the third book the moon s theory and the profession of the equinoxes were
more fully deduced from their principles ; and the theory of the comets
was confirmed by more examples of the calculati >n of their orbits, done
also with greater accuracy.
In this third edition the resistance of mediums is somewhat more largely
handled than before; and new experiments of the resistance of heavy
bodies falling in air are added. In the third book, the argument to prove
that the moon is retained in its orbit by the force of gravity is enlarged
on ; and there are added new observations of Mr. Pound s of the proportion
of the diameters of Jter to each other : there are, besides, added Mr.
Kirk s observations of the comet in 16SO ; the orbit of that comet com
puted in an ellipsis by Dr. Halley ; and the ortit of the comet in
computed by Mr. Bradley,
OOK I.
THE
MATHEMATICAL PRINCIPLES
OF
NATURAL PHILOSOPHY
DEFINITIONS.
DEFINITION I.
77w? quantity of matter is the measure of the same, arising from its
density and hulk conjutictly.
THUS air of a double density, in a double space, is quadruple in quanti
ty ; in a triple space, sextuple in quantity. The same thing is to be un
derstood of snow, and fine dust or powders, that are condensed by compres
sion or liquefaction and of all bodies that are by any causes whatever
differently condensed. I have no regard in this place to a medium, if any
such there is, that freely pervades the interstices between the parts oi
bodies. It is this quantity that I mean hereafter everywhere under the
name of body or mass. And the same is known by the weight of each
body ; for it is proportional to the weight, as I have found by experiments
on pendulums, very accurately made, which shall be shewn hereafter.
DEFINITION II.
The quantity of motion is the measure nf tlie same, arising from the
velocity and quantity of matter corjunctly.
The motion of the whole i<! the sum of the motions of all the parts ; and
therefore in a body double in quantity, with equal velocity, the motion is
iouble ; with twice the velocity, it is quadruple,
DEFINITION III.
The vis insita, or innate force of matter, is a power of resisting, hy
which every body, as much as in it lies, endeavours to persevere in its
present stale, whether it be of rest, or of moving uniformly forward
in a right line.
This force is ever proportional to the body whose force it is ; and differs
nothing from the inactivity of the mass, but in our manner of conceiving
T4 THE MATHEMATICAL PRINCIPLES
it. A body, from the inactivity of matter, is not without difficulty put out
of its state of rest or motion. Upon which account, this vis insita, may,
by a most significant name, be called vis inertia, or force of inactivity.
Hut a body exerts this force only, when another force, impressed upon it,
endeavours to change its condition ; and the exercise of this force may bo
considered both as resistance and impulse ;
it is resistance, in so far as the
body, for maintaining its present state, withstands the force impressed; it
is impulse, in so far as the body, by not easily giving way to the impressed
force of another, endeavours to change the state of that other. Resistance
is usually ascribed to bodies at rest, and impulse to those in motion;
but motion and rest, as commonly conceived, are only relatively distin
guished ; nor are those bodies always truly at rest, which commonly are
taken to be so.
DKFLMTIOX IV.
Ait impressed force is an action exerted upon a body, in order to change
its state, either of rest, or of moving uniformly forward in a right
line.
This force consists in the action only; and remains no longer in the
body, when the action is over. For a body maintains every new state it
acquires, by its vis inertice only. Impressed forces are of origins
as from percussion, from pressure, from centripetal force.
DEFINITION V.
A centripetal force is that by which bodies are drawn or impelled, or any
way tend, towards a point as to a centre.
