Monday, March 26, 2012

Isaac Newton (1643-1727): Peering Inside the Mind of God

by G. Jack Urso
 

Nature and Nature's laws lay hid in night;
God said, Let Newton be! and all was light
                                                         Alexander Pope – 1727 (Tenn)

Isaac Newton's work is the culmination of the Copernican Revolution. He confirmed the validity of the Copernican heliocentric model of the universe (with Kepler's modifications) and in doing so put the last nail into the coffin of Aristotelian science. His major work, Principia Mathematica, also known as Principles of Natural Philosophy, is a cornerstone of modern science. In it Newton introduces the basic laws of physics, provides a mathematical analysis of force and motion, is first to use vectors to describe the direction and size of forces, and describes his law of universal gravitation (Bricker and Hughes, 2-3).  

Early Years 

Fig. 1: Sir Isaac Newton
Born January 4, 1643, in the farm country of Lincolnshire, England, Isaac Newton’s father was an wealthy, illiterate, farmer who owned much land and livestock, but was unable to sign his own name. After his father's death, his mother remarried when little Isaac was only two. Newton did not get along with his stepfather and accounts of his childhood are not happy ones. Idle and inattentive in school, Newton shows little of his genius early on. It is only later when an uncle and the schoolmaster take an interest in him that Newton begins to shine academically. He attended his uncle's alma mater, Trinity College, Cambridge, in order to become a lawyer, yet was attracted by the works of Nicolaus Copernicus, Johannes Kepler, and Galileo Galilei (Field). 

Influences 

"What Des-Cartes did was a good step. You have added much several ways, and especially in taking the colors of thin plates into philosophical consideration. If I have seen further it is by standing on the shoulders of giants."
Isaac Newton to Robert Hooke – 1676 (Bricker and Hughes, 2-3)

During his university years Newton was exposed to works that would greatly influence his thinking. One of these works was Rene Decartes' (1596-1650) Principles of Philosophy, first published in 1644. In it, Decartes dealt with the nature of aether and how it influenced motion in an evolving theory of the time called of corpuscularism, which maintained the aether was comprised of minute particles called corpuscles that influenced motion (Kuhn 237).  

Decartes believed that all change in the universe occurred as a result of the free movement and occasional collisions of these corpuscles. He engaged in a series of logical deductions as a way to comprehend the structure of the Copernican heliocentric model of the universe. His deductions are intuitive, but not supported by quantifiable physical evidence. Indeed, many of his deductions can now be proven wrong. Of the seven laws of collision Decartes introduced, only one was retained by his successors. Although his theories regarding the laws of collision were largely disproved, his idea regarding the collision process itself was retained (Kuhn 238).


The solutions required to address the problems introduced by corpuscularism led to the law of the conservation of momentum and the development of the relationship between force and the change in momentum it creates (Kuhn 239). That many of Decartes' conclusions can be proven wrong doesn't lessen the importance of the contribution of Principles of Philosophy. Tycho Brahe never accepted the Copernican heliocentric universe, yet his observations provided valuable evidence leading to its eventual acceptance. Similarly, while Decartes' conclusions may miss the mark, Principles of Philosophy does lead to Newton's Principia Mathematica 

Corpuscularism was only one of two explanations for the motion of the planets that evolved from Copernicus' work. Kepler advanced the theory of the mechanical solar system to explain the orbits of the planets. The mechanical solar system model relied on recent research regarding magnets to explain the eccentric orbits of the planets. In 1600, English physician William Gilbert published On the Magnet, in which he recognizes that the Earth itself is one large magnet. Kepler took this one step further and suggested that the Sun and other planets were also magnets and it was this force that drove the orbits of the planets; Kepler named this force the anima motrix (Kuhn 247, 249).  

