Science 122 Part 2 Summary

# Part 2 Summary

#### Part two traces the scientific revolution beginning with the Roman Empire, through
the middle ages, The Renaissance, the heliocentric model of Copernicus, the precision
instruments of Tycho Brahe, the mathematical wizardy of Johannes Kepler, the use
of the telescope and studies of motion undertaken by Galileo, and Sir Isaac Newton's
synthesis of the Law of Universal Gravitation.

# Darkness and Dawn

#### With the rise and fall of the Roman Empire, the concept of learning, logic and
speculation changed with the spread of Christianity and Islam. The chaos that followed
Rome's tight hold on Europe and the Mediterranean created a six hundred year period
during which learning was encouraged only if it related to studying the scriptures
or the nature of man. The works of the ancients were closely guarded as they were
read, copied and recopied generation after generation both in Éurope and the
middle east.

#### In the middle east learning was not so stifled. Rules of algebra were formulated,
the decimal system along with arabic numbers and the zero were introduced into mathematics,
and star catalogs were improved with the addition of many new stars.

#### The rediscovery in Europe of works of Aristotle and Ptolemy impressed officials
of the Church in their breadth, depth, and apparent truth. An effort was made to
incorporate the cosmology of aristotle and the astronomy of Ptolemy with the dogma
of the Church, culminating in the publication of Summa Theologica by Thomas Aquinas.

#### Successful as St. Thomas was in incorporating Aristotle's cosmology into Church
doctrine, and it was a major intellectual achievement, it was difficult to reconcile
the geometry of Ptolemy with the cosmology of Aristotle. The predicted positions
of the planets were far from agreement with those predicted by calculations performed
according to the Ptolemaic prescription. And the Ptolemaic formulae themselves had
been modified in unknown ways, intentionally or otherwise, having been bounced around
in three different languages for one thousand years.

#### The calendar was off by some unknown amount and the seasons didn't occur at the
right times, thereby affecting religious holidays such as Christmas and Easter, but
also affecting the timing of planting and harvesting of crops.

#### A more accurate way of calculating the positions was welcomed by the Church,
especially since Copernicus dedicated his book to the Pope and professed it to be
no more than a more convenient and more consistent way to predict the locations of
the planets in the sky and bring the calendar back into synch with the stars.

#### At the same time, the text of the book clearly shows Copernicus' preference for
a cosmology and a geometry which were both heliocentric and parsimonious.

# Brahe and Kepler

#### Tycho Brahe was the first professional astronomer, having convinced the King
of Denmark to give him the money to build and fund a Royal Observatory.

#### From this observatory he would observe and track the planets night after night
for twenty years, The instruments he invented and built allowed observational precision
never before possible.

#### This allowed him to determine that Ptolemy's calculations did not predict the
paths of the planets nearly as well as had been thought previously. The observational
precision also allowed him to determine that two comets and a supernova were heavenly
events and did not occur in the sublunar realm where they were allowed by Aristotle's
cosmoloty, and where it had been previously assumed that they did occur.

#### Brahe hired a young assistant named Jophannes Kepler. Upon Brahe's death Kepler
was given his data and his job.

#### Kepler, being a good Pythagorean mathematician, and a diligent worker, set out
to prove once and for all the Pythagorean harmony and the Platonic perfection of
the universe.

#### Although he failed to do so he did discover that the motion of the planets could
best be described by a heliocentric model which had all of the planets, including
earth, orbiting the sun in elliptical orbits rather than in circles. He showed that
the planetary orbits contained geometric and harmonic relationships previouly undiscovered.

#### Kepler's work was significant not only because of the mathematic relationships,
but also because he drew attention to the concept of a central force which keeps
the planets in orbit.

# Kepler’s Laws

#### Keplers laws of planetary motion are simply stated and concisely defined. They
are empirical laws, which means that they were the most parsimonious ways to represent
Brahe’s data. Kepler could not explain them although he gave an explanation based
on a misunderstanding of magnetism.

#### In his attempt to find the Pythagorean harmony of the universe, and because of
the small discrepency in the orbit of Mars (about 1/15 of a degree) would not fit
into a circle, Kepler was forced to give up the circle and try an ellipse. Having
done this and finding that an ellipse worked for Mars, he discovered that elliptical
orbits work better for all the planets, including earth.

