NEWTON AND GRAVITATION
So all of Kepler's laws could be embodied in a single law of gravitation
toward a central body, whose force of attraction decreases outward in
exact proportion as the square of the distance increases.
Only one farther step had to be taken, and this the most complicated of
all: he must make all the bodies of the sky conform to his third law of
motion. This is: Action and reaction are equal, or the mutual actions of
any two bodies are always equal and oppositely directed. There must be
mutual attractions everywhere: earth for sun as well as sun for earth,
moon for sun and sun for moon, earth for Venus and Venus for earth,
Jupiter for Saturn and Saturn for Jupiter, and so on.
The motions of the planets in the undisturbed ellipses of Kepler must be
impossible. As observations of the planets became more accurate, it was
found that they really did fail to move in exact accord with Kepler's
laws unmodified. Newton was unable, with the imperfect processes of the
mathematics of his day to ascertain whether the deviations then known
could be accounted for by his law of gravitation; but he nevertheless
formulated the law with entire precision, as follows:
Every particle of matter in the universe attracts every other particle
with a force exactly proportioned to the product of their masses, and
inversely as the square of the distance between their centers.
The centuries of astronomical research since Newton's day, however, have
verified the great law with the utmost exactness. Practically every
irregularity of lunar and planetary motion is accounted for; indeed, the
intricacies of the problems involved, and the nicety of their solution,
have led to the invention of new mathematical processes adequate to the
difficulties encountered.
And about the middle of the last century, when Uranus departed from the
path laid out for it by the mathematical astronomers, its orbital
deviations were made the basis of an investigation which soon led to the
assignment of the position where a great planet could be found that
would account for the unexplained irregularities of the motion of
Uranus. And the immediate discovery of this planet, Neptune, became the
most striking verification of the Newtonian law that the solar system
could possibly afford.
The astronomers of still later days investigating the statelier motions
of stellar systems find the Newtonian law regnant everywhere among the
stars where our most powerful telescopes have as yet reached. So that
Newton's law is known as the law of Universal Gravitation, and its
author is everywhere held as the greatest scientist of the ages.
Newton's _Principia_ may be regarded as the culminating research of the
inductive method, and further outline of its contents is desirable. It
is divided into three books following certain introductory sections. The
first book treats of the problems of moving bodies, the solutions being
worked out generally and not with special reference to astronomy. The
second book deals with the motion of bodies through resistant media, as
fluids, and has very little significance in astronomy. The third book is
the all important one, and applies his general principles to the case of
the actual solar system, providing a full explanation of the motions of
all the bodies of the system known in his day. Anyone who critically
reads the _Principia_ of Newton will be forced to conclude that its
author was a genius in the highest sense of the word. The elegance and
thoroughness of the demonstrations, and the completeness of application
of the law of gravitation are especially impressive.
The universality of his new law was the feature to which he gave
particular attention. It was clear to him that the gravitation of a
planet, although it acted as if wholly concentrated at the center, was
nevertheless resident in every one of the particles of which the planet
is composed. Indeed, his universal law was so formulated as to make
every particle attract every other particle; and an investigation known
as the Cavendish experiment--a research of great delicacy of
manipulation--not only proves this, but leads also to a measurement of
the earth's mean density, from which we can calculate approximately how
much the earth actually weighs.
Another way to attack the same problem is by measuring the attraction of
mountains, as Maskelyne, Astronomer Royal of Scotland did on Mount
Schehallien in Scotland, which was selected because of its sheer
isolation. The attraction of the mountain deflected the plumb-lines by
measurable amounts, the volume of the mountain was carefully
ascertained by surveys, and geologists found out what rocks composed it.
So the weight of the entire mountain became pretty well known, and
combining this with the observed deflection, an independent value of the
earth's weight was found.
Still other methods have been applied to this question, and as an
average it is found that the materials composing the earth are about
five and a half times as heavy as water, and the total weight of the
earth is something like six sextillions of tons.
What is the true shape of the earth? And does the earth's turning round
on its axis affect this shape? Newton saw the answer to these questions
in his law of gravitation. A spherical figure followed as a matter of
course from the mutual attraction of all materials composing the earth,
providing it was at rest, or did not turn round on its axis. But
rotation bulges it at the equator and draws it in at the poles, by an
amount which calculation shows to be in exact agreement with the amount
ascertained by actual measurement of the earth itself.
