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Sir Isaac Newton & Astronomy

Posted on July 23rd, 2007 by Admin.
Categories: Uncategorized.

We now reach the period which is the culminating point of interest in
the history of dynamical astronomy.  Isaac Newton was born in
1642. Pemberton states that Newton, having quitted Cambridge to avoid
the plague, was residing at Wolsthorpe, in Lincolnshire, where he had
been born; that he was sitting one day in the garden, reflecting upon
the force which prevents a planet from flying off at a tangent and
which draws it to the sun, and upon the force which draws the moon to
the earth; and that he saw in the case of the planets that the sun’s
force must clearly be unequal at different distances, for the pull out
of the tangential line in a minute is less for Jupiter than for
Mars. He then saw that the pull of the earth on the moon would be less
than for a nearer object. It is said that while thus meditating he saw
an apple fall from a tree to the ground, and that this fact suggested
the questions: Is the force that pulled that apple from the tree the
same as the force which draws the moon to the earth?  Does the
attraction for both of them follow the same law as to distance as is
given by the planetary motions round the sun? It has been stated that
in this way the first conception of universal gravitation arose.[1]

Quite the most important event in the whole history of physical
astronomy was the publication, in 1687, of Newton’s _Principia
(Philosophiae Naturalis Principia Mathematica)_. In this great work
Newton started from the beginning of things, the laws of motion, and
carried his argument, step by step, into every branch of physical
astronomy; giving the physical meaning of Kepler’s three laws, and
explaining, or indicating the explanation of, all the known heavenly
motions and their irregularities; showing that all of these were
included in his simple statement about the law of universal
gravitation; and proceeding to deduce from that law new irregularities
in the motions of the moon which had never been noticed, and to
discover the oblate figure of the earth and the cause of the
tides. These investigations occupied the best part of his life; but he
wrote the whole of his great book in fifteen months.

Having developed and enunciated the true laws of motion, he was able
to show that Kepler’s second law (that equal areas are described by
the line from the planet to the sun in equal times) was only another
way of saying that the centripetal force on a planet is always
directed to the sun. Also that Kepler’s first law (elliptic orbits
with the sun in one focus) was only another way of saying that the
force urging a planet to the sun varies inversely as the square of the
distance. Also (if these two be granted) it follows that Kepler’s
third law is only another way of saying that the sun’s force on
different planets (besides depending as above on distance) is
proportional to their masses.

Having further proved the, for that day, wonderful proposition that,
with the law of inverse squares, the attraction by the separate
particles of a sphere of uniform density (or one composed of
concentric spherical shells, each of uniform density) acts as if the
whole mass were collected at the centre, he was able to express the
meaning of Kepler’s laws in propositions which have been summarised as
follows:–

The law of universal gravitation.–_Every particle of matter in the
universe attracts every other particle with a force varying inversely
as the square of the distance between them, and directly as the
product of the masses of the two particles_.[2]

But Newton did not commit himself to the law until he had answered
that question about the apple; and the above proposition now enabled
him to deal with the Moon and the apple. Gravity makes a stone fall
16.1 feet in a second. The moon is 60 times farther from the earth’s
centre than the stone, so it ought to be drawn out of a straight
course through 16.1 feet in a minute. Newton found the distance
through which she is actually drawn as a fraction of the earth’s
diameter.  But when he first examined this matter he proceeded to use
a wrong diameter for the earth, and he found a serious discrepancy.
This, for a time, seemed to condemn his theory, and regretfully he
laid that part of his work aside. Fortunately, before Newton wrote the
_Principia_ the French astronomer Picard made a new and correct
measure of an arc of the meridian, from which he obtained an accurate
value of the earth’s diameter. Newton applied this value, and found,
to his great joy, that when the distance of the moon is 60 times the
radius of the earth she is attracted out of the straight course 16.1
feet per minute, and that the force acting on a stone or an apple
follows the same law as the force acting upon the heavenly bodies.[3]

The universality claimed for the law–if not by Newton, at least by
his commentators–was bold, and warranted only by the large number of
cases in which Newton had found it to apply. Its universality has been
under test ever since, and so far it has stood the test. There has
often been a suspicion of a doubt, when some inequality of motion in
the heavenly bodies has, for a time, foiled the astronomers in their
attempts to explain it. But improved mathematical methods have always
succeeded in the end, and so the seeming doubt has been converted into
a surer conviction of the universality of the law.

