Galileo's view of science is indebted to the teaching of the Jesuit professors at the Collegio Romano, but Galileo's concept of mathematical physics also corresponds to that of Giovan Battista Benedetti. Lacking documentary evidence that would connect Benedetti directly with the Jesuits, or the Jesuits with Benedetti, I infer a common source: the ‘Spanish connection’, that is, Domingo de Soto. I then give indications that the fourteenth-century work at Oxford and Paris on calculationes was transmitted via Spain and Portugal to Rome and other centers where Jesuits had colleges, and figured in the rise of mathematical physics at the beginning of the seventeenth century. A result of these researches is their vindication of Duhem, as contrasted with Koyré, on the origins of modern mechanics.