| It can be shown that if Euclidean geometry is internally consistent, then the alternative non-Euclidean geometries are consistent as well. |
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| By chapter two he is proving Pythagoras' theorem from first principles and introducing non-Euclidean geometry. |
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| It follows from Kant's view that we know a priori that non-Euclidean geometry cannot be applied in physics. |
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| It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry. |
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| A modern realist would say that what mathematicians hadn't realised is that, as well as Euclidean geometry, there also exists non-Euclidean geometry. |
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| Beltrami in this 1868 paper did not set out to prove the consistency of non-Euclidean geometry or the independence of the Euclidean parallel postulate. |
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| Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing. |
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| The wormhole is invoked as a way of describing the concrete geographies of positionality and their non-Euclidean relationship to the Earth's surface. |
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| The approach that concentrates on non-Euclidean geometry is ideal for students who already have a mastery of Euclidean geometry, but it cannot replace such a mastery. |
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| Ivan Karamazov sees his inability to grasp non-Euclidean geometry as evidence that he can't understand God. |
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| Riemannian geometry is a non-Euclidean geometry that studies local properties of a smooth manifold. |
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| It is also known as a type of non-Euclidean Geometry, being in many respects similar to Euclidean Geometry. |
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| Did people have some inkling of non-Euclidean geometry, some premonition that the parallel postulate might be false? |
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| If this homeomorphism is not Identical transformation, then R will represent a non-Euclidean space. |
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| He also realised that there were an infinite number of non-euclidean geometries and this, Taurinus claimed, was highly significant. |
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| He also studied birational contact transformations and non-euclidean and non-archimedean geometries. |
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| It was not until the 19th century that this postulate was dropped and non-euclidean geometries were studied. |
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| In particular he made contributions of major importance in the theory of polytopes, non-euclidean geometry, group theory and combinatorics. |
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| Geometry had began to lose its 'metric' character with projective and non-euclidean geometries being studied. |
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| His work on non-euclidean geometries was used by Einstein in his general theory of relativity. |
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| This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent. |
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| Much of his career is spent working on physics and non-euclidean geometry. |
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