the orb. Therefore that every orb may persevere uniformly in its motion,
it is necessary that the impressions made upon both sides of the orb should
be equal, and have contrary directions. Therefore since the impressions
are as the contiguous superficies, and as their translations from one another^
the translations will be inversely as the superficies, that is, inversely as the
squares of the distances of the superficies from the centre. But the differ
ences of the angular motions about the axis are as those translations applied
to the distances, or as the translations directly and the distances inversely;
that is, by compounding those ratios, as the cubes of the distances inversely.
Therefore if upon the several parts of the infinite right line SABCDEQ
SEC. IX.j OF NATURAL PHILOSOPHY. 373
there be erected the perpendiculars Aa, B6. Cc, Dd, Ee, c.
; reciprocally
proportional to the cubes of SA5 SB, SO, SD, SE, etc., the sums of the
differences, that is, the whole angular motions will be as the corresponding
sums of the lines A#, B&, Cc, DC/, Ee, <fcc., that is (if to constitute an uni
formly fluid medium the number of the orbs be increased and their thick
ness diminished in infinitum), as the hyperbolic areas AaQ, B&Q,, CcQ,
DtfQ,, EeQ,, etc., analogous to the sums ; and the periodic times being re
ciprocally proportional to the angular motions, will be also reciprocally
proportional to those areas. Therefore the periodic time of any orb DIO
is reciprocally as the area Dt/Q,, that is (by the known methods of quadra
tures), directly as the square of the distance SD. Which was first to be
demonstrated.
CASE 2. From the centre of the sphere let there be drawn a great num
ber of indefinite right lines, making given angles with the axis, exceeding
one another by equal differences
; and, by these lines revolving about the
axis, conceive the orbs to be cut into innumerable annuli; then will every
annulus have four annuli contiguous to it, that is, one on its inside, one on
its outside, and two on each hand. Now each of these annuli cannot be
impelled equally and with contrary directions by the attrition of the inte
rior and exterior annuli, unless the motion be communicated according to
the law which we demonstrated in Case 1. This appears from that dem
onstration. And therefore any series of annuli, taken in any right line
extending itself in infinitum from the globe, will move according to the
law of Case 1, except we should imagine it hindered by the attrition of the
annuli on each side of it. But now in a motion, according to this law, no
such is, and therefore cannot be, any obstacle to the motions persevering
according to that law. If annuli at equal distances from the centre
revolve either more swiftly or more slowly near the poles than near the
ecliptic, they will be accelerated if slow, and retarded if swift, by their
mutual attrition; and so the periodic times will continually approach to
equality, according to the law of Case 1. Therefore this attrition will not
at all hinder the motion from going on according to the law of Case 1
, and
therefore that law will take place ; that is, the periodic times of the several
annuli will be as the squares of their distances from the centre of the globe.
Which was to be demonstrated in the second place.
CASE 3. Let now every annulus be divided by transverse sections into
innumerable particles constituting a substance absolutely and uniformly
fluid ; and because these sections do not at all respect the law of circular
motion, but only serve to produce a fluid substance, the law of circular mo
tion will continue the same as before. All the very small annuli will eithei
not at all change their asperity and force of mutual attrition upon account
of these sections, or else they will change the same equally. Therefore the
proportion of the causes remaining the same, the proportion of the effects
3r4 THE MATHEMATICAL PRINCIPLES [BOOK II.
will remain the same also ; that is, the proportion of the motions and tin
periodic times. Q.E.D. But now as the circular motion, and the centri
fugal force thence arising, is greater at the ecliptic than at the poles, there
must be some cause operating to retain the several particles in their ciicles
;
otherwise the matter that is at the ecliptic will always recede from the
centre, and come round about to the poles by the outside of the vortex,
and from thence return by the axis to the ecliptic with a perpetual circu
lation.
COR. 1. Hence the angular motions of the parts of the fluid about the
axis of the globe are reciprocally as the squares of the distances from the
centre of the globe, and the absolute velocities are reciprocally as the same
squares applied to the distances from the axis.
COR. 2. If a globe revolve with a uniform motion about an axis of a
given position in a similar and infinite quiescent fluid with an uniform
motion, it will communicate a whirling motion to the fluid like that of a
vortex, and that motion will by degrees be propagated onward in infinitnm ;
and this motion will be increased continually in every part of the fluid, till
the periodical times of the several parts become as the squares of the dis
tances from the centre of the globe.
