THE LAW OF GRAVITATION.
Gravitation--The Falling of a Stone to the Ground--All Bodies fall
equally, Sixteen Feet in a Second--Is this true at Great
Heights?--Fall of a Body at a Height of a Quarter of a Million
Miles--How Newton obtained an Answer from the Moon--His Great
Discovery--Statement of the Law of Gravitation--Illustrations of
the Law--How is it that all the Bodies in the Universe do not rush
Together?--The Effect of Motion--How a Circular Path can be
produced by Attraction--General Account of the Moon's Motion--Is
Gravitation a Force of Great Intensity?--Two Weights of 50
lbs.--Two Iron Globes, 53 Yards in Diameter, and a Mile apart,
attract with a Force of 1 lb.--Characteristics of
Gravitation--Orbits of the Planets not strictly Circles--The
Discoveries of Kepler--Construction of an Ellipse--Kepler's First
Law--Does a Planet move Uniformly?--Law of the Changes of
Velocity--Kepler's Second Law--The Relation between the Distances
and the Periodic Times--Kepler's Third Law--Kepler's Laws and the
Law of Gravitation--Movement in a Straight Line--A Body unacted on
by Disturbing Forces would move in a Straight Line with Constant
Velocity--Application to the Earth and the Planets--The Law of
Gravitation deduced from Kepler's Laws--Universal Gravitation.
Our description of the heavenly bodies must undergo a slight
interruption, while we illustrate with appropriate detail an important
principle, known as the law of gravitation, which underlies the whole of
astronomy. By this law we can explain the movements of the moon around
the earth, and of the planets around the sun. It is accordingly
incumbent upon us to discuss this subject before we proceed to the more
particular account of the separate planets. We shall find, too, that the
law of gravitation sheds some much-needed light on the nature of the
stars situated at the remotest distances in space. It also enables us to
cast a glance through the vistas of time past, and to trace with
plausibility, if not with certainty, certain early phases in the history
of our system. The sun and the moon, the planets and the comets, the
stars and the nebulae, all alike are subject to this universal law, which
is now to engage our attention.
What is more familiar than the fact that when a stone is dropped it
will fall to the ground? No one at first thinks the matter even worthy
of remark. People are often surprised at seeing a piece of iron drawn to
a magnet. Yet the fall of a stone to the ground is the manifestation of
a force quite as interesting as the force of magnetism. It is the earth
which draws the stone, just as the magnet draws the iron. In each case
the force is one of attraction; but while the magnetic attraction is
confined to a few substances, and is of comparatively limited
importance, the attraction of gravitation is significant throughout the
universe.
Let us commence with a few very simple experiments upon the force of
gravitation. Hold in the hand a small piece of lead, and then allow it
to drop upon a cushion. The lead requires a certain time to move from
the fingers to the cushion, but that time is always the same when the
height is the same. Take now a larger piece of lead, and hold one piece
in each hand at the same height. If both are released at the same
moment, they will both reach the cushion simultaneously. It might have
been thought that the heavy body would fall more quickly than the light
body; but when the experiment is tried, it is seen that this is not the
case. Repeat the experiment with various other substances. An ordinary
marble will be found to fall in the same time as the piece of lead. With
a piece of cork we again try the experiment, and again obtain the same
result. At first it seems to fail when we compare a feather with the
piece of lead; but that is solely on account of the air, which resists
the feather more than it resists the lead. If, however, the feather be
placed upon the top of a penny, and the penny be horizontal when
dropped, it will clear the air out of the way of the feather in its
descent, and then the feather will fall as quickly as the penny, as
quickly as the marble, or as quickly as the lead.
If the observer were in a gallery when trying these experiments, and if
the cushion were sixteen feet below his hands, then the time the marble
would take to fall through the sixteen feet would be one second. The
time occupied by the cork or by the lead would be the same; and even the
feather itself would fall through sixteen feet in one second, if it
could be screened from the interference of the air. Try this experiment
where we like, in London, or in any other city, in any island or
continent, on board a ship at sea, at the North Pole, or the South Pole,
or the equator, it will always be found that any body, of any size or
any material, will fall about sixteen feet in one second of time.
Lest any erroneous impression should arise, we may just mention that the
distance traversed in one second does vary slightly at different parts
of the earth, but from causes which need not at this moment detain us.
We shall for the present regard sixteen feet as the distance through
which any body, free from interference, would fall in one second at any
part of the earth's surface. But now let us extend our view above the
earth's surface, and enquire how far this law of sixteen feet in a
second may find obedience elsewhere. Let us, for instance, ascend to the
top of a mountain and try the experiment there. It would be found that
at the top of the mountain a marble would take a little longer to fall
through sixteen feet than the same marble would if let fall at its base.
The difference would be very small; but yet it would be measurable, and
would suffice to show that the power of the earth to pull the marble to
the ground becomes somewhat weakened at a point high above the earth's
surface. Whatever be the elevation to which we ascend, be it either the
top of a high mountain, or the still greater altitudes that have been
reached in balloon ascents, we shall never find that the tendency of
bodies to fall to the ground ceases, though no doubt the higher we go
the more is that tendency weakened. It would be of great interest to
find how far this power of the earth to draw bodies towards it can
really extend. We cannot attain more than about five or six miles above
the earth's surface in a balloon; yet we want to know what would happen
if we could ascend 500 miles, or 5,000 miles, or still further, into the
regions of space.
