Principia Mathematica (Motte Translation, 1848) — part 6 of 45

It is indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent ; be- cause the parts of that immovable space, in which those motions are per- formed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate : for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions ; partly from the forces, which are the causes and elFects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavour of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions ; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented ; that is, we might discover their hinder- most faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circu- lar motion, even in an immense vacuum, where there was nothing external or sensible with wdiich the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine from the relative translation of the globes among those bodies, whether the motion did belong to the globes or to the bodies. But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest ; and then, lastly, from the trans- lation of the globes among the bodies, we should find the determination of their motions. But how we are to collect the true motions from their causes, effects, and apparent differences ; and, vice versa, how from the mo- tions, either true or apparent, we may come to the knowledge of their causes and effects, shall be explained more at large in the following tract. For to this end it was that I composed it.

OF NATURAL PHILOSOPHY. 3,3

AXIOMS, OR LAWS OF MOTION.

LAW I. Every body perseveres in its state of rest, or of uniform motion in a right

line, unless it is co'mpelled to change that state by forces impressed

thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is re- tarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both pro- gressive and circular for a much longer time.

LAW IL

The alteration of motion is ever proportional to the 7notive force impress- ed ; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, accordino- as they directly conspire with or are directly contrary to each other ; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

LAW m.

To every action there is always opposed an equal reaction : or the mu- tual actions of two bodies upon each other are ahvays equal, and di- rected to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) 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 as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other.

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If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The- changes made by these actions are equal, not in the velocities but in the motions of bodies ; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally pro- portional to the bodies. • This law takes place also in attractions, as will be proved in the next scholium.

COROLLARY L

A body by two forces conjoined ivill describe the diagonal of a parallelo- gram, in the same tim£ that it ivould describe the sides, by those forces apart.

If a body in a given time, by the force M impressed "^ "

apart in the place A, should with an uniform motion be carried from A to B ; and by the force N impressed apart 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. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line BD. The body therefore will arrive at the line BD in the same time, whether the force N be impressed or not ; and therefore at the end of that time it wiU be found somewhere in the line BD. By the same argument, at the end of the same time it will be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I.

COROLLARY IL

And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD; and, on the contrary, the re- solution of any one direct force AD into two oblique forces A C and CD : which composition and resolution are abundantly confirmed from, mechanics.

As if the unequal radii OM and ON drawn from the centre O of any wheel, should sustain the weights A and P by the cords MA and NP ; and the forces of those weights to move the wheel were required. Through the centre O draw the right line KOL, meeting the cords perpendicularly in K and L ; and from the centre O, with OL the greater of the distances

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85

OK and OL, describe a circle, meeting the cord MA in D : and drawing OD, make AC paral- lel and DC perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be fixed to the plane of the wheel or j^ not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the line AD, and let it be resolved into the forces AC and CD ; of which the force AC, drawing the radius OD directly from the centre, will have no effect to move the wheel : but the other force DC, drawing the radius DO perpendicularly, will have the same effect as if it drew perpendicularly the radius OL equal to OD ; that is, it will have the same effect as the weight P, if that weight is to the weight A as the force DC is to the force DA ; that is (because of the sim- ilar triangles ADC, DOK), as OK to OD or OL. Therefore the weights A and P, which are reciprocally as the radii OK and OL that lie in the same right line, will be equipollent, and so remain in equilibrio ; which is the well known property of the balance, the lever, and the wheel. If either weight is greater 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 the cord Nj9, partly sustained by the oblique plane pG] draw pH, NH, the former perpendicular to the horizon, the latter to the plane j^G; and if the force of the weight p tending downwards is represented by the line j^H, it may be resolved into the forces joN, HN. If there was any plane pGi, perpendicular to the cord joN, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those planes pGi, pG, it would press those planes perpendicularly with the forces p'N, HN ; to wit, the plane pGi with the force jt?N, and the plane pG with the force HN. And therefore if the plane pQ, was taken away, so that the weight might stretch the cord, because the cord, now sustaining the weight, supplies the place of the plane that was removed, it will be strained by the same force p'N which pressed upon the plane before. Therefore, the tension of this oblique cord p'N will be to that of the other perpendic- ular cord PN as pN to pH. And therefore if the weight p is to the weight A in a ratio compounded of the reciprocal ratio of the least distances of the cords PN, AM, from the centre of the wheel, and of the direct ratio of jdH topN, 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 con- sidered as a wedge between the two internal surfaces of a body split by it ; and hence the forces of the wedge and the mallet may be determined ; for

S6 THE MATHEMATICAL PRINCIPLES

because the force with which the weight p presses the plane pQi is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line j!?H towards both the planes, as jt?N to pH. ; and to the force with which it presses the other plane pG, as p'N to NH. And thus the force of the screw may be deduced from a like resolution of forces ; it being no other than a wedge impelled with the force of a lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther con- firmed. For on what has been said depends the whole doctrine of mechan- ics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, pullics, levers, cords, and weights, ascending directly or obliquely, and other mechan- ical powers ; as also the force of the tendons to move the bones of animals.