COR. 3. Because the inward parts of the vortex are by reason of their
greater velocity continually pressing upon and driving forward the external
parts, and by that action are perpetually communicating motion to them,
and at the same time those exterior parts communicate the same quantity
of motion to those that lie still beyond them, and by this action preserve
the quantity of their motion continually unchanged, it is plain that the
motion is perpetually transferred from the centre to the circumference of
the vortex, till it is quite swallowed up and lost in the boundless extent of
that circumference. The matter between any two spherical superficies
concentrical to the vortex will never be accelerated ; because that matter
will be always transferring the motion it receives from the matter nearer
the centre to that matter which lies nearer the circumference.
COR. 4. Therefore, in order to continue a vortex in the same state of
motion, some active principle is required from which the globe may receive
continually the same quantity of motion which it is always communicating
to the matter of the vortex. Without such a principle it will undoubtedly
come to pass that the globe and the inward parts of the vortex, being al
ways propagating their motion to the outward parts, and not receiving any
new motion, will gradually move slower and slower, and at last be carried
round no longer.
COR. 5. If another globe should be swimming in the same vortex at a
certain distance from its centre, and in the mean time by some force revolve
constantly about an axis of a given inclination, the motion of Jiis globe
will drive the fluid round after the manner of a vortex and at first this
SEC. IX.] OF NATURAL PHILOSOPHY. 375
new and small vortex will revolve with its globe about the centre of the
other; and in the mean time its motion will creep on farther and farther,
and by degrees be propagated in iiifinitum, after the manner of the first
vortex. And for the same reason that the globe of the new vortex wat
carried about before by the motion of the other vortex, the globe of this
other will be carried about by the motion of this new vortex, sc that the
two globes will revolve about some intermediate point, and by reason of
that circular motion mutually fly from each other, unless some force re
strains them. Afterward, if the constantly impressed forces, by which the
globes persevere in their motions, should cease, and every thing be left to
act according to the laws of mechanics, the motion of the globes will lan
guish by degrees (for the reason assigned in Cor. 3 arid 4), and the vortices
at last will quite stand still.
COR. 6. If several globes in given places should constantly revolve with
determined velocities about axes given in position, there would arise from
them as many vortices going on in infinitum. For upon the same account
that any one globe propagates its motion in itifinitum, each globe apart
will propagate its own motion in infiidtwtn also ; so that every part of the
infinite fluid will be agitated with a motion resulting from the actions of
all the globes. Therefore the vortices will not be confined by any certain
limits, but by degrees run mutually into each other ; and by the mutual
actions of the vortices on each other, the globes will be perpetually moved
from their places, as was shewn in the last Corollary ; neither can they
possibly keep any certain position among themselves, unless some force re
strains them. But if those forces, which are constantly impressed upon
the globes to continue these motions, should cease, the matter (for the rea
son assigned in Cor. 3 and 4) will gradually stop, and cease to move in
vortices.
COR. 7. If a similar fluid be inclosed in a spherical vessel, and, by the
uniform rotation of a globe in its centre, is driven round in a vortex ; and
the globe and vessel revolve the same way about the same axis, and their
periodical times be as the squares of the semi-diameters ; the parts of the
fluid will not go on in their motions without acceleration or retardation,
till their periodical times are as the squares of their distances from
the centre of the vortex. No constitution of a vortex can be permanent
but this.
COR. 8. If the vessel, the inclosed fluid, and the globe, retain this mo
tion, and revolve besides with a common angular motion about any given
axis, because the mutual attrition of the parts of the fluid is not changed
by this motion, the motions of the parts among each other will not be
changed ;
for the translations of the parts among themselves depend upon
this attrition. Any part will persevere in that motion in which its attri
376 THE MATHEMATICAL PRINCIPLES [BOOK II.
tion on one side retards it just as much as its attrition on the other side
accelerates it.
COR. 9. Therefore if the vessel be quiescent, and the motion of the
globe be given, the motion of the fluid will be given. For conceive a plane
to pass through the axis of the globe, and to revolve with a contrary mo
tion ; and suppose the sum of the time of this revolution and of the revolu
tion of the globe to be to the time of the revolution of the globe as the
square of the semi-diameter of the the square of the semi-diameter
of the globe ; and the periodic times of the parts of the fluid in respect of
this plane will be as the squares of their distances from the centre of the
globe.
