urged in the direction of the tangent of the cycloid, and will have the same
ratio to the length of the pendulum as the force in D has to the force of
gravity. Let that force, therefore, be expressed by that length CD, and
the force of gravity by the length of the pendulum ; and if in DE you
take DK in the same ratio to the length of the pendulum as the resistance
has to the gravity, DK will be the exponent of the resistance. From the
centre C with the interval CA or CB describe a semi-circle BEeA. Let
the body describe, in the least time, the space Dd ; and, erecting the per
pendiculars DE, de, meeting the circumference in E and e, they will be as
the velocities which the body descending in vacuo from the point B would
acquire in the places D and d. This appears by Prop, LII, Book L Let
SEC. VLJ OF NATURAL PHILOSOPHY. 311
therefore, these velocities be expressed by those perpendiculars DE, de ;
arid let DF be the velocity which it acquires in D by falling from B in
the resisting medium. And if from the centre C with the interval OF we
describe the circle F/M meeting the right lines de and AB in / and M,
then M will be the place to which it would thenceforward, without farther
resistance, ascend, and (//"the velocity it would acquire in d. Whence,
also, if FO- represent the moment of the velocity which the body D, in de
scribing the least space DC/, loses by the resistance of the medium ; and
CN be taken equal to Cg ;
then will N be the place to which the body, if
it met no farther resistance, would thenceforward ascend, and MN will be
the decrement of the ascent arising from the loss of that velocity. Draw
F/n perpendicular to dft and the decrement Fg of the velocity DF gener
ated by the resistance DK will be to the incrementfm of the same velo
city, generated by the force CD, as the generating force DK to the gener
ating force CD. But because of the similar triangles F////, Fhg, FDC,
fm is to Fm or Dd as CD to DF ; and, ex ceqno, Fg to Dd as DK to
DF. Also Fh is to Fg as DF to CF ; and, ex ax/uo perturbate, Fh or
MN to Do1
as DK to CF or CM ; and therefore the sum of all the MN X
CM will be equal to the sum of all the Dd X DK. At the moveable
point M suppose always a rectangular ordinate erected equal to the inde
terminate CM, which by a continual motion is drawn into the whole
length Aa ; and the trapezium described by that motion, or its equal, the
rectangle Aa X |aB, will be equal to the sum of all the MN X CM, and
therefore to the sum of all the Dd X DK, that is, to the area BKVTa
O.E.D.
COR. Hence from the law of resistance, and the difference Aa of the
arcs Ca} CB, may be collected the proportion of the resistance to the grav
ity nearly.
For if the resistance DK be uniform, the figure BKTa will be a rec
tangle under Ba and DK; and thence the rectangle under ^Ba and Aa
will be equal to the rectangle under Ba and DK, and DK will be equal to
jAa. Wherefore since DK is the exponent of the resistance, and the
length of the pendulum the exponent of the gravity, the resistance will be
to the gravity as \Aa to the length of the pendulum ; altogether as in
Prop. XXVIII is demonstrated.
If the resistance be as the velocity, the figure BKTa will be nearly an
ellipsis. For if a body, in a non-resisting medium, by one entire oscilla
tion, should describe the length BA, the velocity in any place D would be
as the ordinate DE of the circle described on the diameter AB. There
fore since Ea in the resisting medium, and BA in the non-resisting one,
are described nearly in the same times ; and therefore the velocities in each
of the points of Ba are to the velocities in the correspondent points of the
length BA. nearly as Ba is to BA , the velocity in the point D in the re
312 THE MATHEMATICAL PRINCIPLES [BJOK 11.
sisting medium will be as the ordinate of the circle or ellipsis described
upon the diameter Ba ; and therefore the figure BKVTa will be nearly ac
ellipsis. Since the resistance is supposed proportional to the velocity, le\
OV be the exponent of the resistance in the middle point O ; and an ellip
sis BRVSa described with the centre O, and the semi-axes OB, OV, will
be nearly equal to the figure BKVTa, and to its equal the rectangle Act
X BO. Therefore Aa X BO is to OV X BO as the area of this ellipsis
to OV X BO; that is, Aa is to OV as the area of the semi-circle to the
square of the radius, or as 1 1 to 7 nearly ; and, therefore, T
7
TAa is to the
length of the pendulum as the resistance of the oscillating body in O to
its gravity.
