dium in each of the places thereof, by which a body may describe a
given spiral.
From the centripetal force the velocity in each place must be found ;
then from the retardation of the velocity the density of the medium is
found, as in the foregoing Proposition.
But I have explained the method of managing these Problems in the
tenth Proposition and second Lemma of this Book; and will no longer
detain the reader in these perplexed disquisitions. I shall now add some
things relating to the forces of progressive bodies, and to the density and
resistance of those mediums in which the motions hitherto treated of, and
those akin to them, are performed.
SEC. V.] OF NATURAL PHILOSOPHY. 293
SECTION V.
Of the density and compression offluids ; and of hydrostatics.
THE DEFINITION OF A FLUID.
A fluid is any body whose parts yield to any force impressed on it,
by yielding, are easily moved among themselves.
PROPOSITION XIX. THEOREM XIv
All the parts of a homogeneous and unmovedfluid included in any un
moved vessel, and compressed on every side (setting aside the consider
ation of condensation, gravity, and all centripetal forces], will be
equally pressed on every side, and remain in their places without any
motion arising from that pressure.
CASE 1. Let a fluid be included in the spherical
vessel ABC, and uniformly compressed on every
side : 1 say, that no part of it will be moved by
that pressure. For if any part, as L), be moved,
all such parts at the same distance from the centre
on every side must necessarily be moved at the
same time by a like motion ; because the pressure
of them all is similar and equal ; and all other
motion is excluded that does not arise from that
pressure. But if these parts come all of them nearer to the centre, the
fluid must be condensed towards the centre, contrary to the supposition.
If they recede from it, the fluid must be condensed towards the circumfer
ence ; which is also contrary to the supposition. Neither can they move
in any one direction retaining their distance from the centre, because for
the same reason, they may move in a contrary direction : but the sami
part cannot be moved contrary ways at the same time. Therefore no
part of the fluid will be moved from its place. Q,.E.T).
CASE 2. I say now, that all the spherical parts of this fluid are equally
pressed on every side. For let EF be a spherical part of the fluid
;
if this
be not pressed equally on every side, augment the lesser pressure till it be
pressed equally on every side; and its parts (by Case I) will remain in
their places. But before the increase of the pressure, they would remain
in their places (by Case 1) ; and by the addition of a new pressure they
will be moved, by the definition of a fluid, from those places. Now these
two conclusions contradict each other. Therefore it was false to say that
the sphere EF was not pressed equally on every side. Q,.E.D.
CASE 3. I say besides, that different spherical parts have equal pressures.
For the contiguous spherical parts press each other mutually and equally
in the point of contact (by Law III). But (by Case 2) they are pressed on
every side with the same force. Therefore any two spherical parts lot
391 THE MATHEMATICAL PRINCIPLES [BoOK II.
contiguous, since an intermediate spherical part can touch both, will be
pressed with the same force. Q.E.D.
CASE 4. I say now, that all the parts of the fluid are every where press
ed equally. For any two parts may be touched by spherical parts in any
points whatever ; and there they will equally .press those spherical parts
(by Case 3). and are reciprocally equally pressed by them (by Law III).
Q.E.D.
CASE 5. Since, therefore, any part GHI of the fluid is inclosed by the
rest of the fluid as in a vessel, and is equally pressed on every side ; and
also its parts equally press one another, and are at rest among themselves ;
it is manifest that all the parts of any fluid as GHI, which is pressed
equally on every side, do press each other mutually and equally, and are at
rest among themselves. Q.E.D.
CASE 6. Therefore if that fluid be included in a vessel of a yielding
substance, or that is not rigid, and be not equally pressed on every side,
the same will give way to a stronger pressure, by the Definition of fluidity.
CASE 7. And therefore, in an inflexible or rigid vessel, a fluid will not
Sustain a stronger pressure on one side than on the other, but will give
way to it, and that in a moment of time ; because the rigid side of the
vessel does not follow the yielding liquor. But the fluid, by thus yielding,
will press against the opposite side, and so the pressure will tend on every
side to equality. And because the fluid, as soon as it endeavours to recede
from the part that is most pressed, is withstood by the resistance of the
vessel on the opposite side, the pressure will on every side be reduced to
equality, in a moment of time, without any local motion : and from thence
the parts of the fluid (by Case 5) will press each other mutually and equal
ly, and be at rest among themselves. Q..E.D.