#### The first law relates that for all of the planets the sun occupies one focus
of the ellipse while the other focus is empty.

#### The second law relates the speed of a planet in its orbit to its distance from
the sun. This law is sometimes called the equal areas law because of the way Kepler
stated it. If each planet is connected to the sun by an imaginary line, it sweeps
out equal areas in equal times.

#### The third law is known as the harmonic law because of its Pythgorean nature.
It states that the square of the period of time required for one orbit is proportional
to the cube of the average distance from the sun. The ratio T^{2}/d^{3}
is the same for all planets except the moon, including earth.

#### Kepler’s laws are significant for several reasons. Foremost is that they support
the heliocentric model of Copernicus. Second, although not circular, the ellipse
is in the same family as the circle, and the geometry of the universe is still Euclidian
and Phythagorean. The circular paradigm is not broken, just slightly bent.

#### Third, the fact that earth has the same constant of proportion as the other planets
points out that the earth is indeed just another planet and not the center of the
universe.

#### Finally, the laws show the first mathematical relationship between heavenly objects.
Previously mathematics was only useful for calculations, and required different methods
for different objects. This quantitative connection insists on an explanation for
the relationship.

# Describing Motion

## Aristotle & Scholastic Physics

#### Aristotle's views on motion were closely linked to his cosmology which required
the Prime Mover to keep it running and which held that all motion occurs because
of impure mixtures of elements which were trying to separate and attain perfection.

#### In *Summa Theologica*, Aquinas synthesized a new paradigm for learning which
combined Aristotle's cosmology with Platonic philosophy, Ptoleaic astronmy, and Church
dogma. The universe described by Aquinas was geocentric and moved according the Divine
Plan in the method described by Ptolemy. Mathematics, althought useful for calculating
the location of planets, is of little use in describing change.

#### Matter dominates the universe and causes change as described by Aristotle.

#### In the Scholastic physics, the natural state of matter is at rest and motion
cannot be sustained without cause. Corollaries of this assumption were used to prove
that the earth could not be in motion around the sun or on its axis.

#### Aristotle described four kinds of motion. Alteration included weathering, erosion,
rusting, and other types of chemical changes.

#### Natural or local motion is vertical motion which is controlled and limited by
the matter through which motion occurs and by the weight or gravity of the falling
object. It is always in straight lines.

#### Violent motion is motion which is caused by pushing or pulling. Chariots, projectiles
and ships undergo violent motion when they are moved. According to Aristotle, once
the pushing or pulling force stops, motion ceases. Violent motion also occurs only
in straight lines.

#### Celestial motion is the motions of the heavens. It is perfect, circular, uniform,
and driven by the Prime Mover ( God) through crystal spheres made of imponderable
quintessence. Only in the heavens can motion occur in other than straight lines.

## A Modern View of Motion

#### Our modern concept of motion was formulated by Galileo because he recognized
that it was the weakest of Aristotle's theories.

#### Having seen the phases of Venus and the moons of Jupiter, he was convinced of
the truth of the Copernican system, but could not convince his contemporaries that
the heavens as seen through the telescope were real. He realized that to prove Aristotle's
cosmology to be incorrect he would have to first prove that Aristotle's theory of
motion was incorrect.

#### Galileo introduced the concept of time into motion studies as he defined velocity
as the ratio of distance to time. Adding the concept of uniform acceleration as the
rate of change of velocity gave him all he needed to test whether or not freefall
was a case of uniform acceleration.

Through the use of algebraic logic, Galileo derived the relationship between distance
and time when acceleration is occuring. Distance and time can easily be measured,
so measurements of the location of an object at different times can test the relationship
against the theory.

In modern physical science we use graphs of physical variables to help us visualize
relationships in terms of shapes. The use of graphs to see the shapes of relationships
was also criticial in Newton's analysis of gravity, and in his development of the
calculus.

A graph of distance versus time will be a straight line if motion is constant. The
slope of the line represents the ratio d/t and its numerical value equals the velocity.

If motion is not constant, the graph will be made of line segments of different slopes.