Conceive that a traveller were endowed with some means of soaring aloft
for miles and thousands of miles, still up and up, until at length he
had attained the awful height of nearly a quarter of a million of miles
above the ground. Glancing down at the surface of that earth, which is
at such a stupendous depth beneath, he would be able to see a wonderful
bird's-eye view. He would lose, no doubt, the details of towns and
villages; the features in such a landscape would be whole continents and
whole oceans, in so far as the openings between the clouds would permit
the earth's surface to be exposed.
At this stupendous elevation he could try one of the most interesting
experiments that was ever in the power of a philosopher. He could test
whether the earth's attraction was felt at such a height, and he could
measure the amount of that attraction. Take for the experiment a cork, a
marble, or any other object, large or small; hold it between the
fingers, and let it go. Everyone knows what would happen in such a case
down here; but it required Sir Isaac Newton to tell what would happen in
such a case up there. Newton asserts that the power of the earth to
attract bodies extends even to this great height, and that the marble
would fall. This is the doctrine that we can now test. We are ready for
the experiment. The marble is released, and, lo! our first exclamation
is one of wonder. Instead of dropping instantly, the little object
appears to remain suspended. We are on the point of exclaiming that we
must have gone beyond the earth's attraction, and that Newton is wrong,
when our attention is arrested; the marble is beginning to move, so
slowly that at first we have to watch it carefully. But the pace
gradually improves, so that the attraction is beyond all doubt, until,
gradually acquiring more and more velocity, the marble speeds on its
long journey of a quarter of a million of miles to the earth.
But surely, it will be said, such an experiment must be entirely
impossible; and no doubt it cannot be performed in the way described.
The bold idea occurred to Newton of making use of the moon itself, which
is almost a quarter of a million of miles above the earth, for the
purpose of answering the question. Never was our satellite put to such
noble use before. It is actually at each moment falling in towards the
earth. We can calculate how much it is deflected towards the earth in
each second, and thus obtain a measure of the earth's attractive power.
From such enquiries Newton was able to learn that a body released at the
distance of 240,000 miles above the surface of the earth would still be
attracted by the earth, that in virtue of the attraction the body would
commence to move off towards the earth--not, indeed, with the velocity
with which a body falls in experiments on the surface, but with a very
much lesser speed. A body dropped down from the distance of the moon
would commence its long journey so slowly that a _minute_, instead of a
_second_, would have elapsed before the distance of sixteen feet had
been accomplished.[11]
It was by pondering on information thus won from the moon that Newton
made his immortal discovery. The gravitation of the earth is a force
which extends far and wide through space. The more distant the body, the
weaker the gravitation becomes; here Newton found the means of
determining the great problem as to the law according to which the
intensity of the gravitation decreased. The information derived from the
moon, that a body 240,000 miles away requires a minute to fall through a
space equal to that through which it would fall in a second down here,
was of paramount importance. In the first place, it shows that the
attractive power of the earth, by which it draws all bodies earthwards,
becomes weaker at a distance. This might, indeed, have been anticipated.
It is as reasonable to suppose that as we retreated further and further
into the depths of space the power of attraction should diminish, as
that the lustre of light should diminish as we recede from it; and it is
remarkable that the law according to which the attraction of gravitation
decreases with the increase of distance is precisely the same as the
law according to which the brilliancy of a light decreases as its
distance increases.
The law of nature, stated in its simplest form, asserts that the
intensity of gravitation varies inversely as the square of the distance.
Let me endeavour to elucidate this somewhat abstract statement by one or
two simple illustrations. Suppose a body were raised above the surface
of the earth to a height of nearly 4,000 miles, so as to be at an
altitude equal to the radius of the earth. In other words, a body so
situated would be twice as far from the centre of the earth as a body
which lay on the surface. The law of gravitation says that the intensity
of the attraction is then to be decreased to one-fourth part, so that
the pull of the earth on a body 4,000 miles high is only one quarter of
the pull of the earth on that body so long as it lies on the ground. We
may imagine the effect of this pull to be shown in different ways. Allow
the body to fall, and in the interval of one second it will only drop
through four feet, a mere quarter of the distance that gravity would
cause near the earth's surface.
We may consider the matter in another way by supposing that the
attraction of the earth is measured by one of those little weighing
machines known as a spring balance. If a weight of four pounds be hung
on such a contrivance, at the earth's surface, the index of course shows
a weight of four pounds; but conceive this balance, still bearing the
weight appended thereto, were to be carried up and up, the _indicated_
strain would become less and less, until by the time the balance reached
4,000 miles high, where it was _twice_ as far away from the earth's
centre as at first, the indicated strain would be reduced to the
_fourth_ part, and the balance would only show one pound. If we could
imagine the instrument to be carried still further into the depths of
space, the indication of the scale would steadily continue to decline.
By the time the apparatus had reached a distance of 8,000 miles high,
being then _three_ times as far from the earth's centre as at first, the
law of gravitation tells us that the attraction must have decreased to
one-ninth part. The strain thus shown on the balance would be only the
ninth part of four pounds, or less than half a pound. But let the voyage
be once again resumed, and let not a halt be made this time until the
balance and its four-pound weight have retreated to that orbit which the
moon traverses in its monthly course around the earth. The distance thus
attained is about sixty times the radius of the earth, and consequently
the attraction of gravitation is diminished in the proportion of one to
the square of sixty; the spring will then only be strained by the
inappreciable fraction of 1-3,600 part of four pounds. It therefore
appears that a weight which on the earth weighed a ton and a half would,
if raised 240,000 miles, weigh less than a pound. But even at this vast
distance we are not to halt; imagine that we retreat still further and
further; the strain shown by the balance will ever decrease, but it will
still exist, no matter how far we go. Astronomy appears to teach us that
the attraction of gravitation can extend, with suitably enfeebled
intensity, across the most profound gulfs of space.
