Making sense of absolute measurement: James Clerk Maxwell, William Thomson, Fleeming Jenkin, and the invention of the dimensional formula

- Maxwell and Jenkin invented the dimensional formula in the early 1860s for unit conversion.
- Electrical measurement formed the main context for dimensions from then until at least the 1890s.
- The conceptual-historical development of an operational view of dimensions is described.
- Thomson and Maxwell did not obviously hold a dynamical reductionist theory of dimensions.
- An alliance between energy physics and telegraph engineering was hard to achieve and maintain.

During the 1860s, the Committee on Electrical Standards convened by the British Association for the Advancement of Science (BAAS) attempted to articulate, refine, and realize a system of absolute electrical measurement. I describe how this context led to the invention of the dimensional formula by James Clerk Maxwell and subsequently shaped its interpretation, in particular through the attempts of William Thomson and Fleeming Jenkin to make absolute electrical measurement intelligible to telegraph engineers. I identify unit conversion as the canonical purpose for dimensional formulae during the remainder of the nineteenth century and go on to explain how an operational interpretation was developed by the French physicist Gabriel Lippmann. The focus on the dimensional formula reveals how various conceptual, theoretical, and material aspects of absolute electrical measurement were taken up or resisted in experimental physics, telegraphic engineering, and electrical practice more broadly, which leads to the conclusion that the integration of electrical theory and telegraphic practice was far harder to achieve and maintain than historians have previously thought. This ultimately left a confusing legacy of dimensional concepts and practices in physics.