the fluid always contrary to the motion of the pendulum in its return : and
the resistance arising from this motion, as also the resistance of the thread
by which the pendulum is suspended, makes the whole resistance of a pen
dulum greater than the resistance deduced from the experiments of falling
bodies. For by the experiments of pendulums described in that Scholium,
a globe of the same density as water in describing the length of its semidiameter
in air would lose the -3^-0 part of its motion. But by the
theory delivered in this seventh Section, and confirmed by experiments of
falling bodies, the same globe in describing the same length would lose only
a part of its motion equal to j-Vir? supposing the density of water to be
to the density of air as 8 r>0 to 1. Therefore the resistances were found
greater by the experiments of pendulums (for the reasons just mentioned)
than by the experiments of falling globes ; and that in the ratio of about
4 to 3. Bat yet since the resistances of pendulums oscillating in air, wa
ter, and quicksilver, are alike increased by like causes, the proportion of
the resistances in these mediums will be rightly enough exhibited by th
SEC. VII.J OF NATUKAL PHILOSOPHY. 355
experiments of pendulums, as well as by the experiments of falling bodies.
And from all this it may be concluded, that the resistances of bodies, moving
in any fluids whatsoever, though of the most extreme fluidity, are, cceteris
paribus, as the densities of the fluids.
These things being thus established, we may now determine what part
of its motion any globe projected in any fluid whatsoever would nearly lose
in a given time. Let D be the diameter of the globe, and V its velocity
at the beginning of its motion, and T the time in which a globe with the
velocity V can describe in vacua a space that is, to the space |D as the
density of the globe to the density of the fluid ; and the globe projected
*V
in that fluid will, in any other time t lose the part , the part
1 -p t
TV
r remaining ; and will describe a space, which will be to that de
scribed in the same time in, vacua with the uniform velocity V, as the
T + t
logarithm of the number ~ multiplied by the number 2,302585093 is
to the number
7^, by Cor. 7, Prop. XXXV. In slow motions the resist
ance may be a little less, because the figure of a globe is more adapted to
motion than the figure of a cylinder described with the same diameter. In
swift motions the resistance may be a little greater, because the elasticity
and compression of the fluid do not increase in the duplicate ratio of the
velocity. But these little niceties I take no notice of.
And though air. water, quicksilver, and the like fluids, by the division
of their parts in infinitum, should be subtilized, and become mediums in
finitely fluid, nevertheless, the resistance they would make to projected
globes would be the same. For the resistance considered in the preceding
Propositions arises from the inactivity of the matter; and the inactivity
of matter is essential to bodies, and always proportional to the quantity
of matter. By the division of the parts of the fluid the resistance arising
from the tenacity and friction of the parts may be indeed diminished : but
the quantity of matter will not be at all diminished by this division
; and
if the quantity of matter be the same, its force of inactivity will be the
same ; and therefore the resistance here spoken of will be the sanue, as being
always proportional to that force. To diminish this resistance, the quan
tity of matter in the spaces through which the bodies move must be dimin
ished ; and therefore the celestial spaces, through which the globes of the
planets and comets are perpetually passing towards all parts, with the
utmost freedom, and without the least sensible diminution of their motion,
must be utterly void of any corporeal fluid, excepting, perhaps, some ex
tremely rare vapours and the rays of light.
356 THE MATHEMATICAL PRINCIPLES [BoOK 11.
Projectiles excite a motion in fluids as they pass through them, and this
motion arises from the excess of the pressure of the fluid at the fore parts
of the projectile above the pressure of the same at the hinder parts : and
cannot be less in mediums infinitely fluid than it is in air, water, and quick
silver, in proportion to the density of matter in each. Now this excess of
pressure does, in proportion to its quantity, not only excite a motion in the
fluid, but also acts upon the projectile so as to retard its motion ; and there
fore the resistance in every fluid is as the motion excited by the projectile
in the fluid ; and cannot be less in the most subtile aether in proportion to
the density of that aether, than it is in air, water, and Quicksilver, in pro
portion to the densities of those fluids.
SECTION VIII.
Of motion propagated through fluids.
PROPOSITION XLI. THEOREM XXXII.
A pressure is not propagated through a fluid in rectilinear directions
unless ichere the particles of the fluid lie in a right line.
