LINE, primarily of thread or cord, a succession of objects in a row, a mark or stroke, a course in any particular direction.
In mathematics several definitions of the line may be framed. The synthetical genesis of a line from the notion of a point is the basis of Euclid's definition, (ypailizi 5i /.4KOS 6.7rXars "a line is widthless length"), and in a subsequent definition he affirms that the boundaries of a line are points. The line appears in definition 6 of the Elements as the boundary of a surface. An other synthetical definition, also treated by the ancient Greeks, but not by Euclid, regards the line as generated by the motion of a point, and, in a similar manner, the surface was regarded as the flux of a line, and a solid as the flux of a surface. Analytical definitions, although not finding a place in the Euclidean treat ment, have advantages over the synthetical derivation. Thus the boundaries of a solid may define a plane, the edges a line, and the corners a point ; or a section of a solid may define the surface, a section of a surface the line, and the section of a line the point.
The line has only extension and is unidimensional; and the point, having only position has no dimensions.
The definition of a straight line is a matter of much complex ity. Euclid defines it as the line which lies evenly with respect to the points on itself. Archimedes defines it as the shortest dis tance between two points. The definition, "a straight line is one which when rotated about its two extremities does not change its position," is essentially due to Heron (q.v.).
In analytical geometry the right line is always representable by an equation, or equations, of the first degree; thus in cartesian coordinates of two dimensions the equation is of the form Ax-I-By+C=o, in triangular coordinates, Ax+By+Cz=o. In three-dimensional coordinates, the line is represented by two linear equations. (See ANALYTIC GEOMETRY ; LINE GEOMETRY; CURVE; CURVES, SPECIAL.)