1 / 19

Presentation 3

Presentation 3. Section 8.2. Rene’ Descartes. Works Discours de la Methode La Dioptrique - nature and property of light, law of refraction, eye, telescope La Geometrie – combined methods of algebra and geometry to produce the principles of analytic geometry

ianthe
Download Presentation

Presentation 3

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Presentation 3 Section 8.2

  2. Rene’ Descartes • Works • Discours de la Methode • La Dioptrique - nature and property of light, law of refraction, eye, telescope • La Geometrie – combined methods of algebra and geometry to produce the principles of analytic geometry • Le Monde - encyclopedic treatise on physics (The World) • Published after his death in 1664 because of Galileo’s problems with the Church • La Meteores - meteorology – explanation of atmospheric phenomena

  3. “I think, therefore I am” • Geometrie • Book one: - Problems which can be constructed by means of circles and straight lines only - Apply algebra to geometry • Book two: On the Nature of Curved Lines – distinction between geometrical and mechanical kind of curves • Book three: Algebraic aspects – the nature of equations and the principles underlying their solutions.

  4. Descartes Rule of Sign • Example x6– 10x2 + x + 1 = 0 • 2 sign changes therefore, the equation cannot have more than two positive roots. That is two positive roots or none Change x to –x • (-x)6 – 10(-x)2 + (-x) + 1 = 0 Simplify: x6– 10x2 - x + 1 = 0 • 2 sign changes therefore, the equation cannot have more than two negative roots. That is two negative roots or none You Try: x6 + 3x3 + x - 1 = 0

  5. Section 8.3Newton: The Principia Mathematica • Sir Isaac Newton • Principia Mathematica (1687) • Law of Motion (page 402)

  6. Mathematicians • John Wallis • Father of English Cryptography • Tractatus de SectionibusConicis (1656) • ArithmeticaInfinitorum -  makes initial appearance in print • Pierre de Fermat and Rene Descartes • Coordinate geometry/ analytic geometry • Pierre de Fermat • Number theory • Cardan • Theory of probability (Liber de LudoAleae) • Leibniz • Symbolic logic

  7. Calculus • Isaac Newton and Gottfired Leibniz • And • Cavalieri, Torricelli, Barrow, Descartes, Fermat, Wallis • Newton letter to Hooke • “If I have seen farther than others, it is because I have stood on the shoulders of giants”

  8. Section 8.4Calculus Controversy • The Mechanical World: Descartes and Newton • The Calculus Controversy • The Early Work of Leibniz • Newton’s Fluxional Calculus • The Dispute over Priority • Maria Agnesi and Emilie du Chatelet

  9. Calculus Controversy: • The invention of calculus was one of the greatest intellectual achievements of the 1600’s. • It also introduced a long and bitter controversy of who discovered calculus. • The methods were introduced by Newton in England. • Coincidentally Leibniz introduced similar methods of calculus during the same time. • Inferences of plagiarism became public which was brought before the Royal Society • Newton played a major role in the development of calculus, sharing credit with Gottfried Leibniz. Leibniz Newton

  10. Gottfried Leibniz (1647- 1716) • Brief Biography- • Born: (1646- 17160) in University town of Leipzig two years before the end of the Thirty Years’ War • Father died when he was 6; he was a jurist and professor of moral philosophy • Leibniz had no direction in his studies after his father’s death. • His world was founded in books. He taught himself Latin at 8; studied Greek at 12; became acquainted with a wide range of classical writers; compared and contrasted principles of Aristotle with those of Democritus. • Regarded as something of a prodigy and soon outstripped all his contemporaries.

  11. He later graduated from Leipzig defending a thesis on a point for philosophy; attended mathematics lectures; concentrated on legal studies; earned his masters a year later; He was given a teaching position in philosophical faculty; Applied for degree of doctor of law- refused because he was too young. One year later finished doctorate at University of Altdorf (Nuremberg) • His dissertation made a favorable impression that he was offered a professorship. He preferred politics so he wrote as essay on the study of law gained him a post in the service of the archbishop-elector of Mainz, where he helped reform the current statutes. • The rest of his life was spent in residence at the courts of Mainz and Hanover until his death in 1716.

