PHILOSOPHY OF GRAVITATION
ART. 18. _Gravitation Attraction._--The Law of Gravitation being a
compound law, and not a simple law (Art. 8), it is necessary that the
principles which govern universal attraction should now be considered.
The law which governs Gravitation Attraction may be defined as follows:
Every particle of matter in the universe attracts every other particle
with a force whose direction is that of a line joining the centre of
their masses, whose magnitude is directly as the product of their
masses, and inversely as the square of the distance between them.
This may be divided into four parts.
(1) The Universality of Gravitation.
(2) The Direction of the Forces involved.
(3) The Proportion of these Forces.
(4) The Law of Inverse Squares.
The theory of the Aether, therefore, which will be perfected in this
work, must not only satisfactorily account for the Attraction of
Gravitation on a strictly philosophical basis, but the laws, governing
the pressures or tensions of that physical medium, must harmonize with
each of the parts of the complex Law of Gravitation into which it has
been resolved.
ART. 19. _Universality of the Attractive Force._--The principle upon
which Universal Attraction rests is found in the words: "Every particle
of matter in the universe attracts every other particle." It must,
however, be admitted that this statement has never actually been proved.
The smallest body that Newton used to prove his Law of Attraction was
our satellite the moon.
Cavendish, however, in 1798, by a series of experiments, conclusively
demonstrated that the force of Gravitation existed in small bodies. He
took two small leaden balls of a certain weight, and fixed them at the
ends of a rod about six feet long, the rod being suspended by a piece of
wire in the air. Large leaden balls were then brought near the small
ones, and great care was taken to see if there were any twist in the
wire by which they were suspended. It was found that the wire had become
twisted on the approach of the large leaden balls, and thus he was able
to prove that every particle of the attracted and attracting body are
mutually concerned in the Attraction of Gravitation. There is abundant
evidence of the application of this force in relation to our earth, as
we shall see later on.
The universality of the Attraction of Gravitation is a fact that has
been proved in a thousand ways, and a thousand times. All stars and
suns, and all planets, satellites and comets and nebulae are subject to
this universal law. Astronomy teaches us that its power extends across
the vast abysses of space, and that stars situated at distances that
cannot possibly be measured, are subject to this world-wide law. Some of
the greatest discoveries in astronomical science were due to the
operations of this wonderful law, the gravitating influences of certain
planets indicating their existence, although their discovery had not yet
been made.
The discovery of Neptune through the mathematical calculations of Le
Verrier and Mr. Adams in 1846 was the crowning proof of the Law of
Gravitation. Mr. Adams in England had noticed that the planet Uranus was
being pulled out of the course by some unknown power, and so set to work
to calculate the position of the body which thus influenced the motion
of Uranus in its orbit. He located the position of the supposed
influencing body strictly by mathematical calculations, and then took
his results to the Astronomer Royal. Delay, however, occurred in the
search for the supposed new planet, and nothing was done further in the
matter for many months. Meanwhile Le Verrier in France, unknown to Mr.
Adams, had been making similar calculations with reference to the
perturbations of Uranus, and had arrived at similar results.
These results were sent to the Berlin astronomers, and the heavens were
searched for the supposed new planet. After a time, the planet was
discovered in that part of the heavens indicated by Le Verrier, and for
a time his name stood out as the sole discoverer. Gradually, however,
the claims of Adams were admitted and recognized, and to-day his claims
to participate in the honour of the wonderful achievement are generally
admitted. Thus the discovery of Neptune gave to the Law of Gravitation a
stability and proof that perhaps it had never received before.
Further evidence of the existence of the universality of the attractive
force, is to be found in a certain system of stars known as binary
stars, which revolve around each other, while they gravitate around a
common centre. Recent researches in astronomy only seek more and more to
confirm the universality and effectiveness of this grand law, that seems
to hold the entire universe in its sway.
Any medium, therefore, which is postulated as the physical cause of
Gravitation, must itself be as universal as Gravitation, in order for it
to be able to fulfil this condition of universality. We shall find, as
we proceed, that the only possible medium which can fulfil this
condition, is the universal Aether, whose qualities and properties are
already partly known and partly understood.