Of this sort is gravity, by which bodies tend to the centre of the earth
magnetism, by which iron tends to the loadstone ; and that force, what
ever it is, by which the planets are perpetually drawn aside from the rec
tilinear motions, which otherwise they would pursue, and made to revolve
in curvilinear orbits. A stone, whirled about in a sling, endeavours to re
cede from the hand that turns it
; and by that endeavour, distends the
sling, and that with so much the greater force, as it is revolved with the
greater velocity, and as soon as ever it is let go, flies away. That force
which opposes itself to this endeavour, and by which the sling perpetually
draws back the stone towards the hand, and retains it in its orbit, because
it is directed to the hand as the centre of the orbit, I call the centripetal
force. And the same thing is to be understood of all bodies, revolved in
any orbits. They all endeavour to recede from the centres of their orbits ;
and wore it not for the opposition of a contrary force which restrains them
to, and detains them in their orbits, which I therefore call centripetal, would
tiy off in right lines, with an uniform motion. A projectile, if it was not
for the force of gravity, would not deviate towards the earth, tut would
OF NATUJIAL PHILOSOPHY. 7fl
go off from it in a right line, and that with an uniform motion,, if the re
sistance of the air was taken away. It is by its gravity that it is drawn
aside perpetually from its rectilinear course, and made to deviate towards
the earth, more or less, according to the force of its gravity, and the velo
city of its motion. The less its gravity is, for the quantity of its matter,
or the greater the velocity with which it is projected, the less will it devi
ate from a rectilinear course, and the farther it will go. If a leaden balJ,
projected from the top of a mountain by the force of gunpowder with a
given velocity, and in a direction parallel to the horizon, is carried in a
curve line to the distance of two miles before it falls to the ground ; the
same, if the resistance of the air were taken away, with a double or decuple
velocity, would fly twice or ten times as far. And by increasing the velo
city, we may at pleasure increase the distance to which it might be pro
jected, and diminish the curvature of the line, which it might describe, till
at last it should fall at the distance of 10, 30, or 90 degrees, or even might
go quite round the whole earth before it falls
; or lastly, so that it might
never fall to the earth, but go forward into the celestial spaces, and pro
ceed in its motion in iiifiuitum. And after the same manner that a pro
jectile, by the force of gravity, may be made to revolve in an orbit, and go
round the whole earth, the moon also, either by the force of gravity, if it
is endued with gravity, or by any other force, that impels it towards the
earth, may be perpetually drawn aside towards the earth, out of the r&tilinear
way, which by its innate force it would pursue; and would be made
to revolve in the orbit which it now describes ; nor could the moon with
out some such force, be retained in its orbit. If this force was too small,
it would not sufficiently turn the moon out of a rectilinear course : if it
was too great, it would turn it too much, arid draw down the moon from
its orbit towards the earth. It is necessary, that the force be of a just
quantity, and it belongs to the mathematicians to find the force, that may
serve exactly to retain a body in a given orbit, with a given velocity ; and
vice versa, to determine the curvilinear way, into which a body projected
from a given place, with a given velocity, may be made to deviate from
its natural rectilinear way, by means of a given force.
The quantity of any centripetal force may be considered as of three
kinds; aboolu e, accelerative, and motive.
DEFINITION VI.
The absolute quantity of a centripetal force is the measure f
>f
the same
proportional to the efficacy of the cause that propagates itfrom the cen
tre, through the spaces round about.
Thus the magnetic force is greater in one load-stone and less in another
according to their sizes and strength of intensity.
76 THE MATHEMATICAL PRINCIPLES
DEFINITION VII.
The accelerative quantity of a centripetal force is the measure, of tht
same, proportional to the velocity which it generates in a given time.
Thus the force of the same load-stone is greater at a less distance, and
less at a greater : also the force of gravity is greater in valleys, less on
tops of exceeding high mountains ; and yet less (as shall hereafter be shown),
at greater distances from the body of the earth ; but at equal distan
ces, it is the same everywhere ; because (taking away, or allowing for, the
resistance of the air), it equally accelerates all falling bodies, whether heavy
or light, great or small.
DEFINITION VIII.
TJie motive quantity of a centripetal force, is the measure of the samt\
proportional to the motion which it generates in a given twip.
Thus the weight is greater in a greater body, less in a less body ; and.
in the same body, it is greater near to the earth, and less at remoter dis
tances. This sort of quantity is the centripetency, or propension of the
whole body towards the centre, or, as I may say, its weight ; and it is al
ways known by the quantity of an equal and contrary force just sufficient
to Ifinder the descent of the body.