The principle of gravitation is of far wider scope than we have yet
indicated. We have spoken merely of the attraction of the earth, and we
have stated that this force extends throughout space. But the law of
gravitation is not so limited. Not only does the earth attract every
other body, and every other body attract the earth, but each of these
bodies attracts the other; so that in its more complete shape the law of
gravitation announces that "every body in the universe attracts every
other body with a force which varies inversely as the square of the
distance."
It is impossible for us to over-estimate the importance of this law. It
supplies the clue by which we can unravel the complicated movements of
the planets. It has led to marvellous discoveries, in which the law of
gravitation has enabled us to anticipate the telescope, and to feel the
existence of bodies before those bodies have even been seen.
An objection which may be raised at this point must first be dealt with.
It seems to be, indeed, a plausible one. If the earth attracts the moon,
why does not the moon tumble down on the earth? If the earth is
attracted by the sun, why does it not tumble into the sun? If the sun is
attracted by other stars, why do they not rush together with a frightful
collision? It may not unreasonably be urged that if all these bodies in
the heavens are attracting each other, it would seem that they must all
rush together in consequence of that attraction, and thus weld the whole
material universe into a single mighty mass. We know, as a matter of
fact, that these collisions do not often happen, and that there is
extremely little likelihood of their taking place. We see that although
our earth is said to have been attracted by the sun for countless ages,
yet the earth is just as far from the sun as ever it was. Is not this in
conflict with the doctrine of universal gravitation? In the early days
of astronomy such objections would be regarded, and doubtless were
regarded, as well-nigh insuperable; even still we occasionally hear them
raised, and it is therefore the more incumbent on us to explain how it
happens that the solar system has been able to escape from the
catastrophe by which it seems to be threatened.
There can be no doubt that if the moon and the earth had been initially
placed _at rest_, they would have been drawn together by their mutual
attraction. So, too, if the system of planets surrounding the sun had
been left initially _at rest_ they would have dashed into the sun, and
the system would have been annihilated. It is the fact that the planets
are _moving_, and that the moon is _moving_, which has enabled these
bodies successfully to resist the attraction in so far, at least, as
that they are not drawn thereby to total destruction.
It is so desirable that the student should understand clearly how a
central attraction is compatible with revolution in a nearly circular
path, that we give an illustration to show how the moon pursues its
monthly orbit under the guidance and the control of the attracting
earth.
[Illustration: Fig. 36.--Illustration of the Moon's Motion.]
The imaginary sketch in Fig. 36 denotes a section of the earth with a
high mountain thereon.[12] If a cannon were stationed on the top of the
mountain at C, and if the cannonball were fired off in the direction C E
with a moderate charge of powder, the ball would move down along the
first curved path. If it be fired a second time with a heavier charge,
the path will be along the second curved line, and the ball would again
fall to the ground. But let us try next time with a charge still further
increased, and, indeed, with a far stronger cannon than any piece of
ordnance ever yet made. The velocity of the projectile must now be
assumed to be some miles per second, but we can conceive that the speed
shall be so adjusted that the ball shall move along the path C D, always
at the same height above the earth, though still curving, as every
projectile must curve, from the horizontal line in which it moved at the
first moment. Arrived at D, the ball will still be at the same height
above the surface, and its velocity must be unabated. It will therefore
continue in its path and move round another quadrant of the circle
without getting nearer to the surface. In this manner the projectile
will travel completely round the whole globe, coming back again to C and
then taking another start in the same path. If we could abolish the
mountain and the cannon at the top, we should have a body revolving for
ever around the earth in consequence of the attraction of gravitation.
Make now a bold stretch of the imagination. Conceive a terrific cannon
capable of receiving a round bullet not less than 2,000 miles in
diameter. Discharge this enormous bullet with a velocity of about 3,000
feet per second, which is two or three times as great as the velocity
actually attainable in modern artillery. Let this notable bullet be
fired horizontally from some station nearly a quarter of a million miles
above the surface of the earth. That fearful missile would sweep right
round the earth in a nearly circular orbit, and return to where it
started in about four weeks. It would then commence another revolution,
four weeks more would find it again at the starting point, and this
motion would go on for ages.
Do not suppose that we are entirely romancing. We cannot indeed show the
cannon, but we can point to a great projectile. We see it every month;
it is the beautiful moon herself. No one asserts that the moon was ever
shot from such a cannon; but it must be admitted that she moves as if
she had been. In a later chapter we shall enquire into the history of
the moon, and show how she came to revolve in this wonderful manner.
As with the moon around the earth, so with the earth around the sun. The
illustration shows that a circular or nearly circular motion harmonises
with the conception of the law of universal gravitation.
We are accustomed to regard gravitation as a force of stupendous
magnitude. Does not gravitation control the moon in its revolution
around the earth? Is not even the mighty earth itself retained in its
path around the sun by the surpassing power of the sun's attraction? No
doubt the actual force which keeps the earth in its path, as well as
that which retains the moon in our neighbourhood, is of vast intensity,
but that is because gravitation is in such cases associated with bodies
of enormous mass. No one can deny that all bodies accessible to our
observation appear to attract each other in accordance with the law of
gravitation; but it must be confessed that, unless one or both of the
attracting bodies is of gigantic dimensions, the intensity is almost
immeasurably small.