In 1666, English physician Robert Hooke took this idea to the next level by suggesting the solar system actually moved as a sort of celestial equivalent to a terrestrial mechanism. Hooke got rid of the idea of the anima motrix and used a pendulum to explain his hypothesis. Imagine a simple pendulum, such as a weight attached to a string hung on a hook in the ceiling. The weight at the end of a pendulum has the tendency to move in a straight line, yet, due to the resistance of the string, the pendulum's tendency to move in a straight line is modified (Kuhn 249).  

Observing the movement of the pendulum, Hooke noticed that the weight would settle into circular movements. He rationalized that this must be similar to the movements of the planets. The irregularities of the orbits of the planets could be explained as the planets' tendency to move in a straight line, modified by the action of some unknown force. Newton would identify and explain that force as gravity (Kuhn 251).  

While this section is titled Influences, Isaac Newton would probably take exception to his colleague Dr. Robert Hooke being listed as an influence. Newton could document that he arrived at similar conclusions to the questions Hooke dealt with prior to any of Hooke's publications (De Gandt 6). Furthermore, Hooke, along with Dutch physicist Christaan Huygens, was a critic of Newton's work (Bricker and Hughes 155). To make matters worse, Hooke also claimed that Newton's Law of Universal Gravitation was stolen from him, which was easily shown not to be true (Westfall 206). Despite these events, Hooke's work does show the evolution towards Principia Matematica and presents problems that it would have to address. 

De Motu 

Early on in his life Newton was a proponent of the corpuscular philosophy, for without any physical evidence to support that's all it ever was. However, Newton was aware of the limitations of corpuscularism and the attempt to resolve them led to Principia Mathematica (Westfall 206). The idea of some kind of gravity, or theory of heaviness as Kepler referred to it, was acknowledged by his predecessors. Newton did not invent gravity but he did quantify and define the attractive force that was driving the orbit of the planets. A visit in 1684 with Edmond Halley (who predicted the time interval for the orbit of the comet that bears his name) resulted in a short treatise called De Motu (On Motion) (De Gandt 7). De Motu gives Newton the opportunity to work out his ideas on centrifugal and centripetal force and how they relate to an object's trajectory.  

De Motu considers four theorems and four problems in about ten pages of text, a much smaller book than the Principia, which has almost 200 propositions and over 500 pages. Halley proposed a question, which in effect said, “Given the law of force, how do you determine an object's trajectory?” Newton responded, but the question he responded to in De Motu was, “Given the trajectory, how do you find the law of force?” This is the "inverse problem" of De Motu. Newton presents an innovative idea in response to what he thinks is the question; he introduces and defines centripetal force as "a body which is attracted or impelled towards some point viewed as a center." What that force is, however, is left undefined, and little is discussed about the center point as well (De Gandt 7-8, 15).  

Despite this Newton does manage to answer Halley's question by addressing its inverse, if doing so in a roundabout manner; however, an important question is left. If centripetal force is responsible for the obit of an object (a planet) can it be demonstrated that this force diminishes as the square of the distance from the focal point (the Sun)? Principia Mathematica extends the study of force by examining other areas such as "simple machines, impact, pendular motions, optics, motion in resisting media and fluid dynamics” (De Gandt 55).

Principia Mathematica               

Newton began writing Principia Mathematica after his visit with Halley in 1684. It was the intended follow up to De Motu and its purpose was to "demonstrate the frame of the System of the World." It was initially published in Latin in 1687 and the first English translations appeared just a few years after Newton's death in 1727. Comparing the translations it is evident that translators may miss references, or translate a passage so poorly that interpreting its meaning can be a challenge. Considering the difficulty of the material that can make ascertaining Newton's intention nearly impossible depending on the translation (Newton viii, xiii, xv). This also underscores the necessity to go back to primary sources periodically, particularly with translated texts. 

Newton's law of universal gravitation is the power that drives the Copernican heliocentric world system. In the Principia, Newton expresses it as: 
"Any two bodies in the universe are attracted to each other with a force that is proportional to the masses of the two bodies and inversely proportional to the square of the distance between them."       (Snow 52)  
In other words: 
             Every object attracts every other object, by virtue of their
               having mass.

