places of things fhould be moveable, is abfurd. we come to fome immoveable place, as in the Thefe are therefore the abfolute places; and translations out of those places are the only abfolute motions. But because the parts of space cannot be seen, or diftinguished from one another by the senses, therefore in their ftead we ufe fenfible measures of them. For, from the pofitions and diftances of things from any body, confidered as immove able, we define all places; and then with refpect to fuch places, we eflimate all motions, confidering bodies as transferred from fome of thofe places into others. And fo, inftead of abfolute places and motions, we ufe relative ones; and that without any inconvenience in common affairs: but in philofophical difquifitions we ought to abftract from our fenfes, and confider things themselves diftinct from what are only fenfible measures of them. For it may be, that there is no body really at reft, to which the places and motions of others may be referred. But we may diftinguish REST and MOTION, abfolute and relative, one from the other, by their properties, caufes, and effects. It is a property of reft, that bodies really at reft do rest in respect of each other. And therefore, as it is poffible, that, in the remote regions of the fixed ftars, or perhaps far beyond them, there may be fome body abfolutely at reft, though it be impoffible to know from the pofition of bodies to one another in our regions, whether any of thefe do keep the fame pofition to that remote body; it follows, that abfolute reft cannot be determined from the pofition of bodies in our regions. It is a property of motion, that the parts which retain given pofitions to their wholes do partake of the motion of their wholes. For all parts of revolving bodies endeavour to recede from the axis of motion; and the impetus of bodies moving forwards arifes from the joint impetus of all the parts. Therefore, if furrounding bodies are moved, thofe that are relatively at reft within them will partake of their motion. Upon which account the true and abfolute motion of a body cannot be determined by the translation of it from those only which feem to reft; for the external bodies ought not only to appear at reft, but to be really at reft. For otherwife all included bodies, befide their translation from near the furrounding ones, partake likewise of their true motions; and though that tranflation was not made, they would not really be at reft, but only feem to be fo. For the furrounding bodies ftand in the like relation to the furrounded, as the exterior part of a whole does to the interior, or as the thell does to the kernel; but if the fhell moves, the kernel will alfo move, as being part of the whole, without any removal from near the fhell. A property near akin to the preceding is, that if a place is moved, whatever is placed therein moves along with it; and therefore a body which is moved from a place in motion, partakes alfo of the motion of its place. Upon which account all motions from places in motion are no other than parts of entire and abfolute motions; and every entire mo ion is compofed of the motion of the body out of its first place, and the motion of this place out of its place; and fo on, until above-mentioned example of the failor. Wherefore entire and abfolute motions can be no other wife determined than by immoveable places. Now, no other places are immoveable but those that from infinity to infinity.co all retain the fame given pofitions one to another; and upon this account must ever remain unmoved, and do thereby conftitute what we call IMMOVEABLE SPACE. The caufes, by which true and relative motions are diftinguithed one from the other, are the forces impreffed open bodies to generate motion. True motion is neither generated nor altered, but by fome force impreffed upon the body moved: but relative motion may be generated or altered without any force impreffed upon the body. For it is fufficient only to imprefs fome force on other, bodies with which the former is compared, that, by their giving way, that relation may be changed, in which the relative reft or motion of the other body did confift. Again, true motion fuffers always fome change from any force impreffed upon the moving bedy; but relative motion does not neceffarily undergo any changes by fuch force. For if the fame forces are likewife impreffed on thofe other bodies with which the comparison is made, that the relative pofition may be preserved; then that condition will be preferved, in which the relative motion confifts. And therefore any relative motion may be changed when the true mo tion remains unaltered, and the relative may be preferved when the true motion fuffers fome change. Upon which account true motion does by no means confift in fuch relations. SECT. IV. Of the DIFFERENCE between ABSO LUTE and RELATIVE MOTION. The effects which distinguish abfolute from relative motion are, the forces of receding from the axis of circular motion. For there are no fuch forces in a circular motion purely relative; but, in a true and abfolute circular motion, they are greater