Let us attempt to illustrate how feeble is the gravitation between
masses of easily manageable dimensions. Take, for instance, two iron
weights, each weighing about 50lb., and separated by a distance of one
foot from centre to centre. There is a certain attraction of gravitation
between these weights. The two weights are drawn together, yet they do
not move. The attraction between them, though it certainly exists, is an
extremely minute force, not at all comparable as to intensity with
magnetic attraction. Everyone knows that a magnet will draw a piece of
iron with considerable vigour, but the intensity of gravitation is very
much less on masses of equal amount. The attraction between these two
50lb. weights is less than the ten-millionth part of a single pound.
Such a force is utterly infinitesimal in comparison with the friction
between the weights and the table on which they stand, and hence there
is no response to the attraction by even the slightest movement. Yet, if
we can conceive each of these weights mounted on wheels absolutely
devoid of friction, and running on absolutely perfect horizontal rails,
then there is no doubt that the bodies would slowly commence to draw
together, and in the course of time would arrive in actual contact.
If we desire to conceive gravitation as a force of measurable intensity,
we must employ masses immensely more ponderous than those 50lb. weights.
Imagine a pair of globes, each composed of 417,000 tons of cast iron,
and each, if solid, being about 53 yards in diameter. Imagine these
globes placed at a distance of one mile apart. Each globe attracts the
other by the force of gravitation. It does not matter that buildings and
obstacles of every description intervene; gravitation will pass through
such impediments as easily as light passes through glass. No screen can
be devised dense enough to intercept the passage of this force. Each of
these iron globes will therefore under all circumstances attract the
other; but, notwithstanding their ample proportions, the intensity of
that attraction is still very small, though appreciable. The attraction
between these two globes is a force no greater than the pressure exerted
by a single pound weight. A child could hold back one of these massive
globes from its attraction by the other. Suppose that all was clear, and
that friction could be so neutralised as to permit the globes to follow
the impulse of their mutual attractions. The two globes will then
commence to approach, but the masses are so large, while the attraction
is so small, that the speed will be accelerated very slowly. A
microscope would be necessary to show when the motion has actually
commenced. An hour and a half must elapse before the distance is
diminished by a single foot; and although the pace improves
subsequently, yet three or four days must elapse before the two globes
will come together.
The most remarkable characteristic of the force of gravitation must be
here specially alluded to. The intensity appears to depend only on the
quantity of matter in the bodies, and not at all on the nature of the
substances of which these bodies are composed. We have described the two
globes as made of cast iron, but if either or both were composed of lead
or copper, of wood or stone, of air or water, the attractive power would
still be the same, provided only that the masses remain unaltered. In
this we observe a profound difference between the attraction of
gravitation and magnetic attraction. In the latter case the attraction
is not perceptible at all in the great majority of substances, and is
only considerable in the case of iron.
In our account of the solar system we have represented the moon as
revolving around the earth in a _nearly_ circular path, and the planets
as revolving around the sun in orbits which are also approximately
circular. It is now our duty to give a more minute description of these
remarkable paths; and, instead of dismissing them as being _nearly_
circles, we must ascertain precisely in what respects they differ
therefrom.
If a planet revolved around the sun in a truly circular path, of which
the sun was always at the centre, it is then obvious that the distance
from the sun to the planet, being always equal to the radius of the
circle, must be of constant magnitude. Now, there can be no doubt that
the distance from the sun to each planet is approximately constant; but
when accurate observations are made, it becomes clear that the distance
is not absolutely so. The variations in distance may amount to many
millions of miles, but, even in extreme cases, the variation in the
distance of the planet is only a small fraction--usually a very small
fraction--of the total amount of that distance. The circumstances vary
in the case of each of the planets. The orbit of the earth itself is
such that the distance from the earth to the sun departs but little from
its mean value. Venus makes even a closer approach to perfectly circular
movement; while, on the other hand, the path of Mars, and much more the
path of Mercury, show considerable relative fluctuations in the distance
from the planet to the sun.
It has often been noticed that many of the great discoveries in science
have their origin in the nice observation and explanation of minute
departures from some law approximately true. We have in this department
of astronomy an excellent illustration of this principle. The orbits of
the planets are nearly circles, but they are not exactly circles. Now,
why is this? There must be some natural reason. That reason has been
ascertained, and it has led to several of the grandest discoveries that
the mind of man has ever achieved in the realms of Nature.
In the first place, let us see the inferences to be drawn from the fact
that the distance of a planet from the sun is not constant. The motion
in a circle is one of such beauty and simplicity that we are reluctant
to abandon it, unless the necessity for doing so be made clearly
apparent. Can we not devise any way by which the circular motion might
be preserved, and yet be compatible with the fluctuations in the
distance from the planet to the sun? This is clearly impossible with the
sun at the centre of the circle. But suppose the sun did not occupy the
centre, while the planet, as before, revolved around the sun. The
distance between the two bodies would then necessarily fluctuate. The
more eccentric the position of the sun, the larger would be the
proportionate variation in the distance of the planet when at the
different parts of its orbit. It might further be supposed that by
placing a series of circles around the sun the various planetary orbits
could be accounted for. The centre of the circle belonging to Venus is
to coincide very nearly with the centre of the sun, and the centres of
the orbits of all the other planets are to be placed at such suitable
distances from the sun as will render a satisfactory explanation of the
gradual increase and decrease of the distance between the two bodies.
