An ancient proverb proclaims that "Anything worth knowing is already known." But capitalizing on that knowledge may require rigorous thinking. According to Roche (1998), this is the basis of dimensional analysis, a field which can be fascinating, while also being frustratingly inconsistent linguistically and convoluted, conceptually.
In the nineteenth century, dimensional explorations of mechanical vibration, electrical circuits, and viscous flows involved explicit homogeneity tests, which are checks for invariance under a change in unit size. It was then that researchers came to some startling conclusions about measurement dimensions: "Simple consideration of homogeneity sometimes allows the calculation a priori of a formula, apart from a numerical coefficient" (p. 211). It happens not infrequently that results in the form of laws , i.e., scientific regularities, are put forward as novelties on the basis of elaborate experiments, which might have been predicted a priori after a few minutes' consideration. All that was required was "to specify clearly all the quantities on which the desired result may reasonably be supposed to depend" (p. 211).
According to Roche, the current state of dimensional analysis is full of mysteries, because it has ambiguous terminology, lacks a critical history, and is scattered over a considerable and unsystematic literature. It is subtle and has not been recognized for being what it is, a new and substantial methodology. Roche credits Rayleigh's and Vaschy's homogeneity test as properly exploiting "gauge invariance and dimensional homogeneity for the first time in physics" constituting a "remarkable new methodological tool of mathematical physics" (p. 214).
Since Rasch measurement and dimensional analysis share a common focus on tests of invariance and data homogeneity, they are probably closely related, if not mathematically identical. The establishment of a firm connection between these two areas of research would benefit dimensional analysis by making the advantages and value associated with its subtleties and substantive contributions more apparent. Rasch analysis would benefit by being more firmly connected with mathematical applications that emphasize the identification and delineation of variable dimensions.
William P. Fisher, Jr.
Roche, J. (1998). The Mathematics of Measurement: A Critical History. London: The Athlone Press.
Dimensional Analysis and Rasch Fisher, W.P. Jr. Rasch Measurement Transactions, 2001, 15:2 p. 822
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