tion of the point E, then will PI or PHSi be the time elapsed since the
beginning of the motion of the point F, and PK or PHSA; the time elapsed
since the beginning of the motion of the point G; and therefore Ee, F0,
Gy, will be respectively equal to PL, PM, PN, while the points are going,
and to PI, Ptn, Pn, when the points are returning. Therefore ey or EG
4- Gy Et will, when the points are going, be equal to EG LN
364 THE MATHEMATICAL PRINCIPLES [BOOK II.
and in their return equal to EG + In. But ey is the breadth or ex
pansion of the part EG of the medium in the place ey ; and therefore the
expansion of that part in its going is to its mean expansion as EG
LN to EG; and in its return, as EG -f In or EG + LN to EG.
Therefore since LN is to KH as IM to the radius OP, and KH to EG
as the circumference PHSAP to BC ; that is, if we put V for the
radius of a circle whose circumference is equal to BC the interval of the
pulses, as OP to V and, ex cequo, LN to EG as IM to V ; the expansion
of the part EG, or of the physical point F in the place ey, to the mean ex
pansion of the same part in its first place EG, will be as V IM to V
in going, and as V -f im to V in its return. Hence the elastic force of the
point P in the place ey to its mean elastic force in the place EG is as 11. 11, v fivf
* v m 1 ^s Somo>
an<^ as v i
^ v in lts re^urn. And by V J.1VJL V V -f Iffl V
the same reasoning the elastic forces of the physical points E and G in going
are as . qr
and ^ ==~ to T, ; and the difference of the forces to the
mean elastic force of the medium as T^ VV-V X HL-Vx KN + HL X KN
1 HL KN 1
to ~ ;
that is, as : to
^,
or as HL KN to V ; if we suppose
(by reason of the very short extent of the vibrations) HL and KN to be
indefinitely less than the quantity V. Therefore since the quantity V is
given, the difference of the forces is as HL KN ;
that is (because HL
KN is proportional to HK, and OM to OI or OP ; and because HK
and OP are given) as OM ; that is, if F/ be bisected in ft, as ft</>. And
for the same reason the difference of the elastic forces of the physical points
e and y, in the return of the physical lineola ey, is as ftr/>. But that dif
ference (that is, the excess of the elastic force of the point e above the
elastic force of the point y) is the very force by which the intervening phy
sical lineola ey of the medium is accelerated in going, and retarded in re
turning ; and therefore the accelerative force of the physical lineola ey is
as its distance from ft, the middle place of the vibration. Therefore (by
Prop. XXXVIII, Book 1) the time is rightly expounded by the arc PI ;
and the linear part of the medium sy is moved according to the law abovementioned,
that is, according to the law of a pendulum oscillating ; and
the case is the same of all the linear parts of which the whole medium is
compounded. Q,.E.D.
COR. Hence it appears that the number of the pulses propagated is the
same with the number of the vibrations of the tremulous body, and is not
multiplied in their progress. For the physical lineola ey as soon as it
returns to its first place is at rest
; neither will it move again, unless ii
SEC. V11I.J OF NATURAL PHILOSOPHY. 36
receives a new motion either from the impulse of the tremulous body, or
of the pulses propagated from that body. As soon, therefore, as the pulses
cease to be propagated from the tremulous body, it will return to a state
of rest, and move no more.
PROPOSITION XLVIII. THEOREM XXXVIII.
The velocities of pulses propagated in an elastic fluid are in a ratin
compounded of the subduplicate, ratio of the elastic force directly, and
the subduplicate ratio of the density inversely ; supposing the elastic
Jorce of thefluid to be proportional to its condensation
CASE I. If the mediums be homogeneous, and the distances of the pulses
in those mediums be equal amongst themselves, but the motion in one me
dium is more intense than in the other, the contractions and dilatations of
the correspondent parts will be as those motions ;
not that this proportion
is perfectly accurate. However, if the contractions and dilatations are not
exceedingly intense, the error will not be sensible ; and therefore this pro
portion may be considered as physically exact. Now the motive elastic
forces are as the contractions and dilatations ; and the velocities generated
in the same time in equal parts are as the forces. Therefore equal and
corresponding parts of corresponding pulses will go and return together,
through spaces proportional to their contractions and dilatations, with ve
locities that are as those spaces ; and therefore the pulses, which in the
time of one going and returning advance forward a space dq aal to their
breadth, and are always succeeding into the places of the pulses that im
mediately go before them, will, by reason of the equality of the distances,
go forward in both mediums with equal velocity.
