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 T2/d3 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.