A is fixed to the pulley B, and the cord, by which the weight is
upheld, is annexed by one extremity to a hook C, and at the other end
is held by the power D. Here the weight is supported by a cord doubled;
insomuch that although the cord were not strong enough to hold the
weight single, yet being thus doubled it might support it. If the
end of the cord held by the power D were hung on the hook C, as well
as the other end; then, when both ends of the cord were tied to the
hook, it is evident, that the hook would bear the whole weight; and
each end of the string would bear against the hook with the force of
half the weight only, seeing both ends together bear with the force of
the whole. Hence it is evident, that, when the power D holds one end
of the weight, the force, which it must exert to support the weight,
must be equal to just half the weight. And the same proportion between
the weight and power might be collected from comparing the respective
velocities, with which they would move; for it is evident, that the
power must move through a space equal to twice the distance of the
pulley from the hook, in order to lift the pulley up to the hook.
49. IT is equally easy to estimate the effect, when many pulleys are
combined together, as in fig. 39, 40; in the first of which the under
set of pulleys, and consequently the weight is held by six strings; and
in the latter figure by five: therefore in the first of these figures
the power to support the weight, must be one sixth part only of the
weight, and in the latter figure the power must be one fifth part.
50. THERE are two other ways of supporting a weight by pulleys, which I
shall particularly consider.
51. ONE of these ways is represented in fig. 41. Here the weight being
connected to the pulley B, a power equal to half the weight A would
support the pulley C, if applied immediately to it. Therefore the
pulley C is drawn down with a force equal to half the weight A. But if
the pulley D were to be immediately supported by half the force, with
which the pulley C is drawn down, this pulley D will uphold the pulley
C; so that if the pulley D be upheld with a force equal to one fourth
part of the weight A, that force will support the weight. But, for the
same reason as before, if the power in E be equal to half the force
necessary to uphold the pulley D; this pulley, and consequently the
weight A, will be upheld: therefore, if the power in E be one eighth
part of the weight A, it will support the weight.
52. ANOTHER way of applying pulleys to a weight is represented in fig.
42. To explain the effect of pulleys thus applied, it will be proper to
consider different weights hanging, as in fig. 43. Here, if the power
and weights balance each other, the power A is equal to the weight B;
the weight C is equal to twice the power A, or the weight B; and for
the same reason the weight D is equal to twice the weight C, or equal
to four times the power A. It is evident therefore, that all the three
weights B, C, D together are equal to seven times the power A. But if
these three weights were joined in one, they would produce the case of
fig. 40: so that in that figure the weight A, where there are three
pulleys, is seven times the power B. If there had been but two pulleys,
the weight would have been three times the power; and if there had been
four pulleys, the weight would have been fifteen times the power.
53. THE wedge is next to be considered. The form of this instrument is
sufficiently known. When it is put under any weight (as in fig. 44.)
the force, with which the wedge will lift the weight, when drove under
it by a blow upon the end A B, will bear the same proportion to the
force, wherewith the blow would act on the weight, if directly applied
to it; as the velocity, which the wedge receives from the blow, bears
to the velocity, wherewith the weight is lifted by the wedge.
54. THE screw is the fifth mechanical power. There are two ways of
applying this instrument. Sometimes it is screwed into a hole, as
in fig. 45, where the screw A B is screwed through the plank C D.
Sometimes the screw is applied to the teeth of a wheel, as in fig. 46,
where the thread of the screw A B turns in the teeth of a wheel C D. In
both these cases, if a bar, as A E, be fixed to the end A of the screw;
the force, wherewith the end B of the screw in fig. 45 is forced down,
and the force, wherewith the teeth of the wheel C D in fig. 44 are
held, bears the same proportion to the power applied to the end E of
the bar; as the velocity, wherewith the end E will move, when the screw
is turned, bears to the velocity, wherewith the end B of the screw in
fig. 43, or the teeth of the wheel C D in fig. 46, will be moved.
55. THE inclined plane affords also a means of raising a weight with
less force, than what is equal to the weight it self. Suppose it were
required to raise the globe A (in fig. 47.) from the ground B C up
to the point, whose perpendicular height from the ground is E D. If
this globe be drawn along the slant D F, less force will be required
to raise it, than if it were lifted directly up. Here if the force
applied to the globe bear the same proportion only to its weight, as
E D bears to F D, it will be sufficient to hold up the globe; and
therefore any addition to that force will put it in motion, and draw
it up; unless the globe, by pressing against the plane, whereon it
lies, adhere in some degree to the plane. This indeed it must always
do more or less, since no plane can be made so absolutely smooth as
to have no inequalities at all; nor yet so infinitely hard, as not to
yield in the least to the pressure of the weight. Therefore the globe
cannot be laid on such a plane, whereon it will slide with perfect
freedom, but they must in some measure rub against each other; and this
friction will make it necessary to imploy a certain degree of force
more, than what is necessary to support the globe, in order to give
it any motion. But as all the mechanical powers are subject in some
degree or other to the like impediment from friction; I shall here
only shew what force would be necessary to sustain the globe, if it
could lie upon the plane without causing any friction at all. And I
say, that if the globe were drawn by the cord G H, lying parallel to
the plane D F; and the force, wherewith the cord is pulled, bear the
same proportion to the weight of the globe, as E D bears to D F; this
force will sustain the globe. In order to the making proof of this, let
the cord G H be continued on, and turned over the pulley I, and let
the weight K be hung to it. Now I say, if this weight bears the same
proportion to the globe A, as D E bears to D F, the weight will support
the globe. I think it is very manifest, that the center of the globe A
will lie in one continued line with the cord H G. Let L be the center
of the globe, and M the center of gravity of the weight K. In the first
place let the weight hang so, that a line drawn from L to M shall lie
horizontally; and I say, if the globe be moved either up or down the
plane D F, the weight will so move along with it, that the center of
gravity common to both the weights shall continue in this line L M, and
therefore shall in no case descend. To prove this more fully, I shall
depart a little from the method of this treatise, and make use of a
mathematical proportion or two: but they are such, as any person, who
has read ~EUCLID’S ELEMENTS~, will fully comprehend; and are
in themselves so evident, that, I believe, my readers, who are wholly
strangers to geometrical writings, will make no difficulty of admitting
them. This being premised, let the globe be moved up, till its center
be at G, then will M the center of gravity of the weight K be sunk to
N; so that M N shall be equal to G L. Draw N G crossing the line M L
in O; then I say, that O is the common center of gravity of the two
weights in this their new situation. Let G P be drawn perpendicular to
M L; then G L will bear the same proportion to G P, as D F bears to D
E; and M N being equal to G L, M N will bear the same proportion to G
P, as D F bears to D E. But N O bears the same proportion to O G, as
M N bears to G P; consequently N O will bear the same proportion to
O G, as D F bears to D E. In the last place, the weight of the globe
A bears the same proportion to the other weight K, as D F bears to D
E; therefore N O bears the same proportion to O G, as the weight of
the globe A bears to the weight K. Whence it follows, that, when the
center of the globe A is in G, and the center of gravity of the weight
K is in N, O will be the center of gravity common to both the weights.
After the same manner, if the globe had been caused to descend, the
common center of gravity would have been found in this line M L. Since
therefore no motion of the globe either way will make the common center
of gravity descend, it is manifest, from what has been said above, that
the weights A and K counterpoize each other.
56. I SHALL now consider the case of pendulums. A pendulum is made
by hanging a weight to a line, so that it may swing backwards and
forwards. This motion the geometers have very carefully considered,
because it is the most commodious instrument of any for the exact
measurement of time.
57. I HAVE observed already[61], that if a body hanging perpendicularly
by a string, as the body A (in fig. 48.) hangs by the string A B, be
put so into motion, as to be made to ascend up the circular arch A C;
then as soon as it has arrived at the highest point, to which the
motion, that the body has received, will carry it; it will immediately
begin to descend, and at A will receive again as great a degree of
motion, as it had at first. This motion therefore will carry the
body up the arch A D, as high as it ascended before in the arch A C.
Consequently in its return through the arch D A it will acquire again
at A its original velocity, and advance a second time up the arch
A C as high as at first; by this means continuing without end its
reciprocal motion. It is true indeed, that in fact every pendulum,
which we can put in motion, will gradually lessen its swing, and
at length stop, unless there be some power constantly applied to
it, whereby its motion shall be renewed; but this arises from the
resistance, which the body meets with both from the air, and the string
by which it is hung: for as the air will give some obstruction to the
progress of the body moving through it; so also the string, whereon the
body hangs, will be a farther impediment; for this string must either
slide on the pin, whereon it hangs, or it must bend to the motion of
the weight; in the first there must be some degree of friction, and
in the latter the string will make some resistance to its inflection.
However, if all resistance could be removed, the motion of a pendulum
would be perpetual.
58. BUT to proceed, the first property, I shall take notice of in this
motion, is, that the greater arch the pendulous body moves through, the
greater time it takes up: though the length of time does not increase
in so great a proportion as the arch. Thus if C D be a greater arch,
and E F a lesser, where C A is equal to A D, and E A equal to A F;
the body, when it swings through the greater arch C D, shall take up
in its swing from C to D a longer time than in swinging from E to F,
when it moves only in that lesser arch; or the time in which the body
let fall from C will descend through the arch C A is greater than the
time, in which it will descend through the arch E A, when let fall from
E. But the first of these times will not hold the same proportion to
the latter, as the first arch C A bears to the other arch E A; which
will appear thus. Let C G and E H be two horizontal lines. It has been
remarked above[62], that the body in falling through the arch C A will
acquire as great a velocity at the point A, as it would have gained by
falling directly down through G A; and in falling through the arch E A
it will acquire in the point A only that velocity, which it would have
got in falling through H A. Therefore, when the body descends through
the greater arch C A, it shall gain a greater velocity, than when it
passes only through the lesser; so that this greater velocity will in
some degree compensate the greater length of the arch.
59. THE increase of velocity, which the body acquires in falling from
a greater height, has such an effect, that, if straight lines be drawn
from A to C and E, the body would fall through the longer straight
line C A just in the same time, as through the shorter straight line
E A. This is demonstrated by the geometers, who prove, that if any
circle, as A B C D (fig. 49.) be placed in a perpendicular situation;
a body shall fall obliquely through every line, as A B drawn from the
lowest point A in the circle to any other point in the circumference
just in the same time, as would be imployed by the body in falling
perpendicularly down through the diameter C A. But the time in which
the body will descend through the arch, is different from the time,
which it would take up in falling through the line A B.
60. IT has been thought by some, that because in very small arches this
correspondent straight line differs but little from the arch itself;
therefore the descent through this straight line would be performed
in such small arches nearly in the same time as through the arches
themselves: so that if a pendulum were to swing in small arches,
half the time of a single swing would be nearly equal to the time,
in which a body would fall perpendicularly through twice the length
of the pendulum. That is, the whole time of the swing, according to
this opinion, will be four fold the time required for the body to fall
through half the length of the pendulum; because the time of the body’s
falling down twice the length of the pendulum is half the time required
for the fall through one quarter of this space, that is through half
the pendulum’s length. However there is here a mistake; for the whole
time of the swing, when the pendulum moves through small arches, bears
to the time required for a body to fall down through half the length of
the pendulum very nearly the same proportion, as the circumference of a
circle bears to its diameter; that is very nearly the proportion of 355
to 113, or little more than the proportion of 3 to 1. If the pendulum
takes so great a swing, as to pass over an arch equal to one sixth part
of the whole circumference of the circle, it will swing 115 times,
while it ought according to this proportion to have swung 117 times; so
that, when it swings in so large an arch, it loses something less than
two swings in an hundred. If it swing through 1/10 only of the circle,
it shall not lose above one vibration in 160. If it swing in 1/20 of
the circle, it shall lose about one vibration in 690. If its swing be
confined to 1/40 of the whole circle, it shall lose very little more
than one swing in 2600. And if it take no greater a swing than through
1/60 of the whole circle, it shall not lose one swing in 5800.
61. NOW it follows from hence, that, when pendulums swing in small
arches, there is very nearly a constant proportion observed between
the time of their swing, and the time, in which a body would fall
perpendicularly down through half their length. And we have declared
above, that the spaces, through which bodies fall, are in a two fold
proportion of the times, which they take up in falling[63]. Therefore
in pendulums of different lengths, swinging through small arches, the
lengths of the pendulums are in a two fold or duplicate proportion of
the times, they take in swinging; so that a pendulum of four times the
length of another shall take up twice the time in each swing, one of
nine times the length will make one swing only for three swings of the
shorter, and so on.
62. THIS proportion in the swings of different pendulums not only holds
in small arches; but in large ones also, provided they be such, as the
geometers call similar; that is, if the arches bear the same proportion
to the whole circumferences of their respective circles. Suppose (in
fig. 48.) A B, C D to be two pendulums. Let the arch E F be described
by the motion of the pendulum A B, and the arch G H be described by
the pendulum C D; and let the arch E F bear the same proportion to the
whole circumference, which would be formed by turning the pendulum A
B quite round about the point A, as the arch G H bears to the whole
circumference, that would be formed by turning the pendulum C D quite
round the point C. Then I say, the proportion, which the length of the
pendulum A B bears to the length of the pendulum C D, will be two fold
of the proportion, which the time taken up in the description of the
arch E F bears to the time employed in the description of the arch G H.
