to the motion lost,
was inches.
The number of the os
cillations in water.
The number of the os
cillations in air.
16
. li . 3 . 7 . lH.12f.13j
85i . 287 . 535
SEC. VI.] OF NATURAL PHILOSOPHY. 319
In the experiments of the 4th column there were equal motions lost in
535 oscillations made in the air, and If in water. The oscillations in the
air were indeed a little swifter than those in the water. But if the oscil
lations in the water were accelerated in such a ratio that the motions of
the pendulums might be equally swift in both mediums, there would be
still the same number 1 j of oscillations in the water, and by these the
same quantity of motion would be lost as before ; because the resistance i>
increased, and the square of the time diminished in the same duplicate ra
tio. The pendulums, therefore, being of equal velocities, there were equal
motions lost in 535 oscillations in the air, and 1} in the water; and there
fore the resistance of the pendulum in the water is to its resistance in the
air as 535 to 1 }. This is the proportion of the whole resistances in the
case of the 4th column.
Now let AV + CV 2
represent the difference of the arcs described in the
descent and subsequent ascent by the globe moving in air with the greatest
velocity V ; and since the greatest velocity is in the case of the 4th column
to the greatest velocity in the case of the 1st column as 1 to 8 ; and that
difference of the arcs in the case of the 4th column to the difference in the
2 16
case of the 1st column as ^ to
7, or as 86J to 4280 ; put in these
cases 1 and 8 for the velocities, and 85 1 and 4280 for the differences of
the arcs, and A + C will be S5|, and 8A -f 640 == 4280 or A + SC
= 535 ; and then by reducing these equations, there will come out TC =
449^ and C = 64T\ and A = 21f ; and therefore the resistance, which is
as TVAV + fCV 2
, will become as 13 T
6
TV + 48/^Y 2
. Therefore in the
case of the 4th column, where the velocity was 1, the whole resistance is to
its part proportional to the square of the velocity as 13 T
6
T + 48/F or
61 }f to 48/e ; and therefore the resistance of the pendulum in water is to
that part of the resistance in air, which is proportional to the square of the
velocity, and which in swift motions is the only part that deserves consid
eration, as 61}^ to 4S/g and 535 to 1} conjunctly, that is, as 571 to 1.
If the whole thread of the pendulum oscillating in the water had been im
mersed, its resistance would have been still greater ; so that the resistance
of the pendulum oscillating in the water, that is, that part which is pro
portional to the square of the velocity, and which only needs to be consid
ered in swift bodies, is to the resistance of the same whole pendulum, oscil
lating in air with the same velocity, as about 850 to 1, that is as, the den
sity of water to the density of air, nearly.
In this calculation we ought also to have taken in that part of the re
sistance of the pendulum in the water which was as the square of the ve
locity ; but I found (which will perhaps seem strange) that the resistance
in the water was augmented in more than a duplicate ratio of the velocity.
In searching after the cause, I thought upon this, that the vessel was toe
320 THE MATHEMATICAL PRINCIPLES [BOOK II.
narrow for the magnitude of the pendulous globe, and by its narrowness
obstructed the motion of the water as it yielded to the oscillating globe.
For when I immersed a pendulous globe, whose diameter was one inch only,
the resistance was augmented nearly in a duplicate ratio of the velocity,
I tried this by making a pendulum of two globes, of which the lesser and
lower oscillated in the water, and the greater and higher was fastened to
the thread just above the water, and, by oscillating in the air, assisted the
motion of the pendulum, and continued it longer. The experiments made
by this contrivance proved according to the following table.
Arc descr. in first descent . .16.8. 4.
