LINEs in tHe Plane: An Introduction

One cannot study lines in the plane without mentioning the Greek mathematician, Euclid. He was the author (circa 300 B.C.) of the now very famous geometric and number theoretic treatise called the Elements. It consisted of 13 separate volumes comprising the works of earlier mathematicians, together with Euclid’s own logical system of definitions, postulates (axioms), theorems, geometric constructions and rigorous deductive proofs. For almost 2000 years, Euclidean geometry was the only kind of geometry studied and used. However, by the mid-nineteenth century, both German and Russian mathematicians had created coherent geometric systems which were not Euclidean. For example, in the hyperspherical model of elliptic geometry devised by Bernhard Riemann: there are no parallel lines at all; the sum of the angles of a triangle is always strictly greater than 180 degrees; and the shortest distance between two points is not a straight line as in Euclid’s sense, but an arc along a circle. By contrast, in hyperbolic geometry (sometimes called Bolyai-Lobachevskian geometry), there may be an infinite number of lines which can be drawn parallel to a given line in the same 2-dimensional space through a point not on it.  In Euclid’s geometry, exactly one such line can be drawn.

In these tutorials on lines in the plane, you will discover just how fundamental Euclidean geometry was to the development of the geometry of the Cartesian coordinate plane.

Last Updated by SAS October 07, 2026, 16:00:47