If the particles a, b
} c, d, e, lie in a right line, the pres
sure may be indeed directly propagated from a to e ; but
then the particle e will urge the obliquely posited parti
te) cles / and g obliquely, and those particles / and g will
not sustain this pressure, unless they be supported by the
particles h and k lying beyond them ; but the particles
that support them are also pressed by them ; and those particles cannot
sustain that pressure, without being supported by, and pressing upon, those
particles that lie still farther, as / and m, and so on in itiflnitum. There
fore the pressure, as soon as it is propagated to particles that lie out of
right lines, begins to deflect towards one hand and the other, and will be
propagated obliquely in infinitum ; and after it has begun to be propagat
ed obliquely, if it reaches more distant particles lying out of the right
line, it will deflect again on each hand and this it will do as often as it
lights on particles that do not lie exactly in a right line. Q.E.D.
COR. If any part of a pressure, propagated through a fluid from a given
point, be intercepted by any obstacle, the remaining part, which is not in
tercepted, will deflect into the spaces behind the obstacle. This may be
demonstrated also after the following manner. Let a pressure be propagat
ed from the point A towards any part, and, if it be possible, in rectilinear
SEC, Vlil.l OF NATURAL PHILOSOPHY. 57
directions ; and the obstacle
NBCK being perforated in BC,
let all the pressure be intercepted
but the coniform part A PQ, pass
ing through the circular hole BC.
Let the cone APQ, be divided
into frustums by the transverse
plants, de, fg, Id. Then while
the cone ABO, propagating the
pressure, urges the conic frustum.
degf beyond it on the superficies
de, and this frustum urges the next frustum fgih on the superficies/g", and
that frustum urges a third frustum, and so in infinitum ; it is manifest
(by the third Law) that the first frustum defg is, by the re-action of the
second frustum fghi, as much urged and pressed on the superficies fg, as
it urges and presses that second frustum. Therefore the frustum degf is
compressed on both sides, that is, between the cone Ade and the frustum
fhig; and therefore (by Case 6, Prop. XtX) cannot preserve its figure,
unless it be compressed with the same force on all sides. Therefore with
the same force with which it is pressed on the superficies de,fg, it will
endeavour to break forth at the sides df, eg ; and there (being not in the
least tenacious or hard, but perfectly fluid) it will run out, expanding it
self,- unless there be an ambient fluid opposing that endeavour. Therefore,
by the effort it makes to run out, it will press the ambient fluid, at its sides
df, eg, with the same force that it does the frustum fylti ; and therefore,
the pressure will be propagated as much from the sides df, e~, into the
spaces NO, KL this way and that way, as it is propagated from the srptrficies/
g- towards PQ,. QJE.D.
PROPOSITION XLII. THEOREM XXXIII.
All motion propagated through a fluid diverges from a rectilinear pro*
gress into ///. unmoved spaces.
CASE 1. Let a motion be
propagated from the point A
through the hole BC, and, if it
be possible, let it proceed in the
conic space BCQP according to
right lines diverging from the
point A. And let us first sup
pose this motion to be that of
waves in the surface of standing
water ; and let de,fg, hi, kl, &c.,
be the tops of the several waves,
divided from each other by as
any intermediate valleys or hollows. Then, because the water in tht
358 THE MATHEMATICAL PRINCIPLES [BOOK I.*
ridges of the waves is higher than in the unmoved parts of the fluid KL,
NO, it will run down from off the tops of those ridges, e, g, i, I, &c., dy fj
hj k, &c., this way and that way towards KL and NO ; and because the
water is more depressed in the hollows of the waves than in the unmoved
parts of the fluid KL, NO, it will run down into those hollows out of those
unmoved parts. By the first deflux the ridges of the waves will dilate
themselves this way and that way, and be propagated towards KL and NO.
And because the motion of the waves from A towards PQ is carried on by
a continual deflux from the ridges of the waves into the hollows next to
them, and therefore cannot be swifter than in proportion to the celerity of
the descent ; and the descent of the water on each side towards KL and NO
must be performed with the same velocity ; it follows that the dilatation
of the waves on each side towards KL and NO will be propagated with the
same velocity ;is the waves themselves go forward with directly from A to
PQ,. And therefore the whole space this way and that way towards KL
and NO will be filled by the dilated waves rfgr, shis, tklt, v/nnv, &c.