or lefs according to the quantity of the motion. If a veffel, hung by a long cord, is fo often turned about that the cord is strongly twifted, then filled with water, and let go, it will be whirled about the contrary way; and while the cord is ontwifting itself, the furface of the water will at firft be plain, as before the veffel began to move; but the vessel by gradually communicating its motion to the water, will make it begin fenfibly to revolve, and recede by little and little from the middle, and afcend to the fides of the veffel, forming itself into a concave figure; and the fwifter the motion becomes, the higher will the water rife, till at laft, performing its revolu tions in the fame times with the veffel, it becomes relatively at reft in it. This afcent of the water fhows its endeavour to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, discovers itself, and may be meafured by this endeavour. At first, when the relative motion in the water was greateft, it produced no endeavour to recede from the axis; the water fhowed no tendency to the circumference, nor any afcent towards the fides of the veifel, but remained of a plain surface; and therefore its true circula: circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the afcent thereof towards the fides of the vessel proved its endeavour to recede from the axis; and this endeavour showed the real circular motion of the water perpetually increafing, till it had acquired its greateft quantity, when the water rested relatively in the vessel. And therefore this endeavour does not depend upon any tranflation of the water in respect of the ambient bodies; nor can true circular motion be defined by fuch tranflations. There is only one real circular motion of any one revolving body, correfponding to only one power of endeavouring to recede from its axis of motion, as its proper and adequate effect: but relative motions in one and the fame body are innumerable, according to the various relations it bears to external bodies; and, like other relations, are altogether deftitute of any real effect, otherwife than they may perhaps participate of that only true motion. And therefore, in the fyftem which fuppofes that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them, the feveral parts of thofe heavens and the planets, which are indeed relatively at reft in their heavens, do yet really move. For they change their pofition one to another, which never happens to bodies truly at reft; and being carried together with the heavens, participate of their motions, and, as parts of revolving wholes, endeavour to recede from the axis of their motion. Wherefore relative quantities are not the quantities themselves whose names they bear, but those fenfible measures of them, either accurate or inaccurate, which are commonly used inftead of the measured quantities themfelves. And then, if the meaning of words is to be determined by their ufe, by the names TIME, SPACE, PLACE, and MOTION, their measures are properly to be underftood; and the expreffion will be unufual and purely mathematical, if the measured quantities themselves are meant. It is indeed a matter of great difficulty to difcover, and effectually to diftinguish, the true motions of particular bodies from those that are only apparent: because the parts of that immoveable space in which those motions are performed, do by no means come under the obfervation of our fenfes. Yet we have fome things to direct us in this intricate affair; and these arife partly from the apparent motions, partly from the forces which are the causes and effects of the true motions. For instance, if two globes, kept at a given diftance one from the other by a cord that connects them, were revolved about their common centre of gravity; we might, from the tenfion of the cord, difcover the endeavour of the globes to recede from the axis of motion, and thence we might compute the quantity of their circular motions. And then, if any equal forces should be impreffed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decreafe of the tenfion of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces thofe forces ought to be impreffed, that the motions of the globes might be moft augmented; that is, we might discover their hindermoft faces, or those which follow in the circular motion. But the faces which follow being known, and confequently the oppofite ones that precede, we should likewife know the determination of their motions. And thus we might find both the quantity and determination of this circular motion, even in an immense vacuum, where there was nothing external or fenfible, with which the globes might be compared. But now, if in that space fome remote bodies were placed that kept always a given pofition one to another, as the fixed ftars do in our regions; we could not indeed determine from the relative translation of the globes among thofe bodies, whether the motion did belong to the globes or to the bodies. But if we obferved the cord, and found that its tenfion was that very tenfion which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at reft; and then, laftly, from the translation of the globes among the bodies, we should find the determination of their motions. SECT. V. Of the LAWS of MOTION. HAVING thus explained himself, Sir ISAAC propofes to fhow how we are to collect the true motions from their causes, effects, and apparent differences; and vice verfa, how, from the motions, either true or apparent, we may come to the knowledge of their caufes and effects. In order to this, he lays down the following axioms or laws of motion. 