Arc descr. in last ascent . . 12 . 6 . 3 . li . J . | . T
3
F
Dif. of arcs, proport. to 1 . pi i
motion lost
$
T r T*
Number of oscillations... 3f . 6j . 12^. 211 . 34 . 53 . 62)
In comparing the resistances of the mediums with each other, I also
caused iron pendulums to oscillate in quicksilver. The length of the iron
wire was about 3 feet, and the diameter of the pendulous globe about i of
an inch. To the wire, just above the quicksilver, there was fixed another
leaden globe of a bigness sufficient to continue the motion of the pendulum
for some time. Then a vessel, that would hold about 3 pounds of quick
silver, was filled by turns with quicksilver and common water, that, by
making the pendulum oscillate successively in these two different fluids, I
might find the proportion of their resistances ; and the resistance of the
quicksilver proved to be to the resistance of water as about 13 or 14 to 1
;
that is. as the density of quicksilver to the density of water. When I made
use of a pendulous globe something bigger, as of one whose diameter was
about ^ or | of an inch, the resistance of the quicksilver proved to be to
the resistance of the water as about 12 or 10 to 1. But the former experi
ment is more to be relied on, because in the latter the vessel was too nar
row in proportion to the magnitude of the immersed globe; for the vessel
ought to have been enlarged together with the globe. I intended to have
repeated these experiments with larger vessels, and in melted metals, and
other liquors both cold and hot ; but I had not leisure to try all: and be
sides, from what is already described, it appears sufficiently that the resist
ance of bodies moving swiftly is nearly proportional to the densities of
the fluids in which they move. I do not say accurately ; for more tena
cious fluids, of equal density, will undoubtedly resist more than those that
are more liquid ; as cold oil more than warm, warm oil more than rain
water, and water more than spirit of wine. But in liquors, which are sen
sibly fluid enough, as in air, in salt and fresh water, in spirit of wine, of
turpentine, and salts, in oil cleared of its fseces by distillation and warmed,
in oil of vitriol, and in mercury, and melted metals, and any other such
like, that are fluid enough to retaia for some time the motion impressed
SEC. VI.J OF NATURAL PHILOSOPHY. 321
upon them by the agitation of the vessel, and which being poured out are
easily resolved into drops, I doubt not but the rule already laid down may
be accurate enough, especially if the experiments be made with larger
pendulous bodies and more swiftly moved.
Lastly, since it is the opinion of some that there is a certain ^ethereal
medium extremely rare and subtile, which freely pervades the pores of all
bodies ; and from such a medium, so pervading the pores of bodies, some re
sistance must needs arise; in order to try whether the resistance, which wre
experience in bodies in motion, be made upon their outward superficies only,
or whether their internal parts meet with any considerable resistance upon
their superficies, I thought of the following experiment I suspended a
round deal box by a thread 11 feet long, on a steel hook, by means of a ring
of the s-ime metal, so as to make a pendulum of the aforesaid length. The
hook had a sharp hollowr
edge on its upper part, so that the upper arc of
the ring pressing on the edge might move the more freely ; and the thread
was fastened to the lower arc of the ring. The pendulum being thus pre
pared, I drew it aside from the perpendicular to the distance of about 6
feet, and that in a plane perpendicular to the edge of the hook, lest the
ring, while the pendulum oscillated, should slide to and fro on the edge of
the hook : for the point of suspension, in which the ring touches the hook,
ought to remain immovable. I therefore accurately noted the place to
which the pendulum was brought, and letting it go, I marked three other
places, to which it returned at the end of the 1st, 2d, and 3d oscillation.
This I often repeated, that I might find those places as accurately as pos
sible. Then I filled the box with lead and other heavy metals that were
near at hand. But, first, I weighed the box when empty, and that pnrt of
the thread that went round it, and half the remaining part, extended be
tween the hook and the suspended box ; for the thread so extended always
acts upon the pendulum, when drawn aside from the perpendicular, with half
its weight. To this weight I added the weight of the air contained in the
box And this whole weight was about -fj of the weight of the box when
filled wr ith the metals. Then because the box when full of the metals, by ex
tending the thread with its weight, increased the length of the pendulum,
f shortened the thread so as to make the length of the pendulum, when os
cillating, the same as before. Then drawing aside the pendulum to the
place first marked, and letting it go, I reckoned about 77 oscillations before
the box returned to the second mark, and as many afterwards before it came
to the third mark, and as many after that before it came to the fourth
xnark. From whence I conclude that the whole resistance of the box, when
full, had not a greater proportion to the resistance of the box, when empty,
than 78 to 77. For if their resistances were equal, the box, when full, by
reason of its vis insita, which was 78 times greater than the vis tfuritoof
the same when empty, ought to have continued its oscillating motion so
21
322 THE MATHEMATICAL PRINCIPLES
| BOOK II.
much the longer, and therefore to have returned to those marks at the end
of 78 oscillations. But it returned to them at the end of 77 oscillations.