ART. 20. _Direction of the Forces._--The attraction of Gravitation is
always directed along the straight line which joins the centres of
masses of the attracting and attracted bodies. Thus, if the earth and
moon are taken as examples, an imaginary straight line drawn from the
centre of the earth's mass to the centre of the mass of the moon would
be the direction in which the gravitative force would be exerted. Now a
line which joins the central body to its satellite we shall see when we
come to deal with Kepler's Laws is known as the Radius Vector. Thus the
path of the attraction between the two bodies is along the Radius
Vector. It is a singular coincidence that the path of a ray of light
from the sun also coincides with the Radius Vector, as it is one of the
laws of light that the path of a ray always follows a straight line.
It must not, however, be assumed, that while the attractive power is
being exerted along any one straight line joining the centres of two
bodies, therefore the attractive power is not operative in relation to
any other part of the space, around the body. If our earth, for example,
had four moons instead of one, and they were each in different positions
in relation to the earth, then the law as to the direction of the forces
would still hold good. We have examples of this in the case of Jupiter
with his five moons, and Saturn with his eight moons. So that the
attractive force of Gravitation is again like light, it operates on all
sides equally at one and the same time. A lamp in the middle of a room
sends its light waves on every side at one and the same time, so that
while each ray has for its path a straight line, yet those rays are
emitted equally on every side. In like manner, though the direction of
the forces between two attracting bodies is that of a straight line, yet
the law of universal attraction is equally exerted on every side of the
planet at one and the same time.
In the theory of the Aether, therefore, to be developed in this work, it
will have to be demonstrated that the direction of the forces, which are
originated and transmitted by that physical medium, must philosophically
fulfil the conditions which govern the direction of the forces, as
observed in gravitational phenomena.
ART. 21. _Proportion of the Forces._--Newton proved that the attraction
is proportional to the product of the masses of the bodies concerned.
Hence it is that the sun, which is the centre of the solar system, is
capable of attracting the most remote planets, because the mass of the
sun is greater than the mass of all the planets put together. Or take
another illustration. Suppose that the sun and the earth are at equal
distances from Saturn. Now the sun's mass is about 300,000 times that of
our earth. Therefore if the earth draws Saturn through a certain
distance in one second, the sun would draw Saturn through a distance
which is 300,000 greater than the earth in the same period.
The governing principle, therefore, which decides the proportion of the
attractive forces between two bodies is mass, and not simply density or
volume. The mass of a body is a property which remains the same, as long
as the inertia of the body remains constant. Mass is really a measure of
the inertia of a body, or that property of a body by which it continues
in its state of motion or of rest.
Mass is therefore a compound quantity, being equal to volume multiplied
by density, so that if the volume of any body is halved, the density is
doubled. Thus, the proportion of the attractive force between any two
bodies ever remains the same, so long as the masses of the two bodies
remain the same. Through all the changes of volume and density of any
body, its attractive force remains constant, as long as the mass remains
constant; for the simple reason, that as the volume of a body is
increased, the density is proportionately decreased; or, as the volume
is decreased, the density is increased.
For example, the volume of the sun as compared with the volume of the
earth, is about 1,300,000 times greater, but the proportion of the
attractive forces between the two bodies, is about 324,000 to 1. This
difference is accounted for by the fact, that the density of the sun is
about one quarter the mean density of the earth, hence their masses are
in the proportion of 324,000 to 1. Thus the proportion of the attractive
forces between any two bodies is dependent upon their masses, and not
simply upon their volume or density.
ART. 22. _Law of Inverse Squares._--The Law of Inverse Squares which is
applicable to Gravitation is equally true of Sound, Light, Heat and
Electricity, the Law being that Gravitation acts inversely as the square
of distance. That is to say, if the distance of any body from the sun,
for example, be doubled, then the force of Gravitation is diminished to
one quarter of the intensity which would be exerted on the body in the
first position.
Thus the further a body is from its controlling centre, the weaker the
Attraction of Gravitation upon it becomes. Taking therefore Mercury and
the earth as examples, we find that their mean distances are
respectively 35,000,000 miles and 92,000,000, which is a proportion of
about 1 to 2-1/2. So that the intensity of the sun's attraction on the
earth is about four-twenty-fifths of what it is on Mercury, that being
the inverse square of the relative distances of the two bodies.