These quantities of forces, we may, for brevity s sake, call by the names
of motive, accelerative, and absolute forces ; and, for distinction s sake, con
sider them, with respect to the bodies that tend to the centre ;
to the places
of those bodies ; and to the centre of force towards which they tend ; that
is to say, I refer the motive force to the body as an endeavour and propen
sity of the whole towards a centre, arising from the propensities of the
several parts taken together ; the accelerative force to the place of the
body, as a certain power or energy diffused from the centre to all places
around to move the bodies that are in them : and the absolute force to
the centre, as endued with some cause, without which those motive forces
would not be propagated through the spaces round about ; whether that
cause be some central body (siuh as is the load-stone, in the centre of the
magnetic force, or the earth in the centre of the gravitating force), or
anything else that does not yet appear. For I here design only to give a
mathematical notion of those forces, without considering their physical
causes and seats.
Wherefore the accelerative force will stand in the same relation to the
motive, as celerity does to motion. For the quantity of motion arises from
the celerity drawn into the quantity of matter : and the motive force arises
from the accelerative force drawn into the same quantity of matter. For
the sum of the actions of the accelerative force, upon the several ;
articles
of the body, is the motive force of the whole. Hence it is, that near the
OF NATURAL PHILOSOPHY. 77
surface of the earth, where the accelerative gravity, or force productive of
gravity, in all bodies is the same, the motive gravity or the weight is as
the body : but if we should ascend to higher regions, where the accelerative
gravity is less, the weight would be equally diminished, and would always
be as the product of the body, by the accelerative gravity. So in those re
gions, where the accelerative gravity is diminished into one half, the weight
of a body two or three times less, will be four or six times less.
I likewise call attractions and impulses, in the same sense, accelerative,
and motive ; and use the words attraction, impulse or propensity of any
sort towards a centre, promiscuously, and indifferently, one for another ;
considering those forces not physically, but mathematically : wherefore, the
reader is not to imagine, that by those words, I anywhere take upon me to
define the kind, or the manner of any action, the causes or the physical
reason thereof, or that I attribute forces, in a true and physical sense, to
certain centres (which are only mathematical points) ; when at any time I
happen to speak of centres as attracting, or as endued with attractive
powers.
SCHOLIUM.
Hitherto I have laid down the definitions of such words as are less
known, and explained the sense in which I would have them to be under
stood in the following discourse. I do not define time, space, place and
motion, as being well known to all. Only I must observe, that the vulgar
conceive those quantities under no other notions but from the relation they
bear to sensible objects. And thence arise certain prejudices, for the re
moving of which, it will be convenient to distinguish them into absolute
and relative, true and apparent, mathematical and common.
I. Absolute, true, and mathematical time, of itself, and from its own na
ture flows equably without regard to anything external, and by another
name is called duration : relative, apparent, and common time, is some sen
sible and external (whether accurate or unequable) measure of duration by
the means of motion, which is commonly used instead of true time ; such
as an hour, a day, a month, a year.
II. Absolute space, in its own nature, without regard to anything exter
nal, remains always similar and immovable. Relative space is some mo
vable dimension or measure of the absolute spaces ; which our senses de
termine by its position to bodies ; and which is vulgarly taken for immo
vable space ; such is the dimension of a subterraneous, an aereal, or celestial
space, determined by its position in respect of the earth. Absolute and
relative space, are the same in figure and magnitude ; but they do not re
main always numerically the same. For if the earth, for instance, moves,
a space of our air, which relatively and in respect of the earth remains al
ways the same, will at one time be one part of the absolute space into which
TS THE MATHEMATICAL PRINCIPLES
the air passes ;
at another time it will be another part of the same, and so.
absolutely understood, it will be perpetually mutable.
III. Place is a part of space which a body takes up, and is according to
the space, either absolute or relative. I say, a part of space ; not the situation,
nor the external surface of the body. For the places of equal solids are
always equal ; but their superfices, by reason of their dissimilar figures, are
often unequal. Positions properly have no quantity, nor are they so much
the places themselves, as the properties of places. The motion of the whole
is the same thing with the sum of the motions of the parts ; that is, the
translation of the whole, out of its place, is the same thing with the sum
of the translations of the parts out of their places ; and therefore the place
of the whole is the same thing with the sum of the places of the parts, and
for that reason, it is internal, and in the whole body.
