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Thursday, 21 June 2012

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Branches of Mathematics

Pure mathematics

Applied mathematics

Hitoshi Matsumoto
Born	(1963-09-08) September 8, 1963 (age 48) Amagasaki, Hyōgo, Japan
Medium	Owarai Television
Nationality	Japanese
Years active	1983 – Present
Genres	Owarai

Persondata
Name	Matsumoto, Hitoshi
Alternative names	Matchan
Short description	Comedian
Date of birth	1963-09-08
Place of birth	Amagasaki, Hyōgo, Japan
Date of death
Place of death

Name	Sir Isaac Newton
Alt	Head and shoulders portrait of man in black with shoulder-length grey hair, a large sharp nose, and an abstracted gaze
Birth date	December 25, 1642[NS: 4 January 1643]
Birth place	Woolsthorpe-by-Colsterworth Lincolnshire, England
Residence	England
Nationality	English
Ethnicity	Caucasian
Religion	Christian
Death date	March 20, 1727[NS: 31 March 1727]
Death place	Kensington, Middlesex, England
Fields	Physics, mathematics, astronomy, natural philosophy, alchemy, Christian theology
Workplaces	University of CambridgeRoyal Society Royal Mint
Alma mater	Trinity College, Cambridge
Doctoral advisor
Academic advisors	Isaac Barrow Benjamin Pulleyn
Doctoral students
Notable students	Roger Cotes William Whiston
Known for	Newtonian mechanics Universal gravitation Infinitesimal calculus Optics Binomial series Newton's method Philosophiæ Naturalis Principia Mathematica
Influences	Henry More Polish Brethren
Influenced	Nicolas Fatio de Duillier John Keill
Religion	Arianism; for details see article
Signature	Isaac Newton signature.svg
Signature alt	Is. Newton
Footnotes	His mother was Hannah Ayscough. His half-niece was Catherine Barton. }}

Isaac Newton was born on what is retroactively considered 4 January 1643 [OS: 25 December 1642] at Woolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. At the time of Newton's birth, England had not adopted the Gregorian calendar and therefore his date of birth was recorded as Christmas Day, 25 December 1642. Newton was born three months after the death of his father, a prosperous farmer also named Isaac Newton. Born prematurely, he was a small child; his mother Hannah Ayscough reportedly said that he could have fit inside a quart mug (≈ 1.1 litres). When Newton was three, his mother remarried and went to live with her new husband, the Reverend Barnabus Smith, leaving her son in the care of his maternal grandmother, Margery Ayscough. The young Isaac disliked his stepfather and maintained some enmity towards his mother for marrying him, as revealed by this entry in a list of sins committed up to the age of 19: "Threatening my father and mother Smith to burn them and the house over them." Although it was claimed that he was once engaged, Newton never married.

From the age of about twelve until he was seventeen, Newton was educated at The King's School, Grantham (where his alleged signature can still be seen upon a library window sill). He was removed from school, and by October 1659, he was to be found at Woolsthorpe-by-Colsterworth, where his mother, widowed by now for a second time, attempted to make a farmer of him. He hated farming. Henry Stokes, master at the King's School, persuaded his mother to send him back to school so that he might complete his education. Motivated partly by a desire for revenge against a schoolyard bully, he became the top-ranked student. The Cambridge psychologist Simon Baron-Cohen considers it "fairly certain" that Newton had Asperger syndrome.

In June 1661, he was admitted to Trinity College, Cambridge as a sizar – a sort of work-study role. At that time, the college's teachings were based on those of Aristotle, whom Newton supplemented with modern philosophers, such as Descartes, and astronomers such as Copernicus, Galileo, and Kepler. In 1665, he discovered the generalised binomial theorem and began to develop a mathematical theory that later became infinitesimal calculus. Soon after Newton had obtained his degree in August 1665, the university temporarily closed as a precaution against the Great Plague. Although he had been undistinguished as a Cambridge student, Newton's private studies at his home in Woolsthorpe over the subsequent two years saw the development of his theories on calculus, optics and the law of gravitation. In 1667, he returned to Cambridge as a fellow of Trinity. Fellows were required to become ordained priests, something Newton desired to avoid due to his unorthodox views. Luckily for Newton, there was no specific deadline for ordination and it could be postponed indefinitely. The problem became more severe later when Newton was elected for the prestigious Lucasian Chair. For such a significant appointment, ordaining normally could not be dodged. Nevertheless, Newton managed to avoid it by means of a special permission from Charles II (see "Middle years" section below).

