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Analysis by Its History [Hairer] - mklabgr - 08-17-2026
Analysis by Its History
Authors: Ernst Hairer and Gerhard Wanner
Publisher: Springer
Series:Undergraduate Texts in Mathematics
Analysis by Its History is an unusual introduction to mathematical analysis because it deliberately avoids the standard modern order of presentation. Instead of beginning with rigorous definitions of limits and continuity and then proceeding to differentiation and integration, Hairer and Wanner largely follow the historical development of calculus. The reader encounters problems and ideas much as mathematicians originally did—from Cardano, Descartes, Fermat, Newton, Leibniz and Euler through Cauchy, Riemann, Weierstrass and others. Practical questions about curves, areas, logarithms and infinite processes gradually lead to infinite series, derivatives, integrals and differential equations. Only afterward does the book explain how nineteenth-century mathematicians supplied the rigorous foundations that modern analysis textbooks normally introduce at the beginning.
The book is organized into four major parts: Introduction to Analysis of the Infinite, Differential and Integral Calculus, Foundations of Classical Analysis, and Calculus in Several Variables. Along the way it covers subjects such as the binomial theorem, logarithms, infinite series, differential equations, Stirling's formula, continuity, the mean value theorem, compactness, gradients and differentiation under the integral sign. It also resurrects topics that have largely disappeared from introductory analysis courses, including continued fractions, elliptic integrals and the Euler–Maclaurin summation formula. Historical quotations, original sources, diagrams, worked calculations and exercises help show why mathematical concepts arose rather than presenting them merely as finished theorems.
Despite its title, this is not primarily a history-of-mathematics book. It is a genuine analysis textbook whose historical development provides the organizing principle. That makes it particularly valuable for mathematics students and teachers who already know some calculus but want to understand the intellectual path from intuitive computation to rigorous analysis. The historical approach makes familiar concepts appear less arbitrary: definitions such as continuity, convergence and differentiability emerge as solutions to genuine mathematical difficulties encountered by earlier mathematicians. The result is a demanding but unusually engaging alternative or companion to a conventional real-analysis textbook.
Key takeaways
- Analysis is presented as an evolving subject, rather than as a completed collection of definitions and theorems.
- The book reverses much of the conventional teaching order: calculus and its motivating problems come before the rigorous foundations.
- Historical sources, illustrations and forgotten classical topics provide insight into why modern definitions and techniques developed.
- It is especially worthwhile for mathematics teachers and students interested in connecting calculus, rigorous analysis and the history of mathematical ideas.
Springer — Analysis by Its History