COROLLARY III.

The qua7itity of 'motion^ which is collected by taking the smn of the quo- tions directed towards the same parts ^ and the difference of those that are directed to contrary parts, siffers no change from the action of bodies among themselves.

For action and its opposite re-action are equal, by Law* III, and there- fore, by Law II, they produce in the motions equal changes towards oppo- site parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows ; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both ; and therefore the difference of the motions directed towards opposite parts will remain the same.

Thus if a spherical body A with two parts of velocity is triple of a spherical body B which follows in the same right line with ten parts of velocity, 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 be 16 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 reflexion A will proceed with 9, 10, or 11 parts, and B with 7, 6, or 5 parts; the sum 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 with 15, 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 back with 1 part, having not only lost its whole motion, but (if I may so say) one part more ; or it will go back with 2 parts, because a progressive mo- tion of 12 parts is taken off. And so the sums of the conspiring motions 15 -hi, or 16+0, and the differences of the contrary motions 17 — 1 and

OF NATURAL PHILOSOPHY. 87

18 — 2, will always be equal to 16 parts, as they were before the meeting and reflexion of the bodies. But, the motions being known with which the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of the body A was of 6 parts before reflexion and of 18 parts after, and the velocity was of 2 parts before reflexion, the velocity thereof after reflexion will be found to be of 6 parts ; by saying, as the 6 parts of motion before to 18 parts after, so are 2 parts of velocity before reflexion to 6 parts after. But if the bodies are either not spherical, or, moving in different right lines, impinge obliquely one upon the other, and their motions after re- flexion are required, in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse ; then the motion of each body (by Corol. 11) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, be- cause the bodies act upon each other in the direction of a line perpendicu- lar to this plane, the parallel motions are to be retained the same after reflexion as before ; and to the perpendicular motions we are to assign equal changes towards the contrary parts ; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows ; and it would be too tedious to demonstrate every particular that relates to this subject.

COROLLARY IV. The common centre of gravity of tivo or more bodies does not alter its

state of motion or rest by the actions of the bodies among them^selves ;

and therefore the common centre of gravity of all bodies acting upon

each other [excluding outivard actions and impediments) is either at

rest, or moves uniformly in a right line.

For if two points proceed with an uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated here- after in Lem. XXIII and its Corol., when the points are moved in the same plane ; and by a like way of arguing, it may be demonstrated Avhen the points are not moved in the same plane. Therefore if any number of li'.dies move uniformly in right lines, the common centre of gravity of any tAYO of them is either at rest, or proceeds uniformly in a right line; because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those two and that of a third body will be either at rest or moving uni- formly in a right line: because at that centre the distance between the

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S8 THE MATHEMATICAL PRINCIPLES

common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line ; because the distance between the common centre of the three bodies, and the centre of the fourth is there also divided in a given ratio, and so on in infinitum. Therefore, in a system of bodies where there is neither any mutual action among themselves, nor any foreign force impressed upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest or moves uniformly forward in aright line. Moreover, in a system of two bodies mutually acting upon each other since the distances between their centres and the common centre of gravity of both are reciprocally as the bodies, the relative motions of those bodies whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor re- tarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that ac- tion : and much less the common centre of gravity of the others with which that action does not intervene ; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts re- ciprocally proportional to the total sums of those.bodies whose centres they are : and therefore while those two centres retain their state of motion or rest, the common centre of all does also retain its state : it is manifest that the common centre of all never suffers any change in the state of its mo- tion or rest from the actions of any two bodies between themselves. But in such a system all the actions of the bodies among themselves either hap- pen between two bodies, or are composed of actions interchanged between some two bodies ; and therefore they do never produce any alteration in the common centre of all as to its state of motion or rest. Wherefore since that centre, when the bodies do not act mutually one upon another, either is at rest or moves uniformly forward in some right line, it will, notwithstanding the mutual actions of the bodies among themselves, always persevere in its state, either of rest, or of proceeding uniformly in a right line, unless it is forced out of this state by the action of some power im- pressed from without upon the whole system. And therefore the same law tak«s place in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the centre of gravity.

