Families of courves ad the origins of partial differential

This book provides a detailed description of the main episodes in the emergence of partial differentiation during the period 1690-1740. It argues that the development of this concept - to a considerable degree of perfection - took place almost exclusively in problems concerning families of curves. Thus, the book shows the origins of the ideas and techniques which paved the way for the sudden introduction of partial differential equations in 1750. The main methodological characteristic of the book is its emphasis on a full understanding of the motives, problems and goals of the mathematicians of that time. 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