Euclid's Fifth Postulate on parallel lines was widely debated since its first formulation, and we know that some proofs of it were attempted already in antiquity. The late Renaissance and the Early Modern Age witness an explosion of interest in the topic, and a number of treatises were explicitly dedicated to its demonstration, or to a proof of its unprovability. However, most of these attempts had no philosophical concerns at all, and the discussions about the famous Postulate were limited to its mathematical import and significance in the overall axiomatization of Euclidean geometry. In the 18th century, though, some philosophers and mathematicians began to question the metaphysical relevance of the Postulate toward a description of physical and real space, and these discussions will eventually lead to the famous late-19th- and early-20th-century debate on the nature of space in connection with the developments of non-Euclidean geometries and the General Theory of Relativity. I would like to attempt a short description of the 18th-century transition from a mathematical problem to a metaphysical one, and to relate this transformation with the general conception of a geometrical space that is coming to life in that age. In particular, I will refer to the works of John Wallis, Gottfried Leibniz, Thomas Reid and Immanuel Kant.
