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This lecture explores the foundational geometric propositions of Euclid and Vitruvius, focusing on the construction of architectural figures and the urban planning concepts presented in Vitruvius' De Architectura. The instructor delves into the historical context, highlighting the transition from the Republic to the Roman Empire and the practical applications of geometry in architecture. The lecture also discusses the significance of the gnomon in sundials, the concept of symmetry in ancient mathematics, and the construction of urban layouts to protect cities from adverse winds.