Assignment 3

Sir Isaac Newton was the first man to prove mathematically that Kepler's Laws of nature follow from the square inverse force of universal gravitation [1]. Many astronomers and physicists at the time had the intuition that such an force may well exist, with most believing the force to be one of gravity, while some thought it was due to magnetism. When a Dr. Edmond Halley visited Newton for the first time, his purpose was to inquire about this exact problem. Much to his elated shock, Newton had solved the problem many years before. Amazingly, he had invented a fundamental form of calculus, necessary to prove the relation between universal gravitation and Kepler's Laws [2]. This was exciting news for Halley, for not only was it the solution to an incredibly difficult problem [in some sense an impossible one], but it was also an incredible breakthrough for all of physics and astronomy.

                                              Fig. 1 [9]

Newton's proof was groundbreaking. It demonstrated beyond a shadow of a doubt that the force of gravity was not only a very real thing, but also one that is universal, a force that applies to everything. In this way, Newton was able to cast into the fire, all the old notions of heavenly and earthly bodies being separate, as Earth must experience gravity in the same way as all the other planets [1]. Another more subtle reason for the significance of this proof, is that it was the spark to the great flame that was Newton's Principia. Formalizing the proof caused Newton to become borderline obsessed with the material and though Halley was only expecting a write up of the proof, he instead received a veritable goldmine of brilliant new discoveries [3]. These discoveries would eventually lead to the creation of Principia Newton's book containing all of his work on the laws of gravity. In other words, it was a book containing an entire field of physics that Newton birthed from this one idea [2]. In this way the initial spark of the square inverse force proof was far more important than just its direct corollaries. 

The creation of Principia was not exactly smooth sailing and very nearly never occurred. Interestingly, the  publication of the book  had far more to do with the personalities of two odd men [and one not-so odd] than any mathematical proof. Newton was a surly, sulky recluse, with far more angst built up than your average teenager; he immaturely held grudges and would not take criticism. Robert Hooke however, was the Joker to Newton's Batman; brash, brazen and with an ego to boot, Hooke was a jock of academia if there ever was one [4].  However, unlike the relationship between the Joker and his masked vigilante, Hooke had very little interest in Newton, though the latter had quite the grudge against the former. Years before he had begun working on Principia Newton had sent a paper to the Royal Philosophical Society about his thoughts on color and light. He received a reply containing nothing but scathing criticism from none other than Robert Hooke, who at the time was a member of the Society [2]. 

After a time, Hooke forgot all about his words to Newton, then one fateful day he and two other members at the Royal Philosophical Society would start a conversation that would set up Newton for a permanent place in history. Robert Hooke, Edmond Halley and Christopher Wren sat discussing the inverse square force and how it may relate to universal gravitation, they all agreed that it would be incredible to prove that one induces the other and/or visa versa [1]. With his ego bursting at the seams, Hooke adamantly proclaimed that he would formalize a proof of the relation within two months. Amused and intrigued Wren proposed a gambit, a battle of wits between Hooke and Halley. Two months time they were give, to produce a proof of the square inverse force relation and the winner was to receive 40 shillings [2]. 

As Newton was the first to prove the relation, naturally Halley and Hooke failed. Hooke, embarrassed by his inability to create a proof levied excuse after excuse, claimed his occupation with other more pressing endeavors; Halley however, was truly intrigued. He was an accomplished mathematician himself and though blunt, Hooke was no slouch either. Halley wondered who could solve a problem that he and Hooke could not [2]. Here we return to the beginning of the story, Halley, bested by the problem, sought aid in the form of Isaac Newton. Halley had heard that Newton was an incredibly brilliant mathematician working at Cambridge and upon his arrival, Halley was not disappointed. He did however, leave empty handed, as Newton had not yet formally written up his proof [3].

