other planets, they _gravitate_ or have a tendency to approach the sun;
the earth being larger than the moon _attracts_ her, and she
_gravitates_ towards the earth; the planets are attracted by and
gravitate towards each other; a stone when thrown from the earth, by its
attraction and the gravitating power or weight of the stone, is brought
to the earth again; the waters in the ocean gravitate towards the center
of the earth; and it is by this power we stand on all parts of the earth
with our feet pointing to the center.
PUPIL. This information affords me great pleasure.
TUTOR. Having mentioned attraction of magnetism, electricity, and
gravitation, it may not be amiss to inform you of another kind, called
_attraction of cohesion_.
PUPIL. Any thing which tends to my improvement, I shall be obliged to
you to communicate.
TUTOR. By attraction of cohesion is meant that property in bodies which
connects or firmly unites the different particles of matter of which the
body is composed.
PUPIL. Pray, Sir, inform me what you mean by the _laws_ of attraction?
TUTOR. You are to understand, 1st. That _attraction decreases as the
squares of the distances between the centers of the attracting bodies
increase_.
PUPIL. I must beg you, Sir, to explain to me the meaning of the squares
of the distances.
TUTOR. Any number multiplied into itself is a square number, thus 1 is
the square of 1; 4 is the square of 2; 9 is the square of 3, and so on,
because 1 multiplied into itself is 1; 2 by 2 is 4; 3 by 3 is 9, &c. Now
suppose, that when the planet is at B (Plate II. fig. 4.) it is twice as
far from the sun as it is at A: how much more will it be attracted by
the sun at A than at B?
PUPIL. You say, Sir, that the distance is twice as great at B as at A?
TUTOR. I do.
PUPIL. Then as the square of the distance 2 is 4, the decrease of
attraction at B, the planet at A will be attracted with four times the
force it would be at B.—Am I right, Sir?
TUTOR. Perfectly so. And if the distance at B were three times as great
as at A, it would be attracted with a force nine times as great.
PUPIL. I perceive it must be so.
TUTOR. I shall now give you the 2d law, namely, That _bodies attract one
another with forces proportionable to the quantities of matter they
contain_.
PUPIL. Do all bodies of the same magnitude contain equal quantities of
matter?
TUTOR. No, certainly: For a ball of cork may be as large as one of lead,
and yet not contain the same quantity of matter, because it is more
porous, and not so compact or dense a body as the lead; neither will a
ball of lead of the same magnitude as one of gold contain an equal
quantity of matter.—So the sun, though a million of times as big as the
earth, contains a quantity of matter only 200,000 as great, therefore
attracts the earth with a force 200,000 as great as the earth attracts
him.
PUPIL. I think this is clear.
TUTOR. We will now suppose that in the river are two boats of equal
bulk, at the distance of twenty yards from each other, and that a man in
one boat pulls a rope which is fastened to the other, what effect will
be produced, or where do you think the boats will meet?
PUPIL. Had you not told me that bodies attract one another with forces
which are proportioned to the quantities of matter they contain, I
should say the boat to which the rope is fastened would come to that in
which the man stands: but as I imagine you mean to apply this to
attraction, by the above rule, they will meet at a point which is half
way between them.
TUTOR. If one boat were three times the bulk of the other, how then?
PUPIL. The lightest would move three times as far as the heaviest, or 15
yards whilst the heaviest moved only 5.
TUTOR. Upon my word you reason philosophically. In both cases you are
perfectly right.
PUPIL. As the sun is so immense a body that his quantity of matter is so
much greater than the planets, I am at a loss to know why they are not
by the power of attraction drawn to him.
TUTOR. And so they would if the attractive power were not counteracted
by another of equal force.
PUPIL. Did you not say, Sir, that the planets are kept in their orbits
by attraction?
TUTOR. I did. But you find that by attraction _only_ the sun would draw
all the planets to himself.
PUPIL. That is evident. But I wish to know what this counteracting power
you speak of is?
TUTOR. I will tell you presently.—You must remember that _simple_ motion
is naturally rectilineal, that is, all bodies, if there were nothing to
prevent them, would move in strait lines.
PUPIL. Then as the planetary motion is circular, it cannot be simple?
TUTOR. No. It is a _compound_ of the two forces I have been mentioning:
the one is called the attractive or centripetal force; the other, the
projectile or centrifugal force.
PUPIL. The former I clearly comprehend, but not the latter. I can
conceive, that if two bodies approach each other by attraction they must
move in a right line.
