Line

Line, a word of which the numerous meanings may be deduced from the primary ones of thread or cord, a succession of objects in a row, a mark or stroke, a course or route in any particular direction. The word is derived from the Lat. linea, where all these meanings may be found, but some applications are due more directly to the Fr. ligne. Linea, in Latin, meant originally “something made of hemp or flax,” hence a cord or thread, from linum, flax. “Line” in English was formerly used in the sense of flax, but the use now only survives in the technical name for the fibres of flax when separated by heckling from the tow (see Linen). The ultimate origin is also seen in the verb “to line,” to cover something on the inside, originally used of the “lining” of a garment with linen.

In mathematics several definitions of the line may be framed according to the aspect from which it is viewed. The synthetical genesis of a line from the notion of a point is the basis of Euclid’s definition, γραμμὴ, δὲ μῆκος ἀπλατές (“a line is widthless length”), and in a subsequent definition he affirms that the boundaries of a line are points, γραμμῆς δὲ πέρατα σημεῖα. The line appears in definition 6 as the boundary of a surface: ἐπιφανείας δὲ πέρατα γραμμαἰ (“the boundaries of a surface are lines”). Another synthetical definition, also treated by the ancient Greeks, but not by Euclid, regards the line as generated by the motion of a point (ῥύσις σημείου), and, in a similar manner, the “surface” was regarded as the flux of a line, and a “solid” as the flux of a surface. Proclus adopts this view, styling the line ἀρχή in respect of this capacity. Analytical definitions, although not finding a place in the Euclidean treatment, 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 notion of dimensions follows readily from either system of definitions. The solid extends three ways, i.e. it has length, breadth and thickness, and is therefore three-dimensional; the surface has breadth and length and is therefore two-dimensional; the line has only extension and is unidimensional; and the point, having neither length, breadth nor thickness but only position, has no dimensions.

The definition of a “straight” line is a matter of much complexity. Euclid defines it as the line which lies evenly with respect to the points on itself—εὐθεῖα γραμμή ἐστιν ἥτις ἐξ ἴσου τοῖς ἐφ᾽ ἑαυτῆς σημείοις κεῖται: Plato defined it as the line having its middle point hidden by the ends, a definition of no purpose since it only defines the line by the path of a ray of light. Archimedes defines a straight line as the shortest distance between two points.

A better criterion of rectilinearity is that of Simplicius, an Arabian commentator of the 5th century: Linea recta est quaecumque super duas ipsius extremitates rotata non movetur de loco suo ad alium locum (“a straight line is one which when rotated about its two extremities does not change its position”). This idea was employed by Leibnitz, and most auspiciously by Gierolamo Saccheri in 1733.

The drawing of a straight line between any two given points forms the subject of Euclid’s first postulate—ᾐιτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγάγειν, and the producing of a straight line continuously in a straight line is treated in the second postulate—καὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖν.

For a detailed analysis of the geometrical notion of the line and rectilinearity, see W. B. Frankland, Euclid’s Elements (1905). 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 + By + C = 0, in triangular coordinates Ax + By + Cz = 0. In three-dimensional coordinates, the line is represented by two linear equations. (See Geometry, Analytical.) Line geometry is a branch of analytical geometry in which the line is the element, and not the point as with ordinary analytical geometry (see Geometry, Line).