Now if the resistance DK be in the duplicate ratio of the velocity, the
figure BKVTa will be almost a parabola having V for its vertex arid OV
for its axis, and therefore will be nearly equal to the rectangle under fBa
and OV. Therefore the rectangle under |Ba and Aa is equal to the rec
tangle fBa X OV, and therefore OV is equal to fAa ; and therefore the
resistance in O made to the oscillating body is to its gravity as fAa to the
length of the pendulum.
And I take these conclusions to be accurate enough for practical uses.
For since an ellipsis or parabola BRVSa falls in with the figure BKVTa
in the middle point V, that figure, if greater towards the part BRV or
VSa than the other, is less towards the contrary part, and is therefore
nearly equal to it.
PROPOSITION XXXI. THEOREM XXV.
If the resistance made to an oscillating body in each of the proportional
parts of the arcs described be augmented or diminished in, a given ra
tio, the difference between the arc described in the descent and the arc
described in the subsequent ascent ivill be augmented or diminished in
the same ratio.
For that difference arises from
the retardation of the pendulum
by the resistance of the medium,
and therefore is as the whole re
tardation and the retarding resist
ance proportional thereto. In the
foregoing Proposition the rectan-
M isr u c o .-/ n P gle under the right line ^aB and
the difference Aa of the arcs CB, Ca, was equal to the area BKTa, And
that area, if the length aB remains, is augmented or diminished in the ra
tio of the ordinates DK ; that is, in the ratio of the resistance and is there
fore as the length aB and the resistance conjunctly. And therefore the
rectangle under Aa and |aB is as aB and the resistance conjunctly, anc
therefore Aa is as the resistance. QJE.D.
SEC. VI.l OF NATURAL PHILOSOPHY. 313
COR. 1. Hence if the resistance be as the velocity, the difference of
the arts in the same medium will be as the whole arc described : and the
contrary.
COR. 2. If the resistance be in the duplicate ratio of the velocity, that
difference will be in the duplicate ratio of the whole arc : and the contrary.
COR. 3. And universally, if the resistance be in the triplicate or any
other ratio of the velocity, the difference will be in the same ratio of the.
whole arc : and the contrary.
COR. 4. If the resistance be partly in the simple ratio of the velocity,
and partly in the duplicate ratio of the same, the difference will be partly
in the ratio of the whole arc, and partly in the duplicate ratio of it: and
the contrary. So that the law arid ratio of the resistance will be the
same for the velocity as the law and ratio of that difference for the length
of the arc.
COR. 5. And therefore if a pendulum describe successively unequal arcs,
and we can find the ratio of the increment or decrement of this difference
for the length of the arc described, there will be had also the ratio of the
increment or decrement of the resistance for a greater or less velocity.
GENERAL SCHOLIUM.
From these propositions we may find the resistance of mediums by pen
dulums oscillating therein. I found the resistance of the air by the fol
lowing experiments. I suspended a wooden globe or ball weighing oT^
ounces troy, its diameter CJ London inches, by a fine thread on a firm
hook, so that the distance between the hook and the centre of oscillation of
the globe was 10| feet. I marked on the thread a point 10 feet and 1 inch
distant from the centre of suspension and even with that point I placed a
ruler divided into inches, by the help whereof I observed the lengths of the
arcs described by the pendulum. Then I numbered the oscillations ia
which the globe would lose
-{- part of its motion. If the pendulum was
drawn aside from the perpendicular to the distance of 2 inches, and thence
let go, so that in its whole descent it described an arc of 2 inches, and in
the first whole oscillation, compounded of the descent and subsequent
ascent, an arc of almost 4 inches, the same in 164 oscillations lost j part
of its motion, so as in its last ascent to describe an arc of If inches. If
in the first descent it described an arc of 4 inches, it lost j part of its mo
tion in 121 oscillations, so as in its last ascent to describe an arc of 3^
inches. If in the first descent it described an arc of 8, 16, 32, or 64 inches,
it lost | part of its motion in 69, 35|, 18|-7 9| oscillations, respectively.