TUTOR. If you shoot a marble on a smooth piece of ice, in what direction
will it run?
PUPIL. Strait forward.
TUTOR. This is a projectile force.—Could you, do you think, shoot it in
any other direction?
PUPIL. No, Sir.
TUTOR. Then is not this motion also rectilineal?
PUPIL. It is.
TUTOR. When you strike a ball with your cricket-bat, or throw a stone
with your hand, is it not projected or thrown forward by the force of
the bat or hand?
PUPIL. Certainly.
TUTOR. And does it not move in a strait line?
PUPIL. At first it appears to do so; but afterwards it inclines towards
and falls to the earth.
TUTOR. Cannot you account for this?
PUPIL. I suppose it must be drawn to the earth by attraction.
TUTOR. You are right. The attraction of the earth, and the resistance of
the atmosphere or air through which it moves, retards its progress, or
it would continue moving in a strait line, with a velocity equal to that
which was at first impressed upon it. In like manner the beneficent
Creator of the Universe impressed a force on all the planets which
should be equal to that of the attractive power of the sun, that one
might not overcome the other.
PUPIL. This wants explaining.
TUTOR. I would willingly gratify you, but as I have much more to say on
the subject, I fear it will be too great a burthen on your memory; it
will therefore be better to postpone it.
PUPIL. As you please, Sir.
------------------------------------------------------------------------
DIALOGUE VII.
TUTOR.
Having at our last meeting explained to you the nature of the attractive
and projectile forces, I shall proceed to shew you that it is by the
joint action or combination of these two forces that the planets are
retained in their orbits.
PUPIL. I am all anxiety, as I wish to be informed how, or in what manner
they can act against each other, to produce that effect.
TUTOR. Answer me a few questions, and you will soon know.
PUPIL. As many as you please, Sir.
TUTOR. If you whirl a stone in a sling, what will be its motion?
PUPIL. Circular.
TUTOR. Is you let it suddenly slip out of the sling, will it continue
its circular motion?
PUPIL. No, Sir, but fly off in a strait line.
TUTOR. This line you must remember is what mathematicians call the
tangent of a circle, as A _a_, B _b_, &c. (Plate II. fig. 5.) for all
bodies moving in a circle have a natural tendency to fly off in that
direction. Thus a body at A will tend towards _a_; at B towards _b_, and
so on; but the central force acting against it preserves its circular
motion.
PUPIL. By the central force here you mean the action of the hand, do you
not?
TUTOR. Yes. For, as soon as the stone is released and that power is
lost, it assumes its natural, that is, its rectilineal motion.—Again. If
you are left at liberty, cannot you run strait forward?
PUPIL. Yes, Sir.
TUTOR. Now, suppose one of your companions were to fasten a rope round
your body, and at the extent of it were to stand still and hold it
tight, with a force equal to that with which you run, could you, do you
think, move in a strait line, that is, in a tangent of a circle?
PUPIL. No, Sir. I must run in a circle.
TUTOR. Why?
PUPIL. Because, whilst the rope is extended I am prevented running in
any other direction.
TUTOR. Just so it is with the planets: the attractive or centripetal
force of the sun being equal to that of the projectile or centrifugal
force of the planets, they are by attraction prevented moving on in a
strait line, and, as it were, drawn towards the sun; and by the
projectile force from being overcome by attraction. They must therefore
revolve in circular orbits.
PUPIL. What I have so long wished is now accomplished. I understand it
perfectly.
TUTOR. What I have now explained relates not only to the primary planets
which have the sun for their center of motion; but, you must remember
that the secondary planets are governed by the same laws, in revolving
about their respective primaries; for, as by the attractive power of the
sun combined with the projectile force of the primary planets they are
retained in their orbits; so also the action of the primaries upon their
respective secondaries together with their projectile force, will
preserve them in their orbits.
PUPIL. Pray, Sir, what have you else to observe?
TUTOR. Have I not told you that the orbits of the planets are not true
circles, but a little elliptical?
PUPIL. Yes, Sir; and I shall be glad to know the reason of it.
TUTOR. If the attractive power of the sun were uniformly the same in
every part of their orbits they would be true circles, and the planets
would pass over _equal_ portions of their orbits in _equal_ times; that
is, they would move from B to C, (Plate II. fig. 5.) in the same time as
from A to B, &c.