1. EVERY BODY PERSEVERES IN ITS STATE OF REST, OR OF UNIFORM MOTION IN A RIGHT LINE, UNLESS IT IS COMPELLED TO CHANGE THAT STATE BY FORCES IMPRESSED UPON IT. Sir Ifaac's proof of this axiom is as follows: "Projectiles perfevere in their motions, fo far as they are not retarded by the refiftance of the air, or impelled downwards by the force of gravity. A top, whofe parts, by their cohesion, are perpetually drawn afide from rectilinear motions, does not ceafe its rotation otherwife than as it is retarded by the air. The greater bodies of the planets and comets, meeting with lefs refiftance in more free spaces, preferve their motions, both progreffive and circular, for a much longer time.” Notwithstanding this demonftration, however, the axiom hath been violently difputed. It hath been argued, that bodies continue in their state of motion because they are subjected to the continual impuife of an invifible and fubtile fluid, which always pours in from behind, and of which all places are full. It hath been affirmed that motion is as natural to this fluid as reft is to all other matter; that it is impoffible we can know in what manner a body would be influenced by moving forces if it were entirely deftitute of gravity. According to what we can obferve, the momentum of a body, or its tendency to move, depends very much on its gravity. A heavy cannon ball will fly to a much greater distance than a light one, though both are actuated by an equal force. It is by no means clear, therefore, that a body totally deftitute of gravity would have any proper momentum of its own; and if it had no momentum, it could not continue its motion for the F 2 fmalleft fmalleft space of time after the moving power was withdrawn. Some have imagined that matter was capable of beginning motion of itself, and confequently that the axiom was falfe; because we fee plainly that matter in fome cafes hath a tendency to change from a state of motion to a state of reft, and from a ftate of rest to a state of motion. A paper appeared on this fubject in the firft volume of the Edinburgh Physical and Literary Effays; but the hypothefis never gained any ground. 2. THE ALTERATION OF MOTION IS EVER PROPORTIONAL TO THE MOTIVE FORCE IM PRESSED; AND IS MADE IN THE DIRECTION OF THE RIGHT LINE IN WHICH THAT FORCE IS IMPRESSED. Thus, if any force generates a certain quantity of motion, a double force will generate a double quantity, whether that force be impref. fed all at once, or in fucceffive moments. To this law no objection of confequence has ever been made. It is founded on this felf-evident truth that every effect must be proportional to its caufe. Mr Young, who feems fond of detecting the errors of Newton, finds fault indeed with the expreffions in which the law is ftated; but he owns, that if thus expreffed, The alteration of motion is proportional to the actions or refiftances which produce it, and is in the direction in which the actions or refiftances are made, it would be unexceptionable. 3. TO EVERY ACTION THERE ALWAYS IS OPPOSED AN EQUAL RE-ACTION: OR THE MUTUAL ACTION OF TWO BODIES UPON EACH OTHER ARE ALWAYS EQUAL, AND DIRECTED TO CONTRARY PARTS.-This axiom is alfo difputed by many. In the above-mentioned paper in the phyfical Effays, the author endeavours to make a diftinction between re-action and refiftance; and the fame attempt has been made by Mr Young. "When an action generates no motion (fays he), it is certain that its effects have been deftroyed by a contrary and equal action. When an action generates two contrary and equal motions, it is alfo evident that mutual actions were exerted, equal and contrary to each other. All cafes where one of thefe conditions are not found, are exceptions to the truth of the law. If a finger preffes against a ftone, the stone if it does not yield to the preffure, preffes as much upon the finger: but if the ftone yields, it re-acts lefs than the finger acts; and if it Should yield with all the momentum that the force of the preffure ought to generate, which it would do if it were not impeded by friction, or a medium, it would not re-act at all. So if the ftone drawn by a horse, follows after the horfe, it does not re-act fo much as the horse acts; but only fo much as the velocity of the ftone is diminished by friction, and it is the re-action of friction only, not of the tone. The ftone does not re-act because it does not act; it refifts, but reftance is not action. In the loss of motion from a ftriking body, equal to the gain in the body ftruck, there is a plain folution without requiring any re-action. The motion loft is identically that which