A History of Units and Dimensional Analysis

A modern scientist would write d = vt, meaning distance equals velocity times time. In the works of Galileo, Huygens, and Newton, one never finds a numerical value for a velocity, an acceleration, or a force. Instead, results are stated in terms of ratios (as in the above quotation). Any calculation that one could perform in the modern way, using equations that relate dimensioned quantities, could conceivably be performed Galileo’s way, using ratios. However, the modern way has significant advantages: it allows for conversion between different systems of measurement, e.g., meters and feet; it is much more concise; and it enables dimensional analysis, where one infers the form of a physical equation without doing any physics. This paper chronicles the development of the idea of units and dimensions in physics. There was a gradual transition from Galileo’s ratios to equations that relate dimensioned quantities. The first man to have a modern understanding of dimensioned quantities was Fourier, who stated these principles in the later editions of The Analytic Theory of Heat [1]. We briefly discuss some of the advances that occurred after Fourier, namely the technique of dimensional analysis, where one infers the form of a equation using dimensional considerations on the quantities involved.