Middle years [link]

Mathematics [link]

Newton's work has been said "to distinctly advance every branch of mathematics then studied". His work on the subject usually referred to as fluxions or calculus, seen in a manuscript of October 1666, is now published among Newton's mathematical papers. The author of the manuscript ''De analysi per aequationes numero terminorum infinitas'', sent by Isaac Barrow to John Collins in June 1669, was identified by Barrow in a letter sent to Collins in August of that year as: Newton later became involved in a dispute with Leibniz over priority in the development of infinitesimal calculus. Most modern historians believe that Newton and Leibniz developed infinitesimal calculus independently, although with very different notations. Occasionally it has been suggested that Newton published almost nothing about it until 1693, and did not give a full account until 1704, while Leibniz began publishing a full account of his methods in 1684. (Leibniz's notation and "differential Method", nowadays recognised as much more convenient notations, were adopted by continental European mathematicians, and after 1820 or so, also by British mathematicians.) Such a suggestion, however, fails to notice the content of calculus which critics of Newton's time and modern times have pointed out in Book 1 of Newton's ''Principia'' itself (published 1687) and in its forerunner manuscripts, such as ''De motu corporum in gyrum'' ("On the motion of bodies in orbit"), of 1684. The ''Principia'' is not written in the language of calculus either as we know it or as Newton's (later) 'dot' notation would write it. But his work extensively uses an infinitesimal calculus in geometric form, based on limiting values of the ratios of vanishing small quantities: in the ''Principia'' itself Newton gave demonstration of this under the name of 'the method of first and last ratios' and explained why he put his expositions in this form, remarking also that 'hereby the same thing is performed as by the method of indivisibles'.

Because of this, the ''Principia'' has been called "a book dense with the theory and application of the infinitesimal calculus" in modern times and "lequel est presque tout de ce calcul" ('nearly all of it is of this calculus') in Newton's time. His use of methods involving "one or more orders of the infinitesimally small" is present in his ''De motu corporum in gyrum'' of 1684 and in his papers on motion "during the two decades preceding 1684".

Newton had been reluctant to publish his calculus because he feared controversy and criticism. He was close to the Swiss mathematician Nicolas Fatio de Duillier. In 1691, Duillier started to write a new version of Newton's ''Principia'', and corresponded with Leibniz. In 1693 the relationship between Duillier and Newton deteriorated, and the book was never completed.

Starting in 1699, other members of the Royal Society (of which Newton was a member) accused Leibniz of plagiarism, and the dispute broke out in full force in 1711. The Royal Society proclaimed in a study that it was Newton who was the true discoverer and labelled Leibniz a fraud. This study was cast into doubt when it was later found that Newton himself wrote the study's concluding remarks on Leibniz. Thus began the bitter controversy which marred the lives of both Newton and Leibniz until the latter's death in 1716.

Newton is generally credited with the generalised binomial theorem, valid for any exponent. He discovered Newton's identities, Newton's method, classified cubic plane curves (polynomials of degree three in two variables), made substantial contributions to the theory of finite differences, and was the first to use fractional indices and to employ coordinate geometry to derive solutions to Diophantine equations. He approximated partial sums of the harmonic series by logarithms (a precursor to Euler's summation formula), and was the first to use power series with confidence and to revert power series.

He was appointed Lucasian Professor of Mathematics in 1669 on Barrow's recommendation. In that day, any fellow of Cambridge or Oxford was required to become an ordained Anglican priest. However, the terms of the Lucasian professorship required that the holder ''not'' be active in the church (presumably so as to have more time for science). Newton argued that this should exempt him from the ordination requirement, and Charles II, whose permission was needed, accepted this argument. Thus a conflict between Newton's religious views and Anglican orthodoxy was averted.

Optics [link]

From 1670 to 1672, Newton lectured on optics. During this period he investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multicoloured spectrum into white light. Modern scholarship has revealed that Newton's analysis and resynthesis of white light owes a debt to corpuscular alchemy. .

He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. Newton noted that regardless of whether it was reflected or scattered or transmitted, it stayed the same colour. Thus, he observed that colour is the result of objects interacting with already-coloured light rather than objects generating the colour themselves. This is known as Newton's theory of colour.