COR. Whence neither will a motion of the parts of the fluid among
themselves be changed by a pressure communicated to the external super
ficies, except so far as either the figure of the superficies may be somewhere
altered, or that all the parts of the fluid, by pressing one another more in
tensely or remissly, may slide with more or less difficulty among them
selves.
PROPOSITION XX. THEOREM XV.
Jf all the parts of a sphericalfluid, homogeneous at equal distancesfrom
the centre, lying on. a spherical concentric bottom, gravitate towards
the centre of the whole, the bottom will sustain the weight of a cylin
der, whose base is equal to the superficies of the bottom, and whose al
titude is the same with that of the incumbent fluid.
Let OHM be the superficies of the bottom, and AEI the upper super
ficies of the fluid. Let the fluid be distinguished into concentric orbs of
3qual thickness, by the innumerable spherical superficies *3PK, CGL : and
SEC. V OF NATURAL PHILOSOPHY. 295
conceive the force of gravity to act only in the
upper superficies of every orb, and the actions
to be equal on the equal parts of all the su
perficies. Therefore the upper superficies AE
is pressed by the single force of its own grav
ity, by which all the parts of the upper orb,
and the second superficies BFK, will (by
Prop. XIX), according to its measure, be
equally pressed. The second superficies BFK
is pressed likewise by the force of its own
gravity, which, added to the former force,
makes the pressure double. The third superficies CGL is, according to its
measure, acted on by this pressure and the force of its own gravity besides,
which makes its pressure triple. And in like manner the fourth superfi
cies receives a quadruple pressure, the fifth superficies a quintuple, and so
on. Therefore the pressure acting on every superficies is not as the solid
quantity of the incumbent fluid, but as the number of the orbs reaching
to the upper surface of the fluid
; and is equal to the gravity of the lowest
orb multiplied by the number of orbs : that is, to the gravity of a solid
whose ultimate ratio to the cylinder above-mentioned (when the number of
the orbs is increased and their thickness diminished, ad infiititum, so that
the action of gravity from the lowest superficies to the uppermost may besome
continued) is the ratio of equality. Therefore the lowest superficies
sustains the weight of the cylinder above determined. Q..E.D. And by a
like reasoning the Proposition will be evident, where the gravity of the
fluid decreases in any assigned ratio of the distance from the centre, and
also where the fluid is more rare above and denser below. Q.E.D.
COR. 1. Therefore the bottom is not pressed by the whole weight of the
incumbent fluid, but only sustains that part of it which is described in the
Proposition ; the rest of the weight being sustained archwise by the spheri
cal figure of the fluid.
COR. 2. The quantity of the pressure is the same always at equal dis
tances from the centre, whether the superficies pressed be parallel to the
horizon, or perpendicular, or oblique ;
or whether the fluid, continued up
wards from the compressed superficies, rises perpendicularly in a rectilinear
direction, or creeps obliquely through crooked cavities and canals, whether
those passages be regular or irregular, wide or narrow. That the pressure
is not altered by any of these circumstances, may be collected by applying
the demonstration of this Theorem to the several cases of fluids.
COR. 3. From the same demonstration it may also be collected (by Prop.
XIX), that the parts of a heavy fluid acquire no motion among themselvei
by the pressure of the incumbent veight, except that motion which arises
from condensation.
296 THE MATHEMATICAL PRINCIPLES [BCOK II
Con. 4. And therefore if another body of the same specific gravity, in
capable of condensation, be immersed in this fluid, it will acquire no mo
tion by the pressure of the incumbent weight: it will neither descend nor .
ascend, nor change its figure. If it be spherical, it will remain so, notwith
standing the pressure ;
if it be square, it will remain square; and that,
whether it be soft or fluid : whether it swims freely in the fluid, or lies at
the bottom. For any internal part of a fluid is in the same state with the
submersed body ; and the case of all submersed bodies that have the same
magnitude, figure, and specific gravity, is alike. If a submersed body, re
taining its weight, should dissolve and put on the form of a fluid, this
body, if before it would have ascended, descended, or from any pressure as
sume a new figure, would now likewise ascend, descend, or put on a new
figure ; and that, because its gravity and the other causes of its motion
remain. But (by Case 5, Prop. XIX; it would now be at rest, and retain
its figure. Therefore also in the former case.