63. THUS pendulums, which swing in very small arches, are nearly an
equal measure of time. But as they are not such an equal measure to
geometrical exactness; the mathematicians have found out a method of
causing a pendulum so to swing, that, if its motion were not obstructed
by any resistance, it would always perform each swing in the same time,
whether it moved through a greater, or a lesser space. This was first
discovered by the great ~HUYGENS~, and is as follows. Upon the
straight line A B (in fig. 49.) let the circle C D E be so placed, as
to touch the straight line in the point C. Then let this circle roll
along upon the straight line A B, as a coach-wheel rolls along upon
the ground. It is evident, that, as soon as ever the circle begins to
move, the point C in the circle will be lifted off from the straight
line A B; and in the motion of the circle will describe a crooked
course, which is represented by the line C F G H. Here the part C H of
the straight line included between the two extremities C and H of the
line C F G H will be equal to the whole circumference of the circle C D
E; and if C H be divided into two equal parts at the point I, and the
straight line I K be drawn perpendicular to C H, this line I K will
be equal to the diameter of the circle C D E. Now in this line if a
body were to be let fall from the point H, and were to be carried by
its weight down the line H G K, as far as the point K, which is the
lowest point of the line C F G H; and if from any other point G a body
were to be let fall in the same manner; this body, which falls from
G, will take just the same time in coming to K, as the body takes up,
which falls from H. Therefore if a pendulum can be so hung, that the
ball shall move in the line A G F E, all its swings, whether long or
short, will be performed in the same time; for the time, in which the
ball will descend to the point K, is always half the time of the whole
swing. But the ball of a pendulum will be made to swing in this line by
the following means. Let K I (in fig. 52.) be prolonged upwards to L,
till I L is equal to I K. Then let the line L M H equal and like to K
H be applied, as in the figure between the points L and H, so that the
point which in this line L M H answers to the point H in the line K H
shall be applied to the point L, and the point answering to the point
K shall be applied to the point H. Also let such another line L N C be
applied between L and C in the same manner. This preparation being
made; if a pendulum be hung at the point L of such a length, that the
ball thereof shall reach to K; and if the string shall continually bend
against the lines H M L and L N C, as the pendulum swings to and fro;
by this means the ball shall constantly keep in the line C K H.
64. NOW in this pendulum, as all the swings, whether long or short,
will be performed in the same time; so the time of each will exactly
bear the same proportion to the time required for a body to fall
perpendicularly down, through half the length of the pendulum, that is
from I to K, as the circumference of a circle bears to its diameter.
65. IT may from hence be understood in some measure, why, when
pendulums swing in circular arches, the times of their swings are
nearly equal, if the arches are small, though those arches be of very
unequal lengths; for if with the semidiameter L K the circular arch O
K P be described, this arch in the lower part of it will differ very
little from the line C K H.
66. IT may not be amiss here to remark, that a body will fall in this
line C K H (fig. 53.) from C to any other point, as Q or R in a shorter
space of time, than if it moved through the straight line drawn from
C to the other point; or through any other line whatever, that can be
drawn between these two points.
67. BUT as I have observed, that the time, which a pendulum takes in
swinging, depends upon its length; I shall now say something concerning
the way, in which this length of the pendulum is to be estimated. If
the whole ball of the pendulum could be crouded into one point, this
length, by which the motion of the pendulum is to be computed, would
be the length of the string or rod. But the ball of the pendulum must
have a sensible magnitude, and the several parts of this ball will not
move with the same degree of swiftness; for those parts, which are
farthest from the point, whereon the pendulum is suspended, must move
with the greatest velocity. Therefore to know the time in which the
pendulum swings, it is necessary to find that point of the ball, which
moves with the same degree of velocity, as if the whole ball were to be
contracted into that point.
68. THIS point is not the center of gravity, as I shall now endeavour
to shew. Suppose the pendulum A B (in fig. 54.) composed of an
inflexible rod A C and ball C B, to be fixed on the point A, and lifted
up into an horizontal situation. Here if the rod were not fixed to the
point A, the body C B would descend directly with the whole force of
its weight; and each part of the body would move down with the same
degree of swiftness. But when the rod is fixed at the point A, the
body must fall after another manner; for the parts of the body must
move with different degrees of velocity, the parts more remote from A
descending with a swifter motion, than the parts nearer to A; so that
the body will receive a kind of rolling motion while it descends. But
it has been observed above, that the effect of gravity upon any body
is the same, as if the whole force were exerted on the body’s center of
gravity[64].
[Illustration]
Since therefore the power of gravity in drawing down the body must also
communicate to it the rolling motion just described; it seems evident,
that the center of gravity of the body cannot be drawn down as swiftly,
as when the power of gravity has no other effect to produce on the
body, than merely to draw it downward. If therefore the whole matter of
the body C B could be crouded into its center of gravity, so that being
united into one point, this rolling motion here mentioned might give
no hindrance to its descent; this center would descend faster, than it
can now do. And the point, which now descends as fast, as if the whole
matter or the body C B were crouded into it, will be farther removed
from the point A, than the center of gravity of the body C B.
69. AGAIN, suppose the pendulum A B (in fig. 55.) to hang obliquely.
Here the power of gravity will operate less upon the ball of the
pendulum, than before: but the line D E being drawn so, as to stand
perpendicular to the rod A C of the pendulum; the force of gravity
upon the body C B, now it is in this situation, will produce the same
effect, as if the body were to glide down an inclined plane in the
position of D E. But here the motion of the body, when the rod is fixed
to the point A, will not be equal to the uninterrupted descent of the
body down this plane; for the body will here also receive the same
kind of rotation in its motion, as before; so that the motion of the
center of gravity will in like manner be retarded; and the point, which
here descends with that degree of swiftness, which the body would have,
if not hindered by being fixed to the point A; that is, the point,
which descends as fast, as if the whole body were crouded into it, will
be as far removed from the point A, as before.
70. THIS point, by which the length of the pendulum is to be estimated,
is called the center of oscillation. And the mathematicians have laid
down general directions, whereby to find this center in all bodies. If
the globe A B (in fig. 56.) be hung by the string C D, whose weight
need not be regarded, the center of oscillation is found thus. Let the
straight line drawn from C to D be continued through the globe to F.
That it will pass through the center of the globe is evident. Suppose E
to be this center of the globe; and take the line G of such a length,
that it shall bear the same proportion to E D, as E D bears to E C.
Then E H being made equal to ⅖ of G, the point H shall be the center of
oscillation[65]. If the weight of the rod C D is too considerable to
be neglected, divide C D (fig. 57) in I, that D I be equal to ⅓, part
of C D; and take K in the same proportion to C I, as the weight of the
globe A B to the weight of the rod C D. Then having found H, the center
of oscillation of the globe, as before, divide I K in I, so that I L
shall bear the same proportion to L H, as the line C H bears to K; and
L shall be the center of oscillation of the whole pendulum.
71. THIS computation is made upon supposition, that the center of
oscillation of the rod C D, if that were to swing alone without any
other weight annexed, would be the point I. And this point would be
the true center of oscillation, so far as the thickness of the rod is
not to be regarded. If any one chuses to take into consideration the
thickness of the rod, he must place the center of oscillation thereof
so much below the point I, that eight times the distance of the center
from the point I shall bear the same proportion to the thickness of the
rod, as the thickness of the rod bears to its length C D[66].
72. IT has been observed above, that when a pendulum swings in an
arch of a circle, as here in fig. 58, the pendulum A B swings in the
circular arch C D; if you draw an horizontal line, as E F, from the
place whence the pendulum is let fall, to the line A G, which is
perpendicular to the horizon: then the velocity, which the pendulum
will acquire in coming to the point G, will be the same, as any body
would acquire in falling directly down from F to G. Now this is to be
understood of the circular arch, which is described by the center of
oscillation of the pendulum. I shall here farther observe, that if the
straight line E G be drawn from the point, whence the pendulum falls,
to the lowest point of the arch; in the same or in equal pendulums the
velocity, which the pendulum acquires in G, is proportional to this
line: that is, if the pendulum, after it has descended from E to G, be
taken back to H, and let fall from thence, and the line H G be drawn;
the velocity, which the pendulum shall acquire in G by its descent from
H, shall bear the same proportion to the velocity, which it acquires
in falling from E to G, as the straight line H G bears to the straight
line E G.
73. WE may now proceed to those experiments upon the percussion of
bodies, which I observed above might be made with pendulums. This
expedient for examining the effects of percussion was first proposed
by our late great architect Sir ~CHRISTOPHER WREN~. And it
is as follows. Two balls, as A and B (in fig. 59.) either equal or
unequal, are hung by two strings from two points C and D, so that, when
the balls hang down without motion, they shall just touch each other,
and the strings be parallel. Here if one of these balls be removed to
any distance from its perpendicular situation, and then let fall to
descend and strike against the other; by the last preceding paragraph
it will be known, with what velocity this ball shall return into its
first perpendicular situation, and consequently with what force it
shall strike against the other ball; and by the height to which this
other ball ascends after the stroke, the velocity communicated to this
ball will be discovered. For instance, let the ball A be taken up to
E, and from thence be let fall to strike against B, passing over in
its descent the circular arch E F. By this impulse let B fly up to G,
moving through the circular arch H G. Then E I and G K being drawn
horizontally, the ball A will strike against B with the velocity,
which it would acquire in falling directly down from I; and the ball
B has received a velocity, wherewith, if it had been thrown directly
upward, it would have ascended up to K. Likewise if straight lines be
drawn from E to F and from H to G, the velocity of A, wherewith it
strikes, will bear the same proportion to the velocity, which B has
received by the blow, as the straight line E F bears to the straight
line H G. In the same manner by noting the place to which A ascends
after the stroke, its remaining velocity may be compared with that,
wherewith it struck against B. Thus may be experimented the effects of
the body A striking against B at rest. If both the bodies are lifted
up, and so let fall as to meet and impinge against each other just upon
the coming of both into their perpendicular situation; by observing
the places into which they move after the stroke, the effects of their
percussion in all these cases may be found in the same manner as before.
74. SIR ~ISAAC NEWTON~ has described these experiments;
and has shewn how to improve them to a greater exactness by making
allowance for the resistance, which the air gives to the motion of the
balls[67]. But as this resistance is exceeding small, and the manner
of allowing for it is delivered by himself in very plain terms, I need
not enlarge upon it here. I shall rather speak to a discovery, which
he made by these experiments upon the elasticity of bodies. It has
been explained above[68], that when two bodies strike, if they be not
elastic, they remain contiguous after the stroke; but that if they
are elastic, they separate, and that the degree of their elasticity
determines the proportion between the celerity wherewith they separate,
and the celerity wherewith they meet. Now our author found, that the
degree of elasticity appeared in the same bodies always the same, with
whatever degree of force they struck; that is, the celerity wherewith
they separated, always bore the same proportion to the celerity
wherewith they met: so that the elastic power in all the bodies, he
made trial upon, exerted it self in one constant proportion to the
compressing force. Our author made trial with balls of wool bound up
very compact, and found the celerity with which they receded, to bear
about the proportion of 5 to 9 to the celerity wherewith they met; and
in steel he found nearly the same proportion; in cork the elasticity
was something less; but in glass much greater; for the celerity,
wherewith balls of that material separated after percussion, he found
to bear the proportion of 15 to 16 to the celerity wherewith they
met[69].
75. I SHALL finish my discourse on pendulums, with this farther
observation only, that the center of oscillation is also the center
of another force. If a body be fixed to any point, and being put in
motion turns round it; the body, if uninterrupted by the power of
gravity or any other means, will continue perpetually to move about
with the same equable motion. Now the force, with which such a body
moves, is all united in the point, which in relation to the power of
gravity is called the center of oscillation. Let the cylinder A B C D
(in fig. 60.) whose axis is E F, be fixed to the point E. And supposing
the point E to be that on which the cylinder is suspended, let the
center of oscillation be found in the axis E F, as has been explained
above[70]. Let G be that center: then I say, that the force, wherewith
this cylinder turns round the point E, is so united in the point G,
that a sufficient force applied in that point shall stop the motion of
the cylinder, in such a manner, that the cylinder should immediately
remain without motion, though it were to be loosened from the point E
at the same instant, that the impediment was applied to G: whereas, if
this impediment had been applied to any other point of the axis, the
cylinder would turn upon the point, where the impediment was applied.
If the impediment had been applied between E and G, the cylinder would
so turn on the point, where the impediment was applied, that the end
B C would continue to move on the same way it moved before along with
the whole cylinder; but if the impediment were applied to the axis
farther off from E than G, the end A D of the cylinder would start out
of its present place that way in which the cylinder moved. From this
property of the center of oscillation, it is also called the center of
percussion. That excellent mathematician, Dr. BROOK TAYLOR, has farther
improved this doctrine concerning the center of percussion, by shewing,
that if through this point G a line, as G H I, be drawn perpendicular
to E F, and lying in the course of the body’s motion; a sufficient
power applied to any point of this line will have the same effect, as
the like power applied to G[71]: so that as we before shewed the center
of percussion within the body on its axis; by this means we may find
this center on the surface of the body also, for it will be where this
line H I crosses that surface.
