Abstract : D'Alembert is the author of a fourth and last Treaty on Hydrodynamics, which is little known by historians of sciences: the 57th Memoir of the Volume VIII of his ‘Opuscules mathématiques' (1780). In this Memoir he mostly responds to a text published by Borda in 1766, in which the latter strongly criticized the basic concepts and principles of D'Alembert's ‘Traité des fluides' (1744). This dispute between both scientists arose during the crisis which affected hydrodynamics sciences concerning the concordance between theory and experiments. The partial differential equations obtained by D'Alembert in his ‘Essai d'une nouvelle théorie de la résistance des fluides' (1752) and Euler in its 1755 Memoirs could not have been solved. The one-dimensional approach developed by Daniel Bernoulli in the ‘Hydrodynamica' (1738), then adopted by Jean Bernoulli in the ‘Hydraulica' (1743) and D'Alembert in the ‘Traité des fluides' (1744) is based on an approximation, the hypothesis of parallel slices, which is too far from available experimental results. Borda tried to fill this gap in 1766 to lead this research field out of the impasse.
In the 57th Memoir, D'Alembert, who had been directly criticized by Borda, gives his opinion on the topic, responds implicitly to Borda critics, comes back on its previous texts and proposes new theoretical research approach. The study of this treaty on mathematical and physical point of view leads to a new analysis of the principles and concepts at the basis of his theory on the fluids, most of them have indeed only be partially studied by historians. It also enable us to compare his works with the ones of the main other stakeholders in hydrodynamics at this time, i.e. Daniel Bernoulli, Jean Bernoulli, Euler and Lagrange, and to give an overview of the consideration they have received and their sustainability.