PUPIL. That is clear, but as their orbits are elliptical, when the
planets are farthest from the sun, the velocity with which they move
must be lessened as the attraction is decreased.
TUTOR. And they must consequently pass over _unequal_ parts of their
orbits in _equal_ portions of time. And, as _a double velocity will
balance a quadruple or fourfold power of gravity or attraction_, it
follows, that as the centripetal force is four times as great at A as at
B (Plate II. fig. 4.) the centrifugal force will be twice as great, and
would carry a planet from A to _a_ in the same time it would from B to
_b_, and in its orbit from A to _c_ as soon as from B to _d_, and
thereby describe the area, or space contained between the letters A S
_c_, in the same time as the area or space B S _d_. For according to the
laws of the planetary motions, in their periodical revolutions, _they
always describe equal areas in equal times_.
PUPIL. The orbits of the comets being very elliptical, the irregularity
of their motions must be exceedingly great.
TUTOR. Great, indeed!—One of them passed so near the sun as to acquire a
heat which Sir Isaac Newton computed to be two thousand times hotter
than red hot iron.[12]
[Footnote 12: Dr. Herschel is of opinion, that bodies near the sun
do not acquire so great a degree of heat as has been generally
imagined.]
PUPIL. Astonishing! If they pass so near the sun, the centripetal force
must act powerfully on the body of the comet.
TUTOR. And that force, you know, must be equalled by the projectile
force; so you find they move when near the sun with amazing
celerity.—But when arrived at their aphelion, where the influence of the
sun is weak, what a transition!
PUPIL. Wonderful, indeed!—Their motion is excessively slow, and the sun
must appear little more than a fixed star. Surely they cannot be
inhabited, can they?
TUTOR. We cannot speak positively; but, as they differ so much from the
planets, which we have reason to suppose are so, it is imagined they are
designed for some purpose unknown to us.
PUPIL. When is the earth in its perihelion?
TUTOR. In December; and our summer half year is longer than the winter
half, by about eight days.
PUPIL. I suppose this is occasioned by the inequality of the earth’s
annual motion.
TUTOR. It is; and this inequality is the cause of the difference of time
between the sun and a well regulated clock; the latter keeps equal time,
whilst the former is constantly varying.
PUPIL. I have often seen in the almanack clock fast, clock slow, but did
not know the meaning of it: I imagine it is that the clock should be so
much faster or slower than the time by the sun as is there mentioned.
TUTOR. It is: but there are tables calculated to shew the difference of
time for every day in the year; so that if you know the exact times of
the day by the sun, and have one of these tables, you will see what the
time should be by the clock, to a second, which is not shewn in a common
almanack.
PUPIL. In speaking of the annual or yearly motion of the earth, you have
no where mentioned the cause of the seasons; will it be agreeable to do
it now, Sir?
TUTOR. The vicissitudes of the seasons, the cause of day and night, &c.
shall be the subject of future lessons: we shall find sufficient to
employ us at present.
PUPIL. I think you told me just now that the earth is nearest the sun in
December; that is our winter; this seems a little mysterious.
TUTOR. It may appear so to you now, by-and-by you will be of a different
opinion. I shall explain this matter to you with that of the seasons,
&c.
PUPIL. I fear I have interrupted you.—As you said you had sufficient
employment for us, I shall be glad to know what it is.
TUTOR. Hitherto I have spoken of the sun’s being fixed, and that the
planets revolve about him as a center. Instead of which the sun and
planets move round one common center, called the center of gravity.
PUPIL. What is this center of gravity?
TUTOR. Have you never seen a person raise a heavy weight by means of a
long pole or leaver, which it was not in his power to lift without it?
PUPIL. Yes, Sir, and it excited my astonishment.
TUTOR. Now, suppose the weight to see raised to be 10 Cwt. and the prop
on which the leaver rested 1 foot from the body to be raised; and the
person at the other end of the leaver 10 feet from the prop; with what
weight must he press to raise the 10 Cwt.?
PUPIL. I think that very easy; for, as he is ten times as far from the
prop as the weight is, a pressure of 1 Cwt. which is one-tenth of the
weight to be raised will do it.
TUTOR. To be sure; and yet you say you were astonished when you saw it!
Every thing we do not understand at first appears difficult.—To apply
this to our present purpose. You see that a weight of 1 Cwt. at 10 feet
from a prop, will balance another of 10 Cwt. at one foot from it. Now,
instead of a prop let the two weights be nicely poised on a center,
round which they may freely turn; the heaviest would move in a circle,
whose radius, or distance from the center would be one foot, whilst the
lightest would move in one 10 feet from the center in the same time.