There can be no doubt that the movements of the moon and of the planets
would be, to a large extent, explained by such a system of circular
orbits; but the spirit of astronomical enquiry is not satisfied with
approximate results. Again and again the planets are observed, and again
and again the observations are compared with the places which the
planets would occupy if they moved in accordance with the system here
indicated. The centres of the circles are moved hither and thither,
their radii are adjusted with greater care; but it is all of no avail.
The observations of the planets are minutely examined to see if they can
be in error; but of errors there are none at all sufficient to account
for the discrepancies. The conclusion is thus inevitable--astronomers
are forced to abandon the circular motion, which was thought to possess
such unrivalled symmetry and beauty, and are compelled to admit that the
orbits of the planets are not circular.
Then if these orbits be not circles, what are they? Such was the great
problem which Kepler proposed to solve, and which, to his immortal
glory, he succeeded in solving and in proving to demonstration. The
great discovery of the true shape of the planetary orbits stands out as
one of the most conspicuous events in the history of astronomy. It may,
in fact, be doubted whether any other discovery in the whole range of
science has led to results of such far-reaching interest.
We must here adventure for a while into the field of science known as
geometry, and study therein the nature of that curve which the
discovery of Kepler has raised to such unparalleled importance. The
subject, no doubt, is a difficult one, and to pursue it with any detail
would involve us in many abstruse calculations which would be out of
place in this volume; but a general sketch of the subject is
indispensable, and we must attempt to render it such justice as may be
compatible with our limits.
The curve which represents with perfect fidelity the movements of a
planet in its revolution around the sun belongs to that well-known group
of curves which mathematicians describe as the conic sections. The
particular form of conic section which denotes the orbit of a planet is
known by the name of the _ellipse_: it is spoken of somewhat less
accurately as an oval. The ellipse is a curve which can be readily
constructed. There is no simpler method of doing so than that which is
familiar to draughtsmen, and which we shall here briefly describe.
We represent on the next page (Fig. 37) two pins passing through a sheet
of paper. A loop of twine passes over the two pins in the manner here
indicated, and is stretched by the point of a pencil. With a little care
the pencil can be guided so as to keep the string stretched, and its
point will then describe a curve completely round the pins, returning to
the point from which it started. We thus produce that celebrated
geometrical figure which is called an ellipse.
It will be instructive to draw a number of ellipses, varying in each
case the circumstances under which they are formed. If, for instance,
the pins remain placed as before, while the length of the loop is
increased, so that the pencil is farther away from the pins, then it
will be observed that the ellipse has lost some of its elongation, and
approaches more closely to a circle. On the other hand, if the length of
the cord in the loop be lessened, while the pins remain as before, the
ellipse will be found more oval, or, as a mathematician would say, its
_eccentricity_ is increased. It is also useful to study the changes
which the form of the ellipse undergoes when one of the pins is altered,
while the length of the loop remains unchanged. If the two pins be
brought nearer together the eccentricity will decrease, and the ellipse
will approximate more closely to the shape of a circle. If the pins be
separated more widely the eccentricity of the ellipse will be increased.
That the circle is an extreme form of ellipse will be evident, if we
suppose the two pins to draw in so close together that they become
coincident; the point will then simply trace out a circle as the pencil
moves round the figure.
[Illustration: Fig. 37.--Drawing an Ellipse.]
The points marked by the pins obviously possess very remarkable
relations with respect to the curve. Each one is called a _focus_, and
an ellipse can only have one pair of foci. In other words, there is but
a single pair of positions possible for the two pins, when an ellipse of
specified size, shape, and position is to be constructed.
The ellipse differs principally from a circle in the circumstance that
it possesses variety of form. We can have large and small ellipses just
as we can have large and small circles, but we can also have ellipses of
greater or less eccentricity. If the ellipse has not the perfect
simplicity of the circle it has, at least, the charm of variety which
the circle has not. The oval curve has also the beauty derived from an
outline of perfect grace and an association with ennobling conceptions.
The ancient geometricians had studied the ellipse: they had noticed its
foci; they were acquainted with its geometrical relations; and thus
Kepler was familiar with the ellipse at the time when he undertook his
celebrated researches on the movements of the planets. He had found, as
we have already indicated, that the movements of the planets could not
be reconciled with circular orbits. What shape of orbit should next be
tried? The ellipse was ready to hand, its properties were known, and the
comparison could be made; memorable, indeed, was the consequence of this
comparison. Kepler found that the movement of the planets could be
explained, by supposing that the path in which each one revolved was an
ellipse. This in itself was a discovery of the most commanding
importance. On the one hand it reduced to order the movements of the
great globes which circulate round the sun; while on the other, it took
that beautiful class of curves which had exercised the geometrical
talents of the ancients, and assigned to them the dignity of defining
the highways of the universe.
But we have as yet only partly enunciated the first discovery of Kepler.
We have seen that a planet revolves in an ellipse around the sun, and
that the sun is, therefore, at some point in the interior of the
ellipse--but at what point? Interesting, indeed, is the answer to this
question. We have pointed out how the foci possess a geometrical
significance which no other points enjoy. Kepler showed that the sun
must be situated in one of the foci of the ellipse in which each planet
revolves. We thus enunciate the first law of planetary motion in the
following words:--
_Each planet revolves around the sun in an elliptic path, having
the sun at one of the foci._
We are now enabled to form a clear picture of the orbits of the planets,
be they ever so numerous, as they revolve around the sun. In the first
place, we observe that the ellipse is a plane curve; that is to say,
each planet must, in the course of its long journey, confine its
movements to one plane. Each planet has thus a certain plane
appropriated to it. It is true that all these planes are very nearly
coincident, at least in so far as the great planets are concerned; but
still they are distinct, and the only feature in which they all agree is
that each one of them passes through the sun. All the elliptic orbits of
the planets have one focus in common, and that focus lies at the centre
of the sun.