From this work, he concluded that the lens of any refracting telescope would suffer from the dispersion of light into colours (chromatic aberration). As a proof of the concept, he constructed a telescope using a mirror as the objective to bypass that problem. Their interest encouraged him to publish his notes ''On Colour'', which he later expanded into his ''Opticks''. When Robert Hooke criticised some of Newton's ideas, Newton was so offended that he withdrew from public debate. Newton and Hooke had brief exchanges in 1679–80, when Hooke, appointed to manage the Royal Society's correspondence, opened up a correspondence intended to elicit contributions from Newton to Royal Society transactions, which had the effect of stimulating Newton to work out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see Newton's law of universal gravitation – History and ''De motu corporum in gyrum''). But the two men remained generally on poor terms until Hooke's death.

Newton argued that light is composed of particles or corpuscles, which were refracted by accelerating into a denser medium. He verged on soundlike waves to explain the repeated pattern of reflection and transmission by thin films (Opticks Bk.II, Props. 12), but still retained his theory of ‘fits’ that disposed corpuscles to be reflected or transmitted (Props.13). Later physicists instead favoured a purely wavelike explanation of light to account for the interference patterns, and the general phenomenon of diffraction. Today's quantum mechanics, photons and the idea of wave–particle duality bear only a minor resemblance to Newton's understanding of light.

In his ''Hypothesis of Light'' of 1675, Newton posited the existence of the ether to transmit forces between particles. The contact with the theosophist Henry More, revived his interest in alchemy. He replaced the ether with occult forces based on Hermetic ideas of attraction and repulsion between particles. John Maynard Keynes, who acquired many of Newton's writings on alchemy, stated that "Newton was not the first of the age of reason: He was the last of the magicians." Newton's interest in alchemy cannot be isolated from his contributions to science. This was at a time when there was no clear distinction between alchemy and science. Had he not relied on the occult idea of action at a distance, across a vacuum, he might not have developed his theory of gravity. (See also Isaac Newton's occult studies.)

In 1704, Newton published ''Opticks'', in which he expounded his corpuscular theory of light. He considered light to be made up of extremely subtle corpuscles, that ordinary matter was made of grosser corpuscles and speculated that through a kind of alchemical transmutation "Are not gross Bodies and Light convertible into one another, ...and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition?" After the exchanges with Hooke, Newton worked out a proof that the elliptical form of planetary orbits would result from a centripetal force inversely proportional to the square of the radius vector (see Newton's law of universal gravitation – History and De motu corporum in gyrum). Newton communicated his results to Edmond Halley and to the Royal Society in ''De motu corporum in gyrum'', a tract written on about 9 sheets which was copied into the Royal Society's Register Book in December 1684. This tract contained the nucleus that Newton developed and expanded to form the ''Principia''.

The ''Principia'' was published on 5 July 1687 with encouragement and financial help from Edmond Halley. In this work, Newton stated the three universal laws of motion that enabled many of the advances of the Industrial Revolution which soon followed and were not to be improved upon for more than 200 years, and are still the underpinnings of the non-relativistic technologies of the modern world. He used the Latin word ''gravitas'' (weight) for the effect that would become known as gravity, and defined the law of universal gravitation.

In the same work, Newton presented a calculus-like method of geometrical analysis by 'first and last ratios', gave the first analytical determination (based on Boyle's law) of the speed of sound in air, inferred the oblateness of the spheroidal figure of the Earth, accounted for the precession of the equinoxes as a result of the Moon's gravitational attraction on the Earth's oblateness, initiated the gravitational study of the irregularities in the motion of the moon, provided a theory for the determination of the orbits of comets, and much more.

Newton made clear his heliocentric view of the solar system – developed in a somewhat modern way, because already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the solar system. For Newton, it was not precisely the centre of the Sun or any other body that could be considered at rest, but rather "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and this centre of gravity "either is at rest or moves uniformly forward in a right line" (Newton adopted the "at rest" alternative in view of common consent that the centre, wherever it was, was at rest).