Another curious effect, not at first apparent, was that all bodies
carried from high latitudes toward the equator would get lighter and
lighter, in consequence of the centrifugal force of rotation. This was
unexpectedly demonstrated by Richer when the French Academy sent him
south to observe Mars in 1672. His clock had been regulated exactly in
Paris, and he soon found that it lost time when set up at Cayenne. The
amount of loss was found by observation, and it was exactly equal to the
calculated effect that the reduction of gravity by centrifugal action
should produce.
Also Newton saw that his law of gravitation would afford an explanation
of the rise and fall of the tides. The water on the side of the earth
toward the moon, being nearer to the moon, would be more strongly
attracted toward it, and therefore raised in a tide. And the water on
the farther side of the earth away from the moon, being at a greater
distance than the earth itself, the moon would attract the earth more
strongly than this mass of water, tending therefore to draw the earth
away from the water, and so raising at the same time a high tide on the
side of the earth away from the moon. As the earth turns round on its
axis, therefore, two tidal waves continually follow each other at
intervals of about twelve hours.
The sun, too, joins its gravitating force with that of the moon, raising
tides nearly half as high as those which the moon produces, because the
sun's vaster mass makes up in large part for its much greater distance.
At first and third quarters of the moon, the sun acts against the moon,
and the difference of their tide-producing forces gives us "neap tides";
while at new moon and full, sun and moon act together, and produce the
maximum effect known as "spring tides."
Newton passed on to explain, by the action of gravitation also, the
precession of the equinoxes, a phenomenon of the sky discovered by
Hipparchus, who pretty well ascertained its amount, although no reason
for it had ever been assigned. The plane of the earth's equator extended
to the celestial sphere marks out the celestial equator, and the two
opposite points where it intersects the plane of the ecliptic, or the
earth's path round the sun, are called the equinoctial points, or simply
the equinoxes. And precession of the equinoxes is the motion of these
points westward or backward, about 50 seconds each year, so that a
complete revolution round the ecliptic would take place in about 26,000
years.
Newton saw clearly how to explain this: it is simply due to the
attraction of the sun's gravitation upon the protuberant bulge around
the earth's equator, acting in conjunction with the earth's rotation on
its axis, the effect being very similar to that often seen in a spinning
top, or in a gyroscope. The moon moving near the ecliptic produces a
precessional effect, as also do the planets to a very slight degree; and
the observed value of precession is the same as that calculated from
gravitation, to a high degree of precision.
Newton died in 1727, too early to have witnessed that complete and
triumphant verification of his law which ultimately has accounted for
practically every inequality in the planetary motions caused by their
mutual attractions. The problems involved are far beyond the complexity
of those which the mathematical astronomer has to deal with, and the
mathematicians of France deserve the highest credit for improving the
processes of their science so that obstacles which appeared insuperable
were one after another overcome.
Newton's method of dealing with these problems was mainly geometric, and
the insufficiency of this method was apparent. Only when the French
mathematicians began to apply the higher methods of algebra was progress
toward the ultimate goal assured. D'Alembert and Clairaut for a time
were foremost in these researches, but their places were soon taken by
Lagrange, who wrote the "Mecanique Analytique," and Laplace, whose
"Mecanique Celeste" is the most celebrated work of all. In large part
these works are the basis of the researches of subsequent mathematical
astronomers who, strictly speaking, cannot as yet be said to have
arrived at a complete and rigorous solution of all the problems which
the mutual attractions of all the bodies of the solar system have
originated.
It may well be that even the mathematics of the present day are
incompetent to this purpose. When the brilliant genius of Sir William
Hamilton invented quaternion analysis and showed the marvelous facility
with which it solved the intricate problems of physics, there was the
expectation that its application to the higher problems of mathematical
astronomy might effect still greater advances; but nothing in that
direction has so far eventuated. Some astronomers look for the invention
of new functions with numerical tables bearing perhaps somewhat the
relation to present tables of logarithms, sines, tangents, and so on,
that these tables do to the simple multiplication table of Pythagoras.