COR. 5. Therefore a body that is specifically heavier than a fluid con
tiguous to it will sink ; and that which is specifically lighter will ascend,
and attain so much motion and change of figure as that excess or defect of
gravity is able to produce. For that excess or defect is the same thing as an
impulse, by which a body, otherwise in equilibria with the parts of the
fluid, is acted on: and may be compared with the excess or defect of a
weight in one of the scales of a balance.
COR. 6. Therefore bodies placed in fluids have a twofold gravity the
one true and absolute, the other apparent, vulgar, and comparative. Ab
solute gravity is the whole force with which the body tends downwards ;
relative and vulgar gravity is the excess of gravity with which the body
tends downwards more than the ambient fluid. By the first kind of grav
ity the parts of all fluids and bodies gravitate in their proper places ; and
therefore their weights taken together compose the weight of the whole.
For the whole taken together is heavy, as may be experienced in vessels
full of liquor ; and the weight of the whole is equal to the weights of all
the parts, and is therefore composed of them. By the other kind of grav
ity bodies do not gravitate in their places ; that is, compared with one
another, they do not preponderate, but, hindering one another s endeavours
to descend, remain in their proper places, as if they were not heavy. Those
things which are in the air, and do not preponderate, are commonly looked
on as not heavy. Those which do preponderate are commonly reckoned
heavy, in as much as they are not sustained by the weight of the air. The
Common weights are nothing else but the excess of the true weights above
the weight of the air. Hence also, vulgarly, those things are called light
which are less heavy, and, by yielding to the preponderating air, mount
upwards. But these are only comparatively lig s &mA not truly so, because
hey descend in racuo. Thus, in water, bodies *>icfc. by their greater or
SEC. V.] OF NATURAL PHILOSOPHY. 297
less gravity, descend or ascend, are comparatively and apparently heavy or
light ; and their comparative and apparent gravity or levity is the excess
.or defect by which their true gravity either exceeds the gravity of the
water or is exceeded by it. But those things which neither by preponder
ating descend, nor, by yielding to the preponderating fluid, ascend, although
by their true weight they do increase the weight of the whole, yet com
paratively, and in the sense of the vulgar, they do not gravitate in the wa
ter. For these cases are alike demonstrated.
COR. 7. These things which have been demonstrated concerning gravity
take place in any other centripetal forces.
COR. 8. Therefore if the medium in which any body moves be acted on
either by its own gravity, or by any other centripetal force, and the body
be urged more powerfully by the same force ; the difference of the forces is
that very motive force, which, in the foregoing Propositions, I have con
sidered as a centripetal force. But if the body be more lightly urged by
that force, the difference of the forces becomes a centrifugal force, and is tc
be considered as such.
COR. 9. But since fluids by pressing the included bodies do not
change their external figures, it appears also (by Cor. Prop. XIX) that they
will not change the situation of their internal parts in relation to onf
another ; and therefore if animals were immersed therein, and that all sen
sation did arise from the motion of their parts, the fluid will neither hurt
the immersed bodies, nor excite any sensation, unless so far as those bodies
may be condensed by the compression. And the case is the same of any
system of bodies encompassed with a compressing fluid. All the parts of
the system will be agitated with the same motions as if they were placed
in a vacuum, and would only retain their comparative gravity ; unless so
far as the fluid may somewhat resist their motions, or be requisite to conglutinate
them by compression.
PROPOSITION XXI. THEOREM XVI.
<et the density of any fluid be proportional to the compression, and its
parts be attracted downwards by a centripetal force reciprocally pro
portional to the distances from the centre : I say, that, if those dis
tances be taken continually proportional, the densities of thefluid at
the same distances will be also continually proportional.
Let ATV denote the spherical bottom of the fluid, S the centre, S A, SB.
SC, SD, SE, SF, &c., distances continually proportional. Erect the per
pendiculars AH, BI, CK, DL, EM, PN, &c., which shall be as the densi
ties of the medium in the places A, B, C, D, E, F : and the specific grav
ATT RT f^K"