Newton's postulate of an invisible force able to act over vast distances led to him being criticised for introducing "occult agencies" into science. Later, in the second edition of the ''Principia'' (1713), Newton firmly rejected such criticisms in a concluding General Scholium, writing that it was enough that the phenomena implied a gravitational attraction, as they did; but they did not so far indicate its cause, and it was both unnecessary and improper to frame hypotheses of things that were not implied by the phenomena. (Here Newton used what became his famous expression ''Hypotheses non fingo'').

With the ''Principia'', Newton became internationally recognised. He acquired a circle of admirers, including the Swiss-born mathematician Nicolas Fatio de Duillier, with whom he formed an intense relationship. This abruptly ended in 1693, and at the same time Newton suffered a nervous breakdown.

Later life [link]

In the 1690s, Newton wrote a number of religious tracts dealing with the literal interpretation of the Bible. Henry More's belief in the Universe and rejection of Cartesian dualism may have influenced Newton's religious ideas. A manuscript he sent to John Locke in which he disputed the existence of the Trinity was never published. Later works''The Chronology of Ancient Kingdoms Amended'' (1728) and ''Observations Upon the Prophecies of Daniel and the Apocalypse of St. John'' (1733)were published after his death. He also devoted a great deal of time to alchemy (see above).

Newton was also a member of the Parliament of England from 1689 to 1690 and in 1701, but according to some accounts his only comments were to complain about a cold draught in the chamber and request that the window be closed.

Newton moved to London to take up the post of warden of the Royal Mint in 1696, a position that he had obtained through the patronage of Charles Montagu, 1st Earl of Halifax, then Chancellor of the Exchequer. He took charge of England's great recoining, somewhat treading on the toes of Lord Lucas, Governor of the Tower (and securing the job of deputy comptroller of the temporary Chester branch for Edmond Halley). Newton became perhaps the best-known Master of the Mint upon the death of Thomas Neale in 1699, a position Newton held for the last 30 years of his life. These appointments were intended as sinecures, but Newton took them seriously, retiring from his Cambridge duties in 1701, and exercising his power to reform the currency and punish clippers and counterfeiters. As Master of the Mint in 1717 in the "Law of Queen Anne" Newton moved the Pound Sterling ''de facto'' from the silver standard to the gold standard by setting the bimetallic relationship between gold coins and the silver penny in favour of gold. This caused silver sterling coin to be melted and shipped out of Britain. Newton was made President of the Royal Society in 1703 and an associate of the French Académie des Sciences. In his position at the Royal Society, Newton made an enemy of John Flamsteed, the Astronomer Royal, by prematurely publishing Flamsteed's ''Historia Coelestis Britannica'', which Newton had used in his studies.

In April 1705, Queen Anne knighted Newton during a royal visit to Trinity College, Cambridge. The knighthood is likely to have been motivated by political considerations connected with the Parliamentary election in May 1705, rather than any recognition of Newton's scientific work or services as Master of the Mint. Newton was the second scientist to be knighted, after Sir Francis Bacon.

Towards the end of his life, Newton took up residence at Cranbury Park, near Winchester with his niece and her husband, until his death in 1727. His half-niece, Catherine Barton Conduitt, served as his hostess in social affairs at his house on Jermyn Street in London; he was her "very loving Uncle," according to his letter to her when she was recovering from smallpox.

Newton died in his sleep in London on 31 March 1727 [OS: 20 March 1726], and was buried in Westminster Abbey. Newton, a bachelor, had divested much of his estate to relatives during his last years, and died intestate. After his death, Newton's hair was examined and found to contain mercury, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life. English poet Alexander Pope was moved by Newton's accomplishments to write the famous epitaph:

Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light.

Newton himself had been rather more modest of his own achievements, famously writing in a letter to Robert Hooke in February 1676:

If I have seen further it is by standing on the shoulders of giants.

Two writers think that the above quote, written at a time when Newton and Hooke were in dispute over optical discoveries, was an oblique attack on Hooke (said to have been short and hunchbacked), rather thanor in addition toa statement of modesty. On the other hand, the widely known proverb about standing on the shoulders of giants published among others by 17th-century poet George Herbert (a former orator of the University of Cambridge and fellow of Trinity College) in his ''Jacula Prudentum'' (1651), had as its main point that "a dwarf on a giant's shoulders sees farther of the two", and so its effect as an analogy would place Newton himself rather than Hooke as the 'dwarf'.

In a later memoir, Newton wrote:

I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.

