a lesser cubic space ace ; and the distances of the par- F
tides retaining a like situation with respect to each
other in both the spaces, will be as the sides AB, ab of
the cubes ; and the densities of the mediums will be re
ciprocally as the containing spaces AB 3
, ab 3
. In the
plane side of the greater cube ABCD take the square
DP equal to the plane side db of the lesser cube: and,
by the supposition, the pressure with which the square
DP urges the inclosed fluid will be to the pressure with
which that square db urges the inclosed fluid as the densities of the me
diums are to each other, that is, asa/>
3 to AB 3
. But the pressure with
which the square DB urges the included fluid is to the pressure with which
the square DP urges the same fluid as the square DB to the square DP,
that is, as AB2
to ab z
. Therefore, ex cequo, the pressure with which the
square DB urges the fluid is to the pressure with which the square db
urges the fluid as ab to AB. Let the planes FGH,/V?, U drawn through
the middles of the two cubes, and divide the fluid into tw^/ parts, These
parts will press each other mutually with the same forces with which they
A
THE MATHEMATICAL PRINCIPLES [BOOK II.
are themselves pressed by the planes AC, ac, that is, in the proportion of
ab to AB : arid therefore the centrifugal forces by which these pressures
are sustained are in the same ratio. The number of the particles being
equal, and the situation alike, in both cubes, the forces which all the par
ticles exert, according to the planes FGH,/o7/,, upon all, are as the forces
which each exerts on each. Therefore the forces which each exerts on
each, according to the plane FGH in the greater cube, are to the forces
which each exerts on each, according to the plane fgh in the lesser cube,
us ab to AB,*that is, reciprocally as the distances of the particles from each
other. Q.E.D.
And, vice versa, if the forces of the single particles are reciprocally as
the distances, that is, reciprocally as the sides of the cubes AB, ab ; the
sums of the forces will be in the same ratio, and the pressures of the sides
i)B, db as the sums of the forces ; and the pressure of the square DP to
the pressure of the side DB as ab 2 to AB 2
. And, ex cequo, the pressure of
the square DP to the pressure of the side db as ab* to AB 3
; that is, the
force of compression in the one to the force of compression in the other as
the density in the former to the density in the latter. Q.E.D.
SCHOLIUM.
By a like reasoning, if the centrifugal forces of the particles are recip
rocally in the duplicate ratio of the distances between the centres, the cubes
of the compressing forces will be as the biquadrates of the densities. If
the centrifugal forces be reciprocally in the triplicate or quadruplicate ratio
of the distances, the cubes of the compressing forces will be as the quadratocubes,
or cubo-cubes of the densities. And universally, if D be put for the
distance, and E for the density of the compressed fluid, and the centrifugal
forces be reciprocally as any power Dn of the distance, whose index is the
number ??, the compressing forces will be as the cube roots of the power
En + 2
. whose index is the number n + 2 ; and the contrary. All these
things are to be understood of particles whose centrifugal forces terminate
in those particles that are next them, or are diffused not much further.
We have an example of this in magnetical bodies. Their attractive vir
tue is terminated nearly in bodies of their own kind that are next them.
The virtue of the magnet is contracted by the interposition of an iron
plate, and is almost terminated at it : for bodies further off are not attracted
by the magnet so much as by the iron plate. If in this manner particles repel
others of their own kind that lie next them, but do not exert their virtue
on the more remote, particles of this kind will compose such fluids as are
treated of in this Proposition, If the virtue of any particle diffuse itself
every way in inftnitum, there will be required a greater force to produce
an equal condensation of a greater quantity of the flui 1. But whether
SEC. VI.] OF NATURAL PHILOSOPHY. 303
elastic fluids do really consist of particles so repelling each other, is a phy
sical question. We have here demonstrated mathematically the property
of fluids consisting of particles of this kind, that hence philosophers may
take occasion to discuss that question.
SECTION VI.
Of the motion and resistance offunependulous bodies.
PROPOSITION XXIV. THEOREM XIX.
The quantities of matter i/i funependulous bodies, whose centres of oscil
lation are equally distant from, the centre of suspension, are in a, ratio
compounded of the ratio of the weights and the duplicate ratio of the
times of the oscillations in vacuo.
