How did non-Euclidean geometry lead mathematicians to examine Euclid’s hidden assumptions?

The discovery of non-Euclidean geometries in the 19th century fundamentally challenged the long-held belief that Euclidean geometry was the only system accurately describing physical reality. By demonstrating that altering Euclid’s parallel postulate resulted in perfectly consistent mathematical systems, mathematicians like Nikolay Lobachevsky and János Bolyai broke the absolute identification of mathematics with spatial intuition.12

This realization prompted a shift toward separating abstract mathematics from intuition, leading scholars to scrutinize the Elements for hidden assumptions. Mathematicians identified that Euclid had relied on unstated premises, such as visual diagrams to define concepts like "betweenness" or "inside," and questionable procedures like superposition to prove congruency.34

To address these "blemishes," mathematicians like Moritz Pasch and David Hilbert sought to rebuild geometry on a more rigorous foundation. Hilbert’s The Foundations of Geometry (1899) formalized the system by using undefined concepts and explicit axioms, a development that became central to the study of metamathematics and the broader quest to establish consistent foundations of mathematics.342