Albert Einstein kept a picture of Newton on his study wall alongside ones of Michael Faraday and James Clerk Maxwell. Newton remains influential to today's scientists, as demonstrated by a 2005 survey of members of Britain's Royal Society (formerly headed by Newton) asking who had the greater effect on the history of science, Newton or Einstein. Royal Society scientists deemed Newton to have made the greater overall contribution. In 1999, an opinion poll of 100 of today's leading physicists voted Einstein the "greatest physicist ever;" with Newton the runner-up, while a parallel survey of rank-and-file physicists by the site PhysicsWeb gave the top spot to Newton.

Commemorations [link]

Newton's monument (1731) can be seen in Westminster Abbey, at the north of the entrance to the choir against the choir screen, near his tomb. It was executed by the sculptor Michael Rysbrack (1694–1770) in white and grey marble with design by the architect William Kent. The monument features a figure of Newton reclining on top of a sarcophagus, his right elbow resting on several of his great books and his left hand pointing to a scroll with a mathematical design. Above him is a pyramid and a celestial globe showing the signs of the Zodiac and the path of the comet of 1680. A relief panel depicts putti using instruments such as a telescope and prism. The Latin inscription on the base translates as:

Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colours thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25 December 1642, and died on 20 March 1726/7. — Translation from G.L. Smyth, ''The Monuments and Genii of St. Paul's Cathedral, and of Westminster Abbey'' (1826), ii, 703–4.

From 1978 until 1988, an image of Newton designed by Harry Ecclestone appeared on Series D £1 banknotes issued by the Bank of England (the last £1 notes to be issued by the Bank of England). Newton was shown on the reverse of the notes holding a book and accompanied by a telescope, a prism and a map of the Solar System.

A statue of Isaac Newton, looking at an apple at his feet, can be seen at the Oxford University Museum of Natural History.

In popular culture [link]

Religious views [link]

According to most scholars, Newton was a monotheist who believed in biblical prophecies but was Antitrinitarian. 'In Newton's eyes, worshipping Christ as God was idolatry, to him the fundamental sin'. Historian Stephen D. Snobelen says of Newton, "Isaac Newton was a heretic. But ... he never made a public declaration of his private faith—which the orthodox would have deemed extremely radical. He hid his faith so well that scholars are still unravelling his personal beliefs." Snobelen concludes that Newton was at least a Socinian sympathiser (he owned and had thoroughly read at least eight Socinian books), possibly an Arian and almost certainly an anti-trinitarian. In an age notable for its religious intolerance, there are few public expressions of Newton's radical views, most notably his refusal to receive holy orders and his refusal, on his death bed, to receive the sacrament when it was offered to him.

In a view disputed by Snobelen, T.C. Pfizenmaier argues that Newton held the Arian view of the Trinity rather than the Western one held by Roman Catholics, Anglicans and most Protestants. He also tried unsuccessfully to find hidden messages within the Bible.

Newton wrote more on religion than he did on natural science. He believed in a rationally immanent world, but he rejected the hylozoism implicit in Leibniz and Baruch Spinoza. The ordered and dynamically informed Universe could be understood, and must be understood, by an active reason. In his correspondence, Newton claimed that in writing the ''Principia'' "I had an eye upon such Principles as might work with considering men for the belief of a Deity". He saw evidence of design in the system of the world: "Such a wonderful uniformity in the planetary system must be allowed the effect of choice". But Newton insisted that divine intervention would eventually be required to reform the system, due to the slow growth of instabilities. For this, Leibniz lampooned him: "God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion." Newton's position was vigorously defended by his follower Samuel Clarke in a famous correspondence. A century later, Pierre-Simon Laplace's work "Celestial Mechanics" had a natural explanation for why the planet orbits don't require periodic divine intervention.

Effect on religious thought [link]

Newton and Robert Boyle's mechanical philosophy was promoted by rationalist pamphleteers as a viable alternative to the pantheists and enthusiasts, and was accepted hesitantly by orthodox preachers as well as dissident preachers like the latitudinarians. His spokesman, Clarke, rejected Leibniz' theodicy which cleared God from the responsibility for ''l'origine du mal'' by making God removed from participation in his creation, since as Clarke pointed out, such a deity would be a king in name only, and but one step away from atheism. But the unforeseen theological consequence of the success of Newton's system over the next century was to reinforce the deist position advocated by Leibniz. The understanding of the world was now brought down to the level of simple human reason, and humans, as Odo Marquard argued, became responsible for the correction and elimination of evil.