76. I SHALL now proceed to the last kind of motion, to be treated on
in this place, and shew what line the power of gravity will cause a
body to describe, when it is thrown forwards by any force. This was
first discovered by the great ~GALILEO~, and is the principle,
upon which engineers should direct the shot of great guns. But as in
this case bodies describe in their motion one of those lines, which in
geometry are called conic sections; it is necessary here to premise a
description of those lines. In which I shall be the more particular,
because the knowledge of them is not only necessary for the present
purpose, but will be also required hereafter in some of the principal
parts of this treatise.
77. THE first lines considered by the ancient geometers were the
straight line and the circle. Of these they composed various figures,
of which they demonstrated many properties, and resolved divers
problems concerning them. These problems they attempted always to
resolve by the describing straight lines and circles. For instance, let
a square A B C D (fig. 61.) be proposed, and let it be required to make
another square in any assigned proportion to this. Prolong one side,
as D A, of this square to E, till A E bear the same proportion to A D,
as the new square is to bear to the square A C. If the opposite side B
C of the square A C be also prolonged to F, till B F be equal to A E,
and E F be afterwards drawn, I suppose my readers will easily conceive,
that the figure A B F E will bear to the square A B C D the same
proportion, as the line A E bears to the line A D. Therefore the figure
A B F E will be equal to the new square, which is to be found, but is
not it self a square, because the side A E is not of the same length
with the side E F. But to find a square equal to the figure A B F E
you must proceed thus. Divide the line D E into two equal parts in the
point G, and to the center G with the interval G D describe the circle
D H E I; then prolong the line A B, till it meets the circle in K; and
make the square A K L M, which square will be equal to the figure A B F
E, and bear to the square A B C D the same proportion, as the line A E
bears to A D.
78. I SHALL not proceed to the proof of this, having only here set it
down as a specimen of the method of resolving geometrical problems
by the description of straight lines and circles. But there are some
problems, which cannot be resolved by drawing straight lines or circles
upon a plane. For the management therefore of these they took into
consideration solid figures, and of the solid figures they found that,
which is called a cone, to be the most useful.
79. A CONE is thus defined by EUCLIDE in his elements of geometry[72].
If to the straight line A B (in fig. 62.) another straight line, as A
C, be drawn perpendicular, and the two extremities B and C be joined by
a third straight line composing the triangle A C B (for so every figure
is called, which is included under three straight lines) then the two
points A and B being held fixed, as two centers, and the triangle A C B
being turned round upon the line A B, as on an axis; the line A C will
describe a circle, and the figure A C B will describe a cone, of the
form represented by the figure B C D E F (fig. 63.) in which the circle
C D E F is usually called the base of the cone, and B the vertex.
80. NOW by this figure may several problems be resolved, which cannot
by the simple description of straight lines and circles upon a plane.
Suppose for instance, it were required to make a cube, which should
bear any assigned proportion to some other cube named. I need not here
inform my readers, that a cube is the figure of a dye. This problem
was much celebrated among the ancients, and was once inforced by the
command of an oracle. This problem may be performed by a cone thus.
First make a cone from a triangle, whose side A C shall be half the
length of the side B C Then on the plane A B C D (fig. 64.) let the
line E F be exhibited equal in length to the side of the cube proposed;
and let the line F G be drawn perpendicular to E F, and of such a
length, that it bear the same proportion to E F, as the cube to be
sought is required to bear to the cube proposed. Through the points E,
F, and G let the circle F H I be described. Then let the line E F be
prolonged beyond F to K, that F K be equal to F E, and let the triangle
F K L, having all its sides F K, K L, L F equal to each other, be hung
down perpendicularly from the plane A B C D. After this, let another
plane M N O P be extended through the point L, so as to be equidistant
from the former plane A B C D, and in this plane let the line Q L R
be drawn so, as to be equidistant from the line E F K. All this being
thus prepared, let such a cone, as was above directed to be made, be so
applied to the plane M N O P, that it touch this plane upon the line
Q R, and that the vertex of the cone be applied to the point L. This
cone, by cutting through the first plane A B C D, will cross the circle
F H I before described. And if from the point S, where the surface of
this cone intersects the circle, the line S T be drawn so, as to be
equidistant from the line E F; the line F T will be equal to the side
of the cube sought: that is, if there be two cubes or dyes formed, the
side of one being equal to E F, and the side of the other equal to F T;
the former of these cubes shall bear the same proportion to the latter,
as the line E F bears to F G.
81. INDEED this placing a cone to cut through a plane is not a
practicable method of resolving problems. But when the geometers had
discovered this use of the cone, they applied themselves to consider
the nature of the lines, which will be produced by the intersection
of the surface of a cone and a plane; whereby they might be enabled
both to reduce these kinds of solutions to practice, and also to render
their demonstrations concise and elegant.
82. WHENEVER the plane, which cuts the cone, is equidistant from
another plane, that touches the cone on the side; (which is the case of
the present figure;) the line, wherein the plane cuts the surface of
the cone, is called a parabola. But if the plane, which cuts the cone,
be so inclined to this other, that it will pass quite through the cone
(as in fig. 65.) such a plane by cutting the cone produces the figure
called an ellipsis, in which we shall hereafter shew the earth and
other planets to move round the sun. If the plane, which cuts the cone,
recline the other way (as in fig. 66.) so as not to be parallel to any
plane, whereon the cone can lie, nor yet to cut quite through the cone;
such a plane shall produce in the cone a third kind of line, which
is called an hyperbola. But it is the first of these lines named the
parabola, wherein bodies, that are thrown obliquely, will be carried
by the force of gravity; as I shall here proceed to shew, after having
first directed my readers how to describe this sort of line upon a
plane, by which the form of it may be seen.
83. TO any straight line A B (fig. 67.) let a straight ruler C D be
so applied, as to stand against it perpendicularly. Upon the edge of
this ruler let another ruler E F be so placed, as to move along upon
the edge of the first ruler C D, and keep always perpendicular to it.
This being so disposed, let any point, as G, be taken in the line A B,
and let a string equal in length to the ruler E F be fastened by one
end to the point G, and by the other to the extremity F of the ruler E
F. Then if the string be held down to the ruler E F by a pin H, as is
represented in the figure; the point of this pin, while the ruler E F
moves on the ruler C D, shall describe the line I K L, which will be
one part of the curve line, whose description we were here to teach:
and by applying the rulers in the like manner on the other side of
the line A B, we may describe the other part I M of this line. If the
distance C G be equal to half the line E F in fig. 64, the line M I L
will be that very line, wherein the plane A B C D in that figure cuts
the cone.
84. THE line A I is called the axis of the parabola M I L, and the
point G is called the focus.
85. NOW by comparing the effects of gravity upon falling bodies, with
what is demonstrated of this figure by the geometers, it is proved,
that every body thrown obliquely is carried forward in one of these
lines, the axis whereof is perpendicular to the horizon.
86. THE geometers demonstrate, that if a line be drawn to touch a
parabola in any point, as the line A B (in fig. 68.) touches the
parabola C D, whose axis is Y Z, in the point E; and several lines F
G, H I, K L be drawn parallel to the axis of the parabola: then the
line F G will be to H I in the duplicate proportion of E F to E H,
and F G to K L in the duplicate proportion of E F to E K; likewise
H I to K L in the duplicate proportion of E H to E K. What is to be
understood by duplicate or two-fold proportion, has been already
explained[73]. Accordingly I mean here, that if the line M be taken to
bear the same proportion to E H, as E H bears to E F, H I will bear
the same proportion to F G, as M bears to E F; and if the line N bears
the same proportion to E K, as E K bears to E F, K L will bear the
same proportion to F G, as N bears to E F; or if the line O bear the
same proportion to E K, as E K bears to E H, K L will bear the same
proportion to H I, as O bears to E H.
87. THIS property is essential to the parabola, being so connected with
the nature of the figure, that every line possessing this property is
to be called by this name.
88. NOW suppose a body to be thrown from the point A (in fig. 69.)
towards B in the direction of the line A B. This body, if left to
it self, would move on with a uniform motion through this line A B.
Suppose the eye of a spectator to be placed at the point C just under
the point A; and let us imagine the earth to be so put into motion
along with the body, as to carry the spectator’s eye along the line C D
parallel to A B; and that the eye would move on with the same velocity,
wherewith the body would proceed in the line A B, if it were to be
left to move without any disturbance from its gravitation towards the
earth. In this case if the body moved on without being drawn towards
the earth, it would appear to the spectator to be at rest. But if the
power of gravity exerted it self on the body, it would appear to the
spectator to fall directly down. Suppose at the distance of time,
wherein the body by its own progressive motion would have moved from A
to E, it should appear to the spectator to have fallen through a length
equal to E F: then the body at the end of this time will actually have
arrived at the point F. If in the space of time, wherein the body
would have moved by its progressive motion from A to G, it would have
appeared to the spectator to have fallen down the space G H: then the
body at the end of this greater interval of time will be arrived at
the point H. Now if the line A F H I be that, through which the body
actually passes; from what has here been said, it will follow, that
this line is one of those, which I have been describing under the name
of the parabola. For the distances E F, G H, through which the body
is seen to fall, will increase in the duplicate proportion of the
times[74]; but the lines A E, A G will be proportional to the times
wherein they would have been described by the single progressive motion
of the body: therefore the lines E F, G H will be in the duplicate
proportion of the lines A F, A G; and the line A F H I possesses the
property of the parabola.
89. IF the earth be not supposed to move along with the body, the
case will be a little different. For the body being constantly drawn
directly towards the center of the earth, the body in its motion will
be drawn in a direction a little oblique to that, wherein it would be
drawn by the earth in motion, as before supposed. But the distance to
the center of the earth bears so vast a proportion to the greatest
length, to which we can throw bodies, that this obliquity does not
merit any regard. From the sequel of this discourse it may indeed
be collected, what line the body being thrown thus would be found
to describe, allowance being made for this obliquity of the earth’s
action[75]. This is the discovery of Sir IS. NEWTON; but has no use in
this place. Here it is abundantly sufficient to consider the body as
moving in a parabola.
90. THE line, which a projected body describes, being thus known,
practical methods have been deduced from hence for directing the
shot of great guns to strike any object desired. This work was first
attempted by ~GALILEO~, and soon after farther improved by
his scholar ~TORRICELLI~; but has lately been rendred more
complete by the great Mr. ~COTES~, whose immature death is an
unspeakable loss to mathematical learning. If it be required to throw
a body from the point A (in fig. 70.) so as to strike the point B;
through the points A, B draw the straight line C D, and erect the line
A E perpendicular to the horizon, and of four times the height, from
which a body must fall to acquire the velocity, wherewith the body is
intended to be thrown. Through the points A and E describe a circle,
that shall touch the line C D in the point A. Then from the point
B draw the line B F perpendicular to the horizon, intersecting the
circle in the points G and H. This being done, if the body be projected
directly towards either of these points G or H, it shall fall upon
the point B; but with this difference, that, if it be thrown in the
direction A G, it shall sooner arrive at B, than if it were projected
in the direction A H. When the body is projected in the direction A
G; the time, it will take up in arriving at B, will bear the same
proportion to the time, wherein it would fall down through one fourth
part of A E, as A G bears to half A E. But when the body is thrown in
the direction of A H, the time of its passing to B will bear the same
proportion to the time, wherein it would fall through one fourth part
of A E, as A H bears to half A E.
91. IF the line A I be drawn so as to divide the angle under E A D in
the middle, and the line I K be drawn perpendicular to the horizon;
this line will touch the circle in the point I, and if the body be
thrown in the direction A I, it will fall upon the point K: and this
point K is the farthest point in the line A D, which the body can be
made to strike, without increasing its velocity.
92. THE velocity, wherewith the body every where moves, may be found
thus. Suppose the body to move in the parabola A B (fig. 71.) Erect A
C perpendicular to the horizon, and equal to the height, from which a
body must fall to acquire the velocity, wherewith the body sets out
from A. If you take any points as D and E in the parabola, and draw
D F and E G parallel to the horizon; the velocity of the body in D
will be equal to what a body will acquire in falling down by its own
weight through C F, and in E the velocity will be the same, as would
be acquired in falling through C G. Thus the body moves slowest at
the highest point H of the parabola; and at equal distances from this
point will move with equal swiftness, and descend from that highest
point through the line H B altogether like to the line A H in which it
ascended; abating only the resistance of the air, which is not here
considered. If the line H I be drawn from the highest point H parallel
to the horizon, A I will be equal to ¼ of B G in fig. 70, when the body
is projected in the direction A G, and equal to ¼ of B H, when the body
is thrown in the direction A H provided A D be drawn horizontally.
