LINE

, in Geometry, a quantity extended in length only, without either breadth or thickness.

A Line is sometimes considered as generated by the flux or motion of a point; and sometimes as the limit or termination of a superficies, but not as any part of that surface, however small.

Lines are either right or curved. A right, or <*>aight Line, is the nearest distance between two points, which are its extremes or ends; or it is a Line which has in every part of it the same direc- tion or position. But a curve Line has in every part of it a different direction, and is not the shortest distance between its extremes or ends.

Right Lines are all of the same species; but curves are of an infinite number of different sorts. As many may be conceived as there are different compound motions, or as many as there may be different relations between their ordinates and abscisses. See Curves.

Again, Curve Lines are usually divided into geometrical and mechanical.

Geometrical Lines, are those which may be found exactly in all their parts. See Geometrical Line.

Mechanical Lines are such as are not determined exactly in all their parts, but only nearly, or tentatively. But

Des Cartes, and his followers, define geometrical Lines to be those which may be expressed by an algebraical equation of a determinate or finite degree; called its locus. And mechanical Lines, such as cannot be expressed by such an equation.

But others distinguish the same Lines by the name algebraical and transcendental.

Lines are also divided into orders, by Newton, according to the number of intersections which may be made by them and a right Line, viz, the 1st, 2d, 3d, 4th, &c, order, according as they may be cut by a right Line, in 1, or 2, or 3, or 4, &c, points. In this way of considering them, the right Line only is of the 1st order, being but one in number; the 2d order contains 4 curves only, being such as may be cut from a cone by a plane, viz, the circle, the ellipse, the hyperbola, and the parabola; the lines of the 3d order have been enumerated by Newton, in a particular treatise, who makes their number amount to 72; but Mr. Stirling found 4 others, and Mr. Stone 2 more; though it is disputed by some whether these 2 last ought to be accounted different from some of Newton's, or not. See Newton's Enumer. Lin. Tert<*> Ordin. also Stirling's Lineæ Tert. Ordin. Newtonianæ Oxon. 1717, 8vo. and Philos. Trans. number 456, &c. Again,

Algebraical Lines are divided into different orders according to the power or degree of their equations. So, the simple equation or equation of the 1st degree, denotes the 1st order or right line; the equation , of the 2d degree, denotes the Lines of the 2d order; and the equation of the 3d degree, expresses the Lines of the 3d order; and so on. See Cramer's Introd. à l'Analyse des Lignes Courbes.

Lines, considered as to their positions, are either parallel, perpendicular, or oblique. And the construction and properties of each of these, see under the respective terms.

Line also denotes a French measure of length, being the 12th part of an inch, or the 144th part of a foot

In Astronomy,

Line of the Apses, or Apsides, the Line joining the two apses, or the longer axis of the orbit of a planet.

Fiducial Line, the index line or edge of the ruler, which passes through the middle of an astrolabe, or other instrument, on which the sights are fitted, and marking the divisions.|

Horizontal Line, a Line parallel to the horizon.

Line of the Nodes, that which joins the nodes of the orbit of a planet, being the common section of the plane of the orbit with the plane of the ecliptic.

In Dialling,

Horizontal Line, is the common section of the horizon and the dial-plate.

Horary, or Hour Lines, are the common intersections of the hour-circles of the sphere with the plane of the dial.

Equinoctial Line is the common intersection of the equinoctial and the plane of the dial.

In Fortification, Line is sometimes used for a ditch, bordered with its parapet: and sometimes for a row of gabions, or sacks of earth, extended lengthwise on the ground, to serve as a shelter against the enemy's fire.

When the trenches were carried on within 30 paces of the glacis, they drew two Lines, one on the right, and the other on the left, for a place of arms.

Lines are commonly made to shut up an avenue or entrance to some place; the sides of the entrance being covered by rivers, woods, mountains, morasses, or other obstructions, not easy to be passed over by an army. When they are constructed in an open country, they are carried round the place to be defended, and resemble the Lines surrounding a camp, called Lines of circumvallation. Lines are also thrown up to stop the progress of an army; but the term is most used for the Line which covers a pass that can only be attacked in front.

When lines are made to cover a camp, or a large tract of land, where a considerable body of troops is posted, the work is not made in one straight, or uniformly bending Line; but, at certain distances, the Lines project in saliant angles, called redents, redans, or flankers, towards the enemy. The distance between these angles is commonly between the limits of 200 and 260 yards; the ordinary flight of a musket ball, point blank, being commonly within those limits; though muskets a little elevated will do effectual service at the distance of 360 yards.

Fundamental Line, is the first Line drawn for the plan of a place, and which shews its area.

Central Line, is the Line drawn from the angle of the centre to the angle of the bastion.

Line of Defence, &c. See Defence &c.

Line of Approach, or Attack, signifies the work which the besiegers carry on under cover, to gain the moat, and the body of the place.

Line of Circumvallation, is a Line or trench cut by the besiegers, within cannon-shot of the place, which ranges round the camp, and secures its quarters against any relief to be brought to the besieged.

Line of Contravallation, is a ditch bordered with a parapet, serving to cover the besiegers on the side next the place, and to stop the sallies of the garrison.

Lines of Communication are those which run from one work to another.

Line of the Base, is that which joins the points of the two nearest bastions.

To Line a work, signisies to face it, as with brick or stone; for example, to strengthen a rampart with a firm wall, <*> to encompass a parapet or moat with good <*>urf, &c.

Line

, in Geography and Navigation, is emphatically used for the Equator or Equinoctial Line.

The seamen use to baptize their fresh men, and passengers, the first time they cross the Line: that is, to dip them in the sea, suspended by a rope from the yardarm, unless they compound for it, by giving something to drink.

In Perspective,

The Geometrical Line, is a right Line drawn in any manner on the geometrical plane.

Terrestrial or Fundamental Line, is the common intersection of the geometrical plane and plane of the picture.

Line of the Front, is any Line parallel to the terrestrial Line.

Vertical Line, is the section of the vertical and draft planes.

Visual Line, is the Line or ray conceived to pass from the object to the eye.

Objective Line, is any Line drawn on the geometrical plane, whose representation is sought for in the draught or picture.

Line of Measures, is used by Oughtred, and others, to denote the diameter of the primitive circle, in the projection of the sphere in plano, or that Line in which falls the diameter of any circle to be projected.

Linear Numbers, are such as have relation to length only; such, for example, as express one side of a plane figure; and when the plane figure is a square, the linear number is called a root.

Linear Problem, is one that can be solved geometrically by the intersection of two right lines. This is called a simple problem, and is capable of only one solution.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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