Non-Euclidean geometry Space

spherical geometry similar elliptical geometry. on sphere (the surface of ball) there no parallel lines.

euclid s elements contained 5 postulates form basis euclidean geometry. 1 of these, parallel postulate, has been subject of debate among mathematicians many centuries. states on plane on there straight line l1 , point p not on l1, there 1 straight line l2 on plane passes through point p , parallel straight line l1. until 19th century, few doubted truth of postulate; instead debate centered on whether necessary axiom, or whether theory derived other axioms. around 1830 though, hungarian jános bolyai , russian nikolai ivanovich lobachevsky separately published treatises on type of geometry not include parallel postulate, called hyperbolic geometry. in geometry, infinite number of parallel lines pass through point p. consequently, sum of angles in triangle less 180° , ratio of circle s circumference diameter greater pi. in 1850s, bernhard riemann developed equivalent theory of elliptical geometry, in no parallel lines pass through p. in geometry, triangles have more 180° , circles have ratio of circumference-to-diameter less pi.