93. THUS I have recounted the principal discoveries, which had been
made concerning the motion of bodies by Sir ~ISAAC NEWTON~’S
predecessors; all these discoveries, by being found to agree with
experience, contributing to establish the laws of motion, from whence
they were deduced. I shall therefore here finish what I had to say
upon those laws; and conclude this chapter with a few words concerning
the distinction which ought to be made between absolute and relative
motion. For some have thought fit to confound them together; because
they observe the laws of motion to take place here on the earth, which
is in motion, after the same manner as if it were at rest. But Sir
~ISAAC NEWTON~ has been careful to distinguish between the
relative and absolute consideration both of motion and time[76]. The
astronomers anciently found it necessary to make this distinction in
time. Time considered in it self passes on equably without relation to
any thing external, being the proper measure of the continuance and
duration of all things. But it is most frequently conceived of by us
under a relative view to some succession in sensible things, of which
we take cognizance. The succession of the thoughts in our own minds
is that, from whence we receive our first idea of time, but is a very
uncertain measure thereof; for the thoughts of some men flow on much
more swiftly, than the thoughts of others; nor does the same person
think equally quick at all times. The motions of the heavenly bodies
are more regular; and the eminent division of time into night and day,
made by the sun, leads us to measure our time by the motion of that
luminary: nor do we in the affairs of life concern our selves with any
inequality, which there may be in that motion; but the space of time
which comprehends a day and night is rather supposed to be always the
same. However astronomers anciently found these spaces of time not to
be always of the same length, and have taught how to compute their
differences. Now the time, when so equated as to be rendered perfectly
equal, is the true measure of duration, the other not. And therefore
this latter, which is absolutely true time, differs from the other,
which is only apparent. And as we ordinarily make no distinction
between apparent time, as measured by the sun, and the true; so we
often do not distinguish in our usual discourse between the real, and
the apparent or relative motion of bodies; but use the same words for
one, as we should for the other. Though all things about us are really
in motion with the earth; as this motion is not visible, we speak of
the motion of every thing we see, as if our selves and the earth stood
still. And even in other cases, where we discern the motion of bodies,
we often speak of them not in relation to the whole motion we see, but
with regard to other bodies, to which they are contiguous. If any body
were lying on a table; when that table shall be carried along, we say
the body rests upon the table, or perhaps absolutely, that the body is
at rest. However philosophers must not reject all distinction between
true and apparent motions, any more than astronomers do the distinction
between true and vulgar time; for there is as real a difference between
them, as will appear by the following consideration. Suppose all the
bodies of the universe to have their courses stopped, and reduced to
perfect rest. Then suppose their present motions to be again restored;
this cannot be done without an actual impression made upon some of them
at least. If any of them be left untouched, they will retain their
former state, that is, still remain at rest; but the other bodies,
which are wrought upon, will have changed their former state of rest,
for the contrary state of motion. Let us now suppose the bodies left
at rest to be annihilated, this will make no alteration in the state
of the moving bodies; but the effect of the impression, which was made
upon them, will still subsist. This shews the motion they received to
be an absolute thing, and to have no necessary dependence upon the
relation which the body said to be in motion has to any other body[77].
94. BESIDES absolute and relative motion are distinguishable by their
Effects. One effect of motion is, that bodies, when moved round any
center or axis, acquire a certain power, by which they forcibly
press themselves from that center or axis of motion. As when a body
is whirled about in a sling, the body presses against the sling, and
is ready to fly out as soon as liberty is given it. And this power
is proportional to the true, not relative motion of the body round
such a center or axis. Of this Sir ~ISAAC NEWTON~ gives the following
instance[78]. If a pail or such like vessel near full of water be
suspended by a string of sufficient length, and be turned about till
the string be hard twisted. If then as soon as the vessel and water
in it are become still and at rest, the vessel be nimbly turned about
the contrary way the string was twisted, the vessel by the strings
untwisting it self shall continue its motion a long time. And when the
vessel first begins to turn, the water in it shall receive little or
nothing of the motion of the vessel, but by degrees shall receive a
communication of motion, till at last it shall move round as swiftly
as the vessel it self. Now the definition of motion, which ~DES
CARTES~ has given us upon this principle of making all motion
meerly relative, is this: that motion, is a removal of any body from
its vicinity to other bodies, which were in immediate contact with
it, and are considered as at rest[79]. And if this be compared with
what he soon after says, that there is nothing real or positive in the
body moved, for the sake of which we ascribe motion to it, which is
not to be found as well in the contiguous bodies, which are considered
as at rest[80]; it will follow from thence, that we may consider the
vessel as at rest and the water as moving in it: and the water in
respect of the vessel has the greatest motion, when the vessel first
begins to turn, and loses this relative motion more and more, till
at length it quite ceases. But now, when the vessel first begins to
turn, the surface of the water remains smooth and flat, as before the
vessel began to move; but as the motion of the vessel communicates by
degrees motion to the water, the surface of the water will be observed
to change, the water subsiding in the middle and rising at the edges:
which elevation of the water is caused by the parts of it pressing from
the axis, they move about; and therefore this force of receding from
the axis of motion depends not upon the relative motion of the water
within the vessel, but on its absolute motion; for it is least, when
that relative motion is greatest, and greatest, when that relative
motion is least, or none at all.
95. THUS the true cause of what appears in the surface of this water
cannot be assigned, without considering the water’s motion within the
vessel. So also in the system of the world, in order to find out the
cause of the planetary motions, we must know more of the real motions,
which belong to each planet, than is absolutely necessary for the uses
of astronomy. If the astronomer should suppose the earth to stand
still, he could ascribe such motions to the celestial bodies, as should
answer all the appearances; though he would not account for them in so
simple a manner, as by attributing motion to the earth. But the motion
of the earth must of necessity be considered, before the real causes,
which actuate the planetary system, can be discovered.
CHAP. III.
Of CENTRIPETAL FORCES.
WE have just been describing in the preceding chapter the effects
produced on a body in motion, from its being continually acted upon
by a power always equal in strength, and operating in parallel
directions[81]. But bodies may be acted upon by powers, which in
different places shall have different degrees of force, and whose
several directions shall be variously inclined to each other. The most
simple of these in respect to direction is, when the power is pointed
constantly to one center. This is truly the case of that power, whose
effects we described in the foregoing chapter; though the center of
that power is so far removed, that the subject then before us is most
conveniently to be considered in the light, wherein we have placed it:
But Sir ISAAC NEWTON has considered very particularly this other case
of powers, which are constantly directed to the same center. It is upon
this foundation, that all his discoveries in the system of the world
are raised. And therefore, as this subject bears so very great a share
in the philosophy, of which I am discoursing, I think it proper in this
place to take a short view of some of the general effects of these
powers, before we come to apply them particularly to the system of the
world.
2. THESE powers or forces are by Sir ~ISAAC NEWTON~ called centripetal;
and their first effect is to cause the body, on which they act, to quit
the straight course, wherein it would proceed if undisturbed, and to
describe an incurvated line, which shall always be bent towards the
center of the force. It is not necessary, that such a power should
cause the body to approach that center. The body may continue to
recede from the center of the power, notwithstanding its being drawn
by the power; but this property must always belong to its motion, that
the line, in which it moves, will continually be concave towards the
center, to which the power is directed. Suppose A (in fig. 72.) to be
the center of a force. Let a body in B be moving in the direction of
the straight line B C, in which line it would continue to move, if
undisturbed; but being attracted by the centripetal force towards A,
the body must necessarily depart from this line B C, and being drawn
into the curve line B D, must pass between the lines A B and B C. It is
evident therefore, that the body in B being gradually turned off from
the straight line B C, it will at first be convex toward the line B C,
and consequently concave towards the point A: for these centripetal
powers are supposed to be in strength proportional to the power of
gravity, and, like that, not to be able after the manner of an impulse
to turn the body sensibly out of its course into a different one in
an instant, but to take up some space of time in producing a visible
effect. That the curve will always continue to have its concavity
towards A may thus appear. In the line B C near to B take any point as
E, from which the line E F G may be so drawn, as to touch the curve
line B D in some point as F. Now when the body is come to F, if the
centripetal power were immediately to be suspended, the body would no
longer continue to move in a curve line, but being left to it self
would forthwith reassume a straight course; and that straight course
would be in the line F G: for that line is in the direction of the
body’s motion at the point F. But the centripetal force continuing its
energy, the body will be gradually drawn from this line F G so as to
keep in the line F D, and make that line near the point F to be convex
toward F G, and concave toward A. After the same manner the body may be
followed on in its course through the line B D, and every part of that
line be shewn to be concave toward the point A.
3. THIS then is the constant character belonging to those motions,
which are carried on by centripetal forces; that the line, wherein the
body moves, is throughout concave towards the center of the force. In
respect to the successive distances of the body from the center there
is no general rule to be laid down; for the distance of the body from
the center may either increase, or decrease, or even keep always the
same. The point A (in fig. 73.) being the center of a centripetal
force, let a body at B set out in the direction of the straight line B
C perpendicular to the line A B drawn from A to B. It will be easily
conceived, that there is no other point in the line B C so near to A,
as the point B; that A B is the shortest of all the lines, which can
be drawn from A to any part of the line B C; all other lines, as A D,
or A E, drawn from A to the line B C being longer than A B. Hence it
follows, that the body setting out from B, if it moved in the line B
C, it would recede more and more from the point A. Now as the operation
of a centripetal force is to draw a body towards the center of the
force: if such a force act upon a resting body, it must necessarily put
that body so into motion, as to cause it to move towards the center
of the force: if the body were of it self moving towards that center,
the centripetal force would accelerate that motion, and cause it to
move faster down: but if the body were in such a motion, as being left
to itself it would recede from this center, it is not necessary, that
the action of a centripetal power upon it should immediately compel
the body to approach the center, from which it would otherwise have
receded; the centripetal power is not without effect, if it cause the
body to recede more slowly from that center, than otherwise it would
have done. Thus in the case before us, the smallest centripetal power,
if it act on the body, will force it out of the line B C, and cause it
to pass in a bent line between B C and the point A, as has been before
explained. When the body, for instance, has advanced to the line A D,
the effect of the centripetal force discovers it self by having removed
the body out of the line B C, and brought it to cross the line A D
somewhere between A and D: suppose at F. Now A D being longer than A B,
A F may also be longer than A B. The centripetal power may indeed be
so strong, that A F shall be shorter than A B; or it may be so evenly
balanced with the progressive motion of the body, that A F and A B
shall be just equal: and in this last case, when the centripetal force
is of that strength, as constantly to draw the body as much toward the
center, as the progressive motion would carry it off, the body will
describe a circle about the center A, this center of the force being
also the center of the circle.
4. IF the body, instead of setting out in the line B C perpendicular
to A B, had set out in another line B G more inclined towards the
line A B, moving in the curve line B H; then as the body, if it were
to continue its motion in the line B G, would for some time approach
the center A; the centripetal force would cause it to make greater
advances toward that center. But if the body were to set out in the
line B I reclined the other way from the perpendicular B C, and were to
be drawn by the centripetal force into the curve line B K; the body,
notwithstanding any centripetal force, would for some time recede from
the center; since some part at least of the curve line B K lies between
the line B I and the perpendicular B C.
5. THUS far we have explained such effects, as attend every centripetal
force. But as these forces may be very different in regard to the
different degrees of strength, wherewith they act upon bodies in
different places; I shall now proceed to make mention in general of
some of the differences attending these centripetal motions.
6. TO reassume the consideration of the last mentioned case. Suppose a
centripetal power directed toward the point A (in fig. 74.) to act on
a body in B, which is moving in the direction of the straight line B
C, the line B C reclining off from A B. If from A the straight lines A
D, A E, A F are drawn at pleasure to the line C B; the line C B being
prolonged beyond B to G, it appears that A D is inclined to the line
G C more obliquely, than A B is inclined to it, A E is inclined more
obliquely than A D, and A F more than A E. To speak more correctly, the
angle under A D G is less than that under A B G, the angle under A E G
less than that under A D G, and the angle under A F G less than that
under A E G. Now suppose the body to move in the curve line B H I K.
Then it is here likewise evident, that the line B H I K being concave
towards A, and convex towards the line B C, it is more and more turned
off from the line B C; so that in the point H the line A H will be less
obliquely inclined to the curve line B H I K, than the same line A H
D is inclined to B C at the point D; at the point I the inclination
of the line A I to the curve line will be more different from the
inclination of the same line A I E to the line B C, at the point E;
and in the points K and F the difference of inclination will be still
greater; and in both the inclination at the curve will be less oblique,
than at the straight line B C. But the straight line A B is less
obliquely inclined to B G, than A D is inclined towards D G: therefore
although the line A H be less obliquely inclined towards the curve H B,
than the same line A H D is inclined towards D G; yet it is possible,
that the inclination at H may be more oblique, than the inclination at
B. The inclination at H may indeed be less oblique than the other, or
they may be both the same. This depends upon the degree of strength,
wherewith the centripetal force exerts it self, during the passage of
the body from B to H. After the same manner the inclinations at I and K
depend entirely on the degree of strength, wherewith the centripetal
force acts on the body in its passage from H to K: if the centripetal
force be weak enough, the lines A H and A I drawn from the center A to
the body at H and at I shall be more obliquely inclined to the curve,
than the line A B is inclined towards B G. The centripetal force may
be of that strength as to render all these inclinations equal, or if
stronger, the inclinations at I and K will be less oblique than at
B. Sir ~ISAAC NEWTON~ has particularly shewn, that if the
centripetal power decreases after a certain manner with the increase
of distance, a body may describe such a curve line, that all the
lines drawn from the center to the body shall be equally inclined to
that curve line.[82] But I do not here enter into any particulars, my
present intention being only to shew, that it is possible for a body to
be acted upon by a force continually drawing it down towards a center,
and yet that the body shall continue to recede from that center; for
here as long as the lines A H, A I, &c drawn from the center A to the
body do not become less oblique to the curve, in which the body moves;
so long shall those lines perpetually increase, and consequently the
body shall more and more recede from the center.
