8 Isaac Newton
8.1 Potted
biography
Rural background
Woolsthorpe Manor
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1642: born in Lincolnshire, East Midlands of England.
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Indifferent performance at school until there was an
altercation with another student, after which time N.
applied himself and came top of the class.
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Major interest was making models and mechanical toys,
but also was always absorbed in reading, and developed
a life-long habit of keeping a diary.
He was a small & weak baby.
Father died before birth. Mother remarried when he was
2, and moved to live with new husband, leaving N. with
his grandparents on the farm until he was 12.
School years
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Left school at 16 and was meant to take over running the
farm, but showed no aptitude or interest and was a
disaster.
Cambridge University
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1661: went to Trinity College Cambridge as a sizar.
At this time Cambridge was rather backward intellectually,
and new ideas of Copernicus, Galileo, Kepler, Descartes
largely hadn’t penetrated: scholarship was focused on Aristotle
and on voluminous theological studies.
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His mother realized that Newton should go to
university--with strong encouragement of his school
headmaster, and so arranged for him to finish school.
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This required boarding with a family in nearby town of
Grantham.
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However, Trinity was a hotbed of Cartesian philosophy.
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Husband of this family was a pharmacist and Newton
developed a lifelong interest in chemistry, as well as
doctoring himself with various chemical remedies.
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Also fell in love with a Miss Storey, a stepdaughter of his
host, and became engaged to marry her ca. 1661.
Newton would have studied arithmetic, Euclid, and
trigonometry, classical & medieval Latin, logic & ethics.
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But he also studied optics with his tutor, read Kepler’s Optics
and struggled through Descartes’ Analytical Geometry.
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1665 graduated with B.A. degree.
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The University (and the country) were also just recovering
from the Puritanism of the Protectorate & Charles II was
proclaimed monarch in 1661.
Newton’s telescope
Back in Woolsthorpe
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In 1665 the university was closed due to a resurgence of
the Plague.
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Newton spent much of the next two years back at the
family manor, quietly reading and thinking.
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During these “golden years” he made three classic
discoveries that revolutionized the future of science:
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the mathematical method of “fluxions,” i.e. the
calculus,
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understanding the composition of light, &
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the law of universal gravitation.
The chromatic aberration of refracting telescope led to
his invention of a reflecting telescope.
Cambridge professor
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In 1667 Newton returned to Cambridge as a Fellow of
Trinity.
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Engagement to Miss Storey ended (or faded away).
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A year later he published a letter in the Transactions of the
Royal Society on his “new theory of light and colours.”
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Huygens appears not to have appreciated the significance
of this work, nor completely understood it.
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This was quite a disappointment to Newton.
Scientific disputes
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His publication on light also began a long dispute with
Robert Hooke, who claimed priority in Newton’s
discoveries in optics. (There were more to follow.)
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A result of these disputes was to strengthen Newton’s
opposition to publishing his work, believing that others
would draw him into further disputes which he
considered a waste of time.
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A major difference in outlook between Newton,
Huygens, & Hooke was Newton’s insistence on
quantitative measurement, rather than qualitative
speculation.
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He did not send any further papers to the Royal Society
on optics, and in 1675 offered his resignation as as
Fellow, in part because he believed the Society had not
supported his ideas.
In 1669 he was appointed (2nd) Lucasian Professor.
First lecture course was on optics.
His invention of the reflecting telescope brought
fellowship in the Royal Society in 1671.
Problems with orthodoxy
Life in Cambridge
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Newton had always done much theological reading and
thinking, and as a result became convinced that the doctrine of
the Trinity was not Biblical.
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This (Arian heresy) was a major problem for him: if this were to
become known he would have lost his college fellowship.
At the request of one of his friends, in 1676 Newton
exchanged two letters with Gottfried Leibniz describing
some of his mathematical discoveries.
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The law required that the Lucasian Professor become ordained
in the Church of England, which would mean affirming the 39
Articles of Religion, including the doctrine of the Trinity, which
he refused to do.
These were courteous and open, but later would become
part of a long controversy over priority in discovery of
the calculus.
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During this period (from ca. 1669) Newton did much
experimentation in chemistry & alchemy (i.e.
transmutation of metals and finding an elixir for
immortality), having studied all the ancient alchemical
treatises.
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Newton’s lifestyle was largely solitary. His concentration
was extraordinary: he would become so engaged on a
problem he would forget to eat and sleep.
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By 1675 he had given up hope, but at the last minute the King
issued a royal dispensation, lifting the requirement that the
Lucasian Professor must take holy orders.
