Evaluation of Scientific Sources in Mechanics

In this introductory chapter we present the motivations that prompted the authors and editors to work on this volume. The fil rouge followed in the discussion presented below is based on the following consideration: the study of the genesis of mathematical and mechanical theories does not have a merely philological purpose, but can influence and even inspire the development of new original ideas. We present some examples that clarify our thesis: the development of the model of planetary motion and a historical-critical study of the development of Continuum Mechanics. Clearly understanding the main errors and misunderstandings that other than pure research logics have introduced in the scientific discussion is the only way to learn a serious and rigorous approach to Science. An idea that has guided us in the development of this chapter is that fragmentation of culture brings to inability to deal with complexity. The only way to study and, above all, understand the complexity of our world is through a unified vision of knowledge.

In this chapter, we present the translation of the main excerpts of Heiberg’s Prolegomena to his Archimedes Edition. This text was originally written in Latin [Heiberg, J. L. (1910). Archimedis Opera omnia cum commentariis Eutocii: Vol. 1-3. In aedibus BG Teubneri.] and contains the evidence of interesting phenomena in the transmission of ancient scientific texts. Considering the nature of the present work, which is rather interested in problems concerning the transmission and degradation of scientific knowledge through the centuries, the following translation concerns mainly the pages in which issues related to this type of problem are addressed. We have, however, omitted the translation of extremely technical parts of the philologist's work, which are beyond the scope of the present work.

This chapter contains a side-by-side translation from German to English of E. Hellinger’s fundamental review article “Die allgemeinen Ansätze der Mechanik der Kontinua”, which appeared 1913 in the “Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen”, Bd.~IV-4, Hft. 5. The chapter closes with a thorough discussion about selected topics treated in the article.

In this chapter, we look in detail at the aspects concerning the transmission of scientific knowledge of the Italian school of Continuum Mechanics, mainly headed by Gabrio Piola, which strongly supported the point of view of Archytas of Tarentum as rediscovered by D’Alembert and Lagrange. The process of systematically removing references to the name of Gabrio Piola in Continuum Mechanics (and part of his results) is just one of many examples of how some social groups have, over the centuries and in different cultural fields, rewritten more or less relevant parts of the cultural knowledge of a society. Specifically, Gabrio Piola’s contribution to mechanical sciences has been greatly underestimated in both the more theoretical mathematical-physics literature and in the more applied and engineering oriented one. We remark, in the discussion presented in this Chapter, that at the basis of this phenomenon one can always find common features as a sectarian vision of cultural progress or the conviction that the point of view of the own social group is clearly superior to that of all the others.

In this chapter, we present a historical survey on the Principle of Virtual Work as the guiding principle for constructing mathematical models to describe and predict phenomena. In particular, we want to make the reader aware of the development of two main approaches toward the formulation of new models, the first using the Principle of Virtual Work, while the second using the balance of some quantities to be suitably chosen. This dualism, which was probably already present in Hellenistic times, has been nurtured in modern times by some of the most important scientists of the last centuries. We think it is worthwhile to study their work, not only to attempt a historically-founded authorship attribution of the underlying ideas, but especially so that their work can guide us while developing new theories. Our discussion, which begins from the available fragmentary sources dealing with Hellenistic Mechanics, focuses on the efforts by D'Alembert and Lagrange, which produced a modern comprehensive formulation of Classical Mechanics based on the Principle of Virtual Work. Referring to these historical instances, we advocate the effectiveness of using the Principle of Virtual Work, as opposed to balance laws, as a basic postulate in formulating new models, arguing that it allows using a minimal set of a priori and clear conjectures, avoiding the need for a posteriori ad hoc - hence often logically incompatible - assumptions based on "physical intuition". As a paradigmatic case, we present the formulation of N-th Gradient Continuum Mechanics developed in the pioneering - but not yet widely known - work of Piola, where a wise use of the Principle of Virtual Work leads to a theory that is more general than the one developed starting from Cauchy’s "tetrahedron argument", based on the Law of Balance.