End of the world [link]

In a manuscript he wrote in 1704 in which he describes his attempts to extract scientific information from the Bible, he estimated that the world would end no earlier than 2060. In predicting this he said, "This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail."

It was Newton's conception of the Universe based upon Natural and rationally understandable laws that became one of the seeds for Enlightenment ideology. Locke and Voltaire applied concepts of Natural Law to political systems advocating intrinsic rights; the physiocrats and Adam Smith applied Natural conceptions of psychology and self-interest to economic systems; and sociologists criticised the current social order for trying to fit history into Natural models of progress. Monboddo and Samuel Clarke resisted elements of Newton's work, but eventually rationalised it to conform with their strong religious views of nature.

Counterfeiters [link]

As warden of the Royal Mint, Newton estimated that 20 percent of the coins taken in during The Great Recoinage of 1696 were counterfeit. Counterfeiting was high treason, punishable by the felon's being hanged, drawn and quartered. Despite this, convicting the most flagrant criminals could be extremely difficult. However, Newton proved to be equal to the task. Disguised as a habitué of bars and taverns, he gathered much of that evidence himself. For all the barriers placed to prosecution, and separating the branches of government, English law still had ancient and formidable customs of authority. Newton had himself made a justice of the peace in all the home counties - there is a draft of a letter regarding this matter stuck into Newton's personal first edition of his Philosophiæ Naturalis Principia Mathematica which he must have been amending at the time. Then he conducted more than 100 cross-examinations of witnesses, informers, and suspects between June 1698 and Christmas 1699. Newton successfully prosecuted 28 coiners.

One of Newton's cases as the King's attorney was against William Chaloner. Newton put Chaloner on trial for counterfeiting and had him sent to Newgate Prison in September 1697. But Chaloner had friends in high places, who helped him secure an acquittal and his release.

Laws of motion [link]

The famous three laws of motion (stated in modernised form): ''Newton's First Law'' (also known as the Law of Inertia) states that an object at rest tends to stay at rest and that an object in uniform motion tends to stay in uniform motion unless acted upon by a net external force. The meaning of this law is the existence of reference frames (called inertial frames) where objects not acted upon by forces move in uniform motion (in particular, they may be at rest).

''Newton's Second Law'' states that an applied force, $\vec{F}$ , on an object equals the rate of change of its momentum, $\vec{p}$ , with time. Mathematically, this is expressed as : $\vec F = \frac{\mathrm{d}\vec p}{\mathrm{\mathrm{d}}t} \, = \, \frac{\mathrm{d}}{\mathrm{d}t} (m \vec v) \, = \, \vec v \, \frac{\mathrm{d}m}{\mathrm{d}t} + m \, \frac{\mathrm{d}\vec v}{\mathrm{d}t} \,.$

If applied to an object with constant mass (d''m''/d''t'' = 0), the first term vanishes, and by substitution using the definition of acceleration, the equation can be written in the iconic form

: $\vec F = m \, \vec a \ .$

The first and second laws represent a break with the physics of Aristotle, in which it was believed that a force was necessary in order to maintain motion. They state that a force is only needed in order to ''change'' an object's state of motion. The SI unit of force is the newton, named in Newton's honour.

''Newton's Third Law'' states that for every action there is an equal and opposite reaction. This means that any force exerted onto an object has a counterpart force that is exerted in the opposite direction back onto the first object. A common example is of two ice skaters pushing against each other and sliding apart in opposite directions. Another example is the recoil of a firearm, in which the force propelling the bullet is exerted equally back onto the gun and is felt by the shooter. Since the objects in question do not necessarily have the same mass, the resulting acceleration of the two objects can be different (as in the case of firearm recoil).

Unlike Aristotle's, Newton's physics is meant to be universal. For example, the second law applies both to a planet and to a falling stone.

The vector nature of the second law addresses the geometrical relationship between the direction of the force and the manner in which the object's momentum changes. Before Newton, it had typically been assumed that a planet orbiting the Sun would need a forward force to keep it moving. Newton showed instead that all that was needed was an inward attraction from the Sun. Even many decades after the publication of the ''Principia'', this counterintuitive idea was not universally accepted, and many scientists preferred Descartes' theory of vortices.