CASE 2. If the distances of the pulses or their lengths are greater in one
medium than in another, let us suppose that the correspondent parts de
scribe spaces, in going and returning, each time proportional to the breadths
of the pulses ; then will their contractions and dilatations be equal : and
therefore if the mediums are homogeneous, the motive elastic forces, which
agitate them with a reciprocal motion, will be equal also. Now the matter
to be moved by these forces is as the breadth of the pulses ; and the space
through which they move every time they go and return is in the same
ratio. And, moreover, the time of one going and returning is in a ratic
compounded of the subduplicate ratio of the matter, and the o-uwuupncatc
ratio of the space ; and therefore is as the space. But the pulses advance
a space equal to their breadths in the times of going once and returning
once; that is, they go over spaces proportional to the times, and therefore
are equally swift.
CASE 3. And therefore in mediums of equal density and elastic force,
all the pulses are equally swift. Now if the density or the elastic force of
the medium were augmented, then, because the motive force is increased
366 THE MATHEMATICAL PRINCIPLES [BoOK 11
in the ratio of the elastic force, and the matter to be moved is increased in
the ratio of the density, the time which is necessary for producing the
same motion as before will be increased in the subduplicate ratio of the
density, and will be diminished in the subduplicate ratio of the elastic
force. And therefore the velocity of the pulses will be in a ratio com
pounded of the subduplicate ratio of the density of the medium inversely,
and the subduplicate ratio of the elastic force directly. Q,.E.D.
This Proposition will be made more clear from the construction of the
following Problem.
PROPOSITION XLIX. PROBLEM XL
The. density and elastic force of a medium being given, to find the, ve
locity of the pulses.
Suppose the medium to be pressed by an incumbent weight after the manner
of our air ; and let A be the height, of a homogeneous medium, whose
weight is equal to the incumbent weight, and whose density is the same
with the density of the compressed medium in which the pulses are propa
gated. Suppose a pendulum to be constructed whose length between the
point of suspension and the centre of oscillation is A : and in the time in
which that pendulum will perform one entire oscillation composed of
its going and returning, the pulse will be propagated right onwards
through a space equal to the circumference of a circle described with the
radius A.
For, letting those things stand which were constructed in Prop. X.LV11,
if any physical line, as EF, describing the space PS in each vibration, be
acted on in the extremities P and S of every going and return that it
makes by an elastic force that is equal to its weight, it will perform its
several vibrations in the time in which the same might oscillate in a cy
cloid whose whole perimeter is equal to the length PS ; and that because
equal forces will impel equal corpuscles through equal spaces in the same
or equal times. Therefore since the times of the oscillations are in the
subduplicate ratio of the lengths of the pendulums, and the length of the
pendulum is equal to half the arc of the whole cycloid, the time of one vi
bration would be to the time of the oscillation of a pendulum whose length
is A in the subduplicate ratio of the length ^PS or PO to the length A.
But the elastic force with which the physical lineola EG is urged, when it
Is found in its extreme places P, S, was (in the demonstration of Prop.
XLVII) to its whole elastic force as HL KN to V, that is (since the
point K now falls upon P), as HK to V: and all that force, or which is
the same thing, the incumbent weight by which the lineola EG is com
pressed, is to the weight of the lineola as the altitude A of the incumbent
weight to EG the length of the lineola ; and therefore, ex ctquo, the force
SEC. VII1.I OF NATURAL PHILOSOPHY. 367
with which the lincola EG is urged in the places P and S
is to the weight of that lineola as HK X A to V X EG ;
or
as PO X A to VV; because HK was to EG as PO to V.
