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]).
LINE ENGRAVING, on plates of copper or steel, the method of engraving (q.v.), in which the line itself is hollowed, whereas in the woodcut when the line is to print black it is left in relief, and only white spaces and white lines are hollowed.
The art of line engraving has been practised from the earliest ages. The prehistoric Aztec hatchet given to Humboldt in Mexico was just as truly engraved as a modern copper-plate which may convey a design by Flaxman; the Aztec engraving is ruder than the European, but it is the same art. The important discovery which made line engraving one of the multiplying arts was the discovery how to print an incised line, which was hit upon at last by accident, and known for some time before its real utility was suspected. Line engraving in Europe does not owe its origin to the woodcut, but to the chasing on goldsmiths’ work. The goldsmiths of Florence in the middle of the 15th century were in the habit of ornamenting their works by means of engraving, after which they filled up the hollows produced by the burin with a black enamel made of silver, lead and sulphur, the result being that the design was rendered much more visible by the opposition of the enamel and the metal. An engraved design filled up in this manner was called a niello. Whilst a niello was in progress the artist could not see it so well as if the enamel were already in the lines, yet he did not like to put in the hard enamel prematurely, as when once it was set it could not easily be got out again. He therefore took a sulphur cast of his niello in progress, on a matrix of fine clay, and filled up the lines in the sulphur with lampblack, thus enabling himself to judge of the state of his engraving. At a later period it was discovered that a proof could be taken on damped paper by filling the engraved lines with a certain ink and wiping it off the surface of the plate, sufficient pressure being applied to make the paper go into the hollowed lines and fetch the ink out of them. This was the beginning of plate printing. The niello engravers thought it a convenient way of proving their work—the metal itself—as it saved the trouble of the sulphur cast, but they saw no further into the future. They went on engraving nielli just the same to ornament plate and furniture; nor was it until the 16th century that the new method of printing was carried out to its great and wonderful results. There are, however, certain differences between plate-printing and block-printing which affect the essentials of art. When paper is driven into a line so as to fetch the ink out of it, the line may be of unimaginable fineness, it will print all the same; but when the paper is only pressed upon a raised line, the line must have some appreciable thickness; the wood engraving, therefore, can never—except in a tour de force—be so delicate as plate engraving. Again, not only does plate-printing excel block-printing in delicacy; it excels it also in force and depth. There never was, and there will never be, a woodcut line having the power of a deep line in a plate, for in block-printing the line is only a blackened surface of paper slightly impressed, whereas in plate-printing it is a cast with an additional thickness of printing ink.
The most important of the tools used in line-engraving is the burin, which is a bar of steel with one end fixed in a handle rather like a mushroom with one side cut away, the burin itself being shaped so that the cutting end when sharpened takes the form of a lozenge, point downwards. The burin acts exactly like a plough; it makes a furrow and turns out a shaving of metal as the plough turns the soil of a field. The burin, however, is pushed while the plough is pulled, and this peculiar character of the burin, or graver, as a pushed instrument at once establishes a wide separation between it and all the other instruments employed in the arts of design, such as pencils, brushes, pens and etching needles.