            An object with twice the mass will attract other objects
              with twice the force.

            An object twice as far away will attract other objects with
              one-fourth the force.
                                                                                               (Westfall 1)
The Laws of Motion 

The Principia sets down Newton's laws of motion, the core of the fundamental physics behind the heliocentric model of the universe. These laws explain the basis of how universal gravitation works. 

Law 1: A body at rest tends to stay at rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed on it. 

Law 2: 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. 

Law 3: To every action there is an equal and opposite reaction.

Newton attributes the first two laws to Galileo: 

"By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies varied as the square of the time and that the motion of projectiles was in the curve of the parabola…."
                                                                  Isaac Newton (Westfall 1)       

Newtonian Mechanics 

Principia Mathematica also defines the principles of the mechanics behind the Newton's laws of motion. Over 300 years after their first publication these principles are among the first concepts studied by physics students today. Due to the necessity to accurately state these laws, they are reprinted below as defined by Dr. Joseph S. Tenn, Department of Physics and Astronomy, Sonoma State University: 
 
  1. The position of a body is a function of the time. It can be  expressed in terms of three coordinates, for example, how far East, North, and Up it is from an arbitrarily chosen origin (a distance South is just a negative distance North).   
  2. The rate of change of the position is called the velocity. It includes both magnitude (speed) and direction.  
  3. The rate of change of the velocity is called the acceleration. It also includes both magnitude and direction.  
  4. If you know where a body is at some instant and you also know its velocity, you can calculate where it will be a very short time later.  
  5. If you know the velocity of a body at some instant and you also know its acceleration, you can calculate its velocity a very short time later.  
  6. Acceleration of a body is the net force acting on it divided by its mass.  
  7. The net force acting on a body is the vector sum of all of the  forces acting on the body. For two forces it is the diagonal of the parallelogram formed by representing them as arrows.  

Isaac Newton demonstrated in the Principia that any motion in the sky could be explained by using the above laws plus gravity, a universal force. This gravitational force is proportional to the mass of the objects in question (be they planets, moons or comets) and inversely proportional to the square of the distance between the objects. The same laws apply to the motion of the Earth as well with the inclusion of friction in addition to gravity (Tenn). 

The World According to Newton 

Copernicus, Kepler, Brahe, Decartes, and even Galileo, to a degree, considered a divine force to be at least partially responsible for the motion of the planets and stars. Newtonian mechanics showed that a set of physical laws, not divine intercession, was responsible for motion. To the people of the time this seemed as though the universe was a giant clockwork mechanism with the Earth as just another cog. Perhaps God winded the mechanism up, but it is not an act of the divine that moves the planets and the stars, rather a force of nature that is the result of the interaction of two bodies on one another. 

Newtonian science influenced other scientific areas as well. Consider Newton's Third Law of Motion, for every action there is an equal and opposite reaction. It has been used to illustrate innumerable cause and effect situations completely unrelated to physics.  

Chemistry, biology, psychology, and the social sciences applied the principles of Newtonian science to their own disciplines by reducing the objects of their study into events and "objective entities" whose actions are observed, documented and evaluated. A set of rules or core principles are formulated and then applied to determine the reaction of any subject or object under their study. In the science of human behavior, the effect of this is to remove the moral question from consideration. Behavior becomes an observable, quantifiable event to be documented and evaluated, not judged (Bricker and Hughes 215).  

By the time of the publication of Principia Mathematica in 1687 the Protestant Reformation was already well under way. What Newton contributes to the Reformation, and the evolution of philosophical thought itself, is to move discussion of morality from a debate about standards of absolute right and wrong to a more subjective perspective (Bricker and Hughes 216). Behavior that is wrong for one culture may then be seen as perfectly acceptable to another. This further weakened the Catholic Church's influence by implying that what the Vatican held to be true just might not be relevant to everyone.  

