This publication includes an unabridged and annotated translation of two works by Johann Heinrich Lambert (17281777) written in the 1760s: Vorläufige Kenntnisse für die, so die Quadratur und Rectification des Circuls suchen and Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques. The translations are accompanied by a contextualised study of each of these works and provide an overview of Lamberts contributions, showing both the background and the influence of his work. In addition, by adopting a biographical approach, it allows readers to better get to know the scientist himself.
Lambert was a highly relevant scientist and polymath in his time, admired by the likes of Kant, who despite having made a wide variety of contributions to different branches of knowledge, later faded into an undeserved secondary place with respect to other scientists of the eighteenth century. In mathematics, in particular, he is famous for his research on non-Euclidean geometries, although he is likely best known for having been the first who proved the irrationality of pi. In his Mémoire, he conducted one of the first studies on hyperbolic functions, offered a surprisingly rigorous proof of the irrationality of pi, established for the first time the modern distinction between algebraic and transcendental numbers, and based on such distinction, he conjectured the transcendence of pi and therefore the impossibility of squaring the circle.
Part I: Antecedents.
Chapter 1. From Geometry to Analysis.
Chapter
2. The situation in the first half of the 18th century. Euler and
continued fractions.- Part II: Johann Heinrich Lambert (17281777).
Chapter
3. A biographical approach to Johann Heinrich Lambert.
Chapter 4. Outline of
Lambert's Mémoire (1761/1768).
Chapter 5. An anotated translation of
Lambert's Mémoire (1761/1768).
Chapter 6. Outine of Lambert's Vorläufige
Kenntnisse (1766/1770).
Chapter
6. An anotated translation of Lambert's
Vorläufige Kenntnisse (1766/1770).- Part III: The influence of Lambert's work
and the development of irrational numbers.
Chapter 8. The state of
irrationals until the turn of the century.
Chapter 9. Title to be set up.
Eduardo Dorrego López studied Mathematics (minor in Pure Mathematics) at the University of Santiago de Compostela (USC), where he also obtained his Masters degree in the Department of Algebra. He got enrolled into the Doctorate in Mathematics program at the Institute of Mathematics of the University of Seville (IMUS) obtaining his PhD in History of Mathematics under the supervision of Prof. José Ferreirós (2021). His research focuses on the development of irrational and transcendental quantities in the 18th and 19th centuries, as well as on the work of Johann Heinrich Lambert. He has carried out research stays in Oxford and Seville working primarily on Lambert and Lagrange. He has published a book (in Spanish) on Lambert and his contribution to the irrationality of pi and the circle-squaring problem (with Elķas Fuentes Guillén; College Publications, 2021), as well as a book chapter (in Spanish) on Lamberts work about non-euclidean geometries (with José Ferreirós; Plaza y Valdés, forthcoming). He is also a high school mathematics teacher. Elķas Fuentes Guillén is a Junior Star Research Fellow at the Institute of Philosophy of the Czech Academy of Sciences (FLŚ AV R) who focuses on the transition from mathematical practices that were common in the late 18th century to practices that emerged in the second half of the 19th century, as well as on the work of Bernard Bolzano. In 2018 he became the first Ibero-American researcher to be awarded the Josef Dobrovskż Fellowship by the Czech Academy of Sciences, after which he held postdoctoral positions at the Department of Mathematics of the Faculty of Sciences at UNAM and the FLŚ AV R. Among his recent publications are the forthcoming book Matematické dķlo Bernarda Bolzana ve svtle jeho rukopis (Nakladatelstvķ Filosofia), a chapter for Springers Handbook of the History and Philosophy of Mathematical Practice (2022) and The 1804 examination for the chair of Elementary Mathematics at the University of Prague (with Davide Crippa; Historia Mathematica, 2021). José Ferreirós is professor of Logic and Philosophy of Science at the Universidad de Sevilla, Spain. A former Fulbright Fellow at UC Berkeley, and member of the Académie Internationale de Philosophie des Sciences, he was founding member and first president of the APMP (Association for Philosophy of Mathematical Practices). Among his publications one finds Labyrinth of Thought (Birkhäuser, 1999), a history of set theory and its role in modern maths, the monograph Mathematical Knowledge and the Interplay of Practices (Princeton UP, 2016), an intellectual biography of Riemann (Riemanniana Selecta, CSIC, 2000), and the collective volume The Architecture of Modern Mathematics (Oxford UP, 2006).
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