8. The Birth of a New Science

Kapitel in diesem Buch

1. Frontmatter I
2. Contents IX
3. Preface XI
4. 1. John Napier, 1614 1
5. 2 Recognition 11
6. Computing with Logarithms 18
7. 3 Financial Matters 23
8. 4. To the Limit, If It Exists 28
9. Some Curious Numbers Related to e 37
10. 5. Forefathers of the Calculus 40
11. 6. Prelude to Breakthrough 49
12. Indivisibles at Work 56
13. 7. Squaring the Hyperbola 58
14. 8. The Birth of a New Science 70
15. 9. The Great Controversy 83
16. The Evolution of a Notation 95
17. 10 e^x: The Function That Equals Its Own Derivative 98
18. The Parachutist 109
19. Can Perceptions Be Quantified? 111
20. 11 e^θ Spira Mirabilis 114
21. A Historic Meeting between J. S. Bach and Johann Bernoulli 129
22. The Logarithmic Spiral in Art and Nature 134
23. 12 (e^x + e^-x)/2: The Hanging Chain 140
24. Remarkable Analogies 147
25. Some Interesting Formulas Involving e 151
26. 13 e^ix. “The Most Famous of All Formulas” 153
27. A Curious Episode in the History of e 162
28. 14 e^x+iy: The Imaginary Becomes Real 164
29. A Most Remarkable Discovery 183
30. 15. But What Kind of Number Is It? 187
31. Appendixes
32. Appendix 1. Some Additional Remarks on Napier’s Logarithms 199
33. Appendix 2. The Existence of lim (1+1/n)ⁿ as n→∞ 201
34. Appendix 3. A Heuristic Derivation of the Fundamental Theorem of Calculus 204
35. Appendix 4. The Inverse Relation between lim (b^h-1)/h = 1 and lim (1+h)^1/h as h→0 206
36. Appendix 5. An Alternative Definition of the Logarithmic Function 207
37. Appendix 6. Two Properties of the Logarithmic Spiral 209
38. Appendix 7. Interpretation of the Parameter Hyperbolic Functions 212
39. Appendix 8. e to One Hundred Decimal Places 215
40. Bibliography 217
41. Index 221