Apple incident [link]

Newton himself often told the story that he was inspired to formulate his theory of gravitation by watching the fall of an apple from a tree. Although it has been said that the apple story is a myth and that he did not arrive at his theory of gravity in any single moment, acquaintances of Newton (such as William Stukeley, whose manuscript account, published in 1752, has been made available by the Royal Society) The question was not whether gravity existed, but whether it extended so far from Earth that it could also be the force holding the Moon to its orbit. Newton showed that if the force decreased as the inverse square of the distance, one could indeed calculate the Moon's orbital period, and get good agreement. He guessed the same force was responsible for other orbital motions, and hence named it "universal gravitation".

Various trees are claimed to be "the" apple tree which Newton describes. The King's School, Grantham, claims that the tree was purchased by the school, uprooted and transported to the headmaster's garden some years later. The staff of the [now] National Trust-owned Woolsthorpe Manor dispute this, and claim that a tree present in their gardens is the one described by Newton. A descendant of the original tree can be seen growing outside the main gate of Trinity College, Cambridge, below the room Newton lived in when he studied there. The National Fruit Collection at Brogdale

''De motu corporum in gyrum'' (1684)

''Philosophiæ Naturalis Principia Mathematica'' (1687)

''Opticks'' (1704)

''Reports as Master of the Mint'' (1701–25)

''Arithmetica Universalis'' (1707)

''The System of the World'', ''Optical Lectures'', ''The Chronology of Ancient Kingdoms, (Amended)'' and ''De mundi systemate'' (published posthumously in 1728)

''Observations on Daniel and The Apocalypse of St. John'' (1733)

''An Historical Account of Two Notable Corruptions of Scripture'' (1754)

Notes [link]

References [link]

This well documented work provides, in particular, valuable information regarding Newton's knowledge of Patristics

External links [link]

Newton's Scholar Google profile

ScienceWorld biography by Eric Weisstein

Dictionary of Scientific Biography

The Newton Project

The Newton Project – Canada

Rebuttal of Newton's astrology (via archive.org)

Newton's Religious Views Reconsidered

Newton's Royal Mint Reports

Newton's Dark Secrets NOVA TV programme

from The Stanford Encyclopedia of Philosophy:

*Isaac Newton, by George Smith

*Newton's Philosophiae Naturalis Principia Mathematica, by George Smith

*Newton's Philosophy, by Andrew Janiak

*Newton's views on space, time, and motion, by Robert Rynasiewicz

Newton's Castle Educational material

The Chymistry of Isaac Newton Research on his Alchemical writings

FMA Live! Program for teaching Newton's laws to kids

Newton's religious position

The "General Scholium" to Newton's Principia

Kandaswamy, Anand M. ''The Newton/Leibniz Conflict in Context''

Newton's First ODE – A study by on how Newton approximated the solutions of a first-order ODE using infinite series

The Mind of Isaac Newton Images, audio, animations and interactive segments

Enlightening Science Videos on Newton's biography, optics, physics, reception, and on his views on science and religion

Newton biography (University of St Andrews)

and see at s:Author:Isaac Newton for the following works about him:

*“Newton, Sir Isaac” in A Short Biographical Dictionary of English Literature by John William Cousin, London: J. M. Dent & Sons, 1910.

*“Newton, Isaac,” in Dictionary of National Biography, London: Smith, Elder, & Co., (1885-1900)

*Memoirs of Sir Isaac Newton's life by William Stukeley, 1752

Writings by him [link]

Newton's works – full texts, at the Newton Project

The Newton Manuscripts at the National Library of Israel - the collection of all his religious writings

Newton's Principia – read and search

''Descartes, Space, and Body'' and ''A New Theory of Light and Colour'', modernised readable versions by Jonathan Bennett

Opticks, or a Treatise of the Reflections, Refractions, Inflexions and Colours of Light'', full text on archive.org

Newton Papers, Cambridge Digital Library

and see at s:Author:Isaac Newton for the following works by him:

*Philosophiae Naturalis Principia Mathematica

*Opticks: or, a Treatise of the Reflections, Refractions, Inflections and Colours of Light

*Observations upon the Prophecies of Daniel and the Apocalypse of St. John

*New Theory About Light and Colour

*An Historical Account of Two Notable Corruptions of Scripture

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