Q.E.I). That these things are so, any one may find by making the exper
iment in still water.
CASE 2. Let us suppose that de, fg, hi, kl, mn, represent pulses suc
cessively propagated from the point A through an elastic medium. Con
ceive the pulses to be propagated by successive condensations and rarefactions
of the medium, so that the densest part of every pulse may occupy a
spherical superficies described about the centre A, and that equal intervals
intervene between the successive pulses. Let the lines de, fg. hi, Id, &c..
represent the densest parts of the pulses, propagated through the hole BC ;
and because the medium is denser there than in the spaces on either side
towards KL and NO. it will dilate itself as well towards those spaces KL,
NO, on each hand, as towards the rare intervals between the pulses ; and
thence the medium, becoming always more rare next the intervals, and
more dense next the pulses, will partake of their motion. And because the
progressive motion of the pulses arises from the perpetual relaxation of the
den?er parts towards the antecedentrnre intervals; and since the pulses will
relax themselves on each hand towards the quiescent parts of the medium
KL, NO, with very near the same celerity ; therefore the pulses will dilate
themselves on all sides into the unmoved parts KL, NO, with almost the
same celerity with which they are propagated directly from the centre A;
and therefore will fill up the whole space KLON. Q.E.D. And we find
the same by experience also in sounds which are heard through a mountain
interposed ; and,*if they come into a chamber through the window, dilate
themselves into all the parts of the room, and are heard in every corner;
and not as reflected from the opposite walls, but directly propagated from
the window, as far as our sense can judge.
CASE 3 Let us suppose, lastly, that a motion of any kind is propagated
:C. VIII.j OF NATURAL PHILOSOPHY. 369
from A through the hole BC. Then since the cause of this propagation is
that the parts of the medium that are near the centre A disturb and agitate
those which lie farther from it; and since the parts which are urged are
fluid, and therefore recede every way towards those spaces where they are
less pressed, they will by consequence recede towards all the parts of tht
quiescent medium; as well to the parts on each hand, as KL and NO,
as to those right before, as PQ, ; and by this means all the motion, as soon
as it has passed through the hole BC, will begin to dilate itself, and from
thence, as from its principle and centre, will be propagated directly every
way. Q.E.D.
PROPOSITION XLIII. THEOREM XXXIV.
Every tremulous body in an elastic medium propagates the motion of
the pulses on every side right forward ; but in a non-elastic medium
excites a circular motion.
CASE. 1. The parts of the tremulous body, alternately going and return
ing, do in going urge and drive before them those parts of the medium that
lie nearest, and by that impulse compress and condense Nthem ; and in re
turning suffer those compressed parts to recede again, and expand them
selves. Therefore the parts of the medium that lie nearest to the tremulous
body move to and fro by turns, in like manner as the parts of the tremulous
body itself do ; and for the same cause that the parts of this body agitate
these parts of the medium, these parts, being agitated by like tremors, will
in their turn agitate others next to themselves ; and these others, agitated
in like manner, will agitate those that lie beyond them, and so on in, infinitum.
And in the same manner as the lirst parts of the medium were
condensed in going, and relaxed in returning, so will the other parts be
condensed every time they go, and expand themselves every time they re
turn. And therefore they will not be all going and all returning at the
same instant (for in that case they would always preserve determined dis
tances from each other, and there could be no alternate condensation and
rarefaction) ; but since, in the places where they are condensed, they ap
proach to, and, in the places where they are rarefied, recede from each other,
therefore some of them will be going while others are returning ; and so on
in infinitum. The parts so going, and in their going condensed, are pulses,
by reason of the progressive motion with which they strike obstacles in
their way; and therefore the successive pulses produced by a tremulous
body will be propagated in rectilinear directions
; and that at nearly equal
distances from each other, because of the equal intervals of time in which
the body, by its several tremors produces the several pulses. And though
the parts of the tremulous body go and return .n some certain and deter
minate direction, yet the pulses propagated from thence through the medium
will dilate themselves towards the sides, by the foregoing Proposition : anc7
360 THE MATHEMATICAL PRINCIPLES [BoOK 11
will be propagated on all sides from that tremulous body, as from a com
mon centre, in superficies nearly spherical and concentrical. An example
of this we have in waves excited by shaking a finger in water, which
proceed not only forward and backward agreeably to the motion of the
finger, but spread themselves in the manner of concentrical circles all round
the finger, and are propagated on every side. For the gravity of the water
supplies the place of elastic force.