is found in the other body; this fuppofition accounts for the whole phenomenon in the moft fimple man ner. If it be not admitted, but the solution by ion is infifted upon, it will be incumbent on the party to account for the whole effect of communication of motion; otherwise he will lie under the imputation of rejecting a folution which is fimple, obvious, and perfect; for one complex, unnatural, and incomplete. However this may be determined, it will be allowed, that the cir cumftances mentioned afford no ground for the inference, that action and re-action are equal, fince appearances may be explained in another way.' Thus, If there be a perfect reciprocity betwixt an impinging body and a body at reft fuftaining its impulfe, may we not at our pleasure confider either body as the agent, and the other as the refiftant? Let a moving body, A, pass from north to fouth, an equal body B at reft, which receives the stroke of A, act upon A from fouth to north, and A refift in a contrary direction, both inelastic: let the motion reciprocally communicated be cal led fix. Then B at reft communicates to A lix degrees of motion towards the north, and receives fix degrees towards the fouth. B having no other motion than the fix degrees it communicated, will, by its equal and contrary lofs and gain, remain in equilibrio. Let the original motion of A have been 12, then A having received a contrary action equal to fix, fix degrees of its motion will be deftroyed or in equilibrio; confequently, a motive force as fix will remain to A towards the fouth, and B will be in equilibrio, or at reft. A will then endeavour to move with fix degrees, or half its original motion, and B will remain at rest as before. A and B being equal maffes, by the laws of communication three degrees of motion will be communicated to B, or A with its fix degrees will act with three, and B will re-act also with three, B then will act on A from fouth to north equal to three, while it is acted upon or refifted by A from north to fouth, equal alfo to three, and B will remain at reft as before; A will alfo have its fix degrees of motion reduced to one half by the contrary action of B, and only 3 degrees of motion will remain to A, with which it will yet endeavour to move; and finding B ftili at reft, the fame procefs will be repeated till the whole motion of A is reduced to an infinitely fmall quantity, B all the while remaining at reft, and there will be no communication of motion from A to B, which is contrary to experience. Let a body, A, whofe mafs is 12, at reft, be impinged upon firft by B, having a mass as 12, and à velocity as 4, making a momentum of 48; and adly by C, whofe mafs is fix, and velocity 8, making a momentum of 48 equal to B, the three bodies being inelaftic. In the firft cafe, A will become poffeffed of a momentum of 24, and 24 will remain to B; and, in the 2d cafe, A will become poffeffed of a momentum of 32, and 16 will remain to C, both bodies moving with equal velocities after the fhock, in both cases, by the laws of percuffion. It is required to know, if in both cafes A refifts equally, and if B and C act equally? If the actions and refiftances are equal, how does A in the one cafe destroy 24 parts of B's motion, and in the other cafe 32 parts of C's motion, by an equal refiftance? And how does B communicate in one cafe 24 degrees of motion, and C 32, by equal actions? If the actions and refistances are uncquai, unequal, it is asked how the fame mafs can refift differently to bodies impinging upon it with equal momenta, and how bodies poffeffed of equal momenta can exert different actions, it being admitted that bodies refift proportional to their maffes, and that their power of overcoming refiftance is proportional to their momenta ?-It is incumbent on those who maintain the doctrine of univerfal re-action, to free it from thefe difficulties and ap parent contradictions. Others grant that Sir Ifaac's axiom is very true with respect to terreftrial fubftances; but they affirm, that, in thefe, both action and re-action are the effects of gravity. Subftances void of gravity would have no momentum; and without this they could not act; they would be moved by the leaft force, and therefore could not refift or re-act. If, therefore, there is any fluid which is the caufe of gravity, though fuch fluid could act upon terreftrial fubftances, yet these could not re-act upon it, because they have no force of their own, but depend entirely upon it for their momentum. In this manner, fay they, we may conceive that the planets circulate, and all the operations of nature are carried on by means of a fubtile fluid; which being perfectly active, and the reft of matten altogether paffive, there is neither refistance nor lofs of motion. See MOTION, § 5-7. From the preceding axiom Sir Ifaac draws the following corollaries. 1. A body by two forces conjoined will defcribe the diagonal of a parallelogram in the fame time that it would defcribe the fides by thofe forces apart. 2. Hence we may explain the compofition of any one direct force out of any two oblique ones, viz. by making the two oblique forces the fides of a parallelogram, and the direct one the diagonal. 