What were some Mathematical Contributions that Sir Isaac Newton did?

Question

Please answer only if you know the answer. Thank you very much.

Jake you could you simplify?

Answer 1

Newton and Gottfried Leibniz developed calculus independently, using their own unique notations (as most great mathematicions do.) Although Newton had worked out his method years before Leibniz, he published almost nothing about it until 1693, and did not give a full account until 1704. Meanwhile, Leibniz began publishing a full account of his methods in 1684. Moreover, Leibniz's notation and "differential Method" were universally adopted on the Continent, and after 1820 or so, in the British Empire. Newton claimed that he had been reluctant to publish his calculus because he feared being mocked for it. Starting in 1699, other members of the Royal Society accused Leibniz of plagiarism, and the dispute broke out in full force in 1711. Thus began the bitter calculus priority dispute with Leibniz, which marred the lives of both Newton and Leibniz until the latter's death in 1716. This dispute created a divide between British and Continental mathematicians that may have retarded the progress of British mathematics by at least a century.

Newton is generally credited with the generalized 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 also discovered a new formula for pi.

He was elected Lucasian professor of mathematics in 1669. In that day, any fellow of Cambridge or Oxford had to be 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
From 1670 to 1672, he 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. 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 the colours we observe are the result of how objects interact with the incident already-coloured light, not the result of objects generating the colour. For more details, see Newton's theory of colour. Many of his findings in this field were criticized by later theorists, the most well-known being Johann Wolfgang von Goethe, who postulated his own colour theories.

A replica of Newton's 6-inch reflecting telescope of 1672 for the Royal Society.From this work he concluded that any refracting telescope would suffer from the dispersion of light into colours, and invented a reflecting telescope (today known as a Newtonian telescope) to bypass that problem. By grinding his own mirrors, using Newton's rings to judge the quality of the optics for his telescopes, he was able to produce a superior instrument to the refracting telescope, due primarily to the wider diameter of the mirror. (Only later, as glasses with a variety of refractive properties became available, did achromatic lenses for refractors become feasible.) In 1671 the Royal Society asked for a demonstration of his reflecting telescope. 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. The two men remained enemies until Hooke's death.

In one experiment, to prove that colour perception is caused by pressure on the eye, Newton slid a darning needle around the side of his eye until he could poke at its rear side, dispassionately noting "white, darke & coloured circles" so long as he kept stirring with "ye bodkin."

Newton argued that light is composed of particles, but he had to associate them with waves to explain the diffraction of light (Opticks Bk. II, Props. XII-L). Later physicists instead favoured a purely wavelike explanation of light to account for diffraction. Today's quantum mechanics restores the idea of "wave-particle duality", although photons bear very little resemblance to Newton's corpuscles (e.g., corpuscles refracted by accelerating toward the denser medium).

(dubious; discuss) Newton is believed to have been the first to explain precisely the formation of the rainbow from water droplets dispersed in the atmosphere in a rain shower. Figure 15 of Part II of Book One of the Opticks shows a perfect illustration of how this occurs.

In his Hypothesis of Light of 1675, Newton posited the existence of the ether to transmit forces between particles. Newton was in contact with Henry More, the Cambridge Platonist who was born in Grantham, on alchemy, and now his interest in the subject revived. 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."[5] Newton's interest in alchemy cannot be isolated from his contributions to science.[6] (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 wrote 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?"[7] Newton also constructed a primitive form of a frictional electrostatic generator, using a glass globe (Optics, 8th Query).


Gravity and motion
Newton's own copy of his Principia, with hand written corrections for the second edition.Further information: The writing of Principia Mathematica
In 1679, Newton returned to his work on mechanics, i.e., gravitation and its effect on the orbits of planets, with reference to Kepler's laws of motion, and consulting with Hooke and Flamsteed on the subject. He published his results in De Motu Corporum (1684). This contained the beginnings of the laws of motion that would inform the Principia.

The Philosophiae Naturalis Principia Mathematica (now known as 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 were not to be improved upon for more than two hundred years. He used the Latin word gravitas (weight) for the force that would become known as gravity, and defined the law of universal gravitation. In the same work he presented the first analytical determination, based on Boyle's law, of the speed of sound in air.

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 that lasted until 1693. The end of this friendship led Newton to a nervous breakdown.


Later life
Isaac Newton in 1712For more details on this topic, see Isaac Newton's later life.
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 his only recorded comments were to complain about a cold draft 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 Master Lucas (and finagling Edmond Halley into the job of deputy comptroller of the temporary Chester branch). Newton became perhaps the best-known Master of the Mint upon Lucas' death in 1699, a position Newton held until his death. 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 Newton unofficially moved the Pound Sterling from the silver standard to the gold standard by creating a relationship between gold coins and the silver penny in the "Law of Queen Anne"; these were all great reforms at the time, adding considerably to the wealth and stability of England. It was his work at the Mint, rather than his earlier contributions to science, that earned him a knighthood from Queen Anne in 1705.

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 star catalogue, which Newton had used in his studies.

Newton's grave in Westminster AbbeyNewton died in London on March 20th, 1727, and was buried in Westminster Abbey. His half-niece, Catherine Barton Conduitt,[8] served as his hostess in social affairs at his house on Jermyn Street in London; he was her "very loving Uncle",[9] according to his letter to her when she was recovering from smallpox. Although Newton, who had no children, had divested much of his estate onto relatives in his last years he actually died intestate. His considerable liquid estate was divided equally between his eight half-nieces and half-nephews (three Pilkingtons, three Smiths and two Bartons (including Catherine Barton Conduitt).[10] Woolsthorpe Manor passed to his heir-in-law, a John Newton ("God knows a poor representative of so great a man"), who, after six years of "cock[fight]ing, horse racing, drinking and folly" was forced to mortgage and then sell the manor before dying in a drunken accident.[11]

After his death, Newton's body was discovered to have had massive amounts of mercury in it, probably resulting from his alchemical pursuits. Mercury poisoning could explain Newton's eccentricity in late life.[12]

Answer 2

The Newton-Raphson method is an iterative method to find the zeroes of an arbitrary differntiable function.

Newton's Forward Difference Formula fits a polynomial to a set of points.

Answer 3

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