IV. Absolute motion is the translation of a body from one absolute
place into another ; and relative motion, the translation from one relative
place into another. Thus in a ship under sail, the relative place of a body
is that part of the ship which the body possesses ; or that part of its cavity
which the body fills, and which therefore moves together with the ship :
and relative rest is the continuance of the body in the same part of the
ship, or of its cavity. But real, absolute rest, is the continuance of the
body in the same part of that immovable space, in which the ship itself,
its cavity, and all that it contains, is moved. Wherefore, if the earth is
really at rest, the body, which relatively rests in the ship, will really and
absolutely move with the same velocity which the ship has on the earth.
But if the earth also moves, the true and absolute motion of the body will
arise, partly from the true motion of the earth, in immovable space ; partly
from the relative motion of the ship on the earth ; and if the body moves
also relatively in the ship ;
its true motion will arise, partly from the true
motion of the earth, in immovable space, and partly from the relative mo
tions as well of the ship on the earth, as of the body in the ship ; and from
these relative motions will arise the relative motion of the body on the
earth. As if that part of the earth, where the ship is, was truly moved
toward the east, with a velocity of 10010 parts; while the ship itself, with
a fresh gale, and full sails, is carried towards the west, with a velocity ex
pressed by 10 of those parts ; but a sailor walks in the ship towards the
east, with 1 part of the said velocity ; then the sailor will be moved truly
in immovable space towards the east, with a velocity of 10001 parts, and
relatively on the earth towards the west, with a velocity of 9 of those parts.
Absolute time, in astronomy, is distinguished from relative, by the equa
tion or correction of the vulgar time. For the natural days are tr^y un
equal, though they are commonly considered as equal, and used for a meas
ure of time ; astronomers correct this inequality for their more accurate
deducing of the celestial motions. It may be, that there is no such thing
as an equable motion, whereby time may H accurately measured. All mo
OF NATURAL PHILOSOPHY. 79
tions may be accelerated and retarded; but the true, or equable, progress of
absolute time is liable to no change. The duration or perseverance of the
existence of things remains the same, whether the motions are swift or slow,
or none at all : and therefore it ought to be distinguished from what are
only sensible measures thereof ; and out of which we collect it, by means
of the astronomical equation. The necessity of which equation, for deter
mining the times of a phamomenon, is evinced as well from the experiments
of the pendulum clock, as by eclipses of the satellites of Jupiter.
As the order of the parts of time is immutable, so also is the order of
the parts of space. Suppose those parts to be moved out of their places, and
they will be moved (if the expression may be allowed) out of themselves.
For times and spaces are, as it were, the places as well of themselves as of
all other things. All things are placed in time as to order of succession ;
and in space as _to order of situation. It is from their essence or nature
that they are places ; and that the primary places of things should be
moveable, is absurd. These are therefore the absolute places ; and trans
lations out of those places, are the only absolute motions.
But because the parts of space cannot be seen, or distinguished from one
another by our senses, therefore in their stead we use sensible measures of
them. For from the positions and distances of things from any body con
sidered as immovable, we define all places ; and then with respect to such
places, we estimate all motions, considering bodies as transferred from some
of those places into others. And so, instead of absolute places and motions,
we use relative ones; and that without any inconvenience in common af
fairs ; but in philosophical disquisitions, we ought to abstract from our
senses, and consider things themselves, distinct from what are only sensible
measures of them. For it may be that there is no body really at rest, to
which the places and motions of others may be referred.
But we may distinguish rest and motion, absolute and relative, one from
the other by their properties, causes and effects. It is a property of rest,
that bodies really at rest do rest in respect to one another. And therefore
as it is possible, that in the remote regions of the fixed stars, or perhaps
far beyond them, there may be some body absolutely at rest
; but impossi
ble to know, from the position of bodies to one another in our regions
whether any of these do keep the same position to that remote body; it
follows that absolute rest cannot be determined from the position of bodies
in our regions.
It is a property of motion, that the parts, which retain given positions
to their wholes, do partake of the motions of those wholes. For all the
parts of revolving bodies endeavour to recede from the axis of motion ;
and the impetus of bodies moving forward, arises from the joint impetus
of all the parts. Therefore, if surrounding bodies are moved, those that
are relatively at rest within them, will partake of their motion. Upon
which account, the true and absolute motion of a body cannot be Jeter
8C THE MATHEMATICAL PRINCIPLES
mined by the translation of it from those which only seem to rest
; for the
external bodies ought not only to appear at rest, but to be really at rest.
