Horatio Scott Carslaw was a key figure in the development of the mathematical tools that are now used by thermogeologists and hydrogeologists. Carslaw was a student of Lord Kelvin at the University of Glasgow, and can be regarded as a link between Kelvin’s application of Fourier’s mathematics to geological problems and our present practitioners. Charles Vernon Theis and Clarence Lubin cited Carslaw’s mathematics as a key influence in the development of the Theis well function. Carslaw’s protégé, John Conrad Jaeger, took Carslaw’s mathematics firmly into the geological realm. This paper examines Carslaw’s wide-reaching influence and the development of the line source ‘exponential integral’ function, noting the contributions made by other scientists, including Kelvin, Whitehead and Lord Rayleigh.