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Sir Isaac Newton sneaking a peek at Leibniz's derivative answers.

“ ” I do not know what I may appear to the world. To myself, I seem to have been only like a boy, playing on the seashore, musing myself by now and then finding a smoother pebble, a prettier shell than ordinary, while the great ocean of truth lay all undiscovered before me. —Isaac Newton[1]

Sir Isaac Newton (January 4th, 1642–March 31, 1727) was a demigod mathematician, physicist, mystic, alchemist and philosopher. Credited with revolutionary advances in classical mechanics, optics, and mathematics, he is rated as one of the most important and influential individuals in the history of science and mathematics.

Master of the Enlightenment [ edit ]

See the main article on this topic: Age of Enlightenment

Isaac Newton clearly understood the importance of experiments in scientific research, writing:

In experimental philosophy, propositions gathered from the phenomena by induction are taken as true, whether exactly or approximately, contrary hypotheses notwithstanding, until other phenomena appear through which they are either rendered more accurate or liable to exceptions.[2]

Indeed, he also gave a rather neat description of the scientific method:

The best and safest method of philosophizing seems to be, first to inquire diligently into the properties of things, and to establish those properties by experiences and then to proceed more slowly to hypotheses for the explanation of them. For hypotheses should be employed only in explaining the properties of things, but not assumed in determining them; unless so far as they may furnish experiments.[3]

Newton's cradle is a device that demonstrates the conservation of momentum and energy.

Like many others who studied natural philosophy, the old name for science before the modern disciplines developed, his interests were varied. His early mathematical investigations led to the generalized binomial theorem. While his predecessors and many of his contemporaries thought, following the footsteps of Aristotle, a remarkable philosopher but a poor physicist, that heavenly bodies obeyed a different set of rules from those here on Earth, Newton believed that the same set must be satisfied by both. In other words, the laws of physics must be universal. This motivated his studies in mechanics. He had quite a bit of a brainstorm, writing down over a hundred postulates and definitions before settling down with just three axioms, now called Newton's laws of motion. The first law states that every body maintains the same speed and direction unless acted upon by an external net force. If there is a non-zero net force acting on the body, than that net force is equal to rate of change of a quantity of motion defined to be its mass times its velocity, . In the special case that the body has constant mass, we have the familiar expression . It is crucial to realize that forces are vectors, hence the bold font, quantities with both magnitude and direction. For every force applied, there exists another equal in magnitude and opposite in direction, provided the system is isolated; this is his third law of motion, from which springs the law of (linear) momentum conservation.[4][5] There is, however, a bit of a problem. The mathematics available at the time was simply insufficient for further investigations in mechanics. Newton "simply" worked it out, calling it the "method of fluxions", known as calculus today.[6] Note, however, that the seeds of calculus have already been sowed by their predecessors, especially Archimedes of Syracuse. Newton and one of his contemporaries, Leibniz, are credited with the discovery of calculus in the sense that they recognize the fundamental importance of the concept of a limit and exploited the relationship between the geometric process of finding a tangent line to a curve at a point (differentiation) and the area bounded by a curve (integration), the Fundamental Theorem of Calculus, and developed this mathematical method in a systematic fashion. Unlike some of his contemporaries and successors who developed various techniques for integration, Newton tended to think of functions in terms of infinite series and integrated or differentiated them term-by-term.[note 1][7] Indeed, the use of the infinite series representation of a function automatically allows for an approximation scheme of arbitrary accuracy[note 2]