It is well to illustrate this remarkable law by considering the
circumstances of two or three different planets. Take first the case of
the earth, the path of which, though really an ellipse, is very nearly
circular. In fact, if it were drawn accurately to scale on a sheet of
paper, the difference between the elliptic orbit and the circle would
hardly be detected without careful measurement. In the case of Venus the
ellipse is still more nearly a circle, and the two foci of the ellipse
are very nearly coincident with the centre of the circle. On the other
hand, in the case of Mercury, we have an ellipse which departs from the
circle to a very marked extent, while in the orbits of some of the minor
planets the eccentricity is still greater. It is extremely remarkable
that every planet, no matter how far from the sun, should be found to
move in an ellipse of some shape or other. We shall presently show that
necessity compels each planet to pursue an elliptic path, and that no
other form of path is possible.
Started on its elliptic path, the planet pursues its stately course, and
after a certain duration, known as the _periodic time_, regains the
position from which its departure was taken. Again the planet traces out
anew the same elliptic path, and thus, revolution after revolution, an
identical track is traversed around the sun. Let us now attempt to
follow the body in its course, and observe the history of its motion
during the time requisite for the completion of one of its circuits. The
dimensions of a planetary orbit are so stupendous that the planet must
run its course very rapidly in order to finish the journey within the
allotted time. The earth, as we have already seen, has to move eighteen
miles a second to accomplish one of its voyages round the sun in the
lapse of 365-1/4 days. The question then arises as to whether the rate
at which a planet moves is uniform or not. Does the earth, for instance,
actually move at all times with the velocity of eighteen miles a
second, or does our planet sometimes move more rapidly and sometimes
more slowly, so that the average of eighteen miles a second is still
maintained? This is a question of very great importance, and we are able
to answer it in the clearest and most emphatic manner. The velocity of a
planet is _not_ uniform, and the variations of that velocity can be
explained by the adjoining figure (Fig. 38).
[Illustration: Fig. 38.--Varying Velocity of Elliptic Motion.]
Let us first of all imagine the planet to be situated at that part of
its path most distant from the sun towards the right of the figure. In
this position the body's velocity is at its lowest; as the planet begins
to approach the sun the speed gradually improves until it attains its
mean value. After this point has been passed, and the planet is now
rapidly hurrying on towards the sun, the velocity with which it moves
becomes gradually greater and greater, until at length, as it dashes
round the sun, its speed attains a maximum. After passing the sun, the
distance of the planet from the luminary increases, and the velocity of
the motion begins to abate; gradually it declines until the mean value
is again reached, and then it falls still lower, until the body recedes
to its greatest distance from the sun, by which time the velocity has
abated to the value from which we supposed it to commence. We thus
observe that the nearer the planet is to the sun the quicker it moves.
We can, however, give numerical definiteness to the principle according
to which the velocity of the planet varies. The adjoining figure (Fig.
39) shows a planetary orbit, with, of course, the sun at the focus S. We
have taken two portions, A B and C D, round the ellipse, and joined
their extremities to the focus. Kepler's second law may be stated in
these words:--
"_Every planet moves round the sun with such a velocity at every
point, that a straight line drawn from it to the sun passes over
equal areas in equal times._"
[Illustration: Fig. 39.--Equal Areas in Equal Times.]
For example, if the two shaded portions, A B S and D C S, are equal in
area, then the times occupied by the planet in travelling over the
portions of the ellipse, A B and C D, are equal. If the one area be
greater than the other, then the times required are in the proportion of
the areas.
This law being admitted, the reason of the increase in the planet's
velocity when it approaches the sun is at once apparent. To accomplish a
definite area when near the sun, a larger arc is obviously necessary
than at other parts of the path. At the opposite extremity, a small arc
suffices for a large area, and the velocity is accordingly less.
These two laws completely prescribe the motion of a planet round the
sun. The first defines the path which the planet pursues; the second
describes how the velocity of the body varies at different points along
its path. But Kepler added to these a third law, which enables us to
compare the movements of two different planets revolving round the same
sun. Before stating this law, it is necessary to explain exactly what is
meant by the _mean_ distance of a planet. In its elliptic path the
distance from the sun to the planet is constantly changing; but it is
nevertheless easy to attach a distinct meaning to that distance which is
an average of all the distances. This average is called the mean
distance. The simplest way of finding the mean distance is to add the
greatest of these quantities to the least, and take half the sum. We
have already defined the periodic time of the planet; it is the number
of days which the planet requires for the completion of a journey round
its path. Kepler's third law establishes a relation between the mean
distances and the periodic times of the various planets. That relation
is stated in the following words:--
"_The squares of the periodic times are proportional to the cubes
of the mean distances._"
Kepler knew that the different planets had different periodic times; he
also saw that the greater the mean distance of the planet the greater
was its periodic time, and he was determined to find out the connection
between the two. It was easily found that it would not be true to say
that the periodic time is merely proportional to the mean distance. Were
this the case, then if one planet had a distance twice as great as
another, the periodic time of the former would have been double that of
the latter; observation showed, however, that the periodic time of the
more distant planet exceeded twice, and was indeed nearly three times,
that of the other. By repeated trials, which would have exhausted the
patience of one less confident in his own sagacity, and less assured of
the accuracy of the observations which he sought to interpret, Kepler at
length discovered the true law, and expressed it in the form we have
stated.
