1911 Encyclopædia Britannica/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,
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—
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 ἀ
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).