The Mathematical Principles of Natural Philosophy

by Isaac Newton

*
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 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 retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.

*
The alteration of motion is ever proportional to the motive
force impressed; 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, according 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.

*
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 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. 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 proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

*
A body by two forces conjoined will describe the diagonal of
a parallelogram, in the same time that it would 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; 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 will 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.

*
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 resolution of any one direct force AD into two
oblique forces AC 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 OK and OL, describe a circle, meeting the cord MA in D: and drawing OD, make AC parallel 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 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 similar 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 N*p*, partly sustained by the oblique
plane *p*G; draw *p*H, NH, the former perpendicular
to the horizon, the latter to the plane *p*G; and if the
force of the weight *p* tending downwards is represented by
the line *p*H, it may be resolved into the forces *p*N,
HN. If there was any plane *p*Q, perpendicular to the cord *p*N,
cutting the other plane *p*G in a line parallel to the
horizon, and the weight *p* was supported only by those
planes *p*Q, *p*G, it would press those planes
perpendicularly with the forces *p*N; HN; to wit, the plane
*p*Q with the force *p*N, and the plane *p*G
with the force HN. And therefore if the plane *p*Q 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 perpendicular
cord PN as *p*N to *p*H. 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 *p*H to
*p*N, 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 the forces of the wedge and the mallet
may be determined; for because the force
with which the weight *p* presses the plane *p*Q 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 *p*H
towards both the planes, as *p*N to *p*H; and to the
force with which it presses the other plane *p*G, 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 confirmed. For on what has been said depends the whole
doctrine of mechanics variously demonstrated by different authors.
For from hence are easily deduced the forces of machines, which are
compounded of wheels, pullies, 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.

*
The quantity of motion, which is collected by taking the sum
of the motions directed towards the same parts, and the
difference of those that are directed to contrary parts, suffers
no change from the action of bodies 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 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 motion of 12 parts is taken off. And so the sums of the conspiring motions 15+1, or 16+0, and the differences of the contrary motions 17−1 and 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 reflexion 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. II) is to be resolved into two, one perpendicular 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 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.

*
The common centre of gravity of two or more bodies does not
alter its state of motion or rest by the actions of the bodies
among themselves; and therefore the common centre of gravity of
all bodies acting upon each other (excluding outward 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 hereafter 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 when the points are not moved in the same plane.
Therefore if any number of bodies move uniformly in right lines, the
common centre of gravity of any two 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
uniformly in a right line because at that centre the distance
between the 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 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 gravity of both arc 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 retarded, 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 action: 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 reciprocally 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 motion 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 happen 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 impressed from without upon the whole system. And therefore the same law takes 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.

The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.

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 supposition), 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.

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 same, 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 produce any change in the positions or motions of the bodies among themselves.

Hitherto I have laid down such principles as have been received by
mathematicians, and are confirmed by abundance of experiments. By
the first two 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 agreeing with 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 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 impresses forces and takes off 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
motion
arising from its gravity. As if the body A by its motion of
projection 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
paralellogram
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 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. Huygens, the greatest geometers of our times,
did severally determine the rules of the 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 early 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
equal strings AC, BD, from the
centres C, D. About these centres, with those 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 RV will be the retardation arising from
the resistance of the air. Of this RV let ST be a fourth part,
situated in the middle, to wit, so as RS and TV 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 the 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 reflexion A, without
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 reflexion, suppose the
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 *t*A. 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 *l* 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 *t*A, that we may have its motion 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*l*, 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 reflexion. 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
backwards 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 went 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 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 pendulous bodies themselves, and from the
irregularity of the texture proceeding 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 diminish the reflexion in such a certain proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies absolutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies perfectly 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 perceive) 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 computation 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 something 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.
Suppose 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 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 persevere 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 part 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 indeed 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 carried 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.

So those weights are of equal force to move the arms of a balance; which during the play of the balance are reciprocally as their velocities upwards and downwards; that is, if the ascent or descent is direct, those weights are of equal force, which are reciprocally as the distances of the points at which they are suspended from the axis of the balance; but if they are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will be equipollent, which are reciprocally as the heights of their ascent and descent taken according to the perpendicular; and that on account of the determination of gravity downwards.

And in like manner in the pully, or in a combination of pullies, the force of a hand drawing the rope directly, which is to the weight, whether ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight.

In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are reciprocally as the velocities of the parts of the wheel on which they are impressed, will mutually sustain the one the other.

The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body.

The forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the progress of the wedge in the direction of the force impressed upon it by the mallet is to the velocity with which the parts of the wood yield to the wedge, in the direction of lines perpendicular to the sides of the wedge. And the like account is to be given of all machines.

The power and use of machines consist only in this, that by
diminishing the velocity we may augment the force, and the contrary:
from whence in all sorts of proper machines, we have the solution of
this problem; *To move a given weight with a given power*,
or with a given force to overcome any other given resistance. For if
machines are so contrived that the velocities of the agent and
resistant are reciprocally as their forces, the agent will just
sustain the resistant, but with a greater disparity of velocity will
overcome it. So that if the disparity of velocities is so great as
to overcome all that resistance which commonly arises either from
the attrition of contiguous bodies as they slide by one another, or
from the cohesion of continuous bodies that are to be separated, or
from the weights of bodies to be raised, the excess of the force
remaining, after all those resistances are overcome, will produce an
acceleration of motion proportional thereto, as well in the parts of
the machine as in the resisting body. But to treat of mechanics is
not my present business. I was only willing to show by those
examples the great extent and certainty of the third Law of motion.
For if we estimate the action of the agent from its force and
velocity conjunctly, and likewise the reaction of the impediment
conjunctly from the velocities of its several parts, and from the
forces of resistance arising from the attrition, cohesion, weight,
and acceleration of those parts, the action and reaction in the use
of all sorts of machines will be found always equal to one another.
And so far as the action is propagated by the intervening
instruments, and at last impressed upon the resisting body, the
ultimate determination of the action will be always contrary to the
determination of the reaction.