sity of the globe to the density of the medium, that is, as A to A B, G
the time in which the globe falling with the weight B without resistance
describes the space P, and H the velocity which the body acquires by that
fall. Then H will be the greatest velocity with which the globe can pos
sibly descend with the weight B in the resisting medium, by Cor. 2, Prop
XXXVIII ; and the resistance which the globe meets with, when descend
ing with that velocity, will be equal to its weight B ; and the resistance it
meets with in any other velocity will be to the weight B in the duplicate ra
tio of that velocity to the greatest velocity H, by Cor. 1, Prop. XXXVIII.
This is the resistance that arises from the inactivity of the matter of
the fluid. That resistance which arises from the elasticity, tenacity, and
friction of its parts, may be thus investigated.
Let the globe be let fall so that it may descend in the fluid by the weight
B ; and let P be the time of falling, and let that time be expressed in sec
onds, if the time G be given in seconds. Find the absolute number N
2P
agreeing to the logarithm 0,4342944819 >
and let L be the logarithm of
N 4- 1
the number and *^e velocity acquired in falling will bf
THE MATHEMATICAL PRINCIPLES [BOOK 11
N i 2PF
jj=
H, and the height described will be -^ 1 .38629430 IIP +
4,6051701S6LF. If the fluid be of a sufficient depth, we may neglect the
2PF
term 4,6051 70186LF; and r- 1,3362943611F will be the altitude
described, nearly. These things appear by Prop. IX, Book II, and its Corol
laries, and are true upon this supposition, that the globe meets with no other
resistance but that which arises from the inactivity of matter. Now if it
really meet with any resistance of another kind, the descent will be slower,
and from the quantity of that retardation will be known the quantity of
this new resistance.
That the velocity and descent of a body falling in a fluid might more
easily be known, I have composed the following table ; the first column of
which denotes the times of descent ; the second shews the velocities ac
quired in falling, the greatest velocity being 100000000: the third exhib
its the spaces described by falling in those times, 2F being the space which
the body describes in the time G with the greatest velocity ; and the fourth
gives the spaces described with the greatest velocity in the same times.
2P
The numbers in the fourth column are
-pn
and by subducting the number
1,3962944 4,60517021,, are found the numbers in the third column ;
and these numbers must be multiplied by the space F to obtain the spaces
described in falling. A fifth column is added to all these, containing the
spaces described in the same times by a body falling in vacno with the
force of B its comparative weight,
SEC. VII.
| OF NATURAL PHILOSOPHY. 347
\
SCHOLIUM*
In order to investigate the resistances of lluids from experiments, I pro
cured a square wooden vessel, whose length and breadth on the inside was
9 inches English measure, and its depth 9 feet \ ; this I filled with rain
water: and having provided globes made up of wax, and lead included
therein, I noted the times of the descents of these globes, the height through
which they descended being 112 inches. A solid cubic foot of English
measure contains 76 pounds troy weight of rain water ; and a solid inch
contains if ounces troy weight, or 253 grains: and a globe of water of
one inch in diameter contains 132,645 grains in air, or 132,8 grains in
vacn.o ; and any other globe will be as the excess of its weight in vacuo
above its weight in water.
EXPER. 1. A globe whose weight was 156^ grains in air, and 77 grains
in water, described the whole height of 1 12 inches in 4 seconds. And, upon
repeating the experiment, the globe spent again the very same time of 4
seconds in falling.
The weight of this globe in vacuo is 156^1 grains; and the excess of
this weight above the weight of the globe in water is 79^ f grains. Hence
the diameter of the globe appears to be 0,84224 parts of an inch. Then it
will be, as that excess to the weight of the globe in vacuo, so is the density
of the water to the density of the globe; and so is f parts of the diameter
of the globe (viz. 2,24597 inches) to the space 2F, which will be therefore
4.4256 inches. Now a globe falling in vacuo with its whole weight of
156^f grains in one second of time will describe 193| inches ; and falling
in water in the same time with the weight of 77 grains without resistance,
will describe 95,219 inches ; and in the time G, which is to one second of
time in the subduplicate ratio of the space P, or of 2,2128 inches to 95,219
inches, will describe 2,2128 inches, and will acquire the greatest velocity H
with which it is capable of descending in water. Therefore the time G is
0",15244. And in this time G, with that greatest velocity H, the globe
will describe the space 2F, which is 4,4256 inches; and therefore in 4 sec
onds will describe a space of 1 16,1245 inches. Subduct the space 1,3862944 F,
or 3,0676 inches, and there will remain a space of 113,0569 inches, which
the globe falling through water in a very wide vessel will describe in 4 sec
onds. But this space, by reason of the narrowness of the wooden vessel
before mentioned, ought to be diminished in a ratio compounded of the subduplicate
ratio of the orifice of the vessel to the excess of this orifice above
half a great circle of the globe, and of the simple ratio of the same orifice
to its excess above a great circle of the globe, that is, in a ratio of 1 to
0,9914. This done, we have a space of 112,08 inches, which a globe fall
ing through the water in this wooden vessel in 4 seconds of time ought
nearly to describe by this theory; but it described 112 inches by the ex
periment.