Having once established the law, Newton proceeded to trace some of its
consequences. He saw that the figure of the earth depends partly on
the mutual gravitation of its parts, and partly on the centrifugal
tendency due to the earth’s rotation, and that these should cause a
flattening of the poles. He invented a mathematical method which he
used for computing the ratio of the polar to the equatorial diameter.

He then noticed that the consequent bulging of matter at the equator
would be attracted by the moon unequally, the nearest parts being most
attracted; and so the moon would tend to tilt the earth when in some
parts of her orbit; and the sun would do this to a less extent,
because of its great distance. Then he proved that the effect ought to
be a rotation of the earth’s axis over a conical surface in space,
exactly as the axis of a top describes a cone, if the top has a sharp
point, and is set spinning and displaced from the vertical. He
actually calculated the amount; and so he explained the cause of the
precession of the equinoxes discovered by Hipparchus about 150 B.C.

One of his grandest discoveries was a method of weighing the heavenly
bodies by their action on each other. By means of this principle he
was able to compare the mass of the sun with the masses of those
planets that have moons, and also to compare the mass of our moon with
the mass of the earth.

Thus Newton, after having established his great principle, devoted his
splendid intellect to the calculation of its consequences. He proved
that if a body be projected with any velocity in free space, subject
only to a central force, varying inversely as the square of the
distance, the body must revolve in a curve which may be any one of the
sections of a cone–a circle, ellipse, parabola, or hyperbola; and he
found that those comets of which he had observations move in parabolae
round the Sun, and are thus subject to the universal law.

Newton realised that, while planets and satellites are chiefly
controlled by the central body about which they revolve, the new law
must involve irregularities, due to their mutual action–such, in
fact, as Horrocks had indicated. He determined to put this to a test
in the case of the moon, and to calculate the sun’s effect, from its
mass compared with that of the earth, and from its distance. He proved
that the average effect upon the plane of the orbit would be to cause
the line in which it cuts the plane of the ecliptic (i.e., the line of
nodes) to revolve in the ecliptic once in about nineteen years. This
had been a known fact from the earliest ages. He also concluded that
the line of apses would revolve in the plane of the lunar orbit also
in about nineteen years; but the observed period is only ten
years. For a long time this was the one weak point in the Newtonian
theory. It was not till 1747 that Clairaut reconciled this with the
theory, and showed why Newton’s calculation was not exact.

Newton proceeded to explain the other inequalities recognised by Tycho
Brahe and older observers, and to calculate their maximum amounts as
indicated by his theory. He further discovered from his calculations
two new inequalities, one of the apogee, the other of the nodes, and
assigned the maximum value. Grant has shown the values of some of
these as given by observation in the tables of Meyer and more modern
tables, and has compared them with the values assigned by Newton from
his theory; and the comparison is very remarkable.

                                     Newton.      Modern Tables.
                                degrees ‘  “   degrees ‘ ”
Mean monthly motion of Apses         1.31.28         3.4.0
Mean annual motion of nodes         19.18.1,23     19.21.22,50
Mean value of “variation”              36.10          35.47
Annual equation                        11.51          11.14
Inequality of mean motion of apogee    19.43          22.17
Inequality of mean motion of nodes      9.24           9.0

The only serious discrepancy is the first, which has been already
mentioned. Considering that some of these perturbations had never been
discovered, that the cause of none of them had ever been known, and
that he exhibited his results, if he did not also make the
discoveries, by the synthetic methods of geometry, it is simply
marvellous that he reached to such a degree of accuracy. He invented
the infinitesimal calculus which is more suited for such calculations,
but had he expressed his results in that language he would have been
unintelligible to many.

