Book Review: Irrationality, Transcendence and the Circle-Squaring Problem: An Annotated Translation of J. H. Lambert’s Vorläufige Kenntnisse and Mémoire (2nd ed.)

This review examines Irrationality, Transcendence and the Circle-Squaring Problem (2nd ed., 2024) by Eduardo Dorrego López and Elías Fuentes Guillén, offering the first complete annotated English translations of Johann Heinrich Lambert’s two foundational texts on the irrationality of π: the semi-popular Vorläufige Kenntnisse (1766/1770) and the rigorous Mémoire (1761/1768). The review situates these translations within Lambert’s remarkable and largely underappreciated life—a self-taught polymath who rose from a tailor’s workshop in Mulhouse to the Berlin Academy of Sciences—and gives careful attention to the proof of irrationality of π via continued fractions, placing it in dialogue with the Niven–Hermite integral approach, the modern theory of continued fractions as developed by Niven Niv05, and the broader literature on irrationality and transcendence. The volume is assessed as an essential contribution to the historiography of number theory.