For the velocity which a given force can generate in a given matter in
a given time is as the force and the time directly, and the matter inversely.
The greater the force or the time is, or the less the matter, the greater ve
locity will he generated. This is manifest from the second Law of Mo
tion. Now if pendulums are of the same length, the motive forces in places
equally distant from the perpendicular are as the weights : and therefore
if two bodies by oscillating describe equal arcs, and those arcs are divided
into equal parts ; since the times in which the bodies describe each of the
correspondent parts of the arcs are as the times of the whole oscillations,
the velocities in the correspondent parts of the oscillations will be to each
other as the motive forces and the whole times of the oscillations directly,
and the quantities of matter reciprocally : and therefore the quantities of
matter are as the forces and the times of the oscillations directly and the
velocities reciprocally. But the velocities reciprocally are as the times,
and therefore the times directly and the velocities reciprocally are as the
squares of the times; and therefore the quantities of matter are as the mo
tive forces and the squares of the times, that is, as the weights and the
squares of the times. Q.E.D.
COR. 1. Therefore if the times are equal, the quantities of matter in
each of the bodies are as the weights.
COR. 2. If the weights are equal, the quantities of matter will be as the
pquarcs of the times.
COR. 3. If the quantities of matter are equal, the weights will be recip
rocally as the squares of the times.
COR. 4. Whence since the squares of the times, cceteris paribus, are as
the length* of the pendulums, therefore if both the times and quantities of
matter are equal, the weights will be as the lengths of the pendulums.
J04 THE MATHEMATICAL PRINCIPLES [BOOK 11
COR. 5. And universally, the quantity of matter in the pendulous body
is as the weight and the square of the time directly, and the length of the
pendulum inversely.
COR. 6. But in a non-resisting medium, the quantity of matter in the
pendulous body is as the comparative weight and the square of the time
directly, and the length of the pendulum inversely. For the comparative
weight is the motive force of the body in any heavy medium, as was shewn
above ; and therefore does the same thing in such a non-resisting medium
as the absolute weight does in a vacuum.
COR. 7. And hence appears a method both of comparing bodies one
among another, as to the quantity of matter in each ; and of comparing
the weights of the same body in different places, to know the variation of
its gravity. And by experiments made with the greatest accuracy, I
have always found the quantity of matter in bodies to be proportional to
their weight.
PROPOSITION XXV. THEOREM XX.
Funependulous bodies that are, in, any medium, resisted in the ratio oj
the moments of time, andfunepetidulons bodies that move in a nonresisting
medium of the same specific gravity, perform their oscilla
tions in. a cycloid in the same time, and describe proportional parts oj
arcs together.
Let AB be an arc of a cycloid, which
a body D, by vibrating in a non-re
sisting medium, shall describe in any
time. Bisect that arc in C, so that C
may be the lowest point thereof ; and
the accelerative force with which the
body is urged in any place D, or d or
E, will be as the length of the arc CD,
pressed by that same arc ; and since the resistance is as the moment of the
time, and therefore given, let it ba expressed by the given part CO of the
cycloidal arc, and take the arc Od in the same ratio to the arc CD that
the arc OB has to the arc CB : and the force with which the body in d is
urged in a resisting medium, being the excess of the force Cd above the
resistance CO, will be expressed by the arc Od, and will therefore be to
the force with which the body D is urged in a non-resisting medium in the
place D, as the arc Od to the arc CD ; and therefore also in the place B,
as the arc OB to the arc CB. Therefore if two bodies D, d go from the place
B, and are urged by these forces ; since the forces at the beginning are as
the arc CB and OB, the first velocities and arcs first described will be in
the same ratio. Let those arcs be BD and Ed, and the remaining arcf
SEC. VI. |
OF NATURAL PHILOSOPHY. 305
CD, Odj will be in the same ratio. Therefore the forces, being propor
tional to those arcs CD, Od, will remain in the same ratio as at the be
ginning, and therefore the bodies will continue describing together arcs in
the same ratio. Therefore the forces and velocities and the remaining arcs
CD. Od, will be always as the whole arcs CB, OB, and therefore those re
maining arcs wLl be described together. Therefore the two bodies D and
d will arrive together at the places C and O ; that whicli moves in the
non-resisting medium, at the place C, and the other, in the resisting me
dium, at the place O. Now since the velocities in C and O are as the arcs
CB, OB, the arcs which the bodies describe when they go farther will be
in the same ratio. Let those arcs be CE and Oe. The force with which
the body D in a non-resisting medium is retarded in E is as CE, and the
force with which the body d in the resisting medium is retarded in e, is as
the sum of the force Ce and the resistance CO, that is, as Oe ; and there
fore the forces with which the bodies are retarded are as the arcs CB, OB,
proportional to the arcs CE, Oe ; and therefore the velocities, retarded in
that given ratio, remain in the same given ratio. Therefore the velocities
and the arcs described with those velocities are always to each other in
that given ratio of the arcs CB and OB ; and therefore if the entire arcs
AB, aB are taken in the same ratio, the bodies D andc/ will describe those
aics together, and in the places A and a will lose all their motion together.