3. The quantity of motion, which is collected by taking the fum of the motions directed towards the fame parts, and the difference of those that are directed to contrary parts, fuffers no change from the action of bodies among themfelves; because the motion which one body lofes is communicated to another; and if we fuppofe friction and the refiftance of the air to be abfent, the motion of a number of bodies which mutually impelled one another would be perpetual, and its quantity always equal. 4. The common centre of gravity of two or more bodies does not alter its ftate of motion or reft by the actions of the bodies among themfelves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments), is either at reft, or moves uniformly in a right line. 5. The motions of bodies included in a given space are the fame among themfelves, whether that space is at reft, or moves uniformly forward in a right line without any circular motion. The truth of this is evidently shown by the experiment of a fhip, where all motions happen after the fame manner, whether the fhip is at reft, or proceeds uniformly forward in a ftraight line. 6. If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the fame manner as if they had been urged by no fuch forces. The whole of the mathematical part of the Newtonian philofophy depends on the following lemmas; of which the firft is the principal. LEM. I. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before that time approach nearer the one to the other than by any given difference, become ultimately equal. If you deny it; fuppole them to be ultimately unequal, and let D be. their ultimate difference. Therefore they cannot approach nearer to equality than by that given difference D; which is against the fuppofition. Concerning the meaning of this lemma philofophers are not agreed; and unhappily it is the very fundamental pofition on which the whole of the fyftem refts. Many objections have been raised to it by people who fuppofed themselves capable of understanding it. They fay, that it is impoffible we can come to an end of any infinite fe ries, and therefore that the word ultimate can in this cafe have no meaning. In fome cafes the lemma is evidently falfe. Thus, fuppofe there are two quantities of matter, A and B, the one containing half a pound, and the other a third part of one. Let both be continually divided by 2; and though their ratio, or the proportion of the one to the other, doth not vary, yet the difference between them perpetually becomes lefs, as well as the quantities themselves, until both the difference and quantities themfeives become lefs than any affignable quantity; yet the difference will never totally vanish, nor the quantities become equal, as is evident from the two following feries. Diff. 6 12 24 48 96 192 384 768 1536' I I I I I I I I I &c. I &c. 6 12 24 48 96 192 384 768 1536 3072 Thus we fee, that though the difference is continually diminishing, and that in a very large proportion, there is no hope of its vanifhing, or the quantities becoming equal. In like manner, let us take the proportions or ratios of quantities, and we fhall be equally unfuccefsful. Suppofe two quantities of matter, one containing 8 and the other 10 ponnds: thefe quantities already have to each other the ratio of 8 to 10, or of 4 to ; but let us add 2 continually to each of them, and though the ratios continually come nearer to that of equality, it is in vain to hope for a perfect coincidence. Thus, 8 10 12 14 16 18 20 22 24, &c. Ratio 4 5 6 7 8 9 10 11 12, &c. 5 6 7 8 9 10 11 12 13 For this and his other lemmas Sir Ifaac makes the following apology. "Thefe lemmas are premifed, to avoid the tedioufnefs of deducing perplexed demonftrations ad abfurdum, according to the method of ancient geometers. For demon-` ftrations are more contracted by the method of indivifibles; but because the hypothefis of indi vifibles vifibles seems somewhat harsh, and therefore that method is reckoned lefs geometrical, I chose rather to reduce the demonftrations of the following propofitions to the firft and laft fums and ratios of nafcent and evanefcent quantities, that is, to the limits of those fums and ratios; and fo to premife, as fhort as I could, the demonftrations of thofe limits. For hereby the fame thing is per formed as by the method of indivifibles; and now those principles being demonftrated, we may ufe them with more fafety. Therefore, if hereafter I fhould happen to confider quantities as made up of particles, or fhould ufe little curve lines for right ones, I would not be understood to mean indivifibles, but evanefcent divifible quantities; not the fums and ratios of determinate parts, but always the limits of fums and ratios; and that the force of fuch demonftrations always depends on the method laid down in the foregoing lemmas. "Perhaps it may be objected, that there is no ultimate proportion of evanefcent quantities, becaufe the proportion before, the quantities have vanished, is not the ultimate, and, when they are vanished, is none. But