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Works Cited 
 
Bricker, Phillip and Hughes, R.I.G., eds. Philosophical Perspectives
        on Newtonian Science. Cambridge and London: MIT Press,
        1990. Print.
De Gandt, Francois. Force and Geometry in Newton's Principia.
        Trans. Curtis Wilson. Princeton: Princeton University Press, 
        1995. Print.
Field, J.F. The MacTutor History of Mathematics Archive: Johannes 
        Kepler. The University of St. Andrews, Scotland, School of 
        Mathematics and Statistics, 1997. Web. 7 Dec. 2000. <http://
        www-groups.dcs.st-and.ac.uk/~history/Mathematicians/
        Kepler.html>. 
Kuhn, Thomas, S. The Copernican Revolution: Planetary Astronomy
        in the Development of Western Thought. Cambridge: Harvard
        University Press, 1957. Print.

Newton, Sir Isaac. A Treatise of the System of the World.  London:
        Dawsons of Pall Mall, 1969. Print.

Salgado, Rob. Newton, Galileo, and The Laws of Gravitation.
        Syracuse University, 1995. Web. 16 Dec. 2000. <http://
        physics.hallym.ac.kr/education/syracuse/LIGHTCONE/
        newton-gr.html>. 
Snow, Theodore P. The Dynamic Universe. St. Paul, New York, San
        Francisco and Los Angeles: West Publishing Company, 1987. 
        Print.
Tenn, Joseph S. The Copernican Revolution. Sonoma State
        University, 1997. Web. 7 Dec. 2000. <http://
        www.phys-astro.sonoma.edu/people/faculty/tenn/
        CopernicanRevolution.html>.
Westfall, Richard, S. Force in Newton's Physics. New York:
        American Elsevier, 1971. Print.

 

Friday, March 23, 2012

Galileo Galilei (1564-1642): The Starry Messenger

by G. Jack Urso
 


While Nicolaus Copernicus, Tycho Brahe, and Johannes Kepler avoided major confrontations with the church, Galileo Galilei found himself in a serious struggle with the Vatican as a result of his research. By 1609, when Galileo published his first work based upon his observations with the telescope, Europe had enjoyed nearly 100 years of Protestantism. Perhaps enjoy isn't quite the right word for it for in reaction to Protestantism came the Inquisition. It is with that force that Galileo, unlike his predecessors, had to contend. 


Early Years


Fig. 1: Galileo Galilei
circa 1636.
Born in 1564 in Pisa, Italy, Galileo's father was a musician and his mother quite educated for a woman of her time and class. Galileo's father, in addition to a talent for math, was mistrustful of authority; characteristics he passed on to his son. Galileo received his early education from local monks; however, when his father learned that his son was considering joining the local monastery he immediately withdrew Galileo and sent him off to the University of Pisa School of Medicine. Here, Galileo conducted experiments with a pendulum that helped him form ideas that eventually led to his law of falling bodies (McMullin 52-53).
 

In the early years of his career in Pisa, Galileo lectured and wrote on mathematics and mechanics while criticizing Aristotle's science. Galileo also imprudently criticized an engineering proposal by one of the members of the Medici ruling family, which led to his departure to Padua in 1592. There, Galileo served as chair of the mathematics department at the University of Padua until 1610 and it was here that he became acquainted with an instrument that would literally open the eyes of astronomers everywhere, the telescope (McMullin 54-55).
 

Galileo had long been aware of the inaccuracies of Aristotelian science. For example, he could not help noticing that large and small hailstones hit the ground at the same time, contradicting Aristotle's teaching that heavy objects fall faster than light objects (McMullin 54). Galileo's famous "experiment" from the top of the Leaning Tower of Pisa (circa 1590) did not take place as reported. He instead demonstrated his point by rolling balls down an incline. Similar experiments though had been previously documented. Benedetti Giambattista published his results in 1553, followed by Flemish engineer Simon Stevin who published his findings in 1586 (Hawking 16, 55). Galileo may have been challenged by the conservative academic climate of Pisa to conduct his own experiments.
 

