  In ancient Greek mathematics, magnitudes such as lengths were strictly
distinguished from numbers. In modern quantity calculus, a distinction is made
between quantities and scalars that serve as measures of quantities. It can be
argued that quantities should play a more prominent, independent role in modern
mathematics, as magnitudes earlier.
  The introduction includes a sketch of the development and structure of the
pre-modern theory of magnitudes and numbers. Then, a scalable monoid over a
ring is defined and its basic properties are described. Congruence relations on
scalable monoids, direct and tensor products of scalable monoids, subalgebras
and homomorphic images of scalable monoids, and unit elements of scalable
monoids are also defined and analyzed.
  A quantity space is defined as a commutative scalable monoid over a field,
admitting a finite basis similar to a basis for a free abelian group. The
mathematical theory of quantity spaces forms the basis of a rigorous quantity
calculus and is developed with a view to applications in metrology and
foundations of physics.