Now the intensity of Light and Heat received by the earth is regulated
by the same law of inverse squares, so that the earth would receive
about four-twenty-fifths the intensity of light and heat which Mercury
receives when they are both at their mean distances.
This law of inverse squares is applicable to every body which acts as a
gravitating source throughout the whole of the universe, whether that
body be small or large, and whether it be in the form of meteor,
satellite, planet, sun or star.
Each satellite, planet or sun exerts an attractive influence upon every
body that exists, that attractive influence being regulated by the
masses of the respective bodies, and decreasing inversely as the square
of the distance from the body viewed as the centre of attraction. So
that, the further the attracted body is from the attracting body, the
less is the intensity of the mutual attracting forces, though that
intensity does not vary simply as the distance, but rather as the square
of the distance, and that in its inverse ratio. Thus if we take two
masses of any kind or sort, and place them at various distances as
represented by the numbers 1, 2, 3, 4, 5, 6, the intensity of the
attracting forces between the same masses at the relative distances will
be represented by the numbers 1, 1/4, 1/9, 1/16, 1/25, 1/36, which are
the inverse squares of the respective numbers representing their
distances. As we shall see, the same law holds good in relation to heat,
light and electricity, and indeed to all forms of energy which radiate
out from a centre equally in all directions.
There is no need to apply Newton's Rules of Philosophy to this
Attraction of Gravitation, as it has been demonstrated to exist, times
without number. Moreover its laws are exactly the same as those
governing the phenomena of sound, light, heat, and electricity, so that
apart from being proved by actual experiments in relation to the gravity
of the earth, we have a wider experience of the application of the same
ruling principles of the law in other departments of science.
The Law of Universal Attraction, which is strictly the Centripetal Force
of the compound Law of Gravitation, fully satisfies the three governing
rules of Newton's Philosophy. Not only is it simple in its conception,
but it is borne out by experience, and adequately accounts for the
distinctive phenomena which it seeks to explain. By it, astronomical
observations can be taken with a precision and certainty that defy error
or failure. The motion of a planet in its orbit can be so perfectly
calculated, that its position in space in relation to other planets can
be foretold years in advance. The theory of the Aether, therefore, which
is to be perfected in this work, must philosophically show that the
pressures or tensions of that medium, which are postulated as the cause
of Gravitation Attraction, must themselves fulfil the laws of inverse
squares, which govern light, heat, electricity and the Attraction of
Gravitation. I premise that this will be done in the theory of the
Aether to be submitted to the reader in the after pages of this work.
ART. 23. _Terrestrial Gravity._--Before passing from this phase of the
subject, I should like briefly to look at the question of the Attraction
of Gravitation from the standpoint of our own earth, as by so doing we
shall notice some facts regarding the same, hitherto unnoticed, in the
preceding articles.
Terrestrial Gravity is but a phase of Universal Gravitation. One of the
most familiar facts and phenomena of everyday life is, that when a body,
such as a stone or stick or bullet, is thrown or projected into the air,
it always falls to the earth again. This is due to the attraction of the
earth and the stone for each other. It has been proved experimentally
that if a stone and a weight are let fall from a height of 16 feet, they
would reach the earth in one second of time. Again, a feather, or cork,
or even a piece of iron would take exactly the same time falling through
the same space, provided that the feather or cork could be screened from
the resistance of the air.
The distance, however, through which a body falls in one second varies
at different parts of the earth's surface, being least at the equator,
and greatest at the North and South Poles. This is accounted for by the
fact that the polar diameter is only 7899 miles, while the equatorial
diameter is 7925 miles, thus the distance from the centre of the earth
to either pole is about 3950 miles, or 13 miles less than the equatorial
radius of the earth. Now the force of gravity decreases upwards from the
earth's surface inversely as the square of the distance from the earth's
centre of gravity, but decreases downwards simply as the distance from
the centre decreases. Thus if a ball were taken down 2000 miles, that is
half the distance to the centre, it would only weigh half-a-pound, while
if it were taken to the centre of the earth, it would have no weight at
all; while a pound weight at the equator would not weigh one pound at
the poles, because it would be nearer the centre of the earth by 13
miles.
