Mathematics from 1500 to the Present Day

1. Mathematics from 1500 to the Present Day T J Osler

2. François Viète  (1540 - February 13, 1603), generally known as Franciscus Vieta, was a Frenchmathematician.

3. Vieta’s product of nested radicals (1592) was the first formula for Pi

4. John Wallis (November 23, 1616 - October 28, 1703) was an Englishmathematician who is given partial credit for the development of modern calculus. 

5. Wallis product for Pi - 1656

6. William Brouncker, 2nd Viscount Brouncker, FRS (1620 – 5 April1684) was anEnglishmathematician.

7. Brouncker’s continued fraction for Pi - 1656

8. René Descartes  (31 March 1596 – 11 February 1650), was a French philosopher,mathematician, scientist, and writer. Invented Analytic Geometry

9. Isaac Newton 1642 -1727 • Robert Hooke 1635 – 1703 • Edmund Halley 1656 – 1742 • Gottfried Leibniz 1646 - 1716

10. George Berkeley  (12 March 1685 – 14 January 1753), also known as Bishop Berkeley, was an Irishphilosopher. 

11. Berkley found flaws in the foundations of Newton’s calculus. • Newton spoke of “infinitesimals” numbers not zero, but smaller than any assigned quantity. • These difficulties would not be removed until the 1800s

12. Leonhard Paul Euler (pronounced [ˈɔʏlɐ] in German,  in English;15 April 1707 – 18 September 1783) was a pioneering Swissmathematician andphysicist who spent most of his life in Russia and Germany.

13. Zeta Function

16. Calculus of Variations

17. Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death. In the middle, it shows his polyhedral formulaV − E + F = 2.

18. Euler wrote some 866 Books, papers and letters of ground breaking mathematical content • He is the most prolific mathematician of all time • Even though he went blind in his later years, his mathematical productivity increased

19. Joseph-Louis Lagrange, born Giuseppe Lodovico Lagrangia (25 January1736 –10 April1813) was an Italianmathematician and astronomer, who lived most of his life in Prussia and France, making significant contributions to all fields of analysis, tonumber theory, and to classical and celestial mechanics. 

20. Lagrangian mechanics • Between 1772 and 1788, Lagrange re-formulated Classical/Newtonian mechanics to simplify formulas and ease calculations. These mechanics are called Lagrangian mechanics. • Worked on Celestial Mechanics and the solution of algebraic equations

21. Pierre-Simon, marquis de Laplace (23 March 1749 – 5 March 1827) was a Frenchmathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. 

22. Laplace worked on Celestial Mechanics • Mécanique Céleste • Tried to prove that the solar system was stable

23. Johann Carl Friedrich Gauss. (30 April 1777 – 23 February 1855)“The Prince of Mathematicians”

24. As a teenager, Gauss showed how to construct a regular 17 gon • First major geometric construction in 2000 years

25. Prime numbers of this form are also known as the Fermat primes • Gauss proved that a regular n-gon could be geometrically constructed if the number of sides were a product of distinct Fermat Primes times a power of two

26. Titus Bode Law • To find the mean distances of the planets, beginning with the following simple sequence of numbers: • 0 3 6 12 24 48 96 192 384 • With the exception of the first two, the others are simple twice the value of the preceding number. • Add 4 to each number: • 4 7 10 16 28 52 100 196 388 • Then divide by 10: • 0.4 0.7 1.0 1.6 2.8 5.2 10.0 19.6 38.8

28. In 1800 Astronomers begin the search for the planet between Mars and Jupiter

29. Discovery of Asteroid CeresMakes Gauss Famous • 1801 Italian astronomer Piazzi observes a moving celestial object for 41 days before it disappears behind the sun • The newly-discovered planet had been lost • Laplace declared that the new planet was lost because its orbit could not be calculated from so little data

30. 24 year old Gauss discovered a method for computing the planet's orbit using only three of the original observations and successfully predicted where Ceres might be found. • The prediction catapulted him to worldwide acclaim

31. Jean Baptiste Joseph Fourier (March 21, 1768 – May 16, 1830) was a Frenchmathematician and physicist best known for initiating the investigation of Fourier series and their application to problems of heat flow.

32. Let f(x) have period 2L f(x+2L)=f(x)

33. Example Fourier Series 1

34. Fourier Series Example 2

35. Fourier Series challenged the intuition of the greatest mathematicians • How could a sum of such smooth functions as sine and cosine represent discontinuous functions? • Later Carl Weierstrass showed a Fourier Series that was continuous everywhere, but differentiable nowhere!

36. Augustin Louis Cauchy (21 August 1789 – 23 May 1857; was a Frenchmathematician.

37. Cauchy finally provided the Calculus with a rigorous foundation • He introduced the epsilon – n and epsilon – delta definitions of limit. • From these we can rigorously define continuous functions and differentiable functions • The meat of our Real Analysis course

38. Évariste GaloisOctober 25, 1811 – May 31, 1832) was a Frenchmathematician born in Bourg-la-Reine.

39. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. • He died fighting a duel over a woman at age 21

40. Georg Friedrich Bernhard RiemannSeptember 17, 1826 – July 20, 1866) was an extremely influential Germanmathematician who made important contributions to analysis and differential geometry, some of them paving the way for the later development of general relativity.

41. Riemann

42. Riemann Hypothesis – Most famous unsolved problem in mathematicsThe non-trivial zeros of the zeta function lie on the line x = ½ in the complex plane

43. Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician and theoretical physicist, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

44. Jules Henri Poincaré

46. In 1887 he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies.

47. The two body problem was solved analytically by Newton, and the solution is Kepler’s equations of planetary motion. • The three body problem has never been solve analytically, although approximate computer solutions are easy to generate.

48. Albert Einstein 14 March 1879 – 18 April 1955) was a German-born theoretical physicist. He is best known for his theory of relativity and specifically mass–energy equivalence, expressed by the equation E = mc2. Einstein received the 1921 Nobel Prize in Physics "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.”

50. 1879: Albert Einstein is born to Hermann Einstein • 1889: At age 10, Albert sets into a program of self education and reads as much about science as he can. • 1896: Albert graduates from high school at the age of 17 and enrolls at the ETH (the Federal Polytechnic) in Zurich.