This paper recovers an important, century-old debate regarding the methodological and metaphysical foundations of dimensional analysis. Consideration of Richard Tolman's failed attempt to install the principle of similitude---the relativity of size---as the founding principle of dimensional analysis both clarifies the method of dimensional analysis and articulates two metaphysical positions regarding quantity dimensions. Tolman's position is quantity dimension fundamentalism. This is a commitment to dimensional realism and a set of fundamental dimensions which ground all further dimensions. The opposing position, developed primarily (...) by Bridgman, is quantity dimension conventionalism. Conventionalism is an anti-realism regarding dimensional structure, holding our non-representational dimensional systems have basic quantity dimensions fixed only by convention. This metaphysical dispute was left somewhat unsettled. It is shown here that both of these positions face serious problems: fundamentalists are committed to surplus dimensional structure; conventionalists cannot account for empirical constraints on our dimensional systems nor the empirical success of dimensional analysis. It is shown that an alternative position is available which saves what is right in both: quantity dimension functionalism. (shrink)