If motion is uniformly accelerated, such that velocity is added ad a constant rate,
the graph will be an upward curve if velocity is increasing.

On a graph of velocity vs. time, uniform acceleration will plot as a straight line.
The slope of the line represents the ratio DV/t and its numerical value equals the
acceleration. The area bounded by the plot and the time axis represents the distance
traveled during the measured acceleration.

The relationship between slope and area of a graph is the basis for the relationship
between differential and integral calculus, discovered by Newton.

# Galileo: The First Scientist

#### Galileo’s contributions have earned him the title “Father of Science”.

#### Well schooled in classical Greek and Latin, Galileo was trained in the Priesthood
and familiar with the Scholastic Philosophy. He wrote and delivered lectures and
scholarly papers on such topics as the size and shape of Dante’s Inferno, and gained
a reputation all over Europe.

#### Around the age of thirty, he underwent an amazing transformation from a medieval
to a modern man. During this period he became a convert to the Copernican system
and began systematic studies of motion in an attempt to prove Aristotle’s views incorrect.

#### His use of the telescope to view the moons of Jupiter, the phases of Venus, the
flattening of Saturn, and the mountains and craters of our own moon, marked the first
serious observational challenge to the paradigm of geocentric heavenly perfection.
With the publication of a booklet, The Starry Messenger, Galileo drew attention to
his Copernican views and set the stage for further events which would forever change
the way we study our world

#### In 1617 Galileo was warned by the Church to stop teaching the Copernican theory
at the University of Padua where he was now a professor of mathematics.

#### In 1632, having secured permission from the Pope to publish scholarly critique
of the two systems, he published Dialogues Concerning the Two Chief World Systems.
In this book he used a Platonic dialogue to discuss the merits of the two systems.
The dialogues were cleverly written in such a way that each debate was conceded by
the moderator to have been won by the Ptolemaic advocate, while it was clear to the
reader that the Copernican viewpoint was by far a more logical and probable system.

#### The dialogues fooled the Church censors until Galileo’s detractors, who read
much more critically than the censors, discovered the obvious intent. Galileo was
summoned by the Pope and forced to recant, and as an alternative to even worse punishment
was kept confined under house arrest until his death.

#### While confined he completed the manuscript of Two New Sciences, which was published
in Holland in 1638. In this book he detailed his studies on motion which included
the law of freefall, discovery of inertia and the explanation for projectile motion.

## Freefall & Inertia

#### Galileo’s goal of disproving Aristotle’s views on motion was met successfully.
Along the way he made some other surprizing discoveries of the properties of matter
and motion which set the stage for Newton’s work on gravity and the subsequent development
of conservation laws.

#### In addition to his motion studies, and his telescope observations, Galileo used
logic and mathematics in ways that had not been tried before to produce relationships
which were quantitatively testable in a laboratory. He defined motion in an unambiguous
way by introducing the concept of time and its relationship with space or location.
From the data he collected in cleverly designed, carefully controlled and repeated
experiments, he was able to show that pure freefall acceleration is uniform and constant
for all objects regardless of their own size and weight.

#### The air, regarded by Aristotle as the controlling factor in natural motion, was
shown by Galileo to be a interference to motion rather than a cause of it.

#### After observing the motion of balls of various textures as they rolled on inclined
planes of various angles, he concluded that without gravity to speed the descent
and slow the ascent, and without friction, objects would not start or stop moving
at all.

#### Recognizing that falling objects accelerate at the same uniform rate even if
they are moving horizontally allowed him to understand the motion of projectiles.
In this analysis Galileo foreshadowed Descarte’s analytic geometry and Newton’s vector
algebra.

#### Describing projectile motion as a combination of horizontal and vertical motion
and in the same terms and with the same relationships destroyed Aristotle’s concept
of different types of motion.

#### A new paradigm of motion was established in which the idea that motion ceased
unless actively maintained was replaced by the notion that motion continued unless
interferred with.

# Newton and The Laws

#### Newton's synthesis of mechanics ranks as one of the crowning achievements of
the human mind. With his book "Mathematical Principles of Natural Philosophy",
he dealt the final blow to the authority of Aristotle.

