  Gabrio Piola's scientific papers have been underestimated in the
mathematical-physics literature. Indeed a careful reading of them proves that
they are original, deep and far reaching. Actually -even if his contribution to
mechanical sciences is not completely ignored- one can undoubtedly say that the
greatest part of his novel contributions to mechanics, although having provided
a great impetus and substantial influence on the work of many preminent
mechanicians, is in fact generally ignored. It has to be remarked that authors
[10] dedicated many efforts to the aim of unveiling the true value of Gabrio
Piola as a scientist; however, some deep parts of his scientific results remain
not yet sufficiently illustrated. Our aim is to prove that non-local and higher
gradient continuum mechanics was conceived already in Piola's works and to try
to explain the reasons of the unfortunate circumstance which caused the erasure
of the memory of this aspect of Piola's contribution. Some relevant
differential relationships obtained in Piola [Piola, 1845-6] are carefully
discussed, as they are still nowadays too often ignored in the continuum
mechanics literature and can be considered the starting point of Levi-Civita's
theory of Connection for Riemannian manifolds.