7. BUT we may observe farther, that if the centripetal power, while
the body increases its distance from the center, retain sufficient
strength to make the lines drawn from the center to the body to become
at length less oblique to the curve; then if this diminution of the
obliquity continue, till at last the line drawn from the center to
the body shall cease to be obliquely inclined to the curve, and shall
become perpendicular thereto; from this instant the body shall no
longer recede from the center, but in its following motion it shall
again descend, and shall describe a curve line in all respects like to
that, which it has described already; provided the centripetal power,
every where at the same distance from the center, acts with the same
strength. So we observed in the preceding chapter, that, when the
motion of a projectile became parallel to the horizon, the projectile
no longer ascended, but forthwith directed its course downwards,
descending in a line altogether like that, wherein it had before
ascended[83].
8. THIS return of the body may be proved by the following proposition:
that if the body in any place, suppose at I, were to be stopt, and
be thrown directly backward with the velocity, wherewith it was
moving forward in that point I; then the body, by the action of the
centripetal force upon it, would move back again over the path I H B,
in which it had before advanced forward, and would arrive again at the
point B in the same space of time, as was taken up in its passage from
B to I; the velocity of the body at its return to the point B being
the same, as that wherewith it first set out from that point. To give
a full demonstration of this proposition, would require that use of
mathematics, which I here purpose to avoid; but, I believe, it will
appear in great measure evident from the following considerations.
9. SUPPOSE (in fig. 75.) that a body were carried after the following
manner through the bent figure A B C D E F, composed of the straight
lines A B, B C, C D, D E, E F. First let it be moving in the line A B,
from A towards B, with any uniform velocity. At B let the body receive
an impulse directed toward some point, as G, taken within the concavity
of the figure. Now whereas this body, when once moving in the straight
line A B, will continue to move on in this line, so long as it shall be
left to it self; but being disturbed at the point B in its motion by
the impulse, which there acts upon it, it will be turned out of this
line A B into some other straight line, wherein it will afterwards
continue to move, as long as it shall be left to itself. Therefore
let this impulse have strength sufficient to turn the body into the
line B C. Then let the body move on undisturbed from B to C, but at C
let it receive another impulse pointed toward the same point G, and
of sufficient strength to turn the body into the line C D. At D let a
third impulse, directed like the rest to the point G, turn the body
into the line D E. And at E let another impulse, directed likewise to
the point G, turn the body into the line E F. Now, I say, if the body
while moving in the line E F be stopt, and turned back again in this
line with the same velocity, as that wherewith it was moving forward in
this line; then by the repetition of the former impulse at E the body
will be turned into the line E D, and move in it from E to D with the
same velocity as before it moved with from D to E; by the repetition of
the impulse at D, when the body shall have returned to that point, it
will be turned into the line D C; and by the repetition of the other
impulses at C and B the body will be brought back again into the line
B A, with the velocity, wherewith it first moved in that line.
10. THIS I prove as follows. Let D E and F E be continued beyond E. In
D E thus continued take at pleasure the length E H, and let H I be so
drawn, as to be equidistant from the line G E. Then, by what has been
written upon the second law of motion[84], it follows, that after the
impulse on the body in E it will move through E I in the same time, as
it would have imployed in moving from E to H, with the velocity which
it had in the line D E. In F E prolonged take E K equal to E I, and
draw K L equidistant from G E. Then, because the body is thrown back in
the line F E with the same velocity as that wherewith it went forward
in that line; if, when the body was returned to E, it were permitted
to go straight on, it would pass through E K in the same time, as it
took up in passing through E I, when it went forward in the line E F.
But, if at the body’s return to the point E, such an impulse directed
toward the point D were to be given it, whereby it should be turned
into the line D E; I say, that the impulse necessary to produce this
effect must be equal to that, which turned the body out of the line D E
into E F; and that the velocity, with which the body will return into
the line E D, is the same, as that wherewith it before moved through
this line from D to E. Because E K is equal to E I, and K L and H I,
being each equidistant from G E, are by consequence equidistant from
each other; it follows, that the two triangular figures I E H and K
E L are altogether like and equal to each other. If I were writing to
mathematicians, I might refer them to some proportions in the elements
of EUCLID for the proof of this[85] but as I do not here address my
self to such, so I think this assertion will be evident enough without
a proof in form; at least I must desire my readers to receive it as a
proposition true in geometry. But these two triangular figures being
altogether like each other and equal; as E K is equal to E I, so E L is
equal to E H, and K L equal to H I. Now the body after its return to
E being turned out of the line F E into E D by an impulse acting upon
it in E, after the manner above expressed; the body will receive such
a velocity by this impulse, as will carry it through E L in the same
time, as it would have imployed in passing through E K, if it had gone
on in that line undisturbed. And it has already been observed, that the
time, in which the body would pass over E K with the velocity wherewith
it returns, is equal to the time it took up in going forward from E to
I; that is, equal to the time, in which it would have gone through E H
with the velocity, wherewith it moved from D to E. Therefore the time,
in which the body will pass through E L after its return into the line
E D, is the same, as would have been taken up by the body in passing
through E H with the velocity, wherewith the body first moved in the
line D E. Since therefore E L and E H are equal, the body returns into
the line D E with the velocity, which it had before in that line. Again
I say, the second impulse in E is equal to the first. By what has
been said on the second law of motion concerning the effect of oblique
impulses[86], it will be understood, that the impulse in E, whereby
the body was turned out of the line D E into the line E F, is of such
strength, that if the body had been at rest, when this impulse had
acted upon it, this impulse would have communicated so much motion to
the body, as would have carried it through a length equal to H I, in
the time wherein the body would have passed from E to H, or in the time
wherein it passed from E to I. In the same manner, on the return of the
body, the impulse in E, whereby the body is turned out of the line F
E into E D, is of such strength, that if it had acted on the body at
rest, it would have caused the body to move through a length equal to
K L, in the same time, as the body would imploy in passing through E K
with the velocity, wherewith it returns in the line F E. Therefore the
second impulse, had it acted on the body at rest, would have caused it
to move through a length equal to K L in the same space of time, as
would be taken up by the body in passing through a length equal to H I,
were the first impulse to act on the body when at rest. That is, the
effects of the first and second impulse on the body when at rest would
be the same; for K L and H I are equal: consequently the second impulse
is equal to the first.
11. THUS if the body be returned through F E with the velocity,
wherewith it moved forward; we have shewn how by the repetition of the
impulse, which acted on it at E, the body will return again into the
line D E with the velocity, which it had before in that line. By the
same process of reasoning it may be proved, that, when the body is
returned back to D, the impulse, which before acted on the body at that
point, will throw the body into the line D C with the velocity, which
it first had in that line; and the other impulses being successively
repeated, the body will at length be brought back again into the line B
A with the velocity, wherewith it set out in that line.
12. THUS these impulses, by acting over again in an inverted order
all their operation on the body, bring it back again through the
path, in which it had proceeded forward. And this obtains equally,
whatever be the number of the straight lines, whereof this curve
figure is composed. Now by a method of reasoning, which Sir ~ISAAC
NEWTON~ makes great use of, and which he introduced into geometry,
thereby greatly inriching that science[87]; we might make a transition
from this figure composed of a number of straight lines to a figure
of one continued curvature, and from a number of separate impulses
repeated at distinct intervals to a continual centripetal force, and
shew, that, because what has been here advanced holds universally
true, whatever be the number of straight lines, whereof the curve
figure A C F is composed, and howsoever frequently the impulses at
the angles of this figure are repeated; therefore the same will still
remain true, although this figure should be converted into one of a
continued curvature, and these distinct impulses should be changed
into a continual centripetal force. But as the explaining this method
of reasoning is foreign to my present design; so I hope my readers,
after what has been said, will find no difficulty in receiving the
proposition laid down above: that, if the body, which has moved through
the curve line B H I (in fig. 74.) from B to I, when it is come to I,
be thrown directly back with the same velocity as that, wherewith it
proceeded forward, the centripetal force, by acting over again all its
operation on the body, shall bring the body back again in the line I H
B: and as the motion of the body in its course from B to I was every
where in such a manner oblique to the line drawn from the center to
the body, that the centripetal power acted in some degree against the
body’s motion, and gradually diminished it; so in the return of the
body, the centripetal power will every where draw the body forward, and
accelerate its motion by the same degrees, as before it retarded it.
13. THIS being agreed, suppose the body in K to have the line A K no
longer obliquely inclined to its motion. In this case, if the body
be turned back, in the manner we have been considering, it must be
directed back perpendicularly to A K. But if it had proceeded forward,
it would likewise have moved in a direction perpendicular to A K;
consequently, whether it move from this point K backward or forward, it
must describe the same kind of course. Therefore since by being turned
back it will go over again the line K I H B; if it be permitted to go
forward, the line K L, which it shall describe, will be altogether
similar to the line K H B.
14. IN like manner we may determine the nature of the motion, if
the line, wherein the body sets out, be inclined (as in fig. 76.)
down toward the line B A drawn between the body and the center. If
the centripetal power so much increases in strength, as the body
approaches, that it can bend the path, in which the body moves, to
that degree, as to cause all the lines as A H, A I, A K to remain no
less oblique to the motion of the body, than A B is oblique to B C;
the body shall continually more and more approach the center. But if
the centripetal power increases in so much less a degree, as to permit
the line drawn from the center to the body, as it accompanies the
body in its motion, at length to become more and more erect to the
curve wherein the body moves, and in the end, suppose at K, to become
perpendicular thereto; from that time the body shall rise again. This
is evident from what has been said above; because for the very same
reason here also the body shall proceed from the point K to describe a
line altogether similar to the line, in which it has moved from B to K.
Thus, as it was observed of the pendulum in the preceding chapter[88],
that all the time it approaches towards being perpendicular to the
horizon, it more and more descends; but, as soon as it is come into
that perpendicular situation, it immediately rises again by the same
degrees, as it descended by before: so here the body more and more
approaches the center all the time it is moving from B to K; but thence
forward it rises from the center again by the same degrees, as it
approached by before.
15. IF (in fig. 77.) the line B C be perpendicular to A B; then it has
been observed above[89], that the centripetal power may be so balanced
with the progressive motion of the body, that the body may keep moving
round the center A constantly at the same distance; as a body does,
when whirled about any point, to which it is tyed by a string. If the
centripetal power be too weak to produce this effect, the motion of
the body will presently become oblique to the line drawn from itself
to the center, after the manner of the first of the two cases, which
we have been considering. If the centripetal power be stronger, than
what is required to carry the body in a circle, the motion of the body
will presently fall in with the second of the cases, we have been
considering.
16. IF the centripetal power so change with the change of distance,
that the body, after its motion has become oblique to the line drawn
from itself to the center, shall again become perpendicular thereto;
which we have shewn to be possible in both the cases treated of
above; then the body shall in its subsequent motion return again to
the distance of A B, and from that distance take a course similar
to the former: and thus, if the body move in a space free from all
resistance, which has been here all along supposed; it shall continue
in a perpetual motion about the center, descending and ascending
alternately therefrom. If the body setting out from B (in fig. 78.) in
the line B C perpendicular to A B, describe the line B D E, which in D
shall be oblique to the line A D, but in E shall again become erect to
A E drawn from the body in E to the center A; then from this point E
the body shall describe the line E F G altogether like to the line B D
E, and at G shall be at the same distance from A, as it was at B. But
likewise the line A G shall be erect to the body’s motion. Therefore
the body shall proceed to describe from G the line G H I altogether
similar to the line G F E, and at I have the same distance from the
center, as it had at E; and also have the line A I erect to its motion:
so that its following motion must be in the line I K L similar to I H
G, and the distance A L equal to A G. Thus the body will go on in a
perpetual round without ceasing, alternately inlarging and contracting
its distance from the center.
[Illustration]
17. IF it so happen, that the point E fall upon the line B A continued
beyond A; then the point G will fall on B, I on E, and L also on B;
so that the body will describe in this case a simple curve line round
the center A, like the line B D E F in fig. 79, in which it will
continually revolve from B to E and from E to B without end.
18. IF A E in fig. 78 should happen to be perpendicular to A B, in this
case also a simple line will be described; for the point G will fall on
the line B A prolonged beyond A, the point I on the line A E prolonged
beyond A, and the point L on B: so that the body will describe a line
like the curve line B E G I in fig. 80, in which the opposite points B
and G are equally distant from A, and the opposite points E and I are
also equally distant from the same point A.
19. IN other cases the line described will have a more complex figure.
20. THUS we have endeavoured to shew how a body, while it is constantly
attracted towards a center, may notwithstanding by its progressive
motion keep it self from falling down to that center; but describe
about it an endless circuit, sometimes approaching toward that center,
and at other times as much receding from the same.
21. BUT here we have supposed, that the centripetal power is of equal
strength every where at the same distance from the center. And this is
the case of that centripetal power, which will hereafter be shewn to be
the cause, that keeps the planets in their courses. But a body may be
kept on in a perpetual circuit round a center, although the centripetal
power have not this property. Indeed a body may by a centripetal
force be kept moving in any curve line whatever, that shall have its
concavity turned every where towards the center of the force.
