  Newton in his Principia gives an ingenious generalization of the Hellenistic
theory of ratios and inspired experimentally gives a tensor-like definition of
multiplication of quantities measured with his ratios. An extraordinary feature
of his definition is generality: namely his definition a priori allows non
commutativity of multiplication of measured quantities, which may give a
non-trivial linkage to experimental facts subject to quantum mechanics
discovered some two hundred years later. Mathematical scheme he introduces with
this ingenious definition is closely related to the contemporary approach in
spectral geometry. His definition reveals in particular that commutativity of
the multiplication of quantities with physical dimension has the status of
experimental assumption and does not have to be fulfilled in reality, although
neither the mathematical tools nor experimental evidence could allow Newton to
carry out the case when the multiplication is noncommutative. We present a
detailed analysis of his definition as well as a linkage to the theory of
representations of algebras with involution (and with normal forms of von
Neumann algebras and Jordan Banach algebras).