Therefore the difference between the arcs described in the first descent and
the last ascent was in the 1st, 2d, 3d, 4th, 5th, 6th cases, }, 1. 1, 2, 4, 8
inches respectively. Divide those differences by the number of oscillations
in each case, and in one mean oscillation, wherein an arc of 3 , 7-|, 15, 30
314 THE MATHEMATICAL PRINCIPLES [BOOK Jl.
60, 120 inches was described, the difference of the arcs described in the
descent and subsequent ascent will be |^, ^{^ e\> T
4
r; -sji fir parts of an
inch, respectively. But these differences in the greater oscillations are in
the duplicate ratio of the arcs described nearly, but in lesser oscillations
something greater than in that ratio
; and therefore (by Cor. 2, Prop. XXXI
of this Book) the resistance of the globe, when it moves very swift, is in
the duplicate ratio of the velocity, nearly; and when it moves slowly,
somewhat greater than in that ratio.
Now let V represent the greatest velocity in any oscillation, and let A,
B, and C be given quantities, and let us suppose the difference of the arcs
3^
to be AV + BV2 + CV2
. Since the greatest velocities are in the cycloid
as ^ the arcs described in oscillating, and in the circle as | the chords of
those arcs
; and therefore in equal arcs are greater in the cycloid than in
the circle in the ratio of | the arcs to their chords ;
but the times in the
circle are greater than in the cycloid, in a reciprocal ratio of the velocity ;
it is plain that the differences of the arcs (which are as the resistance and
the square of the time conjunctly) are nearly the same in both curves : for
in the cycloid those differences must be on the one hand augmented, with
the resistance, in about the duplicate ratio of the arc to the chord, because
of the velocity augmented in the simple ratio of the same ; and on the
other hand diminished, with the square of the time, in the same duplicate
ratio. Therefore to reduce these observations to the cycloid, we must take
the same differences of the arcs as were observed in the circle, and suppose
the greatest velocities analogous to the half, or the whole arcs, that is, to
the numbers , 1, 2, 4, 8, 16. Therefore in the 2d, 4th, and 6th cases, put
1,4, and 1 6 for V ; and the difference of the arcs in the 2d case will become
i 2 * = A + B + C; in the4th case, ^- = 4A + SB + 160 ;
in the 6th
121 OOj
case, ^ = 16A + 64B -f- 256C. These equations reduced give A =
9?
0,000091 6, B =-. 0,0010847, and C = 0,0029558. Therefore the difference
of the arcs is as 0,0000916V -f 0,0010847V* + 0,0029558V* : and there
fore since (by Cor. Prop. XXX, applied to this case) the re.-ist;mcc of the
globe in the middle of the arc described in oscillating, where the velocity
is V, is to its weight as T
7
TAV -f- T\BV^ + fCV2 to the length of the
pendulum, if for A, B, and C you put the numbers found, the resistance of
the globe will be to its weight as 0,0000583V + 0,0007593V* + 0,OJ22169V 2
to the length of the pendulum between the centre of suspension and the
ruler, that is, to 121 inches. Therefore since V in the second case repre
sents 1, in the 4th case 4, and in the 6th case 16, the resistance will be to
the weight of the globe, in the 2d case, as 0,0030345 to 121 ;
in the 4th, as
0,041748 to 121 ; in the 6th, as 0,61705 to 121.
SEC. VI.] OF NATURAL PHILOSOPHY. 315
The arc, which the point marked in the thread described in the 6th case,
was of 120 Q^,
or 119/g inches. And therefore since the radius was
y a
121 inches, and the length of the pendulum between the point of suspen
sion and the centre of the globe was 126 inches, the arc which the centre of
the globe described was 124/T inches. Because the greatest velocity of the
oscillating body, by reason of the resistance of the air, does not fall on the
lowest point of the arc described, but near the middle place of the whole
arc, this velocity will be nearly the same as if the globe in its whole descent
in a non-resisting medium should describe 62^ inches, the half of that arc,
and that in a cycloid, to which we have above reduced the motion of the
pendulum; and therefore that velocity will be equal to that which the
globe would acquire by falling perpendicularly from a height equal to the
versed sine of that arc. But that versed sine in the cycloid is to that arc
62/2 as the same arc to twice the length of the pendulum 252, and there
fore equal to 15,278 inches. Therefore the velocity of the pendulum is the
same which a body would acquire by falling, and in its fall describing a
space of 15,278 inches. Therefore with such a velocity the globe meets
with a resistance which is to its weight as 0,61705 to 121, or (if we take
that part only of the resistance which is in the duplicate ratio of the veloc.
ty) as 0,56752 to 121.