348 THE MATHEMATICAL PRINCIPLES [BOOK II
EXPER. 2. Three equal globes, whose weights were severally 76^- grains
in air, and 5 T^ grains in water, were let fall successively -; and every one
fell through the water in 15 seconds of time, describing in its fall a height
of 112 inches.
By computation, the weight of each globe in vacuo is 76 T
5
2 grains ;
the
excess of this weight above the weight in water is 71 grains J ; the diam
eter of the globe 0,81296 of an inch; f parts of this diameter 2,1 67S
inches; the space 2F is 2,3217 inches; the space which a globe of 5 T\
grains in weight would describe in one second without resistance, 12,80
inches, and the time G0",301056. Therefore the globe, with the greatest
velocity it is capable of receiving from a weight of 5^ grains in its de
scent through water, will describe in the time 0",3L)1056the space of 2,3217
inches; and in 15 seconds the space 115,678 inches. Subduct the space
1,3862944F, or 1,609 inches, and there remains the space 114.069 inches,
which therefore the falling globe ought to describe in the same time, if the
vessel were very wide. But because our vessel was narrow, the space ought
to be diminished by about 0,895 of an inch. And so the space will remain
113,174 inches, which a globe falling in this vessel ought nearly to de
scribe in 15 seconds, by the theory. But by the experiment it described
112 inches. The difference is riot sensible.
EXPER. 3. Three equal globes, whose weights were severally 121 grains
in air, and 1 grain in water, were successively let fall
; and they fell
through the water in the times 46", 47", and 50", describing a height oi
112 inches.
By the theory, these globes ought to have fallen in about 40". Now
whether their falling more slowly were occasioned from hence, that in slow
motions the resistance arising from the force of inactivity does really bear
a less proportion to the resistance arising from other causes ;
or whether
it is to be attributed to little bubbles that might chance to stick to the
globes, or to the rarefaction of the wax by the warmth of the weather, or
of the hand that let them fall
; or, lastly, whether it proceeded from some
insensible errors in weighing the globes in the water, I am not certain.
Therefore the weight of the globe in water should be of several grains, that
the experiment may be certain, and to be depended on.
EXPER. 4. I began the foregoing experiments to investigate the resistan
ces of fluids, before I was acquainted with the theory laid down in the
Propositions immediately preceding. Afterward, in order to examine the
theory after it was discovered, I procured a wooden vessel, whose breadth
on the inside was Sf inches, and its depth ] 5 feet and -i. Then I made
four globes of wax, with lead included, each of which weighed 1391
grains
in air, and 7\ grains in water. These I let fall, measuring the times of their
falling in the water with a pendulum oscillating to half seconds. The
globes were cold, and had remained so some time, both when they were
SEC. V1L] OF NATUKAL PHILOSOPHY. 3-1 *J
.reighed and when they were let fall
; because warmth rarefies the wax. and
by rarefying it diminishes the weight of the globe in the water ; and wax,
when rarefied, is not instantly reduced by cold to its former density. Be
fore they were let fall, they were totally immersed under water, lest, by the
weight of any part of them that might chance to be above the water, their
descent should be accelerated in its beginning. Then, when after their
immersion they were perfectly at rest, they were let go with the greatest
care, that they might not receive any impulse from the hand that let them
down. And they fell successively in the times of 47 J, 48^, 50, and 51 os
cillations, describing a height of 15 feet and 2 inches. But the weather
was now a little colder than when the globes were weighed, and therefore 1
repeated the experiment another day ; and then the globes fell in the times
of 49, 49i, 50. and 53; and at a third trial in the times of 49, 50, 51.
and 53 oscillations. And by making the experiment several times over, I
found that the globes fell mostly in the times of 49| and 50 oscillations.
