AXIOMS, OR LAWS OF MOTION
LAW I
Every body continues
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.
Projectiles continue 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 continually
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 freer spaces, preserve their
motions both progressive and circular for a much longer time.
LAW II
The change of motion is 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 subtracted 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.
LAW III
To every action there is always opposed an equal reaction:
or, the mutual actions of two bodies upon each other are always equal,
and directed to contrary parts.
Whatever draws or presses another is as much drawn or
pressed by 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 endeavor 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 the 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 inversely proportional to the bodies. This law takes
place also in attractions, as will be proved in the next Scholium.
COROLLARY I
A body, acted on by two forces simultaneously, will describe
the diagonal of a parallelogram in the same time as it would describe the
sides by those forces separately.
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, let the parallelogram
ABCD
be completed, 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.
COROLLARY II
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
P: A = DC : DA,
but because the triangles ADC and DOK are
similar,
DC : DA = OK: OD
= OK : OL.
Therefore
P : A = radius OK:
radius OL.
As these radii lie in the same right line they will be
equipollent, and so remain in equilibrium; which is the wellknown 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 = P, is partly suspended
by the cord Np, partly sustained by the oblique plane
pG;
draw pH, NH, the former perpendicular to the horizon, the
latter to the plane pG; and if the force of the weight p
tending downwards is represented by the line pH, it may be resolved
into the forces pN, HN. If there was any plane pQ,
perpendicular to the cord pN, cutting the other plane pG
in a line parallel to the horizon, and the weight p was supported
only by those planes pQ, pG, it would press those planes
perpendicularly with the forces pN, HN; to wit, the plane
pQ
with the force pN, 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, supplied
the place of the plane that was removed, it would be strained by the same
force pN which pressed upon the plane before. Therefore, the
tension of pN: tension of PN = line
pN
: line pH.
Therefore, if p is to A in a ratio which
is the product of the inverse ratio of the least distances of their cords
pN
and AM from the centre of the wheel, and of the ratio
pH
to pN, then the weights p and A will have the same
effect towards moving the wheel, and will, therefore, sustain each other;
as anyone 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: because the force with which the weight p presses
the plane pQ 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 pH towards both the planes, as
pN : pH;
and to the force with which it presses the other plane
pG,
as
pN : 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 further 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.
COROLLARY III
The quantity of motion, which is obtained 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 reaction 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 subtracted 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 is 3 times greater
than the spherical body B, and has a velocity = 2, and B
follows in the same direction with a velocity = 10, then the
motion of A : motion of B = 6 :
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 reflection 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 reflection of the bodies. But the motions being known
with which the bodies proceed after reflection, the velocity of either
will be also known, by taking the velocity after to the velocity before
reflection, as the motion after is to the motion before. As in the last
case, where the
motion of A before reflection (6) : motion of
A
after (18)
= velocity of A before (2) : velocity of A
after (x);
that is,
6 : 18 = 2 : x, x = 6.
But if the bodies are either not spherical, or, moving
in different right lines, impinge obliquely one upon the other, and their
motions after reflection are required, in those cases we are first to determine
the position of the plane that touches the bodies in the point of impact,
then the motion of each body (by Cor. 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 reflection
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 reflections sometimes arise also 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 case that relates to this subject.
COROLLARY IV
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 external 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. 23 and Corollary, 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 forwards in a right line.
Moreover, in a system of two bodies 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
accelerated 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 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 inversely 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 one upon
another, either is at rest or moves uniformly forwards in some right line,
it will, notwithstanding the mutual actions of the bodies among themselves,
always continue 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.
COROLLARY V
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 impinge one upon another. Wherefore (by Law 2), 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 motions of the bodies among themselves in the other. A clear
proof of this 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 VI
If bodies, moved in any manner 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 not been urged by those 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 2) move all the bodies equally (as to velocity), and therefore
will never produce any change in the positions or motions of the bodies
among themselves.
SCHOLIUM
Hitherto I have laid down such principles as have been received
by mathematicians, and are confirmed by abundance of experiments. By
the first two Laws and the first two Corollaries, Galileo discovered that
the descent of bodies varied as the square of the time (in duplicata
ratione temporis) 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 intervals
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 product of the velocities and the times; that is, as the squares
of the times. And when a body is thrown upwards, its uniform gravity
impresses forces and reduces velocities proportional to the times; and
the times of ascending to the greatest heights are as the velocities to
be taken away, and those heights are as the product of the velocities and
the times, or as the squares 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.
Thus, 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 parallelogram ABCD, and the body by that compounded
motion will at the end of the time be found in the place D; and
the curved line AED, which that body describes, will be a parabola,
to which the right line
AB will be a tangent at 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
Law 3, Sir Christopher Wren, Dr. Wallis, and Mr. Huygens, the greatest
geometers of our times, did severally determine the rules of the impact
and reflection of hard bodies, and about the same time communicated their
discoveries to the Royal Society, exactly agreeing among themselves as
to those rules. Dr. Wallis, indeed, was somewhat earlier 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 experiments
on pendulums, which M. 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 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 lengths as radii, describe
the semicircles EAF,
GBH, bisected respectively 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, namely, so that
RS = TV,
and
RS : ST = 3 : 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 reflection 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
reflection, 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 tA
represent the velocity which the body A had in the place A
immediately after reflection. 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 reflection; and then by the chord
of the arc tA, that we may have its motion in the place
A
immediately after reflection. And so we are to take the product of the
body B by the chord of the arc Bl, that we may have the motion
of the same immediately after reflection. 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 reflection; and then we may compare the
motions between themselves, and collect the effects of the reflection.
Thus trying the thing with pendulums of 10 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 reflection with 2, the body
B was carried backwards
with those 7 parts. If the bodies concurred with contrary motions, A
with 12 parts of motion, and B with 6, then if A receded
with 2, B receded with 8; namely, with a deduction of 14 parts of
motion on each side. For from the motion of A subtracting 12
parts, nothing will remain; but subtracting 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, subtracting 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 reflection 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 meeting and collision
of bodies, the quantity of motion, obtained 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 impact. 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 alleged
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 reflection 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 impact, 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 reflection, I determined the quantity of their
elastic force; and then, according to this force, estimated the reflections
that ought to happen in other cases of impact. 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 reflections, 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 meeting
of any two bodies A, B, 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 equilibrium:
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 forwards in infinitum
with a motion continually accelerated; which is absurd and contrary to
the first Law. For, by the first Law, the system ought to continue in
its state of rest, or of moving uniformly forwards 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 between the earth and its parts is
mutual. Let the earth FI by 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 between 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 nonresisting ether 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 impact and
reflection, whose velocities are inversely 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 inversely 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 inversely as their
velocities upwards and downwards; that is, if the ascent or descent is
direct, those weights are of equal force, which are inversely 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 inversely 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 pulley, or in a combination
of pulleys, 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 inversely as the velocities of the parts of the wheel
on which they are impressed, will mutually sustain each 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 inversely
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 friction 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 aiming only 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 the product of its force and velocity, and likewise the reaction of
the impediment from the product of the velocities of its several parts,
and the forces of resistance arising from the friction, 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 action will be
always contrary to the reaction.
[End of Axioms, or Laws of Motion]

