"Avanti al fabbricare, nel fabbricare e poi che si è fabbricato" With these words, the Sienese mathematician Teofilo Gallaccini (1564-1641) bequeathed later generations with a concise idea of his vision of architecture as a working tool, a vision that betrays critical affinities with today's theories on architecture. In endowing the practice of building with a scientific foundation based largely on mathematics and geometry, Gallaccini asserts that the architect is able to bring his expertise to bear on each and every aspect of design and construction, thereby endowing his craft with necessary cogency, whereas a lack of method leads to poor construction practices, makeshift alterations, and restoration schemes of dubious workmanship. The tradition of Sienese building both before and during the mathematician's lifetime are the dual theme discussed: from the time of their conception, through to the concerted effort to harmonize alterations and new developments.