On the role of mathematics and mathematical knowledge in the invention of Vannevar Bush's early analog computers

The technological, professional, and intellectual context out of which the development of the continuous integraph or product integraph-as the immediate forerunner of Vannevar Bush's differential analyzer-evolved is outlined. In particular the affinity between transmission line research and teaching at MlT's electrical engineering department under Bush's guidance, on the one hand, and the creation of the product integraph for evaluating integrals, which resulted from the appropriate differential equations of the transmission problems, on the other hand, is detailed. I emphasize Bush's perception of promoting engineering by easing the applied mathematics in this field as it appeared in his contribution to the development of operational circuit analysis as an appropriate engineering mathematics as well as in creating analog machinery that was inspired by the formulation of transmission line problems in terms of that very operational methods after Oliver Heaviside.