parallel lines are lines in a plane which do not meet; that is, two lines in a plane that do not intersect or touch each other at any point are said to be parallel. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel; otherwise they are called skew lines. Parallel planes are planes in the same three-dimensional space that never meet.
Parallel lines are the subject of Euclid‘s parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism.
Given parallel straight lines l and m in Euclidean space, the following properties are equivalent:
- Every point on line m is located at exactly the same (minimum) distance from line l (equidistant lines).
- Line m is in the same plane as line l but does not intersect l (recall that lines extend to infinity in either direction).
- When lines m and l are both intersected by a third straight line (a transversal) in the same plane, the corresponding angles of intersection with the transversal are congruent.
Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are “more complicated” than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry. The other properties are then consequences of Euclid’s Parallel Postulate. Another property that also involves measurement is that lines parallel to each other have the same gradient (slope).
The definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid’s Elements. Alternative definitions were discussed by other Greeks, often as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein. Simplicius also mentions Posidonius’ definition as well as its modification by the philosopher Aganis.
At the end of the nineteenth century, in England, Euclid’s Elements was still the standard textbook in secondary schools. The traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines. These reform texts were not without their critics and one of them, Charles Dodgson (a.k.a. Lewis Carroll), wrote a play, Euclid and His Modern Rivals, in which these texts are lambasted.
One of the early reform textbooks was James Maurice Wilson’s Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, “Two straight lines that meet one another have different directions, and the difference of their directions is the angle between them.” Wilson (1868, p. 2) In definition 15 he introduces parallel lines in this way; “Straight lines which have the same direction, but are not parts of the same straight line, are called parallel lines.” Wilson (1868, p. 12) Augustus De Morgan reviewed this text and declared it a failure, primarily on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson also devotes a large section of his play (Act II, Scene VI § 1) to denouncing Wilson’s treatment of parallels. Wilson edited this concept out of the third and higher editions of his text.
Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better. The main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry, simplified and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles then all transversals must do so. Again, a new axiom is needed to justify this statement.
The three properties above lead to three different methods of construction of parallel lines.