  12. Major Accomplishments • Constructed working model of a new calculating machine improvement from Pascal (already invented) • Further developed the sum of any infinite series by using the sum of the reciprocals of the triangular numbers. • Developed and introduced abbreviations such as omn (Latin for sum) and Integral sign; ∫ ( elongated S for Sum)…improved to ∫ d(x) dx (modern day use) • Known for finding inverse tangent – connection to the direct and inverse of tangent problem • Leibniz found the celebrated alternating series for but the formula was known to James Gregory. • Leibniz’s investigations of calculus were based on “characteristic triangles” from Pascal.

  13. Publications • DisputatioArithmetica de Complexionibus • Expanded into ArsCombinatoria (1666): extensively develops the theory of permutations and combinations for the purpose of making logical deductions. • It established a new mathematics-like language of reasoning in which all scientific concepts could be formed by combinations from a basic alphabet of human thought. • Leibniz suggested calculus could be devised of solutions for all problems that could be expressed in his scientific language.

  14. Leibniz’s Creation of Calculus • KEY POINTS • From 1672 – 1676, Leibniz was characterized as “slowly flowering mathematics genius matured” • Various methods invented for determining the tangent line of certain lines to certain classes of curves • Used abbreviations: omn to mean sum in Latin • Investigated basic algorithm of calculus- determined product rule and gave statement of quotient rule • Investigations into calculus were based on what was called “characteristic triangle” • Liebniz used differential-integral calculus but never founded it on the limit concept; the ratio dy/dx was thought of as a quotient of differences and the integral the sum. dy/dx = lim ( y / x) as x  0 • MAJOR ACCOMPLISHMENTS • Leibniz stated inverse tangent problem: find the locus of the function which determined subtangents; later evolved to theory of inverse method of tangent that are reducible to quadratures or integrations. • Leibniz introduced integral symbol

  15. Newton Fluxional Calculus (p. 417) • In 1676, Leibniz had possession of all the rules and notation of his calculus • Newton had developed a equivalent approach with a more geometrical slant. • During this time, Newton discovered the binomial theorem.- condensed into small treatise of 30 pages covering tangency, curvature, centers of gravity and area.- known as October 1666 Tract. • He was able to show the area of the rectangular hyperbola (x+1)y = 1 was a series expansion for the natural logarithm of 1+x. • Publication: Logarithmotechinia(1668) – by Nicholas Mercator; part1 and 2- tables of logarithms; part 3- contained various approximation formulas for logarithms. One was Newtons reduction of log (1+x) to infinite series. • Publication: De Analysiwhich evolved into De MethodisFluxionum (printed 65 years after written) – state the rule for computing area under the curve; which was a comprehensive account of his calculus methods.

  16. The Dispute over priority (p. 425) • Leibniz’s claimed to have an independent method of calculus separate from Newton. • Recall: Leibniz was the first to publish the particulars of his differential calculus to the world in ActaEruditorum. It contained mechanical rules without the proof for computing differentials • Newton invented the method of fluxions • Investigation with the Royal Society to review the accusation to vindicate Leibniz on the charge of having stolen Newton’s invention of calculus. • It caused Newton to withhold other publications that further developed the methods of calculus. This bitterness of the calculus priority dispute significantly affected the history of mathematics.

  17. You Try • Verify Leibniz’s famous identity (which gives an imaginary decomposition of the real number 6 ) • Problem #3 - p.433

  18. You Try • Given the binomial theorem, as stated by Newton in his letter to Oldenburg, is equivalent to the more familiar form: (1+x)r = 1 + rx + [r(r-1)]/2! x2 + [r(r-1)(r-2)]/3! x3 + … Where r is an arbitrary integral or fractional exponent. You Try: Use the binomial theorem to obtain the following series expansions (1+x)-1 = 1 -x + x2 - x3 + …+ (-1)nxn + … Hint: r = -1

  19. Historical Highlight • Maria GaetanaAgnesi • Italy’s Maria GaetanaAgnesi, a brilliant linguist, philosopher, and mathematician, was the first of 21 children of a professor of mathematics at the University of Bologna. • Woman mathematician • The name “Witch of Agnesi” mistranslation of “Curve of Agnesi” (Problem 13 on p. 434) • Publication: • Famous for: • Google to see what you can find out about her and problem

More Related