Thus a pound weight is not always a pound weight. It varies as we carry
it to different parts of the earth's surface, depending upon its
relation to the centre of the earth for its exact weight. The point
which I wish to make perfectly clear, as it will be necessary for future
reference, is, that there is no such thing as weight apart from the
gravity of the earth; or, if we apply the principle to the solar system,
there is no gravitating force in that system apart from the gravitating
force of the central body, the sun, or the planets and other bodies
which form the solar system.
Let us look at this matter from another standpoint, in order to prove
this truth and make the same perfectly clear. If a pound weight were put
in a spring-balance, then at the surface of the earth it would weigh one
pound. Now, we will suppose that we have taken the weight to a height of
4000 miles above the surface of the earth, that is exactly double the
distance from the centre of the earth, the radius of the earth being
approximately 4000 miles. According to the law of inverse squares, the
force of Gravitation decreases inversely as the square of the distance.
The distance having been doubled, the proportion of the forces at the
two places, _i. e._ the earth's surface and 4000 miles above it, are as
1 to 1/4.
Thus at a distance of 4000 miles the weight which weighed one pound at
the earth's surface, now only weighs a quarter of a pound. At a distance
of 8000 miles, the distance would be trebled, therefore the force of
Gravitation is one-ninth, and the weight would weigh one-ninth of a
pound. If we could take the pound weight to the moon, the attractive
force of the earth would be reduced to 1-3600, as the moon is 240,000
miles distant, that is sixty times the earth's radius. The square of 60
is 3600, and if we invert that we get 1-3600, so that the weight which
weighs a pound at the earth's surface, would only weigh 1-3600 part of a
pound at the distance of the moon. This again proves, that apart from
the Attraction of Gravitation, there is no such thing as weight, and
that the weight so called of any body, such as a planet or satellite,
increases or decreases as its distance increases or decreases from its
central attracting body.
ART. 24. _Centrifugal Force._--I have already shown in Art. 10 that
the Centripetal Force and Universal Attraction are one and the same; as
the Centripetal Force always acts towards the centre, and must therefore
be in its operation and influence a gravitating or attractive power.
I have also pointed out in the same article, the necessity of another
force, which is to be the complement, and the counter part of
Gravitation Attraction. That complement and counter force was conceived
by Newton, and called by him the Centrifugal Force. The very nature of
the Centripetal Force demands and necessitates a force which in its mode
of operation is exactly the opposite of the Centripetal Force. Unless
there were such a force, a repellent and repulsive force, then instead
of there being that harmonious working of the universe that now exists,
there must inevitably be a gradual drawing together of all planets and
satellites, of all stars and suns, into one vast, solitary, and ruinous
body.
There are also other phenomena which demand a Centrifugal Force in the
universe. It is a well-known fact, that there exist between the orbits
of Jupiter and Mars, what are called planetoids, about 500 in number,
which are supposed to be the remnants of a broken or shattered world. As
may be expected from such an accumulation, they present the most
extraordinary diversities and eccentricities in the orbits that can
possibly be conceived. They are of all shapes and sizes, and besides
their orbits round the sun, have orbits among themselves. They are so
clustered together that their orbits intersect each other at numerous
points, and when in conjunction are said to suffer great perturbations,
being pulled great distances this way and that by each other's
attractive influence. It is further stated that their orbits so
intersect each other, that if they were imagined to be material rings,
they would be inseparable, and the whole could be suspended by taking
any one of them up at random. Here, then, is presented to us a kind or
order of celestial phenomena for whose well-being and effectual working
the Centripetal Force or the Attraction of Gravitation cannot possibly
account. In their case another force is demanded which shall be the
exact complement and counterpart of the Centripetal Force. There needs
therefore a force, not an imagined one, simply conceived to fill a want,
but a real Force, as real and as plainly to be understood as the
Centripetal Force. A force existing in each world just like the
Attraction of Gravitation, only the reverse of Gravitation, a repellent,
repulsive Force, acting in the reverse mode, and way, to universal
attraction. This Force must be governed by the same rules and laws that
govern the Centripetal Force, if it is to work in harmony with the same.
