Shaping Modern Mathematics

1. Shaping Modern Mathematics Raymond Flood Gresham Professor of Geometry

2. Lectures At the Museum of London • Ghosts of Departed Quantities: Calculus and its Limits Tuesday 25 September 2012 • Polynomials and their Roots Tuesday 6 November 2012 • From One to Many Geometries Tuesday 11 December 2012 • The Queen of Mathematics Tuesday 22 January 2013 • Are Averages Typical? Tuesday 19 February 2013 • Modelling the World Tuesday 19 March 2013

3. Ghosts of Departed Quantities: Calculus and its Limits Raymond Flood Gresham Professor of Geometry

4. What is the Calculus? Integration is used to find areas of shapes in two-dimensional space or volumes in three dimensions.

5. Archimedes (c287 – 212BC) the volume of a cylinder is 11/2times that of the sphere it surrounds. Archimedes, by Georg Andreas Böckler, 1661

6. What is the Calculus? Differentiation is concerned with how fast things move or change. It is used to find speeds and the slopes of tangents to curves.

7. Apollonius’s Conics

8. David Gregory’s 1703 edition of Euclid’sElements with Propositions from Euclid this time drawn on the sand Halley’s 1710 edition of Apollonius with conic sections drawn in the sand Torelli ‘s 1792 edition of Archimedes and this time with a spiral, in fact an Archimedean spiral, drawn in the sand.

9. Characterising the family of ideas called the calculus • A systematic way of finding tangents • A systematic way of finding areas • Connecting tangents and areas

10. Isaac Newton 1642 - 1727

12. d = 4.9 t2

13. d = 4.9 t2 Average speed = distance travelled / time taken

14. Finding instantaneous speed from average speed • Distance travelled in time tequals 4.9 t2 • At a later time, t + o, distance travelled is: 4.9 (t + o)2

15. Finding instantaneous speed from average speed • Distance travelled in time tequals 4.9 t2 • At a later time, t + o, distance travelled is: 4.9 (t + o)2 • In the time interval t to t + 0 distance travelled is 4.9 (t + o)2 – 4.9 t2 = 9.8 t o + 4.9 o2

16. Finding instantaneous speed from average speed • Distance travelled in time tequals 4.9 t2 • At a later time, t + o, distance travelled is: 4.9 (t + o)2 • In the time interval t to t + 0 distance travelled is 4.9 (t + o)2 – 4.9 t2 = 9.8 t o + 4.9 o2 • Divide by o to find average speed over the interval = 9.8 t + 4.9 o

17. Finding instantaneous speed from average speed • Distance travelled in time tequals 4.9 t2 • At a later time, t + o, distance travelled is: 4.9 (t + o)2 • In the time interval t to t + 0 distance travelled is 4.9 (t + o)2 – 4.9 t2 = 9.8 t o + 4.9 o2 • Divide by o to find average speed over the interval = 9.8 t + 4.9 o • Shrink the interval i.e. allow o to approach zero • Then this average speed, 9.8 t + 4.9 o, approaches the instantaneous speed 9.8 t

18. Instantaneous speed is the slope of the tangent at t = 5 Average speeds are the slopes of the lines passing through time t = 5

19. Finding the speed from the distance Distance = 4.9 t2

20. Finding the speed from the distance Distance = 4.9 t2 Speed = 9.8 t

21. Finding the acceleration from the speed Speed = 9.8 t

22. Finding the acceleration from the speed Speed = 9.8 t Acceleration = 9.8 Find the slopes of the tangents

23. DIFFERENTIATION Distance = 4.9 t2 Speed = 9.8 t Acceleration = 9.8 Find the slopes of the tangents Find the slopes of the tangents

