When Leon Battista Alberti arrived in Florence in the early fifteenth century, he was struck by the Brunelleschi–designed dome of Santa Maria del Fiore. The size of the dome, never before achieved, would have been impressive enough, but Alberti would have also seen the large machinery required for its construction: cranks, cranes, pulleys, levers, and so on. The knowledge required for engineering a dome of that magnitude and for devising the machinery necessary for its construction, was based on considerable awareness of Archimedean statics, Aristotelian and Vitruvian mechanics, as well as Euclidean mathematics. Alberti recognized these expertise in Brunelleschi’s work and set about the task of persuading other architects, builders, designers, and his patrons, of the value of such work to the new and traditional institutional settings of princely courts, universities, and academies. In other words, Alberti located theoretical mathematics and the empirical work of master–builders at the nexus of cultural, political, scientific, and technological change. His ability to do this successfully set the tone for the innovations in military technology and hydraulic engineering in the sixteenth and seventeenth centuries, especially the ways in which practical and theoretical mathematics combined and gained the social status for their long–term sustainability in Europe’s cultural, political, and intellectual settings.