COR. 10. Therefore if the vessel move about the same axis with the globe,
or with a given velocity about a different one, the motion of the fluid will
be given. For if from the whole system we take away the angular motion
of the vessel, all the motions will remain the same among themselves as
before, by Cor. 8, and those motions will be given by Cor. 9.
COR. 11. If the vessel and the fluid are quiescent, and the globe revolves
with an uniform motion, that motion will be propagated by degrees through
the whole fluid to the vessel, and the vessel will be carried round by it,
unless violently detained ; and the fluid and the vessel will be continually
accelerated till their periodic times become equal to the periodic times of
the globe. If the vessel be either withheld by some force, or revolve with
any constant and uniform motion, the medium will come by little and
little to the state of motion defined in Cor. 8, 9, 10, nor will it ever perse
vere in any other state. But if then the forces, by which the globe and
vessel revolve with certain motions, should cease, and the whole system be
left to act according to the mechanical laws, the vessel and globe, by means
of the intervening fluid, will act upon each other, and will continue to
propagate their motions through the fluid to each other, till their periodic
times become equal among themselves, and the whole system revolves to
gether like one solid body.
SCHOLIUM.
In all these reasonings I suppose the fluid to consist of matter of uniform
density and fluidity ;
I mean, that the fluid is such, that a globe placed
any where therein may propagate with the same motion of its own, at dis
tances from itself continually equal, similar and equal motions in the fluid
in the same interval of time. The matter by its circular motion endeavours
to recede from the axis of the vortex, and therefore presses all the matter
that lies beyond. This pressure makes the attrition greater, and the
Separation of the parts more difficult
; and by consequence diminishes
the fluidity of the matter. Again ;
if the parts of the fluid are in any one
place denser or larger than in the others, the fluidity will be less in that
[lace, because there are fewer superficies where the parts can be separated
fclC IX.] Or NATURAL PHILOSOPHY. 3?<
from each other. In these cases I suppose the defect of the fluidity to be
supplied by the smoothness or softness of the parts, or some other condi
tion ; otherwise the matter where it is less fluid will cohere more, and be
more sluggish, and therefore will receive the motion more slowly, and pro
pagate it farther than agrees with the ratio above assigned. If the vessel
be riot spherical, the particles will move in lines not circular, but answer
ing to the figure of the vessel ; and the periodic times will be nearly as the
squares of the mean distances from the centre. In the parts between the
centre and the circumference the motions will be slower where the spaces
are wide, and swifter where narrow ; but yet the particles will not tend to the
circumference at all the more for their greater swiftness ;
for they then
describe arcs of less curvity, and the conatus of receding from the centre is
as much diminished by the diminution of this curvature as it is augment
ed by the increase of the velocity. As they go out of narrow into wide
spaces, they recede a little farther from the centre, but in doing so are re
tarded ; and when they come out of wide into narrow spaces, they are again
accelerated ; and so each particle is retarded and accelerated by turns for
ever. These things will come to pass in a rigid vessel ; for the state of
vortices in an infinite fluid is known by Cor. 6 of this Proposition.
I have endeavoured in this Proposition to investigate the properties of
vortices, that I might find whether the celestial phenomena can be explain
ed by them; for the phenomenon is this, that the periodic times of the
planets revolving about Jupiter are in the sesquiplicate ratio of their dis
tances from Jupiter s centre ; and the same rule obtains also among the
planets that revolve about the sun. And these rules obtain also with the
greatest accuracy, as far as has been yet discovered by astronomical obsertion.
