  This article includes a survey of the historical development and theoretical
structure of the pre-modern theory of magnitudes and numbers. In Part 1, work,
insights and controversies related to quantity calculus from Euler onward are
reviewed. In Parts 2 and 3, we define scalable monoids and, as a special case,
quantity spaces; both can be regarded as universal algebras. Scalable monoids
are related to rings and modules, and quantity spaces are to scalable monoids
as vector spaces are to modules. Subalgebras and homomorphic images of scalable
monoids can be formed, and tensor products of scalable monoids can be
constructed as well. We also define and investigate congruence relations on
scalable monoids, unit elements of scalable monoids, basis-like substructures
of scalable monoids and quantity spaces, and scalar representations of elements
of quantity spaces. The mathematical theory of quantity spaces is presented
with a view to metrological applications. This article supersedes
arXiv:1503.00564 and complements arXiv:1408.5024.
