Focusing on Vitruvius' De Architectura Libri Decem, the oldest extant architectural treatise available today, this chapter traces the ways in which mathematical concepts were embedded in the architectural design process in general and more particularly in the creation of Doric, Ionic, and Corinthian temples. In essence, Vitruvius provides a way of understanding how mathematics was used by the architects of the Classical Greek and Roman worlds to both solve practical problems and to create buildings which conformed to the highest aesthetic aspirations of the era. The study of the later typology of Roman amphitheaters provides a clear example of the changing role of geometry in architectural planning, as it shows innovative patterns with respect to Greek tradition and close connections between progress in mathematics and modernity in architecture.