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Thus Charles II saved from oblivion one of the greatest scientists
of all time.
Publication of the Principia
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Parliamentary service
In 1684 Edmund Halley visited Newton and asked what
the shape of a planetary orbit would be if the force
between it and the Sun varied as the inverse square of
the distance between them.
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In 1687 James II became King.
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Newton replied that it would be an ellipse, and that he
had worked it out years before.
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He later produced a proof for Halley, in the course of
which he began developing a book-length discussion on
the subject of motion of bodies in orbit.
As an example the monarch sent a letter to Cambridge
commanding that a Benedictine monk should be given
an M.A. degree without requiring him to sign the 39
Articles of Religion of the Anglican Church.
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This was written out during 1685 & 1686 into his
magnum opus.
Newton encouraged the University to resist, and in the
end the King backed down.
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In 1687 Newton published the Philosophiae Naturalis
Principia Mathematica (Mathematical Principles of
Natural Philosophy).
In 1688 the University chose Newton to represent them
in the House of Commons for two years.
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He seems to have especially enjoyed this period in
London and participation in the halls of government.
Newton’s breakdown
Newton:
portrait of
1689
(aged 46)
Life in London
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He was a devout Roman Catholic and eventually
declared his intent to overthrow the English Church.
In 1694 one of Newton’s former Trinity students, Charles
Montague (later Lord Halifax), became Chancellor of the
Exchequer.
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In 1689 Newton’s mother died.
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In this period Newton also became obsessed with
obtaining a government position, and felt let down by
his friends when a position was not forthcoming.
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In 1692 he suffered a major mental breakdown.
In 1692 a young Swiss mathematician, Fatio de Duillier
who was an admirer and had proposed to edit a new
edition of the Principia, abruptly ended his friendship.
By 1693 he seems to have recovered, but had no interest
in creative scientific work, despite the fact that the
Principia was out of print and others were following up
his ideas.
The Great Man
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In 1699 the dispute between Leibniz and Newton became
a major international conflict.
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Yet another major quarrel erupted between Newton &
John Flamsteed, the Astronomer Royal.
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One of his decisions was the recoinage of English
currency.
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In 1696 Newton was appointed Warden of the Mint, and
the recoinage project was executed with great success.
In 1703 Newton was elected President of the Royal
Society, and continued so until he died.
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He moved to London and set up house with his niece,
Catharine Barton.
In 1704 Newton published his Opticks (in English), & the
Latin translation in 1706. It was wildly popular.
In 1705 he was knighted.
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In 1699 Newton was appointed Master of the Mint, and
held this position until his death.
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There is no doubt he carried out his duties as a civil
servant with distinction.
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In 1727 Newton died and was buried in a spectacular
tomb at Westminster Abbey.
By this time Newton had become one of the most
celebrated persons of his time.
Newton, 1726
Newton’s tomb,
Westminster Abbey
8.2 The Principia
Newton’s title page changes for the
2nd edition of the Principia
Principia summary
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Book I
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Book II
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The Principia
Laws of motion
Hydrostatics and hydrodynamics, in which Newton
demolishes Descartes’ vortex model for the planets.
Book III
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Law of universal gravitation
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Tides
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Accounts of oblate shapes of the Earth & other planets
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Masses of celestial objects in terms of mass of the
Earth.
The Principia, cont’d
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Using his laws of motion and gravitation, Newton
showed that he could deduce mathematically all three of
Kepler’s laws of planetary motion (which Kepler had of
course worked out painstakingly by years of analysis of
decades of observations).
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In short, Newton’s laws of motion and the law of
universal gravitation not only described precisely the
motions of planets orbiting the Sun, they enabled people
calculate things they had not previously thought were
possible.
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He also showed how his laws of motion and gravitation
could explain new phenomena:
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Comparison of predictions of Newton’s theory with
observations confirmed the validity of Newton’s ideas.
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Newton could predict from his theory all the observational
work of centuries.
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This was a spectacular achievement, something no one
had ever done before.
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And so it is not surprising that Newton was regarded as
one of the great intellectual giants of all time.
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the tides
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the oblate shapes of Earth and other planets.
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He calculated the mass of a planet in terms of the
Earth’s mass.
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And he showed that the Great Comet of 1680 was in
a Keplerian orbit about the Sun.
The First Law: Inertia
8.2.1 Newton’s
Laws of Motion
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Newton’s First Law of Motion (Inertia): “Every body
continues in its state of rest, or of uniform motion in a
right line, unless it is compelled to change that state by
forces impressed upon it.”