The Starry Messenger
 

Fig. 2: Galileo's telescope
In 1604, a nova appeared that gave Galileo an opportunity to show that Aristotle was wrong, yet again, in suggesting the heavens are unchanging (a belief endorsed by the Catholic Church). Tycho Brahe's studies of the nova of 1574 gave credence to what Galileo theorized, so, in 1609, Galileo constructed his first telescope which allowed him to support his propositions with evidence. The original Dutch patent for the telescope indicates that it was intended for land observation, not the night sky. Hearing about the device from a clergyman friend, Galileo made one of his own with the ability to magnify a distant object nine times, far exceeding the original Dutch design. The following year, in 1610, Galileo published Sidereus Nuncius (The Starry Messenger), which detailed his observations of the mountains of the Moon, four moons of Jupiter, and "innumerable" stars unseen by the naked eye (Hawking 57-58). All first seen and documented by Galileo.
Fig. 3: On left - Drawing of the Moon by Galileo
On right - Photograph of the Moon from the same angle
Kepler read Sidereus Nuncius and endorsed its findings before ever seeing a telescope. That he accepted Galileo's observations, even though he lacked the instrumentation to duplicate the results, is a testament to the influence of Sidereus Nuncius. Kepler wrote a long letter to Galileo discussing his impressions and even wrote the Duke of Tuscany, Giuliano de' Medici, to praise his most noted resident's work (Kepler 3, 9). This letter was published in Prague in 1610 and lent crucial support to Galileo's findings in the secular world. Predictably, the Catholic Church had a different opinion on the matter.
 

On Floating Bodies and Sunspots
 

In 1612, Galileo was once again on the attack against Aristotelian science. His Discourses on Floating Bodies discussed what causes an object to float as well as the physical nature of ice. In the book Galileo also provides a number of simple experiments that contradict the Aristotelian position on the nature of buoyancy. Galileo would defend this work against numerous academic attacks (McMullin 59-60).
 

Also in 1612, a German Jesuit priest published observations of sunspots and explained them as small planets orbiting the sun. Although this is in keeping with the church's view that the sun was perfect and therefore free of such imperfections as a blemish on its surface, the Jesuit was nonetheless compelled to publish under a pseudonym in order to protect himself from prosecution. Galileo, who had observed sunspots sometime prior to 1612, published his own conclusions in a series of letters in 1613. Not only did Galileo correctly assert that sunspots were an ever-changing manifestation of the Sun itself, but he also signed his name, giving those who opposed any challenge to the Church a target. In 1614, he was denounced from the pulpit in Florence and the Inquisition soon took up the matter on referral from a Dominican priest (McMullin 59-60).
 

Inquisition
 

In 1615 the Church begins its counterattack against the Copernican Revolution. Pope Paul V declared the Sun and Earth did not move and Galileo was ordered to cease espousing views to the contrary (Ollney 89). Additionally, the pope effectively banned Copernicus' De Revolutionibus by suspending it “pending correction.” Galileo, in Rome to answer his accusers, defended his views only to be commanded by the pope to remain silent. Recalled to Florence, Galileo entered a period of scientific inactivity for the next two years (McMullin 61).
 

The Dialogue
 

It was not until Pope Urban VIII succeeded Pope Paul V in 1624 that was Galileo able to resume public discourse of his ideas on cosmology. The new pope gave Galileo permission to discuss his ideas in a hypothetical context. The result was Dialogue on the Two Great World Systems (1632) (Dialogue on the Two xviii). Rather than a text of observations, calculations, diagrams and maps, Galileo presents the discussion of the difference between the two competing world systems as a discourse between three "interlocutors," Salviatus, Sagredus and Simplicius (Dialogue Concerning 6).
 