#### In formulating the law of universal gravitation he synthesized Galileo's work
on motion and freefall with Kepler's laws of planetary motion. To this he added clear
definitions of mass, space, and time, listed rules for scientific inquiry, invented
methods of mathematical analysis and clearly stated three laws of motion.

#### Newton said "If I have seen further than others it is because I have stood
on the shoulders of giants." Even with his genius and legendary ability to concentrate,
Newton could not have done what he did without the work of his predecessors. Newton's
work is often referred to as "The Newtonian Synthesis".

#### The seventeenth century was an especially vital period and was marked by a burst
of creative activity in the arts and the sciences.

#### The question of how to describe planetary motion in circular terms was now passe.
The question of the times was how to explain planetary motion in a way which was
consistent with Kepler’s laws and with earthly gravitation.

#### Isaac Newton was born on Christmas 1642 at Woolsthorpe. He was not a precocious
child, although he had a knack for mechanical toys such as windmills and sundials.
He was a difficult child and suffered from behavior problems. Nonetheless he was
accepted at Cambridge University, with the help of an uncle, where his mathematical
abilities blossomed.

#### At Cambridge Newton was clasically educated in the ideas of the ancients and
in Scholastic Philosophy. He also became acquainted with the ideas of Copernicus,
Kepler, and Galileo.

#### After graduation, the university was closed as a wave of the Black Plague swept
once again through England. He retired to Woolsthorpe where he spent two years contemplating
the universe.

#### Among his accomplishments during this time were his studies in optics, alchemy,
mathematics and natural philosophy. He proved the binomial theorem, invented integral
and differential calculus, formulated our modern theory of color, invented and used
a new type of telescope, and solved the gravity problem.

#### Upon his return to Cambridge in 1664 he began teaching mathematics. The following
year his mentor, Isaac Barrow resigned so that Newton could hold the Lucasian chair
of mathematics.

#### Later Newton presented his theory of color at a meeting of the Royal Society
of London. The Royal Society had been formed by Robert Boyle as a means of communicating
scientific discoveries and theories. Among its members were Edmund Halley, Christopher
Wren, and Robert Hooke.

#### It happened that Newton's theory of color was contrary to Hooke's and Hooke let
him know. Newton was so upset by the criticism that he decided not to participate
in further science at the Royal Society.

#### His greatest work, *Mathematical Priniples of Natural Philosophy* came into
existence many years later when Halley mentioned that Hooke thought the inverse square
central force would create elliptical orbits, but couldn't prove it. Newton proved
it later that night, ostensibly to spite Hooke. Upon hearing Newton's proof, Halley
encouraged him to publish. In fact, Halley agreed not only to pay for publication
of the book, but also to pay Neton's salary while he wrote it.

#### Newton retreated to Woolsthorpe where he worked maniacally for eighteen months.
The "*Principia*", as his work is often called, was an overnight hit,
despite the fact that it contained over three hundred pages of sophisticated geometric
proofs, and was written in scholarly Latin. Apparently Newton wanted to "overkill"
on the proofs, but he also wanted to be sure that the book would only be reviewed
by those with sufficient background to read and understand it.

Newton became an instant hero, having solved one of the riddles of the ages, and
with an elegance not seen before or since.

#### Included in the Principia were the three laws of motion, now known simply as
"Newton's Laws" along with the Law of Universal Gravitation and its derivation,
and explanations for many phenomena such as the tides, weight, projectile motion,
satellite motion, instructions on how to launch artificial satellites, how to track
the orbit of comets and other celestial bodies, and more.

#### Newton's laws stand today as the defining axioms of all of physics. They are
simple, elegant, logically consistent, and easy to apply mathematically to a variety
of situations involving forces.

Simply stated the laws are:

1. An object will continue in a state of motion or at rest unless acted upon by a
non-zero net force.

#### 2. A non-zero net force will produce a change velocity which is proportional
to the force and inversely proportional to the mass.

#### 3. All forces act in pairs with equal and opposite magnitudes on two interacting
objects.

#### In Part 3 we will explore the Law of Universal Gravitation and the Newtonian
Synthesis along with their influences on social and scientific paradigms, their implications
and their impact on the study of matter and energy.