22. TO make this evident I shall first propose the case of a body
moving through the incurvated figure A B C D E (in fig. 81.) which is
composed of the straight lines A B, B C, C D, D E, and E A; the motion
being carried on in the following manner. Let the body first move in
the line A B with any uniform velocity. When it is arrived at the point
B, let it receive an impulse directed toward any point F taken within
the figure; and let the impulse be of that strength as to turn the body
out of the line A B into the line B C. The body after this impulse,
while left to itself, will continue moving in the line B C. At C let
the body receive another impulse directed towards the same point F, of
such strength, as to turn the body from the line B C into the line C D.
At D let the body by another impulse, directed likewise to the point F,
be turned out of the line C D into D E. And at E let another impulse,
directed toward the point F, turn the body from the line D E into E
A. Thus we see how a body may be carried through the figure A B C D E
by certain impulses directed always toward the same center, only by
their acting on the body at proper intervals, and with due degrees of
strength.
23. BUT farther, when the body is come to the point A, if it there
receive another impulse directed like the rest toward the point F, and
of such a degree of strength as to turn the body into the line A B,
wherein it first moved; I say that the body shall return into this line
with the same velocity, as it had at first.
24. LET A B be prolonged beyond B at pleasure, suppose to G; and from G
let G H be drawn, which if produced should always continue equidistant
from B F, or, according to the more usual phrase, let G H be drawn
parallel to B F. Then it appears, from what has been said upon the
second law of motion[90], that in the time, wherein the body would have
moved from B to G, had it not received a new impulse in B, by the means
of that impulse it will have acquired a velocity, which will carry it
from B to H. After the same manner, if C I be taken equal to B H,
and I K be drawn equidistant from or parallel to C F; the body will
have moved from C to K with the velocity, which it has in the line C
D, in the same time, as it would have employed in moving from C to I
with the velocity, it had in the line B C. Therefore since C I and B
H are equal, the body will move through C K in the same time, as it
would have taken up in moving from B to G with the original velocity,
wherewith it moved through the line A B. Again, D L being taken equal
to C K and L M drawn parallel to D F; for the same reason as before the
body will move through D M with the velocity, which it has in the line
D E, in the same time, as it would imploy in moving through B G with
its original velocity. In the last place, if E N be taken equal to D M,
and N O be drawn parallel to E F; likewise if A P be taken equal to E
O, and P Q be drawn parallel to A F: then the body with the velocity,
wherewith it returns into the line A B, will pass through A Q in the
same time, as it would have imployed in passing through B G with its
original velocity. Now as all this follows directly from what has above
been delivered, concerning the effect of oblique impulses impressed
upon bodies in motion; so we must here observe farther, that it can be
proved by geometry, that A Q will always be equal to E G. The proof of
this I am obliged, from the nature of my present design, to omit; but
this geometrical proportion being granted, it follows, that the body
has returned into the line A B with the velocity, which it had, when
it first moved in that line; for the velocity, with which it returns
into the line A B, will carry it over the line A Q in the same time, as
would have been taken up in its passing over an equal line B G with
the original velocity.
25. THUS we have found, how a body may be carried round the figure A
B C D E by the action of certain impulses upon it which should all be
pointed toward one center. And we likewise see, that when the body is
brought back again to the point, whence it first set out; if it there
meet with an impulse sufficient to turn it again into the line, wherein
it moved at first, its original velocity will be again restored; and by
the repetition of the same impulses, the body will be carried again in
the same round. Therefore if these impulses, which act on the body at
the points B, C, D, E, and A, continue always the same, the body will
make round this figure innumerable revolutions.
26. THE proof, which we have here made use of, holds the same in any
number of straight lines, whereof the figure A B D should be composed;
and therefore by the method of reasoning referred to above[91] we are
to conclude, that what has here been said upon this rectilinear figure,
will remain true, if this figure were changed into one of a continued
curvature, and instead of distinct impulses acting by intervals at the
angles of this figure, we had a continual centripetal force. We have
therefore shewn, that a body may be carried round in any curve figure
A B C ( fig. 82.) which shall every where be concave towards any one
point as D, by the continual action of a centripetal power directed to
that point, and when it is returned to the point, from whence it set
out, it shall recover again the velocity, with which it departed from
that point. It is not indeed always necessary, that it should return
again into its first course; for the curve line may have some such
figure as the line A B C D B E in fig. 83. In this curve line, if the
body set out from B in the direction B F, and moved through the line B
C D, till it returned to B; here the body would not enter again into
the line B C D, because the two parts B D and B C of the curve line
make an angle at the point B: so that the centripetal power, which at
the point B could turn the body from the line B F into the curve, will
not be able to turn the body into the line B C from the direction, in
which it returns to the point B; a forceable impulse must be given the
body in the point B to produce that effect.
27. IF at the point B, whence the body sets out, the curve line return
into it self (as in fig. 82;) then the body, upon its arrival again at
B, may return into its former course, and thus make an endless circuit
about the center of the centripetal power.
28. WHAT has here been said, I hope, will in some measure enable my
readers to form a just idea of the nature of these centripetal motions.
29. I HAVE not attempted to shew, how to find particularly, what kind
of centripetal force is necessary to carry a body in any curve line
proposed. This is to be deduced from the degree of curvature, which
the figure has in each point of it, and requires a long and complex
mathematical reasoning. However I shall speak a little to the first
proportion, which Sir ~ISAAC NEWTON~ lays down for this
purpose. By this proposition, when a body is found moving in a curve
line, it may be known, whether the body be kept in its course by a
power always pointed toward the same center; and if it be so, where
that center is placed. The proposition is this: that if a line be drawn
from some fixed point to the body, and remaining by one extream united
to that point, it be carried round along with the body; then, if the
power, whereby the body is kept in its course, be always pointed to
this fixed point as a center, this line will move over equal spaces in
equal portions of time. Suppose a body were moving through the curve
line A B C D (in fig. 84.) and passed over the arches A B, B C, C D
in equal portions of time; then if a point, as E, can be found, from
whence the line E A being drawn to the body in A, and accompanying the
body in its motion, it shall make the spaces E A B, E B C, and E C D
equal, over which it passes, while the body describes the arches A B, B
C, and C D: and if this hold the same in all other arches, both great
and small, of the curve line A B C D, that these spaces are always
equal, where the times are equal; then is the body kept in this line by
a power always pointed to E as a center.
30. THE principle, upon which Sir ~ISAAC NEWTON~ has
demonstrated this, requires but small skill in geometry to comprehend.
I shall therefore take the liberty to close the present chapter with
an explication of it; because such an example will give the clearest
notion of our author’s method of applying mathematical reasoning to
these philosophical subjects.
31. HE reasons thus. Suppose a body set out from the point A (in fig.
85.) to move in the straight line A B; and after it had moved for some
time in that line, it were to receive an impulse directed to some point
as C. Let it receive that impulse at D; and thereby be turned into the
line D E; and let the body after this impulse take the same length of
time in passing from D to E, as it imployed in the passing from A to
D. Then the straight lines C A, C D, and C E being drawn, Sir ~ISAAC
NEWTON~ proves, that the and triangular spaces C A D and C D E are
equal. This he does in the following manner.
32. LET E F be drawn parallel to C D. Then, from what has been said
upon the second law of motion[92], it is evident, that since the
body was moving in the line A B, when it received the impulse in the
direction D C; it will have moved after that impulse through the line
D E in the same time, as it would have taken up in moving through D
F, provided it had received no disturbance in D. But the time of the
body’s moving from D to E is supposed to be equal to the time of its
moving through A D; therefore the time, which the body would have
imployed in moving through D F, had it not been disturbed in D, is
equal to the time, wherein it moved through A D: consequently D F is
equal in length to A D; for if the body had gone on to move through
the line A B without interruption, it would have moved through all
parts thereof with the same velocity, and have passed over equal parts
of that line in equal portions of time. Now C F being drawn, since
A D and D F are equal, the triangular space C D F is equal to the
triangular space C A D. Farther, the line E F being parallel to C D, it
is proved by EUCLID, that the triangle C E D is equal to the triangle C
F D[93]: therefore the triangle C E D is equal to the triangle C A D.
33. AFTER the same manner, if the body receive at E another impulse
directed toward the point C, and be turned by that impulse into the
line E G; if it move afterwards from E to G in the same space of time,
as was taken up by its motion from D to E, or from A to D; then C G
being drawn, the triangle C E G is equal to C D E. A third impulse at
G directed as the two former to C, whereby the body shall be turned
into the line G H, will have also the like effect with the rest. If the
body move over G H in the same time, as it took up in moving over E
G, the triangle C G H will be equal to the triangle C E G. Lastly, if
the body at H be turned by a fresh impulse directed toward C into the
line H I, and at I by another impulse directed also to C be turned into
the line I K; and if the body move over each of the lines H I, and I K
in the same time, as it imployed in moving over each of the preceding
lines A D, D E, E G, and G H: then each of the triangles C H I, and C
I K will be equal to each of the preceding. Likewise as the time, in
which the body moves over A D E, is equal to the time of its moving
over E G H, and to the time of its moving over H I K; the space C A D
E will be equal to the space C E G H, and to the space C H I K. In the
same manner as the time, in which the body moved over A D E G is equal
to the time of its moving over G H I K, so the space C A D E G will be
equal to the space C G H I K.
34. FROM this principle Sir ~ISAAC NEWTON~ demonstrates the proposition
mentioned above, by that method of arguing introduced by him into
geometry, whereof we have before taken notice[94], by making according
to the principles of that method a transition from this incurvated
figure composed of straight lines, to a figure of continued curvature;
and by shewing, that since equal spaces are described in equal times
in this present figure composed of straight lines, the same relation
between the spaces described and the times of their description will
also have place in a figure of one continued curvature. He also deduces
from this proposition the reverse of it; and proves, that whenever
equal spaces are continually described; the body is acted upon by
a centripetal force directed to the center, at which the spaces
terminate.
CHAP. IV.
Of the RESISTANCE of FLUIDS.
BEFORE the cause can be discovered, which keeps the planets in motion,
it is necessary first to know, whether the space, wherein they move, is
empty and void, or filled with any quantity of matter. It has been a
prevailing opinion, that all space contains in it matter of some kind
or other; so that where no sensible matter is found, there was yet a
subtle fluid substance by which the space was filled up; even so as
to make an absolute plenitude. In order to examine this opinion, Sir
~ISAAC NEWTON~ has largely considered the effects of fluids upon bodies
moving in them.
2. THESE effects he has reduced under these three heads. In the
first place he shews how to determine in what manner the resistance,
which bodies suffer, when moving in a fluid, gradually increases in
proportion to the space, they describe in any fluid; to the velocity,
with which they describe it; and to the time they have been in motion.
Under the second head he considers what degree of resistance different
bodies moving in the same fluid undergo, according to the different
proportion between the density of the fluid and the density of the
body. The densities of bodies, whether fluid or solid, are measured by
the quantity of matter, which is comprehended under the same magnitude;
that body being the most dense or compact, which under the same bulk
contains the greatest quantity of solid matter, or which weighs most,
the weight of every body being observed above to be proportional
to the quantity of matter in it[95]. Thus water is more dense than
cork or wood, iron more dense than water, and gold than iron. The
third particular Sir ~IS. NEWTON~ considers concerning the
resistance of fluids is the influence, which the diversity of figure in
the resisted body has upon its resistance.
3. FOR the more perfect illustration of the first of these heads, he
distinctly shews the relation between all the particulars specified
upon three different suppositions. The first is, that the same body
be resisted more or less in the simple proportion to its velocity; so
that if its velocity be doubled, its resistance shall become threefold.
The second is of the resistance increasing in the duplicate proportion
of the velocity; so that, if the velocity of a body be doubled, its
resistance shall be rendered four times; and if the velocity be
trebled, nine times as great as at first. But what is to be understood
by duplicate proportion has been already explained[96]. The third
supposition is, that the resistance increases partly in the single
proportion of the velocity, and partly in the duplicate proportion
thereof.
4. IN all these suppositions, bodies are considered under two respects,
either as moving, and opposing themselves against the fluid by
that power alone, which is essential to them, of resisting to the
change of their state from rest to motion, or from motion to rest,
which we have above called their power of inactivity; or else, as
descending or ascending, and so having the power of gravity combined
with that other power. Thus our author has shewn in all those three
suppositions, in what manner bodies are resisted in an uniform fluid,
when they move with the aforesaid progressive motion[97]; and what the
resistance is, when they ascend or descend perpendicularly[98]. And
if a body ascend or descend obliquely, and the resistance be singly
proportional to the velocity, it is shewn how the body is resisted in
a fluid of an uniform density, and what line it will describe[99],
which is determined by the measurement of the hyperbola, and appears
to be no other than that line, first considered in particular by Dr.
~BARROW~[100], which is now commonly known by the name of the
logarithmical curve. In the supposition that the resistance increases
in the duplicate proportion of the velocity, our author has not given
us the line which would be described in an uniform fluid; but has
instead thereof discussed a problem, which is in some sort the reverse;
to find the density of the fluid at all altitudes, by which any given
curve line may be described; which problem is so treated by him, as
to be applicable to any kind of resistance whatever[101]. But here
not unmindful of practice, he shews that a body in a fluid of uniform
density, like the air, will describe a line, which approaches towards
an hyperbola; that is, its motion will be nearer to that curve line
than to the parabola. And consequent upon this remark, he shews how to
determine this hyperbola by experiment, and briefly resolves the chief
of those problems relating to projectiles, which are in use in the art
of gunnery, in this curve[102]; as ~TORRICELLI~ and others
have done in the parabola[103], whose inventions have been explained at
large above[104].