To illustrate the nature of this law, we shall take for comparison the
earth and the planet Venus. If we denote the mean distance of the earth
from the sun by unity then the mean distance of Venus from the sun is
0.7233. Omitting decimals beyond the first place, we can represent the
periodic time of the earth as 365.3 days, and the periodic time of
Venus as 224.7 days. Now the law which Kepler asserts is that the square
of 365.3 is to the square of 224.7 in the same proportion as unity is to
the cube of 0.7233. The reader can easily verify the truth of this
identity by actual multiplication. It is, however, to be remembered
that, as only four figures have been retained in the expressions of the
periodic times, so only four figures are to be considered significant in
making the calculations.
The most striking manner of making the verification will be to regard
the time of the revolution of Venus as an unknown quantity, and deduce
it from the known revolution of the earth and the mean distance of
Venus. In this way, by assuming Kepler's law, we deduce the cube of the
periodic time by a simple proportion, and the resulting value of 224.7
days can then be obtained. As a matter of fact, in the calculations of
astronomy, the distances of the planets are usually ascertained from
Kepler's law. The periodic time of the planet is an element which can be
measured with great accuracy; and once it is known, then the square of
the mean distance, and consequently the mean distance itself, is
determined.
Such are the three celebrated laws of Planetary Motion, which have
always been associated with the name of their discoverer. The profound
skill by which these laws were elicited from the mass of observations,
the intrinsic beauty of the laws themselves, their widespread
generality, and the bond of union which they have established between
the various members of the solar system, have given them quite an
exceptional position in astronomy.
As established by Kepler, these planetary laws were merely the results
of observation. It was found, as a matter of fact, that the planets did
move in ellipses, but Kepler assigned no reason why they should adopt
this curve rather than any other. Still less was he able to offer a
reason why these bodies should sweep over equal areas in equal times, or
why that third law was invariably obeyed. The laws as they came from
Kepler's hands stood out as three independent truths; thoroughly
established, no doubt, but unsupported by any arguments as to why these
movements rather than any others should be appropriate for the
revolutions of the planets.
It was the crowning triumph of the great law of universal gravitation to
remove this empirical character from Kepler's laws. Newton's grand
discovery bound together the three isolated laws of Kepler into one
beautiful doctrine. He showed not only that those laws are true, but he
showed why they must be true, and why no other laws could have been
true. He proved to demonstration in his immortal work, the "Principia,"
that the explanation of the famous planetary laws was to be sought in
the attraction of gravitation. Newton set forth that a power of
attraction resided in the sun, and as a necessary consequence of that
attraction every planet must revolve in an elliptic orbit round the sun,
having the sun as one focus; the radius of the planet's orbit must sweep
over equal areas in equal times; and in comparing the movements of two
planets, it was necessary to have the squares of the periodic times
proportional to the cubes of the mean distances.
As this is not a mathematical treatise, it will be impossible for us to
discuss the proofs which Newton has given, and which have commanded the
immediate and universal acquiescence of all who have taken the trouble
to understand them. We must here confine ourselves only to a very brief
and general survey of the subject, which will indicate the character of
the reasoning employed, without introducing details of a technical
character.
Let us, in the first place, endeavour to think of a globe freely poised
in space, and completely isolated from the influence of every other body
in the universe. Let us imagine that this globe is set in motion by some
impulse which starts it forward on a rapid voyage through the realms of
space. When the impulse ceases the globe is in motion, and continues to
move onwards. But what will be the path which it pursues? We are so
accustomed to see a stone thrown into the air moving in a curved path,
that we might naturally think a body projected into free space will
also move in a curve. A little consideration will, however, show that
the cases are very different. In the realms of free space we find no
conception of upwards or downwards; all paths are alike; there is no
reason why the body should swerve to the right or to the left; and hence
we are led to surmise that in these circumstances a body, once started
and freed from all interference, would move in a straight line. It is
true that this statement is one which can never be submitted to the test
of direct experiment. Circumstanced as we are on the surface of the
earth, we have no means of isolating a body from external forces. The
resistance of the air, as well as friction in various other forms, no
less than the gravitation towards the earth itself, interfere with our
experiments. A stone thrown along a sheet of ice will be exposed to but
little resistance, and in this case we see that the stone will take a
straight course along the frozen surface. A stone similarly cast into
empty space would pursue a course absolutely rectilinear. This we
demonstrate, not by any attempts at an experiment which would
necessarily be futile, but by indirect reasoning. The truth of this
principle can never for a moment be doubted by one who has duly weighed
the arguments which have been produced in its behalf.
Admitting, then, the rectilinear path of the body, the next question
which arises relates to the velocity with which that movement is
performed. The stone gliding over the smooth ice on a frozen lake will,
as everyone has observed, travel a long distance before it comes to
rest. There is but little friction between the ice and the stone, but
still even on ice friction is not altogether absent; and as that
friction always tends to stop the motion, the stone will at length be
brought to rest. In a voyage through the solitudes of space, a body
experiences no friction; there is no tendency for the velocity to be
reduced, and consequently we believe that the body could journey on for
ever with unabated speed. No doubt such a statement seems at variance
with our ordinary experience. A sailing ship makes no progress on the
sea when the wind dies away. A train will gradually lose its velocity
when the steam has been turned off. A humming-top will slowly expend its
rotation and come to rest. From such instances it might be plausibly
argued that when the force has ceased to act, the motion that the force
generated gradually wanes, and ultimately vanishes. But in all these
cases it will be found, on reflection, that the decline of the motion is
to be attributed to the action of resisting forces. The sailing ship is
retarded by the rubbing of the water on its sides; the train is checked
by the friction of the wheels, and by the fact that it has to force its
way through the air; and the atmospheric resistance is mainly the cause
of the stopping of the humming-top, for if the air be withdrawn, by
making the experiment in a vacuum, the top will continue to spin for a
greatly lengthened period. We are thus led to admit that a body, once
projected freely in space and acted upon by no external resistance, will
continue to move on for ever in a straight line, and will preserve
unabated to the end of time the velocity with which it originally
started. This principle is known as the _first law of motion_.
