Transcendental Curves in the Leibnizian Calculus

Chapter 1: Preliminary matters

• Abstract
• 1.1 Purpose and scope of the present work
• 1.2 Previous scholarship related to the present work
• 1.3 What is new in the present work
• 1.4 Conventions adopted in this work
• 1.5 Some key terms

Chapter 2: Introduction

• Abstract
• 2.1 The problem of transcendental curves
• 2.2 Fundamental conflict between analytic and construction-based paradigms
• 2.3 Implications of this story for the general historiography of the period

Chapter 3: The classical basis of 17th-century philosophy of mathematics

• Abstract
• 3.1 Introduction
• 3.2 Greek geometry
• 3.3 17th-century philosophy of geometry

Chapter 4: Mathematical context

• Abstract
• 4.1 Introduction
• 4.2 The early calculus according to Leibniz
• 4.3 Transcendental curves in the early work of Huygens
• 4.4 Johann Bernoulli’s lectures on the calculus

Chapter 5: Transcendental curves by curve tracing

• Abstract
• 5.1 Introduction
• 5.2 The tractrix
• 5.3 Johann Bernoulli’s generalised tractrix
• 5.4 Leibniz’s construction by tractional motion of any curve given by dy/dx
• 5.5 Jacob Bernoulli’s tractional method
• 5.6 Johann Bernoulli’s crawling curves

Chapter 6: Transcendental curves analytically: exponentials and power series

• Abstract
• 6.1 Introduction
• 6.2 The problem with power series
• 6.3 Exponentials

Chapter 7: Transcendental curves by the reduction of quadratures

• Abstract
• 7.1 Introduction
• 7.2 Computational reduction of quadratures
• 7.3 The rectification of quadratures

Chapter 8: Transcendental curves in physics

• Abstract
• 8.1 Introduction
• 8.2 The elastica
• 8.3 The paracentric isochrone
• 8.4 The brachistochrone
• 8.5 Forces and tangents

Chapter 9: A view from the 18th century

• Abstract
• 9.1 Introduction
• 9.2 The epistemological miracle of analytical methods
• 9.3 Euler’s analytical calculus
• 9.4 Lagrange’s analytical calculus
• 9.5 Lagrange’s analytical mechanics

Chapter 10: Concluding overview

• Abstract