumof the conspiring and the difference of the contrary motions may remainthe same as before. From such kind of reflexions also sometimes arisethe circular motions of bodies about their own centres. But these arecases which I do not consider in what follows ; and it would be too tediousto demonstrate every particular that relates to this subject.COROLLARY IV.The common centre of gravity of two or more bodies does not alter itsstate of motion or rest by the actions of the bodies among themselves ;and therefore the common centre of gravity of all bodies acting uponeach other (excluding outward actions and impediments) is either atrest, or moves uniformly in a right line.For if two points proceed with an uniform motion in right lines, andtheir distance be divided in a given ratio, the dividing point will be eitherat rest, or proceed uniformly in a right line. This is demonstrated hereafter in Lem. XXIII and its Corol., when the points are moved in the sameplane ; and by a like way of arguing, it may be demonstrated when thepoints are not moved in the same plane. Therefore if any number ofKdies move uniformly in right lines, the common centre of gravity of anytwo of them is either at rest, or proceeds uniformly in a right line ; becausethe line which connects the centres of those two bodies so moving is divided atthat common centre in a given ratio. In like manner the common centreof those two and that of a third body will be either at rest or moving uniformly in aright line because at that centre the distance 1 etween th?88 THE MATHEMATICAL PRINCIPLEScommon centre of the two bodies, and the centre of this last, is divided ina given ratio. In like manner the common centre of these three, and of afourth body, is either at rest, or moves uniformly in a right line ; becausethe distance between the common centre of the three bodies, and the centreof the fourth is there also divided in a given ratio, and so on m itifinitum.Therefore, in a system of bodies where there is neither any mutual actionamong themselves, nor any foreign force impressed upon them from without,and which consequently move uniformly in right lines, the common centre ofgravity of them all is either at rest or moves uniformly forward in a right line.Moreover, in a system of two bodies mutually acting upon each other,since the distances between their centres and the common centre of gravityof both are reciprocally as the bodies, the relative motions of those bodies,whether of approaching to or of receding from that centre, will be equalamong themselves. Therefore since the changes which happen to motionsare equal and directed to contrary parts, the common centre of those bodies,by their mutual action between themselves, is neither promoted nor retarded, nor suffers any change as to its state of motion or rest. But in asystem of several bodies, because the common centre of gravity of any twoacting mutually upon each other suffers no change in its state by that action : and much less the common centre of gravity of the others with whichthat action does not intervene ; but the distance between those two centresis divided by the common centre of gravity of all the bodies into parts reciprocally proportional to the total sums of those bodies whose centres theyare : and therefore while those two centres retain their state of motion orrest, xhe common centre of all does also retain its state : it is manifest thatthe common centre of all never suffers any change in the state of its motion or rest from the actions of any two bodies between themselves. Butin such & system all the actions of the bodies among themselves either happen between two bodies, or are composed of actions interchanged betweensome two bodies ; and therefore they do never produce any alteration inthe comrrv n centre of alias to its state of motion or rest. Whereforetiince that centre, when the bodies do not act mutually one upon another,Oilier is nt rest or moves uniformly forward in some right line, it will,:v。>U7ithst?nding the mutual actions of the bodies among themselves, alwaysjAY-jevere in its state, either of rest, or of proceeding uniformly in a rightliiv,, unless it is forced out of this state by the action of some power imprev^-d from without upon the whole system. And therefore the same lawtake*1 place in a system consisting of many bodies as in one single body,with wsgard to their persevering in their state of motion or of rest. Forthe pi 。。jressive motion, whether of one single body, or of a whole system ofbodies us always to be estimated from the motion of the centre of gravity.COROLLARY V.The motions cf bcdies included in a given space a ~e Ike same amongOF NATURAL PHILOSOPHY. 89themselves, whether that space is at rest, or moves uniformly forwardsin a right line without any circular motion.For the differences of the motions tending towards the same parts, andthe sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same ; and it is from those sums and differencesthat the collisions and impulses do arise with which the bodies mutuallyimpinge one upon another. Wherefore (by Law II), the effects of thosecollisions will be equal in both cases ; and therefore the mutual motionsof the bodies among themselves in the one case will remain equal to themutual motions of the bodies among themselves in the other. A clearproof of which we have from the experiment of a ship ; where all motionshappen after the same manner, whether the ship is at rest, or is carrieduniformly forwards in a right line.COROLLARY VI.If bodies, any how moved among themselves, are urged in the direct-tonof parallel lines by equal accelerative forces, they will all continue tomove among themselves, after the same manner as if they had beenurged by no such forces.For these forces acting equally (with respect to the quantities of theDO dies to be moved), and in the direction of parallel lines, will (by Law II)move all the bodies equally (as to velocity), and therefore will never produce any change in the positions or motions of the bodies among themselves.SCHOLIUM.Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the firsttwo Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola; experience agreeingwith both, unless so far as these motions are a little retarded by the resistance of the air. When a body is falling, the uniform force of itsgravity acting equally, impresses, in equal particles of time, equal forcesupon that body, and therefore generates equal velocities; and in the wholetime impresses a whole force, and generates a whole velocity proportionalto the time. And the spaces described in proportional times are as thevelocities and the times conjunctly ; that is, in a duplicate ratio of thetimes. And when a body is thrown upwards, its uniform gravity impresses forces and takes off velocities proportional to the times ; and thetimes of ascending to the greatest heights are as the velocities to be takenoff, and those heights are as the velocities and the times conjunetly, or ir,the duplicate ratio of the velocities. And if a body be projected in anydirection, the motion arising from its projection jS compounded with the90 THE MATHEMATICAL PRINCIPLESmotion arising from its gravity. As if the body A by its motion of piojectionalone could describe in a given time the right lineAB, and with its motion of falling alone could describe inthe same time the altitude AC ; complete the paralellogramABDC, and the body by that compounded motionwill at the end of the time be found in the place D ; andthe curve line AED, which that body describes, will be aparabola, to which the right line AB will be a tangent inA ; and whose ordinate BD will be as the square of the line AB. On thesame Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks. By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huvgens,the greatest geometers of our times, did severally determine the rulesof the congress and reflexion of hard bodies, and much about the sametime communicated their discoveries to the Royal Society, exactly agreeingamong themselves as to those rules. Dr. Wallis, indeed, was somethingmore early in the publication ; then followed Sir Christopher Wren, and,lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth ofthe thing before the Royal Society by the experiment of pendulums, whichMr. Mariottc soon after thought fit to explain in a treatise entirely uponthat subject. But to bring this experiment to an accurate agreement withthe theory, we are to have a due regard as well to the resistance of the airbodies. Let the spherical bodiesCD F IIas to the clastic force of the concurrinA, B be suspended by the parallel andequal strings AC, Bl), from the centresC, D. About these centres, with thoseintervals, describe the semicircles EAF,GBH, bisected by the radii CA, DB.Bring the body A to any point R of the