ities in those places will be aa -r-, ,
-
&c., or, which is all one, a&-
298 THE MATHEMATICAL PRINCIPLES [BOOK II.
AH BI CK
ATT BC CD Suppose, first, these gravities to be uniformly continued
from A to B, from B to C, from C to D, (fee., the decrements in the points
B, C, D, (fee., being taken by steps. Arid these gravi
ties drawn into the altitudes AB, BC, CD, (fee., will
give the pressures AH, BI, CK, (fee., by which the bot
tom ATV is acted on (by Theor. XV). Therefore the
particle A sustains all the pressures AH, BI, CK, DJL,
(fee., proceeding in infinitum ; and the particle B sus
tains the pressures of all but the first AH ; and the par
ticle C all but the two first AH, BI ; and so on : and
therefore the density AH of the first particle A is to
the density BI of the second particle B as the sum of
all AH -f- BI + CK + DL, in infinitum, to the sum of
all BI -f- CK -f DL, (fee. And BI the density of the second particle B is
to CK the density of the third C, as the sum of all BI -f CK + DL, (fee.,
to the sum of all CK -f- DL, (fee. Therefore these sums are proportional
to their differences AH, BI, CK, (fee., and therefore continually propor
tional (by Lem. 1 of this Book) ; and therefore the differences AH, BI,
CK, (fee., proportional to the sums, are also continually proportional.
Wherefore since the densities in the places A, B, C, (fee., are as AH, BI,
CK, (fee., they will also be continually proportional. Proceed intermissively,
and, ex ccquo, at the distances SA, SC, SE, continially proportional,
the densities AH, CK, EM will be continually proportional. And by the
same reasoning, at any distances SA, SD, SG, continually proportional,
the densities AH. DL, GO, will be continually proportional. Let now the
points A, B, C. D, E, (fee., coincide, so that the progression of the specif.c
gravities from the bottom A to the top of the fluid may be made continual ;
and at any distances SA, SD, SG, continually proportional, the densities
AH, DL, GO, being all along continually proportional, will still remain
continually proportional. Q.E.D.
COR. Hence if the density of the fluid in two places,
as A and E, be given, its density in any other place Q,
may be collected. With the centre S, and the rectan
gular asymptotes SQ, SX, describe an hyperbola cut
ting the perpendiculars AH, EM, QT in , e, and q}
as also the perpendiculars HX, MY, TZ, let fall upon
the asypmtote SX, in //, m, and t. Make the area
Y////Z to the given area YmAX as the given area
EeqQ to the given area EeaA ; and the line Z produced will cut off the
line Q,T. proportional to the density. For if the lines SA, SE, SQ are
continually proportional, the areas ReqQ., fyaA will be equal, and thence
X
SEC. V. OF NATURAL PHILOSOPHY. 299
the areas YwYZ. X/zwY, proportional to them, will be also equal ; and
the lines SX, SY, SZ, that is, AH, EM, Q,T continually proportional, as
they ought to be. And if the lines SA, SE, SQ,5 obtain any other order
in the series of continued proportionals, the lines AH, EM, Q,T, because
of the proportional hyperbolic areas, will obtain the same order in another
series of quantities continually proportional.
PROPOSITION XXII THEOREM XVII.
Let the density of any fluid be proportional to the compression, and its
parts be attracted downwards by a gravitation reciprocally propor
tional to the squares of the distancesfrom the centre : I say, that if
the distances be taken in harmonic progression, the densities of the
fluid at those distances will be in a geometrical progression.
Let S denote the centre, and SA,
SB, SC, SD, SE, the distances in
geometrical progression. Erect the
perpendiculars AH, BI, CK, (fee.,
which shall be as the densities of c
the fluid in the places A, B, C, D,
E, (fee., and the specific gravities
thereof in those places will be as
AH BI
,^-, (fee. Suppose these SA2 SB 2 SC 2
gravities to be uniformly continued, the first from A to B, the second from
B to C, the third from C to I), &c. And these drawn into the altitudes
AB, BC, CO, DE, (fec.
; or, which is the same thing/into the distances SA,
ATT r>T OT7"
SB, SC, (fee., proportional to those altitudes, will give -~-r-, ^=5, -~~, (fee..
the exponents of the pressures. Therefore since the densities are as th^
sums of those pressures, the differences AH BI, BI CK, (fee., of tb,
densities will be as the differences of those sums ~-r~, ^, ~~, (fee. With
the centre S, and the asymptotes SA, S#, describe any hyperbola, cutting
the perpendiculars AH, BI, CK, (fee., in a, 6, c, (fee., and the perpendicu
lars H/, I//,, K?#, let fall upon the asymptote Sv, in h, i, k ; and the dif
ferences of the densities tu, uw, (fee., will be as A , ^^, (fee. And the SA; SB;
rectangles tu X th, uw X uij (fee., or tp, uq, (fee., as
that is, as Aa, Bb, (fee.
