This monograph delves into the early development of the calculus of variations in Eighteenth Century continental Europe, showcasing the mathematics of its pioneers. By closely examining the original papers and correspondences of figures like Euler, Lagrange, and the Bernoullis, readers are engaged in the complexities of theory building. The text reveals the challenges, errors, and successes faced by these founders. Accompanied by insightful commentaries and modern perspectives, the authors enhance the original proofs and guide readers through significant advancements in the field. Beginning with Johann Bernoulli's 1697 work on the brachystochrone problem, the narrative traces key developments leading to Lagrange’s δ-calculus, addressing topics such as isoperimetrical problems, shortest lines, geodesics, and Euler’s Methodus Inveniendi and its Additamenta. The authors also reflect on the expansive growth of the calculus of variations over the centuries, hinting at areas beyond the book's scope while discussing the current state of the field. This work appeals to those studying the calculus of variations seeking deeper insights, as well as historians of science and mathematics interested in the evolution of modern calculus and analysis.