COROLLARY Y.

' The motions of bodies included in a given space are the sam^ among

OF NATURAL PHILOSOPHY. 89

themselves, whetlier that space is at rest, or moves uniformly forwards

in a right line without any circular motio7i.

For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are, at first (by sup- position), in both cases the same ; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law II), the effects of those collisions will be equal in both cases ; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship ; where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line.

• COROLLARY YL

ijT bodies, any how moved among them^selves, are urged in the direction of parallel lines hy equal accelerative forces, they will all continue to move among themselves, after the sa^ne manner as if they had been urged by no such forces.

For these forces acting equally (with respect to the quantities of the bodies 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 pro- duce 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 math- ematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the de- scent of bodies observed the duplicate ratio of the time, and that the mo- tion of projectiles was in the curve of a parabola ; experience agreeing with both, unless so far as these motions are a little retarded by the re- sistance of the air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal particles of time, equal forces upon that body, and therefore generates equal velocities ; and in the whole time impresses a whole force, and generates a whole velocity proportional to the time. And the spaces described in proportional times are as the velocities and the times conjunctly ; that is, in a duplicate ratio of the times. And when a body is thrown upwards, its uniform gravity im- presses forces and takes oft' velocities proportional to the times ; and the times of ascending to the greatest heights are as the velocities to be taken off, and those heights are as the velocities and the times conjunctly, or in the duplicate ratio of the velocities. And if a body be projected in any direction, the motion arising from its projection is compounded with the

90

THE MATHEMATICAL PRINCIPLES

motion arising from its gravity. As if the body A by its motion of pro- jection alone could describe in a given time the right line AB, and with its motion of falling alone could describe in the same time the altitude AC ; complete the paralello- gram ABDC, and the body by that compounded motion will at the end of the time be found in the place D ; and the curve line AED, which that body describes^ will be a parabola, to which the right line AB will be a tangent in A ; and whose ordinate BD will be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demon- strated concerning the times of the vibration of pendulums, and are con- firmed by the daily experiments of pendulum clocks. By the same, to- gether with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huy- gens, the greatest geometers of our times, did severally determine the rules of tlie congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more eai*ly in the publication ; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air as to the elastic force of the concurring bodies. Let the spherical bodies A, B be suspended by the parallel and eg c D F H

equal strings AC, BD, from the centres C, D. About these centres, with those pj intervals, describe the semicircles EAF, ^GBH, bisected by the radii CA, DB. Bring the body A to any point R of the arc EAF, and (withdrawing the body B) let it go from thence, and after one oscillation suppose it to return to the point V : then RY will be the retardation arising from the resistance of the air. Of this RY let ST be a fourth part, situated in the middle, to wit, so as RS and TY may be equal, and RS may be to ST as 3 to 2 : then will ST represent very nearly the retardation during the descent from S to A. Restore th^ body B to its place : and, supposing the body A to be let fall from the point S, the velocity thereof in the place of re- flexion A, Avithout sensible error, will be the same as if it had descended in vacuo from the point T. Upon which account this velocity may be represented by the chord of the arc TA. For it is a proposition well known to geometers, that the velocity of a pendulous body in the lowest point is as the chord of the arc which it has described in its descent. After

OF NATURAL PHILOSOPHY. 91

reflexion, suppose tlie body A comes to the place s, and the body B to the place k. Withdraw the body B, and find the place v, from which if the body A, being let go, should after one oscillation return to the place r, st may be a fourth part of rv, so placed in the middle thereof as to leave rs equal to tv, and let the chord of the arc tA. represent the velocity which the body A had in the place A immediately after reflexion. For t will be the true and correct place to which the body A should have ascended, if the resistance of the air had been taken off. In the same way we are to correct the place k to which the body B ascends, by finding the place I to which it should have ascended in vacuo. And thus everything may be subjected to experiment, in the same manner as if we were really placed in vacuo. These things being done, we are to take the product (if I may so say) of the body A, by the chord of the arc TA (which represents its velocity), that we may have its motion in the place A immediately before reflexion ; and then by the chord of the arc /A, that we may have its mo- tion in the place A immediately after reflexion. And so we are to take the product of the body B by the chord of the arc B/, that we may have the motion of the same immediately after reflexion. And in like manner, when two bodies are let go together from different places, we are to find the motion of each, as well before as after reflexion ; and then we may compare the motions between themselves, and collect the effects of the re- flexion. Thus trying the thing with pendulums of ten feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and, of consequence, that the action and reaction were always equal. As if the body A impinged upon the body B at rest with -9 parts of motion, and losing 7, proceeded after reflexion with 2, the body B was carried back- wards with those 7 parts. If the bodies concurred with contrary motions, A with twelve parts of motion, and B with six, then if A receded with 2, B receded with 8 ; to wit, with a deduction of 14 parts of motion on each side. For from the motion of A subducting twelve parts, nothing will remain ; but subducting 2 parts more, a motion will be generated of 2 parts towards the contrary way ; and so, from the motion of the body B of 6 parts, subducting 14 parts, a motion is generated of 8 parts towards the contrary way. But if the bodies were made both to move towards the same way. A, the swifter, with 14 parts of motion, B, the slower, with 5, and after reflexion A vfent on with 5, B likewise went on with 14 parts ; 9 parts being transferred from A to B. And so in other cases. By the congress and collision of bodies, the quantity of motion, collected from the sum of the motions directed towards the same way, or from the difference of those that were directed towards contrary ways, was never changed. For the error of an inch or two in measures may be easily ascribed to the