Therefore since the times in which equal bodies are impelled
through equal spaces are reciprocally in the subduplicate
ratio of the forces, the time of one vibration, produced by
the action of that elastic force, will be to the time of a vi
bration, produced by. the impulse of the weight in a subdu
plicate ratio of VV to PO X A, and therefore to the time
of the oscillation of a pendulum whose length is A in the
subduplicate ratio of VV to PO X A, and the subdupli
cate ratio of PO to A conjunctly ; that is, in the entire ra
tio of V to A. But in the time of one
vibration composed of the going and re
turning of the pendulum, the pulse will
be propagated right onward through a
space equal to its breadth BC. There
fore the time in which a pulse runs over
the space BC is to the time of one oscillation composed of
the going and returning of the pendulum as V to A, that is,
as BC to the circumference of a circle whose radius is A.
But the time in which the pulse will run over the space BC
is to the time in which it will run over a length equal to
that circumference in the same ratio; and therefore in the
time of such an oscillation the pulse will run over a length
equal to that circumference. G,.E.D.
COR. 1. The velocity of the pulses is equal to that which
heavy bodies acquire by falling with an equally accele
rated motion, and in their fall describing half the alti
tude A. For the pulse will, in the time of this fall, sup
posing it to move with the velocity acquired by that fall, run over a
space that will be equal to the whole altitude A ; and therefore in the
time of one oscillation composed of one going and return, will go over a
space equal to the circumference of a circle described with the radius A ;
for the time of the fall is to the time of oscillation as the radius of a circle
to its circumference.
COR. 2. Therefore since that altitude A is as the elastic force of the
fluid directly, and the density of the same inversely, the velocity of the
pulses will be in a ratio compounded of the subduplicate ratio of the den
sity inversely, and the subduplicate ratio of the clastic force directly.
368 THE MATHEMATICAL PRINCIPLES |BoOK IL
PROPOSITION L. PROBLEM XII.
Tofind the distances of the pulses.
Let the number of the vibrations of the body, by whose tremor the pulses
are produced; be found to any given time. By that number divide the
space which a pulse can go over in the same time, and the part found will
be the breadth of one pulse. Q.E.I.
SCHOLIUM.
The last Propositions respect the motions of light and sounds ;
for since
light is propagated in right lines, it is certain that it cannot consist in ac
tion alone (by Prop. XLI and XLIl). As to sounds, since they arise from
tremulous bodies, they can be nothing else but pulses of the air propagated
through it (by Prop. XLIII) ; and this is confirmed by the tremors which
sounds, if they be loud and deep, excite in the bodies near them, as we ex
perience in the sound of drums ;
for quick and short tremors are less easily
excited. But it is well known that any sounds, falling upon strings in
unison with the sonorous bodies, excite tremors in those strings. This is
also confirmed from the velocity of sounds; for since the specific gravities
of rain-water and quicksilver are to one another as about 1 to 13f, and
when the mercury in the barometer is at the height of 30 inches of our
measure, the specific gravities of the air and of rain-water are to one
another as about 1 to 870, therefore the specific gravity of air and quick
silver are to each other as 1 to 11890. Therefore when the height of
the quicksilver is at 30 inches, a height of uniform air, whose weight would
be sufficient to compress our air to the density we find it to be of, must be
equal to 356700 inches, or 29725 feet of our measure ; and this is that
very height of the medium, which I have called A in the construction of
the foregoing Proposition. A circle whose radius is 29725 feet is 186768
feet in circumference. And since a pendulum 39} inches in length com
pletes one oscillation, composed of its going and return, in two seconds of
time, as is commonly known, it follows that a pendulum 29725 feet, or
356700 inches in length will perform a like oscillation in 190f seconds.
Therefore in that time a sound will go right onwards 186768 feet, and
therefore in one second 979 feet.
But in this computation we have made no allowance for the crassitude
of the solid particles of the air, by which the sound is propagated instan
taneously. Because the weight of air is to the weight of water as 1 tc
870, and because salts are almost twice as dense as water ;
if the particles
of air are supposed to be of near the same density as those of water or salt,
and the rarity of the air arises from the intervals of the particles ;
the
diameter of one particle of air will be to the interval between the centres
SEC. VIIL] OF NATURAL PHILOSOPHY. 369
of the particles as 1 to about 9 or 10, and to the interval between the par
ticles themselves as 1 to 8 or 9. Therefore to 979 feet, which, according to
the above calculation, a sound will advance forward in one second of time,
\ve may add ^- 9
-, or about 109 feet, io compensate for the cra-ssitude of the
particles of the air : and then a sound will go forward about 1088 feet in
one second of time.
