![]() ![]() This can be proved using a familiar construction: given a line l and a point P not on l, drop the perpendicular m from P to l, then erect a perpendicular n to m through P. More precisely, given any line l and any point P not on l, there is at least one line through P which is parallel to l. As the proof only requires the use of Proposition 27 (the Alternate Interior Angle Theorem), it is a valid construction in absolute geometry. Proposition 31 is the construction of a parallel line to a given line through a point not on the given line. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Legendre theorem, which states that the sum of the measures of the angles in a triangle has at most 180°. ![]() Indeed, in Euclid's Elements, the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. It might be imagined that absolute geometry is a rather weak system, but that is not the case. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The term was introduced by János Bolyai in 1832. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not sufficient as a basis of Euclidean geometry, other systems, such as Hilbert's axioms without the parallel axiom, are used. Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives.
0 Comments
Leave a Reply. |