PUPIL. Is the center round which they move the center of gravity?
TUTOR. It is; and round an imaginary point as a center the sun and
planets move, always preserving an equilibrium. If the earth were the
only attendant on the sun, as his quantity of matter is 200,000 times as
great as that of the earth, he would revolve in a circle a 200,000th
part of the earth’s distance from him, in the same time as the earth is
making one revolution in its orbit, or in one year; but, as the planets
in their orbits must vary in their positions, the center of gravity
cannot be always at the same distance from the sun.
PUPIL. If it were, the balance could not be preserved.
TUTOR. Clearly so. But you must know that the quantity of matter in the
sun so far exceeds that of all the planets together, that even if they
were all in a line on one side of him he would never be more than his
own diameter distant from his center of gravity; therefore, astronomers
consider the sun as the center of the system, and express themselves
accordingly.
PUPIL. As you told me the secondary planets are governed by the same
laws as the primaries, I imagine they also with their primaries move
round a center of gravity.
TUTOR. They do so.—The earth and moon, Jupiter with his satellites,
Saturn and his attendants, revolve about their respective centers;
these, with the sun and the rest of the planetary system, make their
circuits round their center; every system in the universe is supposed to
revolve in like manner; and all these together to move round one _common
center_.—How are we lost in contemplating the omniscience of the Deity!
How difficult to conceive so many millions of bodies of dead matter
constantly in motion, so nicely balanced and governed by such unerring
laws!—Well may we say with the Psalmist, “Lord! how manifold are thy
works, in wisdom hast thou made them all.”
------------------------------------------------------------------------
DIALOGUE VIII.
TUTOR.
I shall now, agreeably to my promise, explain to you the cause of day
and night, and then proceed with the vicissitudes of the seasons.
PUPIL. That is what I much wish to know; and had you not told me that
the earth moved round the sun every year, I should have found no
difficulty in accounting for the succession of day and night, since the
sun appears to rise and set every day.
TUTOR. That is true; but I think I must have convinced you that so
immense a body as the sun cannot revolve about the earth; as well may
you suppose that in roasting a bird it is necessary that the fire should
move round it.
PUPIL. That I think would be very absurd, as it is much easier for the
bird on the spit to turn to the fire, than for the fire to go round the
bird.
TUTOR. You are certainly right, and if the earth revolve on its axis
every twenty-four hours, will not the different parts of it be
alternately turned to the sun, as the bird on the spit is to the fire?
PUPIL. I do not clearly comprehend what you mean by the axis of the
earth; for, as it moves in open space and has no support, it can have
nothing to resemble the spit on which it turns.
TUTOR. Certainly not. By the earth’s axis is meant an imaginary line
passing through its center, on which it is supposed to turn; as your
ball if rolled on the ground would revolve on an axis whilst it was
moving forward.
PUPIL. I can now answer your question in the affirmative: and, as our
year consists of 365 days, I imagine the earth must make as many
revolutions on its axis whilst it is going once round the sun.
TUTOR. Undoubtedly: and as only one half of a spherical body can at any
time be enlightened by a luminous body, that part of the earth only
which is turned to the sun, can receive the benefit of his enlivening
rays, when it will be day; whilst the opposite part will be involved in
darkness, and it will be night.
PUPIL. I perceive it must be so. But, if the earth move in the manner
you describe, I cannot conceive how it is that we are not sensible of
its motion.
TUTOR. If the motion of the earth were irregular it would be
perceptible; but as it meets with no obstruction the motion must be so
uniform as not to be perceived.
PUPIL. Had I recollected this, I need not have given you this
trouble.—But I am continually meeting with fresh difficulties.
TUTOR. You have only to mention what they are, and I shall take a
pleasure in removing them.
PUPIL. I thank you, Sir; and shall be obliged to you to inform me, how
the motion of the earth can cause the sun to appear to move?
TUTOR. When in a carriage which went smoothly on the road, or in a boat
whose motion was scarcely perceptible on the water, did you never fix
your attention on the objects you passed?
PUPIL. Yes, often, Sir.
TUTOR. And had you not known that you really moved, and that the trees,
&c. were immoveable in the ground, what then would have been your
opinion?
PUPIL. That the trees, &c. moved in a direction contrary to that in
which I was moving.