Moreover, the vapours floating in the air being of another spring, and a
different tone, will hardly, if at all, partake of the motion of the true air
in which the sounds are propagated. Now if these vapours remain unmov
ed, that motion will be propagated the swifter through the true air alone,
and that in the subduplicate ratio of the defect of the matter. So if the
atmosphere consist of ten parts of true air and one part of vapours, the
motion of sounds will be swifter in the subduplicate ratio of 11 to 10, or
very nearly in the entire ratio of 21 to 20, than if it were propagated
through eleven parts of true air : and therefore the motion of sounds above
discovered must be increased in that ratio. By this means the sound will
pass through 1 142 feet in one second of time.
These things will be found true in spring and autumn, when the air is
rarefied by the gentle warmth of those seasons, and by that means its elas
tic force becomes somewhat more intense. But in winter, when the air is
condensed by the cold, and its elastic force is somewhat remitted, the mo
tion of sounds will be slower in a subduplicate ratio of the density ; and,
on the other hand, swifter in the summer.
Now by experiments it actually appears that sounds do really advance
in one second of time about 1142 feet of English measure, or 1070 feet of
French measure.
The velocity of sounds being known, the intervals of the pulses are known
also. For M. Sauveur, by some experiments that he made, found that an
open pipe about five Paris feet in length gives a sound of the same tone
with a viol-string that vibrates a hundred times in one second. Therefore
there are near 10J pulses in a space of 1070 Paris feet, which a sound runs
over in asecond of time ; and therefore one pulse fills up a space of about 1 T
7-
Paris feet, that is, about twice the length of the pipe. From whence it is
probable that the breadths of the pulses, in all sounds made in open pipes,
are equal to twice the length of the pipes.
Moreover, from the Corollary of Prop. XLVIt appears the reason why
the sounds immediately cease with the motion of the sonorous body, and
why they are heard no longer when we are at a great distance from the
sonorous bodies than when we are very near them. And besides, from the
foregoing principles, it plainly appears how it comes to pass that sounds are
so mightily increased in speaking-trumpets ; for all reciprocal motion usea
to be increased by the generating cause at each return. And in tubes hin
dering the dilatation of the sounds, the motion decays more slowly, and
24
370 THE MATHEMATICAL PRINCIPLES [BOOK II.
recurs more forcibly ; and therefore is the more increased by the new mo
tion impressed at each return. And these are the principal phasr. )mena oi
sounds.
SECTION IX.
Of the circular motion offluids.
HYPOTHESIS.
The resistance arisingfrom the want of lubricity in the parts of afluid,
is, casteris paribus, proportional to the velocity with which the parts of
thefluid are separated fro?n each other.
PROPOSITION LI. THEOREM XXXIX.
If a solid cylinder infinitely long, in an uniform and infinite fluid, revolve
with an uniform motion about an axis given in position, and the fluid
be forced round by only this impulse of the cylinder, and every part
of the fluid persevere uniformly in its motion ; I say, that the periodic
times of the parts of thefluid are as their distances Jrom the axis of
the cylinder.
Let AFL be a cylinder turning uni
formly about the axis S, arid let the
concentric circles BGM, CHN, DIO,
EKP, &c., divide the fluid into innu
merable concentric cylindric solid orbs
of the same thickness. Then, because
the fluid is homogeneous, the impres
sions which the contiguous orbs make
upon each other mutually will be (by
the Hypothesis) as their translations
from each, other, and as the contiguous
superficies upon which the impressions
are made. If the impression made upon any orb be greater or less on its
concave than on its convex side, the stronger impression will prevail, and
will either accelerate or retard the motion of the orb, according as it agrees
with, or is contrary to, the motion of the same. Therefore, that every orb
may persevere uniformly in its motion, the impressions made on both sides
must be equal and their directions contrary. Therefore since the impres
sions 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 distances of the superficies from the axis. But the differences of
SEC. IX] OF NATURAL PHILOSOPHY. 371
the angular motions about the axis are as those translations applied to the
distances, or as the translations ctly arid the distances inversely ; that
is, joining these ratios together, as the squares of the distances inversely.
