How did William Thomson use mathematical analysis to interpret experimental results?

William Thomson, Baron Kelvin, utilized mathematical analysis to interpret experimental results primarily by identifying and applying mathematical analogies between disparate physical phenomena.12

Influenced early on by Jean-Baptiste-Joseph Fourier’s The Analytical Theory of Heat, Thomson recognized that the abstract mathematical techniques used to describe heat flow could be extended to other areas of physics. He famously established a mathematical analogy between the flow of heat in solid bodies and the flow of electricity through conductors.2

Thomson employed this analytical approach to solve complex practical problems, such as the delay of electric currents in long-distance submarine cables. By applying equations originally derived for heat flow, he successfully modeled the velocity of currents in telegraph cables, which contradicted the prevailing, less accurate practical assumptions of his time. Furthermore, he used these analytical generalizations to synthesize theories across thermodynamics, hydrodynamics, and electromagnetism, viewing physical changes as energy-related phenomena. His ability to bridge theoretical mathematics and experimental observation was fundamental to his role in the unification of 19th-century scientific thought.21