5. OUR author has also handled distinctly that particular sort of
motion, which is described by pendulums[105]; and has likewise
considered some few cases of bodies moving in resisting fluids round a
center, to which they are impelled by a centripetal force, in order to
give an idea of those kinds of motions[106].
6. THE treating of the resistance of pendulums has given him
an opportunity of inserting into another part of his work some
speculations upon the motions of them without resistance, which have
a very peculiar elegance; where in he treats of them as moved by a
gravitation acting in the law, which he shews to belong to the earth
below its surface[107]; performing in this kind of gravitation, where
the force is proportional to the distance from the center, all that
HUYGENS had before done in the common supposition of its being uniform,
and acting in parallel lines[108].
7. HUYGENS at the end of his treatise of the cause of gravity[109]
informs us, that he likewise had carried his speculations on the
first of these suppositions, of the resistance in fluids being
proportional to the velocity of the body, as far as our author. But
finding by experiment that the second was more conformable to nature,
he afterwards made some progress in that, till he was stopt, by not
being able to execute to his wish what related to the perpendicular
descent of bodies; not observing that the measurement of the curve
line, he made use of to explain it by, depended on the hyperbola.
Which oversight may well be pardoned in that great man, considering
that our author had not been pleased at that time to communicate to
the publick his admirable discourse of the QUADRATURE or MEASUREMENT
OF CURVE LINES, with which he has since obliged the world: for without
the use of that treatise, it is I think no injury even to our author’s
unparalleled abilities to believe, it would not have been easy for
himself to have succeeded so happily in this and many other parts of
his writings.
8. WHAT HUYGENS found by experiment, that bodies were in reality
resisted in the duplicate proportion of their velocity, agrees with the
reasoning of our author[110], who distinguishes the resistance, which
fluids give to bodies by the tenacity of their parts, and the friction
between them and the body, from that, which arises from the power of
inactivity, with which the constituent particles of fluids are endued
like all other portions of matter, by which power the particles of
fluids like other bodies make resistance against being put into motion.
9. THE resistance, which arises from the friction of the body
against the parts of the fluid, must be very inconsiderable; and the
resistance, which follows from the tenacity of the parts of fluids, is
not usually very great, and does not depend much upon the velocity of
the body in the fluid; for as the parts of the fluid adhere together
with a certain degree of force, the resistance, which the body receives
from thence, cannot much depend upon the velocity, with which the body
moves; but like the power of gravity, its effect must be proportional
to the time of its acting. This the reader may find farther explained
by Sir ~ISAAC NEWTON~ himself in the postscript to a discourse
published by me in THE PHILOSOPHICAL TRANSACTIONS, N^o 371. The
principal resistance, which most fluids give to bodies, arises from the
power of inactivity in the parts of the fluids, and this depends upon
the velocity, with which the body moves, on a double account. In the
first place, the quantity of the fluid moved out of place by the moving
body in any determinate space of time is proportional to the velocity,
wherewith the body moves; and in the next place, the velocity with
which each particle of the fluid is moved, will also be proportional
to the velocity of the body: therefore since the resistance, which
any body makes against being put into motion, is proportional both
to the quantity of matter moved and the velocity it is moved with;
the resistance, which a fluid gives on this account, will be doubly
increased with the increase of the velocity in the moving body; that
is, the resistance will be in a two-fold or duplicate proportion of the
velocity, wherewith the body moves through the fluid.
10. FARTHER it is most manifest, that this latter kind of resistance
increasing with the increase of velocity, even in a greater degree than
the velocity it self increases, the swifter the body moves, the less
proportion the other species of resistance will bear to this: nay that
this part of the resistance may be so much augmented by a due increase
of velocity, till the former resistances shall bear a less proportion
to this, than any that might be assigned. And indeed experience shews,
that no other resistance, than what arises from the power of inactivity
in the parts of the fluid, is of moment, when the body moves with any
considerable swiftness.
11. THERE is besides these yet another species of resistance, found
only in such fluids, as, like our air, are elastic. Elasticity belongs
to no fluid known to us beside the air. By this property any quantity
of air may be contracted into a less space by a forcible pressure, and
as soon as the compressing power is removed, it will spring out again
to its former dimensions. The air we breath is held to its present
density by the weight of the air above us. And as this incumbent
weight, by the motion of the winds, or other causes, is frequently
varied (which appears by the barometer;) so when this weight is
greatest, we breath a more dense air than at other times. To what
degree the air would expand it self by its spring, if all pressure
were removed, is not known, nor yet into how narrow a compass it is
capable of being compressed. Mr. BOYLE found it by experiment capable
both of expansion and compression to such a degree, that he could cause
a quantity of air to expand it self over a space some hundred thousand
times greater, than the space to which he could confine the same
quantity[111]. But I shall treat more fully of this spring in the air
hereafter[112]. I am now only to consider what resistance to the motion
of bodies arises from it.
12. BUT before our author shews in what manner this cause of resistance
operates, he proposes a method, by which fluids may be rendered
elastic, demonstrating that if their particles be provided with a power
of repelling each other, which shall exert it self with degrees of
strength reciprocally proportional to the distances between the centers
of the particles; that then such fluids will observe the same rule in
being compressed, as our air does, which is this, that the space, into
which it yields upon compression, is reciprocally proportional to the
compressing weight[113]. The term reciprocally proportional has been
explained above[114]. And if the centrifugal force of the particles
acted by other laws, such fluids would yield in a different manner to
compression[115].
13. WHETHER the particles of the air be endued with such a power,
by which they can act upon each other out of contact, our author
does not determine, but leaves that to future examination, and to
be discussed by philosophers. Only he takes occasion from hence to
consider the resistance in elastic fluids, under this notion; making
remarks, as he passes along, upon the differences, which will arise,
if their elasticity be derived from any other fountain[116]. And this,
I think, must be confessed to be done by him with great judgment;
for this is far the most reasonable account, which has been given of
this surprizing power, as must without doubt be freely acknowledged
by any one, who in the least considers the insufficiency of all the
other conjectures, which have been framed; and also how little reason
there is to deny to bodies other powers, by which they may act upon
each other at a distance, as well as that of gravity; which we shall
hereafter shew to be a property universally belonging to all the bodies
of the universe, and to all their parts[117]. Nay we actually find
in the loadstone a very apparent repelling, as well as an attractive
power. But of this more in the conclusion of this discourse.
14. BY these steps our author leads the way to explain the resistance,
which the air and such like fluids will give to bodies by their
elasticity; which resistance he explains thus. If the elastic power
of the fluid were to be varied so, as to be always in the duplicate
proportion of the velocity of the resisted body, it is shewn that
then the resistance derived from the elasticity, would increase in
the duplicate proportion of the velocity; in so much that the whole
resistance would be in that proportion, excepting only that small
part, which arises from the friction between the body and the parts
of the fluid. From whence it follows, that because the elastic power
of the same fluid does in truth continue the same, if the velocity of
the moving body be diminished, the resistance from the elasticity, and
therefore the whole resistance, will decrease in a less proportion,
than the duplicate of the velocity; and if the velocity be increased,
the resistance from the elasticity will increase in a less proportion,
than the duplicate of the velocity, that is in a less proportion, than
the resistance made by the power of inactivity of the parts of the
fluid. And from this foundation is raised the proof of a property of
this resistance, given by the elasticity in common with the others from
the tenacity and friction of the parts of the fluid; that the velocity
may be increased, till this resistance from the fluid’s elasticity
shall bear no considerable proportion to that, which is produced by the
power of inactivity thereof[118]. From whence our author draws this
conclusion; that the resistance of a body, which moves very swiftly in
an elastic fluid, is near the same, as if the fluid were not elastic;
provided the elasticity arises from the centrifugal power of the
parts of the medium, as before explained, especially if the velocity
be so great, that this centrifugal power shall want time to exert it
self[119]. But it is to be observed, that in the proof of all this our
author proceeds upon the supposition of this centrifugal power in the
parts of the fluid; but if the elasticity be caused by the expansion
of the parts in the manner of wool compressed, and such like bodies,
by which the parts of the fluid will be in some measure entangled
together, and their motion be obstructed, the fluid will be in a manner
tenacious, and give a resistance upon that account over and above what
depends upon its elasticity only[120]; and the resistance derived from
that cause is to be judged of in the manner before set down.
15. IT is now time to pass to the second part of this theory; which
is to assign the measure of resistance, according to the proportion
between the density of the body and the density of the fluid. What
is here to be understood by the word density has been explained
above[121]. For this purpose as our author before considered two
distinct cases of bodies moving in mediums; one when they opposed
themselves to the fluid by their power of inactivity only, and another
when by ascending or descending their weight was combined with that
other power: so likewise, the fluids themselves are to be regarded
under a double capacity; either as having their parts at rest, and
disposed freely without restraint, or as being compressed together by
their own weight, or any other cause.
16. IN the first case, if the parts of the fluid be wholly disingaged
from one another, so that each particle is at liberty to move all ways
without any impediment, it is shewn, that if a globe move in such
a fluid, and the globe and particles of the fluid are endued with
perfect elasticity; so that as the globe impinges upon the particles
of it, they shall bound off and separate themselves from the globe,
with the same velocity, with which the globe strikes upon them; then
the resistance, which the globe moving with any known velocity suffers,
is to be thus determined. From the velocity of the globe, the time,
wherein it would move over two third parts of its own diameter with
that velocity, will be known. And such proportion as the density of the
fluid bears to the density of the globe, the same the resistance given
to the globe will bear to the force, which acting, like the power of
gravity, on the globe without intermission during the space of time now
mentioned, would generate in the globe the same degree of motion, as
that wherewith it moves in the fluid[122]. But if neither the globe nor
the particles of the fluid be elastic, so that the particles, when the
globe strikes against them, do not rebound from it, then the resistance
will be but half so much[123]. Again, if the particles of the fluid and
the globe are imperfectly elastic, so that the particles will spring
from the globe with part only of that velocity wherewith the globe
impinges upon them; then the resistance will be a mean between the two
preceding cases, approaching nearer to the first or second, according
as the elasticity is more or less[124].
17. THE elasticity, which is here ascribed to the particles of the
fluid, is not that power of repelling one another, when out of
contact, by which, as has before been mentioned, the whole fluid may be
rendred elastic; but such an elasticity only, as many solid bodies have
of recovering their figure, whenever any forcible change is made in it,
by the impulse of another body or otherwise. Which elasticity has been
explained above at large[125].
18. THIS is the case of discontinued fluids, where the body, by
pressing against their particles, drives them before itself, while
the space behind the body is left empty. But in fluids which are
compressed, so that the parts of them removed out of place by the body
resisted immediately retire behind the body, and fill that space, which
in the other case is left vacant, the resistance is still less; for a
globe in such a fluid which shall be free from all elasticity, will
be resisted but half as much as the least resistance in the former
case[126]. But by elasticity I now mean that power, which renders
the whole fluid so; of which if the compressed fluid be possessed,
in the manner of the air, then the resistance will be greater than
by the foregoing rule; for the fluid being capable in some degree
of condensation, it will resemble so far the case of uncompressed
fluids[127]. But, as has been before related, this difference is most
considerable in slow motions.
19. IN the next place our author is particular in determining the
degrees of resistance accompanying bodies of different figures; which
is the last of the three heads, we divided the whole discourse of
resistance into. And in this disquisition he finds a very surprizing
and unthought of difference, between free and compressed fluids.
He proves, that in the former kind, a globe suffers but half the
resistance, which the cylinder, that circumscribes the globe, will
do, if it move in the direction of its axis[128]. But in the latter
he proves, that the globe and cylinder are resisted alike[129]. And
in general, that let the shape of bodies be ever so different, yet if
the greatest sections of the bodies perpendicular to the axis of their
motion be equal, the bodies will be resisted equally[130].
20. PURSUANT to the difference found between the resistance of the
globe and cylinder in rare and uncompressed fluids, our author gives us
the result of some other inquiries of the same nature. Thus of all the
frustums of a cone, that can be described upon the same base and with
the same altitude, he shews how to find that, which of all others will
be the least resisted, when moving in the direction of its axis[131].
And from hence he draws an easy method of altering the figure of any
spheroidical solid, so that its capacity may be enlarged, and yet the
resistance of it diminished[132]: a note which he thinks may not be
useless to ship-wrights. He concludes with determining the solid, which
will be resisted the least that is possible, in these discontinued
fluids[133].
21. THAT I may here be understood by readers unacquainted with
mathematical terms, I shall explain what I mean by a frustum of a cone,
and a spheroidical solid. A cone has been defined above. A frustum is
what remains, when part of the cone next the vertex is cut away by a
section parallel to the base of the cone, as in fig. 86. A spheroid is
produced from an ellipsis, as a sphere or globe is made from a circle.