TUTOR. Is not this sufficient to convince you that the apparent motion
of the sun may be occasioned by the revolution of the earth on its axis?
PUPIL. It is:—But if so large a body as the earth make a revolution on
its axis in 24 hours, it must move with great velocity.
TUTOR. It does so; and the inhabitants of London by this motion are
carried at the rate of 560 miles an hour[13].
[Footnote 13: The hourly motion under the equator is 900 miles.]
PUPIL. What an astonishing rapidity!
TUTOR. Now, the sun with the rest of the heavenly bodies must move round
the earth, or the earth must revolve on its axis in 24 hours, to cause
that appearance.
PUPIL. That is plain.
TUTOR. Well then, great as you may suppose the velocity of the earth on
its axis to be, if the sun move round the earth his hourly motion will
be nearly 25 millions of miles; and beyond conception would be that of
the fixed stars. Which now do you think is most probable, that the sun
and stars should move round the earth, or that they, by the simple
motion of the earth, should appear to be in motion?
PUPIL. The latter, to be sure, Sir.—I have one difficulty remaining,
which is this; if a lark rise from a field near London and remain in the
air a quarter of an hour, if the earth move at the rate of 560 miles an
hour, it will go 140 miles whilst the lark is suspended, and yet it
continues over the field,—how can this be?
TUTOR. This objection to the motion of the earth has been made by those
who were older and who thought themselves wiser too than yourself. They
either did not know or did not consider, that the atmosphere which
surrounds the earth is a part of itself, and gravitates towards it, and
therefore partakes of the earth’s motion and carries the lark along with
it. Besides, as the Sun, Venus, Mars, and Jupiter are known to revolve
on their axes, we have reason to suppose that the other planets,
together with the earth, must have the same motion[14].
[Footnote 14: Dr. Herschell says that several of the fixed stars
revolve on their axes.]
PUPIL. How is it known that they do revolve on their axes; and in what
time do they perform their revolutions?
TUTOR. By the assistance of telescopes dark spots have been seen on the
disc of the sun, by the motion of which it is found that he revolves on
his axis in 25-1/4 days; Venus performs her diurnal revolution in about
23 ho. 21 min.; Mars goes round his axis in 24 ho. 39 min.; and Jupiter
in 9 ho. 56 min.; as to the rest, no spot or any fixed point has been
discovered to ascertain the length of their day; Mercury being too near
the sun, and Saturn and the Georgium Sidus too remote for our
observations.
PUPIL. I can no longer doubt of the earth’s motion: and, if it will not
be improper, a description of the atmosphere will give me pleasure.
TUTOR. That I can have no objection to. The atmosphere is a thin,
invisible fluid, most dense or heavy near the earth, but grows gradually
rarer or lighter the higher we ascend, so much so, that at the tops of
some high mountains it is difficult to breathe. It serves not only to
suspend the clouds, furnish us with wind and rain, and answer the common
purposes of breathing, but is also the cause of the morning and evening
twilight, and of all the glory and brightness of the firmament.
PUPIL. How, pray?
TUTOR. If there were no atmosphere, the sun would yield no light but
when our eyes were directed towards him; and the heavens would appear
dark and as full of stars as on a dark winter’s night; but the
atmosphere being strongly illuminated by the sun, reflects the light
back upon us, and makes the whole heavens to shine so strongly, that the
faint light of the stars is obscured, and they are rendered invisible.
PUPIL. I find then the atmosphere is of more use than I imagined. But
how is it the cause of the twilight?
TUTOR. The atmosphere is about 45 miles above the surface of the earth,
therefore the sun’s rays falling upon the higher parts of it before
rising, by reflection causes a faint light, which increases till he
appears above the horizon; and in the evening it decreases after he
sets, till he is 18 degrees below the horizon, where the morning
twilight begins, and the evening twilight ends.
PUPIL. By the horizon, I think you mean that distant boundary of our
sight where the heavens and the earth seem to join all around us, as it
appears from an eminence.
TUTOR. The very same. ’Tis that imaginary circle which intercepts from
our view the sun, moon, and stars each night; and when, by the rotation
of the earth, they appear to descend below it, we say they are set; as
on the contrary, each morning, when they appear above it, we say they
rise.
“To find the spacious line, cast round thine eyes,
“And where the earth’s high surface joins the skies,
“Where stars first set, and first begin to shine,
“There draw the fancy’d image of this line.”
PUPIL. A very pleasing description, indeed.