Therefore the whole oscillations are isochronal, or are performed in equal
times ; and any parts of the arcs, as BD, Ed, or BE, Be, that are described
together, are proportional to the whole arcs BA, B. Q,.E.D.
COR. Therefore the swiftest motion in a resisting medium does not fall
upon the lowest point C, but is found in that point O, in which the whole
arc described Ba is bisected. And the body, proceeding from thence to a,
is retarded at the same rate with which it was accelerated before in its de
scent from B to O.
PROPOSITION XXVI. THEOREM XXI.
Funependulous bodies, that are resisted in the ratio of the velocity, have
their oscillations in a cycloid isochronal.
For if two bodies, equally distant from their centres of suspension, de
scribe, in oscillating, unequal arcs, and the velocities in the correspondent
parts of the arcs be to each other as the whole arcs ; the resistances, pro
portional to the velocities, will be also to each other as the same arcs.
Therefore if these resistances be subducted from or added to the motive
forces arising from gravity which are as the same arcs, the differences or
sums will be to each other in the same ratio of the arcs ; and since the in
crements and decrements of the velocities are as these differences or sums,
the velocities will be always as the whole arcs; therefore if the velocities
are in any one case as the whole arcs, they will remain always in the same
20
306 THE MATHEMATICAL PRINCIPLES [BOOK. 11
ratio. But at the beginning of the motion, when the bodies begin to de
scend and describe those arcs, the forces, which at that time are proportional
to the arcs, will generate velocities proportional to the arcs. Therefore
the velocities will be always as the whole arcs to be described, and there
fore those arcs will be described in the same time. Q,.E.D.
PROPOSITION XXVII. THEOREM XXII.
If fnnependulous bodies are resisted in the duplicate ratio of their
velocities, the differences between the times of the oscillations in a re
sisting medium, and the times of the oscillations in a non-resisting
medium of the same specific gravity, will be proportional to the arcs
described in oscillating nearly.
For let equal pendulums in a re
sisting medium describe the unequal
arcs A, B ; and the resistance of the
body in the arc A will be to the resist
ance of the body in the correspondent
part of the arc B in the duplicate ra
tio of the velocities, that is, as, AA to
BB nearly. If the resistance in the
arc B were to the resistance in the arc
A as AB to AA, the times in the arcs A and B would be equal (by the last
Prop.) Therefore the resistance AA in the arc A, or AB in the arc B,
causes the excess of the time in the arc A above the time in a non-resisting
medium ; and the resistance BB causes the excess of the time in the arc B
above the time in a non-resisting medium. But those excesses are as the
efficient forces AB and BB nearly, that is, as the arcs A and B. Q.E.D.
COR, 1. Hence from the times of the oscillations in unequal arcs in a
resisting medium, may be knowrn the times of the oscillations in a non- re
sisting medium of the same specific gravity. For the difference of the
times will be to the excess of the time in the lesser arc above the time in a
non-resisting medium as the difference of the arcs to the lesser arc.
COR. 2. The shorter oscillations are more isochronal, and very short
ones are performed nearly in the same times as in a non-resisting medium.
