  The first of the two related papers analising and explaining the origin,
manifestations and parodoxical features of the quantum potential (QP) from the
non-relativistic and relativistic point of view. QP arises in the quantum
Hamiltonian, under various procedures of quantization of the systems whose
Hamilton functions are the positive-definite quadratic forms in momenta with
coefficients depending on the coordinates in (n-dimensional) configurational
space (natural systems). Owing to the Riemannian structure thus introduced in
the space, the result of quantization is considered as quantum mecanics (QM) of
a particle. Contradiction of QP to the Principles of General Covariance and
Equivalence is discussed. It is found that actually the historically first
Hilbert space based quantization by E. Schr\"odinger (1926), after revision in
the modern framework of QM, also leads to QP in the form that B. DeWitt had
been found 26 years later. Efforts to avoid QP or reduce its drawbacks are
discussed. The general conclusion is that some form of QP and a violation of
the principles of general relativity induced by it are apparently inevitable in
the non-relativistic QM. It is shown also that Feynman quantization singles out
two versions of QP, which both determine two bi-scalar propagators which fix
two different algorithms of path integral calculation. In the accompanying
paper under the same general title and the subtitle "The Relativistic Point of
View", relation of the non-relativistic QP to the quantum theory of the scalar
field non-minimally coupled to the curved space-time metric is considered.
