Irrationality, Transcendence and the Circle-Squaring Problem

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 (1728-1777).- 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 Master's 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 Elías Fuentes Guillén; College Publications, 2021), as well as a book chapter (in Spanish) on Lambert's work about non-euclidean geometries (with José Ferreirós; Plaza y Valdés, forthcoming). He is also a high school mathematics teacher. 
Elías Fuentes Guillén is a Junior Star Research Fellow at the Institute of Philosophy of the Czech Academy of Sciences (FLÚ AV CR) 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 Dobrovský 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 FLÚ AV CR. Among his recent publications are the forthcoming book Matematické dílo Bernarda Bolzana ve svetle jeho rukopisu (Nakladatelství Filosofia), a chapter for Springer's Handbook of the History and Philosophy of Mathematical Practice (2022) and "The 1804 examination for the chair of Elementary Mathematics at the University of