When Halley finally received word from Newton, after many months of waiting, he got much more than he asked. Newton had sent Halley the groundwork for Principia, something Halley immediately recognized as an incredible breakthrough [3]. He fervently urged Newton to publish his work. Being the hermit that he was, Newton was hesitant to publish for fear of criticism, though he pressed on in the continuation of his work. This trend repeated countless times, Halley would plea Newton to publish, but Newton would refuse yet strangely keep on working [2]. As word of Newton's work began to spread within the circle of Halley's associates, Hooke was naturally one who caught wind. Perhaps jealous of the fact that he himself was unable to solve the problem, he began putting down Newton's work, inventing mistakes and eventually even claiming that the proof idea was his to begin with. Infuriated with his claims in addition to the age old grudge, Newton decided to give Hooke a well deserved back hand to the face, in the form of Principia, the completed publication of all his work on the laws of gravity [1].

It is also possible that Hooke may have had more to do with the proof of the inverse-square law than it may seem initially. It is acknowledged that Hooke sent a letter to Newton about the possible inverse square relation of gravity, though it is unclear whether Newton came up with the idea first. Furthermore, in his Principia, Newton notes that the idea of the inverse square relation was independently discovered by Hooke, Wren and Halley, but this was little comfort to Hooke. [5]

His rivalry with Hooke was not the greatest and most earthshaking conflict that Newton had been a part of. Indeed his infamous controversy against Leibniz would result in Continental mathematics speeding ahead of their once close rival Britain [7]. Ultimately, the conflict arose when Newton and Leibniz both independently discovered Calculus. At first they coexisted happily, as Newton was not one to seek credit, however eventually, accusations from third parties spurred the two men into a furious debate. Leibniz desired the credit for his powerful notation, notation that is still for the most part used today, while Newton simply wanted to defend his honor as an honest man. Naturally, Britain sided with Newton, not only because it was his homeland, but also because all of the studies of calculus in Britain were Newtonian, many British mathematicians could not even decipher work done using Leibniz superior notation [6]. Unfortunately for the Brits, Leibniz notation would prove to be so far ahead of Newton's, that Continental mathematics grew exponentially, leaving Britain in the dust [7].

        Fig. 2 [12]                                                          Fig. 3 [11]                                                             

The events involving Newton, Leibniz, Hooke, Halley and Wren dig up some interesting topics for discussion. Who should receive more credit(?), someone who happens upon and sets up the ground work for a new idea or someone who fully realises that same idea; additionally, can it really be considered a discovery if said discover-er refuses to release the information to the public. I personally believe that the initial spark, the one who starts a new idea and sets the foundation for it should receive far more praise than one who simply fleshes out the details. I find it analogous to the general proof method of popular unsolved equations in the mathematics community. Often times, a well known mathematician will provide a road map of sorts to the solution of a difficult problem; they give a list of simpler ideas that if proven, give rise to the one in question. This can be thought of as the first realization of the idea. From there the rest is in some sense grunt work. Once the plans have been laid out, hundreds of mathematicians can now tackle these simpler problems, this being akin to the fleshing out of a theory. Without the initial intuition, the fleshing out could never occur, making it far more necessary, once the idea is out there, eventually there will be those who will flesh it out. Furthermore, with regards to who should claim credit for the discovery or generation of a theory or idea, I believe the glory should go only to those who share the knowledge. Glory, fame, credit and praise should be given to those deserving and how can one who hides his knowledge from all be deserving? They have not given anything, therein by the property of reciprocation, they deserve nothing. The first individual to publicly disclose the information should be credited, for they are the ones who actually contribute to the general knowledge of the public. 

Of course, as with every opinion, there are the extreme cases where credit should go to one who hides the information out of necessity. A simple example of this is how Alan Turing cracked the Enigma machine during WWII, but could not release said information at the time without jeopardizing the war effort [8].