by the fame argument it may be alleged, that a body arriving at a certain place, and there ftopping, has no ultimate velocity; because the velocity before the body comes to the place is not its ultimate velocity; when it is arrived, it has none. But the answer is eafy: for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its place and the motion ceases, nor after, but at the very inftant it arrives; that is, that velocity with which the body arrives at its laft place, and with which the motion ceafes. And in like manner, by the ultimate ratio of evanefcent quantities is to be understood the ratio of the quantities, not before they vanish, nor afterwards, but with which they vanish. In like manner, the first ratio of nafcent quantities is that with which they begin to be. And the first or laft fum is that with which they begin and cease to be (or to be augmented and diminished). There is a limit which the velocity at the end of the motion may attain, but not exceed; and this is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And, fince fuch limits are certain and definite, to determine the fame is a problem ftrictly geometrical. But whatever is geometrical we may be allowed to make ufe of in determining and demonftrating any other thing that is likewife geometrical. "It may also be objected, that if the ultimate ratios of evanefcent quantities are given, their ultimate magnitudes will be alfo given; and fo all quantities will confift of indivifibles, which is contrary to what Euclid has demonftrated concerning incommenfurables, in the roth book of his elements. But this objection is founded on a falfe fuppofition. For thofe ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities, decreasing continually approach." LEM. II. If in any figure A ac E(Pl. CCXLVI. N° 1.) terminated by the right line Aa, AE, and the curve a c E, there be infcribed any number of parallelograms Ab, Bc, Cd, &c. comprehended under equal bases AB, BC, CD, &c. and the fides Bb, Cc, Dd, &c. parallel to one fide Aa of the figure; and the parallelograms a K b l, b L cm, c M dn, &c. are completed. Then if the breadth of thefe parallelograms be fuppofed to be diminifhed, and their number augmented in infinitum; the ultimate ratios which the infcribed figure AK b Lc MdD, the circumfcribed figure Aalbmendo E, and curvilinear figure Aabcd E, will have to one another, are ratios of equality.— For the difference of the infcribed and circumfcribed figures is the fum of the parallelograms K, Lm, Mn, Do; that is, (from the equality of all their bafes,) the rectangle under one of their bafes K b, and the fum of their altitudes A a, that is, the rectangle AB la. But this rectangle, becaufe its breadth AB is fuppofed diminished in infinitum, becomes lefs than any given space. And therefore, by lem. 1. the figures infcribed and circumfcribed become ultimately equal the one to the other; and much more will the intermediate curvilinear figure be ultimately equal to either. LEM. III. The fame ultimate ratios are also ratios of equality, when the breadths AB, BC, CD, &c. of the parallelograms are unequal, and are all diminished in infinitum.-The demonstration of this differs but little from that of the former. In his fucceeding lemmas, Sir Ifaac goes on to prove, in a manner fimilar to the above, that the ultimate ratios of the fine, chord, and tangent of arcs infinitely diminished, are ratios of equality; and therefore, that in all our reafonings about these we may fafely use the one for the other :-that the ultimate form of evanefcent triangles made by the arc, chord, and tangent, is that of fimilitude, and their ultimate ratio is that of equality; and hence, in reafonings about ultimate ratios, we may fafely use these triangles for each other, whether made with the fine, the arc, or the tangent. He then fhows fome properties of the ordinates of curvilinear figures; and proves that the spaces which a body defcribes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are, in the very beginning of the motion, one to the other in the duplicate ratio of the powers. And, laftly, having added fome demonftrations concerning the evanefcence of angles of contact, he proceeds to lay down the mathematical part of his fyftem, and which depends on the following theorems: THEOR. I. The areas which revolving bodies defcribe, by radii drawn to an immoveable centre of force, lie in the fame immoveable planes, and are proportional to the times in which they are defcribed.-For, suppose the time to be divided into equal parts, and in the firft part of that time, let the body by its innate force defcribe the right line AB (N° 2.); in the second part of that time, the fame would, by law. 1. if not hindered, proceed directly to c along the line B c=AB; fo that by the radii AS, BS, cS, drawn to the centre, the equal areas ASB, BSc, would be described. But, when the body is arrived at B, suppose the centripetal force acts at once with a great impulse, and, turning aside the body from the right line Bc, compels it afterwards to continue its motion along the right line BC. Draw c C parallel to BS, |