I found, by an hydrostatical experiment, that the weight of this wooden
globe was to the weight of a globe of water of the same magnitude as 55
to 97: and therefore since 121 is to 213,4 in the same ratio, the resistance
made to this globe of water, moving forwards with the above-mentioned
velocity, will be to its weight as 0,56752 to 213,4, that is, as 1 to 376^.
Whence since the weight of a globe of water, in the time in which the
globe with a velocity uniformly continued describes a length of 30,556
inches, will generate all that velocity in the falling globe, it is manifest
that the force of resistance uniformly continued in the same time will take
away a velocity, which will be less than the other in the ratio of 1 to 376^- ,
that is, the rr^-r part of the whole velocity. And therefore in the time
37VSG
Jiat the globe, with the same velocity uniformly continued, would describe
the length of its semi-diameter, or 3 T\ inches, it would lose the 3^42 part
of its motion.
I also counted the oscillations in which the pendulum lost j part of its
motion. In the following table the upper numbers denote the length of the
arc described in the first descent, expressed in inches and parts of an inch ;
the middle numbers denote the length of the arc described in the last as
cent ; and in the lowest place are the numbers of the oscillations. I give
un account of this experiment, as being more accurate than that in which
316 THE MATHEMATICAL PRINCIPLES [BOOK ll
only
1
part of the motion was lost. I leave the calculation to such as are
disposed to make it.
First descent ... 2 4 8 16 32 64
Last ascent . . , 1| 3 6 12 24 48
NoscilL . .374 272 162i 83J 41f 22|
I afterward suspended a leaden globe of 2 inches in diameter, weighing
26 1 ounces troy by the same thread, so that between the centre of the
globe and the point of suspension there was an interval of 10^ feet, and 1
counted the oscillations in which a given part of the motion was lost. The
iirst of the following tables exhibits the number of oscillations in which Jpart
of the whole motion was lost ; the second the number of oscillations
in which there was lost \ part of the same.
First descent .... 1 2 4 8 16 32 64
Last ascent .... f J 3^ 7 14 28 56
Numb, of oscilL . . 226 228 193 140 90^ 53 30
First descent .... 1 2 4 8 16 32 64
Last ascent .... 1^ 3 6 12 24 4S
Nunib. of oscill. . .510 518^ 420 318 204 12170
Selecting in the first table the 3d, 5th, and 7th observations, and express
ing the greatest velocities in these observations particularly by the num
bers 1, 4, 16 respectively, and generally by the quantity V as above, there
will come out in ihe 3d observation ~- = A + B + C, in the 5th obser-
2 8
vation ^ = 4A 4- 8B + 16C. in the 7th observation ^-- == 16A 4- 64B t-
,t(j j oU
256C. These equations reduced give A = 0,001414, B == 0,000297, C
0,000879. And thence the resistance of the globe moving with the velocity
V will be to its weight 26^ ounces in the same ratio as 0,0009V +
0,000208V* + 0,000659V 2 to 121 inches, the length of the pendulum.
And if we regard that part only of the resistance which is in the dupli
cate ratio of the velocity, it will be to the weight of the globe as 0,000659V 2
to 121 inches. But this part of the resistance in the first experiment was
to the weight oi the wooden globe of 572-
7
2 ounces as 0,002217V 2 to 121 ;
and thence the resistance of the wooden globe is to the resistance of the
leaden one (their velocities being equal) as 57/2- into 0,002217 to 26 Jinto
0,000659, that is, as 7|- to 1. The diameters of the two globes were
6f and 2 inches, and the squares of these are to each other as 47 and 4,
or 11-J-f and 1, nearly. Therefore the resistances of these equally swift
globes were in less than a duplicate ratio of the diameters. But we have
not yet considered the resistance of the thread, which was certainly very
considerable, and ought to be subducted from the resistance of the pendu
lums here found. I could not determine this accurately, but I found il
SEC. VI.J OF NATURAL PHILOSOPHY. 3 1/
greater than a third part of the whole resistance of the lesser pendulum ;
and thence I gathered that the resistances of the globes, when the resist
ance of the thread is subducted, are nearly in the duplicate ratio of their
diameters. For the ratio of 7} } to 1 , or l(H to 1 is not very
different from the duplicate ratio of the diameters 1 L}f to I.
Since the resistance of the thread is of less moment in greater globes, I
tried the experiment also with a globe whose diameter was ISf inches.
