WE come now to the story of what is by common
consent the greatest of scientific achievements.
The law of universal gravitation is the most far-reaching
principle as yet discovered. It has application
equally to the minutest particle of matter and to the
most distant suns in the universe, yet it is amazing in
its very simplicity. As usually phrased, the law is
this: That every particle of matter in the universe attracts
every other particle with a force that varies directly
with the mass of the particles and inversely as
the squares of their mutual distance. Newton did not
vault at once to the full expression of this law, though
he had formulated it fully before he gave the results
of his investigations to the world. We have now to
follow the steps by which he reached this culminating
achievement.
At the very beginning we must understand that the
idea of universal gravitation was not absolutely original
with Newton. Away back in the old Greek days, as
we have seen, Anaxagoras conceived and clearly expressed
the idea that the force which holds the heavenly
bodies in their orbits may be the same that operates
upon substances at the surface of the earth. With Anaxagoras
this was scarcely more than a guess. After
his day the idea seems not to have been expressed by
any one until the seventeenth century's awakening
of science. Then the consideration of Kepler's Third
Law of planetary motion suggested to many minds
perhaps independently the probability that the force
hitherto mentioned merely as centripetal, through the
operation of which the planets are held in their orbits
is a force varying inversely as the square of the
distance from the sun. This idea had come to Robert
Hooke, to Wren, and perhaps to Halley, as well as to
Newton; but as yet no one had conceived a method by
which the validity of the suggestion might be tested.
It was claimed later on by Hooke that he had discovered
a method demonstrating the truth of the theory
of inverse squares, and after the full announcement of
Newton's discovery a heated controversy was precipitated
in which Hooke put forward his claims with
accustomed acrimony. Hooke, however, never produced
his demonstration, and it may well be doubted
whether he had found a method which did more
than vaguely suggest the law which the observations
of Kepler had partially revealed. Newton's great
merit lay not so much in conceiving the law of inverse
squares as in the demonstration of the law. He
was led to this demonstration through considering the
orbital motion of the moon. According to the familiar
story, which has become one of the classic myths of
science, Newton was led to take up the problem through
observing the fall of an apple. Voltaire is responsible
for the story, which serves as well as another; its
truth or falsity need not in the least concern us. Suffice
it that through pondering on the familiar fact of
terrestrial gravitation, Newton was led to question
whether this force which operates so tangibly here at
the earth's surface may not extend its influence out
into the depths of space, so as to include, for example,
the moon. Obviously some force pulls the moon constantly
towards the earth; otherwise that body would
fly off at a tangent and never return. May not this
so-called centripetal force be identical with terrestrial
gravitation? Such was Newton's query. Probably
many another man since Anaxagoras had asked the
same question, but assuredly Newton was the first man
to find an answer.
The thought that suggested itself to Newton's mind
was this: If we make a diagram illustrating the orbital
course of the moon for any given period, say one
minute, we shall find that the course of the moon departs
from a straight line during that period by a
measurable distance—that: is to say, the moon has been
virtually pulled towards the earth by an amount that
is represented by the difference between its actual
position at the end of the minute under observation
and the position it would occupy had its course been
tangential, as, according to the first law of motion, it
must have been had not some force deflected it towards
the earth. Measuring the deflection in question—which
is equivalent to the so-called versed sine of
the arc traversed—we have a basis for determining
the strength of the deflecting force. Newton constructed
such a diagram, and, measuring the amount
of the moon's departure from a tangential rectilinear
course in one minute, determined this to be, by his
calculation, thirteen feet. Obviously, then, the force
acting upon the moon is one that would cause that
body to fall towards the earth to the distance of thirteen
feet in the first minute of its fall. Would such
be the force of gravitation acting at the distance of the
moon if the power of gravitation varies inversely as
the square of the distance? That was the tangible
form in which the problem presented itself to Newton.
The mathematical solution of the problem was simple
enough. It is based on a comparison of the moon's
distance with the length of the earth's radius. On
making this calculation, Newton found that the pull of
gravitation—if that were really the force that controls
the moon—gives that body a fall of slightly over
fifteen feet in the first minute, instead of thirteen feet.
Here was surely a suggestive approximation, yet, on
the other band, the discrepancy seemed to be too
great to warrant him in the supposition that he had
found the true solution. He therefore dismissed the
matter from his mind for the time being, nor did he return
to it definitely for some years.
It was to appear in due time that Newton's hypothesis
was perfectly valid and that his method of
attempted demonstration was equally so. The difficulty
was that the earth's proper dimensions were
not at that time known. A wrong estimate of the
earth's size vitiated all the other calculations involved,
since the measurement of the moon's distance depends
upon the observation of the parallax, which cannot
lead to a correct computation unless the length of the
earth's radius is accurately known. Newton's first
calculation was made as early as 1666, and it was not
until 1682 that his attention was called to a new and
apparently accurate measurement of a degree of the
earth's meridian made by the French astronomer
Picard. The new measurement made a degree of the
earth's surface 69.10 miles, instead of sixty miles.