But the times of those which are performed in greater arcs are a little
greater, because the resistance in the descent of the body, by which the
time is prolonged, is greater, in proportion to the length described in the
descent than the resistance in the subsequent ascent, by which the time is
contracted. But the time of the oscillations, both short arid long, seems to
be prolonged in some measure by the motion of the medium. For retard
ed bodies are resisted somewhat less in proportion to the velocity, and ac
celerated bodies somewhat more than those that proceed uniformly forwards ;
SEC. VI.] OF NATURAL PHILOSOPHY. 307
because the medium, by the motion it has received from the bodies, going
forwards the same way with them, is more agitated in the former case, and
less in the latter
; and so conspires more or less with the bodies moved.
Therefore it resists the pendulums in their descent more, and in their as
cent less, than in proportion to the velocity; and these two causes concur
ring prolong the time.
PROPOSITION XXVIII. THEOREM XXIII.
If afunependvlous body, oscillating in a cycloid, be resisted in the rati >
of the moments of the time, its resistance will be to the force of grav
ity as the excess of the arc described in the whole descent above the
arc described in the subsequent ascent to twice the length of the pen
dulum.
Let BC represent the arc described
in the descent, Ca the arc described in
the ascent, and Aa the difference of
the arcs : and things remaining as they
were constructed and demonstrated in
Prop. XXV, the force with which the
oscillating body is urged in any place
D will be to the force of resistance as
the arc CD to the arc CO, which is
half of that difference Aa. Therefore the force with which the oscillating
body is urged at the beginning or the highest point of the cycloid, that is,
the force of gravity, will be to the resistance as the arc of the cycloid, be
tween that highest point and lowest point C, is to the arc CO ;
that is
(doubling those arcs), as the whole cycloidal arc, or twice the length of the
pendulum, to the arc Aa. Q.E.D.
PROPOSITION XXIX. PROBLEM VI.
Supposing that a body oscillating in a cycloid is resisted in a duplicate
ratio of the velocity: to find the resistance in each place.
Let Ba be an arc described in one entire oscillation, C the lowest point
C O
K
O ,S P rR Q M
of the cycloid, and CZ half the whole cycloidal arc, equal to the length of
the pendulum ; and let it be required to find the resistance of the body is
30S THE MATHEMATICAL PRINCIPLES [BOOK 1L
any place D. Cut the indefinite right line OQ in the points O, S, P, Q,,
so that (erecting the perpendiculars OK, ST, PI, QE, and with the centre
O, and the aysmptotcs OK, OQ, describing the hyperbola TIGE cutting
the perpendiculars ST, PI, QE in T. I, and E, and through the point I
drawing KF. parallel to the asymptote OQ, meeting the asymptote OK i i
K, and the perpendiculars ST and QE in L and F) the hyperbolic area
PIEQ may be to the hyperbolic area PITS as the arc BC, described in the
descent of the body, to the arc Ca described in the ascent
; and that the
area IEF may be to the area ILT as OQ to OS. Then with the perpen
dicular MN cut off the hyperbolic area PINM, and let that area be to the
hyperbolic area PIEQ as the arc CZ to the arc BC described in the de
scent. And if the perpendicular RG cut off the hyperbolic area PIGR,
which shall be to the area PIEQ as any arc CD to the arc BC described
in the whole descent, the resistance in any place D will be to the force of
OR
gravity as the area IEF IGH to the area PINM.
For since the forces arising from gravity with which the body is
urged in the places Z, B, D, a, are as the arcs CZ. CB, CD, Ca and those
arcs are as the areas PINM, PIEQ, PIGR, PITS; let those areas be the
exponents both of the arcs and of the forces respectively. Let DC? be a
very small space described by the body in its descent : and let it be expressed
r
>y
the very small area RGor comprehended between the parallels RG, rg ;
and produce r<? to //, so that GYlhg- and RGr may be the contemporane
ous decrements of the areas IGH, PIGR. And the increment Gllhg
IEF, or Rr X HG -^ IEF, of the area ~IEF IGH will be , OQ OQ
IFF
to the decrement RGr, or Rr X RG, of the area PIGR, as HG - -
OR
to RG ; and therefore as OR X HG IEF to OR X GR or OP X
PL that is (because of the equal quantities OR X HG, OR X HR OR
X GR, ORHK OPIK, PIHR and PIGR + IGH), as PIGR + IGH
OR OR
IEF to OPIK. Therefore if the area - IEF IGH be called OQ
Y, and RGgr the decrement of the area PIGR be given, the increment of
the area Y will be as PIGR Y.
