Newton's Next Mathematical Work example essay topic

Sir Isaac Newton (1642-1727) Isaac Newton was born in Lincolnshire, on December 25, 1642. He was educated at Trinity College in Cambridge, and resided there from 1661 to 1696 during which time he produced the majority of his work in mathematics. During this time New ton developed several theories, such as his fundamental principles of gravitation, his theory on optics otherwise known as the Lection es Optical, and his work with the Binomial Theorem. This is only a few theories that that Isaac Newton contributed to the world of mathematics. Newton contributed to all aspects of mathematics including geometry, algebra, and physics.

Isaac Newton was born into a poor farming family in 1642 with no father. Newton's father had passed away just a few months before he was born. His mother intended Newton to become a farmer but his lack of interest and the encouragement of John Stokes, Master of the Grantham grammar school and that of his uncle, William Ays cough, led to his eventual admission to his uncle's college. Trinity College, Cambridge, as a student on June 5, 1661. As a boy in Grantham, Newton had been intolerable to his servants and found it difficult to get along with his fellow grammar school peers. As a student, he bought his own food and paid a reduced fee in return for domestic service, a situation that appears unnecessary in view of his mother's wealth.

In the summer of 1662, Newton experienced, some sort of religious crisis which led him to write, in Shelton ian shorthand, his many sins, such as his threat to burn his mother and step-father. As a student at Cambridge Newton found himself among surroundings which were likely to develop and enhance his powers. In his first semester Newton happened to discover a book on astrology, but couldn't understand it very well on account of the geometry and trigonometry. He therefore bought a book by Euclid, and learned very quickly how obvious th propositions seemed.

Later he read and mastered Oughtred's Clav is, and Descartes' Geometry, which led him to take up mathematics rather than chemistry as a serious study. As a result of the Plague, from 1665 threw 1666 Newton had spent a great deal of time at home. During this time it seems evident that a great deal of his best work was accomplished. He thought out the fundamental principles of his theory of gravitation.

He determined that every particle of matter attracts every other particle. Yet he suspected that the attraction varied depending on the product of their masses. He suspected that the force, which retained the moon in its orbit around the earth, was the same as the terrestrial gravity. And to prove this hypothesis he proceeded by doing this. He knew that if a stone wall were allowed to fall near the surface of the earth, the attraction of the earth caused the stone wall to move though sixteen feet in one second. The moon's orbit relative to the earth is nearly a circle, and as a rough approximation assuming so, he knew the distance of the moon, and therefore the length of its path.

He also knew the time it took the moon to go around the earth once, a month. The following diagram is a copy of his experiment. Therefore Newton could find its veloisty at any point such as M. Then he could find the distance MT through which it would move in the next second if it were not pulled by the earth's attraction. At the end of the second it was at M', and therefore the earth E must have pulled it through the distance TM' in one second.

This experiment concluded that his estimate of the distance of the moon was inaccurate. Newton determined his calculation TM' was approximately one-eight less than he thought it would have been in his hypothesis. In October of 1669, Newton was chosen as a professor in replace of resigned Professor Barrow's. Newton chose Optics for the subject of his first research topic. Newton discovered the decomposition of white light into rays of different colored light by means of a prism Newton invented the method for determining the coefficients of refraction of different bodies. This is done by making a ray pass through a prism, so that deviation is minimal.

Let the angle of a prism be I and the deviation of the ray be &, the reflective index will be sin (I+&) cosec: 1/2 I. Later Newton failed at completing a few parts of his experiments and abandoned his hopes of making a refracting telescope, which should be achromatic. Instead he designed a reflecting telescope and later the reflecting microscope. Newton wrote a paper on fluxions in October 1666. This was a work which was not published at the time but seen by many mathematicians and had a major influence on the direction the calculus was to take. Newton thought of a particle tracing out a curve with two moving lines, which were the coordinates. The horizontal velocity x' and the vertical velocity y' were the fluxions of x and y associated with the flux of time.