Therefore if those planets are carried round in vortices revolving
about Jupiter and the sun, the vortices must revolve according to that
law. But here we found the periodic times of the parts of the vortex to
be in the duplicate ratio of the distances from the centre of motion ; and
this ratio cannot be diminished and reduced to the sesquiplicate, unless
either the matter of the vortex be more fluid the farther it is from the cen
tre, or the resistance arising from the want of lubricity in the parts of the
fluid should, as the velocity with which the parts of the fluid are separated
goes on increasing, be augmented with it in a greater ratio than that in
which the velocity increases. But neither of these suppositions seem rea
sonable. The more gross and less fluid parts will tend to the circumfer
ence, unless they are heavy towards the centre. And though, for the sake
of demonstration, I proposed, at the beginning of this Section, an Hypoth
esis that the resistance is proportional to the velocity, nevertheless, it is in
truth probable that the resistance is in a less ratio than that of the velo
city ; which granted, the periodic times of the parts of the vortex will be
in a greater than the duplicate ratio of the distances from its centre. If,
as some think, the vortices move more swiftly near the centre, then slower
378 THE MATHEMATICAL PRINCIPLES [BOOK IT
to a certain limit, then again swifter near the circumference, certainly
neither the sesquiplicate, nor any other certain and determinate ratio, can
obtain in them. Let philosophers then see how that phenomenon of the
sesquiplicate ratio can be accounted for by vortices.
PROPOSITION LIII. THEOREM XLI.
Bodies carried about in a vortex, and returning- in the same orb, are of
the same density with the vortex, and are moved according to the
same law with the parts of the vortex, as to velocity and direction oj
motion.
For if any small part of the vortex, whose particles or physical points
preserve a given situation among each other, be supposed to be congealed,
this particle will move according to the same law as before, since no change
is made either in its density, vis insita, or figure. And again ;
if a congealed
or solid part of the vortex be of the same density with the rest of the vortex,
and be resolved into a fluid, this will move according to the same law as
before, except in so far as its particles, now become fluid, may be moved
among themselves. Neglect, therefore, the motion of the particles among
themselves as not at all concerning the progressive motion of the whole, and
the motion of the whole will be the same as before. But this motion will be
the same with the motion of other parts of the vortex at equal distances
from the centre; because the solid, now resolved into a fluid, is become
perfectly like to the other parts of the vortex. Therefore a solid, if it be
of the same density with the matter of the vortex, will move with the same
motion as the parts thereof, being relatively at rest in the matter that sur
rounds it. If it be more dense, it will endeavour more than before to re
cede from the centre ; and therefore overcoming that force of the vortex,
by which, being, as it were, kept, in equilibrio, it was retained in its orbit,
it will recede from the centre, and in its revolution describe a spiral, re
turning no longer into the same orbit. And, by the same argument, if it
be more rare, it will approach to the centre. Therefore it can never con
tinually go round in the same orbit, unless it be of the same density with
the fluid. But we have shewn in that case that it would revolve accord
ing to the same law with those parts of the fluid that are at the same or
equal distances from the centre of the vortex.
COR. 1. Therefore a solid revolving in a vortex, and continually going
round in the same orbit, is relatively quiescent in the fluid that carries it.
COR. 2. And if the vortex be of an uniform density, the same body may
revolve at any distance from the centre of the vortex.
SCHOLIUM.
Hence it is manifest that the planets are not carried round in corporeal
vortices ; for, according to the Copernican hypothesis, the planets going
SEC. IX.] OF NATURAL PHILOSOPHY. 379
round the sun revolve in ellipses, having the sun in their common focus ;
and by radii drawn to the sun describe
areas proportional to the times. But
now the parts of a vortex can never re
volve with such a motion. Let AD,
BE, CF, represent three orbits describ
ed about the sun S, of which let the
utmost circle CF be concentric to the
sun ; and let the aphelia of the two in
nermost be A, B j and their perihelia
D, E. Therefore a body revolving in
the orb CF, describing, by a radius
drawn to the sun, areas proportional to
the times, will move with an uniform motion. And, according to the laws
of astronomy, the body revolving in the orb BE will move slower in its
aphelion B, and swifter in its perihelion E ; whereas, according to the
laws of mechanics, the matter of the vortex ought to move more swiftly in
the narrow space between A and C than in the wide space between D and
F ; that is, more swiftly in the aphelion than in the perihelion. Now these
two conclusions contradict each other. So at the beginning of the sign of
Virgo, where the aphelion of Mars is at present, the distance between the*
orbits of Mars and Venus is to the distance between the same orbits, at the
beginning of the sign of Pisces, as about 3 to 2 ; and therefore the matter
of the vortex between those orbits ought to be swifter at the beginning of
Pisces than at the beginning of Virgo in the ratio of 3 to 2 ; for the nar
rower the space is through which the same quantity of matter passes in the
same time of one revolution, the greater will be the velocity with which it
passes through it. Therefore if the earth being relatively at rest in this
celestial matter should be carried round by it, and revolve together with it
about the sun, the velocity of the earth at the beginning of Pisces
would be to its velocity at the beginning of Virgo in a sesquialteral ratio.