After some persuasion by his friend Edmond Halley, Newton published his Philosophiæ Naturalis Principia Mathematica, thought to be one of the most influential scientific treatises of all time.[8] It was in this book that he laid down the foundations for calculus, classical mechanics, and his theory of gravity. He demonstrated that an object's force of gravity weakens as the reciprocal of square of the distance from it (an inverse-square law) and derived Kepler's laws of orbital motion, which were previously obtained from extensive empirical data collected by Tycho Brahe's painstaking astronomical observations.[9] Strictly speaking, Newton's law of universal gravitation applies only to point masses, not extended bodies such as the Earth.[note 3] In addition, one wonders what happens when the distance shrinks to zero; it is clear that the force cannot be infinite. Fortunately, Newton showed that a spherically symmetric body can be treated as if all of its mass was concentrated at its center. This means that when the mutual force of gravity between the Earth and the Sun, say, is calculated, the distance in question is that between their centers. Furthermore, inside a spherically symmetric shell, the force vanishes.[2][note 4] Since a ball consists of a stack of concentric spheres, the force at its center is precisely zero. He gave a theory of tides using his newfound understanding of gravitational physics. In an ingenious calculation, Newton showed that the Earth is not exactly spherically symmetric, but had an equatorial bulge due to its rotation about an axis.[10] But since this oblateness is very small, the Earth remains a ball to a very good approximation. It is interesting to note that even though the first lemmas contain the core ideas behind calculus, his subsequent development of mechanics uses only Euclidean geometry.[9][2] At the time, calculus remained a controversial topic and Newton simply detested controversy. We do not know whether he obtained his results using calculus first then confirmed them using older mathematics or did so the other way around. Either way, this is a phenomenal achievement,[9] demonstrating that his geometric ingenuity was on the same level as that of Archimedes. He did, however, provide an account of his work on calculus in a separate publication.[6]

While Johannes Kepler was the first to suggest that a physical force rather than some occult influence is responsible for orbital motion[9], physical cosmology arguably began with Isaac Newton. Newton's law of universal gravitation enables the determination of the motion of heavenly bodies. But he went further than that. Newton suggested that since comets occasionally hit the Sun, perhaps the Earth was originated from the hot debris scattered into space from such a collision, then pulled together by means of gravitational attraction. This means the Earth started out extremely hot, and that this information could be used to determine how old Earth is. He himself estimated that it must be older than 50,000 years.[9] In light of the fact that he is credited with Newton's law of cooling (or heating), which states that the rate of change of temperature of a body with respect to time is directly proportional to the difference between the instantaneous temperature of that body and the ambient temperature, assuming the latter is kept constant,[7] this does not come at a surprise.

Newton's rings generated using 650nm red laser light.

Newton held a keen interest in light. He produced a well-received paper on the theory of colors and built a new and compact telescope which eliminated the unwanted color patterns on the fringe. He gave a description of Newton's rings. His famous prism experiment revealed that white light could be decomposed into various colors, which, if refocused, produce white light once again. He performed a variation on the experiment to test the nature of light. The current idea at the time was that the colors were made by white light being bent, so he contrived a way to pass a red beam of light onto another prism, to see if it would be bent into another spectrum of new colors. The result was light of uniform and undifferentiated red, proving that white light was composed of all the colors. Once again, this flew in the face of common wisdom at the time, when most people thought that white light was a symbol of purity, or even of divine power.[1] In his monograph Opticks, Newton elaborated his corpuscular theory of light and explained pretty much all that was already known about light. Some of his contemporaries, such as Robert Hooke and Christian Huygens, preferred the wave theory. However, at this time no one could really tell if light behaved as a stream of particles or as traveling waves.[9] During his investigations in optics and vision, Newton performed a number of alarming experiments on himself, risking the loss of his vision. He stared directly at the Sun for some time and poked the back of his eye with a bodkin (a fat, blunt needle) to study the resulting colored patterns he saw.[1]

A true mathematical genius, in 1669 Newton was elected as the second Lucasian Professor of Mathematics at Cambridge, a post that would later be held by Paul Dirac and Stephen Hawking. Indeed, he is often considered to be one of the greatest mathematicians in history, alongside Archimedes and Karl Friedrich Gauss.[8] In the early 1700s, Newton, Leibniz, Johann Bernoulli, Jacob Bernoulli, and Guillaume de L'Hôpital independently solved the brachistochrone problem, first stated by Johann Bernoulli. It asks for the curve between two points on a plane on which a particle will slide frictionlessly from one point to the other in the least possible time, assuming gravity is uniform. The solution is not a straight line but a cycloid. Newton received a letter detailing the problem on his way home from the Royal Mint, and sent the solution anonymously back to Bernoulli the next day. However, when Bernoulli read the letter, he immediately recognized its author, reportedly exclaiming, "I know the Lion from his paw!" The brachistochrone problem is a classic in the calculus of variations, which plays an inordinately important role in the further developments of (theoretical) physics. Newton did some work on this subject, but again, did not publish.[10]

Misadventures in alchemy and theology [ edit ]

“ ” Newton was not the first in the Age of Reason. He was the last of the magicians. —John Maynard Keynes, as quoted in [1]

