Families of Curves and the Origins of Partial Differentiation (eBook)

Front Cover 1
Families of Curves and the Origins of Partial Differentiation 4
Copyright Page 5
Contents 7
Acknowledgements 6
CHAPTER 1. INTRODUCTION 12
§1.1 "Monde inconnu" 12
§1.2 Criteria for partial differentiation 13
§1.3 Two Dimensional problem situations 18
§1.4 Differentials versus derivatives and validity of theorems 20
§1.5 Policy of transcription and interpretation 24
§1.6 Transcendental curves and transcendental expressions 29
§1.7 Conventions 31
CHAPTER 2. FAMILIES OF CURVES IN THE 1690s 33
§2.1 Envelopes 33
§2.2 The brachystochrone and its aftermath 41
§2.3 Conclusions 68
CHAPTER 3. ORTHOGONAL TRAJECTORIES 1694–1720 70
§3.1 Introduction 70
§3.2 The problem posed 71
§3.3 Orthogonal trajectories of the brachystochrones 73
§3.4 The limits of Leibniz's method 74
§3.5 Logarithmic curves 76
§3.6 The break–through to transcendental Curves 78
§3.7 Jakob Bernoulli's reaction 80
§3.8 Renascence of the problem 82
§3.9 First reactions to the challenge 84
§3.10 The final test–case 86
§3.11 Johann Bernoulli's alternatives 90
§3.12 Johann Bernoulli 's comparison of methods 98
CHAPTER 4. NICOLAUS I BERNOULLI AND ORTHOGONAL TRAJECTORIES 103
§4.1 Biography and bibliography 103
§4.2 Nicolaus I Bernoulli's partial differential calculus 108
§4. 3 Nicolaus I Bernoulli's resolution of the variable parameter equation 123
CHAPTER 5. EULER'S THEORY OF MODULAR EQUATIONS I N THE 1730s 135
§5.1 Introduction 135
§5.2 Euler's exposè of partial differential calculus in De differentiatione 137
§5.3 Early applications of partial differentiation 144
§5.4 Euler 's theory of modular equations 151
§5.5 Modular equations and ordinary differential equations 161
§5.6 Euler's view of the infinitesimal calculus around 1740 167
EPILOGUE 172
FOOTNOTES CHAPTER 1 174
FOOTNOTES CHAPTER 2 177
FOOTNOTES CHAPTER 3 187
FOOTNOTES CHAPTER 4 196
FOOTNOTES CHAPTER 5 201
APPENDIX 1: NICOLAUS I BERNOULLI'S "DEMONSTRATIO ANALYTICA CONSTRUCTIONIS CURVARUM, QUAE ALIAS POSITIONE DATAS AD ANGULOS RECTOS SECANT, TRADITAE IN ACTIS LIPS. 1719 PAG 295 ET SEQQ." 210
Introduction 210
Text 211
Translation 213
APPENDIX 2 : LEONHARD EULER'S "DE DIFFERENTIATIONE FUNCTIONUM DUAS PLURESVE VARIABILES QUANTITATES INVOLVENTIUM" 215
Introduction 215
Text 216
Marginalia 224
Translation 225
APPENDIX 3: NEWTON'S RULE FOR THE RADIUS OF CURVATURE OF MAY 21ST, 1665 234
BIBLIOGRAPHY 238
Preamble 238
List of letters 239
Books, articles, manuscripts 241