Using H. Whitney's algebra of physical quantities and his definition of a similarity transformation, a family of similar systems (R. L. Causey [3] and [4]) is any maximal collection of subsets of a Cartesian product of dimensions for which every pair of subsets is related by a similarity transformation. We show that such families are characterized by dimensionally invariant laws (in Whitney's sense, [10], not Causey's). Dimensional constants play a crucial role in the formulation of such laws. They are represented (...) as a function g, known as a system measure, from the family into a certain Cartesian product of dimensions and having the property gφ =φ g for every similarity φ . The dimensions involved in g are related to the family by means of certain stability groups of similarities. A one-to-one system measure is a proportional representing function, which plays an analogous role in Causey's theory, but not conversely. The present results simplify and clarify those of Causey. (shrink)

Suppose that entities composed of two independent components are qualitatively ordered by a relation that satisfies the axioms of conjoint measurement. Suppose, in addition, that each component has a concatenation operation that, together either with the ordering induced on the component by the conjoint ordering or with its converse, satisfies the axioms of extensive measurement. Without further assumptions, nothing can be said about the relation between the numerical scales constructed from the two measurement theories except that they are strictly monotonic. (...) An axiom is stated that relates the two types of measurement theories, seems to cover all cases of interest in physics, and is sufficient to establish that the conjoint measurement scales are power functions of the extensive measurement scales. (shrink)

This paper presents a representational theory of derived physical measurements. The theory proceeds from a formal definition of a class of similar systems. It is shown that such a class of systems possesses a natural proportionality structure. A derived measure of a class of systems is defined to be a proportionality-preserving representation whose values are n-tuples of positive real numbers. Therefore, the derived measures are measures of entire physical systems. The theory provides an interpretation of the dimensional parameters in a (...) large class of physical laws, and it accounts for the monomial dimensions of these parameters. It is also shown that a class of similar systems obeys a dimensionally invariant law, which one may safely subject to a dimensional analysis. (shrink)