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

Transcendental Curves in the Leibnizian Calculus analyzes a mathematical and philosophical conflict between classical and early modern mathematics. In the late 17th century, mathematics was at the brink of an identity crisis. For millennia, mathematical meaning and ontology had been anchored in geometrical constructions, as epitomized by Euclid's ruler and compass.

As late as 1637, Descartes had placed himself squarely in this tradition when he justified his new technique of identifying curves with equations by means of certain curve-tracing instruments, thereby bringing together the ancient constructive tradition and modern algebraic methods in a satisfying marriage. But rapid advances in the new fields of infinitesimal calculus and mathematical mechanics soon ruined his grand synthesis.

Descartes's scheme left out transcendental curves, i.e. curves with no polynomial equation, but in the course of these subsequent developments such curves emerged as indispensable. It was becoming harder and harder to juggle cutting-edge mathematics and ancient conceptions of its foundations at the same time, yet leading mathematicians, such as Leibniz felt compelled to do precisely this. The new mathematics fit more naturally an analytical conception of curves than a construction-based one, yet no one wanted to betray the latter, as this was seen as virtually tantamount to stop doing mathematics altogether. The credibility and authority of mathematics depended on it.

• Brings to light this underlying and often implicit complex of concerns that permeate early calculus
• Evaluates the technical conception and mathematical construction of the geometrical method
• Reveals a previously unrecognized Liebnizian programmatic cohesion in early calculus
• Provides a beautifully written work of outstanding original scholarship

PhD students and tenured mathematicians and historians of mathematics with an interest in the early history of calculus and geometry. Leibniz scholars will naturally be most attracted to the content