As was common at the time, Newton also devoted much time to biblical scholarship, alchemy and other theological questions. Newton was a reclusive man and no accounts of him being in any kind of relationship exists, leading to suggestions that he was asexual, or possibly gay.[11][12] There is, however, no credible evidence supporting either claim.[9] He was also Arian in his religious beliefs. He stood up to James II's attempt to appoint a Roman Catholic monk named Alban Francis to an academic post at Cambridge and the appointment was stopped. In his popular book A Brief History of Time, Stephen Hawking has criticized Newton for this as an example of religious intolerance. However it is to be borne in mind that the Roman Catholic Church had burnt Giordano Bruno for heresy, mostly on account of his Arianism (i.e., the belief that Jesus Christ was of similar not same substance to and created by God.) and that this appointment was the thin end of the wedge, leading to similar persecution that had stifled scientific inquiry in Catholic countries after Galileo. In his own lifetime, Newton wrote more on religion than he did on natural science. Worse still, Newton at least once made an argument involving divine powers in his scientific work. It appeared to Newton that the Solar System is unstable, and he thought that every 200 years or so, God would have to intervene by putting the planets back in their place. Fortunately, the 'God hypothesis' turned out to be unnecessary. Pierre-Simon de Laplace demonstrated that to first order, the Solar System is indeed stable; the mean motion of all bodies remain invariant with time.[13]

Unfortunately, for all his marvelous achievements Newton's life was mixed in with some frightening bouts of near insanity. John Maynard Keynes described Newton as not the first of the rationalists, but rather the "last of the magicians" due to his obsession with alchemy. "In modern vulgar terms," Keynes said, "Newton was profoundly neurotic."[1] Indeed, discovering the nature of optics was probably a side effect of trying to turn lead into gold. Later in his life, he suffered a nervous breakdown, which might have been caused by mercury poisoning, overwork, and the end of his friendship with the Swiss mathematician Nicholas Fatio de Duillier.[13][9]

Master of the Royal Mint [ edit ]

Newton's research career was interrupted by a spell in politics, in which the most notable period was his appointment as Master of the Royal Mint. This lucrative post turned him into a wealthy man. (It was also during this period that Newton received and solved the brachistochrone problem.) His obsession with detail meant that under him, the institution operated more effectively than it had ever before. Newton conducted a careful study of the history and methods of counterfeiting. He also operated a network of paid informants and sent money to friendly witnesses who were short on funds so that they could dress nicely for court appearances. In the end, Newton saw to it that multiple counterfeiters were hanged.[14]

Disputes and criticisms [ edit ]

“ ” If I have seen farther, it is by standing on the shoulders of Giants.[15] If I have seen farther, it is by standing on the shoulders of Giants.

This famous quote is typically interpreted as a gesture of modesty by the great scientist in a letter to his colleague Robert Hooke. There is, however, an alternative approach that better matches Newton's personality, which was famously unpleasant toward those who angered him. It did not help that some of his colleagues and potential collaborators got into disputes with him. Now, the standards of grammar (and spelling) back then was rather strange. But Newton's decision to capitalize the word "Giants" was probably a dig at Hooke, who was short and had a twisted back. Robert Hooke, known for his work in microscopy, optics and elasticity, was responsible for getting the Royal Society up and running. In a discussion involving Hooke, Edmond Halley and Christopher Wren, the three concluded that an inverse square law of gravity would give rise to elliptical orbits (Kepler's first law), but none of them could give a mathematical demonstration. Newton did. Not only that, he systematically worked out the rest of Kepler's laws. When Hooke died, Newton succeeded him and transformed it onto a respectable scientific institution. However, under Newton, a man obsessed with details, the only portrait of Hooke was lost.[9]

It is quite unfortunate that Newton got into a very bitter and prolonged dispute with Gottfried Wilhelm Leibniz over who discovered calculus first. We now know that they both did so independently. Newton got there first but published later while Leibniz got there later but published first. We also know that the two corresponded respectfully before the dispute. Newton explained how he came up with the generalized binomial theorem to Leibniz back in the 1660s.[7] Leibniz's notations for the derivative and integral remain in use today even though they seem to be somewhat illogical.[13] Newton's dot notation for the derivative is mainly used in physics to represent how something changes with respect to time. In some cases, it is advantageous to use both of their notations.