Learning of this materially altered calculation as to
the earth's size, Newton was led to take up again his
problem of the falling moon. As he proceeded with
his computation, it became more and more certain that
this time the result was to harmonize with the observed
facts. As the story goes, he was so completely
overwhelmed with emotion that he was forced to ask
a friend to complete the simple calculation. That
story may well be true, for, simple though the computation
was, its result was perhaps the most wonderful
demonstration hitherto achieved in the entire
field of science. Now at last it was known that the
force of gravitation operates at the distance of the
moon, and holds that body in its elliptical orbit, and
it required but a slight effort of the imagination to
assume that the force which operates through such a
reach of space extends its influence yet more widely.
That such is really the case was demonstrated presently
through calculations as to the moons of Jupiter and by
similar computations regarding the orbital motions of
the various planets. All results harmonizing, Newton
was justified in reaching the conclusion that gravitation
is a universal property of matter. It remained, as we
shall see, for nineteenth-century scientists to prove
that the same force actually operates upon the stars,
though it should be added that this demonstration
merely fortified a belief that had already found full
acceptance.
Having thus epitomized Newton's discovery, we
must now take up the steps of his progress somewhat in
detail, and state his theories and their demonstration
in his own words. Proposition IV., theorem 4, of his
Principia is as
follows:
"That the moon gravitates towards the earth and by
the force of gravity is continually drawn off from a
rectilinear motion and retained in its orbit.
"The mean distance of the moon from the earth, in
the syzygies in semi-diameters of the earth, is, according
to Ptolemy and most astronomers, 59; according
to Vendelin and Huygens, 60; to Copernicus, 60 1/3; to
Street, 60 2/3; and to Tycho, 56½. But Tycho, and all
that follow his tables of refractions, making the refractions
of the sun and moon (altogether against the
nature of light) to exceed the refractions of the fixed
stars, and that by four or five minutes
near the horizon, did thereby increase the moon's
horizontal parallax by
a like number of minutes, that is, by a twelfth or
fifteenth part of the whole parallax. Correct this
error and the distance will become about 60½
semi-diameters of the earth, near to what others have
assigned. Let us assume the mean distance of 60
diameters in the syzygies; and suppose one revolution
of the moon, in respect to the fixed stars, to be completed
in 27d. 7h. 43', as astronomers have determined;
and the circumference of the earth to amount to
123,249,600 Paris feet, as the French have found by
mensuration. And now, if we imagine the moon, deprived
of all motion, to be let go, so as to descend
towards the earth with the impulse of all that force
by which (by Cor. Prop. iii.) it is retained in its orb,
it will in the space of one minute of time describe in
its fall 15 1/12 Paris feet. For the versed sine of that arc
which the moon, in the space of one minute of time,
would by its mean motion describe at the distance of
sixty semi-diameters of the earth, is nearly 15 1/12 Paris
feet, or more accurately 15 feet, 1 inch, 1 line 4/9. Wherefore,
since that force, in approaching the earth, increases
in the reciprocal-duplicate proportion of the distance,
and upon that account, at the surface of the earth, is
60 x 60 times greater than at the moon, a body in our
regions, falling with that force, ought in the space of
one minute of time to describe 60 x 60 x 15 1/12 Paris
feet; and in the space of one second of time, to describe
15 1/12 of those feet, or more accurately, 15 feet, 1 inch,
1 line 4/9. And with this very force we actually find
that bodies here upon earth do really descend; for
a pendulum oscillating seconds in the latitude of
Paris will be 3 Paris feet, and 8 lines ½ in length, as
Mr. Huygens has observed. And the space which a
heavy body describes by falling in one second of time
is to half the length of the pendulum in the duplicate
ratio of the circumference of a circle to its diameter
(as Mr. Huygens has also shown), and is therefore 15
Paris feet, 1 inch, 1 line 4/9. And therefore the
force by which the moon is retained in its orbit is
that very same force which we commonly call gravity;
for, were gravity another force different from that,
then bodies descending to the earth with the joint
impulse of both forces would fall with a double
velocity, and in the space of one second of time
would describe 30 1/6 Paris feet; altogether
against experience.''
[43]
All this is beautifully clear, and its validity has never
in recent generations been called in question; yet it
should be explained that the argument does not
amount to an actually indisputable demonstration. It
is at least possible that the coincidence between the
observed and computed motion of the moon may be a
mere coincidence and nothing more. This probability,
however, is so remote that Newton is fully justified in
disregarding it, and, as has been said, all subsequent
generations have accepted the computation as demonstrative.
Let us produce now Newton's further computations
as to the other planetary bodies, passing on to his final
conclusion that gravity is a universal force.