The length of the pendulum between the point of suspension and the cen
tre of oscillation was 122| inches, and between the point of suspension and
the knot in the thread 109| inches. The arc described by the knot at the
first descent of the pendulum was 32 inches. The arc described by the
same knot in the last ascent after five oscillations was 2S inches. The
sum of the arcs, or the whole arc described in one mean oscillation, was 60
inches. The difference of the arcs 4 inches. The y
1
,,- part of this, or the
difference between the descent and ascent in one mean oscillation, is f of
an inch. Then as the radius 10(
J| to the radius 122^, so is the whole arc
of 60 inches described by the knot in one mean oscillation to the whole arc
of 67} inches described by the centre of the globe in one mean oscillation ;
and so is the difference | to a new difference 0,4475. If the length of the
arc described were to remain, and the length of the pendulum should be
augmented in the ratio of 126 to 122}, the time of the oscillation would
be augmented, and the velocity of the pendulum would be diminished in
the subduplicate of that ratio
; so that the difference 0,4475 of the arcs de
scribed in the descent and subsequent ascent would remain. Then if the
arc described be augmented in the ratio of 124 3
3
T to 67}, that difference
0.4475 would be augmented in the duplicate of that ratio, and so would
become 1,5295. These things would be so upon the supposition that the
resistance of the pendulum were in the duplicate ratio of the velocity.
Therefore if the pendulum describe the whole arc of 1243
3
T inches, and its
length between the point of suspension and the centre of oscillation be 126
inches, the difference of the arcs described in the descent and subsequent
ascent would be 1,5295 inches. And this difference multiplied into the
weight of the pendulous globe, which was 208 ounces, produces 318,136.
Again ;
in the pendulum above-mentioned, made of a wooden globe, when
its centre of oscillation, being 126 inches from the point of suspension, de
scribed the whole arc of 124 /T inches, the difference of the arcs described
in the descent and ascent was ^^ into ^. This multiplied into the
i/wi y^
weight of the globe, which was 57-2
7
2 ounces, produces 49,396. But I mul
tiply these differences into the weights of the globes, in order to find their
resistances. For the differences arise from the resistances, and are as the
resistances directly and the weights inversely. Therefore the resistances
are as the numbers 318,136 and 49,396. But that part of the resistance
31 S THE MATHEMATICAL PRINCIPLES [BOOK 1L
of the lesser globe, which is in the duplicate ratio of the velocity, was to
the whole resistance as 0,56752 to- 0,61675, that is, as 45,453 to 49,396 ;
whereas that part of the resistance of the greater globe is almost equal to
its whole resistance
; and so those parts are nearly as 318,136 and 45,453,
that is, as 7 and 1. But the diameters of the globes are 18f and 6| ; and
their squares 351 T
9 and 47 J are as 7,438 and 1, that is, as the resistances
of the globes 7 and 1, nearly. The difference of these ratios is scarce
greater than may arise from the resistance of the thread. Therefore those
parts of the resistances which are, when the globes are equal, as the squares
of the velocities, are also, when the velocities are equal, as the squares of
the diameters of the globes.
But the greatest of the globes I used in these experiments was not per
fectly spherical, and therefore in this calculation I have, for brevity s sake,
neglected some little niceties
; being not very solicitous for an accurate
calculus in an experiment that was not very accurate. So that I could
wish that these experiments were tried again with other globes, of a larger
size, more in number, and more accurately formed ; since the demonstra
tion of a vacuum depends thereon. If the globes be taken in a geometrical
proportion, as suppose whose diameters are 4, 8, 16, 32 inches; one may
collect from the progression observed in the experiments what would hap
pen if the globes were still larger.
In order to compare the resistances of different fluids with each other, 1
made the following trials. I procured a wooden vessel 4 feet long, 1 foot
broad, and 1 foot high. This vessel, being uncovered, 1 filled with spring
water, and, having immersed pendulums therein, I made them oscillate in
the water. And I found that a leaden globe weighing 166| ounces, and in
diameter 3f inches, moved therein as it is set down in the following table
;
the length of the pendulum from the point of suspension to a certain
point marked in the thread being 126 inches, and to the centre of oscilla
tion 134f inches.
The arc described in }
the first descent, by
a point marked in \ 64 . 32 . 16 . $ . 4 . 2 . 1 . . J
the thread was \
inches.
The arc described in )
the last ascent was V 48 . 24 . 12 . 6 . 3 . 1| . . f . T\
inches. \
The difference of the
arcs, proportional