When they fell slower, I suspect them to have been retarded by striking
against the sides of the vessel.
Now, computing from the theory, the weight of the globe in vacno is
139| grains; the excess of this weight above the weight of the globe in
water 132|i grains ; the diameter of the globe 0,99868 of an inch :
|- parts
of the diameter 2,66315 inches; the space 2F 2,8066 inches; the space
which a globe weighing 7| grains falling without resistance describes in a
second of time 9,88164 inches; and the time G0",376843 Therefore the
globe with the greatest velocity with which it is capable of descending
through the water by the force of a weight of 7} grains, will in the time
0",376843 describe a space of 2,8066 inches, and in one second of time a
space of 7,44766 inches, and in the time 25", or in 50 oscillations, the space
186,1915 inches. Subduct the space 1,386294F, or 1,9454 inches, and
there will remain the space 184,2461 inches which the globe will describe
in that time in a very wide vessel. Because our vessel was narrow, let this
space be diminished in a ratio compounded of the subduplicate ratio of the
orifice of the vessel to the excess of this orifice above half a great circle of
the globe, and of the simple ratio of the same orifice to its excess above a
great circle of the globe ; and we shall have the space of 181,86 inches,
which the globe ought by the theory to describe in this vessel in the time
of 50 oscillations, nearly. But it described the space of 182 inches, by
experiment, in 49^ or 50 oscillations.
EXPER. 5. Pour globes weighing 154| grains in air, and 21 1 grains in
water, being let fall several times, fell in the times of 28^, 29, 29 , and 30,
and sometimes of 31, 32, and 33 oscillations, describing a height of 15 feet
and 2 inches.
They ought by the theory to have fallen in the time of 29 oscillations,
nearly.
350 THE MATHEMATICAL PRINCIPLES
| BOOK IL
EXPER. 6. Five globes, weighing 212f grains in air, and 79^ in water,
being several times let fall, fell in the times of 15, 15^, 16, 17, and 18 os
cillations, describing a height of 15 feet and 2 inches.
By the theory they ought to have fallen in the time cf 15 oscillations,
nearly.
EXPER. 7. Four globes, weighing 293 f grains in air, and 35| grains in
water, being let fall several times, fell in the times of 29 30, 301 31, 32,
and 33 oscillations, describing a height of 15 feet and 1 inch and .
By the theory they ought to have fallen in the time of 28 oscillations,
nearly.
In searching for the cause that occasioned these globes of the same weight
and magnitude to fall, some swifter and some slower, I hit upon this ; that
the globes, when they were first let go and began to fall, oscillated about
their centres; that side which chanced to be the heavier descending first,
and producing an oscillating motion. Now by oscillating thus, the globe
communicates a greater motion to the water than if it descended without
any oscillations ; and by this communication loses part of its own motion
with which it should descend
; and therefore as this oscillation is greater
or less, it will be more or less retarded. Besides, the globe always recedes
from that side of itself which is descending in the oscillation, and by so
receding comes nearer to the sides of the vessel, so as even to strike against
them sometimes. And the heavier the globes are, the stronger this oscil
lation is
; and the greater they are, the more is the water agitated by it.
Therefore to diminish this oscillation of the globes, 1 made new ones of
lead and wax, sticking the lead in one side of the globe very near its sur
face; and I. let fall the globe in such a manner, that, as near as possible,
the heavier side might be lowest at the beginning of the descent. By this
means the oscillations became much less than before, and the times in which
the globes fell were not so unequal: as in the following experiments.
EXPER. 8. Four globes weighing 139 grains in air, and 6| in water,
were let fall several times, and fell mostly in the time of 51 oscillations,
never in more than 52, or in fewer than 50, describing a height of 182
inches.
By the theory they ought to fall in about the time of 52 oscillations
EXPER. 9. Four globes weighing 273^ grains in air, and 140 in water,
being several times let fall, fell in never fewer than 12, and never more
than 13 oscillations, describing a height of 182 inches.
These globes by the theory ought to have fallen in the time of 1 1 } os
cillations, nearly.