Therefore if there be erected the lines A", B&, Cc, !.)</, Ee, &c., perpendic
ular to the several parts of he infinite right line SABCDEQ,, and recip
rocally proportional to the squares of SA, SB, SO, SO, SE, &c., and
through the extremities of those perpendiculars there be supposed to pass
an hyperbolic curve, the sums of the differences, that is, the whole angular
motions, will be as the correspondent sums of the lines Ati, B6, Cc1
, DC/, Ed?,
that is (if to constitute a medium uniformly fluid the number of the orbs
be increased and their breadth diminished in infinitum\ as the hyperbolic
areas AaQ, B6Q,, CcQ,, Dc/Q,, EeQ, &c., analogous to the sums ; and the
times, reciprocally proportional to the angular motions, will be also recip
rocally proportional to those areas. Therefore the periodic time of any
particle as I), is reciprocally as the area Dc/Q,, that is (as appears
from the known methods of quadratures of curves), directly as the dis
tance SD. Q.E.D.
COR. 1. Hence the angular motions of the particles of the fluid are re
ciprocally as their distances from the axis of the cylinder, and the absolute
velocities are equal.
COR. 2. If a fluid be contained in a cylindric vessel of an infinite length,
and contain another cylinder within, and both the cylinders revolve about
one common axis, and the times of their revolutions be as their semidiameters,
and every part of the fluid perseveres in its motion, the peri
odic times of the several parts will be as the distances from the axis of the
cylinders.
COR. 3. If there be added or taken away any common quantity of angu
lar motion from the cylinder and fluid moving in this manner; yet because
this new motion will not alter the mutual attrition of the parts of the fluid,
the motion of the parts among themselves will not be changed; for the
translations of the parts from one another depend upon the attrition.
Any part will persevere in that motion, which, by the attrition made
on both sides with contrary directions , is no more accelerated than it is re
tarded.
COR. 4. Therefore if there be taken away from this whole system of the
cylinders and the fluid all the angular motion of the outward cylinder, we
shall have the motion of the fluid in a quiescent cylinder.
COR. 5. Therefore if the fluid and outward cylinder are at rest, and the
inward cylinder revolve uniformly, there will be communicated a circular
motion to the fluid, which will be propagated by degrees through the whole
fluid
; and will go on continually increasing, till such time as the several
parts of the fluid acquire the motion determined in Cor. 4.
COR. 6. And because the fluid endeavours to propagate its motion stil!
372 THE MATHEMATICAL PRINCIPLES [BOOK 11.
farther, its impulse will carry the outmost cylinder also about with it, Tinless
the cylinder be violently detained; and accelerate its motion till the
periodic times of both cylinders become equal among themselves. But if
the outward cylinder be violently detained, it will make an effort to retard
the motion of the fluid ; and unless the inward cylinder preserve that mo
tion by means of some external force impressed thereon, it will make it
3ease by degrees.
All these things will be found true by making the experiment in deep
standing water.
PROPOSITION LIL THEOREM XL.
If a solid sphere, in an uniform and infinite fluid, revolves about an axis
given in position with an uniform motion., and thejiuid be forced round
by only this impulse of the sphere ; and every part of the fluid perse
veres uniformly in its motion ; I say, that the periodic times of the
parts of thefluid are as the squares of their distances from the centre
of the sphere.
CASE 1. Let AFL be a sphere turn
ing uniformly about the axis S, and let
the concentric circles BGM, CHN, DIO,
EKP, &cv divide the fluid into innu
merable concentric orbs of the same
thickness. Suppose those orbs to be
solid ; and, because the fluid is homo
geneous, the impressions which the con
tiguous orbs make one upon another
will be (by the supposition) as their
translations from one another, and the
contiguous superficies upon which the
impressions are made. If the impression upon any orb be greater or less
upon its concave than upon its convex side, the more forcible impression
will prevail, and will either accelerate or retard the velocity of the orb, ac
cording as it is directed with a conspiring or contrary motion to that of