Case 2. If the medium be not elastic, then, because its parts cannot be
condensed by the pressure arising from the vibrating parts of the tremulous
body, the motion will be propagated in an instant towards the parts where
the medium yields most easily, that is; to the parts which the tremulous
body would otherwise leave vacuous behind it. The case is the same with
that of a body projected in any medium whatever. A medium yielding
to projectiles does not recede in infinitum, but with a circular motion comes
round to the spaces which the body leaves behind it. Therefore as often
as a tremulous body tends to any part, the medium yielding to it comes
round in a circle to the parts which the body leaves ; and as often as the
body returns to the first place, the medium will be driven from the place it
came round to, and return to its original place. And though the tremulous
bod} be not firm and hard, but every way flexible, yet if it continue of a
given magnitude, since it cannot impel the medium by its treniors any
where without yielding to it somewhere else, the medium receding from the
parts of the body where it is pressed will always come round in a circle to
the parts that yield to it. Q.E.D.
COR. It is a mistake, therefore, to think, as some have done, that the
agitation of the parts of flame conduces to the propagation of a pressure in
rectilinear directions through an ambient medium. A pressure of that
kind must be derived not from the agitation only of the parts of flame, but
from the dilatation of the whole.
PROPOSITION XL1V. THEOREM XXXV.
If water ascend a/id descend alternately in the erected legs KL, MN, of
a canal or pipe ; and a pendulum be constructed whose length between
the point of suspension and the centre of oscillation is equal to half
the length of the ivater in the canal ; I say, that the water will ascend
and descend in the same times in which the pendulum oscillates.
I measure the length of the water along the axes of the canal and its legs,
and make it equal to the sum of those axes; and take no notice of the
resistance of the water arising from its attrition by the sides of the canal.
Let, therefore, AB, CD, represent the mean height of the water in both
legs ; and when the water in the leg KL ascends to the height EF, the
water will descend in the leg MN to the height GH. Let P be a pendulou/
SEC. Vlll.J OF NATURAL PHILOSOPHY. VJ61
body, VP the thread, V the point of suspension, RPQS the cycloid whicL
ii
L N
the pendulum describes, P its lowest point, PQ an arc equal to the neight
AE. The force with which the motion of the water is accelerated and re
tarded alternately is the excess of the weight of the water in one leg above
the weight in the other; and, therefore, when the water in the leg KL
ascends to EF, and in the other leg descends to GH, that force is double
the weight of the water EABF, and therefore is to the weight of the whole
water as AE or PQ, to VP or PR. The force also with which the body P
is accelerated or retarded in any place, as Q, of a cycloid, is (by Cor. Prop.
LI) to its whole weight as its distance PQ, from the lowest place P to the
length PR of the cycloid. Therefore the motive forces of the water and
pendulum, describing the equal spaces AE, PQ, are as the weights to be
moved ; and therefore if the water and pendulum are quiescent at first,
those forces will move them in equal times, and will cause them to go and
return together with a reciprocal motion. Q.E.D.
COR. 1. Therefore the reciprocations of the water in ascending and de
scending are all performed in equal times, whether the motion be more or
less intense or remiss.
COR. 2. If the length of the whole water in the canal be of 6J feet oi
French measure, the water will descend in one second of time, and will ascond
in another second, and so on by turns in infinitum ; for a pendulum
of Sy-j such feet in length will oscillate in one second of time.
COR. 3. But if the length of the water be increased or diminished, the
time of the reciprocation will be increased or diminished in the subduplicate
ratio of the length.
PROPOSITION XLY. THEOREM XXXVI.
The velocity of waves is in the subduplicate ratio of the breadths.
This follows from the construction of the following Proposition.
PROPOSITION XLVI. PROBLEM X.
Tofind the velocity of waves.