Then if V represent the force arising from the gravity, proportional to
the arc CD to be described, by which the body is acted upon in D, and R
be put for the resistance, V R will be the whole force with which the
body is urged in D. Therefore the increment of the velocity is as V R
and the particle of time in which it is generated conjunctly. But the ve
locity itself is as the contempo] aueous increment of the space described di
SEC. VI.J OF NATURAL PHILOSOPHY. 309
rectly and the same particle of time inversely. Therefore, since the re
sistance is, by the supposition, as the square of the velocity, the increment
of the resistance will (by Lem. II) be as the velocity and the increment of
the velocity conjunctly, that is, as the moment of the space and V R
conjunctly ; and, therefore, if the moment of the space be given, as V
11
; that is, if for the force V we put its exponent PIGR, and the resist
ance R be expressed by any other area Z; as PIGR Z. v
Therefore the area PIGR uniformly decreasing by the subduction of
given moments, the area Y increases in proportion of PIGR Y, and
the area Z in proportion of PIGR Z. And therefore if the areas
Y and Z begin together, and at the beginning are equal, these, by the
addition of equal moments, will continue to be equal and in like man
ner decreasing by equal moments, \vill vanish together. And, vice versa,
if they together begin and vanish, they will have equal moments and te
always equal ; and that, because if the resistance Z be augmented, the ve
locity together with the arc C, described in the ascent of the body, will be
diminished ; and the point in which all the motion together with the re
sistance ceases coming nearer to the point C, the resistance vanishes sooner
than the area Y. And the contrary will happen when the resistance is
diminished.
Now the area Z begins and ends where the resistance is nothing, that is,
at the beginning of the motion where the arc CD is equal to the arc CB,
K /IK
O S P /~R Q M
and the right line RG falls upon the right line Q.E ; and at the end of
the motion where the arc CD is equal to the arc Ca, and RG falls upon
the right line ST. And the area* Y or IEF IGH begins and ends
also where the resistance is nothing, and therefore where IEF and
IGH are equal ; that is (by the construction), where the right line RG
falls successively upon the right lines Q,E and ST. Therefore those areas
begin and vanish together, and are therefore always equal. Therefore the area
OR
IEF IGH is equal to the area Z, by which the resistance is ex
pressed, and therefore is to the area PINM, by which the gravity is ex
pressed, as the resistance to the gravity. Q.E.D.
310 THE MATHEMATICAL PRINCIPLES [BOOK 11.
COR. 1 . Therefore the resistance in the lowest place C is to the force
OP
of gravity as the area ^ ~ IEF to the area PINM.
COR. 2. But it becomes greatest where the area PIHR is to the area
IEF as OR to OQ. For in that case its moment (that is, PIGR Y)
becomes nothing.
COR. 3. Hence also may be known the velocity in each place, as being
in the subduplicate ratio of the resistance, and at the beginning of the mo
tion equal to the velocity of the body oscillating in the same cycloid with
out any resistance.
However, by reason of the difficulty of the calculation by which the re
sistance and the velocity are found by this Proposition, we have thought
fit to subjoin the Proposition following.
PROPOSITION XXX. THEOREM XXIV.
If a right line aB be equal to the arc of a cycloid which an oscillating
body describes, and at each of its points D the perpendiculars DK be
erected, which shall be to the length of the pendulum as the resistance
of the body in the corresponding points of the arc to the force of grav
ity ; I say, that the difference between the arc described in the whole
descent and the arc described in the whole subsequent ascent drawn
into half the sum of the same arcs will be equal to the area BKa
which all those perpendiculars take up.
Let the arc of the cycloid, de
scribed in one entire oscillation, be
expressed by the right line aB,
equal to it, and the arc which
would have been described in vaciw
by the length AB. Bisect AB in
C, and the point C will represent
the lowest point of the cycloid, and
CD Mill be as the force arising from gravity, with which the body in D i,s