It must be universal in its character, having a proportion of forces
equal to the product of the masses of the two bodies which are
concerned, and its path must coincide with the path of gravitational
attraction, that is, in the straight line which joins the centres of
gravity of the two bodies. Further, and what is perhaps the most
important of all, it must act as a repelling or repulsive force which
shall be in the same proportion in regard to distance, as the law
governing Centripetal Force, that is, inversely as the square of the
distance.
Again, and briefly, there are also in existence small bodies called
meteors, which are said to exist by myriads, which float in space, and
circle round the sun. They are of all shapes and sizes, from one ounce
to a ton or even tons, thousands of them coming into contact with our
earth's atmosphere every year, especially in August and November. All of
these small bodies have orbits among themselves, and gravitate round one
another, as they revolve round the sun. Now if the orbits of the
planetoids be such an entangled mass, what must be the orbits of these
meteors? What an indescribable, unimaginable mass of labyrinthian
motions must exist among these myriads of little bodies! How they must
intersect, cross and intermingle each other's orbits! What attraction
and counter-attraction they must exert upon each other! Let me ask any
man to sit down and try to imagine how the present recognized
Centripetal and Centrifugal Forces can account for the effectual working
of these meteors. As illustrating the necessity of a real and physical
Centrifugal Force which is to be the exact counterpart of the
Centripetal Force, I would call the attention of the reader to
Herschel's view of this matter. In dealing with the phenomena of comets'
tails he writes:[1] "Beyond a doubt, the widest and most interesting
prospect of future discovery, which this study holds out to us, is, that
distinction between gravitating and levitating matter, that positive and
irrefutable demonstration in nature of a repulsive force, co-extensive
with, but enormously more powerful than the attractive force we call
gravity which the phenomena of their tails afford." I premise that this
prophecy of Herschel's will be fully demonstrated and proved in the
succeeding pages of this work. For, in the theory of the Aether that is
to be afterwards perfected, it will be philosophically proved that the
physical medium so conceived will satisfactorily account for a force or
motion from the centre of all bodies; which motions fulfil all the
conditions required by that Centrifugal Force, which is the complement
and counterpart of the Attraction of Gravitation. At the present time,
with the conception of a frictionless Aether, it is impossible to
harmonize the existence of such a force or motion with our theory of the
Aether. Yet Professor Lebedew of Moscow, and Nichols and Hull of
America, have incontrovertibly demonstrated by actual experiments the
existence of such a force. Therefore it follows, that if our present
theory of the Aether fails to agree with experimental evidence, such a
theory must be reconstructed in order that our philosophy may be made to
agree with our experiments and our experience.
[Footnote 1: _Lectures on Scientific Subjects._]
ART. 25. _Kepler's Laws._--A long time before Newton had discovered the
Law of Gravitation, Kepler had found out that the motions of the planets
were governed by certain laws, and these came to be known as Kepler's
Laws.
These laws which were given to the world by Kepler, simply represented
facts or phenomena which had been discovered by observation, as Kepler
was unable to account for them, or to give any mathematical basis for
the same.
On the discovery, however, of Universal Gravitation, Newton saw at once
that these laws were simply the outcome of the application of the Law of
Gravitation to the planets, and that they could be accounted for on a
mathematical basis by the Law of Gravitation, as they seemed to flow
naturally from that law.
Kepler's Laws are three in number and may be thus stated--
_1st Law._ Each planet revolves round the sun in an elliptic orbit, with
the sun occupying one of the Foci.
_2nd Law._ In the revolution of a planet round the sun, the Radius Vector
describes equal areas in equal times.
_3rd Law._ The squares of the periodic times of planets are proportional
to the cubes of their mean distances.
Now the question arises, whether it is possible to form a theory of the
Aether which shall satisfactorily and philosophically account for all
the phenomena associated with Kepler's Laws in their relation to the
motions of planets, satellites, or other solar bodies? On the present
conception of the Aether such a result is an absolute impossibility.
With the theory of the Aether, however, to be submitted to the reader in
this work, the result is possible and attainable. If, therefore, such a
result is philosophically proved, as I submit will be done, then we
shall have greater evidence still that the theory so propounded is a
more perfect theory than the one at present recognized by scientists
generally.