Newton’s method of calculating the precession of the equinoxes,
already referred to, is as beautiful as anything in the _Principia_.
He had already proved the regression of the nodes of a satellite
moving in an orbit inclined to the ecliptic. He now said that the
nodes of a ring of satellites revolving round the earth’s equator
would consequently all regress. And if joined into a solid ring its
node would regress; and it would do so, only more slowly, if
encumbered by the spherical part of the earth’s mass. Therefore the
axis of the equatorial belt of the earth must revolve round the pole
of the ecliptic. Then he set to work and found the amount due to the
moon and that due to the sun, and so he solved the mystery of 2,000
years.

When Newton applied his law of gravitation to an explanation of the
tides he started a new field for the application of mathematics to
physical problems; and there can be little doubt that, if he could
have been furnished with complete tidal observations from different
parts of the world, his extraordinary powers of analysis would have
enabled him to reach a satisfactory theory. He certainly opened up
many mines full of intellectual gems; and his successors have never
ceased in their explorations. This has led to improved mathematical
methods, which, combined with the greater accuracy of observation,
have rendered physical astronomy of to-day the most exact of the
sciences.

Laplace only expressed the universal opinion of posterity when he said
that to the _Principia_ is assured “a pre-eminence above all the
other productions of the human intellect.”

The name of Flamsteed, First Astronomer Royal, must here be mentioned
as having supplied Newton with the accurate data required for
completing the theory.

The name of Edmund Halley, Second Astronomer Royal, must ever be held
in repute, not only for his own discoveries, but for the part he
played in urging Newton to commit to writing, and present to the Royal
Society, the results of his investigations. But for his friendly
insistence it is possible that the _Principia_ would never have
been written; and but for his generosity in supplying the means the
Royal Society could not have published the book.

[Illustration: DEATH MASK OF SIR ISAAC NEWTON.
Photographed specially for this work from the original, by kind
permission of the Royal Society, London.]

Sir Isaac Newton died in 1727, at the age of eighty-five. His body
lay in state in the Jerusalem Chamber, and was buried in Westminster
Abbey.

FOOTNOTES:

[1] The writer inherited from his father (Professor J. D. Forbes) a
small box containing a bit of wood and a slip of paper, which had been
presented to him by Sir David Brewster. On the paper Sir David had
written these words: “If there be any truth in the story that Newton
was led to the theory of gravitation by the fall of an apple, this bit
of wood is probably a piece of the apple tree from which Newton saw
the apple fall. When I was on a pilgrimage to the house in which
Newton was born, I cut it off an ancient apple tree growing in his
garden.” When lecturing in Glasgow, about 1875, the writer showed it
to his audience. The next morning, when removing his property from the
lecture table, he found that his precious relic had been stolen. It
would be interesting to know who has got it now!

[2] It must be noted that these words, in which the laws of
gravitation are always summarised in histories and text-books, do not
appear in the _Principia_; but, though they must have been composed by
some early commentator, it does not appear that their origin has been
traced. Nor does it appear that Newton ever extended the law beyond
the Solar System, and probably his caution would have led him to avoid
any statement of the kind until it should be proved.

With this exception the above statement of the law of universal
gravitation contains nothing that is not to be found in the
_Principia_; and the nearest approach to that statement occurs in the
Seventh Proposition of Book III.:–

Prop.: That gravitation occurs in all bodies, and that it is
proportional to the quantity of matter in each.

Cor. I.: The total attraction of gravitation on a planet arises, and
is composed, out of the attraction on the separate parts.

Cor. II.: The attraction on separate equal particles of a body is
reciprocally as the square of the distance from the particles.

[3] It is said that, when working out this final result, the
probability of its confirming that part of his theory which he had
reluctantly abandoned years before excited him so keenly that he was
forced to hand over his calculations to a friend, to be completed by
him.