This book contains d’Alembert’s mathematical texts on pure mathematics written between 1740 and 1752, when he was one of the most important members of the Academy of Sciences in Paris. The book is the fourth volume of the first series of the critical edition of d’Alembert’s collected papers prepared by a group of mathematicians, historians of sciences and philosophers (directed by Christian Gilain, a specialist in the history of mathematical analysis and a professor at Université Pierre et Marie Curie, Paris). The book starts with a comprehensive introduction, where d’Alembert’s mathematical ideas, works and results are presented and their role in the development of mathematics is analysed. The next part contains three texts with the title “Recherches sur le calcul intégral” (1745, 1746, 1747) containing d’Alembert’s fundamental contributions to integral calculus (e.g. integration of algebraic functions of a real variable, integration of rational and irrational functions, the Riccati equation and its solution, the d’Alembert equation and its singular solutions, rectification of ellipse and hyperbola, methods for solution of some systems of differential equations) together with many notes and remarks. The paper “Observation sur quelques mémoires imprimés dans le volume de l’Academie 1749” (1752) contains d’Alembert’s theory of complex numbers and his contributions to the fundamental theorem of algebra, which were inspired by Euler’s works. The fifth text “Sur les logarithmes des quantités négatives” (1752) shows d’Alembert’s concept of the logarithm of negative, and complex, numbers. The sixth of d’Alembert’s published articles “Additions aux recherches sur le calcul intégral” (1752) contains some corrections and an extension of his text on integral calculus from 1747. At the end of the book, there are three appendices containing some proofs of the fundamental theorem of algebra inspired by d’Alembert’s works, a large list of references, indexes and a list of illustrations. The book can be recommended to a wide audience; it is suitable for mathematicians, historians of mathematics and science, students and teachers.

Reviewer:

mbec