ART. 26. _Kepler's First Law._--Each planet revolves round the sun in an
elliptic orbit, the sun occupying one of the Foci.
The ancients thought that the paths of the planets around the sun were
circular in form, because they held that circular motion was perfect. A
system of circular orbits for the paths of the planets round the sun
would be very simple in its conception, and would be full of beauty and
harmony. But exact calculations reveal to us that the path of a planet
is not exactly that of a circle, as the distance of a planet from the
sun in various parts of its orbit is sometimes greater, and sometimes
less, than its mean distance.
The planet Venus has the nearest approach to a circular orbit, as there
are only 500,000 miles between the mean, and greatest and least
distances, but both Mercury and Mars show great differences between
their greatest and least distances from the sun.
If, therefore, the orbits of a planet are not exactly circular, what is
their exact shape? Kepler solved this problem, and proved that the exact
path of a planet round its central body the sun was that of an ellipse,
or an elongated circle. Thus he gave to the world the first of his
famous laws which stated that each planet revolves round the sun in an
orbit which has an elliptic form, the sun occupying one of the Foci.
Not only is the orbit of a planet round the sun elliptic in form, but
the path of the moon round the earth, or the path of any satellite, as
for example a satellite of Mars or Jupiter or Saturn, is also that of an
ellipse, the planet round which it revolves occupying one of the Foci.
It has also been found that certain comets have orbits which cannot be
distinguished from that of an elongated ellipse, the sun occupying one
of the Foci.
Now let us apply the Law of Gravitation to Kepler's First Law, and note
carefully its application.
[Illustration: Fig: 1.]
Let _A_, _B_, _C_, _D_ be an ellipse representing the orbit of the earth,
and let _S_ represent the sun situated at one of the Foci.
We will suppose that the earth is projected into space at the point _A_,
then according to the First Law of Motion, it would proceed in a
straight line in the direction of _A_ _E_, if there were no other force
acting upon the earth. But it is acted upon by the attraction of the
sun, that is the Centripetal Force which is exerted along the straight
line _S_ _A_ (Art. 20), which continues to act upon it according to the
principle already explained in Arts. 21 and 22.
Now, according to the Second Law of Motion and the Parallelogram of
Forces, instead of the earth going off at a tangent in the direction of
_A_ _E_, it will take a mean path in the direction of _A_ _B_, its path
being curved instead of being a straight line.
If the sun were stationary in space, then the mean distance, that is,
the length of the imaginary straight line joining the sun _S_ _A_ to the
earth, would remain unaltered. The Radius Vector _S_ _A_, or the
straight line referred to, would then be perpendicular to the tangent,
and the velocity of the earth round the sun would be uniform, its path
being that of a circle.
The Radius Vector _S_ _A_, however, is not always perpendicular to the
tangent _F_ _E_, and therefore the velocity of the earth is not always
uniform in its motion in its orbit, as sometimes it travels at a lesser
or greater speed than its average speed, which is about 18 miles per
second.
It has to be remembered that the sun itself is in motion, having a
velocity through space of about 4-1/2 miles per second, so that, while
the earth is travelling from _A_ to _B_ the sun is also travelling in
the direction of _S_ _B_. Thus the orbital velocity of the earth, and the
orbital velocity of the sun, together with the Centripetal Force or
universal Gravitation Attraction, are all acting in the same direction
when the earth is travelling from _A_ to _B_, that is, in the direction
of the orbit situated at _B_. This point of the orbit is known as the
perihelion, and at that point the velocity of the earth is at its
greatest, because the earth is then nearest the sun.
According to Newton, the planet when at _B_ would still have a tendency
to fly off into space owing to its Centrifugal Force, but it is held in
check by the Centripetal Force, so that instead of it flying off into
space, it is whirled round and starts off on its journey away from the
sun in the direction of _B_, _C_. The sun, however, is still continuing
its journey in the direction of _S_, _H_, so that not only is the
increased orbital velocity of the earth, which it obtained at its
perihelion, urging the earth away from the sun, but the sun itself in
its advance through space is leaving the earth behind it. The combined
effect of the two motions, the advancing motion of the sun, and the
receding motion of the earth, due to its increased orbital velocity,
drives the earth towards the aphelion, where its distance from the sun
is greatest, and its orbital velocity is the least.