References:

1.  http://en.wikipedia.org/wiki/Isaac_Newton
2.  http://faculty.wcas.northwestern.edu/~infocom/Ideas/newton.html
3.  http://www.pbs.org/wgbh/nova/newton/principia.html
4.  http://www.physics.org/interact/physics-evolution/text-only/03.html
5.  http://en.wikipedia.org/wiki/Inverse-square_law
6.  http://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy
7.  Hyman, Anthony. Charles Babbage, Pioneer of the Computer. Princeton, NJ: Princeton UP, 1982. Print.
8.  http://en.wikipedia.org/wiki/Alan_Turing

Images:

     9.   http://hydrogen.physik.uni-wuppertal.de/hyperphysics/hyperphysics/hbase/forces/imgfor/isqg.gif
    10.  http://upload.wikimedia.org/wikipedia/commons/f/ff/Newton_-                                                                      _Principia_%281687%29,_title,_p._5,_color.jpg 
    11.  https://whatsnewwithnewton.files.wordpress.com/2008/10/dsc01537.jpg
    12. https://classconnection.s3.amazonaws.com/33333/flashcards/644875/jpg/ftc1-leibniz-a.jpg

Assignment 2

Nicholaus Copernicus was a polish mathematician and astronomer who was also a man of the church. Though he was not the first, Copernicus proposed an idea that would, eventually, revolutionize the entire field of astronomy as well as our understanding the solar system; he proposed a heliocentric model of the solar system. Dissatisfied with the geocentric model proposed by Pytolemy, Copernicus yearned for a model of the solar system that would please him, not with respect to accuracy, but with respect to simplicity and beauty. Copernicus ventured to create a system that he could intuitively feel was correct, a system where he could see the beauty of God's work.

Copernicus' model relies on several key ideas:

The planets revolve around the Sun, with Earth making a complete cycle in one year:

The most jarring and perhaps most important change that Copernicus made to the model of the solar system was making it heliocentric, that is he believed that it was Sun centered not Earth centered. As if only to add salt to the metaphorical wound, not only did Copernicus proclaim that the Earth was not special enough to be the center of the system, but that it was just like all the other planets and that they all revolved around the Sun.   

The Earth has a daily rotation around its tilted axis:

As opposed to the common belief at the time, that the starry sky revolved around the Earth to create night and day, Copernicus asserted that the actual cause was the rotation of the Earth around its axis with the stars as a fixed background. Furthermore he believed that the axis of the Earth was not parallel to a tangent of the Sun, but rather that it was tilted towards or away from the Sun and that the axis tilt would also make one full revolution per year.

The motions of the heavenly bodies are circular:

Copernicus believed that all the orbits of the heavenly bodies had to be circular. Circles are in some sense a perfect shape, circles have no beginning nor end and in this way they are eternal. As a man of God, Copernicus would not have himself believe that God would create anything less than such eternal perfection. Ironically, this belief would become one of the root causes for the inaccuracies in Copernicus' calculations and would make the acceptance of his ideas much more difficult. This is not to look down on Copernicus of course, in some sense it may have been necessary to accept this assumption; for it would have been unlikely for the Church to allow blasphemy in the form of elliptical orbits.

Ultimately, Copernicus lived in an age where proving his ideas would prove to be impossible. Not only were the tools and technology lacking, the philosophical implications of his work were so outrageous that he was heavily reluctant to share what work he had. The geocentric model of the solar system was so ingrained in the minds of the educated public that even suggesting otherwise was almost surely a one way ticket to ridicule.

    This of course, is not to say that there was no evidence to support Copernicus' claims. In fact, from the modern perspective, there is much more good in Copernicus' heliocentric model than the bad, however the physics at the time was not sophisticated enough to register relative correctness of his assertions.

    Copernicus' belief in the Sun centered system with an orbiting Earth had the very enticing benefit of easily explaining the retrograde motion of the planets, a mysterious phenomenon that was torturously worked into the Pytolemic model using complex and very specific epicycles. Retrograde motion is simply when another heavenly body, from the perspective of a viewer on Earth, reverses the direction of its orbit. In the heliocentric system if the planets orbit at different speeds, then whenever one passes another, the slower one would appear to experience retrograde motion from the perspective of the faster one. Furthermore, the tilt of Earth's axis while revolving around the Sun would explain the seasons, since the area of the Earth tilted towards the Sun experiences more sunlight during the day, it is warmer producing summer and visa versa for winter. 