Let, therefore, A represent the resistance of the box upon its external
superficies, and B the resistance of the empty box on its internal superficies ;
and if the resistances to the internal parts of bodies equally swift be as the
matter, or the number of particles that are resisted, then 78B will be the
resistance made to the internal parts of the box, when full
; and therefore
the whole resistance A + B of the empty box will be to the whole resist
ance A + 7SB of the full box as 77 to 78, and, by division, A + B to 77B
as 77 to 1
; and thence A + B to B as 77 X 77 to 1, and, by division
again, A to B as 5928 to 1. Therefore the resistance of the empty box in
its internal parts will be above 5000 times less than the resistance on its
external superficies. This reasoning depends upon the supposition that the
greater resistance of the full box arises not from any other latent cause,
but only from the action of some subtile fluid upon the included metal.
This experiment is related by memory, the paper being lost in which I
had described it
;
so that I have been obliged to omit some fractional parts,
which are slipt out of my memory ; and I have no leisure to try it again.
The first time I made it, the hook being weak, the full box was retarded
sooner. The cause I found to be, that the hook was not strong enough to
bear the weight of the box : so that, as it oscillated to and fro, the hook
was bent sometimes this and sometimes that way. I therefore procured a
hook of sufficient strength, so that the point of suspension might remain
unmoved, and then all things happened as is above described.
SEC. VI I.] OF NATURAL PHILOSOPHY. 323
SECTION VII.
Of the, motion offluids, and the resistance made to projected bodies.
PROPOSITION XXXII. THEOREM XXVI.
Suppose two similar systems of bodies consisting of an equal number of
particles, and let the correspondent particles be similar and propor
tional, each in, one system to each in the other, and have a like situa
tion among themselves, and the same given ratio of density to each
other ; and let them begin to move anwng themselves in proportional
times, and with like motions (that is, those in one system among one
another, and those in the other among one another). And if the par
ticles that are in the same system do not touch otte another, except ir
the moments of reflexion ; nor attract, nor repel each other, except with
that are as the diameters of the correspondent parti
cles inversely, and the squares of the velocities directly ; I say, that the
particles of those systems will continue to move among themselves witIt
like motions and in proportional times.
Like bodies in like situations are said to be moved among themselves
with like motions and in proportional times, when their situations at the
end of those times are always found alike in respect of each other ; as sup
pose we compare the particles in one system with the correspondent parti
cles in the other. Hence the times will be proportional, in which similar
and proportional parts of similar figures will be described by correspondent
particles. Therefore if we suppose two systems of this kind; the corre
spondent particles, by reason of the similitude of the motions at their
beginning, will continue to be moved with like motions, so long as they
move without meeting one another ;
for if they are acted on by no forces,
they will go on uniformly in right lines, by the 1st Law. But if they do
agitate one another with some certain forces, and those forces are as the
diameters of the correspondent particles inversely and the squares of the
velocities directly, then, because the particles are in like situations, and
their forces are proportional, the whole forces with which correspondent
particles are agitated, and which are compounded of each of the agitating
forces (by Corol. 2 of the Laws), will have like directions, and have the
same effect as if they respected centres placed alike among the particles ;
and those whole forces will be to each other as the several forces which
compose them, that is, as the diameters of the correspondent particles in
versely, and the squares of the velocities directly : and therefore will cans**
3^4 THE MATHEMATICAL PRINCIPLES [BOOK 11.
correspondent particles to continue to describe like figures. These things
will be so (by Cor. 1 and S, Prop. IV.; Book 1), if those centres are at rest
but if they are moved, yet by reason of the similitude of the translations,
their situations among the particles of the system will remain similar , so
that the changes introduced into the figures described by the particles will
still be similar. So that the motions of correspondent and similar par
ticles will continue similar till their first meeting with each other
; and
thence will arise similar collisions, and similar reflexions; which will again
beget similar motions of the particles among themselves (by what was just
now shown), till they mutually fall upon one another again, and so on ad
infinitum.