By the time the planet has arrived at point _C_, its motion through
space has gradually decreased, and the Centripetal Force begins to
re-assert itself, with the result that the earth is slowly made to
proceed towards the point _D_ of the ellipse, at which point its motion
is the slowest in orbital velocity, only travelling about 16 miles per
second, while the distance of the earth from the sun is the greatest and
has increased from 91,000,000 miles at the perihelion to 94,500,000.
This point of the orbit is known as its aphelion.
After rounding this point, the orbital velocity of the earth begins to
increase again, owing to the diminishing distance of the earth from the
sun, which according to the law of inverse squares (Art. 22) gives an
added intensity to the Centripetal Force.
Thus by the combination of the Laws of Motion and the Law of Gravitation
discovered by Newton, he was able to satisfactorily account for and
explain on a mathematical basis, the reason why the earth and all the
other planets move round the sun in elliptic orbits, according to
Kepler's First Law.
In the development of the physical cause of gravitation, therefore, the
same physical medium, which accounts for that law, must also give a
satisfactory explanation of the first of Kepler's Laws.
ART. 27. _Kepler's Second Law._--This law states that the Radius Vector
describes equal areas in equal times. The Radius Vector is the imaginary
straight line joining the centres of the sun and the earth or planet.
While the First Law shows us the kind of path which a planet takes in
revolving round the sun, the Second Law describes how the velocity of
the planet varies in different parts of its orbit.
If the earth's orbit were a circle, it can be readily seen that equal
areas would be traversed in equal times, as the distance from the sun
would always be the same, so that the Radius Vector being of uniform
length, the rate of motion would be uniform, and consequently equal
areas would be traversed in equal times. Take as an illustration the
earth, which describes its revolution round the sun in 365-1/4 days. Now
if the orbit of the earth were circular, then equal parts of the earth's
orbit would be traversed by the Radius Vector in equal times. So that
with a perfectly circular orbit, one half of the orbit would be
traversed by the Radius Vector in half a year, one quarter in one
quarter of a year, one-eighth in one-eighth of a year, and so on; the
area covered by the Radius Vector being always exactly proportionate to
the time.
From Kepler's First Law, however, we know that the planet's distance
does vary from the sun, and therefore the Radius Vector is sometimes
longer and sometimes shorter than when the earth is at its mean
distance; the Radius Vector being shortest at the perihelion of the
orbit, and longest at the aphelion. We learn from Kepler's Second Law
that when the Radius Vector is shortest, that is, when the planet is
nearest the sun, it acquires its greatest orbital velocity; and when the
Radius Vector is longest, that is, when the planet is farthest from the
sun, the orbital velocity of a planet is the slowest.
Let _A_, _B_, _D_, _C_ represent the elliptic orbit of a planet, with
_S_ sun at one of the Foci, and let the triangles _A_, _S_, _B_ and _D_,
_S_, _C_ be triangles of equal area. Then, according to Kepler's Second
Law, the time taken for the Radius Vector to traverse the area _A_, _S_,
_B_ is equal to the time that the Radius Vector takes to traverse the
area _D_, _S_, _C_. So that the planet would take an equal time in going
from _A_ to _B_ of its orbit, as it would take in going from _D_ to _C_.
Thus the nearer the planet is to the sun, the greater is its orbital
velocity, and the farther it is away from the sun the slower is its
velocity, the velocity being regulated by the distance. The manner in
which the difference of velocity is accounted for by the Law of
Gravitation has already been explained in the preceding article. Thus
Newton proved that Kepler's Second Law was capable of being
mathematically explained, and accounted for, by the universal Law of
Gravitation.
[Illustration: Fig: 2.]
If, therefore, a physical cause can be given for Newton's Law of
Gravitation, then such physical cause must also be able to account for,
and that on a strictly philosophical basis, the second of Kepler's Laws
as well as the first.