92 THE MATHEMATICAL PRINCIPLES

difficulty of executing everything with accuracy. It was not easy to let go the two pendulums so exactly together that the bodies should impinge one upon the other in the lowermost place AB ; nor to mark the places s, and k, to which the bodies ascended after congress. Nay, and some errors, too, might have happened from the unequal density of the parts of the pen- dulous bodies themselves, and from the irregularity of the texture pro- ceeding from other causes.

But to prevent an objection that may perhaps be alledged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic (whereas no such bodies are to be found in nature), I must add, that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to di- minish the reflexion in such a certain proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies abso- lutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies per- fectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force ; because that force (except when the parts of bodies are bruised by their congress, or suffer some such extension as happens under the strokes of a hammer) is (as far as I can per- ceive) certain and determined, and makes the bodies to return one from the other with a relative velocity, which is in a given ratio to that relative velocity with which they met. This I tried in balls of wool, made up tightly, and strongly compressed. For, first, by letting go the pendulous bodies, and measuring their reflexion, I determined the quantity of their elastic force ; and then, according to this force, estimated the reflexions that ought to happen in other cases of congress. And with this computa- tion other experiments made afterwards did accordingly agree ; the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met as about 5 to 9. Balls of steel returned with almost the same velocity : those of cork with a velocity some- thing less ; but in balls of glass the proportion was as about 15 to 16. And thus the third Law, so far as it regards percussions and reflexions, is proved by a theory exactly agreeing with experience.

In attractions, I briefly demonstrate the thing after this manner. Sup- pose an obstacle is interposed to hinder the congress of any two bodies A, B, mutually attracting one the other : then if either body, as A, is more attracted towards the other body B, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of the body A than by the pressure of the body B, and therefore will not remain in equilibrio : but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly

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towards the parts on which B lies ; and in free spaces, to go forward in infinitum with a motion perpetually accelerated ; which is absurd and contrary to the first Law. For, by the first Law, the system ought to per- severe in its state of rest, or of moving uniformly forward in a right line ; and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. I made the experiment on the loadstone and iron. If these, placed apart in proper vessels, are made to float by one another in standing water, neither of them will propel the other ; but, by being equally attracted, they will sustain each other's pressure, and rest at last in an equilibrium.

So the gravitation betwixt the earth and its parts is mutual. Let the earth FI be cut by any plane EG into two parts EGF ^ and EGI, and their weights one towards the other will be mutually equal. For if by another plane HK, parallel to the former EG, the greater partF| EGI is cut into two parts EGKH and HKI. whereof HKI is equal to the part EFG, first cut off, it is evident that the middle part EGKH, will have no propension by its proper weight towards either side, but will hang as it were, and rest in an equilibrium betwixt both. But the one extreme part HKI will with its whole weight bear upon and press the middle part towards the other extreme part EGF ; and therefore the force with which EGI, the sum of the parts HKI and EGKH, tends towards the third part EGF, is equal to the weight of the part HKI, that is, to the weight of the third part EGF. And therefore the weights of the two parts EGI and EGF, one towards the other, are equal, as I was to prove. And in- deed if those weights were not equal, the whole earth floating in the non- resisting aether would give way to the greater weight, and, retiring from it, would be ca,rried off in infinitum.

And as those bodies are equipollent in the congress and reflexion, whose velocities are reciprocally as their innate forces, so in the use of mechanic instruments those agents are equipollent, and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the determination of the forces, are reciprocally as the forces.