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As we have seen, both Descartes and Galileo were
working toward an idea of inertia.
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Descartes argued that matter, once in motion, continued
to move in a straight line until it collided with another
bit of matter, although this assumption is based on purely
metaphysical reasoning.
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Galileo’s ideas on inertia were derived from his
experiments with balls rolling on inclined planes, and on
projectile motion.
Newton’s Law of Inertia
Astronomical application
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Newton’s First Law effectively says that the natural
motion of an object at rest is to remain at rest, or, if it is
moving, to move in a straight line at constant speed.
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If we accept Newton’s First Law, what would we expect
the trajectory of the Moon or a planet would be, if there
were no forces on it?
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In other words, if there are no forces on an object, it will
either remain at rest or move with constant speed in a
straight line: it takes an external force to change either
the speed or direction of motion.
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A straight line!
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Note that Newton is adopting the idea of natural motion
that goes back to Aristotle, but with a rather different
formulation.
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Note also that the same natural motion applies to objects
on Earth and in the heavens.
The Second Law
Since the Moon and planets orbit about the Sun, a
consequence of the First Law is that there must be a
force which pulls them toward the Sun, as Kepler
argued.
Newton’s Second Law
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Newton’s Second Law of Motion: “The change of motion is
proportional to the motive force impressed; and is made in the
direction of the right line in which that force is impressed.”
Repeating: Force ! change in quantity of motion per unit time.
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Note that Newton defines “the quantity of motion” as “the
velocity and quantity of matter conjointly,” so that “quantity of
motion” is not velocity but what we would call momentum =
mass x velocity.
Force ! (mass) x (change in velocity per unit time),
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We see that, whatever force is, it can be measured
quantitatively by its effect on the momentum of an object.
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Note also that “quantity of motion” has a direction, and the
change in quantity of motion is in the same direction as the
direction of the force.
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This can be written: Force is proportional to change in
quantity of motion in a given time.
If mass is constant, then this becomes:
and since acceleration = (change in velocity)/time,
Newton’s Second Law becomes F ! (mass) x (acceleration).
With appropriate choice of units this is the famous
F = ma.
This shows that the mass of an object is its resistance to being
accelerated, i.e. a 5 lb force in a slingshot will accelerate a BB
to high speed, but will hardly move a bowling ball.
The idea of gravitation
8.2.2. The path to
gravitation
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During the two years back at Woolsthorpe Newton was
puzzling just what force could pull planets into orbit
round the Sun.
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The story goes that he was in the garden at Woolsthorpe
and he saw an apple fall from the apple tree.
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He realized that the same force that pulled the apple to
earth should extend much further could be the force that
pulled the circling Moon toward the Earth.
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Although this story is sometimes said to be apocryphal, it
was written down by John Conduitt, who married
Newton’s niece Catharine Barton, and who was his
assistant at the Mint for many years, and is stated by
several independent sources.
Drawing from 1820 of Woolsthorpe Manor and the (only)
apple tree in the Manor garden.
A photograph from 1998 taken with the same view as the
1820 drawing shown previously. The present tree is rooted
off the fallen trunk of the original tree.
Discovery of Gravitation
Gravity & the planets
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Newton had worked out that centrifugal force is
proportional to v2/r.
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He used Kepler’s third law of planetary motion to find an
expression for v in terms of r:
(period)2 = (radius of orbit)3.
The period = 2!r/v & therefore (2!r/v)2 = r3,
so v2 = 4!2/r.
Substituting this into the expression for the force
required to keep a planet (or the Moon) in its orbit:
Force of gravity: mv2/r
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m/r2.
This is how Newton convinced himself that the force
required to keep a planet in orbit is proportional to 1/r2.
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Gravity is the force
between a planet and
the Sun that continually
accelerates the planet
both in speed (Kepler’s
second law) and in
direction (Kepler’s first
law).
Newton’s theory of gravity
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Newton realized that to make a planet take an elliptical
(or circular) orbit required an attractive force between
the planet and the Sun.
He formulated this idea in his law of universal
gravitation: the attractive force between any two bodies
is proportional to the product of their masses (kg) divided
by the square of the distance between them:
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Mm
11
m1 m2
F =G 2
r
r
m2
Newton’s gravitational force
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In Newton’s gravitational force equation:
F = Gm1m2/r2
G is the proportionality constant.
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In MKS units (mass in kg, distance r in meters, force in
Newtons)
G = 6.67x10-11.