EXPER. 10. Four globes, weighing 384 grains in air, and 119^ in water,
oeing let fall several times, fell in the times of 17f 18, 18^, and 19 oscilla
tions, descril ing a height of 181| inches. And when they fell in the time
SEC. VI1.J OF NATURAL PHILOSOPHY. 351
of 19 oscillations, I sometimes heard them hit against the sides of tl.e ves
sel before they reached the bottom.
By the theory they ought to have fallen in the time of 15f oscillations,
nearly.
EXPER. 11. Three equal globes, weighing 48 grains in the air, and 3||
in water, being several times let fall, fell in the times of 43J, 44, 44 1, 45,
and 46 oscillations, and mostly in 44 and 45. describing a height of 182*
inches, nearly.
By the theory they ought to have fallen in the time of 46 oscillations
and f, nearly.
EXPER. 12. Three equal globes, weighing 141 grains in air, and 4| in
water, being let fall several times, fell in the times of 61, 62, 63, 64, and
65 oscillations, describing a space of 182 inches.
And by the theory they ought to have fallen in 641 oscillations
nearly.
From these experiments it is manifest, that when the globes fell slowly,
as in the second, fourth, fifth, eighth, eleventh, and twelfth experiments;
the times of falling are rightly exhibited by the theory but when the
globes fell more swiftly, as in the sixth, ninth, and tenth experiments, the
resistance was somewhat greater than in the duplicate ratio of the velocity.
For the globes in falling oscillate a little : and this oscillation, in those
globes that are light and fall slowly, soon ceases by the weakness of the
motion ; but in greater and heavier globes, the motion being strong, it con
tinues longer, and is not to be checked by the ambient water till after sev
eral oscillations Besides, the more swiftly the globes move, the less are
they pressed by the fluid at their hinder parts; and if the velocity be. per
petually increased, they will at last leave an empty space behind them,
unless the compression of the fluid be increased at the same time. For the
compression of the fluid ought to be increased (by Prop. XXXII and
XXXIII) in the duplicate ratio of the velocity, in order to preserve the re
sistance in the same duplicate ratio. But because this is not done, the
globes that move swiftly are not so much pressed at their hinder parts as
the others; and by the defect of this pressure it comes to pass that their
resistance is a little greater than in a duplicate ratio of their velocity.
So that the theory agrees with the phenomena of bodies falling in water
It remains that we examine the phenomena of bodies falling in air.
EXPER. 13. From the top of St. Paul s Church in London, in Juiib
1710, there e let fall together two glass globes, one full of quicksilver,
the other of air; and in their fall they described a height of 220 English
feet. A wooden table was suspended upon iron hinges on one sidi> and the
other side of the same was supported by a wooden pin. The twn globes
lying upon this table were let fall together by pulling out the pin bj
means of an iron wire reaching from thence quite down to the ground ;
s<
352 THE MATHEMATICAL PRINCIPLES [BOOK II,
that, the pin being removed, the table, which had then no support but the
iron hinges, fell downward, and turning round upon the hinges, gave leave
to the globes to drop off from it. At the same instant, with the same pull
of the iron wire that took out the pin, a pendulum oscillating to seconds
was let go, and began to oscillate. The diameters and weights of the
globes, and their times of falling, are exhibited in the following table.
But the times observed must be corrected
;
for the globes of mercury (by
Galileo s theory), in 4 seconds of time, will describe 257 English feet, and
220 feet in only 3"42 ". So that the wooden table, when the pin was taken
out, did not turn upon its hinges so quickly as it ought to have done; and
the slowness of that revolution hindered the descent of the globes at the
beginning. For the globes lay about the middle of the table, and indeed
were rather nearer to the axis upon which it turned than to the pin. And
hence the times of falling were prolonged about 18
"; and therefore ought
to be corrected by subducting that excess, especially in the larger globes,
which, by reason of the largeness of their diameters, lay longer upon the
revolving table than the others. This being done, the times in which the
six larger globes fell will come forth 8" 12 ",
7" 42% 7" 42 ",
7" 57 ",
8" 12 "
and 7" 42 ".
