In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is rejected. The parallel postulate in Euclidean geometry states, for two ...

Mathematics is distinguished from the sciences by the freedom it enjoys in choosing basic assumptions from which consequences can be deduced by applying the laws of logic. We call the basic ...

Many ancient societies knew important mathematical facts, but only one discovered mathematics—which is not a collection of accurate rules of thumb, but a body of knowledge organized deductively, by ...

If you took Geometry in High School, you almost definitely learned it as a subject based on rules and axioms discovered by the ancient Greeks. The details of this subject, which I must admit was ...

MR. FRANKLAND (NATURE, September 7) has raised the old problem of Bertrand's proof of the parallel-axiom by a consideration of infinite areas. This is perhaps the most subtle and the most specious of ...

The claim is often made that mathematical results are immutable. Once proven, they remain forever valid. But things are not so simple. There are problems at the very core of mathematics that cast a ...

“The treatise itself, therefore, contains only twenty-four pagesthe most extraordinary two dozen pages in the whole history of thought!” “How different with BolyaiJnos and Lobachvski, who claimed at ...