This paper main goal is to depict the generative principles of 16th -18th century Portuguese colonial urban design described from its Pythagorean-Euclidean geometrical genesis and correspondent logical rules and operations. These origins were found in Portuguese compendiums and treaties on practical geometry, architectural and military engineering from the sixteenth through the eighteenth century. The study attempts to show that rigorous operative geometrical discourse is inseparable from social knowledge, whereby form is the operative result of abstract mental processes and logical visual reasoning associated to ideas of growth and reproduction of order.