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This proportionality constant, and experimental
measurements to confirm Newton’s theory, were made
by Henry Cavendish in 1797 & 1798.
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(This enabled Cavendish to then calculate the mass of
the Earth.)
Problems with Newtonian Gravity
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How is the gravitational force transmitted between the Sun
and the planet?
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Most people in this period could only imagine forces being
applied by contact (e.g. Descartes and Huygens).
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Newton himself was reluctant to accept the idea that gravity
was an innate property of matter and could be transmitted
without the mediation of some intervening medium:
"...that one body may act upon another at a distance
through a vacuum without the mediation of anything else, by
and through which their action and force may be conveyed
from one to another, is to me so great an absurdity that, I
believe no man, who has in philosophic matters a competent
faculty of thinking, could ever fall into it."
8.2.3 Applications
of Newton’s
theories
Overview
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We should not let these comments on loose ends in
Newtonian gravity distract from the fact that Newton
used his dynamics and theory of gravitation to solve
problems or provide quantitative explanations no one
else had every been able to do before.
In Book III, “The System of the World,” Newton applies
this theories to a variety of astronomical situations, and
is able to make quantitative predictions which were
revolutionary.
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He checks them against measurements when possible to
see how well his theory does.
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For the many people who could not understand the
Principia itself, confirmation of these quantitative
predictions was simply astounding.
Orbits
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Newton adopted the helio-centric model for the solar
system.
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He was able to show purely from calculation Kepler’s
three laws of planetary motion--which, remember,
Kepler had discovered empirically from decades of
precise observations made by Tycho.
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This marks the absolute demise of the old geo-centric
picture of the heavens.
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It is also a striking confirmation of the validity of
Newton’s theories of dynamics and gravitation.
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We have now, for the first time, a theory of dynamics
which provides quantitative predictions of phenomena.
Comets
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Tides
More generally, Newton showed that the orbits of
objects moving under the influence of gravity may be
ellipses, parabolas, or hyperbolas (these are all conic
sections).
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Newton’s calculations showed that gravitational force
from the Moon is the major cause of the tides.
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(The Sun contributes about 25% of the effect of the
Moon.)
Newton showed that the Great Comet of 1680 was in an
elliptical orbit, and that two apparitions of a comet were
in fact the same comet.
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He calculated the expected heights of tides.
That all this worked proved that comets are “solid,
compact, fixed, and durable” rather than vapors emitted
by the Earth, the Sun, or the planets.
Centrifugal bulges of planets
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Newton argued that, since planets are rotating about an
axis, self-gravity in a direction perpendicular to the spin
axis is somewhat offset by centrifugal force, compared to
the direction along the spin axis.
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This makes the planets into oblate spheroids (pumpkins)
with the equatorial diameter larger than the polar
diameter.
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He compared his theoretical predictions with
measurements of the slightly slower movement of
pendulum clocks at the Earth’s equator compared to the
polar regions, and found the difference implied that the
equatorial diameter is 27 km (the modern value is about
40 km).
In summary
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What Newton showed in Book III is that there are
universal principles or “laws” that apply to a wide range
of physical phenomena.
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These laws can be expressed in mathematical form, and
can make quantitative predictions of the outcomes of
experiments.
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This set the paradigm for physical science which remains
with us to today.
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This also gave people the realization that there are likely
other universal laws or principles which can similarly be
found by creative thinking of a similar sort to Newton’s
use of reason for his discoveries.
He also showed that one gets the strong spring tides
when the Sun and Moon are lined up and the weaker
neap tides when the Sun and Moon are perpendicular.
Masses of planets
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To calculate the masses of planets Newton could use his
form of Kepler’s third law:
P2 = 4!2 R3/GM,
but he did not know the value for G.
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So he used the periods, P, and orbit radii, R, for the
satellites of a planet and the Earth to get the product GM
for each, and thereby the ratio of the planet’s mass to
that of the Earth.
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Example: Jupiter’s moon Callisto.
Period ~ 16 days and R ~ 4.9 x Moon’s orbit radius.
These give MJup/MEarth ~ 330. Modern value is 318.
Ode to Newton
Here is the last stanza of Halley’s Ode to Newton which prefixed
the Principia:
Then ye who now on heavenly nectar fare,
Come celebrate with me in song the name
Of Newton, to the Muses dear; for he
Unlocked the hidden treasuries of Truth:
So richly through his mind had Phoebus cast
The radiance of his own divinity.
Nearer the gods no mortal may approach.
Nothing else can say so clearly the profound effect Newton’s
achievements had on intellectual life in the 17th C.