the Week of Proper 28 / Ordinary 33
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Bible Encyclopedias
Line
1911 Encyclopedia Britannica
or 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, ypaµ,ut), SE /KOs lorXarES ("a line is widthless length"), and in a subsequent definition he affirms that the boundaries of a line are points, ypaµµ6s 8E irEpara vrtµ€ia. The line appears in definition 6 as the boundary of a surface: 7rcq5aveias $E irEpaTa ypaµµai ("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 (kacs vrtµ€iov), 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 ap X 'i 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 threedimensional; 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 - EUBE7a ypaµµrt 1 TLS ivov Tois E4' EaUTis o-f,u€iocs Karat: 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 - titrim) are) IravrOs mnp.cLov 7rav €i OELav 'ypaµµ v &y t'yaay, and the producing of a straight line continuously in a straight line is treated in the second postulate - Kai 71-E7repac,ubnv a av Kara TO KOaXELv. 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-PC =0, in triangular coordinates Ax+By-1-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).
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Chisholm, Hugh, General Editor. Entry for 'Line'. 1911 Encyclopedia Britanica. https://www.studylight.org/encyclopedias/eng/bri/l/line.html. 1910.