COR. 1. Hence if any two bodies, which are similar and in like situations
to the correspondent particles of the systems, begin to move amongst them
in like manner and in proportional times, and their magnitudes and densi
ties be to each other as the magnitudes and densities of the corresponding
particles, these bodies will continue to be moved in like manner and in
proportional times: for the case of the greater parts of both systems and of
the particles is the very same.
COR. 2. And if all the similar and similarly situated parts of both sys
tems be at rest among themselves ; and two of them, which are greater than
the rest, and mutually correspondent in both systems, begin to move in
lines alike posited, with any similar motion whatsoever, they will excite
similar motions in the rest of the parts of the systems, and will continue
to move among those parts in like manner and in proportional times ; and
will therefore describe spaces proportional to their diameters.
PROPOSITION XXXIII. THEOREM XXVII.
The same things biting supposed, I say, that the greater parts of the
systems are resisted in a ratio compounded of the duplicate ratio of
their velocities, and the duplicate ratio of their diameters, and Ihe sim
ple ratio of the density of the parts of the systems.
For the resistance arises partly from the centripetal or centrifugal, forces
with which the particles of the system mutually act on each other, partly
from the collisions and reflexions of the particles and the greater parts.
The resistances of the first kind are to each other as the whole motive
forces from which they arise, that is, as the whole accelerative forces and
the quantities of matter in corresponding parts ; that is (by the sup
position), as the squares of the velocities directly, and the distances of the
corresponding particles inversely, and the quantities of matter in the cor
respondent parts directly : and therefore since the distances of the parti
cles in one system are to the correspondent distances of the particles of the
;ther S3 the diameter of one particle or part in *he former system to the
SEC. VII.] OF NATURAL PHILOSOPHY. C>2"
diameter of the correspondent particle or part in the other, and since the
quantities of matter are as the densities of the parts and the cubes of the
diameters ; the resistances arc to each other as the squares of the velocities
and the squares of the diameters and the densities of the parts of the sys
tems. Q.E.D. The resistances of the latter sort are as the number of
sorrespondent reflexions and the forces of those reflexions conjunctly ; but
the number of the reflexions are to each other as the velocities of the cor
responding parts directly and the spaces between their reflexions inversely.
And the forces of the reflexions are as the velocities and the magnitudes
and the densities of the corresponding parts conjunctly ; that is, as the ve
locities and the cubes of the diameters and the densities of the parts. And,
joining all these ratios, the resistances of the corresponding parts are to
each other as the squares of the velocities and the squares of the diameters
and the densities of the parts conjunctly. Q.E.T).
COR. 1. Therefore if those systems are two elastic fluids, like our air,
and their parts are at rest among themselves ; and two similar bodies pro
portional in magnitude and density to the parts of the fluids, and similarly
gituated among those parts, be any how projected in the direction of lines
similarly posited ; and the accelerative forces with which the particles of
the fluids mutually act upon each other are as the diameters of the bodies
projected inversely and the squares of their velocities directly ; those bodies
will excite similar motions in the fluids in proportional times, and will de
scribe similar spaces and proportional to their diameters.
COR. 2. Therefore in the same fluid a projected body that moves swiftly
meets with a resistance that is, in the duplicate ratio of its velocity, nearly.
For if the forces with which distant particles act mutually upon one
another should be augmented in the duplicate ratio of the velocity, the
projected body would be resisted in the same duplicate ratio accurately ;
and therefore in a medium, whose parts when at a distance do not act mu
tually with any force on one another, the resistance is in the duplicate ra
tio of the velocity accurately. Let there be, therefore, three mediums A,
B, C, consisting of similar and equal parts regularly disposed at equal
distances. Let the parts of the mediums A and B recede from each other
with forces that are among themselves as T and V ; and let the parts of
the medium C be entirely destitute of any such forces. And if four equal
bodies D, E, P7 G, move in these mediums, the two first D and E in the
two first A and B, and the other two P and G in the third C ; and if the
velocity of the body D be to the velocity of the body E, and the velocity
of the body P to the velocity of the body G, in the subduplicate ratio of
the force T to the force V ; the resistance of the body D to the resistance
of the body E, and the resistance of the body P to the resistance of the
body G, will be in the duplicate ratio of the velocities ; and therefore the
resistance of the body D will be to the resistance of the body P as the re
326 THE MATHEMATICAL PRINCIPLES [BOOK II
sistance of the body E to the resistance of the body G. Let the bodies 1)
and F be equally swift, as also the bodies E and G ; and, augmenting the
velocities of the^bodies D arid F in any ratio, and diminishing the forces
of the particles of the medium B in the duplicate of the same ratio, the
medium B will approach to the form and condition of the medium C at
pleasure ; and therefore the resistances of the equal and equally swift
bodies E and G in these mediums will perpetually approach to equality
so that their difference will at last become less than any given. There
fore since the resistances of the bodies D and F are to each other as the
resistances of the bodies E and G, those will also in like manner approach
to the ratio of equality. Therefore the bodies 1) and F, when they move
with very great swiftness, meet with resistances very nearly equal; and
therefore since the resistance of the body F is in a duplicate ratio of the
velocity, the resistance of the body D will be nearly in the same ratio.