If a circle turn round on its diameter, it describes by its motion a
sphere; so if an ellipsis (which figure has been defined above, and
will be more fully explained hereafter[134]) be turned round either
upon the longest or shortest line, that can be drawn through the middle
of it, there will be described a kind of oblong or flat sphere, as
in fig. 87. Both these figures are called spheroids, and any solid
resembling these I here call spheroidical.
22. IF it should be asked, how the method of altering spheroidical
bodies, here mentioned, can contribute to the facilitating a ship’s
motion, when I just above affirmed, that the figure of bodies, which
move in a compressed fluid not elastic, has no relation to the
augmentation or diminution of the resistance; the reply is, that what
was there spoken relates to bodies deep immerged into such fluids, but
not of those, which swim upon the surface of them; for in this latter
case the fluid, by the appulse of the anterior parts of the body, is
raised above the level of the surface, and behind the body is sunk
somewhat below; so that by this inequality in the superficies of
the fluid, that part of it, which at the head of the body is higher
than the fluid behind, will resist in some measure after the manner
of discontinued fluids[135], analogous to what was before observed to
happen in the air through its elasticity, though the body be surrounded
on every side by it[136]. And as far as the power of these causes
extends, the figure of the moving body affects its resistance; for
it is evident, that the figure, which presses least directly against
the parts of the fluid, and so raises least the surface of a fluid
not elastic, and least compresses one that is elastic, will be least
resisted.
23. THE way of collecting the difference of the resistance in rare
fluids, which arises from the diversity of figure, is by considering
the different effect of the particles of the fluid upon the body moving
against them, according to the different obliquity of the several
parts of the body upon which they respectively strike; as it is known,
that any body impinging against a plane obliquely, strikes with a less
force, than if it fell upon it perpendicularly; and the greater the
obliquity is, the weaker is the force. And it is the same thing, if the
body be at rest, and the plane move against it[137].
24. THAT there is no connexion between the figure of a body and its
resistance in compressed fluids, is proved thus. Suppose A B C D (in
fig. 88.) to be a canal, having such a fluid, water for instance,
running through it with an equable velocity; and let any body E, by
being placed in the axis of the canal, hinder the passage of the water.
It is evident, that the figure of the fore part of this body will have
little influence in obstructing the water’s motion, but the whole
impediment will arise from the space taken up by the body, by which it
diminishes the bore of the canal, and straightens the passage of the
water[138]. But proportional to the obstruction of the water’s motion,
will be the force of the water upon the body E[139]. Now suppose both
orifices of the canal to be closed, and the water in it to remain at
rest; the body E to move, so that the parts of the water may pass by
it with the same degree of velocity, as they did before; it is beyond
contradiction, that the pressure of the water upon the body, that
is, the resistance it gives to its motion, will remain the same; and
therefore will have little connexion with the figure of the body[140].
25. BY a method of reasoning drawn from the same fountain is determined
the measure of resistance these compressed fluids give to bodies, in
reference to the proportion between the density of the body and that of
the fluid. This shall be explained particularly in my comment on Sir
~IS. NEWTON~’s mathematical principles of natural philosophy;
but is not a proper subject to be insisted on farther in this place.
26. WE have now gone through all the parts of this theory. There
remains nothing more, but in few words to mention the experiments,
which our author has made, both with bodies falling perpendicularly
through water, and the air[141], and with pendulums[142]: all which
agree with the theory. In the case of falling bodies, the times
of their fall determined by the theory come out the same, as by
observation, to a surprizing exactness; in the pendulums, the rod, by
which the ball of the pendulum hangs, suffers resistance as well as the
ball, and the motion of the ball being reciprocal, it communicates such
a motion to the fluid, as increases the resistance, but the deviation
from the theory is no more, than what may reasonably follow from these
causes.
27. BY this theory of the resistance of fluids, and these experiments,
our author decides the question so long agitated among natural
philosophers, whether all space is absolutely full of matter. The
Aristotelians and Cartesians both assert this plenitude; the Atomists
have maintained the contrary. Our author has chose to determine this
question by his theory of resistance, as shall be explained in the
following chapter.
[Illustration]
[Illustration]
~BOOK II.~
CONCERNING THE
SYSTEM of the WORLD.
CHAP. I.
That the Planets move in a space empty of all sensible matter.
I HAVE now gone through the first part of my design, and have
explained, as far as the nature of my undertaking would permit, what
Sir ~ISAAC NEWTON~ has delivered in general concerning the motion
of bodies. It follows now to speak of the discoveries, he has made
in the system of the world; and to shew from him what cause keeps
the heavenly bodies in their courses. But it will be necessary for
the use of such, as are not skilled in astronomy, to premise a brief
description of the planetary system.
2. THIS system is disposed in the following manner. In the middle is
placed the sun. About him six globes continually roll. These are the
primary planets; that which is nearest to the sun is called Mercury,
the next Venus, next to this is our earth, the next beyond is Mars,
after him Jupiter, and the outermost of all Saturn. Besides these there
are discovered in this system ten other bodies, which move about some
of these primary planets in the same manner, as they move round the
sun. These are called secondary planets. The most conspicuous of them
is the moon, which moves round our earth; four bodies move in like
manner round Jupiter; and five round Saturn. Those which move about
Jupiter and Saturn, are usually called satellites; and cannot any of
them be seen without a telescope. It is not impossible, but there may
be more secondary planets, beside these; though our instruments have
not yet discovered any other. This disposition of the planetary or
solar system is represented in fig. 89.
3. THE same planet is not always equally distant from the sun. But
the middle distance of Mercury is between ⅕ and ⅖ of the distance of
the earth from the sun; Venus is distant from the sun almost ¾ of the
distance of the earth; the middle distance of Mars is something more
than half as much again, as the distance of the earth; Jupiter’s
middle distance exceeds five times the distance of the earth, by
between ⅕ and 1/6 part of this distance; Saturn’s middle distance is
scarce more than 9½ times the distance between the earth and sun; but
the middle distance between the earth and sun is about 217⅛ times the
sun’s semidiameter.
[Illustration]
4. ALL these planets move one way, from west to east; and of the
primary planets the most remote is longest in finishing its course
round the sun. The period of Saturn falls short only sixteen days of 29
years and a half. The period of Jupiter is twelve years wanting about
50 days. The period of Mars falls short of two years by about 43 days.
The revolution of the earth constitutes the year. Venus performs her
period in about 224½ days, and Mercury in about 88 days.
5. THE course of each planet lies throughout in one plane or flat
surface, in which the sun is placed; but they do not all move in the
same plane, though the different planes, in which they move, cross each
other in very small angles. They all cross each other in lines, which
pass through the sun; because the sun lies in the plane of each orbit.
This inclination of the several orbits to each other is represented in
fig. 90. The line, in which the plane of any orbit crosses the plane of
the earth’s motion, is called the line of the nodes of that orbit.
6. EACH planet moves round the sun in the line, which we have mentioned
above[143] under the name of ellipsis; which I shall here shew more
particularly how to describe. I have there said how it is produced in
the cone. I shall now shew how to form it upon a plane. Fix upon any
plane two pins, as at A and B in fig. 91. To these tye a string A C B
of any length. Then apply a third pin D so to the string, as to hold
it strained; and in that manner carrying this pin about, the point of
it will describe an ellipsis. If through the points A, B the straight
line E A B F be drawn, to be terminated at the ellipsis in the points
E and F, this is the longest line of any, that can be drawn within the
figure, and is called the greater axis of the ellipsis. The line G H,
drawn perpendicular to this axis E F, so as to pass through the middle
of it, is called the lesser axis. The two points A and B are called
focus’s. Now each planet moves round the sun in a line of this kind, so
that the sun is found in one focus. Suppose A to be the place of the
sun. Then E is the point, wherein the planet will be nearest of all to
the sun, and at F it will be most remote. The point E is called the
perihelion of the planet, and F the aphelion. In G and H the planet is
said to be in its middle or mean distance; because the distance A G or
A H is truly the middle between A E the least, and A F the greatest
distance. In fig. 92. is represented how the greater axis of each orbit
is situated in respect of the rest. The proportion between the greatest
and least distances of the planet from the sun is very different in the
different planets.
[Illustration]
In Saturn the proportion of the greatest distance to the least is
something less, than the proportion of 9 to 8, but much nearer to
this, than to the proportion of 10 to 9. In Jupiter this proportion
is a little greater, than that of 11 to 10. In Mars it exceeds the
proportion of 6 to 5. In the earth it is about the proportion of 30 to
29. In Venus it is near to that of 70 to 69. And in Mercury it comes
not a great deal short of the proportion of 3 to 2.
[Illustration]
7. EACH of these planets so moves through its ellipsis, that the line
drawn from the sun to the planet, by accompanying the planet in its
motion, will describe about the sun equal spaces in equal times, after
the manner spoke of in the chapter of centripetal forces[144]. There is
also a certain relation between the greater axis’s of these ellipsis’s,
and the times, in which the planets perform their revolutions through
them. Which relation may be expressed thus. Let the period of one
planet be denoted by the letter A, the greater axis of its orbit by
D; let the period of another planet be denoted by B, and the greater
axis of this planet’s orbit by E. Then if C be taken to bear the same
proportion to B, as B bears to A; likewise if F be taken to bear the
same proportion to E, as E bears to D; and G taken to bear the same
proportion likewise to F, as E bears to D; then A shall bear the same
proportion to C, as D bears to G.
8. THE secondary planets move round their respective primary, much
in the same manner as the primary do round the sun. But the motions
of these shall be more fully explained hereafter[145]. And there is,
besides the planets, another sort of bodies, which in all probability
move round the sun; I mean the comets. The farther description of which
bodies I also leave to the place, where they are to be particularly
treated on[146].
9. FAR without this system the fixed stars are placed. These are all so
remote from us, that we seem almost incapable of contriving any means
to estimate their distance. Their number is exceeding great. Besides
two or three thousand, which we see with the naked eye, telescopes open
to our view vast numbers; and the farther improved these instruments
are, we still discover more and more. Without doubt these are luminous
globes, like our sun, and ranged through the wide extent of space; each
of which, it is to be supposed, perform the same office, as our sun,
affording light and heat to certain planets moving about them. But
these conjectures are not to be pursued in this place.
10. I SHALL therefore now proceed to the particular design of this
chapter, and shew, that there is no sensible matter lodged in the space
where the planets move.
11. THAT they suffer no sensible resistance from any such matter, is
evident from the agreement between the observations of astronomers in
different ages, with regard to the time, in which the planets have
been found to perform their periods. But it was the opinion of DES
CARTES[147], that the planets might be kept in their courses by the
means of a fluid matter, which continually circulating round should
carry the planets along with it. There is one appearance that may seem
to favour this opinion; which is, that the sun turns round its own
axis the same way, as the planets move. The earth also turns round its
axis the same way, as the moon moves round the earth. And the planet
Jupiter turns upon its axis the same way, as his satellites revolve
round him. It might therefore be supposed, that if the whole planetary
region were filled with a fluid matter, the sun, by turning round on
its own axis, might communicate motion first to that part of the fluid,
which was contiguous, and by degrees propagate the like motion to the
parts more remote. After the same manner the earth might communicate
motion to this fluid, to a distance sufficient to carry round the moon,
and Jupiter communicate the like to the distance of its satellites.
Sir ~ISAAC NEWTON~ has particularly examined what might be
the result of such a motion as this[148]; and he finds, that the
velocities, with which the parts of this fluid will move in different
distances from the center of the motion, will not agree with the motion
observed in different planets: for instance, that the time of one
intire circulation of the fluid, wherein Jupiter should swim, would
bear a greater proportion to the time of one intire circulation of the
fluid, where the earth is; than the period of Jupiter bears to the
period of the earth. But he also proves[149], that the planet cannot
circulate in such a fluid, so as to keep long in the same course,
unless the planet and the contiguous fluid are of the same density,
and the planet be carried along with the same degree of motion, as
the fluid. There is also another remark made upon this motion by
our author; which is, that some vivifying force will be continually
necessary at the center of the motion[150]. The sun in particular, by
communicating motion to the ambient fluid, will lose from it self as
much motion, as it imparts to the fluid; unless some acting principle
reside in the sun to renew its motion continually. If the fluid be
infinite, this gradual loss of motion would continue till the whole
should stop[151]; and if the fluid were limited, this loss of motion
would continue, till there would remain no swifter a revolution in the
sun, than in the utmost part of the fluid; so that the whole would turn
together about the axis of the sun, like one solid globe[152].
12. IT is farther to be observed, that as the planets do not move in
perfect circles round the sun; there is a greater distance between
their orbits in some places, than in others. For instance, the distance
between the orbit of Mars and Venus is near half as great again in one
part of their orbits, as in the opposite place. Now here the fluid,
in which the earth should swim, must move with a less rapid motion,
where there is this greater interval between the contiguous orbits; but
on the contrary, where the space is straitest, the earth moves more
slowly, than where it is widest[153].
13. FARTHER, if this our globe of earth swam in a fluid of equal
density with the earth it self, that is, in a fluid more dense than
water; all bodies put in motion here upon the earth’s surface must
suffer a great resistance from it; where as, by Sir ~ISAAC
NEWTON~’s experiments mentioned in the preceding chapter, bodies,
that fell perpendicularly down through the air, felt about 1/860 part
only of the resistance, which bodies suffered that fell in like manner
through water.