Exegesis of Sect. II and III.A from “Fundamentals of the Mechanics of Continua” by E. Hellinger

This is a second exegetic essay on the fundamental review article DIE ALLGEMEINEN ANSÄTZE DER MECHANIK DER KONTINUA in the Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. IV-4, Hft. 5 (1913) by Ernst Hellinger which concerns the translation and the commentary of pp. 629–662. Within these pages, the setting of the basic constitutive equations for field theories, whose formulations are based on the principle of virtual work or on the principle of stationary action, are discussed. The interest for a contemporary reader for the herein presented subject is still substantial, as this article clearly contains some considerations and visions being still topical. However, there is also an epistemological interest in examining it from the point of view of a historian of science. Indeed, it represents an available but forgotten source of an important piece of mechanical sciences. Available, because it is still present in our libraries in its complete form, but forgotten because, being written in German by a Jewish refugee escaped to the United States, has been ignored by the main stream of the dominant groups in continuum mechanics. The ideas by Hellinger and the German school of continuum mechanics (remark that this school includes, as we have discovered reading this article, even Gauss himself!) were lost or dramatically deformed in translation. We believe that the destiny of Hellinger's paper is an evidence supporting Lucio Russo's view about history of science. Our aim is to trace the origins of current ideas of mechanical sciences to their original sources.