Ideas of Space: Euclidean, Non-Euclidean, and Relativistic

absolute algebra angle sum argument assumption asymptotic axes axis Beltrami Bolyai and Lobachevskii Bonola called centre circle clock co-ordinates common perpendicular consider constant construction cos² cosh curvature curve deduce defined discussed distance draw Einstein elliptic geometry equal equations equidistant Euclid Euclidean geometry existence figure follows formulae Gauss geodesics given Greek grid horocycle horosphere hyperbolic hypothesis incommensurable intrinsic Jean Lambert length light cone magnitudes mathematicians mathematics measure meet metric motion move non-Euclidean geometry numbers observer obtain origin parallel lines parallel postulate path pencil plane principal curves Proclus proof properties proved pseudosphere Pythagorean Pythagorean triple quadrilateral radius ratio refutation result Riemann right angles right-angled triangle rotating Saccheri segment sides similar sinh space space-time special relativity sphere spherical geometry spherical trigonometry square straight line surface tangent Taurinus theorem theory triangle ABC velocity