The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter usually applies to a form of argument that does not comply with the valid inference rules of logic, whereas the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (e.g., the introduction of Pasch's axiom of Euclidean geometry, the five colour theorem of graph theory). "Pseudaria", an ancient lost book of false proofs, is attributed to Euclid.