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Chapter 1: Preliminary matters
- 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
- 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
- 3.1 Introduction
- 3.2 Greek geometry
- 3.3 17th-century philosophy of geometry
Chapter 4: Mathematical context
- 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
- 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
- 6.1 Introduction
- 6.2 The problem with power series
- 6.3 Exponentials
Chapter 7: Transcendental curves by the reduction of quadratures
- 7.1 Introduction
- 7.2 Computational reduction of quadratures
- 7.3 The rectification of quadratures
Chapter 8: Transcendental curves in physics
- 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
- 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
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
- No. of pages:
- © Academic Press 2017
- 20th April 2017
- Academic Press
- Paperback ISBN:
- eBook ISBN:
"One of the general philosophical teachings that I have learned from this book is to cast doubt on certain modes of thinking that are so engrained in our everyday practice (as scientists and philosophers) to become accepted standards of rationality." --Zentralblat MATH
"This is the story of the right historical context for the development of the Leibnizian calculus.To present a clear picture of the time and of the type of problem animating research in the 17th century, the author first delves into the philosophical context, all practitioners claiming allegiance to the ancient Greeks, but with a continental vs. British difference." -- MathSciNet
"Blåsjö does an excellent job. His study is informative, well organized, thought provoking, clearly written and thoroughly researched. … Blåsjö’s work provides a great service to the mathematical community." -- MAA Reviews
"This book tells the unique and fascinating story of the early calculus, focusing on Leibniz’s publications." -- zbMATH
Viktor Blåsjö works at the Mathematical Institute of Utrecht University. He has published widely on the history of geometry and astronomy as well as historiography. He has published award-winning papers in the American Mathematical Monthly (2006 Lester R. Ford award) and in Historia Mathematica.
Mathematical Institute of Utrecht University
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