    In the end, Copernicus decided that these benefits alone were not enough to warrant a publication of his ideas, and it was not until a student of his named Rheticus insistently urged him to release his work that it would finally be seen by the eyes of the public. Unfortunately, this most important piece of knowledge that Copernicus imparted upon the world [in the form of the book De Revolutionibus] would fly so far below the radar it was practically a resident of the Marianas Trench. This may in part be due to the fact that Copernicus' model did not provide significantly improved results in terms of accuracy over the Pytolemic model, as he stubbornly held onto the idea of epicycles. This led to many academics passing off Copernicus' system as a mathematical tool as opposed to a physical reality. It would not be until hundreds of years later that the heliocentric model would come to be accepted.

    Though Copernicus started the revolution towards heliocentrism by being the first individual to combine both the mathematics and the physics of such a system, the ideas he incorporated into his model were actually much older. The idea that the Earth was not stationary in the universe was first hatched by Philolaus of Magna Graecia, though he did not specifically believe in a heliocentric system. Before Philolaus the was  Aristarchus of Samos, the earliest recorded individual to propose a heliocentric system, back in the days of Aristotle who naturally and promptly stomped the idea back into the supposedly stationary Earth. Furthermore, Aristarchus had a surprisingly significant following so it is probable that at least a few of them worked on the heliocentric system; unfortunately, if such work ever existed, it has long been lost. Beyond the ancient astronomers, a number of Medieval astronomers also toyed with similar ideas as Copernicus including Nicholas of Cusa, who queried whether or not the Sun was the center of the universe. Another was Nicole Oresme, who was also a man of the Church. He believed that the Earth had an axial rotation to cause night and day prior to Copernicus. Even more generally, outside of Europe, there were quite a few astronomers in Indian and Islamic regions who worked on results involving the motion of the Earth and position of the planets. Given the surprisingly large number of individuals who worked on heliocentrism before Copernicus, it is almost miraculous that it took so long for the idea to catch on. Such is the power of the inertia of belief.

    In the end, what is likely the most important belief held by Copernicus was his belief in the Sun being the unmoving center of the Solar System. This fundamental idea was the main concept of Copernicus' system, and without a solid foundation he could not have put his entire model together. It was his precise fusion of mathematics and physics that really pushed the idea of heliocentrism, giving it the initial jolt of energy needed to eventually overtake geocentrism. The spinning of Earth around its axis, though interesting in itself, is not fundamentally a key part of heliocentrism, furthermore, the Earth orbiting the Sun like the other planets is not a hard leap to make, once you assume that the Sun is the center. Ultimately, it is the foundational idea that should be most important, the other two points are interesting and related but are not as vital. Fundamentally the importance is in the name of system itself, helio meaning Sun and centric meaning centered at.

Resources:
-http://en.wikipedia.org/wiki/Philolaus
-http://en.wikipedia.org/wiki/Georg_Joachim_Rheticus
-http://en.wikipedia.org/wiki/Copernican_heliocentrism
-http://starchild.gsfc.nasa.gov/docs/StarChild/whos_who_level2/copernicus.html
-http://en.wikipedia.org/wiki/Heliocentrism
-http://www.space.com/15684-nicolaus-copernicus.html
-http://astronomy.nju.edu.cn/~lixd/GA/AT4/AT402/HTML/AT40203.htm
-http://csep10.phys.utk.edu/astr161/lect/retrograde/copernican.html

Images:
-http://www.splung.com/cosmology/images/retro.gif
-http://www.icr.org/i/articles/imp/imp-261.gif
-http://www.mhhe.com/physsci/astronomy/fix/student/images/04f08.jpg
http://www.sussexvt.k12.de.us/science/The%20History%20of%20the%20World%201500-1899/Copernicus%20and%20the%20heliocentric%20solar%20system_files/image005.jpg