The fluents or flowing quantities were x and y themselves. With this fluxion notation y' / x' was the tangent to f (x, y) = 0. In his 1666 paper Newton discusses the converse problem, given the relationship between x and y' / x' find y. For that reason the slope of the tangent was given for each x and when y' / x' = f (x) then Newton solves the problem by antidifferentiation. He also calculated areas by antidifferentiation and this work contains the first clear statement of the Fundamental Theorem of the Calculus. Newton had problems publishing his mathematical work.

Barrow was in some way to blame for this since the publisher of Barrow's work had gone bankrupt and publishers were, after this, careful of publishing mathematical works. Newton's work on Analysis with infinite series was written in 1669 and circulated in manuscript. It wasn't published until 1711. Similarly his Method of fluxions and infinite series was written in 1671 and published in English translation in 1736. Newton's next mathematical work was Tractatus de Quadrature Cur varum, which he wrote in 1693 but it wasn't published until 1704 when he published it as an Appendix to his Optics.

This work contains another approach, which involves taking limits. Newton says: "In the time in which x by flowing becomes x+o, the quantity x becomes (x+o) i.e. by the method of infinite series, x + nox + (nn-n) /2 oo +... At the end he lets the increment o vanish by 'taking limits'". A well known mathematician Leibniz, learned much on a European tour, which led him to meet Huygens in Paris in 1672. He also met Hooke and Boyle in London in 1673 where he bought several mathematics books, including Barrow's works.

Leibniz had a lengthy correspondence with Barrow. On returning to Paris Leibniz did some very fine work on the calculus, thinking of the foundations very differently from Newton. Newton considered variables changing with time. Leibniz thought of variables x, y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy / dx gives the tangent but he did not use it as a defining property.

For Newton integration consisted of finding fluents for a given fluxion so the fact that integration and differentiation were inverses was implied. Leibniz used integration as a sum. He was also happy to use 'infinitesimal' dx and dy where Newton used x' and y' which were finite velocities. Of course neither Leibniz nor Newton thought in terms of functions, however, but both always thought in terms of graphs.

For Newton the calculus was geometrical while Leibniz took it towards analysis. Leibniz was very conscious that finding a good notation was of fundamental importance and thought a lot about it. Newton, on the other hand, wrote more for himself and, as a consequence, tended to use whatever notation he thought of on the day. Leibniz's notation of d and highlighted the operator aspect which proved important in later developments. By 1675 Leibniz had settled on the notation y dy = y/2 written exactly as it would be today.

His results on the integral calculus were published in 1684 and 1686 under the name 'calculus summitries'; Jacob Bernoulli suggested the name integral calculus in 1690. After Newton and Leibniz the development of the calculus was continued by Jacob Bernoulli and Johann Bernoulli. However when Berkeley published his Analyst in 1734 attacking the lack of rigor in the calculus and disputing the logic on which it was based much effort was made to tighten the reasoning. Mac laurin attempted to put the calculus on a rigorous geometrical basis. Newton explained a wide range of previously unrelated phenomena, the eccentric orbits of comets; the tides and their variations; the precession of the Earth's axis; and motion of the Moon as perturbed by the gravity of the Sun. After suffering a nervous breakdown in 1693, Newton retired from research to take up a government position in London becoming Warden of the Royal Mint (1696) and Master (1699).

In 1703 he was elected president of the Royal Society and was re-elected each year until his death. He was knighted in 1708 by Queen Anne, the first scientist to be so honored for his work.

Bibliography

Andrade, E.N. da C. Sir Isaac Newton. Greewood Pub., 1979.
Gjertsen, D. The Newton Handbook London: Routledge, 1986.
Hall, A.R. Issac Newton Adventurer In Thought. New York: Free Press, 1984.
Issac Newton, [online] web Sir Isaac Newton (1642-1727) Bibliography References Andrade, E.N. da C. Sir Isaac Newton. Greewood Pub., 1979.