ART. 28. _Kepler's Third Law._--The Third Law of Kepler gives the
relation between the periodic time of a planet, and its distance from
the sun. The periodic time of any planet is the time which it takes to
go once round the sun. Thus the periodic time of the earth is 365-1/4
days. The periodic time of Venus is 224.7 days, while that of Mars is
686.9 days.
Kepler had found out that different planets had different periodic
times; he also found out that the greater the mean distance of the
planet, the greater was the time which the planet took to perform its
journey round the sun, and so he set to work to find out the
relationship of the periodic time to the planet's mean distance.
After many trials and many failures he arrived at the right conclusion,
and at last discovered the true law which is known as Kepler's Third
Law, which states that for each and every planet, the squares of their
periodic times are proportional to the cubes of their mean distances.
For purposes of illustration let us take the earth and the planet Venus
and compare these two. The periodic time of the earth is 365 days,
omitting the quarter day. The periodic time of Venus is 224 days
approximately. Now, according to Kepler's Third Law, the square of 365
is to the square of 224, as the cube of the earth's mean distance is to
the cube of Venus's mean distance, which are 92.7 millions of miles and
67 millions of miles respectively. The problem may be thus stated--
As 365^2: 224^2:: 92.7^3: 67^3:
This worked out gives--
133,225: 50,176: 796,597.982: cube of Venus's mean distance.
So that by Kepler's Third Law, if we have the periodic time of any two
planets, and the mean distance of either, we can find out the mean
distance of the other by simple proportion.
In making astronomical calculations, the distances of the planets are
generally obtained by means of Kepler's Third Law, as the periodic time
of the planet is a calculation that may be made by astronomers with
great certainty, and when once the periodic times are found, and the
mean distance of a planet, as our earth for example, is known, the mean
distances of all the other planets in the solar system may soon be
obtained.
In like manner this Third Law of Kepler's is equally applicable to the
satellites of any planet. For example, when the periodic time of both of
Mars' satellites, Phobos and Deimos, are known, being about 8 hours and
30 hours respectively, and the distance of either is known, as Phobos
with a mean distance of 5800 miles, then the mean distance of Deimos can
easily be calculated by this law, and is found to be 14,500 miles.
As discovered by Kepler, the Third Law was simply the result of
observation. He was unable to give any mathematical basis for its
existence. The Laws as they were given to the world by Kepler were
simply three great truths which had been discovered by observation. It
rested with Newton to show how these laws could be accounted for on a
mathematical basis, and to show how they all sprang from one and the
same source, namely the universal Law of Gravitation. In his
_Principia_, he proved that all Kepler's Laws were fully expounded and
explained by his great discovery of Universal Gravitation.
Now what Newton has done for Kepler's Laws from the mathematical
standpoint, we propose to do from the physical standpoint. In the
development of the physical agency or cause of Gravitation, therefore,
among the phenomena and laws, which have to be satisfactorily accounted
for on a physical basis, are these three Laws of Kepler's just referred
to.
So that in addition to the satisfactory explanation of a physical cause
for the Laws of Motion, and the Centripetal and Centrifugal Forces, the
hypothesis of a physical cause of Gravitation must fully and
satisfactorily account for the Laws of Kepler, whose mathematical
explanation was given by Newton.
Further, and what is as equally important, the explanation so given must
be strictly in harmony with the Rules of Philosophy as laid down in Art.
3. That is, the explanation must be simple in character, must not be
contrary to experience or observation, and must satisfactorily account
for the laws which the hypothesis of the physical cause of Gravitation
seeks to explain. This I premise we will do as we pass from stage to
stage in the development of the theory.
I can safely premise that it will be simple in character and conception,
that it will be entirely in harmony with all experience and observation,
and that the physical cause so advanced will give as physical a basis to
Kepler's Laws as Newton's mathematical calculations gave them a
mathematical basis.
In summing up, I need hardly point out, that if all that I have premised
in this and the preceding chapter is accomplished in the after chapters
of this book, then for the first time since the discovery of Universal
Gravitation by Sir Isaac Newton, his great discovery will have received
the long-expected and long-desired physical explanation, that
explanation and cause being founded on his own Rules of Philosophy given
in his immortal _Principia_, and for the first time our Philosophy will
be brought strictly into harmony with our universal experience.