24. INTEGRATION Acceleration = 9.8 Area equals 9.8 t Find the areas

25. INTEGRATION Speed = 9.8 t Acceleration = 9.8 Find the areas

26. INTEGRATION Speed = 9.8 t Acceleration = 9.8 Area =4.9 t 2 Find the areas

27. INTEGRATION Distance = 4.9 t2 Speed = 9.8 t Acceleration = 9.8 Find the areas Find the areas

28. Find the Slopes Distance = 4.9 t2 Acceleration = 9.8 Speed = 9.8 t Find the areas Find the areas

29. DIFFERENTIATION

30. DIFFERENTIATION

31. INTEGRATION

32. Gottfried Leibniz 1646 - 1716

33. Binary Arithmetic It is possible to use … a binary system, so that as soon as we have reached two we start again from unity in this way: (0) (1) (2) (3) (4) (5) (6) (7) (8) 0 1 10 11 100 101 110 111 1000 … what a wonderful way all numbers are expressed by unity and nothing.

34. Leibniz’s Calculating machine The machine’s crucial innovation was a stepped gearing wheel with a variable number of teeth along its length, which allowed multiplication on turning a handle.

35. Leibniz notation d (or dy/dx) notation for differentiation: referring to the change in y divided by the change in x ∫ notation for the integration: finding areas under curves by summing lines. He defined omnia l (all the ls), which he then represented by an elongated S for sum, the integral sign, ∫.

36. First appearance of the Integral sign, ∫on October 29th 1675

37. Leibniz’s 1684 account of his Differential Calculus

38. Leibniz’s rules for differentiation • For any constant a: d(a) = 0, d(ay) = a dy • d(v + y) = dv + dy • d(vy) = vdy + y dv • d(v/y) = (ydv − vdy) / y2

39. The Priority dispute Developed Calculus 1665 – 1667 Published 1704 -1736 Developed Calculus 1673– 1676 Published 1684 - 1686

40. Brachistochrone problem Suppose that you roll a ball down a ramp from a point A to another point B. Which curve should the ramp be if the ball is to reach B in the shortest possible time? Johann Bernoulli 1667 – 1748

41. A cycloidis the curve traced by a fixed point on a circle rolling along a straight line; one can think of a cycloid as the curve traced out by a piece of mud on a bicycle tyre when the bicycle is wheeled along.

42. Model to illustrate that the cycloid gives the path of quickest descent

43. Bishop Berkeley 1685 - 1753 If to be is to be perceived?

44. If to be is to be perceived? There was a young man who said God, Must find it exceedingly odd When he finds that the tree Continues to beWhen no one's about in the Quad. Dear Sir, your astonishment's odd I'm always about in the Quad And that's why the tree Continues to be Since observed by, yours faithfully, God Ronald Knox

46. Bishop Berkeley’s Queries • Query 64 Whether mathematicians, who are so delicate in religious points, are strictly scrupulous in their own science? Whether they do not submit to authority, take things upon trust, and believe points inconceivable’? Whether they have not their mysteries, and what is more, their repugnancies and contradictions?

47. And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

48. Finding instantaneous speed from average speed • Distance travelled in time tequals 4.9 t2 • At a later time, t + o, distance travelled is: 4.9 (t + o)2 • In the time interval t to t + 0 distance travelled is 4.9 (t + o)2 – 4.9 t2 = 9.8 t o + 4.9 o2 • Divide by o to find average speed over the interval = 9.8 t + 4.9 o • Shrink the interval i.e. allow o to approach zero • Then this average speed, 9.8 t + 4.9 o, approaches the instantaneous speed 9.8 t

49. Dividing by Zero To Prove that 5 = 8 0 x 5 = 0 x 8 as they are both 0. If we are able to divide by 0 and do so we get 5 = 8

50. Finding instantaneous speed from average speed • Distance travelled in time tequals 4.9 t2 • At a later time, t + o, distance travelled is: 4.9 (t + o)2 • In the time interval t to t + 0 distance travelled is 4.9 (t + o)2 – 4.9 t2 = 9.8 t o + 4.9 o2 • Divide by o to find average speed over the interval = 9.8 t + 4.9 o • Shrink the interval i.e. allow o to approach zero • Then this average speed, 9.8 t + 4.9 o, approaches the instantaneous speed 9.8 t