Let us apply this principle to the important question of the movement of
the planets. Take, for instance, the case of our earth, and let us
discuss the consequences of the first law of motion. We know that the
earth is moving each moment with a velocity of about eighteen miles a
second, and the first law of motion assures us that if this globe were
submitted to no external force, it would for ever pursue a straight
track through the universe, nor would it depart from the precise
velocity which it possesses at the present moment. But is the earth
moving in this manner? Obviously not. We have already found that our
globe is moving round the sun, and the comprehensive laws of Kepler have
given to that motion the most perfect distinctness and precision. The
consequence is irresistible. The earth cannot be free from external
force. Some potent influence on our globe must be in ceaseless action.
That influence, whatever it may be, constantly deflects the earth from
the rectilinear path which it tends to pursue, and constrains it to
trace out an ellipse instead of a straight line.
The great problem to be solved is now easily stated. There must be some
external agent constantly influencing the earth. What is that agent,
whence does it proceed, and to what laws is it submitted? Nor is the
question confined to the earth. Mercury and Venus, Mars, Jupiter, and
Saturn, unmistakably show that, as they are not moving in rectilinear
paths, they must be exposed to some force. What is this force which
guides the planets in their paths? Before the time of Newton this
question might have been asked in vain. It was the splendid genius of
Newton which supplied the answer, and thus revolutionised the whole of
modern science.
The data from which the question is to be answered must be obtained from
observation. We have here no problem which can be solved by mere
mathematical meditation. Mathematics is no doubt a useful, indeed, an
indispensable, instrument in the enquiry; but we must not attribute to
mathematics a potency which it does not possess. In a case of this kind,
all that mathematics can do is to interpret the results obtained by
observation. The data from which Newton proceeded were the observed
phenomena in the movement of the earth and the other planets. Those
facts had found a succinct expression by the aid of Kepler's laws. It
was, accordingly, the laws of Kepler which Newton took as the basis of
his labours, and it was for the interpretation of Kepler's laws that
Newton invoked the aid of that celebrated mathematical reasoning which
he created.
The question is then to be approached in this way: A planet being
subject to _some_ external influence, we have to determine what that
influence is, from our knowledge that the path of each planet is an
ellipse, and that each planet sweeps round the sun over equal areas in
equal times. The influence on each planet is what a mathematician would
call a force, and a force must have a line of direction. The most simple
conception of a force is that of a pull communicated along a rope, and
the direction of the rope is in this case the direction of the force.
Let us imagine that the force exerted on each planet is imparted by an
invisible rope. Kepler's laws will inform us with regard to the
direction of this rope and the intensity of the strain transmitted
through it.
The mathematical analysis of Kepler's laws would be beyond the scope of
this volume. We must, therefore, confine ourselves to the results to
which they lead, and omit the details of the reasoning. Newton first
took the law which asserted that the planet moved over equal areas in
equal times, and he showed by unimpeachable logic that this at once gave
the direction in which the force acted on the planet. He showed that the
imaginary rope by which the planet is controlled must be invariably
directed towards the sun. In other words, the force exerted on each
planet was at all times pointed from the planet towards the sun.
It still remained to explain the intensity of the force, and to show how
the intensity of that force varied when the planet was at different
points of its path. Kepler's first law enables this question to be
answered. If the planet's path be elliptic, and if the force be always
directed towards the sun at one focus of that ellipse, then mathematical
analysis obliges us to say that the intensity of the force must vary
inversely as the square of the distance from the planet to the sun.
The movements of the planets, in conformity with Kepler's laws, would
thus be accounted for even in their minutest details, if we admit that
an attractive power draws the planet towards the sun, and that the
intensity of this attraction varies inversely as the square of the
distance. Can we hesitate to say that such an attraction does exist? We
have seen how the earth attracts a falling body; we have seen how the
earth's attraction extends to the moon, and explains the revolution of
the moon around the earth. We have now learned that the movement of the
planets round the sun can also be explained as a consequence of this law
of attraction. But the evidence in support of the law of universal
gravitation is, in truth, much stronger than any we have yet presented.
We shall have occasion to dwell on this matter further on. We shall show
not only how the sun attracts the planets, but how the planets attract
each other; and we shall find how this mutual attraction of the planets
has led to remarkable discoveries, which have elevated the law of
gravitation beyond the possibility of doubt.
Admitting the existence of this law, we can show that the planets must
revolve around the sun in elliptic paths with the sun in the common
focus. We can show that they must sweep over equal areas in equal times.
We can prove that the squares of the periodic times must be proportional
to the cubes of their mean distances. Still further, we can show how the
mysterious movements of comets can be accounted for. By the same great
law we can explain the revolutions of the satellites. We can account for
the tides, and for other phenomena throughout the Solar System. Finally,
we shall show that when we extend our view beyond the limits of our
Solar System to the beautiful starry systems scattered through space, we
find even there evidence of the great law of universal gravitation.