from its axis of motion, as its proper and adequate effect
; but relative
motions, in one and the same body, are innumerable, according to the various
relations it bears to external bodies, and like other relations, arc altogether
destitute of any real effect, any otherwise than they may perhaps par
take of that one only true motion. And therefore in their system who
suppose that our heavens, revolving below the sphere of the fixed stars,
carry the planets along with them ; the several parts of those heavens, and
the planets, which are indeed relatively at rest in their heavens, do yet
really move. For they change their position one to another (which never
happens to bodies truly at rest), and being carried together with their
heavens, partake of their motions, and as parts of revolving wholes,
endeavour to recede from the axis of their motions.
Wherefore relative quantities are not the quantities themselves, whose
names they bear, but those sensible measures of them (either accurate cr
inaccurate), which arc commonly used instead of the measured quantities
themselves. And if the meaning of words is to he determined bv their
82 THE MATHEMATICAL PRINCIPLES
use, then by the names time, space, place and motion, their measures arv
properly to be understood ; and the expression will be unusual, and purely
mathematical, if the measured quantities themselves are meant. Upon
which account, they do strain the sacred writings, who there interpret
those words for the measured quantities. Nor do those less defile the
purity of mathematical and philosophical truths, who confound real quan
tities themselves with their relations and vulgar measures.
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 effects
of the true motions. For instance, if tAvo 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 decr ase of the tensicn of 1 le 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 hindermost
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 which 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 oi
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 theii
causes and effects, shall be explained more at large in the following tra<;t
For to this end it was that I composed it.
OF NATURAL PHILOSOPHY.
AXIOMS, OR LAWS OF MOTION.
LAW I.
Hvery body perseveres in its state of rest, or of uniform motion in a ri^ht
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 re
tarded by the air. The greater bodies of the planets and comets, meeting
with less resistance in more free spaces, preserve then jDotions both pro
gressive and circular for a much longer time.
LAW II.
The alteration of motion is ever proportional to the motive force impreus
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, 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 mu
tual actions of two bodies upon each other are always 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.
84 THE MATHEMATICAL PRINCIPLES
If a body impinge upon another, and by its force change the motion of (It*
other, that body also (because of the equality of the mutual pressure) 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 I.
A body by two forces conjoined will describe the diagonal of a parallelo
gram, in the same time that it wovld 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
rorce N be impressed or not ; and therefore at the end of that time it will
he found somewhere in the line BD. By the same argument, at the end
of the same time it AY ill 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
;i 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 re
solution 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
rentre O draw the right line KOL, meeting the cords perpendicularly in
A and L; and from the centre O, with OL the greater of the distances
OF NATURAL PHILOSOPHY.
OK arid OL, describe a circle, meeting the cord
MA in D : and drawing OD, make AC paral- "^
lei and DC perpendicular thereto. Now, it
being indifferent whether the points K, L, D, of
the cords be lixed 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 w ill 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 NJO, partly sustained by the oblique plane pG ; draw p}i, 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
/?H, it may be resolved into the forces joN, HN. If there was any plane
/?Q, perpendicular to the cord y?N, cutting the other plane pG in a 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 joQ, with the force joN, and the plane pG with
the force HN. And therefore if the plane pQ was taken away, so thnt
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 joN which pressed upon the plane before. Therefore,
the tension of this oblique cord joN will be to that of the other perpendic
ular cord PN as jt?N to joH. 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
pH tojoN, the weights will have the same effect towards moving the wheel,
and will therefore sustain each other : as any one may find by experiment.
But the weight p pressing upon those two oblique planes, may be con
sidered as a wedge between the two internal surfaces of a body split by it;
and hence tlif ft IV.P* of th^ v, ^dge and the mallet may be determined; foi
8G 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 joH towards both the
planes, as joN to pH ; and to the force with which it presses the other
plane pG, as joN 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 (/uaittity of motion, which is collected by taking the sum of the mo
tions directed towards the same parts, and the difference of those that
are directed to contrary parts, suffers no change from the action oj
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 1 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 ,1, or 16-1-0, and the differences of the contrary i otions 17 1 and
OF NATURAL PHILOSOPHY.
[S 2, will always be equal to 16 parts, as they were before tie meeting
and reflexion of the bodies. But, the motions being known with whicli
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 tho
body A was of parts before reflexion and of IS 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 parts of motion before
to 18 parts after, so are 2 parts of velocity before reflexion to (5 parts after.
But if the bodies are cither not spherical, or, moving in different right
lines, impinge obliquely one upon the other, and their mot ons 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. II) 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 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 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 when the
points are not moved in the same plane. Therefore if any number of
Kdies 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 uni
formly in aright line because at that centre the distance 1 etween th?
88 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 m itifinitum.
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 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, xhe 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 & 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 comrrv n centre of alias to its state of motion or rest. Wherefore
tiince that centre, when the bodies do not act mutually one upon another,
Oilier is nt rest or moves uniformly forward in some right line, it will,
:v\>U7ithst?nding the mutual actions of the bodies among themselves, always
jAY-jevere in its state, either of rest, or of proceeding uniformly in a right
liiv,, unless it is forced out of this state by the action of some power imprev^-
d from without upon the whole system. And therefore the same law
take*1 place in a system consisting of many bodies as in one single body,
with wsgard to their persevering in their state of motion or of rest. For
the pi \\jressive motion, whether of one single body, or of a whole system of
bodies us always to be estimated from the motion of the centre of gravity.
COROLLARY V.
The motions cf bcdies included in a given space a ~e Ike same among
OF NATURAL PHILOSOPHY. 89
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 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 VI.
If bodies, any how moved among themselves, are urged in the direct-ton
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
DO dies to be moved), and in the direction of parallel lines, will (by Law II)
move all the bodies equally (as to velocity), and therefore will never 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 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 conjunetly, or ir,
the duplicate ratio of the velocities. And if a body be projected in any
direction, the motion arising from its projection jS compounded with the
90 THE MATHEMATICAL PRINCIPLES
motion arising from its gravity. As if the body A by its motion of piojection
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 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. Huvgens,
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. Mariottc 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
bodies. Let the spherical bodies
CD F II
as to the clastic force of the concurrin
A, B be suspended by the parallel and
equal strings AC, Bl), 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