Con. 3. The resistance of a body moving very swift in an elastic fluid
is almost the same as if the parts of the fluid were destitute of their cen
trifugal forces, and did not fly from each other; if so be that the elasti
city of the fluid arise from the centrifugal forces of the particles, and the
velocity be so great as not to allow the particles time enough to act.
COR. 4. Therefore, since the resistances of similar and equally swift
bodies, in a medium whose distant parts do not fly from each other, are as
the squares of the diameters, the resistances made to bodies moving with
very great and equal velocities in an elastic fluid will be as the squares of
the diameters, nearly.
COR. 5. And since similar, equal, and equally swift bodies, moving
through mediums of the same density, whose particles do not fly from each
other mutually, will strike against an equal quantity of matter in equal
times, whether the particles of which the medium consists be more and
smaller, or fewer and greater, and therefore impress on that matter an equal
quantity of motion, and in return (by the 3d Law of Motion) suffer an
equal re-action from the same, that is, are equally resisted ;
it is manifest,
also, that in elastic fluids of the same density, when the bodies move with
extreme swiftness, their resistances are nearly equal, whether the fluids
consist of gross parts, or of parts ever so subtile. For the resistance of
projectiles moving with exceedingly great celerities is not much diminished
by the subtilty of the medium.
COR. G. All these things are so in fluids whose elastic force takes its rise
from the centrifugal forces of the particles. But if that force arise from
some other cause, as from the expansion of the particles after the manner
of wool, or the boughs of trees, or any other cause, by which the particles
are hindered from moving freely among themselves, the resistance, by
reason of the lesser fluidity of the medium, will be greater than in the
Corollaries above.
SEC. VII. OF NATURAL PHILOSOPHY. 32?
K
L, P
O
PROPOSITION XXXIV. THEOREM XXV1I1.
If iu a rare medium, consisting of equal particles freely disposed at
equal distances from each other, a globe and a cylinder described on
equal diameters move with equal velocities in the. direction of the axis
of the cylinder, the resistance of the globe ivill be but half so great an
that of the cylinder.
For since the action of the medi
um upon the body is the same (by
Cor. 5 of the Laws) whether the body
move in a quiescent medium, or
whether the particles of the medium
impinge with the same velocity upon
the quiescent body, let us consider
the body as if it were quiescent, and
see with what force it would be impelled
by the moving medium. Let, therefore, ABKI represent a spherical
body described from the centre C with the semi-diameter CA, and let the
particles of the medium impinge with a given velocity upon that spherical
body in the directions of right lines parallel to AC : and let FB be one of
those right lines. In FB take LB equal to the semi-diameter CB, and
draw BI) touching the sphere in B. Upon KG and BD let fall the per
pendiculars BE, LD ; and the force with which a particle of the medium,
impinging on the globe obliquely in the direction FB, would strike the
globe in B, will be to the force with which the same particle, meeting the
cylinder ONGQ, described about the globe with the axis ACI, would strike
it perpendicularly in b, as LD to LB, or BE to BC. Again ; the efficacy
of this force to move the globe, according to the direction of its incidence
FB or AC, is to the efficacy of the same to move the globe, according to
the direction of its determination, that is, in the direction of the right line
BC in which it impels the globe directly, as BE to BC. And, joining