Salviatus is a proponent of the Copernican system. Sagredus and Simplicius argue for the Aristotelian view. Over the course of four days various issues with both systems are discussed. Salviatus is imbued with Galileo's persuasive prowess. Sagredus has a mind open to civilized conjecture while Simplicius, on the other hand, is obnoxious and rude (Dialogue Concerning 7). Although the book ends with affirmations of the glory of God, it led to a serious encounter with the Inquisition. Naming the defender of the Aristotelian world-view "Simple" probably didn't earn Galileo any favor with the inquisitors.
 

Later Years
 

In 1633, almost seventy, Galileo went to Rome to stand trial. The case centered around the injunction originally filed against Galileo in 1616 after his last trial. The inquisitors had an unsigned memorandum from the Church's records suggesting that Galileo did indeed exceed the injunction's commands. Galileo, however, produced a signed letter from the original trial that his prosecutors did not know existed. This letter, from the Cardinal in charge of the 1616 trial, left room for doubt regarding Galileo's guilt (McMullin 63).
 

Sentenced to life in prison and prohibited from writing about Copernicanism, Galileo was allowed to return to his home near Florence under the close scrutiny of the Inquisition. All of Galileo's works, published and unpublished, were banned. To their credit, it should be noted that three of the ten cardinal-judges conducting the trial refused to sign the decree of sentence (Langford 153).
 

The Two New Sciences
 

Despite the ban on the publication of his works Dialogue Concerning the Two Chief World Systems was published in Strasbourg in 1635 out of reach of the Vatican's control (Langford 157). Galileo did not cease writing in the face of the Inquisition. Using the same characters from Dialogue Concerning the Two Chief World Systems, Galileo finished work on The Two New Sciences in 1636. In it, he establishes laws of accelerated motion and of falling bodies as well as basic theorems regarding projectile motion - ballistics (McMullin 64).
 

Similar in format to the Dialogue of 1632, it also uses the same interlocutors Salviatus, Sagredus, and Simplicius. Galileo presents his theory regarding ballistics, which deals with resistance, cohesion, and acceleration in bodies in motion as well as proof of parabolic trajectories in projectiles. This theory was in opposition to Aristotelian science which taught that projectiles went straight out, then straight down (Dibner and Drake 37).
 

Galileo's sight continues to fail until he finally goes blind in 1637, just after finishing The Two New Sciences. The following year, in 1638, The Two New Sciences is published in Leyden, Holland. Galileo got around the ban on publication by reporting that the book was published without his permission. He spent his final years dictating new chapters for The Two New Sciences and occasionally teaching and lecturing until passing away in 1642, the same year Isaac Newton was born (Langford 158).


The Catholic Church, by the way, never got around to officially forgiving Galileo until 1981 (Ross 21).



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 Works Cited
 
Dibner, Bern and Stillman Drake. A Letter from Galileo. Norwalk:
              Burndy Library, 1967. Print. 

Galilei, Galileo. Dialogue Concerning the Two Chief World 
              Systems-Ptolemaic and Copernican. Trans. Stillman Drake.
              Berkeley and Los Angeles: University of California Press,
              1967. Print.

---. Dialogue on the Two Great World Systems. Ed. Giorgio de
              Santillana. Chicago, Toronto and London: The University
              of Chicago, 1955. Print.

Hawking, Stephen W. A Brief History of Time. New York, Toronto
              and London: Bantam Books, 1988. Print.

Kepler, Johannes. Kepler's Conversation with Galileo's Sideral
              Messenger. Trans. Edward Rosen. New York and London:
              Johnson Reprint Corporation, 1965. Print.

Langford, Jerome, J. Galileo, Science and the Church. New York:
              Desclee Company, 1966. Print.
 

McMullin, Ernan, ed. Galileo: Man of Science. New York and
              London: Basic Books, Inc., 1967. Print.

Ollney, Allan. The Private Life of Galileo. London: MacMillian &
              Co., 1870. Print.