Therefore the fifth in order among the globes that were full of air being
5 inches in diameter, and 483 grains in weight, fell in 8" 12 ", describing a
space of 220 feet. The weight of a bulk of water equal to this globe is
1 6600 grains; and the weight of an equal bulk of air is l||f- grains, or I9 r
3
o
grains ; and therefore the weight of the globe in vacuo is 502T
3
?r grains;
and this weight is to the weight of a bulk of air equal to the globe as
502T
;v to 19 T
3
o- ; and so is 2P to | of the diameter of the globe, that is, to
13i inches. Whence 2F becomes 28 feet 11 inches. A globe, falling in
vacuo with its whole weight of 502T
3
grains, will in one second of time
describe 193| inches as above ; and with the weight of 483 grains will de
scribe 185,905 inches; and with that weight 483 grains in vacuo will de
scribe the space F, or 14 feet 5i inches, in the time of 57 "
58"", and ac
quire the greatest velocity it is capable of descending with in the air.
With this velocity the globe in 8" 12 " of time will describe 245 feet and
5i inches. Subduct 1,3863F, or 20 feet and | an inch, and there remain
225 feet 5 inches. This space, therefore, the falling globe ought by the
SEC. VIIJ OF NATURAL PHILOSOPHY
theory to describe in 8" 12 ". But* by the experiment it descrioed a space
of 220 feet. The difference is insensible.
By like calculations applied to the other globes full of air, I composed
the following table.
EXPER. 14. Anno 1719, in the month of July, Dr. Desaguliers made
some experiments of this kind again, by forming hogs bladders into spheri
cal orbs ; which was done by means of a concave wooden sphere, which the
bladders, being wetted well first, were put into. After that being blown
full of air. they were obliged to fill up the spherical cavity that contained
them ; and then, when dry, were taken out. These were let fall from the
lantern on the top of the cupola of the same church, namely, from a height
of 272 feet ; and at the same moment of time there was let fall a leaden
globe, whose weight was about 2 pounds troy weight. And in the mean
time some persons standing in the upper part of the church where the
globes were let fall observed the whole times of falling ; and others stand
ing on the ground observed the differences of the times between the fall
of the leaden weight and the fall of the bladder. The times were measured
by pendulums oscillating to half seconds. And one of those that stood
upon the ground had a machine vibrating four times in one second ; and
another had another machine accurately made with a pendulum vibrating
four times in a second also. One of those also who stood at the top of the
church had a like machine ; and these instruments were so contrived, that
their motions could be stopped or renewed at pleasure. Now the leaden
globe fell in about four seconds and i of time; and from the addition of
this time to the difference of time above spoken of, was collected the \Vhole
time in which the bladder was falling. The times which the five bladders
spent in falling, after the leaden globe had reached the ground, were, tn*e
first time, 14", 12f, 14f, 17 f, and 16J-" ; and the second time, 14i", 14}",
14", 19", and 16 J". Add to these 4", the time in which the leaden globe
was falling, and the whole times in which the five bladders fell were, the
first fane, 19* 17", 18J", 22", and 21}"; and the second time, 18f, 18i",
ISj", 23{", and 21". The times observed at the top of the church were,
the first time, 19 f", 17f , 18f, 22f , and 21f"; and the second time, 19",
ISf", ISf, 24". and 211". But the bladders did not always fall directly
down, but sometimes fluttered a little in the air, and waved to and fro, aa
354 THE MATHEMATICAL PRINCIPLES [BOOK Jl
they were descending. And by these motions the times of their falling
were prolonged, and increased by half a second sometimes, and sometimes
by a whole second. The second and fourth bladder fell most directly the
first time, and the first and third the second time. The fifth bladder was
wrinkled, and by its wrinkles was a little retarded. I found their diame
ters by their circumferences measured with a very fine thread wound about
them twice. In the following table I have compared the experiments with
the theory ; making the density of air to be to the density of rain-water as
1 to 860, and computing the spaces which by the theory the globes ought
to describe in falling.
Our theory, therefore, exhibits rightly, within a very little, all the re
sistance that globes moving either in air or in water meet with ; which^appears
to be proportional to the densities of the fluids in globes of equal ve
locities and magnitudes.
In the Scholium subjoined to the sixth Section, we shewed, by experi
ments of pendulums, that the resistances of equal and equally swift globes
moving in air, water, and quicksilver, are as the densities of the fluids.
We here prove the same more accurately by experiments of bodies falling
in air and water. For pendulums at each oscillation excite a motion in