AH X th BI X ui
,
(fee. SA SB
For, by the nature of the hyperbola, SA is to AH
or St as th to Ar, and therefore pri is equal to Aa. And, by a like SA
300 THE MATHEMATICAL PRINC. PLES [BOOK II.
reasoning, ^n~~ *s e(
lua^ to ^, &c- But Aa> B^> ^c, &cv are continu
ally proportional, and therefore proportional to their differences Aa B&,
B6 Cc; &c., therefore the rectangles fy?, nq, &c., are proportional to those
differences ; as also the sums of the rectangles tp + uq, or tp + uq -f w
to the sums of the differences Aa Cc or Aa Da7
. Suppose several of
these terms, and the sum of all the differences, as Aa F/, will be pro
portional t? the sum of all the rectangles, as zthn. Increase the number
of terms, and diminish the distances of the points A, B, C, (fee., in iiijinitum,
and those rectangles will become equal to the hyperbolic area zthn.
and therefore the difference Aa F/ 19 proportional to this area. Take
nowT
any distances, as SA, SD, SF, in harmonic progression, and the dif
ferences Aa Da7
, Da1 F/ will be equal ; and therefore the areas thlx,
xlnz, proportional to those differences will be equal among themselves, and
the densities St, S:r, Sz, that is, AH, DL, FN, continually proportional.
Q.E.D.
COR. Hence if any two densities of the fluid, as AH and BI, be given,
the area thiu, answering to their difference tu, will be given; and thence
the density FN will be found at any height SF, by taking the area thnz to
that given area thiu as the difference Aa F/ to the difference Aa Eh.
SCHOLIUM.
By a like reasoning it may be proved, that if the gravity of the particles
of a fluid be diminished in a triplicate ratio of the distances from the centre ;
and the reciprocals of the squares of the distances SA, SB, SC, &c., (namely,
SA 3 SA 3 SA 3
.
opt ^e ta^en m an arithmetical progression, the densities AH.
BI, CK, &c., will be in a geometrical progression. And if the gravity be
diminished in a quadruplicate ratio of the distances, and the reciprocals of
the cubes of the distances (as ^-r^, SRS sps ^c ^ ^e ta^cn ^ n ai> itnmeticai
progression, the densities AH, BI, CK, &c., will be in geometrical pro
gression. And so in irtfinitum. Again ;
if the gravity of the particles of
the fluid be the same at all distances, and the distances be in arithmetical
progression, the densities will be in a geometrical progression as Dr. Halley
has found. If the gravity be as the distance, and the squares of the
distances be in arithmetical progression, the densities will be in geometri
cal progression. And so in infinitum. These things will be so, when the
density of the fluid condensed by compression is as the force of compres
sion ; or, which is the same thing, when the space possessed by the fluid is
reciprocally as this force. Other laws of condensation may be supposed,
as that the cube of the compressing force may be as the biquadrate of the
SEC. V.] OF NATURAL PHILOSOPHY. 301
de isity ; or the triplicate ratio of tlie force the same with the quadruplicate
ratio of the density : in which case, if the gravity he reciprocally as the
square of the distance from the centre, the density will be reciprocally at
the cube of the distance. Suppose that the cube of the compressing force
be as the quadrato-cube of the density ; and if the gravity be reciprocally
as the square of the distance, the density will be reciprocally in a sesquiplicate
ratio of the distance. Suppose the compressing force to be in a du
plicate ratio of the density, and the gravity reciprocally in a duplicate ra
tio of the distance, and the density will be reciprocally as the distance.
To run over all the cases that might bo offered would be tedious. But as
to our own air, this is certain from experiment, that its density is either
accurately, or very nearly at least, as the compressing force ; and therefore
the density of the air in the atmosphere of the earth is as the weight of
the whole incumbent air, that is, as the height of the mercury in the ba
rometer.
PROPOSITION XXIII. THEOREM XVIII.
If a fluid be composed of particles mutually flying each other, and the
drnsity be as the compression, the centrifugal forces of the particles
will be reciprocally proportional to tlie distances of their centres. And,
vice versa, particlesflying each otli,er, with forces that are reciprocally
proportional to the distances of their centres^ compose an elastic fluid,
whose density is as the compression.
Let the fluid be supposed to be included in a cubic
space ACE, and then to be reduced by compression into
