In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate): In a plane, given a line...

fifth postulate (Playfair's axiom). Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary...

succeeded by Hymn Playfair Project TS Playfair, a Canadian sail training vessel Playfair's axiom named after John Playfair Playfair cipher, a manual encryption...

wide audience. Playfair's textbook Elements of Geometry made a brief expression of Euclid's parallel postulate known now as Playfair's axiom. In 1783 he...

postulate: The sum of the angles of a triangle is two right angles. Playfair's axiom: Given a straight line and a point not on the line, exactly one straight...

is given by Playfair's axiom and Euclid I.30 (see Playfair's axiom#Transitivity of parallelism). For instance, Hilbert uses Playfair's axiom while Birkhoff...

the axioms, an affine plane is an object of study belonging to incidence geometry. They are non-degenerate linear spaces satisfying Playfair's axiom. The...

to which is associated a set of lines, which satisfy some axioms (such as Playfair's axiom). Affine geometry can also be developed on the basis of linear...

geometries". These theorems along with their alternative postulates, such as Playfair's axiom, played an important role in the later development of non-Euclidean...

however, modern treatments of geometry use Playfair's axiom instead. To prove proposition 29 assuming Playfair's axiom, let a transversal cross two parallel...