Therefore the sun s apparent diurnal motion at the beginning of Virgo
ought to be above 70 minutes, and at the beginning of Pisces less than 48
minutes; whereas, on the contrary, that apparent motion of the sun is
really greater at the beginning of Pisces than at the beginning of Virgo;
as experience testifies
; and therefore the earth is swifter at the beginning
of Virgo than at the beginning of Pisces ; so that the hypothesis of vor
tices is utterly irreconcileable with astronomical phenomena, and rather
serves to perplex than explain the heavenly motions. How these mo
tions are performed in free spaces without vortices, may be understood
by the first Book j and I shall now more fully treat of it in the following
Book.
BOOK III
BOOK III.
IN the preceding Books I have laid down the principles of philosophy ,
principles not philosophical, but mathematical : such, to wit, as we may
build our reasonings upon in philosophical inquiries. These principles are
the laws and conditions of certain motions, and powers or forces, which
chiefly have respect to philosophy : but, lest they should have appeared of
themselves dry and barren, I have illustrated them here and there with
some philosophical scholiums, giving an account of such things as are of
more general nature, and which philosophy seems chiefly to be founded on ;
such as the density and the resistance of bodies, spaces void of all bodies,
and the motion of light and sounds. It remains that, from the same prin
ciples, I now demonstrate the frame of the System of the World. Upon
this subject I had, indeed, composed the third Book in a popular method,
that it might be read by many ; but afterward, considering that such as
had not sufficiently entered into the principles could not easily discern the
strength of the consequences, nor lay aside the prejudices to which they had
been many years accustomed, therefore, to prevent the disputes which might
be raised upon such accounts, I chose to reduce the substance of this Book
into the form of Propositions (in the mathematical way), which should be
read by those only who had first made themselves masters of the principles
established in the preceding Books : not that I would advise any one to the
previous study of every Proposition of those Books ; for they abound with
such as might cost too much time, even to readers of good mathematical
learning. It is enough if one carefully reads the Definitions, the Laws of
Motion, and the first three Sections of the first Book. He may then pass
on to this Book, and consult such of the remaining Propositions of the
first two Books, as the references in this, and his occasions, shall require.
384 THE MATHEMATICAL PRINCIPLES [BOOK III.
RULES OF REASONING IN PHILOSOPHY,
RULE I.
We are Io admit no more causes of natural things than such as are both
true and sufficient to explain their appearances.
To this purpose the philosophers say that Nature does nothing in vain,
and more is in vain when less will serve ;
for Nature is pleased with sim
plicity, and affects not the pomp of superfluous causes.
RULE II.
Therefore to the same natural effects we must, as far as possible, assign
the same causes.
As to respiration in a man and in a beast; the descent of stones in Europe
and in America ; the light of our culinary fire and of the sun ; the reflec
tion of light in the earth, and in the planets.
RULE III.
The qualities of bodies, which admit neither intension nor remission oj
degrees, and which are found to belong to all bodies within the reach
of our experiments, are to be esteemed the universal qualities of all
bodies whatsoever.
For since the qualities of bodies are only known to us by experiments, we
are to hold for universal all such as universally agree with experiments ;
nnd such as are not liable to diminution can never be quite taken away.
We are certainly not to relinquish the evidence of experiments for the sake
of dreams and vain fictions of our own devising ;
nor are we to recede from
the analogy of Nature, which uses to be simple, and always consonant to
itself. We no other way know the extension of bodies than by our senses,
nor do these reach it in all bodies; but because we perceive extension in
all that are sensible, therefore we ascribe it universally to all others also.
That abundance of bodies are hard, we learn by experience ; and because
the hardness of the whole arises from the hardness of the parts, we therefore
justly infer the hardness of the undivided particles not only of the bodies
we feel but of all others. That all bodies are impenetrable, we gather not
from reason, but from sensation. The bodies which we handle we find im
penetrable, and thence conclude impenetrability to be an universal property