Since the publication of Newton's Opticks, almost all physicists and mathematicians adhered to his corpuscular theory of light, which is respectable in its own right. However, the minority who preferred the wave theory were needlessly marginalized and criticized, all because of Newton's mighty name.[9] Moreover, Newton gave a tentative derivation of Boyle's law in the Principia, in which he assumed that a gas consists of static atoms of negligible sizes compared that its volume that repelled each other via a force that weakened as the inverse of the distances between them. Tentative is the key word here. Newton merely suggested a hypothesis that might theoretically explain Boyle's law. Again, physicists for many years ignored the alternative approach pioneered by Daniel Bernoulli, which also yielded Boyle's law. Like Newton, Bernoulli assumed that gases consisted atoms of tiny dimensions. Unlike Newton, he conceived them as moving about randomly, colliding elastically with each other and with the walls of the container, which gives rise to pressure. Only much later, when physicists realized that heat is a kind of motion did they become interested in Bernoulli's approach. This became the basis for the kinetic molecular theory of gases.[16][note 5] This is not really a criticism against Newton himself, but rather of the physicists who took his words uncritically. In the case of the corpuscular theory of light, it was rather fitting that a sophisticated theory of diffraction, formulated by Augustin Fresnel based on the wave theory, was experimentally verified by François Arago, who tested the predictions extracted by Simeon-Denis Poisson, when both Arago and Poisson were declared supporters of the corpuscular theory.[9][note 6] However, at the start of the 1800s, physicists recognized that Newton's law of cooling (or heating) faltered when more precise measurements were made. This sparked the experimental studies of the behavior of gases. These proved crucial in the development of thermodynamics. Along the way, physicists stumbled upon the rule of Dulong and Petit, which favors the atomic hypothesis.[16] Questioning authority, that is, to really stand upon the shoulders of giants rather than in their shadows, turned out to be a great idea.

For seventeen years, Newton lectured to an essentially empty hall at Cambridge. He eventually gave up teaching.[13] Given the fact that his reputation was enormous even in his own lifetime, one can scarcely avoid the conclusion that Newton was an abstruse professor.

Myths [ edit ]

A popular myth associated with Newton is that he was born on the same day Galileo died. As a matter of fact, Newton was born on January 4 1642, in the Gregorian calendar, first introduced in various countries in the Continent and is the one we use today, and on December 25, 1641, in the Julian calendar, used in England at the time while Galileo died on January 8 1642 in the Gregorian calendar and December 31, 1641 in the Julian calendar.[note 7] However, it is a genuine coincidence that Newton was born exactly a century after Copernicus published his heliocentric theory.[9]

Another well-publicized myth is that Newton came up with the inverse-square law of gravity after an apple fell and hit him on the head. Even though Newton himself mentioned this in his magnum opus, it is in all probability a just a motivating passage. The actual process of discovery took much longer than this. The idea did not come to him out of the blue but gradually. Indeed, it took him quite a bit of work before he was able to give the mathematical proof that eluded Hooke, Wren, and Halley.[9]

Rumor has it that Newton invented the pet door, but this does not seem to be the case.[17] However, he was a talented mechanical inventor, like Hooke. When he was a child he constructed flying machines that carried candles, thereby creating some early episodes of UFO scares.[8][9] He also built a clock that ran using water power and a miniature mill that grinds wheat operated by a mouse turning the wheel.[8]

Famous quotes [ edit ]

“ ” I do not define time, space, place, and motion, as being well known to all.

“ ” We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.

“ ” I feign no hypotheses.

See also [ edit ]

Notes [ edit ]

↑ If you are worried that the linearity of differentiation and integration does not apply to convergent infinite series, Cauchy later proved rigorously that this is in fact acceptable. ↑ This is most often done using Taylor's theorem (with the Lagrange remainder). ↑ In order to derive the gravitational field an extended body, one must add up the fields of all the individual points making up that body. This is where integral calculus comes in. ↑ Newton's shell theorem also holds in electrostatics. ↑ . By taking a detour, first obtaining the Boltzmann factor, of fundamental importance in statistical mechanics, from purely probabilistic considerations, then proving the equipartition theorem before returning to Bernoulli's derivation, one obtains the famous ideal gas law ↑ This is where the Poisson spot came from. ↑ If you want to, you can use this information to mark the beginning and end of your Winter holidays.

References [ edit ]