Let a pendulum be constructed, whose length between the point of sus
pension and the centre of oscillation is equal to the breadth of the waves
362 THE MATHEMATICAL PRINCIPLES (BOOK 1L
and in the time that the pendulum will perform one single oscillation the
waves will advance forward nearly a space equal to their breadth.
That which I call the breadth of the waves is the transverse measure
lying between the deepest
part of the hollows, or the
tops of the ridges. Let
ABCDEF represent the surface of stagnant water ascending and descend
ing in successive waves ; and let A, C, E, &c., be the tops of the waves ;
and let B, D, F, &c., be the intermediate hollows. Because the motion of
the waves is carried on by the successive ascent and descent of the water,
so that the parts thereof, as A, C, E, &c., which are highest at one time
become lowest immediately after ; and because the motive force, by which
the highest parts descend and the lowest ascend, is the weight of the eleva
ted water, that alternate ascent and descent will be analogous to the recip
rocal motion of the water in the canal, and observe the same laws as to the
times of its ascent and descent; and therefore (by Prop. XLIV) if the
distances between the highest places of the waves A, C, E, and the lowest
B, D, F, be equal to twice the length of any pendulum, the highest parts
A, C, E, will become the lowest in the time of one oscillation, and in the
time of another oscillation will ascend again. Therefore between the pas
sage of each wave, the time of two oscillations will intervene ; that is, the
wave will describe its breadth in the time that pendulum will oscillate
twice; but a pendulum of four times that length, and which therefore is
equal to the breadth of the waves, will just oscillate once in that time.
Q.E.L
COR. 1. Therefore waves, whose breadth is equal to 3 7\ French feet,
will advance through a space equal to their breadth in one second of time;
and therefore in one minute will go over a space of 1S3J feet
; and in an
hour a space of 11000 feet, nearly.
COR. 2. And the velocity of greater or less waves will be augmented or
diminished in the subduplicatc ratio of their breadth.
These things are true upon the supposition that the parts of water as
cend or descend in a right line; but, in truth, that ascent and descent is
rather performed in a circle ; and therefore I propose the time denned by
this Proposition as only near the truth.
PROPOSITION XLVIL THEOREM XXXVII.
Ifpulses are propagated through a fluid, the .ve eral particles of the
Jluid, goittff and returning with the shortest reciprocal motion, are al
ways accelerated or retarded according to the law of the oscillating
pendulum.
Let AB, BC, CD, &c., represent equal distances of successive pulses,
ABC the line of direction of the motion of the successive pulses propagated
SEC. VIII.] OF NATURAL PHILOSOPHY.
from A to B ; E, F, G three physical points of the quiescent medium sit
uate in the right line AC at equal distances from each other ; Ee, F/, G^,
equal spaces of extreme shortness, through which those
points go and return with a reciprocal motion in each vi
bration ; e, </>, y, any intermediate places of the same points ;
EF, FG physical lineolae, or linear parts of the medium
lying betAveen those points, and successively transferred into
the places t0, 0y, and ef, fg. Let there be drawn the
right line PS equal to the right line Ee. Bisect the same
in O, and from the centre O, with the interval OP, describe
the circle SIPi. Let the whole time of one vibration
; with
its proportional parts, be expounded by the whole circumlerence
of this circle and its parts, in such sort, that, when
any time PH or PHS/i is completed, if there be let fall to
PS the perpendicular HL or hi, and there
be taken E equal to PL or PI, the physi
cal point E may be found in e. A point,
as E, moving acccording to this law with
a reciprocal motion, in its going from E
through e to e, and returning again through
e to E, will perform its several vibrations with the same de
grees of acceleration and retardation with those of an oscil
lating pendulum. We are now to prove that the several
physical points of the medium will be agitated with such a
kind of motion. Let us suppose, then, that a medium hath
such a motion excited in it from any cause whatsoever, and
consider what will follow from thence.
In the circumference PHSA let there be taken the equal
arcs, HI, IK, or hi, ik, having the same ratio to the whole
circumference as the equal right lines EF, FG have to BC,
the whole interval of the pulses. Let fall the perpendicu
lars IM, KN, or wi, kn ; then because the points E, F, G are
successively agitated with like motions, and perform their en tire vibrations
composed of their going and return, while the pulse is transferred from B
to C ;
if PH or PHS/t be the time elapsed since the beginning of the mo
