13 Facts About Parallel postulate

In geometry, the parallel postulate, called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.

Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates.

However, if the definition is taken so that parallel lines are lines that do not intersect, or that have some line intersecting them in the same angles, Playfair's axiom is contextually equivalent to Euclid's fifth postulate and is thus logically independent of the first four postulates.

Ibn al-Haytham, an Arab mathematician, made an attempt at proving the parallel postulate using a proof by contradiction, in the course of which he introduced the concept of motion and transformation into geometry.

Parallel postulate formulated the Lambert quadrilateral, which Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral", and his attempted proof contains elements similar to those found in Lambert quadrilaterals and Playfair's axiom.

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Parallel postulate recognized that three possibilities arose from omitting Euclid's fifth postulate; if two perpendiculars to one line cross another line, judicious choice of the last can make the internal angles where it meets the two perpendiculars equal .

Parallel postulate showed that the acute and obtuse cases led to contradictions using his postulate, but his postulate is known to be equivalent to the fifth postulate.

Parallel postulate considered the cases of what are now known as elliptical and hyperbolic geometry, though he ruled out both of them.

Parallel postulate worked with a figure that today we call a Lambert quadrilateral, a quadrilateral with three right angles .

Parallel postulate quickly eliminated the possibility that the fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle.

Parallel postulate had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of a model of the acute case on a sphere of imaginary radius.

However, the argument used by Schopenhauer was that the Parallel postulate is evident by perception, not that it was not a logical consequence of the other axioms.

Parallel postulate is equivalent, as shown in, to the conjunction of the Lotschnittaxiom and of Aristotle's axiom.