For otherwise, all included bodies, beside their translation from near the
surrounding ones, partake likewise of their true motions ; and though that
translation were not made they would not be really at rest, but only seem
to be so. For the surrounding bodies stand in the like relation to the
surrounded as the exterior part of a whole does to the interior, or as the
shell does to the kernel ; but, if the shell moves, the kernel will also
move, as being part of the whole, without any removal from near the shell.
A property, near akin to the preceding, is this, that if a place is moved,
whatever is placed therein moves along with it
; and therefore a body,
which is moved from a place in motion, partakes also of the motion of its
place. Upon which account, all motions, from places in motion, are no
other than parts of entire and absolute motions ; and every entire motion
is composed of the motion of the body out of its first place, and the
motion of this place out of its place ; and so on, until we come to some
immovable place, as in the before-mentioned example of the sailor. Where
fore, entire and absolute motions can be no otherwise determined than by
immovable places : and for that reason I did before refer those absolute
motions to immovable places, but relative ones to movable places. Now
no other places are immovable but those that, from infinity to infinity, do
all retain the same given position one to another ; and upon this account
must ever remain unmoved ; and do thereby constitute immovable space.
The causes by which true and relative motions are distinguished, one
from the other, are the forces impressed upon bodies to generate motion.
True motion is neither generated nor altered, but by some force impressed
upon the body moved : but relative motion may be generated or altered
without any force impressed upon the body. For it is sufficient only to
impress some force on other bodies with which the former is compared,
that by their giving way, that relation may be changed, in which the re
lative rest or motion of this other body did consist. Again, true motion
suffers always some change from any force impressed upon the moving
body ; but relative motion docs not necessarily undergo any change by such
forces. For if the same forces are likewise impressed on those other bodies,
with which the comparison is made, that the relative position may be pre
served, then that condition will be preserved in which the relative motion
consists. And therefore any relative motion may be changed when the
true motion remains unaltered, and the relative may be preserved when the
true suffers some change. Upon which accounts; true motion does by no
means consist in such relations.
The effects whicli distinguish absolute from relative motion arc, the
forces of receding from the axis of circular motion. For there are no such
forces in a circular motion purely relative, but in a true and absolute cir
cular motion., they are greater or less, according t the quantity of the
OF NATURAL PHILOSOPHY. 1
motion. If a vessel, hung: by & }ong cord, is so often turned ubout that the
cord is strongly twisted, then filled with water, and held at rest together
with the water ; after, by the sudden action of another force, it is whirled
about the contrary way, and while the cord is untwisting itself, the vessel
continues for some time in this motion ; the surface of the water will at
first be plain, as before the vessel began to move : but the vessel; by grad
ually communicating its motion to the water, will make it begin sensibly
^to revolve, and recede by little and little from the middle, and ascend to the
sides of the vessel, forming itself into a concave figure (as I have experi
enced), and the swifter the motion becomes, the higher will the water rise,
till at last, performing its revolutions in the same times with the vessel,
it becomes relatively at rest in it. This ascent of the water shows its en
deavour to recede from the axis of its motion ; and the true and absolute
circular motion of the water, which is here directly contrary to the relativej
discovers itself, and may be measured by this endeavour. At first,
when the relative motion of the water in the vessel was greatest, it pro
duced no endeavour to recede from the axis ; the water showed no tendency
to the circumference, nor any ascent towards the sides of the vessel, but
remained of a plain surface, and therefore its true circular motion had not
yet begun. But afterwards, when the relative motion of the water had
decreased, the ascent thereof towards the sides of the vessel proved its en
deavour to recede from the axis ; and this endeavour showed the real cir
cular motion of the water perpetually increasing, till it had acquired its
greatest quantity, when the water rested relatively in the vessel. And
therefore this endeavour does not depend upon any translation of the water
in respect of the ambient bodies, nor can true circular motion be defined
by such translation. There is only one real circular motion of any one
revolving body, corresponding to only one power of endeavouring to recede
