Book review: Irrationality, Transcendence and the Circle-Squaring Problem: An Annotated Translation of J. H. Lambert's Vorlaufige Kenntnisse and Memoire

This review examines Irrationality, Transcendence and the Circle-Squaring Problem (2nd ed., 2024) by Eduardo Dorrego Lopez and El & imath;as Fuentes Guillen, offering the first complete annotated English translations of Johann Heinrich Lambert's two foundational texts on the irrationality of pi: the semi-popular Vorlaufige Kenntnisse (1766/1770) and the rigorous Memoire (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 pi 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.